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Professional SLitarg
EDITED
w NICHOLAS
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MURRAY BUTLER
4
i
-
THE TEACHING OF
ELEMENTARY MATHEMATICS
THE TEACHING OP
ELEMENTARY MATHEMATICS
BY
DAVID EUGENE SMITH PROFESSOR OF MATHEMATICS IN TEACHERS' COLLEGE, COLUMBIA UNIVERSITY, NEW YORK.
THE MACMILLAN COMPANY LONDON MACMILLAN & :
IQO2 All rights reserved
CO., LTD.
QA hi.
.
_
/
1974
/
COPYRIGHT, 1900,
BY
THE MACMILLAN COMPANY.
Set up and electrotyped February, 1900. 1901
;
Reprinted January,
April, 1902.
Korfaooti J. S.
Cashing
&
Co.
Berwick & Smith
Norwood Mass.
U.S.A.
AUTHOR'S PREFACE IT
is
evident that the problem of preparing a work
upon the teaching of elementary mathematics may be attacked from any one of various standpoints. A writer
may
confine himself to model lessons, for example
to the explanation of the
subject matter
;
most
difficult
;
or
portions of the
or to the psychology of the subject; or
comparison of historic methods or to the exploiting of some hobby which he has ridden with success or to those devices which occupy so much time in the ordito the
;
;
nary training of teachers.
He may
that elementary mathematics
now
say,
and with
truth,
includes trigonom-
etry, analytic geometry, and the calculus; and that therefore a work with this title should cover the ground
of Dauge's " La
work, cally,
"
Methodologie," or of Laisant's masterly
Mathematique." He may proceed dogmatiand may lay down hard and fast rules for teaching,
excusing this destruction of the teacher's independence
by the thought that the end justifies the means. But with a limited amount of space at his disposal, whatever point of others
more
attack
or less
he selects he must leave the
untouched
an encyclopedia of the subject
;
he cannot condense
in three
hundred pages.
AUTHOR'S PREFACE
vi
Several years ago the author set about to find something of what the world had done in the way of making
and of teaching mathematics, and
to
know
valuable literature of the subject.
He
found, however,
no manual
to guide his reading,
of a library
the really
and so the accumulation
upon the teaching of the subject was a slow
and often discouraging work.
This
handbook
little
is
who care to take a shorter, clearer know something of these great questions Whence came this subject ? Why am I
intended to help those route,
and
to
of teaching,
teaching I
How
it ?
has
it
been taught
my work
read to prepare for
?
What
?
The
subject
should is
thus
considered as in a state of evolution, while comparative
method rather than dogmatic statement
is
the keynote.
It is true that certain
types are suggested,
they are often called
but these are given as represent-
;
methods,
ing the present development of the subject, and not as finalities.
The
effort
has been, throughout, to set forth
the subject as in a state of progress to which forward
movement the teacher enough
is
to contribute
;
we have
quite
literature representing the static element.
Considerable attention has been given to the bibliog-
raphy of the
subject.
At
the risk of being accused of
going beyond the needs of teachers, the author has suggested the most helpful works in French and German, as well as in English, and has not hesitated to quote
from them. in English,
The body
of the
page is, however, always the footnotes may be used or not, as the
AUTHOR'S PREFACE
Where
reader wishes.
vii
a quotation seemed to lose some-
thing by being put into English, the original has been
placed in a footnote.
By
put in touch with those
these references the reader
is
works which the author has
The references might found of great value to him. this but has not seemed desirable. be multiplied, easily There are many books on the teaching of mathematics, some of them quite pretentious in their claims, a few published in America, a few in England and France,
and a large number
in
Germany.
To
even
cite all, or
might be positively harmful it is hoped that the selection made has been reasonably
a majority of these,
;
judicious. If this
a wider
work
field,
even in a small way, to open or to offer a better point of view, to someshall help,
one just entering the profession, the author
will feel
repaid for his labors.
DAVID EUGENE SMITH. STATE NORMAL SCHOOL, BROCKPORT, January, 1900.
N.Y.,
EDITOR'S INTRODUCTION PERHAPS no
single subject of elementary instruction
much from lack of scholarship on the who teach it as mathematics. Arithmetic
has suffered so part of those is
universally taught in schools, but almost invariably
as the art of mechanical computation only.
The
true
significance and the symbolism of the processes employed are concealed from pupil and teacher alike.
This
is
the inevitable result of the teacher's lack of
mathematical scholarship.
The
subtlety, delicacy,
processes direct tine,
and
have
the
indirect.
and accuracy of mathematical
highest
To
treat
educational value,
both
them as mechanical
rou-
not susceptible of explanation or illumination from
a higher point of view,
is
to destroy in large
measure
the value of mathematics as an educational instrument,
and
to aid in arresting the
mental development of the
pupil.
As
long ago as the time of Aristotle it was pointed out that mathematics should not be defined in terms of the content with
which
it
deals, but rather in
terms
EDITOR'S INTRODUCTION
X of
its
method and degree
mathematics, in the
"
Kant says of Pure Reason," " The
of abstractness.
Critique of
science of mathematics presents the most brilliant ex-
ample of how pure reason may successfully enlarge 1 He then its domain without the aid of experience." goes on to point out the ground of the distinction between philosophical and mathematical knowledge,
and adds
" :
Those who thought they could distinguish
philosophy from mathematics by saying that the former was concerned with quality only, the latter with quan-
mistook effect for cause. It is owing to the form of mathematical knowledge that it can refer to quanta only, because it is only the concept of quantities tity only,
that admits of construction, that
of a priori repre-
is,
sentation in intuition, while qualities cannot be repre-
sented in any but empirical intuition." Mr. Charles S. Peirce has recently
2
cism that Kant was not
supposing that
mathematical
justified
in
made
the
criti-
and philosophical necessary reasoning
are distinguished
by the circumstance that the former
uses construction or diagrams.
Mr. Peirce holds that
necessary reasoning whatsoever proceeds by con-
all
structions,
and that we overlook the constructions
in
3 philosophy because they are so excessively simple. He goes on to show that mathematics studies nothing
but pure hypotheses, and that 1
Miiller's Translation
3
Educational Review,
(New York, 15, 214.
it
is
the only science
1896), p. 572.
2
Ibid., p. 573.
EDITOR'S INTRODUCTION
XI
which never inquires what the actual facts are. It is "the science which draws necessary conclusions." This acute argument
tention that construction
reasoning, but
is
I think, at fault in its
is,
is
to point out clearly these facts 1.
The human mind
in
employed
otherwise sound.
of
however,
:
see every perception in a time-relation
and every perception
philosophical
It fails,
so constructed that
is
con-
in
it
must
an order
an object in a space-relation
as outside or beside our perceiving selves.
These necessary time-relations are reducible to Number, and they are studied in the theory of number, 2.
arithmetic and algebra. 3.
These necessary space-relations are reducible to and Form, and they are studied in geometry.
Position
Mathematics,
therefore,
studies
an aspect of
all
knowing, and reveals to us the universe as it presents To apprehend this and itself, in one form, to mind.
be conversant with the higher developments of mathematical reasoning, are to have at hand the means of vitalizing all teaching of elementary mathematics. to
In the present book, the purpose of which present in
simple and succinct form
results of mathematical scholarship, to
them and applied
in
their
is
to
to teachers the
be absorbed by
class-room
teaching,
the
author has wisely combined the genetic and the anaHe shows how the elementary mathelytic methods. matics has developed in history,
how
it
has been used
EDITOR'S INTRODUCTION
Xll
in education,
may
safely
and what
inner nature really
is.
It
be asserted that the elementary mathe-
new
matics will take on a this
its
book and apply
its
reality for those
who
study
teachings.
NICHOLAS MURRAY BUTLER. COLUMBIA UNIVERSITY, February
i,
NEW
1900.
YORK,
CONTENTS CHAPTER
I
PAGE
HISTORICAL REASONS FOR TEACHING ARITHMETIC. tance of the question.
trading peoples.
As a remunerative trade. As a mere show of As an amusement. As a quickener of the
ture value.
knowledge.
Scientific investigation of reasons
CHAPTER
WHY
ARITHMETIC
reasons. value.
The
evolution of reasons.
Early correlation. Utilitarian among Tradition and examinations. The cul-
utilitarian.
beginning
wit.
The
Impor-
The
is
of arithmetic overrated. fail
.
here.
Two
general
The culture
Recognition of the
What
chapters bring out the culture value. well be omitted. Relative value of culture and
19-41
utility
CHAPTER How
1-18
.
II
TAUGHT AT PRESENT.
utility
Teachers generally
culture value.
What may
.
III
ARITHMETIC HAS DEVELOPED.
Reasons for studying Extent of the subject. The first step The second step notation. The next great
the subject.
counting. The twofold nature of ancient arithstep in arithmetic. The period of metic. Arithmetic of the middle ages. the Renaissance.
....
Arithmetic since the Renaissance.
present status of school arithmetic
The 42-70
CONTENTS
xiv
CHAPTER
IV PAGE
How
The value of the ARITHMETIC HAS BEEN TAUGHT. investigation. The departure from object teaching. Rhym-
Form
ing arithmetics.
Instruction
instead of substance.
....
method. Pestalozzi, Tillich. Grube. Recent writers lozzi.
Reaction against Pesta71-9?
in
CHAPTER V THE PRESENT TEACHING OF ARITHMETIC. at.
The number
The writing of numbers. The time for beginning
Objects aimed
The great question The work of the
concept.
the
study.
Oral
of method. first year. arithmetic.
Treating the processes simultaneously. The spiral method. Common vs. decimal fractions. Improvements in algorism.
The
Ratio and Longitude and time. Square root. The metric system. The apMensuration. Text-books. Explanaplied problems. tions. 98-144 Approximations. Reviews formal solution.
proportion.
....
CHAPTER THE GROWTH OF ALGEBRA.
Egyptian
Growth of symbolism.
algebra.
Number
systems.
tions
CHAPTER ALGEBRA,
Greek
Sixteenth century algebra.
Oriental algebra.
algebra.
VI
Higher equa145-160
VII
WHAT AND WHY TAUGHT. Algebra defined. The Why studied. Training in logic. Ethical
function. value.
When
studied.
Arrangement of text-books
CHAPTER TYPICAL PARTS OF ALGEBRA. awakening of interest. gation.
The
161-174
VIII
Outline.
Definitions.
Stating a problem.
negative number.
.
Checks.
The
Signs of aggreFactoring.
The
CONTENTS
XV PAGE
The quadratic equation. Equivaremainder theorem. Extraneous roots. lent equations. Simultaneous equations
and graphs.
numbers.
The
Methods of elimination.
The
applied problems.
Complex
interpretation of
solutions
175-223
CHAPTER THE GROWTH OF GEOMETRY. dawn
of geometry.
cent geometry.
IX
The
Its historical position.
Geometry
in
Egypt Non-Euclidean geometry ;
in Greece. .
.
Re224-233
CHAPTER X WHAT
is
GEOMETRY?
GENERAL SUGGESTIONS FOR TEACH-
Limits of plane geometry. reasons for studying. Geometry in the lower grades. Intermediate grades. Demonstrative geometry. The use of text-books 234-256 ING.
Geometry defined.
The
CHAPTER THE BASES OF GEOMETRY. Axioms and
The
TYPICAL PARTS OF GEOMETRY. strative geometry.
verse theorems.
definitions.
257-270
XII
The
introduction to
demon-
Symbols.
....
Ratio and proportion. Solid geometry
CHAPTER THE TEACHER'S BOOK-SHELF.
INDEX
The
Reciprocal theorems. ConGeneralization of figures. Loci of points.
attack.
sible in geometry.
etry.
bases.
postulates
CHAPTER
Methods of
XI
The impos271-296
XIII
....
Arithmetic.
History and general method
Algebra.
Geom297-305
307-312
THE
TEACHING OF ELEMENTARY MATHEMATICS CHAPTER
I
HISTORICAL REASONS FOR TEACHING ARITHMETIC
For one who
Importance of the question
is
pre-
any particular branch, and who hopes the most important question is this Why
paring to teach for success, is
:
the subject taught
ods,
of
?
More important than
more important than
text-books, or
devices
all
advice of the masters,
or is
all
meth-
questions this
far-
reaching inquiry. Upon the answer depends the solution of the problems relating to the presentation of the subject, the
time
it
grade in which
the devices,
whole matter
know the
should be begun, the
it
should consume, the text-books, the methods, in
"
the general treatment of
in hand.
not whither
way ?
fine,
Thou
It
is
goest,
Unless the goal
the old, old cry,
the
"We
and how can we know is
known, what hope
has one to find the path?
Of
course the inquiry
chine teacher,
is
the teacher
of no interest to the
who
is
ma-
content to follow
THE TEACHING OF ELEMENTARY MATHEMATICS
2
the book unthinkingly, to see the old curriculum re-
main forever unchanged, and teacher trod,
even though
without interest to the
to follow the
be rough
it
But
eye.
America to-day we have a host siastic
teachers
Saxon
who
educational to
willing
are anxious to
and
teachers this question
The evolution
the
system
inquire
of
is
to
path his
and
to the foot
and
in
England young and enthu-
make
best,
the Anglo-
and
experiment.
who For
are
such
vital.
This search after reasons
of reasons
may be pursued either from the standpoint of a mere inquirer into the conditions of to-day, or from that of one who
is
interested
which are now
in
in favor.
the evolution of the ideas
While
it
is
not possible in
a work of this nature to enter into the details of the de-
velopment of the reason for the presence of arithmetic
some
slight reference to this
of interest,
and should be of value.
in the curriculum to-day,
development
The
may be
beginning
utilitarian
In the far East,
and
in the far past, the reason for teaching arithmetic to
was almost always purely utilitarian. To the philosopher it was more than this, but in the early children
it was given place merely that the have sufficient knowledge of the four funboy might damental processes for the common vocations of life. 1
Chinese curricula
1
Schmid, K. A., Geschichte der Erziehung
unsere Zeit, Stuttgart, 1884-98, Vol.
Schmid.
I,
p. 78.
vom Anfang an
bis
auf
Hereafter referred to as
HISTORICAL REASONS FOR TEACHING ARITHMETIC This was done in the
common
3
schools almost from
but in the middle ages
1
the subject so increased in importance that special schools were estab-
the
first,
lished for the study of
A
arithmetic.
little
later
2
it
was taught as a special course in the high schools, open to those who had a taste in this direction, although even then children must have continued to learn
common reckoning
general, however, for
it
in
the earlier years.
has been taught in the far East
two thousand years, because of the
utilities
possesses, or merely for the purposes
it
tion,
or
because
it
In
correlated with
which
examina-
of
a study of
the
sacred books. 3
In India
Early correlation
little
for arithmetic in the schools.
as summarized in the
man
of education,
book of Manu, was
first
to lead a religious
could be expected
The aim
life.
The reading
to bring
of the Veda,
the giving of alms, these were fundamental features of education. 4
than
two
Even to-day thousand
is this
years
methods have remained
the
quite
the case.
curriculum
For more and
the
unchanged, and even
our day, in the native schools, the boy's work
in
largely that of
1
2 8
memorizing the
Under the Sung dynasty, 961-1280. Schmid, Under the Ming dynasty, 1368-1644. Laurie, S.
S.,
Schmid,
I,
p.
I, p.
80.
Hereafter referred to as Laurie,
105-107.
is
scriptures and
Historical Survey of Pre-Christian Education,
1895, p. 128, 141, 148. 4
Hindu
London,
THE TEACHING OF ELEMENTARY MATHEMATICS
4
other knowledge incidentally, a classical
picking up
extreme
of
example
hence
value
of
is
throws light upon the central subject, and has little place in the curriculum. 1
it
it
The same schools,
For such people,
beyond the mere rudiments,
arithmetic,
only as
correlation.
idea characterized the early
Mohammedan
where the Koran furnished the core of
instruc-
a plan of education still obtaining, on a slightly more liberal scale, in the present schools of Islam. 2 It
tion,
sway in the monastic schools where arithmetic, like everything
also held quite general
of the middle ages, else,
was
either
warped
to correlate with theology, or
confined to the simplest calculations. 3
That arithmetic
was popularly considered merely as having some slight value in trade is shown by a familiar bit of monkish doggerel, as old at least as the beginning of the fifteenth 4
century.
It
thus sets forth the values of the seven rhetoric, music, arith-
liberal arts,
grammar, dialectic, metic, geometry, and astronomy "
Gramm. Mus.
1
loquitur, Dia. vera docet,
canit,
:
Rhe. verba colorat
Ar. numerat, Ge. ponderat, As.
For a description of the arithmetic
in the native
;
colit astra."
Hindu
schools of the
present consult Delbos, L., Les Mathematiques aux Indes Orientales, Paris, 1892, 2 8
"
pamphlet. Schmid, II (i),
p. 599.
Ib., II (i), p. 86.
Omnino
In
this line is the rule attributed to
nullus erit in monasterio, qui
aliquid teneat."
*Ib., II (i), p. 114.
non
Pachomius,
discat literas et de scripturis
HISTORICAL REASONS FOR TEACHING ARITHMETIC
For the mediaeval
cloister schools the
5
computation
day was the one great problem. On this depended the other movable feasts, and every monastery was under the necessity of having someone who knew of Easter
enough
of calculating to determine this date. 1
Utilitarian
among
we
Semitic peoples taught.
trading find
The Semite
not in the thing for
the
Among
peoples
arithmetic
more extensively
has generally interested himself its own sake, but for what it
contained for him in a practical way.
Assyrians and Arabs and related national epos and no enduring
have no
peoples 2
But they found trade, and hence it
art.
in arithmetic a subject usable in
Hence the
was extensively taught in their schools. Among the ruins in and about ancient Babylon it is not uncom-
mon
to
counts, pupils'
find
and
lately
work
Among made
tablets
containing
some
arithmetic
in
bank
extensive
specimens
interesting
have come
to
of
3
light.
the Jews, after elementary instruction was 4
obligatory,
arithmetic formed, with writing and
the study of the Pentateuch, the sole
work from the
sixth to the tenth year of the child's school 1
ac-
life.
Rashdall, H., Universities of Europe in the Middle Ages,
I, p.
35.
Schmid, II (i), p. 117. 2 8
Schmid,
these tablets; zar's 4
I, p.
142.
Ib., I, p. 152, 153. it
The
time on.
A.D. 64.
firm of Egibi
was long famous
Laurie, p. 97.
in
and Sons
is
often mentioned in
banking business from Nebuchadnez-
THE TEACHING OF ELEMENTARY MATHEMATICS
6
Even
in
and
Greece,
among
the
philosophers,
where one would expect something beyond the mere necessities of existence, arithmetic was not in general highly valued.
Socrates,
who recommends
ject in the curriculum, does so with a
carrying
course
it
the
beyond
needs
who
the Spartans,
among
of
the
sub-
warning against
common
life.
Of
trained for war, the
science had no place. 1
In Rome, a city of commerce and of war, the subject was naturally looked upon as of merely utilitarian importance. The vast commercial interests of the
the
to
extending
city,
farthest
corner
of
the
made
a business education imperative Arithmetic flourished, but merely as the drudgery of calculation. So Cicero tells us that in his time the Romans esteemed only practical great empire,
a large class.
for
reckoning, nor was the learned sopher, ecclesiastic, and it
to
any higher plane.
In the
1
J.
not taught for the purposes
Girard, Paul, L'Education Athenienne au
C.,
2.
ed., Paris,
1891, p.
Pedagogiques des Grecs, 2
mathematician, able to raise
2
when
cloisters,
Boethius, the philo-
Laurie, p. 360
York, 1896, p.
MUnchen, 1891,
1
;
136-138
;
Paris, 1881, p. 12
Clarke, G.,
85
Sterner,
17,
p.
73, hereafter
;
;
et
Schmid,
The Education
6,
Ve
au IV e
Martin, Alex., I,
siecle
avant
Les Doctrines
p. 231, 232.
of Children at
Rome,
New
M., Geschichte der Rechenkunst,
referred to as Sterner ;
Schmidt, K.,
Geschichte der Padagogik, Cothen, 1873, I, p. 408 ; Dittes, F., Geschichte der Erziehung und des Unterrichts, 9. Aufl., Leipzig, 1890, p. 73 Schmid, ;
II (i), p. 140.
HISTORICAL REASONS FOR TEACHING ARITHMETIC
7
computing Easter or as a "whetstone of wit," was considered as merely of value in Even Beda, one of the best teachers of his trade. of
arithmetic
1 upon the subject as purely utilitarian. During the middle ages, too, there was a great revival of trade and a corresponding revival of com-
time, looked
mercial arithmetic. the
of
thirteenth
For a long time century
Northern
the close
after
was the
Italy
gateway for trade entering Europe from the Orient.
Thence
it
passed
Niirnberg, and
northern
the
northward,
through Augsburg, Main, to Leipzig and towns on the east, and to
Frankfurt
Hanseatic
am
Cologne and the Netherlands on the west. Similarly in France, Lyons and Paris, and in Austria, Vienna, Linz, tres.
and Ofen, became important commercial cenBut Italy was par excellence the mercantile
nation and the source of commercial arithmetic, and
we
the
find
source
among
all
along this pathway of commerce.
satisfaction
the
of
1
2
It
the thirteenth
century a feeling of
arose
the
Church
against
schools.
had so supplanted
2
supreme, from
the
was
the merchants along this path of trade that
early as
as
influence
utilitarian
arithmetical
Mysticism
religion, to
and
dis-
training
formalism
say nothing of other
Schmid, II (i), p. 140. F., Die Methodik der praktischen Arithmetik in historischer
Unger,
Entwickelung
vom Ausgange
Leipzig, 1888, p. 3 seq.
des Mittelalters bis auf die Gegenwart,
Hereafter referred to as Unger.
THE TEACHING OF ELEMENTARY MATHEMATICS
8
subjects of study, that even the
wont
shame
to point with 1
when
Even
training.
common
people were
to the results of
the
universities
monastic
began
to
2 spring up, about noo, and arithmetic might hope to break away from the bonds of commerce, there was
little
Scholasticism, disputations, philo-
improvement.
these had
sophic hair-splitting
to
use for a sub-
One who had made a
ject like this.
in
little
little
progress
was a mathematician.
Save as leading the calculations of the calendar, and as it might fractions
occasionally touch the Aristotelian philosophy, mathe-
matics had no standing. 3
was during this mediaeval period that the HanThis great trust seatic league became a power. for It
1
Schmid, II (i),
2
rise of Universities, lect. " Omnis hie excluditur, omnis est abiectus,
8
Laurie, S.
Qui non
S.,
p. 312.
The
vi.
Aristotelis venit armis tectus."
Chartular. Univ. Paris,
I,
Introd., p. xviii.
In Cologne in 1447 the outlook for Schmid, II (i), p. 427, 447, 448. mathematics, as indeed for other subjects, was exceedingly poor if one may judge from the verses in Horatian measure of the young Conrad Celtes:
"
Nemo Nee
hie latinam
grammaticam
docet,
explotis rhetoribus studet,
Mathesis ignota est, figuris sacris numeris recludit.
Quidque
Nemo
hie per
axem Candida
sidera
Inquirit, aut quse cardinibus vagis
Moventur, aut quid doctus
alta
Contineat Ptolemaeus arte."
Schmid, II (i),
p. 449.
HISTORICAL REASONS FOR TEACHING ARITHMETIC such
it
may be
tical
soon found that
styled
sary to establish its
own
schools
if
it
9
was neces-
wished a prac-
it
And
education for the rising generations.
so
there was to be found in each town of any size along
the highway dominated by the league, an arithmetic
master (Rechenmeister), the
teaching
subject
who
held the monopoly of
Not unfrequently was
there.
the Rechenmeister also the city accountant, treasurer, sealer of weights
that
therefore,
and measures,
arithmetic
etc.
It
was
natural,
should tend to become a
purely utilitarian subject in these places, and so in It is interesting to recall that great measure it was. the last of the Rechenmeisters, Zacharias Schmidt of 1 Niirnberg, kept his place until I82I.
sixteenth
century,
some thinking
when
the
As
late as the
reformers
in education, in
began to do a school as famous as
the Strassburg gymnasium, Johann Sturm, in his cur-
riculum of 1565, makes no mention of arithmetic in his
entire
ten
years'
course, so
completely commer-
had the subject become. 2 To refer more specifically to the
cial
universities,
even
Cambridge, which already in the middle ages led Oxford in mathematical teaching, arithmetic had
at
3 scarcely any attention.
1
2
Unger,
At Oxford during
this period
p. 26, 33.
Paros, Jules, Histoire universelle de la Pedagogic, p. 126;
Schmid,
II (2), P- 3253
Rashdall, H., Universities of Europe in the Middle Ages, II, p. 556.
THE TEACHING OF ELEMENTARY MATHEMATICS
10
a term in Boethius was
when
all
that
was
required.
Even
a chair of arithmetic was founded in the Uni-
Bologna, a school which owed
versity of
nence in mathematics to Arabo-Greek
was
1
promi-
influence,
it
more than that
little
computer.
its
2
of a surveyor and general In Paris the subject had no hold, and in
Vienna, where more was done than in the Sorbonne, 3 only a nominal amount of arithmetic was required.
In general, mathematics was
looked upon as a light
subject in the mediaeval universities.
The Egyptian reason
Tradition and examinations for teaching arithmetic
may be
seen in the interesting
account of a school of the fourteenth century B.C., given by the late Dr. Ebers in the second chapter of Uarda. 4
Here, where the
life
and thought of the
people, so closely joined to the river with
and
its
periodic
naturally took on regularity, mystery rule, canonical form, and mysticism, educational progof rise
ress could only
the outer world.
fall,
come from renewed
Hence
arithmetic
intercourse with
came
to be taught
merely as a matter of custom, of tradition as fixed as human law can be. It was required for examinations, and the examiner followed a certain line; hence, the
1
Rashdall, H., Universities of Europe in the Middle Ages, II, p. 457.
2
Ib., II, p. 243, 66 1 n.; I, p. 249. For the B. A. degree, " Primum librum Euclidis
8
in arithmetica." 4
Ib., II, p. 240, 674.
See also Schmid,
I, p.
172.
.
.
.
aliquem librum
HISTORICAL REASONS FOR TEACHING ARITHMETIC student must be prepared along that
line. 1
II
This
is
tendency under a centralized examinaalways tion system, or where an inflexible official programme As M. Laisant says, "a promust be followed. the
gramme
is
always bad,
because
essentially
it
is
a
programme."
An
excellent illustration of the petrifying tendency
of such light.
a
is
an examination system has recently come to The oldest deciphered work on mathematics
papyrus
Museum.
It
manuscript
preserved
in
the
British
was copied by one Ahmes (Aahmesu,
the Moonborn), a scribe of the Hyksos dynasty, say
between 2000 and 1700 B.C., from an older work dat2 Without going into details as ing from 2400 B.C. the contents of the work,
to
it
answers the present
purposes to say that the arithmetical part was devoted chiefly to unit fractions. Instead of writing the
modern notation) Ahmes and his 4- TV + TTTNow, within the predecessor write it
fraction -fy (using
^
past decade there have been found in Kahun, near 1
2
Schmid, I, p. 173 ; Laurie, p. 44. That is, from the reign of Amenemhat
III,
2425-2383
M., Vorlesungen iiber Geschichte der Mathematik,
I, p.
B.C.
21, n.
Cantor,
This work,
the standard authority in the history of mathematics, will hereafter be referred to as Cantor; Vol.
I, 2.
Auf., 1894, Vol. II, 1892, Vol. Ill, 1898,
The Ahmes papyrus was translated and published by Eisenlohr, Ein mathematisches Handbuch der alten Aegypter, Leipzig, 1877,
Leipzig.
A.,
and an English edition has recently appeared. A brief summary is given in Gow, J., A short History of Greek Mathematics, Cambridge, 1884, p. 15, hereafter referred to as Gow.
12
THE TEACHING OF ELEMENTARY MATHEMATICS
the
pyramids of
Illahum, two
mathematical
treating fractions exactly after the
manner
of
papyri
Ahmes,
and there has been published in Paris an interesting papyrus found in the necropolis of Akhmim, the ancient
in
Panopolis,
Upper
Egypt,
Christian Greek somewhere from the
fifth to
by a
the ninth
In this latter work, also, fractions are
A.D.
century
written
Ahmes had handled them
treated just as
over two
thousand years before. 1 The illustration is extreme, but it shows the tendency of tradition, of canonical
and of the examination system, which for so many centuries dominated the civil service of Egypt. The culture value Occasionally, however, even in
laws,
ancient times, there appeared a suggestion of a higher
Solon and Plato
reason for the study of arithmetic.
saw
the subject
in
mind
an opportunity for
close thinking, the former
to
the
training
placing here
greatest value, and the latter asserting
its
that even the
most elementary operations contributed to the awakening of the soul and to stirring up "a sleepy and uninstructed spirit.
We
see from the Platonic dialogues
how mathematical problems employed thoughts of young Athenians."
1
Baillet, J.,
Memoires 2
tin,
.
.
.
Le papyrus mathematique de
la
2
the
mind and
Plato even goes so
d' Akhmim, Paris, 1892, in the
mission archeologique fran9aise au Caire.
Browning, Oscar, Educational Theories,
New
York, 1882, p. 6; Mar-
Alexandre, Les Doctrines Pedagogiques des Grecs, Paris, 1881,
Schmid,
I. p.
233.
p.
44;
HISTORICAL REASONS FOR TEACHING ARITHMETIC far as to wish arithmetic taught to girls,
also
13
and Aristotle
champions the higher cause when he
asserts that
"children are capable of understanding mathematics when they are not able to understand philosophy." Still, in Aristotle's scheme of state education we look in vain for
any
details
idea here expressed. first
as to the
beyond
carrying out of the
Naturally, too, Pythagoras, the
mathematical master,
great
something
1
mere
saw
in
calculation.
arithmetic
"Gymnastics,
music, mathematics, these were the three grades of his
educational curriculum.
strengthened
;
by the second
made ready
perfected and gods."
the
By
purified
for
the pupil was
first
the
;
by the
third
the
of
society
2
In the middle ages the same feeling occasionally crops out, as when ^Eneas Sylvius (later Pope Pius
from 1458
II,
1464), the
to
apostle of
humanism
in
Germany, advocated the study of arithmetic for its own sake, provided it should not require too much time.
Humanism
failed,
however, to advance math-
ematics to any great extent in the learned schools. With few exceptions this task was left to the technical schools.
was 1
far-sighted
Occasionally some leader like
Stehn
a
slight
enough
to
appreciate
in
Davidson, Thomas, Aristotle and Ancient Educational Ideals,
New
York, 1892, p. 198.
But see Mahaffy, P. J., Old Greek Education, New Ib., p. 100. York, 1882, p. 89, on the slight influence of Pythagoras on education. 2
THE TEACHING OF ELEMENTARY MATHEMATICS
14
degree the educational value of the subject, but such cases were rare. 1 .
As a remunerative
In
trade
the science there have been
not
uncommon
the
.
development of which it was
periods in
mere problem-solvers to undertake puzzles for pay, and occasionally arith-
arithmetical
for
metic has been studied with this in view, although of course to no great extent.
Adam
Riese the famous
1559), solved
Hans Conrad,
German
problems for pay.
a friend of
arithmetician (1492-
Also in the time of
the early Italian algebraists, Scipione del Ferro, Antonio del Fiore, Tartaglia, of affairs existed;
learning was
it
and Cardan, the same
was a period
state
of secret rules,
and
neither open nor free. 2
As a mere show
This has not unf reknowledge quently been one of the most apparent of reasons, and of
especially so in the Latin schools of the sixteenth century.
Thus Gemma
Frisius,
one of the most famous
text-book writers of his time, presents as the second
number millies
in
his arithmetic,
&
millia,
sexcenta
&
trecenta
quadraginta
3 septuaginta octo."
quinque
Such a display
millia,
of words
Stehn (Johannes Stenius) writes, in Wittenberg in 1594, "Num disnumerorum Methodica iure possit exulare Scholis puram et solidam
ciplina
Philosophiam ambientibus." Schmid, II (2), p. 373. 2
3
ter
millena millia, quadringenta quinquaginta sex
millena
1
23456345678, "vicies
Unger,
p. 33, 34.
Arithmeticae Practicae Methodus Facilis, edn. of 1551, p. A.
v.
HISTORICAL REASONS FOR TEACHING ARITHMETIC
15
cannot be dignified by the term knowledge it is only It has its counterpart in the absurdly exa pretence. ;
tended number names in some of our present metics and in subjects like
compound
arith-
proportion.
As an amusement for
Arithmetic has also been taught amenities, and in the seventeenth century
its
works
several
appeared
with
avowed purpose. in Rouen in
this
Such was one published anonymously " Recreations
1628,
"
d'Arithmetique,
etc."
Physiko-Mathematicae
oder
problemes
plusieurs ter's
Deliciae
composees de Schwen-
mathematiques
und physikalische Erquickstunden
tische
1636) was another.
the
best
mathema-
"
(Altdorf,
known was
Perhaps Bachet de Meziriac's " Problemes plaisants et delectables," which appeared in 1612* the source of several of
which
the problems
float
still
around our
lower
schools.
As a quickener of the wit Closely two of the reasons already mentioned arithmetic
keen,
is
especially
fitted
This was
quick-witted.
to
allied to
the idea that
is
make one
one
one or
of
the
sharp,
leading
reasons in certain of the cloister schools, the subject being there taught for its bearing upon the training of of
the
clergy in
catch-problems,
problems
Such
is
disputation.
Hence
problems
intended for
containing
some
the famous one 1
trick
of the
of
widow
Fifth edition, Paris, 1884.
arose a mass
argument,
language, to
whom
etc.
the
1
6
THE TEACHING OF ELEMENTARY MATHEMATICS
dying husband
two-thirds of his property
left
if
the
and one-third
child should be a
if it girl, posthumous should be a boy, the remainder in either case to the child; the widow giving birth to twins, one of each
This particular problem appeared in a collection of about 1000 A.D., and is traced back even to Hadrian's time and the sex, required to divide
the property.
schools of law. 1 The title of Alcuin's (735-804) book, " Propositiones ad acuendos iuvenes," and of Recorde's
"The Whetstone
Witte" (1557) show that for the space of nearly a thousand years these problems which were largely the product of "the empty disputations
and the vain
of
subtleties
of the
schoolmen
"
had
their
strong advocates.
In the eighteenth century, when the reasons for teaching the subject began to be considered more
was brought prominently to the of leaders of educational thought. a number by Thus Hiibsch, who certainly deserves to rank among scientifically, this idea
front
" these leaders, remarks that arithmetic
and by
stone,
consecutively,
This
its
and
is like
a whet-
study one learns to think distinctly, 2
carefully."
thought by certain conscientious teachers This to be the end in view in teaching arithmetic. being
is still
postulated,
they
seek
to
reasoning unnecessarily obscure
make and
arithmetical
difficult,
allow-
ing the use of no equation forms, however simple and 1
Cantor,
I, p.
523, 788.
2
Arithmetica portensis, 1748.
HISTORICAL REASONS FOR TEACHING ARITHMETIC
They simply
helpful.
17
conceal the equation in a mass
of words, and cut off the direct path for the sake of
the exercise derived from stumbling over a circuitous
This appears in the subject of compound pro-
route.
portion and in certain methods of treating percentage.
The argument upon
point of
this
unnecessarily hard, begun tury ago,
1
if
is,
German
and
tive,
and
making arithmetic
Germany over
we may judge by coming
text-books,
two countries
in
in
a
recent American to
a
settlement
England, more conservaopen minded in her lower
at least. less
France,
attempt to draw a rigid line between
schools,
still
algebra
and arithmetic, thus perpetuating the
culties of
cen-
the
diffi-
latter.
Scientific investigation of reasons
About the
close
the eighteenth century the reasons for studying
of
mathematics began to be more
The
subject in the training of
necessity for the of people
classes
Arithmetic
began
now began
not for the scientist
scientifically considered.
to
be generally recognized.
be looked upon as a subject and the merchant only, but for the to
soldier, the priest, the laborer, the lawyer,
for
men
in all
all
walks of
life,
and generally
and a subject valuable
in
mental equipment of the youth. 2 was to train for business, but not that alone; to be
various It 1
2
ways
in the
Unger, p. 163.
The
reasons as then considered are set forth by Murhard, System der
Elemente (1798), quoted
c
at length
by Unger,
p.
142 seq.
1
THE TEACHING OF ELEMENTARY MATHEMATICS
8
interesting,
but not that alone; to train the child to
accuracy, to correlate with other subjects, to
way
for science, but
none
of these alone.
pave the
The
devel-
opment and strengthening of the mental powers in general, this was Pestalozzi's broad view of the aim in teaching arithmetic.
"
So teach that
at every step
the self-activity of the pupil shall be developed," was 1 Diesterweg's counsel.
Thus with the nineteenth century the and independence of the pupil come education.
the
many
The atmosphere begins
self-activity
to the front in
to clear.
Out
of
reasons for the study of arithmetic two for-
mulate themselves as prominent, reasons as yet hidden from the mechanical teacher, who is content with an
answer reached by some mere rule of memory and with the recital of a few score of ill-understood definitions or useless principles, but reasons which are leavening the
mass and which
will give
us vastly improved work in
the next generation. 1
Diesterweg and Heuser's Methodisches Handbuch fur den Gesammt-
unterricht
im Rechnen, 3
Aufl., 1839.
CHAPTER WHY Two
ARITHMETIC
is
II
TAUGHT AT PRESENT
In Chapter
general reasons
I
a brief survey
of the evolution of the reasons for teaching arithmetic
has been given.
It
not at
settled
that
assigned
it
time
all
now
has there appeared that it is the subject should have the in
the curriculum,
should be taught for the purpose
a consequence) that teach it.
When we come
it
now
that
or
it
in view, or (as
should be taught as
we now
examine the question of the real reason for the study of mathematics to-day, we find that
we seek
which quite
we may case, "
to
a receding and an intangible something
baffles our attempts at capture.
Indeed,
rather congratulate ourselves that this
is
the
and say with one of our contemporary educators,
For one,
I
am
glad
we cannot
express either quanti-
tatively or qualitatively the precise educational value
of
any study."
1
In a general way, however, we may summarize the reasons which to the world seem valuable, by saying 1
Hill,
F. A.,
Review, IX,
The Educational Value
p. 349.
19
of Mathematics,
Educational
THE TEACHING OF ELEMENTARY MATHEMATICS
20 that
arithmetic, like
(i) for
former
utility, or (2) for
its
included the general
is
and
of the subject
training in logic,
its
other subjects,
culture. 1
its
"
we need
it
Under the
bread-and-butter value
under the
"
its
applications
its
bearing upon ethical, religious,
and philosophical thought. No one will deny that arithmetic two reasons.
taught either
is
;
latter,
taught for these has a bread-and-butter value because
It
in daily
is
in our purchases, in
computand in our our accounts It income, ing generally. has a culture value because, if rightly taught, it trains
one
life,
to think closely
The
of
utility
and
and accurately.
logically
arithmetic
Since
overrated
the
school requires the pupil to spend eight or nine years in studying arithmetic, the general impression seems
be that
to
demand
this is
because arithmetic
is
so useful as to
an expenditure of time. This view "The direct utilitarian cannot, however, be justified. value
has
so great
of
arithmetic
much
been
value
to
the
overestimated;
or,
its
breadwinner perhaps,
it
nearer the truth to say that, while accuracy and
is 1
Fitch, Lectures
on Teaching, 6th
ed., 1884, chaps. x,xi;
of Compayre's Lectures on Pedagogy, p. 379
mathem. Unterricht, Berlin, 1886, Methode des Rechen-Unterrichtes,
p.
101
;
;
Payne's trans,
Reidt, F., Anleitung
Fitzga, E.,
Die
zum
natiirliche
I. Theil, Wien, 1898, p. 44, hereafter Stammer, Ueber den ethischen Wert des mathemat. Unterrichts, in Hoffmann's Zeitschrift, XXVIII, p. 487, and other articles
referred to as Fitzga;
in this journal. R.,
The
best of the recent discussions
is
Die naturgemasse Methode des Rechen-Unterrichts
Volksschule,
II. Teil,
Munchen, 1899.
given in Knilling, in der
deutschen
WHY speed
in
ARITHMETIC
IS
TAUGHT AT PRESENT
fundamental
simple
processes
21
have been
underestimated, the value of presenting numerous and varied themes in pure arithmetic, and of pressing each to great
rated."
and
difficult lengths,
has been seriously over-
!
For the ordinary purposes of non-technical daily life we need little of pure arithmetic beyond (i) counting, the
tion
to
knowledge of numbers and billions
(the
their representa-
English thousand millions), (2)
addition and multiplication of integers, of decimal fractions with not more than three decimal places, and
common
of simple
gers and decimal
Of applied
fractions, (3)
fractions,
arithmetic
and
subtraction of inte-
(4) a
we need
to
little
know
of division. (i)
a few
tables of denominate numbers, (2) the simpler prob-
lems in reduction of such numbers, as from pounds to
ounces, (3) a slight
amount concerning addition (4) some simple
and multiplication of such numbers,
numerical geometry, including the mensuration of rec-
and (5) enough of percentage to compute a commercial discount and the simple interest on a note.
tangles and
The
parallelepipeds,
table of troy weight, for example, forms part
of the technical education of the goldsmith, the tables of apothecaries' measures
form part of the technical
education of a drug clerk or a physician, equation of payments may have place in the training of a few 1
Hill, F. A., in
Educational Review, IX, p. 350.
THE TEACHING OF ELEMENTARY MATHEMATICS
22
bookkeepers, but for the great mass of people these
time-consuming
subjects
How many
value.
no
have
business
bread-and-butter
men have any more
occasion to use the knowledge of series which they
may have tial
gained in school, than to use the differencalculus? The same question may be asked con-
cerning cube root, and even concerning square root;
most people who have occasion to extract these roots (engineers and scientists) employ tables, the cumber-
some method
of the text-book having long since passed
A
from their minds.
like
question might be raised
respecting alligation, only this has happily nearly dis-
appeared from American arithmetics, although it still remains a favorite topic in Germany. Equation of payments, compound interest (as taught in school), compound (and even simple) proportion, greatest com-
mon
divisor,
fractions,
and various
other
These sub-
the same
open to inquiry. which are the ones which consume most of the
chapters jects,
complex
are
time in the arithmetic classes of the grades after the fourth, are
so rarely
used in business that the ordi-
nary tradesman or professional man almost forgets their meaning within a few months after leaving school.
Of compound numbers, which occupy a year the
pupil's
time
in
school (a
year
saved
in
of
most
by the use of the metric system), the amount actually needed civilized
countries
except
the
Anglo-Saxon,
WHY in daily
ARITHMETIC is
life
very
of length, of area, of
TAUGHT AT PRESENT
IS
23
slight.
The common measures
volume
(capacity),
dupois weight are necessary. able to reduce and to
One
and of
avoir-
needs to be
also
add compound numbers, but two or three de-
rarely those involving more than
nominations.
For
the following
is
by
5
Most
useless
6 pwt. 12
oz.
purposes a problem like Divide 2 Ibs. 7 oz. 19 pwt.
practical :
gr.
problems of common fractions are very In business and in science, common frac-
of the
uncommon.
tions with denominators
above 100 are
mal fraction (which has
now become
rare, the deci-
the
"common"
one) being generally used.
What, then, should be expected of a
way
of the utilities of arithmetic?
(i)
child in the
A
good work-
ing knowledge of the fundamental processes set forth
on
p. 21
jects
(2) accuracy
;
which
(3) a
will
arithmetic taught for the
this could
time
rapidity, sub-
knowledge of the ordinary problems
Were
life.
all
and reasonable
be discussed later in this work; and
now
The
of
utilities
daily alone,
be accomplished in about a third of the
given to the subject.
culture value
Although
it is
true that a large
part of our so-called applied or practical arithmetic
is
ordinary business, and generally applicable hence is quite impractical, it by no means follows " Hamlet " that it may not serve a valuable purpose.
not
may
to
bring us neither food nor clothing, and yet a
THE TEACHING OF ELEMENTARY MATHEMATICS
24
knowledge of Shakespeare's masterpiece is valuable It is a matter of no moment in the every one.
to
men
business affairs of most
or
flows,
who Cromwell
know where
that they
the Caucasus Mountains are, or which
and
was,
the Rhine
way
we cannot
yet
afford to be ignorant of these facts. then, can the teaching
How,
the mere elements be justified?
on Teaching," already "
says, it
if
Arithmetic,
conventionally
cited,
arithmetic
of
beyond
" Lectures Fitch, in his
puts the case tersely.
He
deserves the high place that
it
holds
in
our
educational
mainly on the
system,
to be ground Bain remarks in the treated as a logical exercise" same tenor: "All this presupposes mathematics in
deserves
its
it
aspect of training;
ods,
and
or,
that
as
ideas, that enter into
is
it
providing forms, meththe whole
mechanism
of reasoning, wherever that takes a scientific
As
culture imposed
But,
justification.
made prominent
upon
if
so,
shape.
every one, this is its highest these fruitful ideas should be
in teaching ; that
is,
the teacher should
be conscious of their all-penetrating influence. Moreover, he should keep in view that nine-tenths of pupils derive their chief benefit from these ideas and forms of thinking
which they can transfer
of knowledge
problems 1
and
is
;
not the highest end."
Bain, A., Education, p. 152. Scheller, Theorie
to other
regions
for the large majority the solution of
und
1
See also Fitzga,
p.
27;
Praxis des Volksschulunterrichts,
Rein, Pickel I,
p. 350.
WHY
ARITHMETIC
In other words,
some
But
training in logic.
Hence
tunity for this
there
that
25
to give the child
logic as a science is too
the school substitutes, that
the time,
at
subject, which,
in
seems advisable
it
abstract for him.
TAUGHT AT PRESENT
IS
offers
This
training.
is
the
best
oppor-
the more valuable,
incidentally accomplished another keeping of the numerical machinery in use
result, the
while the child
is
in school, so that his
leaves.
powers of calbe unimpaired from inactivity when he Arithmetic is well chosen for this training
in logic,
because
is
culating will
of an exact
American
it
furnishes almost the only example
science
below the high school, as the
courses are usually arranged.
more valuable
induction
is
and while
it
to the child
And although than deduction,
must be the keynote of primary arithmetic,
deduction plays an important part in the latter portion The fact that the child finds a posiof the subject.
an immutable law, at the time
tive truth,
ment when he desire
to
is
naturally
investigate,
must put away
and with the feeling that he
childish
"
most
unkindest
Columbus was the he
is
sure,
whatever lives, (a
"
real
is
bad
grammar, or that but discoverer of America is
;
and no argument can shake
may happen
+ #)
2
will
difficult
not sure that every every animal needs oxygen,
has petals, that
flower
things, has a value
He
properly to appreciate.
that
in his develop-
with doubt, with the
filled
to the
his faith, that
universe in which he
2 always equal a
+ 2 ab + ^.
26
THE TEACHING OF ELEMENTARY MATHEMATICS
So arithmetic may, even by obsolete problems, train the mind of the child logically to attack the every-day problems of
If
life.
he has been taught to think in he will think in solving
solving his school problems,
the broader ones which
The same forms
of logic,
exercised
common merce, sions.
gives
the same
in
solving a
may show
divisor,
problem
itself
in
checking
in
greatest
com-
years later in
banking, or in one of the learned profes-
in
Hence, arithmetic, when taught with to
meet.
thereafter
attention to detail,
and the same care
the same patience, results
must
he
this in
the pupil not knowledge of facts
mind,
alone, but
that which transcends such knowledge, namely, power. It
must
not, however,
be thought from
phase of the subject
this culture
is
its
name
that
of value only as
a luxury, like the ability to dabble in music or painting.
Just because
it
is
or moderate circumstances
way
in the world,
this culture
phase
man in poor who must make his own the common people that
the child of the
for
it
is
is
most valuable.
Teachers generally teacher of arithmetic
The lower elementary
fail here is
usually
the one in the higher
grades.
reasons for this
has been
much
been written rare,
the
more successful than There
primary part of
are
several
the subject
better investigated, better books have
about
it,
and the child
in
face the nervous
good
higher arithmetics are
the lower grades has not to
shock which
comes a
little
later;
WHY but one
ARITHMETIC the
of
reasons
chief
knows why she
teacher
TAUGHT AT PRESENT
IS
that
is
the primary
teaching arithmetic, while
is
often the one in the higher grades does not.
grade the subject
first
27
In the
being taught largely for
is
its
utilities, and induction plays the important part; this the teacher knows and hence she succeeds. In the
seventh grade the teacher tion
rise to
is
much poor
teaching.
Recognition of the culture is
apt to think that induc-
plays the leading role, an error which gives
still
brought out
first
This culture value
value
by letting the amount taken on
authority of the book or the teacher be a
"In education the process
be encouraged to the uttermost.
make
led to
own
their
inferences.
own
and induced
possible,
Children should be
investigations
They to
minimum.
of self-development should
and
to
draw
be told as
should
discover as
much
their
as
little
as possi-
knowledge which the pupil Any piece has himself acquired, any problem which he has himself solved, becomes by virtue of the conquest
ble.
.
.
of
.
much more thoroughly This is
to
is
his than
not to be construed to
equals.
equals
result
a triangle
that nothing
must assume,
from adding equals
But when Euclid was
that one side of
mean
We
be taken for granted.
example, that
could else be." 1
it
is
for to
criticised for less
proving than the sum of
the other two, as having proved what even the beasts 1
Spencer, Education.
THE TEACHING OF ELEMENTARY MATHEMATICS
28
know,
were entirely right
his disciples
saying that merely teaching facts, but were enin
they were not gaged in the far more important work of giving the power to prove the facts. As Bain puts it, referring to
the higher grades,
to feel that
"
The
pupil
should be
he has accepted nothing without a clear
and demonstrative reason,
to
the
entire
exclusion
authority, tradition, prejudice, or self-interest"
long
lation to
of
of
1
be said of text-books which give " as a kind of inspired revePrinciples shall
What, then, lists
made
"
So
pupils?
far as these are statements of
business customs they have place; but they are generally theorems,
capable of easy proof, and of no great
value without this proof.
Furthermore, of
if
we would make
a
clear thinker
the pupil, he should not be compelled to learn,
verbatim,
all
the text-book. are true
or even a majority of the definitions of
This does not exclude
those which
and understandable and valuable
quent work;
but
it
refers
to those
in
which are
subsefalse,
unintelligible, and not usable, and to partial definitions in all cases where the memorizing of the same hinders
the comprehension of the complete definition subse-
For example, what teacher of arithmetic can define number in such way as to have the definition quently.
both true and intelligible to young pupils, those below the high school? And if he could do so, of what 1
Education,
p. 149.
WHY
ARITHMETIC
value would the
be
it
Or who would l The quantity ?
term
the
simpler
?
of
definition
TAUGHT AT PRESENT
IS
more
the
29
care to undertake fact
is
that
the
the definition.
difficult
Since a definition must explain terms by the use of terms more simple, it follows that one must sometime
come
to terms incapable of definition. 2
we do
not
learn
verbatim
definitions
;
In daily life if asked to
would probably include the mule and zebra and numerous others of the equine define horse, the definition
The
family.
usual
hindered the work of yet,
even
in the first
While
tion J.
complete ones,
grade he multiplies by the
poor
has
multiplication
a child in fractions, and
many
true
is
it
it is
of
definition
that partial truths
frac-
precede
teaching to impress this partial
truth on the
mind
so indelibly,
ment, as to
make
the complete truth difficult of as-
For example, a teacher
similation.
memorize the proportion
than
less
by a memorized
is
fiction
that
a class to
the second term
if
than the
less
drills
first,
the logical treatment of proportion, and then,
To
come
1
test the
i
matter a
La Mathematique,
or the simple definition of
Wissenschaften, 2
meet
to
Duhamel,
sonnement.
I.
2
:
little
Those who may be ambitious
Laisant,
of
a
the fourth must be
a statement entirely unnecessary in
third,
the pupils
state-
to
liere partie,
2
:
further,
make
when
4, they are let
lost.
any reader
the attempt might
first
read
Paris, 1898, p. 14, hereafter referred to as Laisant,
number
in the
Heft, Leipzig, 1898,
J.-M.-C,
=
Encyklopadie der mathematischen
now
in process of publication.
Des Methodes dans 3^me
d.,
Paris,
les
Sciences
1885, p. 16.
de
Rai-
THE TEACHING OF ELEMENTARY MATHEMATICS
3O
repeat the definition of number, as
it
was once burnt
his memory, and see if TT(= 3.14159 is a ) number according to this definition, or V2, or V i. Or try the definition of arithmetic and see if, by this
into
statement, the table of avoirdupois weight
Does the
of the subject.
is
any part
definition of multiplication,
as usually memorized, cover even the simple case of
f X f
to
,
common
the
By By
V2 x V^
say nothing of
the definition of square
V
or
definition of factor root,
is
i
x
V
3
?
\ a factor of \
?
as usually learned,
have we any right to speak of the square root of 3, since 3 has not two equal factors? Are our arithmetics clear enough in statement so that the memoriz-
ing of their definitions will
simple series
2,
2,
2,
2,
tell
a pupil whether the
is
an arithmetical or a
geometric progression, or neither
The
?
argument that learning definitions strengthens the memory and gives a good vocabulary, has too few old
advocates role
now
of the
make
to
memory,
it
worth consideration.
way It
teaching.
necessary in matters should be reduced in a
certainly
mathematical as elsewhere, general
"The
to
very limited proportions in rational
is
not the images, the figures,
or the
formulae which must be impressed upon the mind, so
much 1
as
it
is
the power of reasoning."
l
" Ce ne sont pas les images, figures ou formules, dont
laisser 1'empreinte
Laisant, p. 191.
dans
le
cerveau
;
c'est la faculte
il
faut surtout
du raisonnement."
WHY
ARITHMETIC
IS
TAUGHT AT PRESENT
31
This opposition, on the part of leaders in education, burdening of children's memories, is not new.
to the
"
Locke voiced the same sentiment
:
leave to take notice of one thing
ordinary method
of education
;
I
And
here give
me
think a fault in the
and that
is,
the charging
of children's memories, upon all occasions, with rules and precepts, which they often do not understand, and
constantly as soon forget as given."
one time believed that the instruction
Of
fact,
the verbal
faculties of our nature
the two, to learn
all of
or none, the latter plan
But while memorized be
first
"
Teachers at of
object
to cultivate the verbal
when, in
pupils,
few
is
1
is
primary of their
memory memory is one
which need no
of the
cultivation."
2
the definitions of a text-book
unquestionably the better. may not unfrequently
definitions
justified, this is rarely true of
the memorized rule.
The glib recitation of rules for long division, greatest common divisor, etc., which one hears in some schools -what is all this but a pretence of knowledge? "If a process of gaming knowledge, that is, a true apprehension of realities, it excludes verbal memis
learning
orizing,
cramming, and everything that resolves
itself
on close scrutiny into a pretence of knowledge getting." 3 But not only is this old-fashioned rule-learning (unhappily not yet extinct) a sham it is wholly unscientific. Tillich, one of the best teachers of arithmetic of the ;
2
1
On
8
Dr. James Sully, in the Educational Times, December, 1890.
Education, Daniel's edn., p. 126.
Tate.
THE TEACHING OF ELEMENTARY MATHEMATICS
32
half of the nineteenth century,
first
dogmatic
"It
rules.
he
is,"
said,
saw the danger of "just as unpsycho-
logical to begin the teaching of arithmetic
of inherited rules as
guage .
.
.
by means
to children
by a mass
senseless to try to teach lan-
it is
mere
of
rules of speech.
Since these rules were not independently worked
out by the child, but are simply the memorized results of others' work,
it
cannot but be true that the
metic of most of the pupils a distasteful one at that."
l
arith-
a mere mechanism, and
is
So, too, Jean Mace, in his
well-known "Arithmetic of a Grand-Papa," remarks that to have a child begin with the abstract rule, following this by the solution of a lot of problems, pletely reverse the order of
There
is
to
human development.
com-
2
are, however, a few rules of operation which
must be learned
for the sake of facility
numerical calculation.
Such
is
and speed
in
the rule for substituting
another and a simpler operation for that of dividing one
a rule is
is
to
child to discover
Roger Ascham
1
"We
this
does not
and it
far
more
that such
Even
for himself.
was
It
valuable, to lead the
as far back as
though seldom pracdo not contemne rewles," said he, "but this
realized,
Lehrbuch der Arithmetik, p. xi. In a similar line, Reidt, Fr., Anzum mathematischen Unterricht an hoheren Schulen, Berlin, 1886,
leitung
p. 103. 2
mean
be given as a kind of inspired dogma.
quite as easy,
tised.
But
by another.
fraction
L'Arithmetique du Grand-Papa, 4^mc
d., p. 12.
WHY we
gladly teach rewles
and
sensiblie,
in
ARITHMETIC
common
method
;
IS
TAUGHT AT PRESENT
and teach them more
plainlie,
orderlie than they be
scholes."
And
l
commonlie taught the best of summaries of
that has recently appeared asserts
would bring
33
his pupils to intelligent
" :
Whoever
computation (zu
Rechnen) should develop no rule, but should wait until the children themselves diseinen verstandnisvollen
cover
(bis die
it
Kinder selbst darauf kommen)."
2
Aside from the fact that we make almost no use of operation in our daily computations, a few rules of business and theorems of but needing mensuration, there is the further consideration that of the
rules
the child does not like to solve by rule.
common zeal for
sense
is
discovery
to is
become a
To
discoverer,
one of the inborn
use his
and
traits of
the
the
human mind. If all mathematical problems were solved, or if we had rules for solving them, all interest in the subject
Of course
even greater measure as to undemonformulae, which are merely rules put in un-
strated
familiar
of
"
think,
To fill the child's mind with a language. formulae for percentage, for example, is to human
take a it.
the same objection which exists as to
exists in
rules
list
If all
soul
and
try to
make a machine
of
one learns only by memory, and does not remains dark." 3
What, then, 1
would vanish.
shall
The Scholemaster.
D
be said of the educative value 2
Fitzga, p. 48.
3 Confucius.
THE TEACHING OF ELEMENTARY MATHEMATICS
34
old-fashioned arithmetic which put its problems in "cases," each preceded by the rule? Surely a more mechanical device could hardly be invented.
the
of
And
books
these
yet
exist
to-day in thousands
And
schools in
England and America.
that these
books in the schools of let it
if
of
be said
it
fifty years back not be forgotten
produced good arithmeticians, more time was then given to the subject. Good arithmeticians were produced in spite of, not
that far
because
so
of,
such books.
What chapters bring out the culture value much the particular chapter as the way it
that brings
A
person
not
taught
out the educational value of arithmetic.
may have
exercise in logic
by studying allian awk-
merely indeterminate equations in
gation
ward mediaeval form.
come from those and
It is is
But the best
results will naturally
parts that appeal to the child's
life
interests.
For example, longitude and time, a subject with may be
but slight utilitarian value to most people, so taught est
as to have
high culture value.
The
inter-
attaching to the "date line" and to the recent
world-movement of "standard time," renders the subject a delightful one to children of a certain age.
But
its
"75
-j-
value 15
=
5
is
lost
when
hrs.," since
it
a book gives the form destroys the child's pre-
conceived and correct ideas of the nature of division
accuracy
of
statement
and
of
thought
have
;
been
WHY sacrificed
ARITHMETIC
IS
TAUGHT AT PRESENT
a mere answer,
for
right sold for a
mess
may be made
and the reasoning may give
But
this,
like
the teacher, gives a
is
wrong idea
to
the
fatal
of business.
interest-
rise to logical
other subjects that at once
open
birth-
of pottage.
Similarly, "true discount" ing,
an arithmetical
35
power.
occur to that
objection
it
However much the
pupil may be warned, the name "true discount" will cling to him, and he must learn, after his school
days have gone by, that the true discount in the life he is to live.
What may well may profitably be
be omitted
is really
the false
In considering what
omitted from the arithmetic of
to-
is, of course, the bugbear of the examinabe taken into account as a practical question.
day, there tion to
But
looking at the subject from the standpoint of
the educator rather than the coach, sider
what there
utilitarian
is
that
appeals
we have
to con-
neither
to
the
is
found
little
or no
nor to the culture value, or that
wanting for other reasons. i.
The
following
may be
utilitarian value for the
said to
have
general citizen, and because
give a false notion of business they
they be rejected as undesirable exercises in
may
also
logic.
(a) Equation of payments.
(now rapidly disappearing from Engand American text-books, although still found in
(b) Alligation lish
the German).
36
THE TEACHING OF ELEMENTARY MATHEMATICS (c)
Insurance, in the form usually presented in text-
books. (d)
" Profit and Loss," the text-book expression not
American
meaning, and the problems being merely ordinary ones of simple perthe
having
business
centage, not worthy of a special chapter. (e)
Exchange as usually taught.
business
chinery
If
the modern ma-
problems are given, with for
the
exchange,
subject
modern
the
is
Of
valuable.
course arbitrated exchange has no value per se for the ordinary citizen;
it
is
part of the technical train-
ing of a few brokers.
(/) Commission and brokerage so far as the subproblems like the following B $1000 with which to buy wheat on a mission: how much can B invest?"
ject relates to
() many
Stocks,
where
text-books,
the
problems
fractional
numbers
A
sends
2^%
com-
" :
require,
of
as
shares,
in like
the buying of 8f shares, or where they call for un-
used quotations like iQQ^f. (h) Partial in the state (z)
payments beyond the common methods in which the pupil lives.
Annual
interest,
beyond the mere elements.
Compound interest, beyond the ability to find such interest. The banker, of course, employs tables .
(/)
whenever he has occasion (k)
Compound
to use the subject.
proportion,
hardly a text-book problem
a
subject
in
which
can be found that has
WHY
ARITHMETIC
subject.
TAUGHT AT PRESENT
37
in spite of the pretensions of the
any practical value,
As
IS
for mathematical explanation,
would be
it
a text-book which makes any attempt
difficult to find
in that direction.
Problems
(/)
more than three denominations
tun,
stone
the
the pipe,
square
the
etc.,
the
perch,
common
troy,
the
barleycorn,
the
quintal,
etc.,
There
schools.
is,
somewhat serio-comic aspect
of the matter
"A
gallon isn't a
as set forth in the Football Field:
a wine gallon, or one of three different
It's
gallon.
life
daily
and the technical measures, the
have no place in the indeed, a
and those
in
America),
(in
shingling),
(in
involving
at a time,
Similarly the semi-obsolete meas-
etc.
apothecaries', ures,
needed
not
tables
involving
numbers
denominate
in
sorts of ale gallon, or a corn gallon, or a gallon of oil;
and a gallon of
pounds other
for
train If
oils.
do
oil,
and
hundred
Teneriffe.
pounds of a lock,
for
pounds
if
Madeira, a
sixteen
living of
some
hundred and
Bucellas, a hundred and three
if
if
eight
half
you buy a pipe of wine, how much Ninety-three gallons if the wine
you get? be Marsala, ninety-two seventeen
means seven and a
oil
What
man,
cheese,
is
if
Port, a
Fourteen
a stone?
eight of a slaughtered' bulfive
of
glass,
thirty-two
hemp, sixteen and three-quarters of flax four and twenty of flax at Downpatrick.
at
of
Belfast,
It is four-
teen pounds of wool as sold by the growers, fifteen
THE TEACHING OF ELEMENTARY MATHEMATICS
38
pounds of wool as sold by the wool-staplers to each other. Our very sailors do not mean the same .
.
.
thing when they of-war
means
it
and a half
Of
course
board a man-
on board a merchantman
on board a fishing vessel
we may
all
that,"
And
yet
changed sense.
six feet,
feet,
On
talk of fathoms.
say that in America
five feet."
a
many
or can know, because
"we have
school
to-day teaches the
nobody knows
it
varied,
and our various
have different laws and customs as to what
states consti-
perch of stone, etc., and are quite as unsettled with respect to many meas-
tutes a bushel
ures as
"Of
is
grain, a
of
Great Britain. has been some reform in this
late years, there
particular (the applications of arithmetic), of
1
and that we have no such non-
children the length of the cubit, which
we
five
the
monstrosities
our ancient enemy,
old
curriculum, notably
duodecimals,
have been thrown
of
But there
overboard.
the
and a few
still
remain
many
things,
as
taught in our schools, which occupy time that could better be devoted to the study of other subjects, or at
least
operations. portion,
a
to .
greater
.
.
degree
Compound
compound
of
practice
interest,
partnership,
in
simple
compound
cube
root
and
proits
applications, equation of payments, exchange, 'similar
and the mensuration of the trapezoid and trapezium, of the prism, pyramid, cone, and sphere, surfaces,'
1
Educational Times, October, 1892.
WHY
ARITHMETIC
TAUGHT AT PRESENT
IS
39
are proposed to be dropped from the course in the
grammar
(Boston) 2.
The
following
have
might
school."
may be
much,
where
treated (b)
because
Series,
it
said
culture
omitted on other grounds. (a)
*
belongs,
value,
but
should
be
subject
can
better
be
2
the
The long form
have some, and
to
in
of
algebra.
common
greatest
before about the eighth grade, because
only for
its
logic,
and
it
much
this logic is too
divisor
is
taught for the
average child below that grade. (c)
Compound
proportion,
cause almost no arithmetic pretends to treat wise than by rule, and an explanation is too for pupils
doubtful
if
as apparently for authors.
the child derives
be-
mentioned,
already
it
other-
difficult
Indeed,
it
is
much good even from
simple proportion as usually presented. Relative
value
that
appears
of
culture
arithmetic
is
and
Since
utility
taught
for
these
it
two
general reasons, a question arises as to their relative
importance.
But
this
it
is
lack a unit of measure. 1
impossible to answer.
Laisant remarks 3 that
We it
is
Walker, F. A., Arithmetic in Primary and Grammar Schools, Boston,
1887, p. 12. 2 " The largely
charge I make against the existing course of study is, that it is of exercises which are not exercises in arithmetic at all, or
made up
principally, but are exercises in logic logic, they 8
;
and, secondly, that, as exercises in
are either useless or mischievous"
La Mathematique,
p. 10.
Walker,
Ib., 17.
THE TEACHING OF ELEMENTARY MATHEMATICS
40 like
asking which
the loss of either
sleeping;
who
the more important, eating or
is
is
The
fatal.
teacher
recognizes in the subject only its applications to
trade,
would better give up teaching; the one who it only an exercise in logic will also fail
sees in
but the
;
failure
greatest
comes from seeing
subject neither utility nor logic, as
the teacher
who
the case with
is
the
blindly follows
the
in
old-style, tradi-
tional text-book.
But what
shall
to omit certain
be said for the teacher
problems which are not
whose culture value
is
who
fears
and
utilitarian
counterbalanced by the fact
that they give a false notion of business, or to omit
those traditional puzzles which depend for their
diffi-
ambiguity of statement? Many a in our will confess teacher, especially country schools, to such a fear of omitting problems, lest he be acculty
upon
their
cused of inability to solve them. for all teachers to assist in
It
would be well
creating a sentiment in
favor of omitting the unquestionably superfluous
dangerous, It
and
thus
to
avoid
this
weak
or
criticism.
should also be understood by timid teachers that
that
no disgrace to be unable to solve every puzzle may be sent in, or even every legitimate problem.
And
for those
it
is
who may
feel inclined to boast that
they
have never seen a problem in arithmetic which they could not solve, it may be interesting and instructive to attempt to prove the following simple statements
:
WHY
ARITHMETIC
The sum
IS
TAUGHT AT PRESENT
41
same powers (above the second) of two integers cannot equal a perfect power of the same degree. (In the case of the second degree there are
of the
any number of examples, as
2
3
+ 42 =
2
5 .)
Fermat's theorem.
Every even number bers.
The
is
the
sum
of
two prime num-
Goldbach's theorem. consecutive
integers
8
and
9
are
powers; are there any other consecutive which are exact powers ? Catalan.
exact
integers
CHAPTER How
III
ARITHMETIC HAS DEVELOPED
The
Reasons for studying the subject
historical de-
velopment of the reasons for teaching arithmetic has For the well-informed already been considered. remain two other historical questions
teacher there
The
of importance.
of the subject
teaching
itself,
the
ing briefly
individual
of
somewhat
had
grow
it
as
if
ematics,
be
it
high or low,
first
as the world learns. 1
own mathematics,
But
do not propose the race had not grown
to do.
When, however, we
too.
In the
arithmetic.
should grow his
just as the race has
that he should
sufficient reasons for consider-
history
place, the child learns
it
development methods of
to the
it.
There are good and
"The
relates to the
first
and the second
set
I
before him math-
in its latest,
and most
generalized, and most compacted form, we are trying to manufacture a mathematician, not to grow one." 2
This does not 1
mean
that the child must go through
Cette longue education de 1'humanite, dont le point de depart est
loin de nous, elle
L 'Arithmetique 2
Jas.
Ward
recommence en chaque
du Grand-Papa,
4ifeme
petit
d., p. II.
in the Educational Review, Vol.
42
enfant.
I, p.
100.
si
Jean Mace,
HOW
of the stages of mathematical history
all
of the
ered
way is
"
"
culture-epoch
the in
world
theory
43
an extreme
but what has both-
;
and the
usually bothers the child,
which the world has overcome
suggestive of the
come us
ARITHMETIC HAS DEVELOPED
way
similar ones in his
in
its
difficulties
which the child may over-
own development.
In the second place, the history of the subject gives a point of view from which we can see with
clearer vision the relative subjects,
future
is
what
is
obsolete in the science, and what the to
likely
importance of the various
demand.
Sterner 1 has compared
the teacher of to-day to a traveller
who by much
toil
has reached an eminence and stops to take breath before attempting further heights
;
he looks over the road
by which he has journeyed and sees how he might have done better here, and made a short cut there,
and saved himself much waste yonder.
ment
of
So one who considers the
of arithmetic
and
its
time and
energy
historical develop-
teaching
will
see
how
enormous has been the waste of time and energy, useless has been much of the journey, and how
how
have crept
certain chapters
tant
and remained long
useless.
He
in
when they were imporbecame relatively
after they
will see the subject as
from a mountain
instead of from the slough of despond which the text-
book often presents, and he
will
to teach with clearer vision, to 1
be
able, as a result,
emphasize the impor-
Geschichte der Rechenkunst.
THE TEACHING OF ELEMENTARY MATHEMATICS
44
and thus
tant and to minimize or exclude the obsolete,
himself and of his pupils.
save the strength of
to
He
also
will
some
learn that
the most valuable
of
parts of arithmetic knocked at the doors of the schools
long centuries before they were admitted,
had
and that
to
struggle long and persistently banish some of the most objectionable matter. As
a
result,
teachers have
while
to
condemn the conservatism
he may
which excludes the metric system and logarithms and certain of the more rational methods of operations today, he will have
more
faith in the
ultimate
success
a good cause and will see more clearly his duty
of
as to its advocacy.
Extent of the subject
It
is
to give
more than a glimpse
metic.
The simple
one.
De Morgan's
the history of arith-
question of numeration, discussed
with any fulness, would 1
manifestly impossible
at
fill
masterly
a volume the size of this little
work, "Arithmetical
Books," hardly more than a catalogue (with notes) of certain important arithmetics in fills
one hundred twenty-four pages.
dent
who
his library,
For the
stu-
some suga subsequent chapter. But for
cares to enter this fascinating field
gestions are given in
the
2
critical
present purpose
few important events
it
in
suffices
to
consider merely a
the general
development of
the subject. 1
See, for example, Conant, L. L.,
2
London, 1847.
The Number Concept, New York,
1896.
HOW The
ARITHMETIC HAS DEVELOPED
first step
The
counting
first
45
step in the his-
development of arithmetic was to count like things, or things supposed to be alike; in the broad sense of the term this is a form of measurement. 1 torical
Arithmetic started
when
it
ceased to be a question of
more than that, group and began to be recognized that this group was three and that two; when it was no longer a matter of a stone axe being worth a handful of arrow heads, but of savage warriors being
this
one of an exchange of one axe for eight arrows. How far back in human history this operation goes is
it
how
impossible to say, just as far
counting
it
back human history is
not
limited
to
is
impossible to say
itself
the
goes.
human
Indeed, for
family,
ducks count their young and crows count their enemies. 2 Any discussion of the nature of this animal counting must lead to the
broader question of the ability to think without words, a matter so foreign to the present subject as to have no place here. 3
The
race has not, however, always counted as
present. 1
It
was a long struggle
2
3
may
not fully
McLellan and Dewey's Psychology of Number,
York, 1895.
This subject of animal counting has often been discussed.
briefly treated in the
and
at
know numbers up
In this connection the teacher should read, though he
indorse, Chap. Ill of
New
to
also in Conant's
For
Thought.
Max
It
is
chapter on Counting in Tylor's Primitive Culture,
Number Concept mentioned on
p. 44.
Muller's side of the case see his lecture on the Simplicity of
THE TEACHING OF ELEMENTARY MATHEMATICS
46
The
to ten.
savage counted on some low
primitive
scale, as that of
"i, 2, many," often says, " i,
two or
or 2,
three.
" i,
3,
2,
many," just as the child lot," and somewhat as we
3,
a
4,
To him numbers were
count up very far and then talk of "infinity." It is evident that there must be some systematic
arrangement of numbers in order that the mind may hold the names. For example, if we had unrelated
names
for even the first
be a very
difficult
hundred numbers,
would
it
matter to teach merely their
quence, to say nothing of the combinations.
se-
But by
counting to ten, and then (or after twelve) combining the smaller numbers with ten, as in three-ten (thirtwice-ten (twenty),
and
number system and the combinations
are
teen), four-ten (fourteen),
so on, the
not
.
.
.
difficult.
We
might take any other number than ten for the
base (radix).
If
we took
three
we should
count,
one, two, three, three-and-one,
three-and-two, two-threes,
.
.
.
,
and (with our present numerals) write these, i,
2,
3,
ii
(i.e.,
one three and one
unit), 12, 20,
...
l .
But most peoples, as soon as they were far enough to form number systems, recognized the
advanced 1
A
brief but interesting
summary of
this subject is
given in Fahrmann,
It K. E., Das rhythmische Zahlen, Plauen i. V, 1896, p. 21. treated in numerous text-books and elementary manuals in English.
is
also
HOW
ARITHMETIC HAS DEVELOPED
47
machine, their fingers, and hence count on the scale of ten (our decimal
natural calculating to
began
system).
" In
Aristotle,
the
'
do
Why to
up
all
10,
suggested,
the book of Problemata, attributed to
asked (XV, 3): men, both barbarians and Hellenes, count following question
and not
'
fingers
number scale/
"
some other number?'
to
It
is
several answers of great absurdity,
among
may be
that the true reason :
is
that
all
men have
ten
using these, then, as symbols of their proper
(viz.,
they count everything else by this
10),
!
To-day
it
common
is
to
hear teachers
allowing a child to count on his fingers.
object to
And
yet
one of our best teachers of arithmetic has just remarked, what is indorsed both by history and by com-
mon
sense, that the fingers are the
most available material.
2
most natural and
It is true that there is
some
ground for the objection, especially on the part of teachers who have not the ability to lead children to rapid oral in
this
work
way we
;
but
should
if
the world had not counted
not have
had our
decimal
system. It is really
sidered, that
1
a
little
man
unfortunate, arithmetically con-
has ten instead of twelve fingers,
J., History of Greek Mathematics, Cambridge, 1884, Chap. I. Die Finger sind also das naturlichste und nachste Versinnlichungsmittel. See also Conant's Number Concept, p. 10, Fitzga, p. 82, 14, 59.
Gow,
2
et pass.
THE TEACHING OF ELEMENTARY MATHEMATICS
48
for the scale of twelve
A
radix
is
the easiest of
must not be too
much
require too
small,
would
that
labor in writing comparatively small
For example, on the
numbers.
the scales.
all
since
would appear as 112
2
(i-3
+
1-3
of
scale
+
fourteen
3,
Neither should
2).
the radix be too large, since there must be ten figures
twenty for the radix twenty, and
for the radix ten,
so
and
on,
too
Twelve, like
many characters are objectionable. is a medium radix; but it is better
ten,
than ten because instance, the i
J>
has more divisors.
it
fractions
most commonly used,
viz.,
J,
These are written
i-
on the scale of
10,
0.5,
0.333,.
on the scale of
12,
0.6,
0.4,
Hence
Consider, for
.
.,
0.25,
0.125;
0.3,
0.16.
the advantage of the duodecimal scale, in
all
work involving fractions, is apparent. Counting must have preceded notation by many generations,
just
as talking
And
preceded writing.
while there are good reasons for teaching the numerals
to
character
a child while "
" 3
while he
things), Pestalozzi
ment on
his
side
he is
is
learning
number
(the
learning to pick out three
had the argument of race develop-
when he advocated teaching
the
characters only after the child could count to ten.
And
in teaching
the
child
number, while
be very logical to introduce the ratio idea idea
which
Newton
crystallized
in
his
it
first,
would the
well-known
HOW definition
of
ARITHMETIC HAS DEVELOPED
not in harmony development of the race; first,
number,
the historical
with
the plan
49
is
counting; second, simple operations; third, a notation; this is the
race
order.
Aside from
all
this,
there
is
the more serious question, discussed in a subsequent chapter, as to the psychological phase of the matter;
whether the for
ratio idea is not altogether too abstract
the mind of the child beginning to study num-
can be taught, but its success means a good teacher with a poor method, a David with a sling. It
ber.
While the introduction of the idea is
to
in the beginning
unwarranted by considerations historical, and seems be so by considerations psychological, it is desir-
able as soon as the child has developed sufficiently to
allow
it.
enough is.
The matter has
to
investigated, however,
Laisant,
affairs,
not yet
who
does
not
lose
questions whether the
tell
his
been just
carefully
when
head
in
this
such
ratio idea, usually rele-
the elementary course,
gated to the later years of
should not enter very early, but after careful con" sideration is forced to the conclusion that number, in its
elementary form, comes to us by the evaluation
of collections of like objects."
The second
step
l
notation
Of course
there de-
veloped in connection with counting a certain amount But the the simplest operations. of calculating
second step of great importance was that of writing 1
La Mathematique,
p. 30, 31.
THE TEACHING OF ELEMENTARY MATHEMATICS
5O
The
numbers.
Hindu
many that
we
plans with which
are familiar, the
(" Arabic ") and the Roman, are only two of which have been used. The primitive one was
simple notches in a stick or scratches on a
of
stone.
But of
scientific
systems there are only a few
types.
The Egyptians had
symbols for
in general plan,
powers of
The
IO.
much
a system
like the
100 and higher
10,
i,
Roman
1
Babylonians,
not
stone possessed by the
abundance
the
having
of
Egyptians, resorted to writ-
bricks, which were then baked. They therefore developed a system which required but a
ing on
soft
few characters such as could a stick upon
Their symbols were three, one for ioo. 2
The
early
words for
5,
easily
be impressed by
clay, the so-called cuneiform numerals.
one for
Greeks used the 10,
ioo, 1000,
i,
initial
one for
10,
and
letters
of
the
10,000, a plan leading to
a system about like the Egyptian and Roman. The late Greeks and the Hebrews used their alphabets, giving
to
each
letter
Greeks used a for
a
number
for 2,
value.
for
i, /3 7 an old form called digamma for
5,
1
Cantor
is,
3,
6,
of course, the standard authority on
all
Thus the
8 for 4, e for for 7,
rj
such matters.
for
A
given in Sterner, p. 17 seq. good summary 2 They are given in Beman and Smith's translation of Fink's History is
of Mathematics, Chicago, 1900.
HOW and 9
8,
for
ARITHMETIC HAS DEVELOPED 9.
The next
extra symbol, stood for tens,
and the
etc.,
hundreds.
rest,
nine i
letters,
10,
K =
51
with one
20,
X
=
30,
with one extra character, for the
The system was a
difficult
one to master,
enabled the computer to write numbers below 1000 with few characters. For example, 387, which
but
the TTrf.
it
Romans wrote CCCLXXXVII,
the Greeks wrote
1
The Romans used a system the essential features which are known to all. The origin of the symbols has long been a matter of dispute, but they are now of
generally recognized to be modified forms of old Greek letters,
not found in the Latin alphabet, which came
2 The Romans inthrough the Chalcidian characters. troduced the "subtractive principle" of writing IV
for
5
1,
successors
XL
for
50
10,
etc.,
but they and their
made
little use of it. The tendency to IV is still seen on our clock faces. The number was rarely used, the number usually
write II 1 1 for
bar over a
being written out in words
if
above thousands, while the
double bar sometimes seen in American examination
and the idea that a period must follow a numeral, may be called stupid excrescences of
questions,
Roman
the nineteenth
century.
1
The
fact
that the
Romans
For more complete discussion see Cantor, I, p. 117, or Sterner, p. 50. Wordsworth, Fragments and Specimens of the Early Latin, p. 8 ; Fink's History of Mathematics, English, by Beman and Smith, p. 12 ; 2
Cantor,
I, p.
486; Sterner,
p. 78.
THE TEACHING OF ELEMENTARY MATHEMATICS
52
did not large
make
practical use of their system in writing
numbers should show us the criminal waste
time in
of
requiring children of our day to bother with
the system beyond thousands.
The Hindu back the
Nana Ghat, in and first made known
Bombay the
to
(or so-called Arabic) system can be traced
certain inscriptions found
to
at
Presidency (India), western world in 1877.
These
inscriptions
probably date from the early part of the third cenl tury B.C. and seem to prove that the numerals from
4
to
9 inclusive were the
Bactrian alphabet.
the ancient that
time,
better than
no
zero,
and
for
zero
many
on.
And
words in
The system was
centuries thereafter,
several
one element of
element,
we cannot
2
others of antiquity, because
without which
the place-value
of
letters
initial
is
while the
hundred
place value was
no
had
superiority,
Without
wanting.
write ten, one
it
at
six,
the
and so
somewhat ap-
preciated as early as the time of the cuneiform nu-
seem
have appeared in the Hindu system before 300 A.D., 3 and the first known use of the symbol in a document dates from merals, the zero does not
to
fdur centuries later, 738 A.D.*
There the 1
much
question as to the first
way
in
Cantor,
XVI,
p.
325
seq.,
564.
p.
especially 347. 8
I, p.
which
entered the western world.
See Journal of the Royal Asiatic Society, 1882, N.S. XIV,
1884, N.S. 2
is
Hindu numerals
Ib., p. 567.
4
Ib., p. 563.
336
;
HOW
ARITHMETIC HAS DEVELOPED
53
Sporadic use of the characters is found before the But about 1200 A.D., Leonardo thirteenth century. Fibonacci, of Pisa, returning from a voyage about the
Mediterranean, brought them to Italy. Being then in use in various Moorish towns, they received the
name "Arabic," although more than
nothing
the
borders of
to
the Arabs
may have done
disseminate them along the
Occident.
as
If,
is
not
probable,
1
they invented the zero, they deserve to have the name "Arabic" continued, but if not, the title "Hindu nu-
much
be preferred. It was nearly a century later than Leonardo's time before the system had penetrated as far north as merals"
Paris,
2
is
and
it
to
was not
to the invention
until
about 1500 that, thanks
it began to get a firm For teachers who await with
of printing,
3 footing in the schools.
impatience the popular use of the metric system, or who are discouraged by the apathy of their co-workers 1
Cantor,
I, p.
569, 576.
2
Henry, Ch., Les deux plus anciens Traites Francais d'Algorisme de Geometric, Boncompagni's Bulletino, February, 1882. The Ms.
anonymous and was written about 1275 A.D. 3 Those who are interested in this period of will
find,
writers atlas
on
zur
besides the discussions in Cantor, history,
Halliwell,
likewise interesting
"The
interesting
is
from 1200 to 1500,
Unger, Sterner, and other
facsimiles in
Konnecke,
G., Bilder-
Geschichte der deutschen Nationallitteratur, Marburg, 1887,
p. 40, et pass. is
some
struggle,
et
Crafts of
Text Society.
J.
O.,
Kara Mathematica, London, 2d.
and valuable, as
Nombrynge,"
is
also the
published in
Boncompagni's Bulletino
is,
ed., 1841,
pamphlet edition
of
1894 by The Early English of course, rich in material.
THE TEACHING OF ELEMENTARY MATHEMATICS
54
with respect to the use of logarithms in physical computations, the story of the struggles of the Hindu of value.
is
system
The awkwardness
the
of
old
Roman
system, in
general use even after the opening of the sixteenth is
century,
well seen
in
Kobel's arithmetic, 1 a work
barely mentions the
which
following
is
write
,
multiplying,
a specimen
" :
Hindu numerals.
The
you would add
- to
If
them crosswise on the abacus; then by times
III
III
is
IV
IX, and II times
VIII; add the VIII and IX getting XVII, and this is the numerator; then multiply the denominais
XVII
the
,
XII as
times
III
tors,
and
is
make
XII; a
when Comenius
picture
book
common
use,
line ."
XII
XII under
between,
Even
thus
as late
published in Niirnberg the
for the instruction of children, the
well-known Orbis Pictus, the in
write the
little
which equals one and
1658,
first
IIII
for
he
says,
Roman numerals were "The peasants count
by crosses and half crosses (X and V)." The next great step in arithmetic, after the writing of integers, was that leading to a knowledge of frac-
The
tions.
historic
;
recognition
of
simple fractions
pre-
but the struggle to compute with fractions
extended for thousands of years after 1
is
Das new Rechepiichlein,
Ahmes
copied
1518, quoted here from Unger, p. 16.
HOW famous
his (p.
ARITHMETIC HAS DEVELOPED
that the
n)
has
It
papyrus.
same
is
been
stated
ancient Egyptians could, in general,
had a numerator
write only such fractions as
the
already
55
and
i,
The
true of other ancient peoples.
later
Greeks wrote the numerator followed by the denominator duplicated, and for
1
J-J.
all
f
accented, thus,
The Romans had
ica
i
n tea",
a fancy for fractions with
a constant denominator as a power of
seen
as
12,
foot), and the Babylonians for (^ fractions with a denominator 60 or 6O2 as seen in our
in
our inch
a
of
,
minute and second
=
!
(i
-^
of a degree, i"
=
2
1
of
(g ^)
a degree).
to
With such a struggle to write fractions, be wondered at that the ancients did
is
it
not
relatively
or that the
child
of to-day has to struggle to master the subject.
The
little
in
arithmetical
computation,
world could solve the simple equation many centuries before it could do much with fractions, and hence it is entirely in
harmony with the world growth
in the first
grade such simple equations as 2
before any work in fractions
The decimal arithmetical
century,
in
fraction
ingenuity.
forms
like
is
It
is
to introduce
+ (?) =
7
attempted.
a very late product of appeared in the sixteenth
-fj^ and
5
7
8
@,
for
and about 1592 a curve was used by Biirgi But in 1612, Pitiscus cut off the decimal part.
0.578, to
actually used the
decimal point, and the system was *
Cantor,
I, p.
118.
THE TEACHING OF ELEMENTARY MATHEMATICS
56
1
was
It
perfected.
however, until well into the
not,
eighteenth century that decimal fractions found footing in the schools, nor was
it
much
until the nineteenth
century that their use became general. During the long struggle for supremacy, the old-style fraction
was
the "
literally
common
fraction
"
survives,
the more
common.
In educational circles plan
But
fractions.
tions
we
teaching decimal
of
first,
name now by
the
;
although the decimal form
is
far
often hear advocated the fractions
before
common
any theory of decimal
to attempt
still
or to exclude the simplest
common
frac-
fractions
year of arithmetic, is unscientific from both the psychological and the historical standpoints.
from the
first
The historical order common fraction (of
is,
(i) the unit fraction, (2) the
course not in
its
complete de-
velopment), and (3) the decimal fraction, and this is also the natural sequence from simple to complex,
from concrete
to abstract.
As has The twofold nature of ancient arithmetic been said, arithmetic was studied by the ancients both The Greeks, as a utilitarian and a culture subject. for example, differentiated the science into Arithmetic (apiOfjLTjTtKri)
ing to
and Logistic
(Xoytcm/e??), the
former hav-
do with the theory of numbers, and the
with the art of calculating. 1
Cantor,
2
Gow,
Hence when, long
566-568. History of Greek Mathematics, p. 22.
II, p.
J.,
2
latter after,
HOW
ARITHMETIC HAS DEVELOPED
57
arithmetic,
came together to form our modern the subject came to be defined as "the
science of
numbers and the
these two branches
art of computation," al-
though the modern arithmetic of the schools includes
much besides this. The apiOfjLrjTifcri
of the
Greeks ran also into the
mystery of numbers, and much was made of this subject by Pythagoras (b. about 580 B.C.) and his fol-
That "there
luck in odd numbers" probably dates back to his school, the Latin aphorism, lowers.
"
much
being
is
Deus imparibus numeris gaudet,"
older than Virgil's line,
"Numero deus impare
The mysticism of
3,
7,
now an
and
9,
as
77.)
we owe
especially significant, forms It
study.
is
deficient, perfect,
even
to this ancient ten-
the study, only recently banished
from our schools, of numbers
The
viii,
of numbers, the universal recognition
interesting
dency that
(Eclogue
gaudet."
redundant,
classified as amicable,
etc.
art of calculating (\oyt,(TTi/ctf)
among
the ancients
ran largely to the use of mechanical devices, such as counters (like our checkers), and the abacus, an in-
strument with pebbles late)
sliding
in
(calculi,
grooves
Chinese
in
or
still
on wires.
America
laundryman calculations on an abacus
Korea the school-boy
whence our word
(his
still
calcu-
To-day the performs his
sivan pan),
carries to school his
and
in
bag
of
THE TEACHING OF ELEMENTARY MATHEMATICS
58
counters (in this case short pieces of bone). the ancients, too, and in the middle ages,
reckoning was
a
recognized
the
of
finger-
necessary
calculator. 1
equipment of the It
part
Among
perhaps, not strange that, in the outburst of
is,
enthusiasm
attendant
Hindu numerals
in
upon the introduction of the the schools of Western Europe,
these mechanical aids should have been relegated to
the curiosity shop.
Neither
is it
back, that there should have
strange to us, looking
come a
result quite un-
foreseen by the educators of that time, namely, a loss of the
power of
number.
real insight into
Rules for
computation existed and results were secured, but the realization of
number was
often sadly lacking.
It
was
not until late in the eighteenth century that this loss was recognized and material aids to a comprehension of
number were
restored
by Busse,
Pestalozzi,
and
their associates.
Arithmetic of the middle ages tian
Europeans north of Italy
arithmetical knowledge.
At
Among
we
find
pre-Chris-
little
trace of
the beginning of our era
learning was
at a very low state throughout this region.
Tacitus
us that writing was
tells
unknown among
the
common people, although was an accomplishment of the priests. As business increased, however, some it
mathematical knowledge became necessary even before Salt and amber were exported from Central
our era.
1
For description, see Gow,
p. 24.
HOW
ARITHMETIC HAS DEVELOPED
59
Europe, and Assyrian inscriptions tell of the purchase of the latter commodity from the North. 1 Tacitus tells
German
us that in his time the
had come
tribes
to
know the Roman weights and coins, and hence they knew enough simple counting for trading purposes. To replace the primitive northern arithmetic, came, with the southern conquerors, the Roman. The dominant power soon made it to the financial interest of
And
the traders to use the Italian numerals.
Rome had done
little
for education,
some
although
of her later
statesmen recognized the value of scholarship, as witness Capella, Cassiodorus, and Boethius, and this fact
made
the northern tribes incline to education.
however, had contributed so
North declined,
little
that,
was hardly
Rome, when her power
be expected that there should be any decided contribution to knowledge among her former subjects. Nevertheless, in
in the
Gaul, where
the
it
Franks
established
to
a well-ordered
monarchy, schools were founded, and the French king, Chilperic (d. 584), devoted himself with earnestness a system of public education. The Merovingian princes erected a kind of Court school, after the man-
to
ner of the Romans, and thus were founded the Castle schools which were ages.
Naturally,
nothing
to
common throughout
the middle
schools
contributed
however,
mathematics;
these the
training
did not require the exact sciences. 1
Sterner, p. 101.
of
a knight
THE TEACHING OF ELEMENTARY MATHEMATICS
60
The Church
more
schools did
for mathematics, as
for learning in general.
Wherever the Church went,
there went the
By whatever name known,
whether
school.
cathedral, or parochial, they existed with every large ecclesiastical foundaEspecially did the schools of St. Benedict of cloister,
in connection tion.
Nursia,
1
starting
from the parent monastery
Monte
at
Cassino (near Naples), spread all over Western Europe, until the Benedictine foundations became the recognized centres of learning from the Mediterranean to the North Sea.
In these Church schools mathematics had some standing.
The quadrivium
of
little
music, ge-
arithmetic,
ometry, and astronomy, was commonly recognized in higher education, and in spite of the low plane on
which arithmetic was usually placed (see were found to assign it a worthy place. 2
Bede the Venerable, to York, and other Church to
It
To
Boniface, to
leaders,
59),
we owe
some
Isidore,
Alcuin of the
little
had during the early middle
standing that arithmetic ages.
St.
p.
was doubtless
at
Alcuin's suggestion that
" Charlemagne decreed that the schools should make 1
480-543.
Called by Gregory the Great, "scienter nesciens, et sapi-
enter indoctus," learnedly ignorant and wisely unlearned. 2
So Isidore of
says: "Tolle
computum ceteris
cap. 4,
et
one of the most
influential of mediaeval writers,
rebus omnibus et omnia pereunt.
cuncta ignorantia caeca complectitur, nee
animalibus 4-
Seville,
numerum
qui
calculi
nescit
rationem."
Adime differi
Origines,
seculo
potest a Lib.
Ill,
HOW
ARITHMETIC HAS DEVELOPED
no difference between the sons of men, so that they might
serfs
come and
sit
61
and of
free
on the same
benches to study grammar, music, and arithmetic," l and that "the ecclesiastics should know enough of arithmetic and astronomy to be able to compute the time of Church festivals."
2
Brief reference has already (p.
15) been
5,
made
to
the fact that men, being trained in the monasteries for ecclesiastical work, could get
which correlated with
things terests.
One was
from arithmetic two
their
professional
in-
the ability to compute the date of
Easter (whence comes the chapter on the calendar), and the other was the training in disputation and in
come many
puzzling an opponent (whence
inherited
and useless puzzles of our arithmetics and algebras
A
of to-day).
Alcuin's time
some swine
example of these puzzles of be of interest: "Two men bought
further
may
for 100 solidi, at the rate of 5 swine for 2
divided the swine, sold them at the same
solidi.
They
rate at
which they bought them, and yet received a
profit.
How
could
that
happen?"
3
The
puzzle
is
unravelled by seeing that the swine were of different There were 120 sold at 2 for i solidus, 120 at values. 3
for
i
solidus, so that 5
went
for 2 solidi as before;
120 good ones therefore brought 60 1
p.
Capitularies of 789, art. 70
;
solidi,
and 120
quoted by Guizot, History of France,
248. 2
Sterner, p.
1
10.
8
Cantor,
I, p.
787
;
Sterner, p.
1
10.
I,
62
THE TEACHING OF ELEMENTARY MATHEMATICS
poorer ones 40 solid!
and
still
To weed
so
solidi,
the dealers had their
had 10 swine
left
by way
100
of profit.
out problems of this kind has taken a
long time, and even the present generation finds now and then some advocate of exercises almost as absurd, as sharpeners of the wit.
The
period from Bede to the tenth century, one
middle ages, saw arithmetic largely given up to the computing of Easter, the computist becoming so prominent that the Germans have of
the darkest
of
the
" designated the period as that of the Computists."
Another movement of importance,
to
1
which allusion
has already been made, followed this period of degenThe Hanseatic League, arising from a union eracy. of
German merchants abroad and
of their important
commercial centres at home, attained inence in the thirteenth century.
its
Although
first it
promhad for
primary object the protection of the trade routes between the allied cities, it soon developed other objects,
its
such as the assertion of town independence against the rapacity of the feudal aristocracy, the establishment of warehouses along the paths of
commerce, the formu-
and the general improvement Among these acts was the
lation of laws of trade,
of commercial intercourse.
establishment of the Rechenschulen (reckoning schools, arithmetic
schools).
The inadequacy
of the business
course in the Church schools, and the unsatisfactory 1
Sterner, p. 115
;
but see Cantor,
I, p.
783.
HOW
ARITHMETIC HAS DEVELOPED
63
attempts at teaching bookkeeping, arithmetic,
led
etc.,
Rechenmeister already The guild of Rechenmeisters included some
to the creation of the office of
described.
of the best teachers of the time,
Ulrich Wagner of who wrote the first German arithmetic (1482), Nurnberg, Christoff Rudolff, who wrote the first German algebra, Grammateus, who wrote the first German work on book-
keeping, and others equally celebrated. did this
monopoly become, that
arithmetic out of the
due to
common
So powerful
for a long time
schools,
and
it is
it
kept
in part
was
this influence that not until Pestalozzi's time
arithmetic taught to children on entering school.
When
at last
it
was decided that arithmetic could
profitably be taught in the earliest grades, the inherited
work
of the Rechenmeisters
lower classes, and
it is
was dropped
chiefly
due
in
upon the
to this fact that
we
have had, even to the present day, a mass of business problems (often representing customs of the days of the Rechenschulen, but long since obsolete, like part-
nership involving time) in the grades,
fifth, sixth,
where they are almost wholly
The period
of the Renaissance
and seventh
unintelligible.
The
period of the
rebirth of learning, the Renaissance, is one of the
interesting
which the historian meets.
contributed to
make
most
Manifold causes
the close of the fifteenth century
an era of remarkable mental
activity.
The
fall
of
Constantinople (1453) turned the stream of Greek culture westward, and it reached the shores of Italy with
THE TEACHING OF ELEMENTARY MATHEMATICS
64
a power far in excess of that which the
of
region
Bosphorus.
was
to
were the
which, by the help overthrow the Ptolemaic theory
new
the discovery of a
;
continent and the consequent
commerce the invention of cheap paper and movable type, two influences which gave wings to
revival of
of
exerted in the this
new astronomy
revelations of that of mathematics,
Joined
it
to
thought
known
;
;
and, not the least of
From
well as believing.
discovery, of invention,
dates arithmetic as It is
all,
that great
as the Reformation, which set
not
movement
thinking as
this period of migration, of
and of independent thought,
we know
difficult
men
to see
it.
what would naturally
place in arithmetic at that time.
find
Crystallized in the
new
printed works would be the arithmetic which the Greeks brought from Constantinople, the theory of
by geometric diagrams. The Roman numerals, which had been used almost exclusively to this time, would find a prominent place. The Arab
numbers and
roots
with the Hindu numerals (already more or less known), would contribute its little share
arithmetic,
in
the
coming
way
proportion),
in
of alligation,
and
series,
Rule of Three (our simple which last was known in
classical times as well.
Together with
this inherited
matter would naturally
be placed the arithmetic demanded by the peculiar
The small states, monetary systems, demanded an
conditions diverse
of the time.
with their elaborate
HOW
ARITHMETIC HAS DEVELOPED
65
method
of exchange,, not merely "simple," but also " arbitrated." The absence of an elaborate banking
system like that of to-day rendered the common draft one payable after, instead of at sight. The various systems of measures in the different states and cities 1 required elaborate tables of denominate numbers, and the lack of decimal fractions explains the need
compound numbers with several denominations. The frequent reductions from one table to another,
of
by these circumstances, encouraged the the Rule of Three (Regula de tri, Regeldetri,
necessitated
use of
Regula aurea), so that this piece of mechanism came be esteemed quite highly in the arithmetics of
to
that
time.
brought
Then,
in
in the
problems
sails,
and those which
ican
text-books
not
panies
problems
finally
as
yet
partnership,
demands
of
commerce
mensuration of masts and
General
as
having
in
the
too,
developed in our Amer-
Stock comAverage. been invented, elaborate
involving different
periods
were a necessary preparation for business. Later, business customs demanded Equation of Payments, a scheme not uncommon in days when long standing accounts were the fashion between whole-
of time,
salers
and
retailers.
tions in the days
Such were some of the condi-
when
printing
was
crystallizing the
science of arithmetic. 1
Thus
tables.
Graffenried's
Arithmetica Logistica,
1619,
has 21
pages of
THE TEACHING OF ELEMENTARY MATHEMATICS
66
There have been
Arithmetic since the Renaissance
methods of calculating since Italy, and the business
several improvements in
the
period
of
revival
in
changes have revolutionized the commercial side of arithmetic.
Among
the improvements in pure arithmetic, the
most important can be stated briefly. The first has to do with the invention of the common symbols of
which may, the century from 1550 operation,
rough way, be placed in
in a to
I65O.
1
Prior
to this time
the statement of the operations was set forth in
full,
and for any material advance some stenography or symbolism was necessary.
The second improvement decimal fractions
of
perhaps as
much
relates
about
to Biirgi
1600,
as to
the invention
to
an invention
any
one. 2
due
But
al-
though these fractions appeared three centuries ago, it was not until about 1750 that they found much footing in the schools, so conservative are schoolmasters,
their
authorities. fraction,
constituents,
and
the various
With the establishment
however,
arithmetic
was
of
examining
the decimal
revolutionized,
per-
centage became synonymous with advanced business calculations, the
1
A brief
greatest
historical note
common
divisor
upon the subject may be found
(necessary in
Beman and
Smith's Higher Arithmetic, Boston, 1896, p. 43. 2
Stevin, Kepler, Pitiscus,
See Cantor,
II, p. 567.
and others had a hand
in the invention.
HOW
ARITHMETIC HAS DEVELOPED
common
the days of extensive
in
obsolete for scientific
67
fractions) became and science found a purposes,
new servant to assist in her vast computations. The third improvement is the invention of logarithms by Napier in 1614* One might expect that a scheme which, by means of a simple table, allowed computers to multiply and divide by mere addition and subtraction, would find immediate recognition in
And
the schools.
yet,
so
conservative
is
the pro-
fession that, even in high schools in English speakfind
countries, logarithms
ing spite
do
tice
almost
no
place,
in
of the fact that neither in theory nor in prac-
they present
with many found
difficulties
any
commensurate In Ger-
in the old-style arithmetic.
the schools are more progressive in this matter.
many The
fourth improvement of moment is seen in our modern methods of multiplication and division. A
serious
division
in
problem
matter.
three hundred years ago was a " " " old scratch or " galley
The
method 2 was cumbersome troduction
monly
of the "Italian
use,
at
the best,
and the
in-
Method," which we com-
was a great improvement.
Nor
is
the
day of change in these operations altogether passed, 1
That
that year. Biirgi 2
is
is,
" his " Descriptio mirifici logarithmorum canonis
The
appeared in
best brief discussion of the relative claims of Napier
and
given in Cantor, II, p. 662 seq.
Well
illustrated in Brooks, E.,
Pa., 1880, p. 55, 59.
Philosophy of Arithmetic, Lancaster,
68
THE TEACHING OF ELEMENTARY MATHEMATICS
for
just
now we have
the
"Austrian methods" of
and of division coming to the front in and we may hope soon to see them comGermany, subtraction
monly used
in the English-speaking world.
The improvement as we know it with its fifth
is
partly algebraic.
present
common
Algebra,
symbolism,
dates only from the early part of the seventeenth century.
metic
With all
establishment there departed from arith-
its
reason for the continuance of such subjects as
alligation (an
awkward form
tions), series (better treated
for
by Greek geometric process, Rule plained rule), and,
in
indeterminate equa-
algebra), roots of
by the
Three (as an unex-
general, the necessity for any
Mathematicians recognize no divid-
mere mechanism.
ing line between school arithmetic and school algebra,
and the simple equation,
in algebraic form,
throws such
a flood of light into arithmetic that hardly any leading educator would
now
see the two separated.
The present status of school arithmetic is one of We have these inheritances from the Renaisunrest. sance,
and with
difficulty
we
are breaking
away from
Only recently have we seen alligation disappear from our text-books, and slowly but surely are
them.
we
driving out "true" discount, equation of payments,
arbitrated exchange, troy and
apothecaries' measures,
compound proportion, and other objectionable matter. Such subjects, are, as already suggested, unworthy of a place in the course which
is to fit
for general
citi-
HOW zenship;
ARITHMETIC HAS DEVELOPED
for they are practically obsolete
69 (like
weight), or useless (like arbitrated exchange), or
mechanism and show
of
knowledge
(like
troy
mere
compound
proportion), or they give a false idea of business (like
"true" discount). Slowly we are opening the door to the simple equation, because it illuminates the practical problems of arithmetic, especially those of percentage and propor" It is evident/' says M. Laisant, " that all tion.
through the course of
arithmetic,
letters
should be
introduced whenever their use facilitates the reasoning or search for solutions." 1
The present tendency
decidedly in favor of elimi-
is
nating the obsolete, of substituting modern business for the ancient, of destroying the arithmetic and algebra,
applied arithmetic.
Ten "
of
that the
artificial barrier
between
and
As
stated the case,
of shortening the course in the report of the " Committee "
The conference recommends
course in arithmetic be at the same
time
abridged and enriched; abridged by omitting entirely those subjects which perplex and exhaust the pupil without affording any really valuable mental discipline,
and enriched by a greater number of exercises in simple calculation and in the solution of concrete problems." 2
Three years
later,
1
La Mathematique,
2
For
full
the "Committee of Fifteen" had this p. 206.
report of the mathematical conference, see Bulletin No. 205,
United States Bureau of Education, Washington, 1893, p. 104.
THE TEACHING OF ELEMENTARY MATHEMATICS
70
further
suggestion:
"Your Committee
believes
that,
with the right methods, and a wise use of time in preparing the arithmetic lesson in and out of school, five years are sufficient for the study of mere arithmetic the five years beginning with the second school year
and ending with the close of the sixth year and that the seventh and eighth years should be given to the ;
algebraic
method
of dealing with those
problems that
involve difficulties in the transformation of quantitative indirect functions into numerical or direct quantitative l
data."
In
all this
present change and suggestion of change,
the radical element in the profession several forces
:
pronounced departure; the author of
restrained
his pupils
on some
most powerful influence
also
is
with the financial result; the teacher failure
is
by
the publisher fears to join in a too
is
official
concerned
fearful of the
examination (a
in hindering progress);
and
the pupil and his parents see terrors in any depart-
ure from established traditions. this,
But
in spite of
all
the improvement in the arithmetics in America
has, within a few years, been very
marked
more so
than in any other country. 1
Report of the Committee of Fifteen, Boston, 1895,
P- 24-
CHAPTER How The value
IV
ARITHMETIC HAS BEEN TAUGHT
of the investigation of the
way
in
which
arithmetic has been taught, especially during. the nine-
teenth century,
is
apparent.
Find the methods
fol-
lowed by the most successful teachers, find the failures
made by those who have experimented on new and the broad question of method "
The
science of education without the history of educa-
tion is like a house without a foundation.
tory
education
of
scientific of all
It is
ment
lines,
largely settled.
is
is
itself
the
The
his-
most complete and
systems of education."
1
impossible at this time to trace the develop-
of the general
methods of teaching the
opening of the nineteenth century.
up
to the
in
Chapter
I,
the development
of
subject,
Already,
the reasons
for
teaching the subject has been outlined, and from this the
general
methods
employed
may be
inferred.
Only a hurried glance at a few of the more interesting details
is
possible.
The departure from object teaching Arithmetic, at least in the Western world, was always based upon object teaching until about 1500, when the Hindu 1
Schmidt, Geschichte der Padagogik,
I, p. 9.
THE TEACHING OF ELEMENTARY MATHEMATICS
72
numerals came into general use. siasm of the
first
schools threw counters,
jective is
true,
the
it
not
came
their
the
use
numerical
Hindu
of
see that
it
was
which
essential
number and
the elementary tables
ob-
old-style
for calculation,
comprehension of of
and
saw that the
while they
they did
in arithmetic
of
into
work was unnecessary
development
Hence
their abacus
and launched out
basis for
the enthu-
in
use of these symbols, the Christian
away
And
figures.
But
of
as a
for the
operation.
to pass that a praiseworthy revolution
brought with
it
a blameworthy method
Although there were better tools for the Hindu numerals, arithmetic became even
teaching.
work
more mechanical than
before,
and
time of Pestalozzi, three centuries
awoke
to the great mistake
books became
filled
of
with
was not
later,
until the
that educators
which had been made
discarding objects as a basis for
With the introduction
it
number
in
teaching.
the Eastern figures, textrules
for
operations,
and
teachers followed books in this mechanical tendency.
To
define the terms, to learn the rules, to repeat the
book, this was the almost universal method for three
hundred years before Pestalozzi, and even yet the method has not entirely died out. 1 A modern math1
Janicke and Schurig's Geschichte der Methodik des Unterrichts in den
mathematischen Lehrfachern, Band III of Kehr's Geschichte der Methodik des deutschen Volksschulunterrichtes, Gptha, 1888. The first part of the volume terrichts,
and
Janicke's
Geschichte
will hereafter
be referred
is
der Methodik to as Janicke.
des RechenunJanicke, p. 21.
HOW
ARITHMETIC HAS BEEN TAUGHT
ematician would fare
amination
hence
will
ill
in passing
days,
mathematician
the
just as
those
of
wonder
before of
a
73
an arithmetic extheir
examiners,
couple of
at the absurdities of
1
centuries
many
of our
questions to-day.
The
arithmetics
Rhyming to
a large
memory
number
difficulty of
of rules
committing
upon the subject
to look for a remedy. Some, and them Ascham and Locke, mildly protested among
educators
led
against so
many
rules,
but for a long time a large
number was considered necessary, and
this
plan
is
even yet advocated by many teachers. Among the remedies suggested was that of putting the rules in rhyme, the argument being that (i) a multitude of rules a necessity, (2) rhymes are easily memorized, (3) hence this multitude of rules should appear in rhyme,
is
a good enough syllogism if we admit the major Hence for a long time rhyming rules were in premise. and vogue, might be to-day had not opinions changed as to the value of the rule last
of
quarter
arithmetic in so
little
the
itself.
nineteenth century, however, an
rhyme appeared
are the
Even during the
lessons
of
in
the
New York history
of
State
methods
known.
Form
instead of substance
of the policy of
So we
find
making
much 1
was a natural outcome
arithmetic purely mechanical.
attention paid to the preparation of
For such a paper see Janicke,
p. 22.
THE TEACHING OF ELEMENTARY MATHEMATICS
74
copybooks with curious arrangements of work.
artistic
The this
may
following
tendency
serve to illustrate the results of
1 :
79745
97548
64789
69457
48
4549 2472 363535
303632
81282528
42451640 5463202056
6775391436
5160119905 It is possible that to this
tendency to prepare
artis-
copybooks rather than to acquire facility in arithis to be attributed the continued use of
tic
metic there the old
long
"scratch" or "galley" method of
after
the
more modern
Italian
division,
2
method was
known. in
Instruction
method, for teachers of arithmetic,
appear in noteworthy form about the middle of the seventeenth century, "like an oasis in a
began
1
2
to
Janicke, p. 27.
This method
named.
is
given in
all
of the histories of mathematics already
HOW
ARITHMETIC HAS BEEN TAUGHT
75
But the plans suggested were counting and writing numbers in
desert," says Janicke. still
mechanical
number
space, then addition in such space, " The teacher/' says subtraction, and so on.
unlimited
then
one of the best works of the time, "is first
to write the
nine numbers, then pronounce them four or
then
times,
let
the boys,
five
one after another, repeat
them."
A
methods employed at the opening of the eighteenth century may be seen in the rules for the celebrated Fran eke Institute at picture
of the best
Halle (I/02), 1 rules not without suggestiveness to certain teachers to-day:
"All children who can read tic."
It
was not
shall
study arithme-
until about a century later that the
subject was taught to children just entering school, and to-day we have quite a pre-Pestalozzian move-
ment
to the old plan,
graphic
"On
akin to pre-Raphaelitism in the
arts.
account of the diverse aptitudes of children,
in the matter of arithmetic,
classes
and
;
it
is
impossible to form
hence the teacher shall use a printed book subject from it. ... He shall
shall teach the
go around among the children and give help where 1 Unger, p. 140 ; Janicke, p. 32. In general it may be said that any one who wishes to follow the development of method in arithmetic must consult these works. There is nothing more systematic than Unger,
nothing so complete as Janicke.
THE TEACHING OF ELEMENTARY MATHEMATICS
76
To-day we hear not a little of "the laboratory method" and "individual teaching," a return to the methods of the past, methods in which it
is
the
necessary."
of
inspiration
methods
community work was
since
long
weighed
in
the
wanting,
balance
and
found wanting.
"The child
teacher must dictate no examples, but each
copy the problems from the book and in silence." This plan is also not
shall
work them out
unknown " It
in the teaching of the subject to-day.
would be a good thing
himself
if
the teacher would
work through (durchrechnet) the book
so that
"
he could help the children It was toward the close of the eighteenth century that the modern treatment of elementary arithmetic !
began to show an institution
we
find in
In the Philanthropin at Dessau,
itself.
to
which education owes not a
1776 very
little
little,
improvement upon the old
plan of pretending to teach all of counting, then all But of addition, then all of subtraction, and so on. 1 in
the following
entirely
new
lines,
year Christian Trapp
and
in
such einer Padagogik," in
began upon he 1780 published his "Verwhich he worked out quite
a scheme of teaching young children
how
to
add and
subtract, objects being employed and the effort being made to teach numbers rather than figures. This he
followed by simple work in multiplication and division, 1
Janicke, p. 44.
HOW
ARITHMETIC HAS BEEN TAUGHT
and he worked out a systematic use illustrating the relation of
we may
here that
box of blocks
tens, to units, a forerunner later. 1
mentioned
of the Tillich reckoning-chest
It is
say, with fair approximation to justice,
modern teaching
the
of a
77
of elementary arithmetic begins.
Trapp's successor was Gottlieb von Busse, whose
first
He was
still
works on arithmetic appeared
wedded
to
tion (to trillions),
and so
in order,
in
the old system of
1786.
teaching numerathen the four fundamental processes
But
on.
first
at the
same time he made
a distinct advance in the systematic use of
number
o
pictures (Zahlenbilder, translated by " number builders "/), points
some genius as
five being associated with the group as here shown. He used special forms for tens (to distinguish them from the unit dots), and also for the
hundreds and the thousands, thus carrying a good 2 thing to a ridiculous extreme. still
have
in our
day not a few
In the same
way we
failures as
a result
This
of carrying objective teaching too far.
Grube's
of
errors,
although few would
is
follow
one
him
enough to be harmed by it. Mention should also be made of the work of a
closely
nobleman,
von
Freiherr
Brandenburg, who
is
Rochow,
known
as
of
Rekan,
near
the reformer of the
3 country schools of Germany, and whose influence led 1
3
Janicke, p. 44.
Unger,
p. 138.
2
Ib., p.
45 seq.
;
linger, p. 165 seq.
THE TEACHING OF ELEMENTARY MATHEMATICS
78
on the part of
to the attempt
assistants to
his
make
arithmetic attractive instead of insufferably dull, and to use
it
for training the
mind
as well as for a prepa-
ration for trade. 1
Trapp, Busse, von Rochow, and a few others whose names and work can hardly be menPestalozzi
"the voice of one crying in there was another who should come.
tioned here, were like
the wilderness
" ;
Johann Heinrich Pestalozzi, a poor Swiss schoolmaster, a man who seemed to make a failure of whatever he undertook, laid the real foundation of primary arithHe wrote no metic as it has since been recognized.
upon the subject, and one who searches for his ideas upon number teaching has to pick a little here and a little there from among his numerous
work
directly
papers and
letters,
and take the testimony of those
who knew him. 2 Number had been taught of
objects
to
children his
before Pestalozzi
indeed, as already stated,
by the aid work.
began was the primitive
This,
plan,
and
was thrown over only with the introduction of printing and the Hindu numerals. Trapp and Busse had not to revive the
tried,
for
all
objects
calculations, but to
with
however, and
1
beginners. it
old
plan of
make Their
was reserved
Janicke, p. 48, 46.
2
using objects
a reasonable use of
for
Ib., p.
plans
were
crude,
Pestalozzi scientif-
63
;
Unger,
p. 176.
HOW
work.
perception the basis for
all
79
number
1
Of the
make
to
ically
ARITHMETIC HAS BEEN TAUGHT
mean
course this does not
to recognize the value of
first
was not
at
all
The
new.
perception.
it
in
This
understood
ancients
well, and Horace even placed things which enter by the ear
was
that Pestalozzi
his verse:
affect the
it
"The
mind more
languidly than such as are submitted to the faithful 2
eyes." Pestalozzi,
value to the teaching.
With
however, was the
and
full,
to
put
first it
to
to
recognize
its
practical use in
3
Pestalozzi,
number came the value of
the formal culture value
too,
definitely
and systematically
of
to the front,
"mental gymnastic" (Geistesgymnastik) unduly so, to be sure, and all daw-
was recognized dling
"busy work" was wanting.
rapidly,
quick in
The
children
worked
They showed themselves cheerfully, orally. number work, wide awake, active, and we can
more to-day from Pestalozzi than from any other one teacher of the subject, and this in spite of all the faults of method which he unquestionably possessed. learn
1
" Die Anschauung
ist
Pestalozzi to Gessner.
das absolute Fundament aller Erkenntniss." " Das Geheimniss
Compare Diesterweg
ganze
:
der Elementarmethode ruht in der Anschaulichkeit." 2 " irritant animos dimissa
per aurem,
Segnius
Quam 3
Shafer,
quae sunt oculis subiecta Fr.,
Geschichte
Geschichte der Methodik,
I,
des p.
fidelibus."
Ars poetica,
Anschauungsunterrichts, in
468.
v. 180.
Kehr's
THE TEACHING OF ELEMENTARY MATHEMATICS
80
him 1
It is related of
that a Niirnberg merchant,
had heard with some doubts
came
arithmetic,
who
of his success in teaching
to the school
one day and asked to
be allowed to question the boys. The request being granted, he proposed a rather complicated business
problem involving fractions. To his astonishment the boys inquired whether he wished it solved in writing or "in the head,"
he began
and upon
his
naming the
latter
plan
for himself to figure out the result on paper;
but before he had half done the boys' answers began " I have to come in, so that he left with the remark, three youngsters at home, and each one shall
you as soon as
to
I
possibly exaggerated,
can get there." The incident, not unique Biber 3 and others
is
;
numerous instances
relate
tended
earnest
Pestalozzi's
come
2
of
the
work
success which* in
oral
at-
arithmetic
founded upon perception. Pestalozzi was not narrow in his ideas as to the objects to be
employed, as Tillich and many other This particular
teachers of later times have been. device (say
some form
of abacus),
or that (as
some
set of cubes, or disks, or other geometric forms), did
He
not appeal to him.
1
By Blockmann,
used, to be sure, an arrange-
" Heinrich Pestalozzi, Ziige aus
dem
Bilde seines
Lebens," Dresden, 1846.
De Guimp's
2
See also
3
Life of Pestalozzi, p. 227 et pass.
cellent
work has become
Pestalozzi,
so rare.
American It
is
ed., p. 214.
unfortunate that this ex-
HOW ment
of
ARITHMETIC HAS BEEN TAUGHT "
marks on a chart
(his
81
units' table," Einheits-
tabelle), but he did not limit himself to any such device he led the child to consider all objects which ;
were of
interest to him, nor did
teacher!) to
device of
let
he fear (O modern
him use the most natural calculating the fingers. 1
all
Pestalozzi's leading contributions
as follows
He
1.
came
:
taught arithmetic to children
to school, basing his
seeking to
and
may be summed up
make
when they
first
work upon perception, and
the child independent of
all
rules
Nevertheless, he did not wholly free He avoided the baser the subject from mechanism. form which depended upon rules and principles, but traditions.
he substituted a mechanism of forms based upon perHis never ending 2x1 + ?xi is ception.
3x1=
very tiresome in spite of 2.
He
insisted that the
its
value for beginners. 2
knowledge of number should
precede the knowledge of figures (Hindu numerals), in the number space from
"a matter of all 1
"
to 10.
Now
The
it
is," said he,
of great importance that this ultimate basis
number should not be obscured best insight into Pestalozzi's ideas
work of
the
i
his
friend
in the
along this line
mind by is
given in
and co-worker, Krusi, Anschauungslehre der
Zahlenverhaltnisse, Zurich, 1803. 2
" Damit
fiihrte er in
der Darbietung
vom
vorpestalozzischen puren
Mechanismus zum anschaulichen Zahlmechanismus, an dem unser elementarer Rechenunterricht auch heute noch krankt." Brautigam, Methodik des Rechen-Unterrichts,
G
2.
Aufl.,
Wien, 1895,
P-
2
-
THE TEACHING OF ELEMENTARY MATHEMATICS
82
arithmetical abbreviations."
l
Tillich, Pestalozzi's
talented follower, agrees with his master in this. figures,"
most "
The
he writes, "are only the symbols for numbers. to be taught to the child until
Hence they ought not
To do
the numbers are familiar to him. to
make
otherwise
make
the same mistake that one would
is
in
could not yet talk," 2 a teaching rather radical statement, but one with a core of truth. letters to a child
who
and foremost the child must conceive of number; figures, operations, applications beyond mere counting First
and selecting of groups, these could
wait.
As one
the modern opponents of Grube's heresy has put " First the 3.
He
number concept, then the
also
the
that
insisted
operations."
should
child
of it,
3
know
the elementary operations before he was taught the
"When
Hindu numerals. in this intuitive
method
a child has been exercised
of calculation as far as these
have acquired so complete a knowledge of the real properties and
tables go
(i.e.
proportions of
from
i
to
number
he
10),
as will
will
enable him to enter
with the utmost facility upon the common abridged methods of calculating by the help of ciphers." 4 4.
pure
The Hindu numerals number.
"
His mind
followed this training in is
above confusion and
1
Letter to Gessner, Biber's Pestalozzi, p. 278.
2
Lehrbuch der Arithmetik, p. 41. Beetz, K. O., Das Wesen der Zahl,
8 *
p.
204.
Letter to Gessner, Biber's Pestalozzi, p. 282.
HOW
ARITHMETIC HAS BEEN TAUGHT his
guesswork;
trifling
arithmetic
memory work
83
a rational pro-
is
cess,
not a mere
it is
the result of a distinct and intuitive apprehen-
number"
sion of
or mechanical routine
;
l
Fractions were treated in the same way; first the concept of fraction, then some exercise in opera5.
tions,
shorthand
the
finally
After
characters.
the
has "such an intuitive knowledge of the real proportions of the different fractions, it is a very easy task to introduce him to the use of ciphers for
child
work."
fraction
following
thing in form."
out
2
After
Ratke's
was
Pestalozzi
all,
well-known
" rule,
simply a
First
and then the way of it; matter before The only question is, Did he postpone the itself,
form too long?
He made
arithmetic the most prominent study " in the curriculum. Sound and form often and in 6.
various
ways bear the seeds
and
error
of
deceit
;
number never; it alone leads to positive results." 3 "I made the remark," said Pere Girard, himself one of the foremost
Swiss educators, "to
Pestalozzi, that the fiable
sway
in
his
my
old
friend
mathematics exercised an unjustiestablishment, and that I
feared
the results of this on the education that was given.
Whereupon he manner, 1
'This
replied to is
me
because
I
with
wish
Letter to Gessner, Biber's Pestalozzi, p. 282. 8 Pestalozzi's
Sammtliche Werken,
spirit,
as
was
children
my 2
Ib., p. 283.
11. Bd., p. -226.
his to
THE TEACHING OF ELEMENTARY MATHEMATICS
84
nothing which
believe
be
demonstrated
them as that two and
to
clearly
cannot
make
two
as
four.'
My reply was in the same strain In that case, if I had thirty sons, I would not intrust one of them '
:
to you,
for
onstrate
to
would be impossible for you to demhim, as you can that two and two
it
am
make
four, that I
right
to his obedience/
to arithmetic
argument
is
Thus
and that did
I
have a
Pestalozzi give v
an exaggerated value (not that the Pere s very convincing), and thus it assumed a in
prominence
his father, " 1
curriculum
the
maintained, and which
is
which
his
followers
only now, after the lapse of
a century, being questioned by leading educators. He emphasized oral arithmetic as a mental 7. gymnastic, but he unquestionably carried too
cises
far.
Knilling,
who
in
his first
the
exer-
work wrote
with more force than judgment, was not wide of the
mark when he
said
" :
The
exercises with Pestalozzi's
Rechentafeln and Einheitstabelle (number and
units'
belong to the most monstrous, most bizarre,
tables)
most extravagant, and most curious that have ever appeared 1
2
in the
realm of teaching." 2
Payne's trans, of Compayre's History of Pedagogy, p. 437. Zur Reform des Rechenunterrichtes, I, p. 58. Those who care to
know
the
dikers,
weak
and
points of Pestalozzi, Grube,
to find
them discussed
and other German Metho-
in vigorous language, should read this
work. The later and more valuable works by the same author are also worthy of study Die naturgemasse Methode des Rechen-Unterrichts in der deutschen Volksschule, I. Teil, Miinchen, 1897; IL Teil > l8 99:
HOW 8.
ARITHMETIC HAS BEEN TAUGHT
He abandoned
reckoning, just
85
the mechanism of the old cipherthree centuries before, the cipher-
as,
reckoners (algorismists) had abandoned the abacus, and put oral arithmetic to the front. Number rather than figures,
was
But while
his cry.
instituting a healthy
reaction against the mechanical rules of his predecessors, like
most reformers he went
much
so
distinct
so that the
from
of
art
his arithmetic.
due time another reaction
to the other extreme,
ciphering became quite
Against
set in and, in
extreme in
this
America, drove
out the "mental arithmetic," which Colburn had done
much
so
to establish, replacing
mechanism.
of
reaction has set
century upon a
is
by the worst form
it
In turn, against this movement another
and the
in,
close of
the nineteenth
seeing arithmetic beginning to be placed
much more
satisfactory foundation than ever
before.
Of
Pestalozzi's
contributions to arithmetic but two
seriously influenced the world, perception as the foun-
dation of aim.
number
teaching, and formal culture as the
Although the creator of a method,
general recognition in Germany, and
it is
found
it
known
little
to-day
almost only by name. 1 1
Hoose's Pestalozzian Arithmetic, Syracuse, 1882, made the method
known,
in
its
most presentable form,
raphy relating to Pestalozzi attempting to mention
it.
A
payre's History of Pedagogy,
to
American
so
extensive
brief
resume of
is
and generally
teachers.
that his
in
it
work
is
is
works of
The
bibliog-
hardly worth
given in
Com-
similar nature.
Janicke gives the most judicial summary of the conflicting views con-
THE TEACHING OF ELEMENTARY MATHEMATICS
86
had a host of followers among writers even though his own method found little favor Tillich
Pestalozzi
with teachers. Tillich,
but
untranslatable
1
the
Among who
was
first
of the prominent ones
took for his motto the well-known
"Denkend rechnen und
words,
rechnend denken," words which might be put into " English as thinkingly to mathematize and mathe:
Acknowledging the inspiring insaw the faults
matically to think."
fluence of his master, 2 he nevertheless of the .latter's system
to rectify
may briefly be summed up as follows much attention to a systematic mastery
His plan
them. 1.
and boldly attempted
He
of the
paid
:
decade of numbers, making
first
advanced work.
for the
"My
know all possible relations number space i-io), and by
Norm
ard (eine
this the basis
method teaches one
in the first this
means
bilden) by which
all
to
to
order (in the
form a stand-
higher numbers
can be treated." 2.
He
did not attempt to bring a child to think of a
number, 85 for instance, as so many cerning his theories. especially in his will,"
he
" says,
first
Knilling
is
units,
but rather as
the most interesting of his recent
critics,
work, Zur Reform des Rechenunterrichtes, 1884; "I
make
it
as clear as day that all the
modern
errors in the
teaching of primary arithmetic take themselves back to Pestalozzi," p. 2.
On
the other hand,
J. Riiefli is
I,
Knilling's most interesting critic, in
work, Pestalozzi's Rechenmethodische Grundsatze im Lichte der Kri-
his tik,
1
Bern, 1890.
Allgemeines Lehrbuch der Arithmetik, oder Anleitung zur Rechen-
kunst fur Jedermann, 1806. 2 " Sein Feuer hat mich entflammt."
HOW so
ARITHMETIC HAS BEEN TAUGHT
many tens and
so
many units, and
similarly for larger
a distinct advance on Pestalozzi,
numbers,
8/
who
failed
to bring out the significance of the decimal system. 3.
To
bring out prominently this relation between
tens and units, and between the various units in the first
decade, Tillich devised what he called a Reckon-
box containing 10 one-inch cubes, 10 parallelepipeds 2 inches high and an inch square on the base, 10 three inches high, and so on up to 10 ten
ing-chest, a
The use
inches high.
to
which these rods were put
apparent, and it is also evident that the ratio idea number was prominent in Tillich's mind. 1
is
of
Of the other followers of Pestalozzi, space permits mention of only two. Tiirk 2 makes much of exercise in thinking, the formal training, 3 and follows Pestalozzi in
taking up arithmetic
(in the
number space
first
without the figures
1-20), but he departs
from the
plan of his master in not having the child begin the subject until his tenth year.
reached
its
The formal
height in the works of
culture idea
Kawerau;
4
his
extreme views provoked the reaction. 1
For a modern treatment of the subject see Brautigam's Methodik des
Rechen-Unterrichts, 2
2. Aufl.,
Wien, 1895,
p.
4
seq.
Leitfaden zur zweckmassigen Behandlung des Unterrichts im Rech-
nen, Berlin, 1816. 3
Uebung im Denken,
die Entwickelung
und Starkung des Denkver-
mogens. 4
Leitfaden fur den Unterricht im Rechnen nach Pestalozzischen Grund-
satzen, Bunzlau, 1818.
THE TEACHING OF ELEMENTARY MATHEMATICS
88
Reaction
against
Pestalozzianism
It
was natural
that protests should arise against the extreme views of Pestalozzi
and
his followers.
were often intemperate
in their
Like
all
reformers they
demands and
injudicious
The reaction was plans and it was led men of eminence in bound to come, by educational affairs, men to whom we are not a little indebted for certain opinions now generally held.
in
for
their
improvement.
For example, it was Friedrich Kranckes, whose first work appeared in 1819, who suggested the four concentric circles
which Grube afterward adopted, exercising
the child in the
number space
i-io, then in the space
i-ioo, then i-iooo, and finally 1-10,000.
had done before him, employed
number
He, as Busse pictures,
and
being one of the best teachers in North Germany, He called his his influence greatly extended their use. plan the Method of Discovery (Erfindungsmethode),
and developed his rules from exercise and observation. His problems, moreover, were not of the abstract they touched the daily life of the child and avoided the endless formalism of the Swiss Pestalozzian type
;
Such common-sense and sympathetic methods did not fail to win favor against Pestalozzi's fragmaster.
mentary method. Denzel l was another master of the moderate school.
He
laid
mary
down
these three aims in the teaching of pri-
arithmetic 1
:
Der
Zahlunterricht, Stuttgart, 1828.
HOW To To
1.
2.
ARITHMETIC HAS BEEN TAUGHT
exercise the thought, perception,
89
memory;
lead the children to the essence and the simple
relations of
number
;
To
3. give the children readiness in applying this knowledge to the concrete problems of daily life.
This is
point
a systematic and terse summary, and the third
is
not one which played any part in the Pestaloz-
zian scheme.
Denzel, too, followed a concentric circle
plan, treating the four operations in the circle
i-io,
then again in the circle 1-20, and so on.
Among
the leaders
who
did the most to establish
moderate and common-sense school of teachers
this
must be mentioned Diesterweg 1 and Hentschel, 2 men whose opinions have done much to mould the educational thought of the last half century.
Grube (i8i6-i884) 3 educator
from
lies
Grube's claim to rank as an
largely in his
power
the writings of others.
He
of judicious selection "
used the
concentric
"
he notion, but this was half a century old made much of objective work, but so had every one
circle
since Pestalozzi
;
;
he
insisted that
"
every lesson in arith-
metic must be a lesson in language as well," but so
had
Pestalozzi.
He
gave, however, one
an extremely doubtful one, 1
Methodisches Handbuch
fur
new
principle,
that the four funda-
den Gesammtunterricht im Rechnen,
Elberfeld, 1829. 2
Lehrbuch des Rechenunterrichtes
8
Leitfaden fur das Rechnen, Berlin, 1842.
and by Soldan (1878).
in Volksschulen, 1842.
Trans, by Seeley (1891),
THE TEACHING OF ELEMENTARY MATHEMATICS
90
mental processes should be taught with each number before the next number was taken up, 1 and this is the essence, the only original feature, of the
Grube method.
The book was happily written it was brief common virtue it was easily translated, and it
not a
;
;
thus be-
came, some years ago, almost the only German "method" known in America. Thus it has come about that Grube has been looked upon as a name to conjure by, and neither the faults nor the virtues (much less the origi-
system seem to have been well considered claim to, for by most of those who claim to use it, nality) of the
nobody actually does.
More than
Its chief virtue lies in its thoroughness.
a year
years are of
ing
given to the number space i-io, and three
is
recommended
for the space i-ioo.
number space i-io he says
the
thorough way in which not too long
I
2
Speak"
:
In the
wish arithmetic taught, one
important part of the In regard to extent the pupil has not, apparhe knows only the numbers ently, gained very much but he knows them." There is, howfrom i to 10,
year work.
is
for
this
;
ever, such
a thing as being too thorough; to
that there
all
the next one
1
know
is
about a number before advancing to
is
as
unnecessary as
it
is
illogical,
as
Zahlenbehandlung. See the 6th (last) edition of the Leitfaden, 1881, p. 25, n. "Always from the educational standpoint one must extend the first course Allseitige
2
(*'.*.,
:
i-ioo) over three years for the majority of pupils."
HOW impossible as
ARITHMETIC HAS BEEN TAUGHT it
is
91
Instead of requir-
uninteresting.
ing more time for the group i-io when he published his sixth edition (1881) than he did when he published
the
quired
less.
first
Grube might well have retraining and the training of the
(1842),
Home
know more about numnow than they did in the first half of the cenThe interesting studies of Hartmann, Tanck,
street are such that children
bers tury.
and Stanley Hall have shown that most children have a very fair knowledge of numbers to five before entering school.
On
the other hand, of course the ability
must not be interpreted
to count
to
mean
that the child
has necessarily any clear notion of number. Children often count to 100, as their elders often read poetry, with
The 1.
attention to or appreciation of the meaning.
little
chief defects of the system are these carries
It
illustration
objective
studying numbers by the aid years, until 100 2.
It
is
reached.
attempts to
ing up the next, as
know
to
:
an extreme,
objects
for
three
1
master each number before takif
it
the factors of
were a matter of importance before
51
anything of 75, or
as
children
when
studying 4
of
to
if
it
the child
were possible
the majority
to
knows keep
know some-
thing of 8 before they enter school. 3. 1
It
On
attempts to treat the four processes simulta-
the proper transition from the concrete to the abstract, see
Payne's trans,
of Compayre's Lectures on Pedagogy, p. 384.
THE TEACHING OF ELEMENTARY MATHEMATICS
92
neously, as
equal
they were of equal importance or of
if
difficulty,
While
all
which they are
not.
must recognize that Grube gives many
val-
uable suggestions to teachers, the system as set forth
no sup-
in the last edition of the Leitfaden has almost
porters.
"While stimulating
to excess,
it
every one
will
Of the
to children
later
not carried
mere mechanism, as
easily degenerates into
agree who has
if
carefully looked into
it."
l
"methods," but two or three can be 2
has criticised his predecessors by saying that they teach a great deal about number, but do not teach the child how to operate with numKaselitz
mentioned.
He
ber.
therefore develops, and with
much
skill,
the
making the number the operator. 3 and Tanck 4 are leaders in the modern Knilling
idea of
1
Dittes,
Methodik der Volksschule, 205.
nur Meister umgehen konnen."
"Ein Instrument mit dem
Bartholomai.
"Unmoglich, langweilig, Die Behandlung jeder Einzelzahl ist vmd ganz unniitz. Kallas, Die Methodik des elemenunmoglich und auch vollig unniitz." taren Rechenunterrichts, Mitau, 1889, p. 20, 22. A good summary of the zeitraubend,
system
is
.
.
.
An
given in Unger, p. 188-195.
whole system
is
set forth in
earnest protest against the
Zwei Abhandlungen
iiber
den Rechenunter-
by Christian Harms, Oldenburg, 1889. The method is known to American teachers through translations of the earlier editions, made by Soldan and by Seeley.
richt,
2
Wegweiser fur den Rechenunterricht and other works.
in deutschen Schulen, Berlin,
1878, 3
Works
XXVIII. 4
already cited.
For brief review see Hoffmann's
Zeitschrift,
Jahrg., p. 514.
Rechnen auf der Unterstufe, 1884
Meldorf, 1887
;
;
Der Zahlenkreis von
Betrachtungen uber das Zahlen, Meldorf, 1890.
I
bis 20,
HOW
ARITHMETIC HAS BEEN TAUGHT movement.
pre-Pestalozzian to
Pestalozzi
the
present
assuming that number tion, which it is not.
is
"
assert
They time
teachers
93
from
that
have been
the subject of sense-percep-
Number
not (psychologi-
is
* put into them." They proceed to base their system upon the counting of things, a process in which three ideas are prominent,
cally) got
things,
counted
the
(i)
from
it
is
how many,
mass, (2) the
sense in which the things are considered.
as of things, men, trees, etc.
natural units
numbers
of
etc.
numbers
(3)
;
measured units
without
back
it
is
;
it
occupies
to count; the
"The fact that no matter how taught,
pleasing.
dependently of objects,
in
as individuals, calling
when
The mathe-
no space
it
;
is
not
exist in the external
rhythm
at least first
of counting
nearly
all
chil-
learn to count in-
which the
series idea gets
that they recognize three or four objects at
ahead, first
(2)
exists only in the mind.
child likes 3
dren,
;
metres, grammes,
of mathematical units.
Such a unit does not
imageable.
The
as of
is
Aristotle
to
world;
2
without quality (color, form, etc.); it extent; it is indivisible, a notion going
matical unit is
Knilling
the numbers of arithmetic as (i) numbers
classifies
of
the
(3)
set aside
by
itself,
the fourth one four even that counting proceeds in-
1
McLellan and Dewey, The Psychology of Number,
2
Die naturgemasse Methodik, I, p. 55. Phillips, D. E., Pedagogical Seminary, V,
3
p. 233.
p. 61.
THE TEACHING OF ELEMENTARY MATHEMATICS
94
dependently of the order of number names, and often consists in a repetition of a few names as a means of
following
the series,
learn
these
steps
presented,
advance of
of
and
desire
the
earlier
the series generally goes in
application to
its
with
become an abstract conception.
The naming
.
.
children
unmistakable evidence that
furnish
the series idea has .
that
such, taken
names,
and
things,
the
ten-
modern pedagogy has been to reverse this. dency Counting is fundamental, and counting that is of
.
.
.
spontaneous, free from sensible observation and from the strain of reason. series
much
to
is
things
In the application of the
where the child
and
difficulty,
...
much
this is
first
encounters
increased because
the teacher, not apprehending the full importance of this
too
entirely
with
many systems and
so
numbers."
Tanck base
natural their
etc.,
work
within the
teaching
desire
to
count,
and
Knilling
first
and
backward by ones, twos,
hundred, leading easily to rapid and even
in addition, subtraction, multiplication,
Mental pictures of numbers are of no value
division.
work
actual
head
for
meet
method, a systematic arrangement
counting forward
of
devices
point
1
this
Upon
in
hurry the child over this It is here that we rapidly. to
tries
step,
is
all
;
calculation
is
figure
work
;
the
never more empty of mental pictures than 1
Phillips,
D.
E.,
Pedagogical Seminary, V, p. 221.
HOW when we
ARITHMETIC HAS BEEN TAUGHT
calculate;
calculation
95
not a matter
is
of
a mechanical affair pure and simple. perception, But given these exercises in running up and down it
is
no nearer being an arithone who can finger the scales on the
the numerical scale, one
metician than
is
is
piano to being a musician. basis for subsequent
One of
Each
work and
furnishes the best
skill.
1
most temperate of writers upon
of the
number work 2 thus summarizes the
this
discussion
phase
:
Since through language number space was
1.
created, and since here
lies
the
source of
all
first
com-
must impress upon the child the sequence of number words as a true, serviceable and lasting sound series (Lautreihe). putation, therefore the teacher
2.
Since with this series must in due time be asso-
ciated things, perception enters.
Since the number words establish only the chronological difference in the appearance of the individual 3.
units, suitable exercises
should be given to
make
the
pupil certain as to his order of the units.
This relation of number to time (sequence) is not new, and the subject has been a ground for debate Sir William Kant first made it prominent. Hamilton takes one side and talks about " the science
since
Herbart 3 on the other hand main-
pure time."
of 1
" Diese
sind so wenig das Rechnen selbst, als Uebungen in den Intervallen die Musik sind." Fitzga, p. 23. Fahrmann, K. Emil, Das rhythmische Zahlen, Plauen i. V, 1896, p. 24.
Uebungen
den Scalen und 2 8
in
Psychologic
als
Wissenschaft, II, p. 162.
96
THE TEACHING OF ELEMENTARY MATHEMATICS number
tains that
is
no more related to time than to
a hundred other concepts.
relates
Lange
"
space rather than to time, saying,
number words
pressions for their
ceptions."
1
to
oldest
ex-
The
relate, so far as
meaning, to objects in space.
braic axioms, like
number
.
.
.
we know The alge-
the geometric, refer to space-per-
"Every number concept
is
originally the
mental picture of a group of objects, be they fingers an abacus." 2 On the other hand,
or the buttons of Tillich,
whose method does not wholly agree with
his
sentiment, thus sets forth his views upon this point " The empirical of arithmetic is to be sought in Time :
alone.
which
is
number arrangement representation to the senses, and
therefore only the
It is
capable of
only the sequence which must be fixed in the
first
from this everything else develops. has nothing spatial about it, it exists only in Time, and not as anything absolute there, but only
exercises, for
.
.
.
Number
as
something
relative.
The sequence
thing, not the magnitude."
is
the
great
3
This return to the pre-Pestalozzian idea of beginbut in a much more ning with exercises in counting systematic
way than any
followed
is
2
8
Pestalozzi's
predecessors
the latest phase of instruction in arith-
metic which has 1
of
commanded very
Logische Studien, p. 140. Geschichte des Materialismus,
Lehrbuch der Arithmetik,
general attention.
II, p. 26.
p. 331, 333.
HOW The
ARITHMETIC HAS BEEN TAUGHT
97
idea has been presented in America by Phillips. 1
working out the method in detail, the German have gone to an extreme, assigning "altoand to counting gether too much value to counting in a narrow sense, mere memory work with the num-
But
in
writers
ber series without reference to real things. is
...
a great overrating of the value of counting.
It .
.
.
Counting should be the servant of number work, not
number work the servant 1
of counting."
Some Remarks on Number and
Monograph, 1898; Number and
its
its
2
Applications, Clark University
Applications Psychologically consid-
ered, Pedagogical Seminary, October, 1897. 2
Grass, J.,
Miinchen, 1896,
H
Die Veranschaulichung beim grundlegenden Rechnen, p. 10.
CHAPTER V THE PRESENT TEACHING OF ARITHMETIC Objects aimed at
In Chapter
IV
the growth of
the teaching of primary arithmetic was briefly traced.
The teaching
of the
more advanced portions was not
In the present chapter a few of the recent
considered.
tendencies in both primary and secondary arithmetic will
be
and
briefly mentioned,
what are a few
ascertaining
chiefly with a
of
view to
the points of
con-
as to
what
troversy.
In the
first
place,
it
is
not at
all settled
we
are seeking in teaching arithmetic to a child. Herbart and his followers would have us bring out
Others equally prominent and more numerous assert that it has no such value. "We en-
the ethical value.
tirely
overrate
arithmetic
we
if
soul-forming ethical power.
.
.
ascribe
The mental
.
(Denkthatigkeit) induced by arithmetic tive
and
heartless
to
l
is
it
any
activity
unproduc-
Grube and
many make it adapt itself to language work, made much of the logical training which (gemiitlos)."
others would Pestalozzi it
gave, and several writers have 1
amused themselves
Korner, Geschichte der Padagogik, 1857.
98
THE PRESENT TEACHING OF ARITHMETIC by giving
extended
lists
of
divers virtues cul-
by the simple science of numbers.
tivated
But
quite
99
it
sometimes seems as
been more harmful than
some second
if
these discussions have
beneficial.
When we
hear
dawdling along through a little simple number work, which no doubt has been elegantly developed, and out of which ethical and class
year
and general culture values have no doubt been duly extracted, we are forced to wonder whether in logical
a maze of secondary purposes there
is
not lost the
that of leading the child to "figure"
primary purpose
common problems
of
The number concept The fundamental principle the method of teaching primary arithmetic has
in
quickly and accurately
in
the
his experience.
root
in
essence
the
number
affirms that
of
number.
No
1
number
is
now
one
an object of sense-perception, 2 inherited notion are based not a is
although upon this few of our present ideas as to method. of
its
"The
notion
not the result of immediate sense-per-
ception, but the product of reflection, of an activity of
our minds. nine
horses,
We
cannot see nine.
feet,
nine dollars,
the horses, the feet, the dollars, to us
;
etc., if
can see nine
that
Wesen der Zahl als Neue Bahnen, VI. Jahrg., 201.
iBeetz, K. O., Das
McLellan and Dewey,
p. 61.
is
we
see
they are presented
that there are exactly nine, however,
unterricht. 2
We
we cannot
Einheitsprinzip
im Rechen-
100 see.
THE TEACHING OF ELEMENTARY MATHEMATICS If
know this we are forced and since we usually do this with
we wish
the things
;
our eyes, the idea has got
of
number."
ing that one
abroad that
we
we would be
see
know The
been so considered.
"a
is
justified in say-
not, primarily, a number, and
is
historically interesting to
number
the help
1
In line with this idea
it
to count
to
collection of
it
is
that only recently has definition
classical
units,"
2
of
a definition scien-
tifically worthless.
But while we put number into objects, on the other hand we derive our idea of number only from the presence of the world external to the mind.
group of people, tion (" people "),
them
thus calling
all
see a
by the one abstract name, even
though the individuals be very different.
shows
observation
We
and we begin by making an abstrac" and we say, " Here are ten people
there
however, that
us,
"A
careful
are
no
objects exactly alike; but by a mental operation of which we are quite unconscious, although it holds
within
itself
straction, 1
we
Fitzga, E.,
the
entire
secret
of
mathematical
Die
natiirliche
Volks- und Biirgerschule,
I.
Methode des Rechen-Unterrichtes
Theil,
Wien, 1898.
This
common-sense books on method that has appeared 2 This is found in most of the older arithmetics. Frisius, in his
dinem ex
ab-
take in objects which seem to be alike,
famous text-book,
unitatibus conflatum.
" says,
Numerum
is
in der
one of the most
in a long time.
For example,
Gemma
autores vocant multitu-
Itaque unitas ipsa numerus non
erit."
Arithmeticae Practicae Methodus Facilis, Witebergae, M.D. LI, pars prima.
THE PRESENT TEACHING OF ARITHMETIC rejecting for the time being their is
to
number
perception of a
1
So the
generated in the mind by the sense2 group of things supposed to be alike. is
Hence while we do not have a
sense-perception of
number, on the other hand few now attempt
number without the help
What
of groups.
Here
differences.
be found the source of calculation."
idea of
IOI
to teach
of objects for the formation
these objects shall be
is
more
of a
In Germany the use dispute to-day than ever before. of numeral frames has been carried to an extent not
known
America, and several forms of apparatus have been devised. But however valuable these aids in
may be
in the first grade,
is
it
doubtful
if
there
is
3 In any excuse for their extensive use thereafter. America the tendency has been along the Pestaloz-
zian
line,
although natural
of
taking
objection
means
of
any material that is at hand, has been made to the most
all,
the fingers. 4
Frequently, how-
La Mathematique, p. 15, 1 8, 19, 31. Jede Zahl ist der Inbegriff einer gewissen Menge von Einheiten. Einheiten im Sinne des ersten Rechnens sind wirkliche Dinge. 1
2
Laisant,
"
.
.
.
Ein grundlegender Rechenunterricht ohne Veranschaulichung ist ... undenkbar." Grass, J., Die Veranschaulichung beim grundlegenden
Rechnen, Munchen, 1896, 3
One
p. 5, 6.
of the best brief historical discussions of numeral frames
in Grass, op.
cit.,
61 seq.
The matter
is
is
given
discussed in Payne's transl. of
Compayre's Lectures on Pedagogy, p. 384-385, the note on p. 385 being misleading, however. 4
Die
mittel.
Finger Fitzga,
sind
I, p.
das
18.
naturlichste
und nachste Versinnlichungs-
THE TEACHING OF ELEMENTARY MATHEMATICS
102
ever, teachers
have fallen into the error of forgetting
Busse's valuable suggestion, that the objects should
not be such as to take the child's attention from the central thought.
At
the same time, they should be
such as relate to his daily
and such as have
life
some interest for him. 1 There has
Grube
follow
there
after
been a tendency in America to the extreme of using objects long
also to is
any
much energy
devoted
nize at a glance the
and
this
need
Some have
them.
for
to bringing children
number
has connected
itself
in
to
recog-
a group, say nine,
with the best form of
grouping to establish number relations and to enable the eye to grasp the group readily.
A
consideration
of the forms
shows how much more readily the eye grasps some forms than others. But after all, this is fundamenthe recognition of a familiar form, which we have learned has a certain number of spots, rather
tally
than the recognition of a number. 1
Was
In a
dutch das Leben in Schule und Haus und ausser
game
of
dem Hause
den Erfahrungskreis des Kindes gekommen ist, auch das kann fur das Rechnen verwertet werden. Alle Teile des Gedankenkreises sollen rech-
in
nerisch durchleutet werden, in spielen.
denen
ihrer
Natur nach Zahlen eine Rolle
Rein, Pickel and Scheller, Theorie
unterrichts,
I,
p. 361.
und
Praxis des Volksschul-
THE PRESENT TEACHING OF ARITHMETIC we
103
form of the nine as we do we do not stop to count the spots, nor could we tell the number on a different 1 arrangement unless we counted. cards
recognize the
the form of the knave
The
;
uselessness of carrying this objective
work
too
is apparent when we consider that we never get our ideas of numbers of any size from thinking of
far
groups;
we
get
them from thinking
of the relative
places which they occupy in the number series, or the time which it takes to reach that place in run-
ning up that
the length of the line which
series, or
would represent that number in comparison with
2
unity.
Recently, sustained by high psychological authority,
the effort has been
made
make prominent the ratio That ratio is number is
to
idea from the very outset.
evident;
that the converse
Newton's well-known
of
first
consider
number
is
in
this
has the authority
true,
definition
;
that a child should
way has
its
advocates.
"The fundamental
thing," says one of these "(in teaching arithmetic), is to induce judgments of relaBut such a scheme substitutes a tive magnitudes." 3 1
If
one cares to enter
this field with
any thoroughness,
psychologically, he should read Grass, op.
cit.,
p. 14 seq.,
historically
and
one of the best
discussions available. 2
Um uns grossere Zahlen
ohne Wiederholung des Zahlens etwas deutdem Auskunftsmittel von
licher zu vergegenwartigen, greifen wir daher zu
Substitutionen.
Das gebrauchlichste
stellungen zu substitutieren. 8
Speer,
W. W., The New
Fitzga,
ist,
I,
fur
Zahlvorstellungen Zeitvor-
p. 16.
Arithmetic, Boston, 1896.
THE TEACHING OF ELEMENTARY MATHEMATICS
IO4
complex for a simple number
idea,
it
the historical sequence (whatever that
and
makes use
it
of a notion of
is
contrary to
may be
number
worth),
entirely dif-
ferent from that of which the child will be conscious
founds the idea of number upon measurement, but in so doing it uses the word measure It
in his daily life.
in its narrowest sense.
It
makes
use, also, of sets of
objects (in the systems thus far suggested)
by which
accomplished no more than Tillich accomplished
is
with his blocks, while their character
is
such as to
take the attention from the central thought of number.
Fundamentally, as Laisant has pointed out, and Comte before him, the two notions of counting and
The
1 measuring are the same.
nitude directly by comparison rare
" ;
it is
however, extremely
is,
the indirect measure of magnitudes which
characterizes mathematics."
the ratio
estimation of a mag-
idea
at
As
some time
there can be no question
to
in
the necessity for
the
pupil's
course,
the argument lies only as where the idea should be brought in. 2 The most temperate and philosophical discussion of the subject ;
to
is
" by McLellan and Dewey in their PsyNumber" (1895), a work which should be
that given
chology of read and owned by every teacher in the elementary grades.
but 1
2
it
makes number depend upon measurement, uses this word in the broader sense indicated
Laisant,
It
La Mathematique,
p. 17.
A brief but very good discussion is given in
Beetz, op.
cit.,
p. 299.
THE PRESENT TEACHING OF ARITHMETIC
105
by Comte, including counting as a special form. In counting, however, it wages war against the "fixed unit" system which the authors brand with Grube's
name, although Grube
method
of
for
things
by no means the father of
do theoretically) substitutes the
It actually (as all
it.
is
method
the
Pestalozzian idea of numbers it
leads a general attack
nesses
the
of
burden of the lead
him
The
primary arithmetic.
to seek to
ratio idea
place upon the child the
at the outset, but rather to
common-sense notion
to a
the
symbols,
against the inherited weak-
traditional
work seems not
of
instead of figures, and
of
number
with-
out fixed unit, of counting in the best form of the
Knilling-Tanck school, of applying the knowledge of
number
To
and lengths. but 3
pieces
used
+
cts.
pieces
-f 5
5
to use
cts.
= how many
when he
says:
relations of
cents
?,
volumes
34-5 =
?,
or 3 five-cent
how many five-cent as the world first number use to is number with a varying unit, to get an pieces
introduction to ratio at the
the matter of
to
count things; not to say
five-cent this
?
it,
instead of
to things
the proper
"It
is
are
best. 1
place
Laisant sums up for
proper to ask
the if
ratio
idea
the idea of
ratio, usually assigned place rather late in the study of arithmetic, does not deserve to be considered early
in the course as
a consequence of the notion of number"* McLellan and Dewey,
1
Fitzga, p. 28
2
La Mathematique,
;
p. 30.
p. 78, 147, 149, etc.
THE TEACHING OF ELEMENTARY MATHEMATICS
106
When
elementary work we are led to feel that a child must not only think of a group of things or a ratio when he is learning about the numbers from in
but that he must continue to think of groups and ratios, and to refer to objects, as he progresses, i
to 10,
we impose upon him what no mathematician takes upon himself. The child must get his first notion of numbers from counting things
may
in
things, as the
themselves be groups
world did ;
in
;
these
counting he
measures the group by the unit with which he working; he gets a ratio, if we please to call it so, although the concept is not simple enough to be thrust really
is
But once the idea of number
upon him. is
is
then largely a matter of the number series
;
there,
it
we have
an idea of forty-seven as lying between forty-six and forty-eight, a little below fifty, and as "being a number about half
way
(distance) to a hundred,
and we have a
it would not take long to count it, about half as long (time) as to count a hundred. Thus we place it in a series, on a line, or in the flow of
vague idea that
and thus we get an idea
magnitude but few people visualize it as a group of objects, and why should a child be asked to do so?
time,
of
its
;
Advocates of the idea that number means merely the
how-many
of
a group,
or
the ratio
merely, are disappearing as such
Grassmann,
coming
to
Hankel, G.
be known.
of
lengths
scientific writers as
Cantor, and Weierstrass are
The
doctrine
of
"one-to-one
THE PRESENT TEACHING OF ARITHMETIC "
correspondence teachers,
work
and
it is
107
being understood by elementary not without suggestiveness in simple
is
To
in arithmetic.
the
number
of a group cor-
responds one name and one symbol, as
**
5
five If
we
establish the laws of these numbers, as that
and
and give
to
and
equal
name and one we may work with and we need have
a certain operation one
symbol (as "addition,"
+), then
symbols according to these laws, no thought of the names or the numbers, but can translate back into numbers at any time we choose. Indeed, our symbols
kinds of numbers, as
may
force
us to establish
when we run up
new
against the
6, or V4> or trv to divide the circumfersymbols 4 ence of a circle by the diameter. This notion of "one-
to-one correspondence," while not consciously one of
elementary arithmetic, exists there just as really as it exists in later work. It does not take long for the child
to
"substitute
for
the
reality
creatures of reason, born of his
ing a problem, be
it
things
the
In solv-
one in the calculus, in algebra,
or in the
second year of arithmetic,
stituting
for
the
of
own mind."
actual
represented by symbols;
things
we
we begin by
certain
sub-
abstractions
think in terms of these
THE TEACHING OF ELEMENTARY MATHEMATICS
108
aided by symbols, and finally from our
abstractions,
we
pass back to the concrete and say that we have solved the problem. It is all a matter of " one-
result
to-one correspondence,"
it
being easier for us to work
with the abstract numbers
mentally the process abstraction
we
By
2.
Thence we pass
many forms
bol,
We
to symbols, in
and we make an
algebra, or
concealed,
This equation we
arithmetic.
being a symbol.
solve, the result 3.
of
Funda-
something pass to numbers.
openly, as
either
corresponding
objects.
like this:
is
1.
as in
their
work with the actual
figures than to
equation,
and
find the
number corresponding
and say that the problem
All this does not
mean
be merely a matter of mathematics we find
is
that
solved.
primary number It
symbols. it
to this
more
sym-
l
means
convenient
is
to
that in to
work
purely with symbols, translating back to the corresponding concrete form as may be desired. And so those teachers
who
fear lest the child shall drift into
thinking in symbols instead of in number, are really fearing that the
In a rough of
child
shall
drift
into
way we may summarize
the writers
to
made, as follows
whom
reference
mathematics.
the
conclusions
has chiefly been
:
Let the child learn to count things, thus getting the notion of number. These things are, for the purI.
1
Laisant,
La Mathematique,
p. 20, 21.
THE PRESENT TEACHING OF ARITHMETIC pose of counting, considered
109
and they may be
alike,
single objects or groups.
Let him acquire the number
2.
with
series, exercising the circle of beyond actually counted things. In the learning of symbols it does not seem to
it
3.
be a matter of moment as to whether these are given with the first presentation of number or not. They must, however, be acquired soon.
and
Unconsciously
4.
gradually
the
child
will
acquire the idea (never expressed to him in words) one-to-one correspondence of number, name,
of the
symbol,
and thereafter the pure concept
of
number
play a small part in his arithmetical calculations.
will 5.
The
early,
number should be introduced the work with fractions.
ratio idea of
and applied
in
M. Laisant has tersely The great question of method expressed what is probably in the minds of most sucteachers of elementary mathematics, in the " There are not, I believe, many following words cessful
:
methods of teaching, stand the ensemble
by teaching we are to underby which we seek to
if
of
efforts
furnish with accurate knowledge a
has not yet reached .
.
.
The problem
degree of development. to interest always the same:
its is
human mind which
full
the pupil, to induce research, to continually give him
you please, that he is dis1 covering for himself that which is being taught him."
the notion, the illusion
1
if
La Mathematique,
p.
1
88, 189.
HO THE As
TEACHING OF ELEMENTARY MATHEMATICS the rest,
for
logical
presentation
question
largely a matter of
and detailed
device.
psycho-
we
Shall
by the diagram or by the formula?
extract square root
The
is
it
is
of
importance in
little
relatively
comparison with the great questions of method and So with most of the psychological presentation.
of
questions to be discussed in this
matters of detail which
chapter; they are
one teacher
may work
out
one way, and another a different way, and the difference in result may be so slight that the world has not been able, after centuries of experiment, to decide
These matters vary with classes, with the advancement of pupils, and with the temperament of the teacher. To give simplicity of form with which
better.
is
depth of thought
is
cult art of teaching,
one of the qualities of the
and
it
diffi-
depends upon the individ-
ual to attain to this simplicity. 1
is
The advance in the modern teaching of arithmetic due much more to the recognition of the definite
aim than
to the discovery of
the other hand, the
Garmo and 1
the
improved methods.
influence of
McMurrys
in
such writers as
On De
America, opening up
" Les moyens materials, les precedes pedagogiques a mettre
ceuvre pour obtenir le resultat desire sont la nature
des
classes,
maniere de voir
et le
1
'avancement
des
eminemment eleves,
temperament du professeur.
tion de la simplicite dans la forme avec
la
des qualites de
de
stitue 1'une
p. 192,
194.
1'art difficile
aussi
et .
en
variables, suivant
.
.
d'apres
la
Cette concilia-
profondeur des idees con1
'enseignement."
Laisant,
THE PRESENT TEACHING OF ARITHMETIC German (and
the
Herbartian) views
particularly the
bases of method,
of the
ill
or the basis of education,
has given a great impetus to
teaching
in
general,
and as a consequence has improved the teaching of arithmetic. For the application of these views to lessons
special
in
number the reader
is
referred to
the works of these writers. 1
The whole
question of the formal steps to be taken
in presenting a new subject to a class should be considered apart from a work like this. 2
by a teacher Suffice
it
say here that Rein, whose presentation
to
of the matter
is
as well
known
as any, sets forth five
formal steps in the development of a lesson 2.
ration; tion;
5.
Presentation;
3.
Association;
Since the
Application.
i.
:
Prepa-
Condensa-
4.
English translations
have given the application of the Herbart method to primary work only, the following translation of a grade lesson may be of value.
How
Aim. i.
we
shall
We
Preparation.
fifth-
write 12 tenths of a litre?
can write f
1.,
{!., etc.
Instead
1 De Garmo, Chas., The Essentials of Method, p. 117; McMurry, C. A. and F. M., The Method of the Recitation, p. 19. For the best working out of the subject, however, one must consult Rein, Pickel and Scheller,
Theorie und Praxis des Volksschulunterrichts,
A
arithmetic 2.
6.
Aufl.,
Leipzig, 1898.
brief statement of the application of the formal steps to elementary
Aufl., 2
is
given in Brautigam's
Wien, 1895,
The matter
tables, in
is
P-
I
^
Methodik des Rechen-Unterrichts,
an(^ "* severa l other similar works.
clearly presented, historically
De Garmo's
Herbart,
New
and with comparative
York, 1896, Chap. V.
112 of
f
THE TEACHING OF ELEMENTARY MATHEMATICS
Also 2.
we can
1.
also write
In what other
etc.
-if
1.
i^
1. ;
way can
instead
we
write
of f
yf
1. ?
ijl.,
1.,
(i-^-l.)
?
^
Presentation of the new.
written another way.
be written
0.2.
We
already
or
can also be
i-f-^
know
that
can
-fa
What does a One after
Further examples.
figure before the decimal point indicate?
the decimal point? 3.
and
Association.
Can we
Compare the way of writing 3.3!. Compare ijl. and
write ijl. as
we write i-j%l. ? we have to write more than
;
9 tenths of a whole
1.
1.2!.
Condensation.
4.
I^Q
3 T%1. and
i.il.
litre
If
we reduce
and
or to wholes
litres,
the tenths of a tenths,
litre to
and we place
a decimal point between the wholes and the tenths (or before the tenths, or after the wholes).
or an
The 6.
eighth of a
litre
we cannot
write
A as
fourth tenths.
figures after the dot always indicate tenths.
Application.
numbers,
2.3
;
4.6.
Read
0.4;
Reduce
0.6.
Read, as mixed
to tenths 2.3
;
4.6.
Write
24 wholes and 7 tenths. Write, as a mixed number, 22 tenths. Read, as tenths, 1.2; 2.3. 1
The writing of numbers Since Pestalozzi's time there has been a controversy among teachers as to whether a child should be taught the Hindu numerals along with
we have 1
the numbers themselves.
seen, postponed
Pestalozzi,
as
this writing until the child
Rein, Pickel and Scheller, Theorie und Praxis, V, p. 237.
THE PRESENT TEACHING OF ARITHMETIC
113
had a knowledge of the first decade. His argument, the limit sometimes being changed to five, meets with
much
among some
approval
our best educators
of
to-day. Many even go so far as to use the common symbols of operation and relation before the Hindu numerals are learned, giving forms like
T +.=
=
X
m
Others ask, and with reason,
X should be used, but not one say,
much
with
also
law of
chological
a symbol like
why like
association
is
Still
4.
reason, that the
others
common
psy-
ample warrant
for
placing before the child, simultaneously, the forms
so he fix
may
the
German
see the "one-to-one correspondence,"
the
idea,
This view
4
Four
Illl
is
name, and
the
symbol
and
together.
taken by Hentschel, one of the leading
writers
method
upon
" have pupils," he says,
now
"The
arithmetic.
in
seen the individual num-
bers represented in three ways, and have so repre-
sented them for themselves,
marks, points,
etc., (2)
There now
figures.
namely, (i) by rows of
by number
pictures,
arises the question
of these three forms shall be used in
their
first i
computations.
by the
Can we
and
as
at
to
by which
(3)
little
ones
once put
THE TEACHING OF ELEMENTARY MATHEMATICS
114
them
work with the
into
swer, yes."
The
For myself
figures?
an-
I
1
as
question,
is
The
portance.
case with
usually the
disputed matters of detail,
of
is
relatively
these
little
im-
a century has left
experience of
it
entirely unsettled, the results being, so far as inves-
tigations
have shown as
case as the other.
a point, but difficulty
it
as
quite
yet,
good
one
in
easy to theorize upon such be worth while to consider the
It is
may
which children have
in connecting the
num-
with the proper symbol and especially with the proper name in the number series, and hence to ber
itself
make
as
much
tion involved
use as possible of the law of associain
presenting the
number
picture, the
name, and the symbol simultaneously.
The work
of the first
writers
the
ing
upon
The
year subject
majority of lead-
limit
the
results
of
number space i-io. Some go to Others take the space 1-20, and the argument a strong one that the foundation of all number
operations to the 12. is
work
lies in
the mastery of the subject in this space. 2
advocate counting by tens during the second part of the year, and then filling in the series, thus
Many
1
Klotzsch, Hentschel's Lehrbuch
des
Rechenunterrichts in Volks-
schulen, 14. Aufl., Leipzig, 1891, p. 10. 2
Die Veranschaulichung beim grundlegenden Rechnen, This work gives a brief but valuable resume of the leading theories of first grade work. E.g. Grass,
J.,
Miinchen, 1896.
THE PRESENT TEACHING OF ARITHMETIC
115
number space beyond that in which Such a plan adds to the working. and allows him to teach himself by
giving the child a
he
is
actively
child's interest,
the talk of the home.
On
the whole, present experi-
ence seems to show that the number space 1-20 for operations, with counting forward and backward in the space i-ioo as
and
recommended by Tanck,
Knilling,
others, forms the limit of the
of the first year.
Whether
working curriculum limit can be reached
this
depends entirely upon the But ability of the teacher.
class
of
pupils
attempt to confine not
to
only the results of operations, but also
number
to
the
space
for
i-io,
not only unnecessary, but
is
it
and the
all
ideas
of
the whole
year,
is
stupid and
tedious
for the children.
The
great desideratum in the in
facility
problems.
enough but
it
at
first
year's
work
is
handling numbers, not in solving applied "Tell me a story about four," is harmless first,
although there
is
no " story
"
told
gets to be a very old story before the year
;
is
Children like rapid work in pure number; one has but to step into a class whose teacher is awake done.
to this idea, to realize the fact
;
and
to
dawdle through
" the year with nothing but story" telling about number not only leaves ungratified a natural desire, but it
sows the seed of poor number work thereafter. There has nothing appeared in America for the last few years that,
considering
its
brevity, has
done so much for the
Il6
THE TEACHING OF ELEMENTARY MATHEMATICS
better teaching of the subject as President Walker's "
little
monograph on
mar Schools."
Arithmetic in Primary and Gram-
He
l
cared
ods, but he went to the root of the subject
"At
of his observations. in accuracy,
very
much
if
to
not in
arithmetical
when he
technical applications of arithmetic, to divisions of time, space, etc.
large
amount
paratively
must is
to
;
coins, to
money
in difficulty
grammar
appli-
through
school, until for a
of so-called arithmetic the pupil gets com-
little
practice in the art of numbers."
not, of course,
;
it is
against the neglect of that thorough
ber necessary to
2
This
be construed to mean that the child
have no applied arithmetic
The time
put at
is
and these technical
number and
the successive years of the
work leave
Scarcely has the child been
taught to count as high as ten,
cations are increased in
a number
in
the present time the results
facility, of
be desired.
and meth-
for theories
little
make
simply a protest
drill in
pure num-
a good calculator.
for beginning the
present a matter of dispute.
study of arithmetic
Should the
is
at
year of the subject, above mentioned, be synchronous with the The " Committee of Fifteen " think first school year ? not, and they recommend beginning with the second first
Before Pestalozzi, as already
school year.
said,
the
subject
was not begun
until the child could read.
Pes-
talozzi,
however, recognized that the child has as
much
taste for
numbers 1
as for letters,
Boston, 1887,
and proceeded 2 P. II,
to gratify
THE PRESENT TEACHING OF ARITHMETIC this taste in the first school year, a
erally been followed since
his time.
117
plan which has genThis idea of post-
poning the formal study of number until the second year is one of several pre-Pestalozzian ideas which have recently appeared, and itself
it has not as yet impressed educators as one of great importance. That upon
the practical results for arithmetic,
if
the child con-
tinues to the seventh grade, will probably be equally good, is true. That the child might put his twenty
minutes a day, now devoted to arithmetic, to better use, may be true but that he would do so is improb;
Until
able.
we
systematize play,
and put the time
gained from primary number to physical exercise, in the open air, under a skilled teacher, it is doubtful if the child should give up the few minutes a day in a line of
work
for
which he has a
taste
and about which
he delights to know. Oral arithmetic
before the easy,
fell,
sance. it
to
The
oral arithmetic, so necessary
Hindu numerals made
as
we have
seen, into disfavor at the Renais-
Revived by Pestalozzi and
had much favor not only Colburn's
written computation
excellent
his contemporaries,
Europe, but also, thanks But the work, in America. in
advent of cheap slates and paper and pencils seems to have driven it out of our schools for a generation.
be hoped that we shall not again cease to secure reasonable facility in It
is
now
rapid oral
reviving,
and
it
is
to
work with the ordinary numbers
of daily
life.
Il8
THE TEACHING OF ELEMENTARY MATHEMATICS
The
subject can easily be carried to an extreme; but
within reasonable limits It lubricates
grade.
it
should be demanded in every
the arithmetical machine, and five
minutes a day to this subject could hardly fail to bring all pupils to reasonable facility with numbers.
This
Treating the processes simultaneously course, as impossible as
occupy the same
it
to
is
space at the
means the
is,
of
have several bodies
same
time.
But the
mastery of a number, the study of the four processes, before the next is expression
studied.
As
so-called
already stated, this
is
the essence of the
Grube method, its fundamental feature as well as its fundamental defect. " It seems absurd, or worse than
number
absurd, to insist on thoroughness, on perfect concepts, at a time
when
knows
perfection
is
....
impossible
he has even an intelligent three, working conception of three, he can proceed in a few lessons to the number ten, and will thus have all higher If the child
if
numbers within comparatively easy reach." 1 A more tedious way of presenting number than that of Grube' s would be hard many, The
to find,
this feature still
spiral
method
books we have had
and
yet, in
has a considerable following. In
the
of the older works.
method" seems 1
to
preparation
various experiments of
result of the restless desire to
bad features
America and Ger-
have been
of
text-
late, all
the
break away from the
The
first
so-called
" spiral
suggested by Ruh-
McLellan and Dewey, The Psychology of Number,
p. 172, 176.
THE PRESENT TEACHING OF ARITHMETIC sam,
1
and
to
have found
119
favor anywhere until
little
it
was recently taken up in America. It consists in taking the class around a circle, say with the topics of
common
fractions, decimal fractions, greatest
common
divisor, and square root; then swinging around again on a broader spiral, taking the same topics, but with
more
difficult
problems; then again, and so on
until
the subjects are sufficiently mastered.
The idea has much to recommend it. A child is now expected to master common fractions by going
not
once over the subject and then leaving
it
And
forever.
yet the older text-books expected him to do just that for greatest
common
idea can easily be
divisor,
square root,
carried to
etc.
But the
an extreme, the
class
swinging around the spirals so frequently as to produce mathematical nausea. It is a question how
scheme should be made, and it has not been sufficiently tried to answer this question.
elaborate the
Common sequence which has
vs.
decimal
fractions
The
question
common and decimal fractions It recently been much discussed.
of
is
one
is
easy
of
whole subject by some such remark as, "Logically the decimal fraction comes first, because it grows naturally out of our number system," and this
to dismiss the
is
frequently done in some educational sheets. 1
Aufgaben
fur das praktischen
tern drei Klassen der Realschulen
Rechnen zum Gebrauch und
in
Another in
den un-
den obern Klassen von BUr-
gerschulen in drei concentrisch sich erweiternden Cursen, 1866.
120
THE TEACHING OF ELEMENTARY MATHEMATICS
say that the Prussian educational decree of 1872 put the decimal fractions first, and -that the experience of these many years has proved the wisdom of the will
But
plan.
an argument can be advanced
just as strong
by saying that psychologically the common fraction should precede, because the concept that historically
was
it
in use
the simpler;
is
long before the decimal
system of writing numbers was known, to say nothing
and that Prussia's experiment
of the decimal fraction;
has been productive of such doubtful results that Baden,
and Bavaria, and Saxony
The
still
follow the older plan. 1
however, one belonging rather to the old-fashioned course than to the modern, to the days
question
when
mon plan.
really,
the pupil was expected to "master" com-
fractions
modern
is
before
the
studying
any standing, follow no such no one ever thinks, practically, of
arithmetics, of
The
fact
is,
teaching 0.5 before %, or 0.25 before fractions J, ^,
forms
Our
decimal.
enter into the
0.5, 0.25, represent a
straction,
work
much
.
of the
The simple
first
year
;
the
greater degree of ab-
and hence should have place considerably
later.
But on the other hand, as between adding 0.5 and 0.25, or ^-|^ and -ff-J-, there can be no question as to which should have
first
place.
clusion will probably be reached 1
For
details
as
to
these
state
And
hence the con-
by most teachers that
systems see Dressier,
Der mathe-
matisch-naturwissenschaftliche Unterricht an deutschen (Volksschullehrer-)
Seminaren, Hoffmann's
Zeitschrift,
XXIII.
Jahrg., p. 15.
THE PRESENT TEACHING OF ARITHMETIC
121
the elementary treatment of simple fractions has the first
place, but that, long before the pupil
common
serious difficulties of the
of United States
to the
fraction, the tables
money, or possibly those of the metric
make him
should
system,
comes
familiar with
the decimal
forms and the simple operations therewith.
Improvements in algorism, that of
work
in the
arrangement performing the elementary operations, are
in
is,
Two
constantly appearing, and some are of real value.
which are now struggling for acceptance, with every prospect of success, may be mentioned here as types. In subtracting 297 from 546,
we have
the
-
^
two old plans, both dating from the time of
2
7
the earliest printed text-books, at
The
least.
249 calculation
is
substantially this:
1.
7 from 16, 9; 9 from 13, 4; 2 from
2.
7 from 16, 9; 10 from 14, 4; 3 from
But we have 3.
7 and
To
this
also a
more recent plan 4, 14; 3 and
16; 10 and
9,
4,
2;
or
2.
5,
:
2,
5.
might be added a fourth plan which has
some advocates 4. 7 from 10, 4; 2 from 4, 2.
:
3
;
3
and
6,
9
;
9 from
10,
i
;
i
and
3,
All four of these plans are easily explained, the first
rather
more
easily
But the
than the others.
third has the great advantage of using only the addition
table
saving
in
both addition and subtraction, and of
much time
in
the operation.
It
is
the
so-
122
THE TEACHING OF ELEMENTARY MATHEMATICS
called
"
Austrian method
"
The
of subtraction.
fourth
plan, while a very old one and possessed of some
good features, is so ill adapted to have no place in the school. sary to
say that the
to practical It is
old expressions,
work as
hardly neces-
"borrow" and
"carry," in subtraction and addition are rapidly going they were necessary in the old days of
out of use
;
rules,
arbitrary
but they have no advocates of any
prominence to-day. In division we have also an "Austrian method," a valuable arrangement.
It is
not long since a prob-
lem like 6.275-^-2.5 was "worked" by a rule which was rarely developed. Now the work is arranged in this
way
:
2.51
2.5)6.275
25)62.75
50 12.75 12.5
0.25 0.25
Such an arrangement leaves no trouble with the decimal point, and the work is easily explained. In the above problem the entire
remainder
is
brought
down, and the decimal point is preserved throughout, as should be done until the process is thoroughly understood
;
then the abridgment should appear.
THE PRESENT TEACHING OF ARITHMETIC The
common
explanations of greatest
of
sion
are
fractions, etc.,
so
divisor, divi-
in
fully given
our recent American text-books that
any of
not worth
is
it
123
while to attempt them in a work of this nature.
The formal solution
of
applied problems
generally recognized as logic
The
work.
ever, but cal
it
result of the is
work
now
is
explanation
;
course
of
recognized
work
of
generally
the
when
Hence the
percentage and in analysis given in step form, the actual
solutions of problems
now
number
the value of a logi-
the pupil has reached the proper grade.
are
now
as important as
is
problem
not all-important
as well as
is
in
elementary
being
operations
omitted.
For example A commission merchant remits $1073.50 as the net :
proceeds of a sale after deducting
5%
required the amount received from the 1.
0.95 of the
2.
/.
the
= amount = amount
$1073.50.
$1073.50
by dividing these equals by
Or
better
(not the
still,
number
by of
commission;
sale.
letting dollars,
-s-
0.95
0.95
2.
.'.
$1130,
0.95.
x
represent this amount
since
we
are preserving
the dollar sign before the other numbers), 1.
=
x= $1073.50 x= $1073.50-^-0.95 = $1130-
THE TEACHING OF ELEMENTARY MATHEMATICS
124
This introduces the equation form in a more pronounced way, but this is now generally approved by educators. 1
There are plan
some advocates
still
of
the
following
:
1.
95%
of the
amount
is
amount
is
amount
is
2.
/.
i%
of the
3.
.*.
100%
of the
$1073.50.
-^ of $1073.50= $11.30. 100 x $11.30
=
$1130.
This, the unitary method, is by some thought to be simpler than the others, though why it is simpler to derive o.oi from 0.95 than to derive i from 0.95, it
is
difBcult to say.
The
following form has
also
an occasional advo-
cate: 1.
Let
2.
Then 100%
3.
If
1
This
is
equal the amount.
95% = i% =
4. 5.
00%
and 100%
=
5% = 95%. $1073. 50,
$
11.30,
$1130.
a relic of the mediaeval method of " false posi-
The 100%
tion," a pre-algebraic device.
and we begin by
letting
this
i
is
equal the
merely
i,
unknown
1 "Alle Padagogen sind hierin einverstanden." Hentschel, p. 81. " Can one any imagine a good teacher, who is also a good algebraist, who
will not train his pupils to use letters for is
completed?"
p. 23.
The
Gleichung
numbers long before arithmetic
Safford, T. H., Mathematical Teaching, Boston, 1887,
question
is
discussed in a broad
in der Schule, in
Hoffmann's
way by
Zeitschrift,
Schuster, M., Die
XXIX.
Jahrg., p. 81.
THE PRESENT TEACHING OF ARITHMETIC Of course x
quantity.
or any other symbol might be
we know very
used to better advantage, for
unknown
the
125
not
is
quantity
does not equal $1073.50;
it
I.
95%
is
well that
Furthermore,
95%
of the amount,
or of x, that equals $1073.50.
following such a plan as the one
By
well-founded
the
mechanism
complaint against
of the past disappears.
mentioned
first
the
thoughtless
Instead of words
and rules without content, there is content with a minimum of words and with no unexplained rule. 1 It
"2 -*-
is
ft.
x
8 sq.
only a few years back that such forms as 3 ft.
=6
ft.
=
sq. ft,"
and the
3 ft.,"
Now, however,
all
"2x3 = 6 like
ft,"
"
24
cu.
ft.
were not uncommon.
careful teachers are insisting that
such inaccuracies of statement
beget inaccuracy of hence should not be tolerated in the
thought and schoolroom.
It
is
true
that
these
all
depend upon
the definitions assumed, and that well-known teachers
have advocated such a change of allow of saying
1
Die Kinder
.
.
.
"4
ft.
X
2 yds.
3456
as will
2 sq. in." ; but,
losen einschlagige Aufgaben, aber alles das geschieht
meistens auf mechanischera Wege. Inhalt.
=
definition
Fitzga, p. 5.
The
Wir
finden
Worte und Regeln ohne
other side of the case, the danger of using
is presented in Supt. Greenwood's Dissent from Dr. Harris's Report of the Committee of Fifteen. 2 This illustration, from an article by Professor A. Lodge in the General
algebra unnecessarily
Report of the Association January, 1888. in recent years.
for the
Improvement of Geometrical Teaching,
Similar articles have appeared in Hoffmann's Zeitschrift
126
THE TEACHING OF ELEMENTARY MATHEMATICS
with our present definitions, such forms lead to great looseness of thought. It is the loose
erated by
manner
of writing out solutions, tol-
teachers, that gives
many
the
rise to half
The
mistakes in reasoning which vitiate pupils' work.
carelessness in form begets that carelessness of thought
which gives point 1.
A
bottle
.-.
2. /.
J
a bottle
to such
=a full = a full
=2 dimes = 4
20 dimes
400
amusing absurdities as these bottle J empty.
:
Divide by
,
bottle empty.
dollars.
Square each member and
dollars. 1
Longitude and time furnish a type of the applied
problems of arithmetic, one in which much carelessness of form and thought is often apparent, and as such it is entitled to some special consideration.
The
subject
is
best presented, perhaps, by a brief
discussion of the question of the relative positions of
the sun and earth at the hour of the class recitation, the globe being held before the class, the northern
hemisphere
and North America being on the
visible,
be recognized easily
lower half so as to
then
"
right side
"
up
located, the question of
noon on the
(it
being
The sun being
to the pupils).
the forenoon and the after-
earth's surface
may be
discussed, then
the position of midnight, then the effect of the revolution of the earth with respect to these periods 1
Adapted from Rebiere,
Paris, 1893, P-
33 1 -
A.,
Mathematiques
et
mathematicians,
;
and 2, ed,,
THE PRESENT TEACHING OF ARITHMETIC finally, for
I2/
one lesson, the number of degrees through vicinity must pass in order
which the schoolhouse and
that the time shall be 24 hours later.
All this leads to the development of two tables, the foundations upon which the subject rests:
TABLE 360 correspond to 24 .-.
i
corresponds to
.-.
i'
corresponds to
/.
i" corresponds to
I
hrs.
$ of -fa
of
4 min. 4
TABLE 24 .-.
i
/. i .*.
i
to
hrs.
correspond
hr.
corresponds to
min. corresponds to sec.
To
=
say that 360
= 24
Ibs.
correspondence, as
such equality as
The
sec.
II
360.
^ of -fa
is
in
= 15. = ^ of = ^ of
360
of
15
corresponds to -fa of
say that $ 4
= -^ hr, = 4 min. = T^ min.= 4 sec. = T^ sec.
J 7 of 24 hrs.
15'
i
i'
24
hrs. is as inaccurate
of
beef
value,
there
;
may
but
etc.,
theory of the subject
their
15".
as to
be some
there
is
now
no
is
best brought out
oral problems of
this nature
the difference in longitude between two ships is
15'.
set forth in the statement.
by numerous simple what
= =
difference in time?
20 min., what
in time
is
tude?
To make such problems
is
their
is
:
If
10,
If their difference
difference in longipractical,
cases of
128
THE TEACHING OF ELEMENTARY MATHEMATICS
ships or observatories should be used, since the recent
rapid development of standard time has shut out local
time in the large majority of places in the civilized world.
Written
may now be
solutions
such form as the following
The IO
=
3"
:
longitude between two ships
difference in
45'
required in some
is
required the difference in time.
1.
10 x 4
2.
45 x T^ min.
min.
= =
40 min. 3 min. (or
45 x 4
sec.
=
180 sec.
3 min.). 3.
sec.
=
2 sec.
difference
in
time
30 x
The
^
43 min. 2
.*.
4.
between two ships
sec.
is
43
min. 2 sec., required the difference in longitude. 1.
43 x J of
2x15" =
2.
Some
their
10
45' (or 43 *
30"-
10
3-
'
only.
adherence to the mechanical 43
15 |io
f
45' hr.
= 43
15
s.,
rule,
serious
and
to
2 sec.
I5_ 645' 10
=
min. 2 sec.
such forms
Explain
all
degrees
divided by an
will,
hr.
custom of
More
mm.
30"
3 min. 2 sec.
we
"2
write
general
:
-)
45' 30".
still
and " for longitude
such forms as these
x 15'=
15 sec., or 2 h. 3 m.
unwise to change the
using the is
=
iof
for 2 hr. 3 min.
is
it
=
the older arithmetics
of
15""
3'
but
i
abstract
tell
30" 45' 30"
the eye
number give
that
hours,
THE PRESENT TEACHING OF ARITHMETIC and that time
129
transformed by some miracle into longitude by multiplying by 15! Text-book makers may argue for brevity, but the astronomer and the
who wish
navigator tables.
is
It
not
is
brevity
use
always
brevity
we
that
longitude
seek;
it
an
is
understanding of the process.
The two
points at which the teacher needs to aim,
the elementary correspondence between
after
tude and
time
(2) the date
is
fixed,
line.
The
longi-
are (i) standard time,
complicated
old-style
and prob-
may well give way to these new and interesting The last decade of the nineteenth century topics. has seen standard time made well-nigh universal in the lems
highly civilized portions of the world, and the recent
events in the Philippines have given to the subject of the date line even greater interest for American pupils. 1
and proportion
Ratio tional
copartnership
still
maintain
most
in
usually setting forth an
array
from some generations
past.
of
of
conven-
their
our
arithmetics,
problems inherited
There
is
just
now
a
good deal said about introducing the ratio concept earlier in the course, and this may happily break up the partnership and show ratio as the important subject
which
At 1
it
really
present,
For a
full
in
is.
the
standard
discussion of these
two
type of
subjects, with late information con-
cerning standard time, and with maps showing the date referred to
Beman and
K
arithmetic,
line,
the reader
Smith's Higher Arithmetic, Boston, 1897.
is
THE TEACHING OF ELEMENTARY MATHEMATICS
I3O
merely as an introduction
has place
ratio
portion.
The
of
as
rule,
if
is
subject
it
rarely
arithmetical
only
found
is
subject
were
used
to
pro-
matter
a
taught as
be used so often as to
to
The
treatment.
unscientific
this
justify
latter
in
is,
the
business, and almost
its
applications
fact
value
of
are
to
be
problems and in problems involving Before simple equations were invented the subject had much more value than at present, and the arbitrary " Rule of Three," as it was called, in physical
similar figures.
may have been subject
At
justifiable.
teach the
present, to
by mere rule, or by any such senseless device and effect" method, is unwarranted.
as the "cause
There
now a growing reform This movement employs
is
proportion.
notation, with which the
common
tiplying these equals
Consider, If a
plumb
Let
line
x=
I
number
the
= the =
or 6ft.
-
x=
familiar,
= T4^,
fractional
and the
to find x.
Mul-
^.
6
shadow 6
yd. long casts a
shadow 84
or -
and
3,
an adjacent
is
Then
by
:
is
presenting
the
example, a single applied problem
for
instant casts a I.
pupil
equation form, thus
how high
in
just
flagstaff ft.
long
which
:
ft.
long,
at the
same
?
of feet required.
ratio of the heights,
the ratio of the shadow lengths.
THE PRESENT TEACHING OF ARITHMETIC
And
2.
shadow
since
the
Multiplying by
3,
/.
the staff
ft.
is
After the class
be
42 is
:
=
3
84
:
x=
42.
high.
familiar with
with the
given
these are needed in
x
the
proportional to
lengths,
3.
should
are
heights
131
the theory, the
other
common
symbols,
scientific
reading, thus
6, or even the antiquated form
Solutions
of
this
with
nature,
the
x
:
3
:
:
84
reasoning
:
:
6.
set
"
"
forth, give us the
work
because
thought reckoning (Denkrechnen) which our best educators demand, in place of the rule-
work
of the old school. 1
that
being the
nation which
formula
is
figure,
inherited
plan
geometrically,
from the Greeks, the
most excelled
preferable on
the square on for the
treated
formerly
in geometry in ancient But the method which follows the algebraic
times. 2
1
was
root
Square
f -f
n
is
many
f + 2
accounts.
2 fn
+
The
^2 where ,
fact that
f
stands
found part of the root and n for the next may profitably be pictured by a geometric
The general
question
of
proportion
is
discussed in
a
valuable
by Dressier, Der mathematisch-naturwissenschaftliche Unterricht an deutschen (Volksschullehrer-) Seminaren, Hoffmann's Zeitschrift,
article
XXIII. Jahrg., 2
Theon
ric plan.
i.
of Alexandria, father of Hypatia, gave the
Gow, History of Greek Mathematics,
p. 55;
common Cantor,
geomet-
I, p.
460.
THE TEACHING OF ELEMENTARY MATHEMATICS
132
But the formula
be preferred to the diagram, as a basis for work, because diagram.
The geometric
1.
to the square
is
to
notion limits the idea of involution
and cube roots;
The formula method makes the cube and higher
2.
roots very simple after square root
We
3.
concepts
understood;
are working with numbers, not with geometric ;
The formula
4.
is
lends
itself
more
easily to a clear
explanation of the process.
One
of the great difficulties in explaining square root
the fact that tradition has encumbered
lies in
superfluous
difficulties.
for
Consider,
with
it
instance,
the
do we separate into periods of two The answer figures each, beginning at the right?" it not do was neces"We need be so; given, might question,
"Why
when square
sary
root
was merely a matter
of rule;
one thinks, such separation is quite unnecessary; furthermore, we would not begin at the right anyway,
if
but rather at the decimal point, this rule having been
framed long before the decimal point was known." Again, "Why do we bring down only one period at a
time?"
For reply we may
say,
better for beginners to bring
each time, because
Of
all
don't;
we may adopt
desire,
it is
makes the explanation this
much
of the remainder
course, after the complete process
stood
we
it
"We
down
is
easier."
fully under-
and other abridgments
and then the explanation
is
not
if
difficult;
THE PRESENT TEACHING OF ARITHMETIC
133
very poor policy to let such unnecessary questions enter at a time when the teacher is seeking but
to
it
is
have the process clearly understood. may be said that these suggestions and the follow-
It
ing solution
make
the process longer than necessary.
But since almost the
sole justification for the
of involution is the fact that this training is of
paramount importance.
purposes the square root
subject
offers training in logic,
it
For
practical
usually extracted
is
by the
help of tables.
A
problem in square root might, then, be arranged
as follows
:
23.4
=root
z 2 547.56 contains some square, f +2fn + n
2/= 40 = 43
147.56 contains
129
2/= 46 = 46.4
2/n
=2/ + ^
+ n*
t
where /= 20
2
18.56 contains 2
fn + n\ where /= 23
18.56 = 2/;z +
2
This arrangement shows what each number equals (exactly or
explain are (i)
taken as the culties.
1
For
and the only things these equalities, and (2) why 2 /
approximately),
"trial divisor," matters offering
no
to is
diffi-
1
full
explanation, and
method, treatment of
for other suggestions as to the factoring
fractions^
the
double sign,
Smith's Higher Arithmetic, Boston, 1897, P- 35-
etc.,
see
Beman and
134
THE TEACHING OF ELEMENTARY MATHEMATICS The common measures
The metric system life
demand
have become thoroughly
of daily
Until they
great attention in arithmetic. familiar, until they
have taken
prominent place in the child's mind, until they have been taught with the actual measures (as far as may be) in hand, and until they have been practically used
hundreds of concrete problems, the metric system has no place. The child can get along for a while without this system indeed, he may never be conin
;
scious of a loss
if
he does not know
mon system he needs
On the other hand,
it;
but the com-
daily.
compared with the apothecaries' and troy measures, or with leagues, furlongs, barleycorns,
pipes,
tuns,
as
the metric
etc.,
quintals,
system
should certainly have precedence. Only two or three bits of advice to the teacher need measures, like all others taught be actually in hand they must be made to seem real by abundant use; merely to learn
be given.
First, these
to the child, should
the tables
is
of
little
;
value.
The French
their little cases of metric units
schools, with
on the front wall of the
always within sight of the children, an example worthy of our attention. 1
recitation rooms, set
Again, the child will probably use the system by itself if at all that is, he will not be translating back ;
and forth with the common system.
grammes
in
4
cwt. 37 Ibs. 2 oz., 1
See also Fitzga,
is
To ask how many
worthless as a practical
I, p.
41, 57.
THE PRESENT TEACHING OF ARITHMETIC problem;
it
gives the child a
little
"figuring," but
destroys his appreciation of the great
A few
the modern system.
be translated, as
of the
135
common
in a question like this
it
advantages of
:
units
may
A traveller
in
Germany is allowed 25 kilos of baggage free; about how many pounds is this ? But such translation should be confined to common cases and to oral work. The pupil should be led to see that the names are not so strange as might at first appear. As a gasmetre measures gas, and a water-metre measures water, so a metre is a unit of meastire ; it is a little longer than our yard.
And
as a mill i
is
is
metre;
metre
is
So
o.ooi of
o.oi of $i, so a centimetre is o.oi of
comes before
as a decimal point
as a
is
metre;
as a cent I
o.ooi of $i, so a millimetre
of
is o.i
dekagon
is
i
tenths, so a deci-
metre;
a loangled figure, so a dekametre
10 metres.
milli-
means
centi-
means
o.ooi, o.oi,
and there are only three new
deci-
means
O.I,
deka-
means
10,
prefixes to learn:
hekto-,
which means
100,
kilo-,
which means
1000,
myria-, which means
10,000.
THE TEACHING OF ELEMENTARY MATHEMATICS
136
With these
mind the
prefixes well in
metric system are practically known. deal of the oral
devoted
and
to
asking
drill
these
in this
prefixes,
great
work may profitably be taking them at random
numerical
their
tables of the
Hence a
equivalents,
and vice
versa.
The grade
which the metric system is taught is determined largely by the science work in the school. Since
all
in
science
now
uses this system,
may be
it
taken
up as soon as simple physical problems are introduced. But reference is so frequently made to the system in the current literature of the day, that to postpone the subject beyond the eighth grade, or to teach
perfunctory manner,
is
it
in a
unwarranted.
The applied problems, and especially the business problems involving percentage, are so well adjusted
and
uses
to
the
in
the modern
capacities
American
text-books, that
But topics
be said upon the subject. count, time,
the various grades,
of
little
need
like true dis-
of
payments, partnership, involving exchange, insurance as it was fifty these subjects have no place in the com-
equation
arbitrated
years ago mon school arithmetic of to-day.
Our
recent books
generally print pictures of drafts, checks, notes,
etc.,
and give such explanations of common business customs as render these intelligible to pupils before they leave the eighth of
the
actual
grade.
documents
Such in
helps,
the
and the study
classroom,
will
si-
THE PRESENT TEACHING OF ARITHMETIC
137
much of the prevalent criticism that we teach much for the school and too little for life. 1 " Short cuts" The short methods so much sought earlier times are now less in demand. The reason
lence too
in
not that time
is considered less precious, but that " " have been found generally to apply the short cuts is
problems of no importance, or that the elaborate use of tables has rendered them unnecessary. For
to
example, pert
it
was once considered a mark
accountant
to
have
methods of reckoning turns
man
of
an ex-
hand numerous short
at
interest;
now
the accountant
once to his interest tables, and the average with no tables at hand has forgotten the rules at;
of his school days.
Formerly the expression "75 -s- 15 = 5 hrs." was allowed on the score that its brevity justified its falsity; now, any one who has occasion to solve problems of this kind in a practical way resorts to Formerly, mere
rule
work was
justified in
tables.
square and
cube root on the plea of brevity; now, for practical purposes, we generally extract such roots by logarithmic or evolution tables.
Mensuration
Even now the
was formerly taught strictly scientific
solely
by
rule.
treatment belongs to
But there are certain propositions that are so commonly needed that they must have place in
geometry.
arithmetic 1
for
those
who may
Vielfach nur fur die Schule
und nicht
not
fur das
study geometry. Leben.
Fitzga,
I, p. 6.
THE TEACHING OF ELEMENTARY MATHEMATICS
138
Such are the propositions which give the formulae for measuring the square, or more generally the rectangle and the parallelogram, the triangle, possibly the trapezoid, the circle, the parallelepiped, the cylinder,
and possibly taken up
and
outlined
is
of reason 3
sq. in. is
X
=
easily
by i
6
in.
altitude,
be
reasonably scientific way, our modern text-
most of
in
by
easily
3 in. is easily
the area
of
made
a matter
using a figure illustrating the statement
sq. sq.
=
in. in.
shown by
6
A
sq. in., or
the statement
2x3
parallelogram cut from paper
the use of the
area the rectangle
in
may
figures
For example, the computation
of a rectangle 2
x
these
of
in arithmetic in a
this
books.
2
and sphere.
also the cone
The mensuration
of
the
scissors
same base and same
a figure already considered.
shown
to equal
By
paper-cut-
be equal to half of a certain parallelogram, and hence to half of the rectangle having the same base and the same altitude. ting the triangle
By
is
to
a few measurements of circumferences and their
corresponding diameters the ratio c : d can be shown to be approximately 3^, a value sufficiently exact for
The
ordinary mensuration. if
thought best, that
closer approximation
it
is
is
teacher
proved
in
may
then
state,
geometry that a
3.1416, or 3.14159.
The
pupil
has thus the interest of a partial discovery, and at the same time the possibilities of the more advanced
mathematics are suggested.
Similarly, as set forth in
THE PRESENT TEACHING OF ARITHMETIC
many
our better
of
of
class
text-books,
necessary propositions in mensuration
139
the
may
other
profitably
be treated. 1 In the days
Text-books
when
and poor there was some excuse
The
orate notes. institution of
for
elab-
copy-book was then an some importance. But at present there
we have good
books,
save the time of pupil and teacher.
mean
dictating
arithmetic
no such excuse;
is
text-books were few
that the
book
shall
but rather a servant to
and they
This does not
be a master to be feared,
assist.
In the lower grades,
while the teacher should seek to follow the general lines of the text-book,
each new demonstration should
be discovered by the class (of course with the teacher's leading) in advance of the assignment of book work. If the author's plan is reasonably satisfactory
it
should
be followed, in order that the pupil may be able to review the discussion without the waste of time in note-taking;
a great
many hours
are squandered by
teachers in attempting to "develop" something along
some
line
when
the author's method
better.
not
followed
by the text-book is
quite as good
in
hand, usually
There are now several excellent text-books
with satisfactory demonstrations and with up-to-date problems, and these should receive the support of the profession. 1
See also Hanus, P. H., Geometry in the Grammar School, Boston,
1893.
THE TEACHING OF ELEMENTARY MATHEMATICS
140
in
But with any text-book we shall do well to keep " mind the words of President Hall American :
seem to me to have spun the simple and immediate relations and properties of numbers over teachers
with pedantic factoring, is
not this
decimals,
smaller
that
I
know do
is
for
compass,
four rules, fractions,
per
proportion,
all
text-books
The
difficulties.
The
essential?
only
this,
they look
pure number relations, which
is
and
cent.,
and
only
roots,
best European are at
in
facility
hindered by the
the in
irrele-
vant material which publishers and bad teachers use as padding."
1
The
Explanations
be
given to
question of the explanations to
and demanded from a child
The primary work
one.
is
is
a serious
preeminently that of lead-
ing the child to discover the relations of number, and to
memorize certain
which he
will
subsequently need.
action suggested " Follow a
tion
:
facts (like the multiplication table)
A
few rules of
by M. Laisant are worthy of attenrigorously experimental method and
do not depart from it; leave the child in the presence of concrete realities which he sees and handles
make
to
his
own
abstractions
;
never
attempt
to
2
demonstrate anything to him; merely furnish to him such explanations as he is himself led to ask; and 1
Letter from G. Stanley Hall to F. A. Walker, in the latter's
on arithmetic, 2
p. 23.
Le.y by a formal, logical demonstration.
monograph
THE PRESENT TEACHING OF ARITHMETIC
141
give and preserve to this teaching an appearance of pleasure rather than of a task which is imfinally,
If cerebral fatigue
posed.
led to
is
and
his attention
fix
is
produced,
if
the child
on matters of no
interest,
master a line of reasoning too much vance for him, then the result is a failure." 1 to
The
in ad-
period of explanation comes later in the course,
say after the fifth grade; but even here the explana-
be by questioning on the part of full and free demonstration by
tion should rather
the teacher than by a
Where complete "explanations"
the pupil.
are re-
quired from the pupil, say of subjects like greatest
common etc.,
the division of fractions, cube root,
divisor,
the result
is
usually a lot of memoriter
work
of
no more value than the repetition of a string of rules. " of the various But by questioning as to the " why steps, the reasoning
that It
set is
is is
essential)
is
(which
in
most such work
is
all
laid bare.
the same with
many
applied problems.
The
analysis sometimes required of pupils
forms of
of very questionable value.
On
the other hand, a
own reasoning is, of course, when he is sufficiently advanced
statement of the pupil's
extremely important, to
give
But for primary children any elaborate
it.
explanation
is
impossible.
Indeed, in the midst of
our theorizing on the subject of explanations, refreshing to
it
all is
read what a psychologist like Professor 1
La Mathematique,
p. 203, 204.
THE TEACHING OF ELEMENTARY MATHEMATICS
142
James has "
upon the subject of primary work
to say
...
It is
in the
mind takes most
child's
Working out results name things when they
delight.
rule of thumb, learning to
by
see them, drawing maps, learning languages,
me
:
of concretes that the
association
seem
to
the most appropriate activities for children under
thirteen to be
engaged
man
dent that no
will
...
in.
I
feel
pretty confi-
be the worse analyst or reasoner
or mathematician at twenty for lying fallow in these
respects during his entire childhood."
There
Approximations teachers that some
is
a feeling
this
rather encouraged
is
among many
virtue attaches to the carrying of
a result to a large number of
hence
1
decimal
places,
and
among
pupils.
As
a matter of fact the contrary is usually the case in If the diameter of a circle has been measpractice.
ured
correctly
to
o.ooi
inch
there
compute attempting than three decimal places, and 3.1416 than 3.14159.
tiplier
at thousandths
The
is
no
circumference
the
to
is
use to
in
more
a better mul-
result should be cut off
and the labor
of extending
it
beyond
that place should be saved.
Now
since
we
rarely
use
decimals
beyond o.ooi
except in scientific work, and since no result can be
more exact than
the data,
and since even our
scientific
measurements rarely give us data beyond three or four decimal places, the practical operations are the contracted 1
Letter to F. A. Walker, in the latter's monograph, p. 22.
THE PRESENT TEACHING OF ARITHMETIC ones,
those which are correct to a given
143
number
of
For
this reason, in this age of science, apare of great value in the higher methods proximate grades which precede the study of physics. The fol-
places.
lowing are types of such work
l :
10.48
10.48
3.1416
3.1416)32.92=
31416)329200 3142 150 126
24 24 32.92
For the same reason the logarithmic tions
of
table
of
is
practical use of
a small
great value in the computa-
Two
elementary physics.
or
suffice to explain the use of the tables
three lessons
and
to justify
the laws of operation, a small working table can be
bought for five cents, and the abundant practice.
stupidity,
reviews,"
physics affords
However much reviews may
Reviews their
field of
a
as
skilful
is
apt
to
teacher
fail
from
be the case with "set is
always reviewing in But there is one
connection with the advance work.
season 1
The
when a review
is
essential,
a
brisk running
explanations are given in any higher arithmetic,
and Smith,
p. 8,
n.
e.g.
Beman
THE TEACHING OF ELEMENTARY MATHEMATICS
144
over of the preceding work that the pupil his bearings,
and
this is at the
Such a refreshening
year.
take
of the mind, such a lubri-
the mental machinery, gets one ready for
cating of
Complaints which teachers generally
the year's work.
make
may
opening of the school
poor work in the preceding grade are not unfrequently due to the one complaining; the effects of the long vacation have been forgotten; the engine of
is
rusty and
is
made.
it
needs oiling before the serious
start
In these reviews the same correctness of statement
necessary as in the original presentation, though To let a child not always the same completeness. is
+ 3x2
say that 2
let
is
10 (instead of 8)
is
to
sow
grow up and choke the good wheat. him see forms like
which
will
2
ft.
V4 or to let
x
3
ft.
sq. ft.
=6 =
sq.
ft.,
2 ft, 2
45
is
contained in $ 10,"
etc.,
is
to take
like
"2 times
away a large part
mathematics should possess.
15=
x 0.50 = 1
him hear expressions
as 2
-j-
tares
To
3 hrs.,
1, etc.,
"As many
times
greater than $3," of the value that
CHAPTER
VI
THE GROWTH OF ALGEBRA algebra Reserving for the following the of the definition of algebra, we chapter question may say that the science is by no means a new one.
Egyptian
Or
rather, to be
tion
is
more
precise, the idea of the equa-
not new, for this
is
only a part of the rather
we
undefined discipline which
call
In the
algebra.
oldest of extant deciphered mathematical manuscripts,
Ahmes papyrus to which reference has already been made, the simple equation appears. It is true that neither symbols nor terms familiar in our day are used, but in the so-called hau computation the the
linear equation with
for
Symbols
unknown
one unknown quantity subtraction,
addition,
quantity
are
equality,
The
used.
its
twenty-fourth:
whole,
bols
it
makes y -+
means
problems
are
thirty-first):
makes
"Hau 19,"
x=
also
"Hau,
Ahmes
i.e.,
gives,
seventh, (literally heap\ which put in modern sym-
more
Somewhat like
given, f,
its
J,
the its
\x + \x + \x + x='&. i4S
difficult
following {,
33," L
an
is
its
19.
its
and the
following
example of the simpler problems which his
solved.
is
its
(his
whole,
it
THE TEACHING OF ELEMENTARY MATHEMATICS
146
must be
It
that
however,
said,
Ahmes had no
notion of solving the equation by any of our present
His was rather a "rule of
algebraic methods. position," as
the
fying
was
called in mediaeval times,
guess-
an answer, finding the error, and then modi-
at
ing
it
false
some work
Ahmes
1
guess
accordingly.
also
gives
and one example
in arithmetical series
in
geometric.
Algebra made no further progress, now known, among the Egyptians. But in
Greek algebra so far as
the declining generations of
"golden age" had passed,
As
tance.
already
Greece,
matics accordingly.
n
odd
the
assumed some imporGreek mind had a
it
it
worked out a wonder-
system of geometry and warped
first
after
stated, the
leaning toward form, and so ful
long
numbers
The
fact n*
is
t
its
other mathe-
sum
that the
for
of
example, was
the dis-
covered or proved by a geometric figure square root was extracted with reference to a geometric diagram; ;
numbers
figurate
tell
their
by
name
that
geometry
entered into their study.
So we
"
"
Elements of Geometry 2 for formulae (a + ) and other simple (B.C., c. 300) algebraic relations worked out and proved by geometric figures. Hence Euclid and his followers knew 1
Besides
p. 38.
p. 18.
A
find
in
Eisenlohr's
short sketch
is
Euclid's
translation
given in
already
mentioned, see Cantor,
I,
Gow's History of Greek Mathematics,
THE GROWTH OF ALGEBRA
147
from the figure that to "complete the square," the 2 geometric square, of ^ + 2 ax, it is necessary to add a2
He
.
also solved, geometrically, quadratic equations
form ax
of the
x*
1
form x
equations of the
With the ever,
it
b,
older
+ x2 =
y
a,
Greek view
was impossible Recognizing
headway.
ax
for
and simultaneous
b,
xyb^ of
mathematics, how-
algebra
the
linear,
to
make much
quadratic,
and
cubic functions of a variable, because these could be
represented by
lines,
squares,
and cubes, the Greeks
of Euclid's time refused to consider the fourth
a
of
because
variable
beyond
the
fourth
power was
dimension
their empirical space.
Algebra had, however, made a beginning before Euclid's time. Thymaridas of Paros, whose personal
unknown, had already solved some simple equations, and had been the first to use the expressions given or defined (cbpLa-fjLevoi), and unknown history
is
.quite
2
and
seems not improbable that the quadratic equation was somewhat familiar before the Alexandrian school was founded. 3 Arisor undefined
totle,
too,
quantities
(ao/atcrrot),
had employed in
the
it
letters
to
statement of a
indicate
unknown
problem,
although
not in an equation. 4 1
2 8 4
Heath, T. L., Diophantos of Alexandria, Cambridge, 1885, p. 140. Cantor,
I, p.
148
;
Gow,
Cantor,
I, p.
301
;
but see Heath's Diophantos, p. 139.
Gow,
p. 105.
p. 97, 107.
THE TEACHING OF ELEMENTARY MATHEMATICS
148
The most
notable advance before the Christian era
was made by Heron
of
Alexandria, about
100
B.C.
Breaking away from the pure geometry of his predecessors, and not hesitating to speak of the fourth he solved the quadratic equation 1 and This was the even ran up against imaginary roots. 2 the downfall of of Greek mathematics, turning-point
power of
lines,
new discipline. we owe the first new science. An
their pure geometry, the rise of a
But
is
it
serious
Diophantus that
to
to
attempt
work out
this
Alexandrian, living in the fourth century, probably in the
first
half,
he wrote a work,
entirely devoted to algebra.
3
'ApiOfjLrjTi/cd,
This work
is
almost
the
first
one known to have been written upon algebra alone Diophantus uses only one unknown (or chiefly). quantity, 6 apiOpfa or o aopunos it
by
5'
or 9'. 4
G symbol S ), the cube
(its
apiOfjids,
The square he
calls 5
/cvfios (/e
),
symbolizing
Swa/it?, poiver
and he also gives
He names to the fourth, fifth, and sixth powers. has symbols for equality and for subtraction, and the modern expression x* 1
Cantor,
I, p.
377
;
Gow,
$x*
+ %x
i
he would write
p. 106.
374 ; Beman, W. W., vice-presidential address, Section A, American Assoc. Adv. Sci., 1897. 2
8
Cantor,
I, p.
L., Diophantos of Alexandria, Cambridge, 1885 Gow, Hankel and Cantor, of course, on all such names. De Morgan has a good article on Diophantus in Smith's Diet, of Gk. and Rom. Biog., p.
Heath, T.
100
;
;
a work containing several valuable biographies of mathematicians. 4
For discussion of the symbol, see Heath,
p. 56-66.
THE GROWTH OF ALGEBRA the form
in
more
ic
z
d^ ol rjjjiS s
z fj,
d
l
a form
t
The
than our own.
difficult
tions will be understood
149
not particularly
nature of his solu-
from the following example,
modern symbols being here used " Find two numbers whose sum is 20 and the difference of whose :
80.
is
squares
x+
Put for the numbers Squaring,
The
x*
difference,
x
Result, greater
10
+ 20 x + 40 x = 80.
we have
Dividing,
differ
10,
is
100,
^
-f-
100
20 x.
2.
less
12,
x.
8."
is
2
This does not
from our own present plan, although being less we would probably say
troubled by negative numbers
:
- xf -x* = 80. 400 40* = 80. 320 = 40^.
(20 /.
.-.
/.
thus
It
8=;r, and 20
appears
x=i2.
Diophantus understood the well. The quadratic, however, he
that
simple equation fairly
solved merely by rule.
Thus he
says,
"84^ 7^=7,
x=
J," giving but one of the two roots. Of the negative quantity he apparently knew nothing,
therefore
and
his
single
work was
easy
degrees. 1
limited,
cubic,
to
His
favorite
Heath,
p. 72.
with the exception of a
equations
of
subject
was 2
the
first
two
indeterminate
Ib., p. 76.
150
THE TEACHING OF ELEMENTARY MATHEMATICS
equations of the second degree, and on this account
indeterminate equations in
One
nated as Diophantine.
are
general of the
often
desig-
most remarkable
work of Diophantus is that, most other algebraists down to about 1700 although A.D., used geometric figures more or less, he nowhere facts connected with the
appeals
to
them. 1
Greeks
in
this
Summing up the work of the field, we may say that they could
and quadratic equations, could represent geometrically the positive roots of the latter, and solve simple
could handle indeterminate equations of the
first
and
second degrees. Oriental algebra
Diophantus, and
It
a
in
was long
after
country well
the time of
removed from
Greece, and among a race greatly differing from the
Hellenic people, that algebra took step forward.
mathematician
It
is
its
next noteworthy
true that Aryabhatta,
(b. 476),
made some
a Hindu
contributions to the
subject not long after Diophantus wrote, but he did not 2 carry the subject materially farther than the Greeks,
and
it
was not
until about
800
A.D.
that the next real
advance was made.
When c.
under the Calif Al-Mansur (the Victorious,
712 -775)
it
1
Gow,
n.
2
Cantor,
p.
114
was decided
;
Hankel,
to build a
new
capital for
p. 162.
575; Hankel, p. 172; Matthiessen, L., Grundziige der antiken und modernen Algebra der litteralen Gleichungen, 2. Ausg., Leipzig, 1896, p.
I,
p.
967.
THE GROWTH OF ALGEBRA the
Mohammedan
rulers, the
site of
151
an ancient
city
dating back to Nebuchadnezzar's time, on the banks of the Tigris,
was chosen.
To
were called scholars from
all
Christians from the West,
new
this
city of
Bagdad
over the civilized world,
Buddhists from the East,
and such Mohammedans as might, in those early days of that religion, be available. With this enlightened educational policy, a policy opposed to in-breeding and to sectarianism,
Bagdad soon grew to be the centre of Under Harun-al-Raschid
the civilization of that period.
(Aaron the
calif
Just,
reached the summit of
Indus to the (786-833),
from 786 its
Sismondi
califate
power, extending from the
Pillars of Hercules.
whom
to 809) the
calls
His son Al-Mamun "the father of
letters
and the Augustus of Bagdad," brought Arab learning It was during his reign, in the first quarter to its height. of the ninth century, that there
came from Kharezm
(Khwarazm), a province of Central Asia, a mathematician known from his birthplace as Al-Khowarazmi. 1
He
wrote the
first
general work of any importance on
algebra, that of Diophantus being largely confined to a single class of equations,
He
present name. qabalah, that equation," a
is,
title
and
he gave its Ilm al-jabr wo? I mu-
to the science
designated
it
"the science of redintegration and which appeared in the thirteenth cen-
tury Latin as Indus algebra almucgrabalceque, in 1 Abu Ja'far Mohammed ben Musa al-Khowarazmi, Abu med son of Moses from Kharezm. Cantor, I, p. 670.
Ja'far
six-
Moham-
THE TEACHING OF ELEMENTARY MATHEMATICS
152
teenth century English as algiebar
modern
in
1 English as algebra.
his writings
and
almachabel, and
So important were
also
on arithmetic, that just as " Euclid "is in
England a synonym
for elementary geometry, so algo-
ritmi (from al-Khowarazmi) was for a long time a syn-
onym
word which has
science of numbers, a
for the
survived in our algorism (algorithm).
Al-Khowarazmi
the
discussed
and quadratic equations
in
of
solution
simple
a scientific manner,
tinguishing six different classes,
much
dis-
as our old-style
on arithmetic distinguished the various "cases" His classes were, in modern notation, of percentage. writers
ax* bx
= bx,
+
2 c,
ax*
=
c,
bx
=
+ bx =
c,
x* +c =
bx,
x*
showing how primitive was the science which
could not
grasp
His method of
2 type ax -f bx 4- c = o. stating and solving a problem may
the general 3
"
same amount
to
Roots and squares are instance, one square and ten
be seen in the following
numbers
equal to
roots of the say,
x*
c,
;
for
:
4
thirty-nine
;
that
is
to
what must be the square which, when increased its own roots, amounts to thirty-nine ? The
by ten of solution
which
is this
:
you halve the number of the roots, five. This you
in the present instance yields
multiply by itself; the product 1
See also Heath,
8
From The Algebra
2
p. 149.
of
Le.,
*2
+
10*
=
39.
twenty-five.
Cantor,
I,
Add
p. 676.
Mohammed-ben-Musa, edited and
by Frederic Rosen, London, 1831. 4
is
translated
THE GROWTH OF ALGEBRA
the root it
sum
thirty-nine; the
this to
1
of
number
half the
remainder
which
this,
This 2
which you sought."
for
forth without
explanation
familiar formula for i.e.,
p
x=
the
V/2
%
Now
sixty-four.
take
eight,
and subtract from
root,
which
the
of
three.
is
is
is
153
five; the
is
the root of the square
is
The
solution
merely
sets
the rule expressed in
our
+px + q
o,
solution of x*
4g, except
that only one root
He
however recognizes the existence of two roots where both are real and positive, as in the is
given.
equation
2 ;r
+
=
21
3
io;tr.
In practice
he
commonly
uses but one root.
Algebra made little adfew special
Sixteenth century algebra vance, save in the cubics,
way
of the solution of a
from the time of
Mohammed ben Musa
to the
Its course century, seven hundred years. had run from Egypt to Greece, and from Greece (and
sixteenth
Grecian Alexandria) to Persia.
It
now
transfers itself
and works slowly northward. In a famous work printed in Niirnberg in 1545,
from Persia
the
to Italy
"Ars magna," 4 Cardan
a cubic equation 1
2
I.e.,
The
;
that
is,
gives a complete solution of
he solves an equation of the
the square root. successive
+ 39 = 64;
\/6j =
8
Rosen, p. n.
4
Hieronymi
steps
8;
are
8-5 =
Cardani,
as
follows: | of 10
=
5;
5.5
= 25;
25
3.
praestantissimi
mathematici,
philosophi,
medici, Artis Magnse, sive de regvlis algebraicis, Lib. unus.
ac
154
THE TEACHING OF ELEMENTARY MATHEMATICS
form
jfi+pxq, to which all other cubics can He mentions, however, his indebtedness
reduced.
be to
though not as generously as seems to have been their due. 1 earlier writers,
This of
is
not the place to consider the relative claims
Cardan,
and Fiori
Tartaglia
(Tartalea),
Cardan seems
(Florido).
Ferro to
(Ferreus),
have obtained
under pledge of But however secrecy and then to have published it. this was, by the middle of the sixteenth century the cubic equation was solved, and Ludovico Ferrari at of
solution
Tartaglia' s
the
cubic
about the same time solved the quartic. Algebra had now reached such a point that mathematicians
were able to
solve, in
one way or another, geneThereafter the
ral equations of the first four degrees.
chief
improvements were
standing the
(i) in symbolism, (2) in under-
number system
of algebra, (3) in finding
approximate roots of higher numerical equations, (4) in simplifying the methods of attacking equations, and (5)
For the purposes of
in the study of algebraic forms.
elementary algebra of the 1
first
we need
inuenit, tradidit uero
cu in certamen
cu Nicolao
iam annis ab hinc
triginta
Tartalea
Brixellense
&
ipse, qui
tibus tradidisset, suppressa demonstratione, freti
hoc
aliquando uenisset, cum nobis roganauxilio,
tionem qusesiuimus, eamque in modos, quod difficillimum subiecimus. Fol. 29,
v.
ferme capit-
Anthonio Mariae Florido Veneto, qui
occasionem dedit, ut Nicolaus inuenerit,
sic
speak only
three.
Scipio Ferreus Bononiensis
ulum hoc
at this time to
demonstra-
fuit,
redactara
THE GROWTH OF ALGEBRA Growth
of
Algebra, as
symbolism
is
155
readily seen, Its
is
history has
very dependent upon symbolism. been divided into three periods, of rhetorical, of syncoThe rhetorical algebra pated, and of symbolic algebra. its
is
that in which the equation
in the
example given on
p.
is
written out in words, as
152 from Al-Khowarazmi
the syncopated, that in which the words
most of the example given on p. 149 Diophantus; the symbolic, that in which an
viated, as
from
;
are abbre-
in
arbitrary shorthand
is
used, as in our
common
algebra
of to-day.
The growth
of symbolism has been slow.
radical sign of
Chuquet
V^
other forms, as
and
to the
more
(1484),
IO to our >
R
4 .
10,
is
symbol,
V 10
only slowly becom-
ing appreciated in elementary schools,
is
a tedious and a
So from Cardan's
wandering path. cubus p
the
through various
common
refined io*, which
From
rebus aequalis 20, for
6.
through Vieta's
iC
-8Q
-f
i6N
3 sequ. 40, forjr
-
2 8;tr
+
16^=40,
and Descartes's x*
XD
ax
bb for x* = ax y
b\
and Hudde's x*
1
Beman and
p. 108.
oo qx.r,
for x*=qx-\-r,
l
Smith's translation of Fink's History of Mathematics,
THE TEACHING OF ELEMENTARY MATHEMATICS
156
has likewise been a long and tiresome journey.
Such
1
simple symbols as the x for multiplication, and the still 2 simpler dot used by Descartes, the = for equality,
x~n
the
for
3 ,
these
Even now
nition.
all
the
had a long struggle for recogsymbol
-5-
has only a limited
acceptance in the mathematical world, and there are three widely used forms for the decimal point. 4 Thus symbolism has been a subject of slow growth, and we are
in the period of unrest.
still
We
5 may, however, assign to the Frenchman Vieta the honor of being the founder of symbolic algebra in His first book large measure as we recognize it to-day.
on algebra, " In artem analyticam isagoge," appeared in 6 Laisant thus summarizes his contribution " He i59i. :
who should be looked upon as the founder of algebra as we conceive it to-day. The powerful impulse which he gave consisted in this, that while unknown it is
had already been represented by letters to writing, it was he who applied the same method
quantities facilitate
to
known
quantities as well.
search for values gave tions to
1
2 4
First
way
is
that day,
when
to the search for the opera-
used by Oughtred in 1631. 8
Wallis.
usually written 2.5 in America, 2-5 in England, 2,5
on the Con-
tinent. 5
6
the
be performed, the idea of the mathematical
Recorde, 1556. 2\
From
Francois Viete, 1540-1603. Cantor, II, p. 577; for a general
summary of
his work, see p. 595.
THE GROWTH OF ALGEBRA function enters into the science, and this its
subsequent progress."
natural
obstacle
number
to
is
its
of
understanding
been, perhaps, the
progress.
The
the positive integer.
is
the source of
1
Number systems The difficulty the number systems of algebra has greatest
157
the world met only problems which
primitive,
So long as
may be
repre-
sented by the modern form ax + b = c, where c > b b is a multiple of a, as in 3^+2=11, these and c
But when problems appeared which involve the form of equation where b is not
numbers
sufficed.
axb
a multiple of
We
mixed number.
have seen (Chap. Ill) how the
world had to struggle for
many
centuries before
it
came
It was only by numbers of this kind. an appeal to graphic methods (the representation of numbers by lines) that the fraction came to be under-
to understand
stood.
When,
further, problems requiring the solution
of an equation like in x*
2, still
a
xn =a, a
new kind
of
not being an n th power, as
number was
necessary, the
number, a form which the Greeks interpreted geometrically for square and cube roots. The next step led to equations like x + a = b, with real
a
and
> b,
irrational
as in
x+
5
=
2,
a form which for
many
baffled mathematicians because they could 1
La Mathematique,
p. 55.
centuries
not bring
THE TEACHING OF ELEMENTARY MATHEMATICS
158
themselves to take the step into the domain of negaIt was not until the genius of Destive numbers. (1637) more
cartes
the
completely grasped
the
idea of
between algebra and
one-to-one
correspondence geometry, that the negative number was taken out of the domain of numerce fictce^- and made entirely
One more
real.
the
solution
of
was,
step
of
equations
however, necessary for the form X" -f- a = o. 1
do with an equation like + 4 = o was still an unanswered question. To say that x 4, or
What
2 .z
to
V
2V
i,
or
2V
the meaning of until
avails nothing unless
i,
the symbol
the close of
the
symbol a
+
i."
made
b^J
It
was not
century that any
eighteenth
considerable progress was of the
"V
we know
in the interpretation
In 1797 Caspar Wessel,
i.
a Norwegian, suggested the modern interpretation, and
published a memoir upon complex
numbers
proceedings of the Royal
of Sciences
Letters of
Denmark
for
Academy I797-
2
in
Not, however,
the
and until
Gauss published his great memoir on the subject (1832) was the theory of the graphic representation of 1
Cardan, Ars magna, 1545, Fol.
2
This has recently been republished in French translation, under the Essai sur la representation analytique de la direction, Copenhague,
title
1897,
w ith
3, v.
a historical preface by H. Valentiner.
For a valuable summary
of the history, see the vice-presidential address of Professor
A
of the American Assoc. Adv.
given in the author's History of
A
brief
Beman, Section
summary is also Modern Mathematics, in Merriman and
Woodward's Higher Mathematics,
Sci.,
New
1897.
York, 1896.
THE GROWTH OF ALGEBRA the complex
number generally known
world.
ical
159
mathemat-
to the
Elementary text-book writers to
indisposed presentation
is
the
give
subject
place,
seem
still
although
its
as simple as that of negative numbers. 1
For the purposes of elementary teaching only a single other historical question demands consideration, the approximate solution of numerical equations, and
one of arithmetic than of algebra. Algebra has proved that there is no way of solving the general equation of degree higher than four that
even
this is rather
;
that
is,
by the
common
operations of algebra
we can
solve the equation
ax* but that
bx*
we cannot ax*
We
+
+
bx*
+ ex* + dx + e = o,
solve the equation
+ ex* + dx* + ex+f =
o.
2
can, however, approximate the real roots of
numerical
algebraic
That
practical work.
and
equation, is,
we can
this
find that
suffices
any for
one root of
the equation
x*
-f-
I2x*
+ 59^r 3 +
150 AT
2
+ 2iox
207
=o
0.638605803+,
is
we have no formula for solving such equations by algebraic operations as we have for solving
but
1
For an elementary treatment, see Beman and Smith's Algebra, Boston,
1900. 2
For
historical resume, see the author's History of
already cited, p. 519.
Modern Mathematics
160
THE TEACHING OF ELEMENTARY MATHEMATICS
The simple method now proximation
is
who published mentary works
due it
generally used for this ap-
an Englishman, W. G. Horner, and it now appears in ele" Horner's method." English as
to
in 1819,
in
Foreign writers have, however, been singularly slow in recognizing its value.
CHAPTER
WHAT AND WHY TAUGHT
ALGEBRA, Algebra
its
Chapter VI
In
defined
algebra was that
VII
the
growth of
considered in a general way, assuming
nature was fairly well
Nor
known.
is
it
without good reason that this order was taken, for the definition of the subject
is
cuss the teaching of the subject
examine more carefully It is
when
best understood
But before proceeding
considered historically.
is
it
to dis-
necessary to
into its nature.
manifestly impossible to draw a definite line be-
tween the various related
sciences, as
between botany
and zoology, between physics and astronomy, between algebra and arithmetic, and so on. The child who meets the expression 2 x ( ? ) = 8, in the first grade, has touched the elements of algebra. The student of algebra
who
called
is
upon
to
simplify
is
facing merely a problem of arithmetic.
a
considerable
erly tion,
parts
of
number algebra,
found lodgment
science
became
arithmetic,
M
like
of
the treatment of
as in
topics
arithmetic
generally
the
In
which are
theory 161
known; of
fact,
prop-
propor-
before
its
while
much
irrational
sister
of
(including
1
THE TEACHING OF ELEMENTARY MATHEMATICS
62
complex) numbers, has found place in algebra simply because it was not much needed in practical arithmetic. 1
Recognizing this laxness of distinction between the two sciences, Comte 2 proposed to define algebra "as
having for taking
which
its
this
the
object
in
expression
signifies
its
full
of
tions into equivalent explicit ones. 3
may be
arithmetic
mination
of
therefore,
we
the
Of
implicit
func-
In the same
way
values
will
of
functions.
Henceforth,
that
Algebra is the say and Arithmetic the Calculus of
briefly
4
course this must not be taken as a definition
universally "
Teachers
As
accepted. "
"
methodology 1
meaning,
defined as destined to the deter-
Calculus of Functions , Values."
equations;
logical
transformation
the
of
resolution
who
says
:
It
a prominent writer upon is
very
difficult
to give a
care to examine one of the best elementary works
upon
arithmetic in the strict sense of the term, should read Tannery, Jules,
Le9ons d'Arithmetique theorique 2
The Philosophy
sophic positive, by 8
zero
and
I.e., ;
in
x2
W.
+ px +
this equation
M.'Gillespie,
q
New
o we have an
may be
this transformation
4
et pratique, Paris, 1894.
of Mathematics, translated from the Cours de Philo-
York, 1851,
p. 55.
implicit function of
x equated
to
so transformed as to give the explicit function
belongs to the domain of algebra.
Laisant begins his chapter L'Algebre (La Mathematique, p. 46) by reference to this definition, and makes it the foundation of his discussion
of the science.
WHAT AND WHY TAUGHT
ALGEBRA,
good
definition of algebra.
We
say that
it
163
is
merely
a generalized or universal arithmetic, or rather that the
science
of
sidered generally' (D'Alembert).
magnitudes conBut as Poinsot has
well observed, this
it
'it
is
view
altogether
distinct parts.
arithmetic.
.
.
.
to consider
for
limited,
under a point of algebra has two
The first part may be called universal The other part rests on the theory and arrangement.
combinations
of
is
too
calculating
give the following definition.
.
.
.
.
We may
.
Algebra has for
.
its
object the generalizing of the solutions of problems relating
to
the
computation of
magnitudes, and of
studying the composition and transformations of for-
mulae
which
to
English and French
recent
of
this generalization leads."
The function
Taking Comte's
it
in the scientific
is
from
definition,
question
advanced teachers. tal,
coordinate
1
Dauge,
I
first
of "
I
tried
is
first
the
necessary this
is,
realized
is
steps
fixing of
apart
by
all
found," says Professor Chrysto teach university students
geometry, that
Felix,
definition as a point
teaching of algebra
How
"when
elementary algebras
evident that one of the
the idea of function. all
best
at defining the subject. 2
make no attempt of departure,
The
1
I
had
to
go
Cours de Methodologie Mathematique,
back and
2. ed.,
Gand
et
Paris, 1896, p. 103. 2
Chrystal, G., 2 vols. 2 ed., Edinburgh, 1889.
d'Algebre elementaire, Paris, 1896.
Bourlet, C., Le9ons
THE TEACHING OF ELEMENTARY MATHEMATICS
164
them algebra over
teach
The fundamental
again.
an integral function of a certain degree, having a certain form and so many coefficients, was to them as much an unknown quantity as the pro-
idea
of
verbial
a-."
Happily first
not only pedagogically one of the practically it is a very easy one
this
is
steps, but
because "
1
Two
abundance
the
of
circumstances
general
that all that
we
and
formation,
see
is
the
are
that
2
the
strike
subjected
other
mutually interdependent." tary illustrations
familiar
of
mind
;
these
changes
we
versa;
call
involving time;
a stone vice
the distance a function of the time,
and the time a function of the journey;
railway
are
the best elemen-
and the distance varies as the time, and
falls,
one,
continual trans-
to
Among
those
illustrations.
the
distance
distance.
again
We
varies
take a as
the
time, and again time and distance are functions of each other. Similarly, the interest on a note is a
function of the time, and also of the rate and the principal.
This notion of function to
common way
the
that here
problems of
x=
2,
3, 1
this
etc.,
is
not necessarily foreign
presenting
algebra, except
emphasized and the name is Teachers always give to beginners Evaluate X* + 2 x + i for nature
the idea
made prominent.
of
which
is
:
is
nothing else than finding the
Presidential address, 1885.
2
Laisant, p. 46.
WHAT AND WHY TAUGHT
ALGEBRA,
165
value of a function for various values of the variable.
=
for a
tion of
f(a,
b
i,
=
2, is
a and for
b),
the value of aB
find
Similarly, to
b,
or,
special
+ 3 a?b
1 3 ati
-f-
-+-
bz
merely to evaluate a certain funcas the mathematician would say, values
of
the
variables.
It
is
thus seen that the emphasizing of the nature of the function and the introduction of the
symbol are not at
difficult for
all
name and
the
beginners, and they
The
constitute a natural point of departure.
introduc-
tion to algebra should therefore include the giving of
the quantities which enter into a function,
values to
and thus the evaluation of the function
Having now functions,
1
itself.
defined algebra as the study of certain
which includes as a large portion the solution
of equations, the question arises as to
its
value in the
curriculum.
Why
studied
Why
should one study this theory of
certain simple functions, or seek to solve the quadratic
equation, or concern himself with the highest factor of
two functions
meets
branches of learning,
all
we study theology, What doth it profit ias,
1
?
It is the
cui bono ?
biology, geology to
know
common
same question which
Why should
God,
life,
earth
?
music, to appreciate Pheid-
to stand before the fagade at
Rheims, or
to
wonder
Certain functions, for functions are classified into algebraic and trans-
cendental, and with the latter elementary algebra concerns itself but
little.
b, but with the transcenE.g., algebra solves the algebraic equation x? b it does not directly concern itself. dental equation ax
=
1
66 THE TEACHING OF ELEMENTARY MATHEMATICS
magic of Titian's coloring ? As Malesherbes remarked on Bachet's commentary on Diophantus, " It at the
won't lessen the price of bread;" 1 or as D'Alembert retorts from the mathematical side, d propos of the Iphi-
gnie
of Racine,
"What
does this prove?"
made answer
Professor Hudson has
because
intellectual study spirit,
to
it
pays
same nature as that
of the
purchase with
money
" :
To pursue an
indicates a sordid
of Simon,
who wanted The
the power of an apostle.
real reason for learning, as
it is
for teaching algebra,
a part of Truth, the knowledge of which
that
it is
own
reward.
"
'
'
Such an answer
He
is
is,
is its
rarely satisfactory to the ques-
and too wide, as it may be used to justify the teaching and the learning of any and every branch of truth and so, indeed, it tioner.
or she considers
it
too vague
;
does.
A true education should
seek to give a knowledge
of every branch of truth, slight perhaps, but sound as far as
it
goes,
sympathize is
and
sufficient to
enable the possessor to
some degree with those whose privilege it for themselves at least, and it may be for
in
to acquire,
the world at large, a fuller and deeper knowledge.
person who
knowledge is
is is
wholly ignorant of like
one who
thereby cut off from
ests of 1
"
is
many
A
great subject of
any born without a limb, and
of the pleasures
and
inter-
life.
Le commentaire de Bachet
prix du pain."
sur Diophante ne fera pas diminuer le
WHAT AND WHY TAUGHT
ALGEBRA, "
I
maintain, therefore, that algebra
on account of
any
167
benefit
but because
its utility,
is not to be taught not to be learnt on account of
which may be supposed is
it
to be got from it ; a part of mathematical truth, and no
one ought to be wholly alien from that important depart-
ment
of
The
human knowledge." 1
sentiments expressed by Professor
meet the approval
of
all
true
Hudson
teachers.
will
Algebra
is
taught but slightly for its utilities to the average citizen. Useful it is, and that to a great degree, in all subsequent
mathematical work the mechanic,
it is
;
but for the merchant, the lawyer,
of slight practical value.
Training in logic
But Professor Hudson
states, in
the above extract, only a part of the reason for teaching that we need to know of it as a branch of human knowledge. This might permit, and sometimes
the subject
seems also as
to give rise to, very poor teaching.
an exercise
in logic,
the teacher's work, raising
mechanical
humdrum
true education. later in his
and it
We
need
it
this gives character to
from the
tedious, barren,
of rule-imparting to the plane of
Professor
Hudson expresses
paper when he says,
this idea
" Rules are always
mischievous so long as they are necessary
:
it
is
only
when they are superfluous that they are useful." Thus to be able to extract the fourth root of x*+4** 4^+ I is a matter of very little moment. The 1 Hudson, W. H. H., On the Teaching of Elementary Algebra, paper before the Educational Society (London), Nov. 29, 1886.
1
THE TEACHING OF ELEMENTARY MATHEMATICS
68
pupil cannot use the result, nor will he be liable to use
the process in his subsequent work in algebra. that he should have
power
to
But
grasp the logic involved
in extracting this root is
very important, for it is this very mental power, with its attendant habit of concentration, with its antagonism to wool-gathering, that we should seek to foster. highest
common
matter of
little
To have
a rule for finding the
factor of three functions
likewise a
is
importance, since the rule will soon fade
from the memory, and in case of necessity a text-book can easily be found to supply it but to follow the logic ;
of the process, to keep the tion while
performing here
of the subject,
mind
it,
herein
is
to
intent lies
upon the opera-
much
be sought
its
of the value
chief raison
d'etre.
Hence the teacher who of
algebraic function
fails
The one who
science.
unreal
a set of
fails to
to
emphasize the idea
reach the pith of the
seeks merely the answers to
problems, usually so manufactured
as to give rational results alone, instead of seeking to
give that power which bra's in
itself
for x*
that the
x
o
is
Practical value
the chief reason for alge-
success.
necessary and that
of great value to see
is
is
being, will fail of
x= why
o,
for
it
i
value
condition ;
but
it
is
such condition.
most people algebra
valuable only for the culture which
same time
little
sufficient
x= i, x =
this is
Although
It is of
it
brings, at the
has never failed to appeal to the
common
WHAT AND WHY TAUGHT
ALGEBRA,
men
sense of practical
169
as valuable for other reasons.
All subsequent mathematics, the theory of astronomy, of physics, and of mechanics, the fashioning of guns, the computations of ship building, of bridge building,
and of engineering
in general, these rest
upon the operaNapoleon, who was not a
tions of elementary algebra.
man
to overrate the impractical, thus
gave a statesman's
estimate of the science of which algebra
"The advancement,
stone:
is
a corner-
the perfecting of mathe-
bound up with the prosperity of the State." l Ethical value There are those who make great claims
matics, are
for algebra, as for other mathematical disciplines, as
a means of cultivating the love for truth, thus giving to the subject a high ethical value. Far be it from teachers of the science to gainsay nize those to bear
we do
who
all this,
follow Herbart in bending
upon the moral building-up
or to antagoall
education
of the child.
But
well not to be extreme in our claims for mathe-
Cauchy, one of the greatest of the French mathematicians of the nineteenth century, has left us matics.
some advice along
this line:
"There
are other truths
than the truths of algebra, other realities than those of Let us cultivate with zeal the mathesensible objects. matical sciences, without seeking to extend them beyond their
own
limits;
attack history 1
let
us not imagine that
by formulae,
L'avancement,
prosperite de
and
1'Etat.
le
or
we can
employ the theorems
perfectionnement des mathematiques sont
lies
of a
la
THE TEACHING OF ELEMENTARY MATHEMATICS
I/O
algebra and the integral calculus in the study of ethics." For illustration, one has but to read Herbart's Psychology to see
how absurd
the extremes to which even a great
thinker can carry the applications of mathematics.
Of course algebra has its ethical value, as has every But the subject whose aim is the search for truth. direct application of the study to the life
When we
slight.
find ourselves
of this kind for algebra,
it
is
we
making
live is
very
great claims
well to recall the words
Mme. de Stae'l, paying her respects to those who, in her day, were especially clamorous to mathematicize all " life Nothing is less applicable to life than matheof
:
matical reasoning.
A
decidedly false or true;
mixed
mathematics
is
everywhere else the true
is
proposition in
in with the false."
When
studied
tion of algebra,
Having framed a tentative definiand having considered the reason for
studying the science, we are led to the question as to the place of algebra in the curriculum.
At
the present
time,
in
America,
it
is
generally
taken up in the ninth school year, after arithmetic and before demonstrative geometry. Since most teachers are tied to a particular local school system, as to matters
of
curriculum,
the question
is
not to
But as a problem of practical one. has such interest as to deserve attention.
them a very education
it
Quoting again from Professor Hudson ginnings
of
all
the
great
divisions
of
" :
The
be-
knowledge
WHAT AND WHY TAUGHT
ALGEBRA, should find education to
later
their
of
something of everything, in order But it is everything of something. all
subjects cannot be taught at once,
cannot be learnt at once
there
;
observed, a certain sequence
is
well be that one sequence
more
My
other.
curriculum
perfect
at first
;
learn
needless to say all
a
place in
171
opinion
is
an order to be
is
necessary, and
it
may
beneficial than an-
that, of this ladder of learning,
is
Algebra should form one of the lowest rungs; and
I
find that in the Nineteenth Century for October, 1886,
the Bishop of Carlisle, Dr.
Comte, the
Positivist
Harvey Goodwin, quotes with approval, to
Philosopher,
the same effect. " The reason is this
Algebra is a certain science, from unimpeachable axioms, and its conproceeds clusions are logically developed from them it has its :
it
;
own
but they
difficulties,
special
are
not those
of
in the balance conflicting probable evidence
weighing which requires mind.
It is
the
for
possible
student to
the
step firmly before proceeding to the left
hazy or in doubt;
and enables
it
thus
give
power,
vigor,
commonly given give is
it
each
plant
next, nothing
is
studies of
it
later.
to
the
a different
Mathematics
mind
this
is
reason for studying them.
I
strength,
as the
a maturer
strengthens the mind
it
better to master
nature that are presented to
of
powers
stronger
;
as the reason for studying Algebra early, that
to say,
for
beginning to study
it
early;
it
is
not
1/2
THE TEACHING OF ELEMENTARY MATHEMATICS
of
not even possible, to finish the study before commencing another. On the
is
it
necessary,
Algebra
other hand,
is
it
not necessary to be always teaching
Algebra; what we have to do, as elementary teachers, is
to guide our
pupils to learn
enough
door open for further progress
to
leave the
we take them over
;
the threshold, but not into the innermost sanctuary.
"The age
at
begin differs in
which the study of Algebra should each individual case. ... It must be than nine years of age
rare that a child younger fit
to
others,
and
the
although be taken up at
begin;
may
superior limit
my own
;
any
opinion
is,
age, there
that
is
most
subject, like
is
no
would be
it
seldom advisable to defer the commencement to
later
than twelve years." This opinion has been quoted not for indorsement, but rather as that of a teacher and a mathematician of such
idea
is
prominence as at
quite
command
to
of beginning at about the age of fourteen
or even later,
the
wisdom
question of
sequence.
and
it
of our
age involved,
Are we wise it
or fifteen,
a serious question as to
raises
Indeed, not only
course.
eight years, dropping
ping that
The
respect.
variance with the American custom
but in
also
that
teaching
of
is
the
general
arithmetic
for
and taking up algebra, drop-
and taking up geometry, with possibly a
brief review of
three
all
high school course
?
later,
at
the
close
of
the
ALGEBRA,
WHAT AND WHY TAUGHT
Fully recognizing
ment
of
what
hesitate
present feels
is
to
the
statement,
express his personal is
plan
that with
a dogmatic
the best course, and hence
any such
avoid
to
the folly of
173
author
state-
desiring
does not
conviction that the
not a wisely considered
elementary arithmetic
He
one.
should go,
as
1 already set forth in Chapter V, the simple equation,
and also metrical geometry with the models in hand; that algebra and arithmetic should run side by side during the eighth and ninth years, and that demonstrative geometry should run side by side with the latter
part
of
algebra.
One
of
the best of
recent
series of text-books, Holzmliller's, 2 follows this general
and the arrangement has abundant justification most of the Continental programmes. It is so scientifically sound that it must soon find larger acceptance plan,
in
English and American schools.
in
Arrangement
ject just discussed, a
arrangement
we
shall
of
As
of text-books
word
is
related to the sub-
in place concerning the
our text-books.
It
is
probable that
long continue our present general
plan of
having a book on arithmetic, another on algebra, and still another on geometry, thus creating a mechanical barrier between these sciences.
We
shall also, doubt-
There is a good article upon this by Oberlehrer Dr. M. Schuster, Die Gleichung in der Schule, in Hoffmann's Zeitschrift, XXIX. Jahrg. (1898), 1
p. 81. 2
Leipzig, B. G. Teubner.
174
THE TEACHING OF ELEMENTARY MATHEMATICS
each book the theory and the exercises for practice, because this is the English and American custom, giving in our algebras a few pages
combine
less,
of theory
The
in
followed by a large
Continental
however,
plan,
toward the separation of
book
the
on
the
number
of
exercises.
inclines
decidedly
the book of
theory,
changes of the former.
It
thus is
exercises
allowing
frequent
doubtful, however,
the plan will find any favor in America,
its
tures.
There
is,
if
advan-
tages being outweighed by certain undesirable 1
from
fea-
perhaps, more chance for the adoption
of the plan of incorporating the necessary arithmetic,
and geometry
two or three grades into a single book, a plan followed by Holzmuller with algebra,
much 1
An
success. interesting set of statistics with respect to
given by p. 410,
for
J.
W.
A.
under the
Young title,
in
Hoffmann's
Zeitschrift,
German
XXIX.
text-books
is
Jahrg. (1898),
Zur mathematischen Lehrbiicherfrage.
CHAPTER
VIII
TYPICAL PARTS OF ALGEBRA
While
Outline
it
not worth while in a work of
is
kind to enter into commonplace explanations of matters which every text-book makes more or less this
may be
lucid, it
topics that are
of value to call attention to certain
somewhat neglected by the ordinary
run of classroom manuals. ent
text-book
his
upon
He
waste of time. for
much
and of students'
is
effort
is
his
likewise
depend-
exercises, is
a
dependent upon the
requires
economy of time him to follow the
some unusual reason
is
But he
it.
teacher
most of
of the theory, since
text unless there
ing from
The
any considerable number
since the dictation of
book
for
is
for depart-
not dependent upon the book
for the sequence of topics, nor for all of the theory,
nor for
all
he precluded the interest possible, and introduc-
of his problems;
from creating
all
ing a flood of
light,
the subject. that
For
may add
neither
is
through his superior knowledge of
this reason this
chapter
to the teacher's interest
is
written,
by throwing and may few portions, light upon typical methods of some treating suggest thereby improved it
some
the entire subject.
a
THE TEACHING OF ELEMENTARY MATHEMATICS
176
The
Definitions
number
able
of
policy
any consider-
learning
of definitions at the beginning of a
new
subject of study has already been discussed in Chapter
The
II.
idea
is
always of vastly more impor-
At the same much danger from the inexact definibe found in many text-books, a danger all
tance than the memorized statement.
time there tions
to
is
the greater because of the pretensions of the science
be exact, and because there
always be found teachers who believe it their duty to burn the definitions indelibly into the mind. to
Whether the
definitions
are
will
learned
not, the teacher at least will need to
they are correct. assistance will
need
For
this
verbatim
or
know whether
purpose he will find
little
from other elementary school-books. He to resort to such works as Chrystal, 1 as
Oliver, Wait,
and Jones, 2 or as Fisher and Schwatt 3
in English, as Bourlet 4 in
French, as the convenient
handbooks of the Sammlung Goschen 6 or the new Sammlung Schubert 6 in German, and Pincherle's little
little
Italian
handbooks. 7
1
Algebra, 2
vols.,
2
A Treatise
on Algebra, Ithaca, N.
8
Text-book of Algebra, part
4 6 6
bra,
2
ed.,
Edinburgh, 1889.
i,
Lemons d'Algebre elementaire,
As
Y., 1887.
Philadelphia, 1898. Paris, 1896.
und Algebra, and Sporer's Niedere Analysis. As Schubert's Elementare Arithmetik und Algebra, and Fund's AlgeSchubert's Arithmetik
Determinanten und elementare Zahlentheorie, both published
in 1899.
Algebra elementare, and Algebra complementare. A good bibliography of this subject, for teachers, is given by T. J. McCormack in his 7
TYPICAL PARTS OF ALGEBRA
A
few
illustrations of the general
common run way
of
of definitions
may
\J J
weakness of the
be of service in the
leading teachers to a more critical examina-
tion of such statements.
The
usual definition of degree of a monomial
A
to think of 3
x;
a and
in
3
so
is
and continues
loosely stated that the beginner thinks
which
as of the fifth degree,
it
is
but for the purposes of algebra, eswith equations, it is quite as often
pecially in dealing
considered as of the third degree in x, a distinction
bling,
A
comes upon
square root
much
the student, after
until
usually ignored
usually defined as one of the two
is
equal factors of an expression, although is
same
taught, almost at the
sion of
which he
no two equal root of
2
.r
Even
+
so
time, that the
E.g.,
2 ;tr
introducing
new
of
practice
the
particular
notes to the
is
prime.
to is
values
edition of
Some books
avoiding
expression
is
it
the
follow
difficulty
by
of
condition," "equation In the algebra of
again!
an equality which
exists
only
of
called
the
certain
De Morgan's
matics, Chicago, 1898, p. 187.
N
I
a fashion entirely inexpressive of
and never referring to-day an equation for
+
simple a concept as that of equation
usually defined in
ancient
expres-
he speaks of the square
and yet says that
the present algebraic meaning.
an
the student
extracting the square root has
is
factors.
I,
stum-
it.
work,
letters
On
the Study of Mathe-
THE TEACHING OF ELEMENTARY MATHEMATICS
178
unknown
As
quantities.
the term
braists of the present time, 2
tion strictly speaking, although
neither
An
+ b = b + a2
a2
is
equation, as
the
is
+3= it
word
is
5
expresses equality; it
although
,
used by algen t an equa-
is
now
is
an
identity.
used, always
con-
an unknown quantity. The term " axiom " is subject to similar abuse. No mathematician now defines it as " a self-evident truth," and no psychology would now sanction such an unscientific statement. Algebraists, those who make the 1
tains
science to-day, agree that an axiom
statement so
commonly
is
merely a general
accepted as to be taken for
granted, and a statement which needs to be considered
with care in the light of the modern advancement of the
For example, no student who thinks would
science.
say that
it
"self-evident
is
are equal."
If
4 = 4,
it
"
is
that "like roots of equals
not "self-evident" that a
square root of 4 equals a square root of not equal
4, for -f-2
does
2.
Again, of
what value
is it
nary definition of addition
to a pupil to learn the ordi-
Text-books commonly say,
?
in substance, that the process of uniting two or more
expressions in a single expression
but what nition
is
meant by
this
"uniting"
would better be omitted, or
some approach
to
scientific
it
is ?
called addition
Either the
would better have
accuracy; the choice of
De Morgan's use of the word is not that of modern writers. Study of Mathematics, 2 ed., Chicago, 1898, p. 57, 91. 1
;
defi-
See The
TYPICAL PARTS OF ALGEBRA these alternatives
may depend upon
the class, or pos-
sibly upon the teacher.
The simple concept
of factor, so vital to the pupil's
progress in algebra, usually suffers with the factor, as
we
so often read, one of several
expressions which
expression will divide
tors
of x
another i ?
If so,
?
it
Possibly
to rational terms in x.
factor x?
we
shall
i,
make
together
multiplied
In other words,
?
will
be said that we are limited
If so,
when we ask
\
?
Are the
then, shall
also excluded
factors Is x*
?
trivial
=
This does not
we say about
x\,
factoring
or are fractions
a factorable, we not knowing in
a=4
advance but that
These are not
i
But possibly we are
and
x+\
3 ^r
and imaginary numbers
expected to exclude irrational
x*
a pupil to
expect him to say that
involve any irrational term in x.
What,
a given
an expression which are ^/x-\- I and VJ i facis it
(^-I)(^+J + JV^3)(^+J-JV^3)?
altogether.
Is a
rest.
numbers or
9 or some other square?
or
"catch" questions.
Upon
the
answers depends the entire notion of factoring, the basis upon which we are to build the greatest part of algebra
Of
less importance,
of highest
factor of
2(b
the theory of equations.
common 2(>
3
-
3
)
a), or simply
the lowest
but
factor.
common
still
and 4( (a
of value,
What is the 2
)?
-a
2
)?
And
multiple of a
questions should not be puzzling
b ;
is
the definition
highest
Is
it
2
(>-),
similarly,
and
common
bal
what
or is
These
the information
is
THE TEACHING OF ELEMENTARY MATHEMATICS
ISO
often needed in the simple reduction of ordinary frac-
and yet our common definitions do not throw much light upon them. The unnecessary and ill-defined term "surd" still clings
tions;
to our algebras.
what
If so,
is
a
synonym for an irrational number ? Is
it
V
rational, say V2, V#, the circulate 0.666- ? Is
VJ, If
or
it
is
is
+ V2
2
These are
irrational
common
all
than algebraic,
is
it
Is
number not
a
it
number ?
7r=3.i4i59---, or
it
a single expressed root like
it
V2 + V3?
a surd? or
merely an
Is
i ?
irrational
number,
is
or
\2 + V3
log 2 a surd
?
?
expressions, arithmetical rather
true,
but conventionally holding a
place in algebra.
In this connection the wonder
how
as to
we
long
shall
may be
expressed
continue to use the terms
"pure" and "affected" (in England adfected) quadratics, instead of the more scientific adjectives "incomplete" and "complete."
The enough
has
been suggested
for care in the
common
The awakening point in
much
inquiry might be extended
all
to
is
show the necessity
definitions of algebra. 1
of interest in
teaching,
the subject, the vital
best accomplished through the
early introduction of the equation. 1
For those who have not access
to the
As soon
works mentioned on
as the
p. 176,
it
Beman and
Smith's Algebra, Boston, 1900,
which the authors have endeavored to
state the necessary definitions
may be in
farther, but
of service to refer to
with some approach to
scientific accuracy.
TYPICAL PARTS OF ALGEBRA pupil
181
can evaluate a few functions, thus becoming
familiar with
the alphabet of algebra, the equation should be introduced with this object prominently in the teacher's mind.
The mere the
solution
first
pupil
of
the simple equation which
presents no
meets
The
difficulty.
teacher will do well to avoid such mechanical phrases as
"clear of
reasoning
is
fractions"
mastered
;
and "transpose" indeed,
may be
it
until
the
questioned
whether these phrases are ever of any value.
Rather
should the processes stand out strongly, thus:
Given -
+ 3=7,
to find the value of x.
Subtracting 3 from each member, 2
member by
Multiplying each
To prove
this
(check the
then 2
4-3
x
2,
result),
= 4+3 =
8.
put 8 for x\
7.
But the greatest difficulty which this time comes from the statement in
algebraic language.
eral is
method
Fortunately there
taken
What
have at
is
no gen-
The
field of traditional rules into
following outline, however,
of value in arranging the statement I.
pupils
of the conditions
of "stating all equations, so that the pupil
forced out of the
of thought.
= 4.
shall
x
represent?
to represent the
number
is
that
usually
:
In general, in question.
x may be E.g., in
1
THE TEACHING OF ELEMENTARY MATHEMATICS
82
"The
the problem,
and the sum
Here x taken to 2.
is
difference of
what
50,
is
two numbers
the smaller
40
number ?
"
some other such symbol) may best be represent "the smaller number."
(or
For what number described
Thus
two expressions be found? lem, the larger
number
is
in the problem in the
evidently 50
two expressions may be found for the x x. 40, and 50 3.
is
How
may
above probx, and hence
difference, viz.,
do you state the equality of these expres-
sions in algebraic language?
With these
directions, thus
quence for the little
x=
x
50
pupil,
4O.
1
outlining a logical
the statements
se-
usually offer
difficulty.
Signs
of
often trouble a pupil
aggregation
than the value of
the
subject warrants.
The
more fact
mathematics we never find any such complicated concatenations as often meet the student almost
is,
in
on the threshold of algebra. Nevertheless the subject consumes so little time and is of so little difficulty as
hardly to justify any serious protest.
Two
points may, however, be mentioned as typical. First, it is a waste of time, and often a serious
waste, to require classes to read aloud expressions like
a + (b-c*)*-\b-{a 1
Beman and
Smith, Algebra.
TYPICAL PARTS OF ALGEBRA
There
is
no value
Mathematicians,
in if
183
such an exercise in oral reading. by strange chance they should
meet such an array of symbols, would never think of reading it aloud. Such a notion, frittering away time and energy and interest, is allied to that which a called " negative a " instead of " "a divided by b being #," which frets about
labors to " minus called
have
"a over b"
(a
mathematical expression well
recognized by the best writers and teachers in several n languages), and which objects to calling a~
the
minus nth
power have meaning) of
power" since
long
that
(forgetful
broadened
"a
to
minus and
their
primitive
petty nothings born of the narrow views
some schoolmaster.
The second place in
many
point refers to a rule which text-books.
It
asserts
still
finds
that in remov-
ing parentheses one should innermost, proceeding ple, these solutions
always begin with the outward. Consider, for exam-
:
Beginning without
Beginning within
a-\a + b-(c-d-e)+c\
a-\a + b--(c-J^~e)+c\
=a
= It
-b-d+e is
evident that there
=
a
b+cdec -b-d+e
are fewer changes of sign
in the second (4) than in the first (8),
and
also that
1
84
THE TEACHING OF ELEMENTARY MATHEMATICS
the second and fourth lines in the second could have
The
been omitted even by a beginner. for the
first
plan
is
that
it
only excuse
more exercise; but would do well to per-
affords
on the same reasoning a child form all multiplications by addition.
The negative number crux
serious
for
through algebra. time for it
its
the
is
supposed to be the
first
bear in his journey has been written as to the
pupil to
Much
Some
introduction.
should find place with the
teachers assert that
first
algebraic concepts.
Others go to the opposite extreme and teach the four fundamental processes with positive integers, and then go over them again with the negative number. Each teacher, like each text-book, has some peculiar
As hobby, and rides it more or less successfully. has been stated, some make much of the idea that a should be read " negative a erally recognized
the
confusion
"minus
thought " minus
senses in which
#," to
"
is
"
instead of the gen-
hoping thereby to avoid be incident to the two used
;
others (and most
of the world's
best writers) recognize that this two" " fold meaning of minus has become so generally
accepted as to render
any attempt at change. shows how unimportant is the question of the time and method of presenting the subject, and of the language in question.
The very
The
writer
difficulty
futile
diversity of view
in
has not been conscious of any great presenting the matter to classes, and
TYPICAL PARTS OF ALGEBRA after trying the various sequences has
followed this plan
first
:
for
185
some time
teach a working knowledge
of the alphabet of algebra, through the evaluation of
simple functions; then awaken the pupil's interest by the introduction of some easy equations, including such as
+2=
^Jx
sity for
8,
V* +1=3,
then show the neces-
etc.;
a kind of number not commonly met in arith-
metic, developing the negative
number and the
The explanation cannot be very scientific at The teacher will depend largely upon graphic
zero. first.
illus-
and upon matters familiar to the pupil. The symbol for 2 below zero, for 50 years before Christ,
tration
the symbols for opposite latitudes or longitudes, these lead to the general symbol for a
a
side of
numbers.
comes
zero
The
into
point
play
the capital of a
common
in
(positive)
and pupils then
the way of illustrations; when empty, when full of
man who, having
the
gas;
$5000, loses $3000,
and then the combined weight of a block and a balloon which pulls upward with $6000
$5000, Ib.
;
Ib.
and the advantage of the expression and minus 20 Ib."
With
this introduction the graphic representation of
a force of 20 "
number on the other
the
ingenuity of teacher
weight of a balloon
10
from
10
Ib.,
and negative numbers on a line is a matter of no difficulty. After this the more scientific pro-
positive
cedure, showing the necessity of the negative if
we
are to solve an equation like
x+3=
i,
number and the
1
THE TEACHING OF ELEMENTARY MATHEMATICS
86
numbers and
definition of negative
complete with
elementary theory.
must not be supposed that the negative number
It is
difficulty the
little
This
this out,
and
theory now
recognize
it.
which have given
rise
all
advanced works on the
"As so
to
to
many
sions, as irrational as useless," says
distinguish between their concrete
single
abstract
their
interpretation,
always confounded up
tities
method.
from
ago pointed
can be established algebraical
impossible
to
in
misplaced discus-
Comte,
"we must
signification
present
and
Under
day.
theory of
negative quana complete manner by a
consideration."
enter into
negative numbers,
which have been almost
the
to
point of view, the
first
graphic
more psychological, but not the more an algebraic standpoint. Comte long
the
is
scientific
by the
approached
necessarily
the
of absolute values,
x
It
is,
however,
any extensive discussion of
the theory at this time. 1
Comte, The Philosophy of Mathematics, translated by
Gillespie,
N.
Y.,
1851, p. 81. 2
Most teachers have access to
Chrystal's Algebra, or Fine's
Number
System of Algebra, and these works give satisfactory discussions of the subject. it
is
For a resume of the matter from the educational standpoint
well to read the Considerations generates sur la theorie des quan-
objections que 1'on y a opposees, in Dauge's Cours de Methodologie mathematique, 2. ed. p. 125. But the best works for
tites negatives, et
the advanced student are the comparatively recent Stolz, Baltzer,
Biermann,
et a!.,
or
Schubert's
German
treatises
Grundlagen der
metik in the Encyklopadie der mathematischen Wissenschaften, Leipzig, 1898.
by
Arithi.
Heft,
TYPICAL PARTS OF ALGEBRA
Of
course the teacher will not
187
leave the
subject
without having the pupil understand that the signs -f-
and
-
have each two
-
distinct
symbols of operation, as in 10 as
quality,
and
As Cauchy
8.
modify the
uses,
one that of
the other that of " The signs puts it, 8,
+
which
quantity before
placed as the adjective modifies the noun."
they
Similarly,
the words plus and minus have (as noted on
two
distinct
uses, as
It is true that
plus #."
"
expressions give place to
"a
in it
p.
184)
quantity" and "a
plus
has been suggested that the
"
and " plus quantities " should " and " positive quantities," positive a
plus a "
these terms being more precise.
upon the
theorize
are
desirability of
But much as we may such usage, the fact
remains that colloquially the shorter expressions are generally used by the world's great mathematicians,
and
probably continue to be so used. older text-books often contain a great deal of
will
The
" worthless matter, and worse, about proving that minus
a minus
is
Of course novo.
a
.
plus," it
is
and "minus
into
minus
plus," etc.
impossible to prove any such thing de
Mathematicians recognize
b-\-ab because we
perfectly well that
define multiplication involv-
ing negatives so that this shall be true.
change the
is
definition
we might change
If
we
should
the result of the
be expected of the teacher multiplication. All that is that it should be shown why the mathematical world is
defines
a
-
b to
mean
to
the same as
+a +
b,
why any
THE TEACHING OF ELEMENTARY MATHEMATICS
I&8
These things
other definition would be inconsistent.
explained, but the text-book "proofs"
are easily
have now been discarded.
the last generation
"
favorite one of these
by a gives
b
plying
the multiplier
product must
The
this: Since multi-
ab, therefore if the sign of
changed, of course the sign of the be changed. As a proof, it is like
is
also
A, a white man, wears black shoes, follows that B, a black man, must wear
saying that therefore
proofs" was
of
if
it
shoes of an opposite color.
When
Checks
since ran
long
a
large
the
upon
transatlantic
rocks near
few
a
miles
in
ized,
hundreds of
lost,
And
checked!
yet one of the
computer learns a
operation,
is
first
check
necessity
which
should
It is
"
or
"
"
prove
"
"
or
verify
most often
moment
that
we
checks
the
teacher
somewhat upon the
mon the
impressed
pupils.
not exhaustive.
;
;
but
first
mathematicians it is
a matter of
see that each step
ones will be suggested, list is
be
day of his a matter of no moment whether we say
first
What
jeopard-
the necessity for checking each
probably use the greatest
and
things that every
upon the student of algebra from the "
lives
because a simple calculation had not been
just
course.
error of
Thousands
calculations.
his
Southampton,
made an
the captain announced that he had
thousands of dollars
steamer not
is right.
shall
A it
require depends few of the more com-
being understood that
TYPICAL PARTS OF ALGEBRA
189
In solving an equation the one and only complete check is that of substituting the result in the original equation (in the statement of the problem
how
makes no matter what axioms we use
It
one).
carefully
we proceed; a
"checks," and wrong "
Chrystal says
:
The
if
how
when
elaborate
hand, no
do
test,
it
matter
stand
this
substituted
Simplifications of
it
Professor is
solution
how simply is
it
a
;
No by
process if
it
do not
and, on the other
obtained, solution."
provided 1
it
Professor
Henrici expresses the same thought in another "
or
if
therein.
been obtained,
no
is
test,
right
As
does not.
or ingenious the
which the solution has stand this
is
assigns to the variables shall
it
the equations
matter
it
result
ultimate test of every solution
that the values which satisfy
there be
if
way
:
equations follow in senseless mo-
poor fellow really thinks that solving a simple equation does not mean the finding of a certain number which satisfies the equation, but the notony, until the
going mechanically through a certain regular process which at the end yields some number. The connection of that to his
To
number with the
mind somewhat illustrate,
pose we
original equation remains
doubtful."
consider the equation
multiply these equals by
be equal, and 1
2
^4=3^6,
Algebra, Vol.
I, p.
x
2,
whence
#+2 = 3.
^3^+ 2 =
286.
2 Presidential address,
Sup-
the results must
Section A, British Assn., 1883.
0.
THE TEACHING OF ELEMENTARY MATHEMATICS
190
Solving, ;tr=2 or
axioms
strictly,
But although we have followed
i.
x
2 will not satisfy the original equa-
So with any equation, the pupil who checks his work is master of the situation answer books are only tion.
;
way, save in the case of unusually complicated
in the
and the pupil knows as well as the teacher (perhaps better) whether his result is right or wrong. "A habit of constant verification cannot be too soon encour-
results,
aged, and the earlier
it is
almost automatically
it is
A very
practised."
t
2
= 2, b =
is
x +
5
3
;
is
an
error.
i
2
i ?
i,
Since
it
+2
2
Or suppose i)
3
+ 32
,
a pupil
= ^ + 5^ + x*
does not, there i
exponents, since any power of
1
22
? Substitute any arbiand the question reduces to this,
arbitrary value
are not usually
1895.
=
the result correct
unless zero enters somewhere;
a case like
-f-
substitute arbitrarily
z
=
The
always equal cP
the two forms agree.
;
2
trary value for x, say
Does
if
O + 3* - 5) (x + 2.x -
asserts that 3
b,
we may
and see
then (2 + s)2 true because each is 25.
E.g., let a
1
+ P? must
In other words,
.
any values for a and which
Whatever values
that of arbitrary values.
are assigned to a and b (a 2 ab 4- b
l
useful check, applicable to the operations of
is
algebra,
acquired the more swiftly and
(^
made
there.
i)/(x
is it i
is
evidently
usually a good one
does not check the is
i,
but mistakes
Of course
= x* +x + i)
i,
in it
checking will
not
Heppel, G., Algebra in Schools, the Mathematical Gazette, February,
TYPICAL PARTS OF ALGEBRA
do
to use the value
i
for x\
and
values should be avoided which
in
1
91
general those
make any
expression
zero.
Another check extensively used by mathematicians
is
The name is long, but the check that of homogeneity. " At present, although simple. homogeneous is '
'
is
somewhere
usually defined
in the first three
a school algebra, the school-boy never
pages of
knows anything
1 meaning, as he has not been used to apply it." The check simply recognizes the fact that if two inte-
about
its
gral functions are homogeneous, their sum, difference,
product, and powers, are homogeneous.
uct of
a
fl
3
-h
ab2 and a2
-f-
ab
may
E.g., the prod-
be a 5
4-
M
2
+ cfib +
2
because the product of a homogeneous function of the third degree and one of the second must be one of fi,
the
a
2
lfi
fifth
;
but
the result
if
is
5 given as a
+
M
2
+
there must be an error, because the result
cPb -f is
not
functions play such
homogeneous. Since homogeneous an important part in mathematics, this check value than at
first
is
of
more
appears.
another check, less extensively used, but so easily applied as to be valuable, is that of symmetry. If two functions are symmetric with respect to cerStill
tain letters, their product, for example,
metric with respect to those letters.
and x2
2
+ xy+y
y, since these 1
are symmetric with
may change
must be sym-
x2 xy+}P respect to x and E.g.,
places without changing
Heppel, G., in the Mathematical Gazette, February, 1895.
THE TEACHING OF ELEMENTARY MATHEMATICS the
forms
the
of
Hence
functions.
+ x^y*
jfi
may be their product, but not x^x^y+x^ although it checks as to homogeneity and for the x=
arbitrary values,
The
i,
y
i.
two of the checks mentioned should be
first
in constant use
by the student; the others are
valu-
able, but not indispensable.
Factoring has already been mentioned as a subject of
supreme
importance
in
waste
Pupils
algebra.
much
time in performing unnecessary multiplications and in not resorting more often to simple factored
For example, the
forms.
student
who
the
begins
solution of the equation
X-
I
by clearing of fractions, gets into trouble both theoretically and practically; he introduces a root which does not belong to the equation, and he causes himself
some unnecessary work.
glance that
and can
x
i
easily
is
He
should
a factor of 2x*
do so
if
+
see
at
4^
3 x*
a i,
he understands the elements
of the subject.
While books
it
must be admitted that the recent
have improved upon
matter of factoring, there ment.
The
subject
often with almost no
is
is
the
room
often
older
ones
in
text-
the
for further improve-
divided
into
difference, as with x*
"cases,"
+ ax +
b,
TYPICAL PARTS OF ALGEBRA
ax +
x*
+ ax
x*
b,
treatment that
of
arrangement of
b,
is
etc.,
193
thus leading to a style It
depressing.
true that the
is
x2
a page of exercises like
ax
-f
+
b,
ax + b, etc., has followed by another of the type x* educational value, but it is also true that the arrange-
ment
is
not a good one.
It
reminds one of the
six-
teenth century plan of having one rule for the quadratic
for
x*+px+q = o, another x*+px = q, and so on.
all this
is
px+q = o,
for x^
The
favorite
+ ax + b,
positive or negative
;
answer
to
and take the
that pupils cannot generalize
single type x^
another
where a or b may be
either
but the experience of the best
teachers shows that pupils can generalize
much
earlier
than some of our text-books would seem to indicate.
Some
special forms
must always precede the general;
but to give only special forms, never referring to the general type, is a serious error.
The
fact
is,
there are only a few distinct types of
much The most important are
factored expressions that are of
value in subse-
quent work.
(i)
ab
+ ac,
the
type involving a monomial factor; (2) ax^+bx+c, the
x; form x
general trinomial quadratic in
(3) cases involving
binomial factors of the
a.
the beginner
these
must be
still
Of course
further
for
differenti-
ated; but problems not involving these three cases,
such as the factoring of
and ** +y*
+
-
3
THE TEACHING OF ELEMENTARY MATHEMATICS
194
have value rather as mental gymnastics than as cases to be used in subsequent work.
The type ax2 + bx + c
includes certain special cases
which must be considered one, such as
x
2
+ 2 ax
a
-f-
briefly before the
2 ,
x2
a2
general
x2 + (a + b)x + ab,
,
which a and b may be either positive or negative. These special cases are satisfactorily discussed in in
The
most text-books. not,
2 general type, ax
+ bx +
is
c,
There are numerous
however, so well treated.
methods of attacking it, but only two are valuable enough for mention here. The first will be understood from the following:
172:4- i2
That
is,
the
17
=
separated into two
is
whose
parts
product is 6-12, and the rest of the work is simple. In general, in ax2 + bx 4- c, the b is separated into
two parts whose product is
is
The reason
ac.
for this
easily seen by considering that
(mx that
4-
is,
parts,
n)(m'x
+
2 n')= mm'x
+ (mn + m'n)x 4- nn*
that the coefficient of
f
x
is
mn' and m'n, whose product
The other plan consists of x2 a square, thus
in
made up is
mm' nn
making the
of
12
= ^(36^4-
two
r .
coefficient
:
6x2 + i?x+
;
17-6^4-72).
TYPICAL PARTS OF ALGEBRA
Now
let
6 x, and we have
z
17 *
Which
+ 72) = K* + 9)0 + 8 )
of these plans
of
rationale
195
followed
is
immaterial, the
is
But
each being easily explained.
it
is
needless to say that the cut-and- try method often given,
and
of taking all possible factors of 6 x*
ing at the proper combination, has
The
little
of to
1
2
and guess-
recommend
it.
cases involving binomial factors of the form
x
a are perhaps the most important of any which the pupil meets in his elementary work, since they r
enter
so
extensively into
They
are
best
treated
the
theory of
equations.
remainder
by the
theorem,
which has long found place in the closing pages of many advanced algebras, where it could not be used to
any
extent.
The theorem
asserts that the remain-
der arising from dividing an integral function of
by x
x
+
in
a can be found in advance by putting a for the
given function.
i6x+ n by x
$x*
be no remainder, for
x
2 24
x
is
I
1
dividing x^
in
E.g., i
we know
+
16
5
+
that 11
= o;
but
the divisor, there will be a remainder
2 3 +5 -2 2
Similarly,
put for x,
x11 x11
l617 jj/
17
j/
2+11 = 16
is
divisible
=y7_yr _
8
x^
there will if
7, for
+ 20 32+11 = 7.
by
xy,
o; but
it
for is
if
not
y
is
divisi-
1
96 THE TEACHING OF ELEMENTARY MATHEMATICS
by x +y,
ble
by x
i.e.,
(7),
for
y7 = _y7 _yr = _ 2 /?,
(-y)V
y
if
is
put for x,
the remainder.
The
easily proved, and its usefulness in eleThe mentary algebra can hardly be overestimated. proof, condensed more than advisable for beginners,
theorem
is
is
as follows
:
x
Let f(x) be the dividend,
a the
divisor,
q the
quotient, r the remainder.
Then f(x) = (x
a)q
-f-
which r cannot con-
r
in
is
true for
y
tain x.
This being an identity and hence for x = a.
But
if
I.e.,
the remainder
a
is
put for x, is
values of x,
all
we have f(a)=
r.
the same as f(x) with a put
for x.
A
have no
teacher will
in
difficulty
putting this
form easily comprehended by beginners. theory is not difficult, and the practice is into a
The very
simple. It
unfortunate
is
time upon
books garret
factor,
as
use of
in
next
to
so
factoring,
many
text-
it
to
the
mathematical
is
usually
upon highest
which the pupil
factoring
sidering the lowest
having spent considerable
of
thereupon relegate The next chapter
common little
that,
the subject
as
common
is
possible
led
After
!
multiple, the
and
here
to
the
make con-
text-books
fractions, pupil is proceed led to use the highest common factor in his reduc-
TYPICAL PARTS OF ALGEBRA
which we
tions,
wise
the
do
in
subject
of
rarely
important
197 but
practice,
other-
sinks
factoring
into
disuse.
What
the remedy
is
for
this
evil
The answer
?
appears when we consider the common uses to which He has two the mathematician puts the subject. uses for
the
it,
being in the solution of equa-
first
and the second
tions,
shortening his work, as in
in
more
the reduction of fractions to forms
Hence
easily han-
proper to follow factoring at once with some simple work in equations, and as soon as fractions are met to use factoring in all simple redled.
ductions,
is
it
the
reserving
highest
common
factor
for
solution
of
cases of real difficulty.
The
equations it
of
application
means
is
an equation
xn + axn ~
l
+
member
makes x
make
x*
That
zero.
=o
3^
4)(x+ i)=
(x if
a
is
x which
o,
or,
is,
what
= 4 we have 0-5=0, and 5-0 = 0. Similarly, the values a*x
The
a. is
= o,
or
x(x
if
the
same
x
i, i
which make
+ a) (x
the
x which
values which
are evidently 4 and
;r
x*
make
shall
the value of
evidently
4 = 0,
like
+n = o,
"
namely, to find a value of first
knows what
the pupil
if
very simple,
to solve
the
to
factoring
a)
= o,
thing,
because
we have
THE TEACHING OF ELEMENTARY MATHEMATICS
198
are evidently
number
able
o,
a,
+
In this
a.
of equations
a consider-
way
with commensurable roots
be given, together with problems involving equations of degree above the first, thus at the same should
time adding to the interest in the subject, giving in factoring, and laying a rational foundation quadratics.
A
drill
for
pupil so trained would not, on reach-
ing the chapter on quadratics, waste his time "completing the square" in the solution of such equations as x*
+ x = o, 2.
or
x2
+ 5 x + 6 = o.
It takes
but
time to introduce this work, whatever text-book use,
and Ihe benefit derived
is
little
is
in
evident.
In the treatment of fractions, to apply the Eucli-
dean method of highest common factor 1 tion of forms like
^2 +7^+ ^ + 9^+ is
to
IQ 14
and
to the reduc-
*3
+ ^ + 5*-
14
encourage the pupil to waste time and
to forget
his elementary work in factoring. The quadratic equation, often looked upon as the
chapter of elementary algebra, seems peculiarly open to mechanical treatment. Add the square of half the coefficient of x> extract the square root, transpose final
this is the rule 1
;
the validity of the result
is
not consid-
" Then there are processes, like the finding of the G. C. M., which
most boys never have any opportunity of using, excepting perhaps in the examination room." Henrici, Presidential address, British Association, Section A, 1883.
TYPICAL PARTS OF ALGEBRA
The reason
ered essential. less historical
;
199
for this procedure
is
doubt-
the early mathematicians were forced to
solve in this way,
and the
tradition has
endured
to
the present.
But
we
if
are to follow this mechanical route,
we may
For practical purposes the pupil be able to write down at sight the eventually needs to roots of equations like x* + 2 x + 3 = o, without stop-
well go even farther.
ping to "complete the square"; for this purpose the formula
P^\ should be as familiar to him as the multiplication table.
To
use the method of the completion of the square in a
thoughtless
way
with every equation has neither a cul-
ture value (since the logic
value (since
it is
is
concealed) nor a utilitarian
an unnecessarily tedious way of reach-
ing the result).
The
best plan of attacking the quadratic equation
as already intimated, through factoring. simple,
it
is
The
plan
is,
is
general (not being limited to quadratics),
can be introduced with factoring and continually reviewed until the chapter on quadratics is reached, and
it
at the
same time
in mind.
When
the student
is
it
keeps the subject of factoring fresh
the chapter on quadratics
is
reached,
already able to handle the ordinary run
manufactured problems, those which "come out " with small integers for roots. Those involving even of
200 THE TEACHING OF ELEMENTARY MATHEMATICS large
numbers, however, require other methods, and completion of the square, an expression
this leads to the
derived from the old geometric method of solving the quadratic equation.
The outcome
be the proof of the fact that
or, if preferred,
complete the square
equation like x*
him
to
and
13.
" I
13*5
it
+
bx
+
c
o.
so important as to in
mind
for use in
That a pupil should
every time he runs against an
+x + =
add three
Some
is
sufficient application to fix
the subsequent parts of algebra. "
method should
the formula for solving ax*
This formula, logically developed,
demand
of this
if
o
as senseless as to require
is
when he wishes
the product of 3
text-books give one or two other methods of
solving the quadratic, but these serve to confuse rather
than assist the pupil. Their interest is more historical than educational. That the teacher may see that the not the only one, however, a few historical devices may be of service
standard solution
is
:
Method (b.
IH4).
of
Brahmagupta
598)
(b.
and
Bhaskara
1
Given
ax*
+ bx =
c.
Then 1
Matthiessen, Grundziige der antiken
Leipzig, 1896, p. 282.
und modernen Algebra,
2.
Ausg.
TYPICAL PARTS OF ALGEBRA
+ ^abx +
d2
=
^ac
2OI
+ IP,
2a This plan, here given in complete form with modern It has symbols, is sometimes called the Hindu method. the advantage of avoiding fractions until the last step.
Method
of
Mohammed ben Musa
(about 800,
see
p. 151) and Omar Khayyam (d. 1123, the author of the Rubaiyat), one of several given by them, and based on geometric considerations. 1
Given
Then
.
and or
This plan
Method
is
essentially the
Given
x*
4-
ax
-+-
b
= o.
x = u + 2.
Let
Then 1
one now in general
of Vieta (i6is). a
^2
+ (23 + d)u + O2 + az +
Matthiessen, p. 309.
2
b}
= o.
Matthiessen, p. 311.
use.
202
THE TEACHING OF ELEMENTARY MATHEMATICS
Since but one condition has been placed upon u
we may impose
+ z,
another, and let
+ a = o,
22
whence 2
and
u*
x=u+z = %a^ V#
whence
2
Here there has been no "completion Method of Grunert (186s). 1 Given
Let
But
4 b.
of the square."
+ ax b = o. x = u + z. 2 - 2 = z ) o. (u + zf-2 u (u + z) + (u 2 2 u, and b = z /. a = -> and s = u= x*
-J-
tfi
.
.-.
2
from which
;r
is
easily found.
method (1856)
Fischer's trigonometric
is
one of sev-
eral of this class.
Given
x*
px + q
Let
x\
P
and
Then and But
'
^2 =/
cos2 $>
one
2 / > 4 q.
root,
2
sin c, the other.
x^+x^p (cos x- x%
=/
x-^x^
q 1
with
o,
2
2
(sin
t
.'.
-
+
sin2 0)
cos
2 c^>)
= /,
= J /2
sin2
Grunert's Archiv, Bd. 40.
-
sin 2 2
^>.
TYPICAL PARTS OF ALGEBRA
For example,
20 =
Here
to solve x*
4-
=
35
57' 44-"6,
71
^ = 61.3607,
whence
93.7062^
The problem shows
*
.'-
^2 =
1984.74
= o.
58' 52."3,
32.3454.
that trigonometry
to assist in the solution of
203
is
able materially
certain kinds of quadratic
equations.
There are many other devices for solving the quadratic, for which the reader must, however, be referred to the great
of Matthiessen.
compendium
Enough
of
these plans have been suggested to show that a de-
parture from the single one in general use, for the
purpose of emphasizing the method of factoring and is not a novelty to be feared;
the use of the formula,
merely to make a judicious selection from the abundance of material at hand.
it
is
To
Equivalent equations
the student
who has not
been taught that there is no escape from the checking of the roots of an equation, and that extraneous roots are liable to enter with any one of several com-
mon the is
so far
little
it
operations,
axioms
until
from the
upon the it is
sufficient
solution case,
subject,
cerning the matter
While
seems a
may
to
blindly follow
reached.
and the text-books that a brief
But
this
offer so
statement con-
be of service to teachers.
true that the solutions of equations de-
pend upon a few well-known
may
is
axioms, these axioms
lead the student into difficulty.
For example:
204 THE TEACHING OF ELEMENTARY MATHEMATICS
x
Let Then, multiplying by 2 Subtracting a
(x
But xa
a*
+ a) (x
a)
2
Dividing by
x
a(x
ax.
= ax
x*
.'.
y
xz =
x,
,
Factoring,
a.
a2
.
a (x
a).
a) = a(x
a).
a,
2a
= a,
2=1.
or
Here every step follows from the preceding one by the
result
is
common
a
of
application
axiom, and
yet
absurd.
undue weight upon demon-
Pupils are apt to place valid
but
strations
apparently
But as
" Bertrand, the French algebraist, says,
mon
J.
sense never loses
its
in
man
that he
is
reality
to set
rights;
evidence a demonstrated formula a
the
is
fallacious.
Com-
up against
about like telling
dead because you happen
to
have
a physician's certificate to that effect."
This tendency of pupils and this testimony of M. What limitations are Bertrand suggest the question :
on the use of
there
question
all
of
E.g.,
the definition
requires
Two
tions.
3-^
each.
But x =
either
and x
i
lent equations, for
of
?
of
To answer
x 3
2
is
are 4-
i
roots
= 3 (x
this
equivalent equa-
equations are said to be equivalent
the roots of
# + 3=
the axioms
of
when
the other.
i) are
equiva-
a root, and the only root,
and x2
9 are not equivalent,
TYPICAL PARTS OF ALGEBRA for
x
one root of the second, but
3 is
root of the
E.g.
does not follow that the
it
:
x=i
If
x*=
then
y
i.
Adding,
x*+x=2.
Solving,
x
2
nor of x^
i
if
it
is
=
i
as
=x+
i
i,
it
does not follow
equivalent to the others.
general,
if
duce
the equation
it
x
i
i,
while
=
They
i.
has two roots, 2 and first
equation.
multiply by a function of is
+
does not follow that
that of
not a root of the
is
(if
is
x-\-
same
i
we
I
not equiva-
is
and we multiply by x i
are the
are not, for x*
but
i,
true that x*
roots
its
i
x
equals are multiplied by
But
equals the results are equal.
x
not of
the others.
an axiom that
that the resulting equation if
2.
+ x = 2, but The equation x* + x=2
lent to either of
E.g.,
or
a root of x*
is i.
It is also
not a
equivalent to either of the orig-
is
resulting equation
The
is
equals are added to equals
if
the results are equal, but
inal ones.
it
first.
axiomatic that
It is
20$
i,
And
x we
in
intro-
integral) one or more new
"
extraneous roots" as they are called. = 25, x8 = 125, xt = 625,-..; 5, then x* Similarly, if x
roots,
but the second equation has one root which the
first
2O6 THE TEACHING OF ELEMENTARY MATHEMATICS has not,
5
the third has two which the
;
5( | neous to the not,
|V
3);
and so
first,
equals needs watching.
dividing
i
x
x
of
When we
customary
V4+ VQ i,
to
or
none
if
That
plus sign.
i,
one or
loses
+3=
be 2
This
5.
we
consider
and not
5,
is
deal with radicals
expressed to understand
is
is,
difficulty is
the sign expressed
to consider only
before them, or
the
V^.
I
roots.
even more pronounced. is
by
then by
original equation, but they = o is also a root. And in
In dealing with radical equations the
it
extra-
equals
+ 2 x* x o, = o, or x =
by a function
dividing
general,
more
of
If x*
;
has
first
three
on.
dividing by x, x* + 2 x These are roots of the
are not the only ones
has
fourth
axiom
the
Furthermore,
the
(
the
value of
+ (3)=
2)
purely conventional;
5,
has
it
simply been agreed that in elementary work the stuunless it is dent shall not be bothered with the expressed, as
the plan
is
VQ=
V~4
square root of 4
is
i,
5,
either 2 or
2,
i,
it
are dealing with the radical equation
seek the root which
=
3
evident that
not very scientific; but so long as
understood no great harm can come from
we
Since the
5.
is
and not
Vx
satisfies
'
x
1
=
3,
it.
V^r
the equation
although
of
it
is
So when i
=
we
3,
+V'x
i
course the
both plus and minus. With the following solution consider this understanding, square root of
i
is
:
TYPICAL PARTS OF ALGEBRA
Given
207
THE TEACHING OF ELEMENTARY MATHEMATICS
2O8
from dividing,"
macy of the " How," and
with no thought as to the
etc.,
He
process.
legiti-
with each step the
gives
content
teachers are often
;
but this
of relatively minor importance, the great questions be asked at each step being, " Why is this true "
and,
Is the process reversible
is
to " ?
" ?
There
Simultaneous equations and graphs
is
often
an objection raised against the introduction of graphs in elementary algebra, that there is no reason for thus
anticipating
that
algebra
and
the
of
history
barriers
we
try
the epigram
bra
"
1
!
to
of
build
a striking
impassable
The
distinctive
geometry merely flood
teachers
rarely abandon it. more fully why two
A
of
is
pupil can
only alge-
upon simultaneous the
understand
equations with
simultaneous,
brought to the eye, by the two
is
so simple,
who have used
linear
are in general
light
is
pictured
introduction of the graph
that
names,
Sophie Germain, "Algebra
geometry
equations,
1
recent event in
What
who would
told
sciences,
separate
really a
separate by
and throws such a
knowns
are
between the branches of mathematics which
vainly
written
is
the two subjects.
rebuke to those
little
are
geometry
although this separation
We
geometry.
analytic
lines
if
plan
much
two un-
the matter
is
which represent
L'algebre n'est qu'une geometric ecrite, la geometric n'est qu'une " frozen It recalls Goethe's description of architecture as
algebre figuree.
music," eine erstarrte Musik, which struck
Mme. de
Stael as so felicitous.
TYPICAL PARTS OF ALGEBRA these equations, than he can by an
He
why
sees, too,
2OQ
analytic
proof.
the attempt to solve the set
2^ + 67=5, 3^ + 97=7, If
fails.
he
is
are not
tions
told that in general three linear equa-
simultaneous, the reason
when supplemented by the equations.
When
the
more
is
clear
pictures, the graphs,
he finds that
in
spite
of
of
the
can
be
general fact just stated, the special equations
are
simultaneous,
added
and
that,
others
indeed,
to the set, as
47^+
15^+15^=30,
i3jj/= 26,
etc.,
the mystery of the matter vanishes as soon as the
graphs are plotted. Similarly for an equation of the second degree combined with another of that degree or with a linear equation. eral
two
While there simultaneous
is
a simple proof that in gen-
quadratics
cannot be solved
without recourse to a quartic equation, most students fail to
appreciate the fact until they have the assist-
ance of graphs.
Most pupils who have "finished"
quadratics would expect to be able to solve the set 5
2
xy
2 j/
x + 6y +
13^
17 y
7
= o,
20
= o,
2IO THE TEACHING OF ELEMENTARY MATHEMATICS
and would wonder
their
at
They cannot understand
inability
why
handle
to
it.
such an innocent look-
ing set as
by quadratics if one makes the lucky give them trouble. They are satisfied
(partly soluble
should
hit)
with one or two roots of the set x*
+ 2xy +/ -7 = 0,
or with half a dozen,
if
x*
+
-$xy
+
23?
- 8 = o,
by the introduction of
extra-
neous ones they can get together that number. " The curious thing is that many examination candidates
who show
great facility in reducing exceptional equa-
tions to quadratics
idea beforehand
appear not to have the remotest
of
the
number
of
solutions
to
be
expected! and that they will very often produce for you by some fallacious mechanical process a solution
which
A a
is
none
at all."
1
valuable exercise for a class which has devoted
little
time to graphs,
nificance of
is
to consider the graphic sig-
each new equation obtained in the solutwo
tion of a pair of simultaneous equations involving
unknowns.
Each equation properly derived must
resent a graph passing through
the graphs corresponding to the 1.
Given
2.
and
first
two.
+73 = 9, x+y = 3-
^8
l
rep-
the intersections of
Chrystal, Presidential address, 1885.
E.g.
:
TYPICAL PARTS OF ALGEBRA
Then
3.
X*
x2 +
4.
=
xy +y* 2,xy *
5-
Equation (3)
3,
211
division.
by
+y2 = 9, by squaring (2). xy = 2, by subtracting and dividing.
represents
an
which
passes the intersections of the graphs (i) and (2) except the point at infinity; (4) represents two
through
parallel
ellipse
all
only one of which passes through the
lines,
and
intersections of (i)
(2); (5) is
the parallels
(4).
intersection
of
allels
to
The the
(x-yf =
i.
an hyperbola passthe ellipse (3) and
ing through the intersections of
solution then passes on to the
straight
line
(2)
with
the
par-
1
In general, the question of the number of roots be expected, the entrance of complex roots in the conditions
pairs,
ous,
or
inconsistent,
and not particularly to stand
out
much
rendering or
equations simultane-
impossible,
these
necessary
theory are made more clearly by the use of the difficult bits of
graph.
Methods
of
Elementary text-books
elimination
ways distinguish
cases
several
respect to linear equations. (2)
subtraction,
possibly
(5)
novelty
only
1
(3)
of
These
elimination are,
al-
with
(i) addition,
comparison, (4) substitution, and If those who love
Bezout's method.
knew
it,
there
A problem used by Professor Beman in
are
numerous
other
his teachers' course in algebra.
THE TEACHING OF ELEMENTARY MATHEMATICS
212
methods which might be brought to the subject.
But are
in to give this turn
1
for the practical purposes of a beginner there
two
only
methods of much value, (i) under which subtraction is merely a
distinct
that of addition,
special case, because the sign of the proper multiplier to
be employed
will
always
one of addition; (2) that of comparison
x=y
2
x
as
just
teaching
is
and
process
to
under which
substitution,
merely a special case, for in equating
^=3^ + 4,
much the
reduce the
as
we substitute the value we compare values. Hence it
subject,
that especial attention
is
to
two
these
to
is
of in
methods
be given, the other plans
suggested by the text-book being shown to be special Indeed, before the pupil leaves the subject it not be going too far to show that the method might
cases.
of substitution
method
is
a special case of the one general
of addition.
Of equal importance with methods mentioned,
The do
is
the existence of the two
the question as to their use.
pupil will easily find for himself,
so,
that the addition
method
is
if
permitted to
usually preferable,
the other being the easier only in special cases, as in that of unit coefficients, or in finding
one of two
values after the other has been ascertained.
When
both equations are of the second degree, the
student should early be led to see that in general no 1
See Matthiessen,
for
example.
TYPICAL PARTS OF ALGEBRA solution
is
possible
cases which
213
by quadratics, and that the only
he can
handle with
any certainty are
those involving homogeneous or symmetric functions.
The methods
of attacking these cases are well
and need not be discussed equations
symmetric
should
it
But
here.
known
in the case of
be noted that most
text-books lose sight of one of the essential features.
the very nature of symmetry the roots must be
By
Consider, for example, the set
identical.
the usual method
By 5,
or
i,
four results,
x
32
= 0.
found to be J( I9 "^329), as should have been anticipated. is
follows, without substituting or applying
It therefore
has identically the same values because of the symmetry of each function as
any special devices, that
to
x and
Of
y.
y
course the particular value of
y
to
be taken with a given value of x is not yet determined, but this is usually seen at once by looking at the two equations. The failure to recognize all this results in
serious loss of time;
the student gets
he might more profitably get it by solving another set of equations than by failing to appreciate one of the essential parts of the theory. exercise,
it
is
true, but
Complex numbers since ical
Gauss, in
world
at
As
already
stated,
it
is
only
1832, brought before the mathemat-
large
the
Argand had developed,
theory which Wessel
that the complex
and
number has
214 THE TEACHING OF ELEMENTARY MATHEMATICS
been well understood. finding
Even now
is
it
only slowly
usually saying (of course "between the lines"),
V
is
to
and we do not know what
i,
do with
and we
it,
trouble
little
works
into elementary text-books, such
way
its
as
possible."
it
means
hasten over
will
Where
in
"Here
or
what
with
it
as
the course in
algebra this perfunctory treatment shall be given has
been the subject of not a
made any
difference.
If
little
the
discussion, as
student
to
is
if
it
receive
nothing, what matter whether that nothing comes this month or next?
What,
When
explained It
should
then,
should
it
be
done
with
the
subject?
be introduced, and how should
it
be
?
an educational maxim already several times
is
invoked in these pages, that a subject is best introAs soon as we duced just before it is to be used. reach
quadratic
equations
as
a
distinct
subject
we
meet complex numbers. Equations like x* -f- i = o, = o where x* + 2 x + 5 o, and in general x* -f- px + q
/ <4q Hence it 2
bers
to
roots
involving
follows that the chapter on
logically
Whether
rise
give
t
it
imaginaries.
complex num-
precedes that on quadratic
equations.
psychologically precedes depends upon
its
difficulty.
The because
difficulty it
is
have known
of
the chapter
has been
overrated
only recently that teachers as a class anything about the subject. In reality
TYPICAL PARTS OF ALGEBRA the graphic treatment of the complex
more
for
difficult
who
the
215
number
is
no
to
is
pupil begin ready is than that of the number to negative quadratics the one about to take up the theory of subtraction.
Teachers are therefore urged, even at the expense of a week's work outside the text-book if that be a
treatment.
The even
the
to present
hardship
elements
applied
problems
of
this
graphic
are
usually
algebra
more objectionable than those
When
of
arithmetic.
the science began to find place in the schools
there had accumulated a large
which
by
offered
the
of
1
arithmetic
little
new
were
number
of
examples
puzzles, but
by algebra These were incorporated in and they have remained there by the
difficulty.
science,
usual influence of two powerful agents
the conserv-
atism of teachers and the various kinds of state ex-
To
aminations.
this latter influence is to
the greatest amount of blame the individuals
to
who
set
in
be charged
the matter, not as
the examinations, but to
the inherent evil (possibly a necessary one) of
system.
know
Certain of
1
;
the
a country
wasted over some particular line they would like to omit them, but their
that time
of problems
the best teachers of
is
One of the best elementary presentations of the subject is given in Number System of Algebra, Boston, 1890, a book which should be
Fine's
upon the shelves of every teacher of this subject. For a classroom ment, see Beman and Smith's Algebra, Boston, 1900.
treat-
2l6 THE TEACHING OF ELEMENTARY MATHEMATICS
hands are
tied
by the necessity that
pass a certain these are
for
examination often
regents', teachers',
among
On
etc.).
educators.
would
too
They
service,
most
the the
like
to
college
objectionable,
other
many
hand,
the most
examiners are among
the
of
(civil
their pupils shall
progressive
see the
mathe-
matical field weeded and conservatively sown anew.
But
hands are also tied by the system. progressive English examiner once remarked
As
their
writer,
"
I
know
that this
a
to the
problem should have no
cannot replace it by a modern one because the schools are not up to such place in the examination, but
I
a change; their text-books do not prepare for
Speaking of
effect
this
of
it."
the examination,
Pro-
fessor Chrystal has not hesitated to express his views
with perfect frankness.
"The
history of this matter
of problems, as they are called, illustrates in a singularly
instructive
way
the
weak
point of our English
system of education. They originated, I fancy, in the Cambridge Mathematical Tripos Examination, as a reaction against
and
the abuses of
science teaching
At
cramming book-work,
they have spread into almost every
first
they
may
witness
test-tubing
in
branch of chemistry.
have been a good thing;
at
events the tradition at Cambridge was strong in
all
my
he who could work the most problems in three or two and a half hours was the ablest man,
day that
and,
be
he ever so ignorant of
his
subject
in
its
TYPICAL PARTS OF ALGEBRA
2 1/
width and breadth, could afford to despise those less gifted with this particular kind of superficial sharp-
for
prepared
way
the end,
But, in
ness.
as
in
problem-working
We
book-work.
for
came
all
it
through old problem papers,
same
to the
exactly
we
the
same
to
work
were directed
and study the
:
style
and
of the day and of the examiner. The in and the examiner much to do with had, truth, day fashion reigned in problems as in everything it, and else." 1 more pointedly he says: "All men Still peculiarities
in
engaged
practically
teaching,
who have learned of their own early
enough, in spite of the defects training, to enable them to take a broad view of the matter,
are
everything to evil. petitive
as
agreed that
It is
is
good
of
which
canker
the
turns
our educational practice
in
the absurd prominence of written com-
examinations
The end
to
all
that
works
all
education
pupil to be examined
;
mischief.
this
fit the nowadays the end of every examination
is
not to be an educational instrument,
to
but to be an
examination which a creditable number of men, however badly taught, shall pass. We reap, but we omit to sow. Consequently our examinations, to be what that is, beyond criticism in the newsmust contain nothing that is not to be found papers in the most miserable text-book that any one can is
called fair
cite
bearing on the subject. 1
.
.
.
The
result of
Presidential address, British Association, Section A, 1885.
all
21 8
THE TEACHING OF ELEMENTARY MATHEMATICS hands of
this is that science, in the
specialists, soars
higher and higher into the light of day, while educators and the educated are left more and more to 1 primeval darkness." This evil, which we have not yet the ingenuity to avoid, stares the teacher in the face when he would
wander
in
replace obsolete matter by problems which have the of the generation in
stamp
which we
It is
live.
not
that these problems about the pipes filling the cistern,
the hound chasing the hare, the age of Demochares,
and the number of
nails in the horse's shoe, are not
good wit-sharpeners, and possessed of a kind of interest; but we have now a large number of equally
good wit-sharpeners problems
possessed
relating to the life
simple science the pupil
is
of
a
living
interest,
we now live, and to the now studying. "I some-
however," says a recent writer, whether boys really enjoy being introduced to such
times feel a doubt, "
as
exercises '
you got
?
or
reply;
'A
and to
says to
B makes the
again as his head tail
B,
how much money have
a very singular hypothetical
fish
whose body
and
tail
is
have given relations of magnitude.
suspect that there
problems."
2
is
These
half
as
long
together, while head and I
cannot but
something unpractical in these problems have some
historical
value as history and some interest from their very 1
Presidential address, 1885.
2
Heppel, G., in the Mathematical Gazette, February, 1895.
TYPICAL PARTS OF ALGEBRA absurdity, but
it
is
ration of teachers
may
treatment
rational
to
219
be hoped that the rising gene-
them
see
of the
"
laid aside.
subject,
A
more
introducing from
the beginning reasoning rather than calculation, and
applying the results obtained to taken from all parts of science,
various as
problems as from
well
would be more interesting to the student, give him really useful knowledge, and would be at the same time of true educational value." 1 everyday
life,
It is a serious question
England's
lead,
has
whether America, following
not
gone
producing no text-books sented
many
the
in
the
of
in
delightful
it
problem-solving is
we
that
are
which the theory is prestyle which characterizes
French works
or those
Bourlet),
into
Certain
altogether too extensively.
example, that
(for
of the recent
Italian
of
school (like
Pincherle's handbooks), or, indeed, those of the conti" In nental writers in general. short, the logic of the
which, both
subject,
speaking,
is
educationally and
the most important part of
The whole
neglected.
training
scientifically is
it,
consists
in
wholly
example
What
should have been merely the help to attain the end has become the end itself. The grinding.
result
is
that algebra, as
we
teach
it,
whose
is rules,
ject is the solution of examination problems.
.
.
.
ob-
The
problems worked in examinations go, very miserable, as the reiterated com-
result, so far as is, 1
after
all,
Henrici, O., Presidential address, British Association, Section A, 1883.
220 THE TEACHING OF ELEMENTARY MATHEMATICS of
plaints
aminee
is
show
examiners a
well-known
;
the effect
enervation
on the
ex-
mind,
an
of
almost incurable superficiality, which might be called Problematic Paralysis a disease which unfits a man to
follow an argument extending beyond the length
of
a printed octavo page.
sional
aid to
.
.
Against the occa-
.
working and propounding of problems as an the comprehension of a subject, and to the
starting of
new
a
idea,
no one
objects,
and
it
has
always been noted as a praiseworthy feature of English methods, but the abuse to which it has run is
most pernicious." l The interpretation says D'Alembert;
And
of solutions
it
often gives
Algebra is generous, more than is asked. 2
one of the mysteries which teachers and text-books usually draw about the science, that some it
the
of
is
solutions
of
the
applied
problems
not
are
usable, are meaningless.
But there should be no mystery about fact, easily explained, that
it
is
not at
this.
It is
all difficult to
a
put
physical limitations on a problem that shall render the result mathematically correct but practically impossible.
can look out of the window 9 times in 2 seconds, how many times can I look out in i second,
For example, at the
1
2
if I
same rate?
The answer, 4^
times,
is
all
right
Chrystal, Presidential address of 1885.
L'algebre
demande.
est
genereuse
;
elle
donne souvent plus
qu'on ne
lui
TYPICAL PARTS OF ALGEBRA mathematically, but physically time.
if
Similarly,
5
men
221
cannot look out half a
I
are to ride in 2 carriages,
how many will go in each, the carriages to contain the same number ? Mathematically the solution is simple, but a physical condition has been imposed, "the carriages to contain the same number," which makes the
A
problem practically impossible.
few such absurd
all the mystery attaching to results of and show how easy it is to impose restricthat exclude some or all results.
cases take
away
this nature,
tions
For example, the number of students class
such
is
140
=o
;
as
to
satisfy the
how many
are there
but
mate,
has
been
the
other,
generous;
The
?
2.r2
has
given
33^
conditions of
root, 20, legiti-
Algebra
meaningless.
|, it
equation
make one
the problem are such as to
in a certain
more than was
asked.
A
Consider also the problem, father is 53 years old after how many years will the father his son 28
and
;
son
be twice as old as the 53 +;tr= 2(28 +-*)
?
From
the
^=3. We
we have
equation are
now
under the necessity of either (i) interpreting the ap3 years after this time,
parently meaningless answer,
or (2) changing the statement of the problem to avoid
such a
Either plan
result.
"
terpret
3
years after
before," which of negative
is
"
is
feasible.
We may
as equivalent to
entirely in
in-
"
3 years accord with the notion
numbers; or we may change the problem
THE TEACHING OF ELEMENTARY MATHEMATICS
222
"How many
to read,
as the
old
years ago was the father twice as
Most algebras require
son."
this
latter
from the days when the negative understood than now.
plan, one inherited
number was
less
"Unlike other characteristic
method
instead of
insoluble,
some other enriches
number
is
impossible,
hesitating
question, algebra seizes
its is
employs
algebra has a special and
of handling impossibilities.
problem of algebra
this is
sciences,
the symbol."
If
that equation
and passing on to these solutions and
The means which
province by them. *
if
The symbol
"
it
3," for the
of years after the present time, without sense
in itself, is seized
and turned
into a
means
for enriching
the domain of algebra by the introduction and interpretation of negative numbers.
The
further interpretation of negative results, and
the discussion of the results of problems involving eral equations, is a field of considerable interest
value;
lit-
and
but since most text-books furnish a sufficient
treatment of the subject,
it
need not be considered
here.
Conclusion ter
might
to dwell
two
The few
easily
topics mentioned in this chap-
be extended.
upon the absurdity
It
would be suggestive
of drilling a pupil
upon the
chapters on surds and fractional
artificially distinct
exponents, as our ancestors used to separate the "rule of three" from proportion 1
matters explainable only
De Campou.
TYPICAL PARTS OF ALGEBRA
by reviewing
common
the
orem
223
The
their history.
theory of fractions, fallacy in the proof of the binomial the-
for general exponents, the use of determinants,
the complete explanation of division or involution, the questions of zero, of infinity, and of limiting values
these and various other topics will suggest themselves
But the
as worthy a place in a chapter of this kind. limitations of this
The
work are such
as to exclude them.
topics already discussed are types,
that they
algebra to
may lead some see how meagre
and
it is
hoped
of our younger teachers of is
the view offered by
of our elementary text-books,
how much
many
interest
can
easily be aroused by a broader treatment of the simpler chapters, and how necessary it is to guard against the
of the
dangers
slipshod methods
which are so often seen is
often taught, there
tion, that
is
and narrow views
in the schools.
As
algebra
force in Lamartine's accusa-
mathematical teaching makes
man
a machine
1
and degrades thought, and there is point to the French epigram, " One mathematician more, one man less." 2 1
L'enseignement mathematique
pensee. 2
fait
Phomme machine
et
degrade
la
Rebiere's Mathe"matiques et mathematiciens, p. 217.
Un mathematicien de plus, un homme de moins.
in Rebiere,
ib.,
p. 217.
Dupanloup.
Quoted
CHAPTER
IX
THE GROWTH OF GEOMETRY Its historical position
Roughly dividing elementary mathematics into the science of number, the science of form, and the science of functions, the subject has
Hence the chrono-
developed historically in this order.
sequence would lead to the consideration of geometry before algebra, not only in the curriculum,
logical
The somewhat
but in a work of this nature. relation of arithmetic
the order here
if
followed,
for a matter of so
closer
and algebra, however, explains
little
explanation
is
necessary
moment.
Reserving for the following chapter, as was done with algebra, the question of the definition of geometry,
we may
sumed more
its
consider by what steps the science as-
We
present form.
clearly the limitations
which the
seen to place upon the subject,
which the science plainly
is
thus understand
shall
taking,
we
definition will
and we
comprehend the nature
of
be
shall see the trend shall
the
the more
work
to
be
undertaken by the next generation of teachers. The world has always The dawn of geometry tended to deify the mysterious. fire,
the
sea,
life,
death,
The
number
224
sun, the stars,
these
have
all
THE GROWTH OF GEOMETRY
225
played parts in the great religious drama. Whether it be that the plains of Babylon were especially adapted to the care of flocks, or that the purity of the tian
atmosphere
led
to
the
study
the
of
Egyp-
heavenly
bodies, or that both of these causes played their parts,
Mesopotamia and along the Nile a primitive astronomy developed at an early period and
certain
took
its
it
is
that in
place as a part of the store of ancient
gious mysteries.
With
it
reli-
went some rude knowledge of practical life also creating from
geometry, the demands of time to time an empirical science of simple mensuration.
Thus among the Babylonians we
find the circle of
the year early computed at 360 days (whence the circle v/as divided into that
number
of degrees),
as astronomical observation improved, at
the correct number. 1
and
later,
more nearly
The Babylonian monuments
so
often picture chariot wheels as divided into sixths, that it
is
probable that the method of dividing the circum-
ference into sixths by means of striking circles was early
known, a method which
tion of the regular hexagon.
circumference
is
seems generally
a to
little
carries with
it
the inscrip-
This would show that the
more than 6 r
have been taken as
or 3 d, but 3
TT
by them and
their neighbors. 2 1
Hankel, Zur Geschichte der Mathematik,
p. 71, for the pre-scientific
geometry. 2 i
around
Kings vii, 23; 2 Chron. iv, 2. "What is one handbreadth through." Talmud.
Q
is
three handbreadths
THE TEACHING OF ELEMENTARY MATHEMATICS
226
The Egyptians were
orientation of their temples, a
moment by a
as
particular
custom
to
the
proper
considered of
still
The
large part of the religious world.
meridian line was established by the pole star, and for the east and west line the temple builders were early
aware of a rule perpendicular.
used by surveyors in laying
still
The
is
present plan
off
a
to take eight links
of a surveyor's chain, place the ends of the chain four links apart,
and stretch
it
with a pin at the
forms a right-angled triangle with sides
this
Egyptians did the
same
the harpedonaptae or as a
"
3, 4, 5.
in building their temples,
The and
rope-stretchers" laid out the plans,
1 engineer lays out those for a building to-day. scholars of the Nile valley also possessed some
civil
The
knowledge of the rudiments their
approximation
proved for TT =
1
2
f5
(- Q -)
=
for a period suration.
for
fifth link,
of the
trigonometry,
the value of
when geometry was in
TT
2
and
was not im-
Ahmes gave
centuries.
3.1605, a remarkably
He was
example
isosceles
many
to
of
the value
good approximation more than men-
little
not so fortunate in
all
of his rules,
the one for finding the area of an
triangle,
which required the
multiplication
measure of half the base by that of one
of
the equal sides. 1
This interpretation of the Greek harpedonaptae
Cantor's ingenious discoveries. 2
A
p. 128.
brief
summary
is
Cantor,
I,
is
one of Professor
p. 62.
given in Gow, History of Greek Mathematics,
THE GROWTH OF GEOMETRY The
indebtedness
the
of
227
who
Greeks,
were
the
next to take up geometry, to the Egyptians is well " It remains only to cite the summarized by Gow :
universal testimony of
the
in
etry was,
first
Greek
writers, that
Greek geom-
instance, derived from Egypt,
and that the latter country remained for many years afterward the chief source of mathematical teaching.
The statement been
already
So
cited.
made
is
also
in
this
has
subject
Plato's
'
Phaedrus,'
to say that the
Aristotle also ('Metaphysics,'
etry
on
Herodotus
Egyptian god Theuth invented arithmetic and geometry and astronomy.
Socrates first
of
was
I,
originally invented in
declares
expressly
i)
admits that geom-
Egypt; and Eudemus
that Thales studied there.
Much
B.C.) reports an Egyptian tradition that geometry and astronomy were the inventions of Egypt, and says that the Egyptian priests claimed
later
Diodorus (70
Solon,
Pythagoras,
Chios,
and Eudoxus as
Democritus, CEnopides
Plato,
their
.
it
.
Strabo
pupils.
further details about the visits of Plato
of
gives
and Eudoxus.
Beyond question, Egyptian geometry, such as was, was eagerly studied by the early Greek phi.
losophers,
grew
man
and was the germ from which
in their
hands
that magnificent science to which every Englishis
indebted for his
thinking."
first
lessons in right seeing and
1
The Greeks 1
were, however,
the
first
History of Greek Mathematics, p. 131.
to
create a
THE TEACHING OF ELEMENTARY MATHEMATICS
228
Thales
science of geometry.
640, ( 548), having through trade secured the financial means for study, travelled in
the
for
Egypt
mathematical lore of the as he
and
received,
Asia
purpose of
acquiring the
priests, giving quite as
a
established
finally
where the
first
much
school in
inimportant vestigations in geometry were made. The most noted pupil of Thales was Pythagoras, who was with him for a short time at least and who
Minor,
scientific
was advised by him to continue his studies in Egypt. The school which Pythagoras afterward opened in Croton, in Southern Italy, was one of the most famous of
all
antiquity,
ously
propositions, is filled
by
among six
and here geometry was
seri-
Here were proved the following
cultivated.
others
equilateral
:
the plane about a point triangles,
four
squares or
the sum of the interior hexagons angles of a triangle is two right angles the sum of the squares on the sides of a right-angled triangle three
regular
;
;
equals the square on the hypotenuse, a fact known to the Egyptians but first proved by the Pythagoreans.
From now on Greek
1
G.
until the third
geometry
passed
century before Christ
through
its
golden
age.
1
For detailed notes as to the discoveries of the Greeks see Allman, J.,
Greek Geometry
from Thales to
Euclid;
Bretschneider,
Geometric und die Geometer vor Eukleides, Leipzig, 1870;
Die
Gow,
J.,
History of Greek Mathematics, Cambridge, 1884; Beman and Smith's translation of Fink's History of Mathematics, Chicago, 1900 ; Chasles,
THE GROWTH OF GEOMETRY The
were made
and were taught -
300.
(
in
elementary
geometry 500 to 300,
form by Euclid, who Alexandria about
logical
famous
the
school at
this period,
During
owing
to the vast extent
opened up by the study of conic sections, 348) placed a definite limit upon elemen-
429,
allowing only the compasses and the
tary geometry,
unmarked
in
the two centuries from
crystallized in
in
of the field
Plato
discoveries
principal
22Q
straight-edge
instruments for the
as
con-
struction of figures.
So complete as a specimen
was Euclid's
of logic
treatment of elementary geometry, that it has been used as a text-book, with slight modifications, for
This use has not, however,
over two thousand years.
been general.
men
in
has needed the exertions of
it
Indeed,
like Hoiiel
France and Loria 1
in
and
Italy,
other Continental writers, to recall from time to time
the merits of
England Euclid
in
tically absolute.
The
still
holds
a
sway that
is
prac-
2
influence of the
Greek writers
M., Apercu historique sur 1'origine and of course Cantor and Hankel.
.
1
But
Euclid to the educational world.
.
.
is still
seen in the
de Geometric, Paris,
Delia varia fortuna di Euclide in relazione con
i
2. ed.,
problemi
1875;
dell'
In-
segnamento Geometrico Elementare, Rome, 1893. 2
who care to enter into the merits of the controversy over may make a pleasant beginning, and at the same time may the mean between Dodgson the mathematician and Carroll the Teachers
Euclid see
writer of children's stories (as Alice in son, C. L., Euclid
and
his
Modern
Wonderland) by reading DodgLondon.
Rivals,
230 THE TEACHING OF ELEMENTARY MATHEMATICS nomenclature of the science the world over. the ancients had no printing, and found to
have the
rolls,
it
"book" came to apply Thus we have the books of the
treatise.
convenient
which made their volumes, somewhat
the word
brief,
Because
to part of a " yEneid," of
the " Iliad," and of treatises on geometry, astronomy, etc.
The word has been preserved
in the divisions of
most elementary geometries as a matter of interesting
Thus
history.
Euclid's
first
book
is
upon
chiefly
straight lines
and the congruence
the second
devoted to the next subject of which the
is
of rectilinear figures
student has already some knowledge
and so
third to circles,
some
of our
modern
on.
rectangles
;
;
the
With doubtful judgment
writers have followed
Legendre
in
reversing the order in the second and third books,
known and
placing circles before rectangles, the less
more
concept before the more familiar and
difficult
simple.
Many Greek, lium
other
as,
words, unlike
"book," are distinctly " for example, "theorem," axiom," "scho-
"
(happily going
"parallelogram," unscientific
out
of
"
fashion),
"parallelepiped"
(often
"
spelling
parallelepiped
"),
trapezoid,"
given
the
"hypotenuse"
with an k, though unscien(still occasionally spelled In many cases, however, the Latin tifically so), etc.
forms have (rather
displaced
more
Latin
the
than
Greek, as Greek),
in
"
" triangle
"quadrilateral,"
"base," "circumference," "vertex," "surface,"
etc.
THE GROWTH OF GEOMETRY After the death of Archimedes
owe the
first fruitful scientific
tion of the circle, itself.
212), to
(
attempts at the
geometry seems
to
whom we mensura-
have exhausted
Excepting a few sporadic discoveries,
stagnant for nearly
231
two thousand years.
it
remained
It
was not
until the seventeenth century that
any great advance was made, a century which saw the discovery of analytic geometry at the hands of Descartes, the revival of pure geometry through the labors of Pascal and his contemporaries, and which saw but failed to recognize the foundation of
projective
geometry
in
the works of
Desargues.
The nineteenth century has seen Recent geometry a notable increase of interest in the geometry of the circle and straight-edge, a geometry which can, however, hardly
be called elementary in the ordinary sense.
France has been the leader
in this
phase of the subject,
with England and the suggestion
Germany following. Carrying out made by Desargues in the seventeenth
century, Chasles, about the middle of the nineteenth century,
developed the theory of anharmonic ratio, making what may be designated modern geomBrocard, Lemoine, and Neuberg have been largely
this the basis of etry.
instrumental in creating a geometry of the circle and the triangle, with special reference to certain interesting angles and points.
way
How much
of all this will find its
into the elementary text-books of the next genera-
tion, replacing, as it
might safely
do,
some
of the
work
THE TEACHING OF ELEMENTARY MATHEMATICS
232
which we now
who this
give,
impossible to say.
it is
The teacher
wishes to become familiar with the elements of
modern advance could hardly do
better than read
1 Casey's Sequel to Euclid.
Along more advanced
lines the progress of
The
has been very rapid.
and Von Staudt,
geometry
labors of Mobius, Pliicker,
Germany, have led to regions undreamed of by the ancients. This work is not, however, in the line of elementary geometry, and therefore Steiner,
in
has no place in the present discussion. 2
Among
the improvements which affect the teaching
of the elementary geometry of to-day, a brief
mention.
these
Among
is
few deserve
the contribution of
"
Mobius on the opposite senses of lines, angles, surfaces, and solids; the principle of duality as given by
Gergonne and Poncelet; the contributions of De Mor-
gan
to the logic of the subject;
the theory of trans-
worked out by Monge, Brianchon, Servois, Carnot, Chasles, and others the theory of the radical
versals as
;
axis, a property discovered by the Arabs, but intro-
definite concept by Gaultier (1813) and used by Steiner under the name of 'line of equal power'; the researches of Gauss concerning inscrip-
duced as a
polygons, adding the 17- and 257-gon to the list and the researches of Muir below the looo-gon; on stellar polygons.
tible
.
.
1
2
.
.
.
.
London, fifth edition, 1888. For a brief review of the subject, see the author's
and Woodward's Higher Mathematics,
New
article in
York, 1896,
p. 558.
Merriman
THE GROWTH OF GEOMETRY
233
"In recent years the ancient problems of trisecting an angle, doubling the cube, and squaring the circle have all been settled by the proof of their insolubility 1 through the use of compasses and straight-edge." "The non-Euclidean geNon-Euclidean geometry ometry is a natural result of the futile attempts which
had been made from the time
of Proklos to the
opening
of the nineteenth century to prove the fifth postulate (also called the twelfth axiom,
and sometimes the
elev-
enth or thirteenth) of Euclid." This is essentially the postulate that through a point one and only one line can be drawn parallel to a given line. The first scientific investigation of this part of the foundation of geometry
was made by Saccheri to its final stage
and the
(1733).
The matter was brought
by Lobachevsky and Bolyai about 1825,
result is a perfectly consistent
geometry denying
the validity, or the necessity, of the postulate in question. 1
2
Smith, D. E., History of
Woodward's work
Modern Mathematics,
cited, p. 564.
lems mentioned, see
Beman and
On the
in
Merriman and
impossibility of solving the prob-
Smith's translation of Klein's
Problems of Elementary Geometry, Boston, 1896. 2 Smith, D. E., History of Modern Mathematics,
p. 565.
Famous
CHAPTER X WHAT
is
GEOMETRY? GENERAL SUGGESTIONS FOR TEACHING
The etymology of a word is often Geometry defined from giving its present meaning. We have already seen this in the case of "algebra" and "algorism"
far
Geometry means earth-measure (777, the earth, measure), and probably took this name be-
(p. 151).
H- fjierpelv, to
was what we would
cause, in
its
now
by the unexpressive term
came
call
prescientific stage,
it
It
"surveying."
mean, among the Greeks, the science of figures or of extent, and this general idea still obtains. to
More
of objects about us in
we may say " By the observation we arrive at the concept of the space :
specifically
which we
live
and
We
are aware at the
tain extent.
have a form.
regularity
is
which these objects have a
These forms are
them
certain of
in
strike us
by
same time
cer-
that they
but
infinitely varied,
their regularity."
l
This
rather apparent than real, and the appear-
ance leads us to make certain abstractions, as of straight line, circle,
as
square,
etc.,
the abstractions 2
objects
forms not met in nature.
made concerning
"
Just
collections
of
are the basis of our arithmetical ideas, so the 1
2
Laisant, p. 89.
234
See p. 100.
WHAT abstractions
IS
GEOMETRY
235
made concerning forms are the origin l Hence the science
our conceptions of geometry."
geometry
is
of
the science of certain abstractions which the
mind makes concerning form.
As Laplace
"
says
order to understand the properties of bodies, first
of
to cast aside their particular properties
:
In
we have
and
to see
them only an extended figure, movable and impenetrable. We must then ignore these last two general in
properties and consider the extent only as a figure.
numerous
The
relations presented under this last point of
view form the object of geometry."
2
Elementary geometry, however, limits itself to comAs already stated, the paratively few of these forms. great
plane
field
opened by the study of conies and higher
curves
led
Plato
to
limit
elementary
plane
which can be constructed by the use of the compasses and the unmarked straightgeometry
edge.
As
to those figures
solid
geometry has gradually developed,
it
has been looked upon as limited to figures analogous to those of plane geometry, the sphere analogous to the circle, the plane to the straight line, etc., with the addition of the prism, pyramid, cone, and cylinder.
of
Euclid, caring
mensuration,
little
paid
for the
almost
no
practical
demands
attention
to
solid
geometry; but the subject has assumed much prominence in the nineteenth century, without, however, 1
2
Laisant, p. 89.
Dauge,
F.,
M6thodologie,
p. 161.
236
THE TEACHING OF ELEMENTARY MATHEMATICS
having its limits clearly defined. For example, whether a cone with a non-circular directrix shall be admitted is
an unsettled question
suration of volume
men-
for purposes of simple
;
might deserve a place, but hardly so unless the mensuration of a non-circular curvilinear plane figure
it
also admitted.
is
base)
(its
But elementary geom-
Limits of plane geometry etry
not only quite uncertain with respect to the
is
devoted to solids
extent of the portion additions
made
IX, have science, as
to
the
limits
sometimes
beyond lum.
the
is
it
far
curricu-
the notions of orthocentre, centroid,
but
what
just is
expected in view of so recent.
the
as
secondary be excluded, for we have long
all
the next generation
is
the triangle,"
of
of
possibilities
ex-centre, etc.,
the
of
extent of the subject
called, the
since introduced
Chapter
portion
"elements," even more undefined.
its
cannot
It
that
of
With the recent "geometry is
the recent
;
plane geometry, referred to in
to
Suffice
shall
be admitted
quite uncertain, as
by would be
the fact that the development it
to say that at present there is
no general agreement as
to
what constitutes element-
that it shall cover substanary geometry, save this " Elements," plus a little tially the ground of Euclid's
work on
loci,
the mensuration of the
suration of certain
common
ment, the
attempting a
futility of
circle,
solids.
and the men-
From
this state-
scientific definition of
the elementary geometry of the schools
is
apparent.
WHAT The reasons
GEOMETRY
IS
237
for studying geometry, as for studying
We
have the practical side simple mensuration, and we have the culture side in the logic which enters into it to arithmetic, are twofold.
the
of
subject in
such a marked degree.
The most taught
mere
part of mensuration
practical
connection
in
now
rule,
with
arithmetic,
its
chief
what the English schools do with
solid
a mistake also often
geometry
by
few necessary formulae.
drop the science there, would be to lose
value, to do
usually
formerly
with the models in hand and with a
semi-scientific deduction of a
To
is
states, though not in the West.
made in our Eastern The danger of doing
nothing with solid geometry save in the way of mensuration, is suggested by Professor Henrici in these
words (referring all,
perhaps,
Euclid's
"Most of English schools) geometry has suffered, because
to the
solid
treatment
of
:
it
is
scanty,
almost incredible that a great part of suration of
solids
teaching.
The
it
seems
the men-
it
simple curved surfaces and* of
areas of
volumes of
and
is
not included in ordinary school
subject
possibly,
is,
mentioned
in
arithmetic, where, under the name of mensuration, a number of rules are given. But the justification of
these rules
who
not
except
supplied,
to
the
student
reaches the application of the integral calculus
and what of
is
points,
is
almost worse
lines,
and
is
;
that the general relation
planes,
in
space,
is
scarcely
238
THE TEACHING OF ELEMENTARY MATHEMATICS
touched upon, instead of the student's mind." 1
The
culture
value,
being
which
fully
impressed on
almost the only one
is
which formal, demonstrative geometry has, includes two In the first place, we need to know geometry phases. for general information, because the rest of the world
knows something that this
of
it.
It
must be admitted, however,
not a very determining reason, for
is
which would
justify
it is
one
keeping any traditional subject in
the curriculum.
The second and
important culture phase is Before Euclid, probably that of the logic of geometry. most of his propositions were known but it was he vitally
;
who arranged them strations
in the order
which have made
his
and with the demon-
work one
of the most
admired specimens of logic that have ever been produced. And this logic has given added significance
and beauty to the truths themselves. " They enrich us by our mere contemplation of them. In this connection I
the Student/ by "
*
wish to quote the beautiful poem
To Archimedes
Schiller
Archimedes and
:
once came a youth,
who
for
knowledge was
thirst-
ing, *
Saying,
Which
Initiate
for
my
me
into the science divine,
country has borne forth
fruit
of such wonderful
value,
And which
the walls of the town 'gainst the
1
Sambuco
Presidential address, 1883.
7
protects.
WHAT 1
Call'st
*
But
it
thou the science divine
was
so,
my
son, ere
Would'st thou have provide thee
fruit
it
GEOMETRY
IS
? It is so,'
239
the wise
man responded
;
availed for the town.
from her, only? even mortals with that
;
Would'st thou the goddess obtain ? seek not the
is
Here, then,
woman
in her
'"
1
!
the dominating value of geometry,
its
value as an exercise in logic, as a means of mental
"as a discipline in the habits of neatness, The order, diligence, and, above all, of honesty. fact that a piece of mathematical work must be definitely
training,
right or wrong,
be discovered,
and that
wrong the mistake can may be made a very effective means of if
it
is
2 conveying a moral lesson."
Without
fixed in mind, the teacher is like a
he knows not whither he
aim well
this
mariner without a
to go or, if he compass has some confused idea of the haven, he has not the ;
means
to find his
way
is
;
thither.
Having now considered the nature of elementary geometry, and the reasons for teaching the science, the question arises as to the general method of presenting
it.
Geometry
the lower grades
jn materially as to the
method
While educators differ
of presenting the subject of
demonstrative geometry, this being for the
1
coming
Schwatt,
I. J.,
still
an open question
generation to consider,
Some
it
is
generally
Considerations showing the Importance of Mathe-
matical Study, Philadelphia, 1895. 2
Mathews, G.
B., in
The School World,
Vol.
I, p.
129 (April, 1899).
240 THE TEACHING OF ELEMENTARY MATHEMATICS agreed that some of the elementary concepts of the science should be acquired in the lower grades. This
view was long ago held by Rousseau. " I have said," he remarks, " that geometry is not adapted to children
;
but this their
is
our
method
fault.
is
We
seem not
to
comprehend that
not ours, and that what should be for us
the art of reasoning should be for them merely the art
Instead of thrusting our method upon them,
of seeing.
we would do
better to adopt theirs.
.
.
For
.
my pupils,
merely the art of handling the rule and 1 Lacroix, one of the best teachers of compasses." mathematics at the opening of the nineteenth century, 2
geometry
is
recognized the same principle is
when he
said
" :
Geometry which
possibly of all the branches of mathematics that to
me
a subject
well adapted to interest children, provided
it is
presented
should be understood
to
them
It
first.
chiefly with respect to
its
seems
applications.
.
.
operations of drawing and of measuring cannot
.
The
fail to
be pleasant, leading them, as by the hand, to the science of reasoning."
Such was
also the
scheme
laid out
by
the mathematician Clairaut and approved by Voltaire,
but in practice
it
has not been systematically followed
by the teaching profession. Laisant, whose rank as a mathematician and an 1
Rebiere, A., Mathematiques et mathematicians, p. 103.
2
His Essais sur 1'enseignement en general, et sur celui des mathematiques en particulier, Paris, 1805, was one of the earliest works of any value on the teaching of mathematics.
WHAT educator
justifies
IS
GEOMETRY
24!
the frequent reference to his name,
thus expresses his views
" :
The
first
notions
of
ge-
ometry should be given to the child along with the notions
first
beginning
of
of
algebra, following theoretical
raisonnee).
But
just as there
preparation
for
arithmetic,
so
tion,
the
theoretical
practice
childhood, of
of
upon the
closely
arithmetic
(Farithmetique
must be a preliminary
namely
practical
calcula-
geometry should be preceded by
drawing.
The
habit
acquired
neatly and with
drawing figures would be of great assistance
sible exactness,
in
sen-
later in
the development of the various chapters of geometry.
The one who
defined geometry as the art of correct
We
reasoning on bad figures, was altogether wrong.
never reason
save
abstractions, and
on
figures
are
never exact; but when the inaccuracy is too manifest, when the drawings are poorly executed and appear confused, this confusion of form readily leads to that of reasoning
there
by
are
logical
and tends
Indeed
to conceal the truth.
cases where
a
reasoning to
poorly drawn
manifest
leads
figure
absurdities.
The
1
education in geometry should therefore be undertaken, as in the case of practical computation, with first
the
child
who knows how to read and that is, who knows drawing.
language tage should be taken in 1
Two
.
this
drawing of
write .
.
Advan-
figures, to
interesting illustrations of this fact are given in Ball's
matical Recreations, London, 1892, p. 32.
the
Mathe-
THE TEACHING OF ELEMENTARY MATHEMATICS
242
number
give to the child the nomenclature of a large of geometric concepts, but always without definitions."
The views
of Hoiiel, one
of
"Let us imagine," he
a
graduated
the best teachers of
of
France, also deserve recogni-
the last generation in tion.
any formal
l
teaching
"the
says,
of
possibility
geometry
elementary
carried on at every step according to one invariable
governed by the rules of severe logic, but with the difficulties always commensurate with plan, always
For such a
the development of the pupil's mind.
scheme the study of geometry would need to be considered from various points of view corresponding to the various degrees of initiation
beginners iarize
their
it
would be necessary
For
the pupil.
of
of all to famil-
first
them with the various geometric figures and names, to lead them to know facts and to
understand their more simple and immediate applications to matters
of
daily
axioms
the
life.
and
to
to
multiply
of
demonstrations, experimental
duction,
treatment
remembering
always is
essentially
We
ought at in
employ, truths,
that
provisional.
.
in-
method
of
The
.
teaching should be purely experimental, and little
the
pupil
should come to
see
place
analogy,
this .
first
that
first
little
all
by
truths
need not necessarily be derived from experience, but that
some are consequences 1
of a certain
La Mathematique,
p. 220.
number
of
WHAT others, a
IS
GEOMETRY
243
number which becomes smaller and smaller
as one advances in the science until he reaches the
fundamental axioms."
The mere
l
ideas above set forth are not the thoughts of theorizers;
more or
less
they
have been carried
The
can schools.
some
of
outline
given in the subsequent pages. said
for the
for
It
of
work
this
is
may, however, be
apply
new schemes
admission
lacking in this particular.
The
to
Intermediate grades
The next
taken in the so-called
mensuration of the
"
common
in
is
of draw-
schools
be
common
too valuable to
step
grammar
in
the work
The
grades."
surfaces and solids should,
of course, never be a matter of
best text-books
the
study of the
geometric forms in the early years be neglected.
is
with
lower grades, in passing, that teachers
should insist that none of the ing which
out
many European and Ameri-
success in
arbitrary rule.
elementary arithmetic
development of the
at
Our
present
rules
for
all
necessary cases not involving irrational numbers.
A
give
satisfactory
pair of shears
and some cardboard enable the teacher
to pass from the rectangle to the parallelogram, and thence to the trapezoid and the triangle, developing
the formulae or rules with
little
difficulty.
Similarly
the formulae for the circle can be developed by cutting this figure into 1
sectors
which are approximately
Rebiere, Mathematiques et mathematiciens, p. 102.
244 THE TEACHING OF ELEMENTARY MATHEMATICS
Only a
triangles.
models of
pasteboard solids,
and
for filling
nish the
these,
labor
little
needed
to
common
prepare
geometric
together with a pail of dry sand
measuring such bodies.
in
comparing volumes,
1
Nor should we regard
this
method
of investigation
merely follows the
of
historic
first
acquired
line
development, the line in which truth
Comte
fur-
developing the formulae for
materials for
It
most
the
some of them
unscientific.
is
is
an interesting illustration by of this method, showing the way in which Galileo determined the ratio of the area of an ordinary induction.
cites
cycloid to that of the generating circle. try of his time
solution
of
was as yet
such
"
The geome-
insufficient for the rational
conceived
Galileo
problems.
idea of discovering that ratio
by a
the
direct experiment.
as exactly as possible two plates of
Having weighed same material and
the
of
equal
thickness,
one of
them having the form of a circle and the other that of the generated cycloid, he found the weight of the latter
always
triple
the
generating
circle,
a
result
is triple
agreeing
veritable solution subsequently obtained
1
For directions as to
this
metic, Boston, 1896, p. 66.
work
see
Beman and
that of
with
the
by Pascal and
Smith's Higher Arith-
Reference should also be made to a valuable
pamphlet by Professor Hanus, Geometry 1893.
whence he
that of the former;
inferred that the area of the cycloid
in the
Grammar
School, Boston,
WHAT Wallis."
more tive
1
would be
It
GEOMETRY
IS
we had even
well, indeed, if
of this induction along with our later demonstra-
One
geometry.
of the
common
especially in the discovery of loci
certain other problems,
make
245
is
sources of failure,
and the solution
correct inductions from carefully
drawn
figures.
Along with this work in mensuration should the
tinue
geometric
The
grades.
drawing
subject has been
siderable success
of
the inability of the pupil to
by several
the
in
con-
earlier
begun worked out with con-
writers. 2
Spencer's Inventional Geometry, while not an ideal
was a noteworthy
text-book,
step in this direction of
based upon accurately drawn figures. Dr. Shaw, speaking of his experiments with children
scientific induction
A
" few along the lines suggested by Spencer, says months' work proved the incalculable value of inven:
tional
years'
geometry
in a school course of study;
experience
in
many
classes
and
and eleven different
in
schools confirms that judgment. "
In these classes the pleasure experienced in the
study has
made
to taught; 1
p.
1
the
work
delightful both to teacher
and
and there has always been a continuous
Philosophy of Mathematics, English by Gillespie,
New
York, 1851,
86.
2
Spencer,
W.
G., Inventional
Geometry,
New
York, 1876
Erste Stufe des mathematischen Unterrichts, II. Abt.
;
Harms,
Oldenburg, 1878, along the same lines as a work by Gille (1854); Schuster, M., Aufgaben fiir den Anfangsunterricht in der Geometric, Program, Oldenburg, 1897. Campbell, Observational Geometry, New York, 1899. 3. Aufl.,
246 THE TEACHING OF ELEMENTARY MATHEMATICS
from the beginning to the end of the term. This pleasure and interest came, not from any environ-
interest
ment, not from the peculiar individuality of the
class,
but because the problems are so graded and stated that the pupil's progress becomes one of self-development
a realization of the highest law in education. "
The
.
.
.
pupil should not be told or shown, but thrown
back upon himself for, in inventional geometry, the knowledge is to be gained by growth and experience, through the powers he possesses and the method of ;
acquirement peculiar to his mind. pupil is
is
put to
not a its
little baffled,
the instructor
tive,
Occasionally the skill of
the teacher
best test to gain the solution without show-
ing or telling him. "
and the
Telling or showing
not the teacher.
.
.
is
the
method
of
.
Inventional geometry should precede the demonstraso as to give the pupil
upon when he takes up
concepts to draw
many
syllogistic demonstration.
De-
monstrative geometry then becomes an easier subject, and he is surer of what he is doing, because he has
more general notions Speaking
as a basis."
of Spencer's work, Mr. Langley, one of the
best teachers of elementary mathematics in England,
confirms the views already expressed
" :
It
has not been
usual for students, at any rate in schools, to approach the study of geometry in this experimental way, though there have probably always been individual teachers
who have used
it
to
varying extents.
Of
late years,
WHAT in fact since
however,
GEOMETRY
IS
more
the theory and practice
to
My
strongly advocated.
day by day
attention has been given
of education,
own
is
it
has been
me
the best method
students, though
able to dispense with
it
experience confirms
in the opinion that
majority of
for the
247
a few
may be
it.
"It has two advantages: (i) It leads to clear conceptions of the truths to be established (2) it may be ;
used to introduce the student naturally to a different method of establishing such truths the deductive method."
1
In America Professor Hanus has been prominent in putting the work on a practical basis. 2 He rec-
ommends two
recitation
seventh and eighth
per week
periods
grades, and one
for
the
for
the ninth,
The
the periods to be at least thirty minutes long. following are his guiding principles for teachers " i. Early instruction in geometry should be :
ob-
ject teaching.
The
"2.
make and keep an
pupil should
accurate
record of his observations, and of the definitions or
1
Langley, E. M.,
How
(London), Vol. VIII, O. of
list
German
2
Zeitschrift,
Outline of
Geometry, The Educational Review
The
subject
text-books, in Dressler's
schaftliche Unterricht
Hoffmann's
to learn
S., p. 3.
work
is
also discussed, with a brief
Der mathematisch-naturwissen-
an deutschen (Volksschullehrer-) Seminaren, XXIII. Jahrg., p. 18. in
Geometry
for the Seventh, Eighth,
in
and Ninth
Grades of the Cambridge Public Schools, Boston, 1893; Geometry in the
Grammar
School, Boston, 1893.
THE TEACHING OF ELEMENTARY MATHEMATICS
248
propositions which his objects has
In
"3.
developed.
work the
his
all
himself
to express
words, as
in
examination of the object or
fully
pupil
should be taught
by drawing, by construction, and and accurately as possible. The
accepted by the teacher should be the language of the science, and not a temporary
language
finally
phraseology to be set aside later. "4. The pupil is to convince himself of geometrical
truths primarily through
measurement, drawing, and superposition, not by a logical demBut gradually (especially during the last
construction, onstration.
year of the work) the pupil should be led to attempt the general demonstration of all the simpler propositions.
" 5.
bined
The
subject should be developed
teacher and
of
effort
and the pupil are
to
This
is
the teacher
best accomplished by
questioning without the use of a text-book con-
taining the definitions, sohttions, "6.
i.e.
cooperate in reconstructing the
subject for themselves. skilful
pupil,
by the com-
The
and
demonstrations.
.
.
.
subject-matter of each lesson should be
considered
in
occurrence
in
its
to
relation
nature and
life,
i.e.
the
actual
the structures of ma-
in
made by man of the geometrical forms studied, and the application of the propositions to the ordinary affairs of life should be the basis and the chines
outcome of every
exercise.
.
.
.
WHAT " tial in
In
249
a course extending through what " our " grammar school has been out-
Germany
corresponds
to
by several
Without going into details, suggested by Rein may serve
writers.
the following course to
GEOMETRY
Accuracy and neatness are absolutely essenall work done by the pupils." l
7.
lined
IS
show what ground the modern Herbartians
pro-
pose to cover. A. Geometric form (Geometrische Formenlehre). Fourth school year The cube, square prism, oblong prism, triangular prism, quadrangular pyramid. In addition to these solids the pupil considers the
measurement
point, straight line, surface, direction,
the straight
and
square
the right angle
line, its
construction,
construction,
the triangle,
the
and
of
parts, the
its
rectangle
and
and the diagonals
of
its
the
rectangle.
The hexagonal
Fifth school year
prism, hexagonal and
octagonal
prism, octagonal
pyramid, truncated
pyramid, cylinder, cone, truncated cone, and sphere.
The
following
plane
figures
are
studied
also
:
the
regular hexagon and octagon, the obtuse angle, the trapezoid and circle. B. Geometry.
Sixth school year
Properties of magnitudes (Eigen-
schaften, Gesetze, der Raumgrossen), constructions, and
mensuration. 1
Hanus.
Size
The
and measurement course
is
of
angles,
outlined in both pamphlets.
the
THE TEACHING OF ELEMENTARY MATHEMATICS
250
Kinds and properties
Division of angles.
protractor.
triangles and parallelograms, with constructions. Mensuration of surfaces, the square, rectangle, paral-
of
and
lelogram,
The
trapezoid.
and segments, and the value of
sectors
its
The
triangle.
circle,
Reg-
TT.
ular polygons.
Measurement and drawing
Seventh school year
of
solids.
C. Practical geometry.
Eighth school year tions.
2.
Similarity.
The congruence
i.
proposi-
Pythagorean theorem.
3.
cations to practical mensuration.
The
Demonstrative geometry
Appli-
x
next step brings the
student to demonstrative geometry, the geometry of or
Euclid,
once
its
Here the educator
equivalent.
confronted
by the
question,
When
is
shall
at
this
work be begun? In England Euclid prises
is
an age which surIn the lycees of France
begun
American educators.
at
and the Gymnasien (or Realschulen, etc.) of Germany, as well as in most of the other preparatory schools of
Europe,
demonstrative
Euclid, also finds
With us the
much
subject usually begins
eleventh school year,
in
the tenth or
and the "Committee
recommends no change 1
geometry, although not than in America.
earlier place
in
this
plan.
of
Ten"
To begin
a
Rein, Pickel and Scheller, Theorie und Praxis des Volksschulunter-
richts;
Das
vierte Schuljahr, 3. Aufl., Leipzig, 1892, p. 232.
WHAT work
GEOMETRY
IS
251
any earlier than this be sanctioned American teachers; the hardly by hard Euclidean method must change, or the subject of the difficulty of Euclid
will
must remain thus object to
seems
as
were,
late
cram the memory
to
for
the
in
curriculum.
be the case
in
an examination,
the
If
England, could be
it
But the considerable
attained here as easily as there.
personal experience of the writer, as well as the far
more extended researches
of
that as a valuable training in
him
convinces
others,
logic, as
a stimulus to
mathematical study, and as a foundation for future research, the study of Euclid as undertaken in England
is
not a success.
1
If
one has any doubt as to
should be removed by this recent judgment, testimony of Professor Minchin, a man thoroughly familiar with the system, and an excellent math-
this
it
ematician and teacher in spite of
the fact
that
he
was brought up on Euclid.
"Why,
then," he says, "is
he comes to the teaching with such rationality
great of
difficulties
human
beings
it
of
that the teacher,
Euclid,
that
confronted
is
belief
his
when
in
the
almost disappears with
the last vestiges of that good temper which he himself
1
once
possessed?
The
reason
simply
that
Holzmiiller, G., Notwendigkeit eines propadeutisch-mathematischen
Unterrichts in den Unterklassen hoherer
Lehranstalten vor
senschaftlich-systematischen, Hoffmann's Zeitschrift, 334-
is
XXVI.
dem
wis-
Jahrg., p. 321,
252 THE TEACHING OF ELEMENTARY MATHEMATICS
book
Euclid's
is
not suitable to the understanding of as regards both
young boys. It fails signally guage and its arrangement.
.
.
fess that, to the best of
my
its
lan-
For myself, I conhad been through
.
belief, I
the six books of Euclid without really understanding
the meaning of an angle"
1
however, a series of text-books should appear which carried the essential part of the first three If,
books of Euclid along with the arithmetic and algebra work of the seventh, eighth, and ninth school years,
thus
connecting the severe demonstrative ge-
ometry with that outlined for the lower grades, it would then be entirely feasible to begin demonstra-
We have, however, geometry earlier than now. no such books in English, at least none which have
tive
succeeded lent
in
any such
as
way
Holzmiiller's excel-
That a child in the has in Germany. 2 grade can handle the pons asinorum of
series
seventh Euclid
quite
as
easily
as
meets in arithmetic, has admit of
dispute.
But
the
been in
problems
he
often
shown too often
America we have
to
been
sporadic cases, without formushowing only lating a well-ordered scheme of work which should this
in
spread the geometry out, along with the algebra and It is reasonable to expect that this the arithmetic.
on
1
The School World (London),
2
In this connection the conclusion of Holzmiiller's
p.
251
is
of interest.
Vol.
I,
1899, p. 161. article
mentioned
WHAT
IS
GEOMETRY
plan will materialize before labors
skilful
"That
of text-books.
years, through the
many
some educated
of
writer
is
it
indispensable for
and coordination science loses
of
algebra, arithmetic,
try should be taught side by side ful;
253
a series
and geome-
not merely use-
is
maintaining
that
unity
mathematics, without which the
in
all interest
taken his arithmetic
A
and value.
first,
and then
his
boy who has algebra, and
then his geometry, has his mental powers less developed (J' esprit moins form^ than they would have
been with three or four years of intelligently pursued."
Naturally a child
parallel
teaching
1
loves
form
as
quite
much
as
Practically he needs number more often, and hence the elements of computation have been
number.
taught to him at an early age.
But when we come greatest com-
into the theoretical part of arithmetic
mon
divisor,
roots,
proportion, etc.
is
it
merely an has
accident (historically explainable) that education the
carried
child to the study of
number and
func-
tions rather than to the study of form.
Hence
in general
it
may be
said that the study of
demonstrative geometry might profitably begin earlier it does in the American schools, but that this
than
would
require, for the best results, a style of presen-
tation
modern
quite
different
from
that
of
Euclid or
followers. 1
Laisant,
La Mathematique,
p. 227.
his
254 THE TEACHING OF ELEMENTARY MATHEMATICS
The use as
it
stands
method
of
But taking the curriculum America at present, what general
of text-books
in
presentation shall be followed,
and what
The
kind of text-book shall be recommended?
great
majority of teachers take some text-book, require the pupils to prove the theorems substantially as therein set
forth,
and demand the demonstration
of
a con-
siderable number of propositions which the English " riders " a name quite as good (and bad) as call
our " original exercises."
The
fall into the habit of merely
result
is
a tendency to
memorizing the
solutions,
thus losing sight of the greatest value of the subject
the training which
To
it
gives in logic.
avoid this danger, numerous plans have been
One
devised.
is
dictating the
that of
propositions,
giving a few suggestions, and requiring the pupil to
work out open it
in
there
his
own
several
to
proofs.
This
a great waste of
usual sequence of propositions
have the
ability to
make
In the
is
condemn first
place
time in the dictation of
a return to medievalism.
notes
however,
objections so serious as to
the minds of most educators. is
plan,
is
Furthermore,
if
the
varied, few teachers
this variation without destroy-
ing something of the logic or symmetry of the subject
;
if
secures
the usual sequence
is
followed, the pupil simply
some text-book on geometry, often a poor one,
and memorizes from
Again, the pupil loses the advantage of having constantly before him a standard that.
WHAT
IS
GEOMETRY
of excellence in logic, in drawing,
of
work, and he
fails
to
to his subsequent progress in
of
the
first
and
in
arrangement
acquire the power to read
and assimilate mathematical
To meet
255
a serious bar
literature,
more advanced
lines.
of the above objections, the waste
time in dictation, text-books have been prepared
containing merely the definitions, postulates, axioms,
But while free from the
enunciations, etc. tion, they are
open
to the others,
first
objec-
and hence have met
with only slight favor.
Text-books have also been prepared which, in place of the proofs, submit series of questions, the answers to
which lead
heuristic
to
method
the
dead printed page for a substitution
is
This
demonstrations.
put into
book form
;
it
the
is
substitutes a
The
live intelligent teacher.
necessarily a poor one, for the printed
questions usually admit of but a single answer each,
and hence they merely disguise the usual formal proof. They give the proof, but they give no model of a logical statement.
The kind
of text-book
which the world has found
most usable, and probably rightly possesses these elements: (i)
A
so,
is
that which
sequence of proposi-
tions which is not only logical, but psychological not merely one which will work theoretically, but one in which the arrangement is adapted to the mind of the ;
pupil; (2) Exactness of statement, avoiding such slip-
shod expressions
as,
"A
circle is a
polygon of an
in-
256 THE TEACHING OF ELEMENTARY MATHEMATICS finite
number
of sides,"
"
Similar figures are those with
and equal angles," without other Proofs given in a form which shall
sides
proportional
explanation; (3)
be a model of excellence for the pupil to pattern after (4)
Abundant
tical
suggestions as to
(5) Propaedeutic cises,
;
exercises from the beginning, with prac-
inserted
work
methods of attacking them; form of questions or exer-
in the
long enough
before
concerned to demand thought
that
the
propositions
is,
not immedi-
ately preceding the author's proof.
Such a book gives the best opportunity for successful work at the hands of a good instructor. But no book can ever take the place of an enthusiastic, reIn the hands of a dull, mechanical, sourceful teacher. gradgrind person with a teacher's license, no book can be successful. The teacher who does not anticipate difficulties which would otherwise be discouraging to the pupil,
tempering these
removing them) by
On
best work.
who
develops,
does fault.
all
of
difficulties (but
skilful questions, is not
all
over-
difficulties,
who
the thinking for the class,
Youth takes
little
doing the
who
the other hand, the teacher
seeks to eliminate
not wholly
interest in that
is
equally at
which
offers
no opportunity for struggle, whether it be on the playground, in the home games of an evening, or in the classroom.
CHAPTER
XI
THE BASES OF GEOMETRY The bases
Geometry as a science starts from cerIt is hardly definitions, axioms, and postulates.
tain
work
the province of this cal
discussion
the
of
science rests,
first
a volume
require
A
pupils.
upon which the
foundations
because such a discussion would
some
of
practically the teacher
is
to
happens
1
size,
and
because
also
unable materially to change
axioms, and
the definitions,
book which
to enter into a philosophi-
postulates
be
in
the
the
of
hands
text-
of
his
brief consideration of these bases of the
science may, however, be of service.
The what
geometry occupy a position somefrom that held by the definitions of
definitions of
different
algebra
and
arithmetic.
There
is
not
the
necessity for exactness in the definition of 1
The
teacher
Cambridge, 1891
may ;
Cambridge, 1897;
consult Dixon, E. T.,
The Foundations
same
monomial of Geometry,
An Essay on the Foundations of Geometry, Poincare, On the Foundations of Geometry, The
Russell,
Monist, October, 1898
;
Hilbert, D., Grundlagen der Geometrie, in Fest-
schrift zur Feier der Enthiillung
Leipzig, 1899; Veronese, G.,
des Gauss- Weber-Denkmals in Gottingen,
Fondamenti
di Geometria,
Padova, 1891;
Koenigsberger, L., Fundamental Principles of Mathematics, Smithsonian Report, 1896, p. 93. s
257
258
THE TEACHING OF ELEMENTARY MATHEMATICS
as in that of right angle, for the latter
a control-
is
ling factor in several logical demonstrations, while the
In the same way more care must be shown in the definition of similar figures than in that
former
not.
is
of simultaneous equations, of isosceles triangle than of
incomplete quadratic, of parallelepiped than of binomial ;
not that
of these terms
all
must not be well under-
stood and properly used, and not that algebra
is
less
than geometry, but that the geometric terms enter into logical proofs in such way as to make their exact
exact statement a matter of greater moment.
Hence
made
Chapter VIII upon accuracy of definition in algebra, apply with even the suggestions, already
greater force
attend so
to
much
in
Nor should the teacher geometry. to the idea that all the truth cannot
be taught at once, as to acquire the dangerous habit of teaching partial truths only, or (as too often happens)
of teaching
mere words, sometimes unintelligible, someA few selections from our elemen-
times wholly false.
tary text-books will illustrate these points.
We
often
straight points."
line
Now
line,
shortest
the
in
first
a
definition,
"A
between
two
distance
place this
is
absurd, beis
measured
and usually on a curved one.
Further-
cause a line
on a
the
is
example, as
for
see,
is
not distance ;
distance
more, the statement merely gives one property of a straight line; it is a theorem, and by no means an
easy one to prove.
A
definition
should be stated in
THE BASES OF GEOMETRY
259
terms more simple than the term defined but distance is one of the most difficult of the elementary con1 to define. Mathematicians have cepts long since ;
"
abandoned the statement.
It is
a definition almost
universally discarded, and it represents one of the most remarkable examples of the persistence with
which an absurdity can perpetuate In the
centuries.
incomprehensible
first
to
through the
itself
place, the idea expressed
beginners, since
it
is
presupposes
the idea of the length of a curve; and further,
is
it
a
case of reasoning in a circle (c'est un cercle vicieux), the length of a curve
for
sum
the limit of a
as
not
is
it
finally,
a
of
definition
demonstrable proposition."
The ary;
hence tion.
fact
it
is,
can
only be
rectilinear at
all,
understood
lengths.
but
And a
rather
2
the concept straight line
is
element-
not capable of satisfactory definition, and
is
should be given merely some brief explanaFrom Plato's time to our own, attempts have it
been made to define such fundamental concepts as As straight line and angle, but with no success. 1
Pascal's
connection
:
rules
for
"
Do
(i)
definitions
are worthy of consideration in
not attempt to define any terms so well
this
known
in
themselves that you have no clearer terms by which to explain them (2) Admit no terms which are obscure or doubtful, without definition (3)
Employ
in definitions
only terms which
or which have already been explained."
mathematiciens, p. 23. 2
Laisant, p. 223.
are
perfectly well
Rebiere,
;
;
known
Mathematiques
et
260
THE TEACHING OF ELEMENTARY MATHEMATICS
" If you ask me what Augustine said of time, is, I cannot tell you; but if you do not ask me,
St. it
I
Pascal said of geometry " It be thought strange that geometry is unable to
know
may
And
too well."
:
define
any of
define
movement, or number, or space, and yet these
its
principal
are the very things which
not surprising, however,
admirable simple
makes
is
and
concepts,
of
that
considers
the
it
cannot
most.
It
consider that this
very
quality
which
them
objects renders
its
is
only to the most
itself
Hence
definition.
rather a
it
when we
attaches
these worthy of being
incapable fine
science
for
concepts;
the inability to de-
merit than a defect, since
it
arises
not from the obscurity of the concepts, but rather from their extreme evidence." 1
Text-books are also liable to err on the side
redundancy rectangle
is
in
definition,
a parallelogram
1
rectangle
all
of
statement,
of
"
A
whose angles are
would be thought absurd to say, a four-sided parallelogram whose op-
It
right angles."
"A
the
as in
is
Rebiere, Mathematiques et mathematicians, p. 16.
For those who
wish thoroughly to investigate the matter of the elementary definitions (straight line, angle, etc.), it will be of value to know that Schotten has
compiled all of the typical definitions of these concepts which have appeared from the time of the Greeks to the present, and has set them forth with critical notes in his valuable treatise, Inhalt und Methode des planimetrischen Unterrichts, Bd. Professor
Newcomb
to his Geometry.
I,
1890;
Bd.
II,
1893;
Bd. Ill, in press.
has also considered the matter briefly in the Appendix
THE BASES OF GEOMETRY and
posite sides are equal
angles
are
But
proper place,
called
a rectangle."
one of the
the
definition
" If say,
to
is
whose
of
all
the
manifest
given at the
one angle of a
a right angle, the parallelogram
is
parallelogram
if
suffices
it
and
because of
angles,"
right
redundancy.
parallel,
26 1
The same
criticism
applies to
common
"A
rec-
if
two
definitions of a square, " sides are all equal ; it suffices
tangle whose
adjacent sides
are
The
equal.
is
definition
commonly
given of similar figures is an illustration of the teaching of a half truth, the whole truth being thereby
permanently excluded, and If a student
two similar circles, or
all
this
with no excuse.
beginning geometry were asked to name figures, he would probably suggest two
two spheres, or two straight lines, or two But when he comes right.
squares, and he would be to the definition he finds
that, of
the four classes of
named, only the squares are similar. It is, however, an easy matter to define similar systems of figures
points,
and then
to
say,
"Two
figures
are
be similar when their systems of points are thus admitting
circles, spheres, similar
cones,
said
to
similar," etc.,
all
which are excluded by the usual text-book definiand all of which deserve to be considered. 1
of
tion,
The
introduction of the
and minima, 1
in
many
modern chapter on maxima makes
of our elementary works,
For further discussion see Beman and Smith's
Geometry, Boston, 1899, p. 182.
New
Plane and Solid
THE TEACHING OF ELEMENTARY MATHEMATICS
262 it
worth while
maximum
greatest value a variable can take,
as the is
to say that the definition of
misleading at the time, but also
is
not only conducive to sub-
sequent misunderstanding. Every teacher of geometry must be aware that, in general, a variable may
have several maxima.
The
laxness
mentary work polyhedral
of is
lines
angle
and
well illustrated in the
We
angle.
defined as "the
difference of
quite
the
as
direction
ele-
case of the
between two
definition because the
elementary as the
polyhedral
into
not unfrequently find angle
which meet" (a poor is
which creeps
definitions
angle
defined
word
direction),
"the
as
word
angle
more planes meeting in a point." formed by The absurdity appears when we substitute the defi" A polyhedral angle nition of angle for the word three or
:
is
'
the difference of direction between two lines which
meet' formed by three or more planes," we teach mathematics as an exact science tration is not a
far to find
"man
of straw";
etc., !
and yet
This
illus-
one need not look
it.
Axioms and postulates nature and the r61e of
In considering briefly the the axioms and postulates of
geometry, we may well begin by asking the meaning of the terms themselves.
Of course
it
is
true that these words
mean
to
any
generation just what the world at that time agrees
they shall mean, and hence
it
is
not a valid argu-
THE BASES OF GEOMETRY ment the
say that Euclid did not employ them in sense understood by his early English transto
lators.
At
number
of years, a
tions of
and
263
same time there
the
has
feeling that the
been, for a
common
defini-
postulate and axiom are absurd in statement
unscientific
in
Heiberg,
historically.
the Elements, oughly, and
is
as
thought, 1
well
as
unjustifiable
most scholarly editor
the
of
has considered the matter very thorconvinced that Euclid used axiom for
a general mathematical truth accepted without proof,
and postulate for
Thus the
nature. to
the
equals
to
in
ever,
axiom
is
similar
statement,
sums
are
truth
"If
equal,"
a
of
geometric
equals are added is
an axiom
but,
;
a given point but one line can be drawn
"Through parallel
a
a given line,"
Euclid's
is
a postulate (not,
The
language).
a "self-evident
notion
how-
that
an
theorem," and a postulate
a problem too simple for solution, torically incorrect, as well as
is
therefore his-
absurd in substance.
A
return to Euclid's use of the words would seem desirable,
although the single word axiom for both classes
would simplify matters. The definition of axiom as "a self-evident truth" has already been characterized as absurd.
For what
evident to one mind is not at all so to another. " be " self-evident to a very good student that
only number whose cube 1
is
i,
until
he
Euclidis elementa, Leipzig, 1883-88.
tries '
is self-
It I
is
may the
cubing
264 THE TEACHING OF ELEMENTARY MATHEMATICS 3; or that 2
JV
J
is
the only fourth root of 16,
some one suggests three others
until
theory of groups.
"axiom"
or that ab
must
he studies quaternions or the
ba, until
always equal
;
The
fact
is,
in
geometry the word
used merely to designate certain general is assumed. Our senses
is
statements the truth of which
seem
to indicate that they are true
false,
we
;
but whether true or
take them for granted and see whither they
lead us. Similarly, in geometry, with the
word "postulate."
A
postulate
is
a statement, referring to geometry, the truth
of which
is
assumed.
may be
false,
not be
it
homogeneous seems true, but it but we assume it true and see whither we is
So we may be able
are led. point,
;
true or
although our senses seem to indicate the
That space
former.
may
The statement may be
more than one
to draw,
line parallel to a
through a given
given
line,
although
our senses, especially as biassed by our early training,
seem to
to indicate not.
deny
logical
this or
But any one
is
entirely at liberty
any other postulate, and to build up a
geometry accordingly,
if
he can.
In the case of
the postulate of parallel lines this was done by Loba-
chevsky and Bolyai, and their geometries are entirely 1 Mathematicians generally agree that the postlogical. 1
For references,
p. 565.
The
Smith,
and Engel, Die Theorie der Leipzig, 1895.
D.
E.,
History of Modern Mathematics,
best historical treatment of the subject Parallellinien
is
von Euklid
that
by Stackel
bis
auf Gauss,
THE BASES OF GEOMETRY ulate
not at
is
"
all
self-evident."
"
As Klein, the wellAs mathematicians
known Gottingen professor, says, we must array ourselves as opponents that the parallel axiom
is
to
265
of Kant's idea
be considered an a priori
l
Lobachevsky and Bolyai postulate that through a given point more than one line can be drawn parallel
truth."
to a given line,
and on
axioms, postulates,
and
this,
together with most of the
definitions of Euclid, they build
up a perfectly consistent geometry. Similarly, as in
plane geometry
we
postulate that
space has three dimensions and that a plane figure
be revolved about an
axis,
may
through three-dimensional
space, so as to coincide with a symmetric figure, so in solid
geometry we might postulate that a
solid
may be
revolved through a four-dimensional space so as to coincide with a symmetric solid, e.g., a right-hand glove
with a left-hand one.
A
could be constructed with
A
postulate
ment;
it
is
is
perfectly consistent geometry this as a postulate. 2
not, therefore, a "self-evident" state-
a geometric assumption.
mentary geometry we
In ordinary
ele-
postulate only certain relations
which most people are willing
to say agree with their do not sense-perceptions. They entirely agree with them, for we have no sense-perception of a straight
1
Vergleichende Betrachtungen, Erlangen, 1872. For a brief and popular statement concerning the fourth dimension, see the recent translation of Schubert, H., Mathematical Essays and Rec2
reations, Chicago, 1898, p. 64.
266 THE TEACHING OF ELEMENTARY MATHEMATICS
a
nor,
line,
fortiori, of
concepts are 1
concepts.
two
Our geometric made from our physical
parallels.
abstractions
all
As
D' Alembert says,
are a kind of asymptote of
which they
limit
"
Geometric truths
physical truths,
indefinitely
the
i.e.,
approach without ever
exactly reaching."
As tion
to the
number
of postulates or axioms, the ques-
wholly unsettled.
is
Practically, the teacher of
the elements will follow those given in his text-book.
But as has been truly said, the list usually given is both insufficient and superabundant, since on the one
hand we use postulates not laid down in the ordinary text-books, and on the other hand we can demon-
some
strate
those which are given, so that
of
it
is
2
unnecessary to assume them. The most recent examination of the postulates of 3 and is here set rectilinear figures is that of Hilbert,
some
because of the high mathemati" In geomecal authority with which it comes to us. forth in
try
1
we Les
detail
consider
three
figures geometriques sont
different
systems of
de pures conceptions de
things.
1'esprit.
Com-
pagnon.
Les Mathematiques, etc., p. 21. He adds, " The axioms of geometry can be reduced to three, that of distance and its essential properties, that of the indefinite increase of distance, and that 2
De
Tilly, in Rebiere,
of unique parallelism." 8
Hilbert, D.,
Grundlagen der Geometrie, in the Gauss- Weber-DenkSee the author's review in The Educa-
mals Festschrift, Leipzig, 1899. tional Review, January, 1900.
THE BASES OF GEOMETRY The
things of the
nating them A, B,
system we c,
call
system we
first
C,
call
points, desig-
the things of the second
;
them
straight lines, designating
we
the things of the third system
;
267
designating them
a,
/3,
.
The
call
a,
b,
planes,
we may
points
the elements of linear geometry; the points and
call
lines
straight
the elements of
straight
points,
lines,
plane geometry; the planes the elements of
and
spatial geometry or of space.
"We
consider the points,
mutual
tain
relations,
by the words,
'lie,'
and planes
lines,
and we designate these
in cer-
relations
'between,' 'parallel,' 'congruent,'
continuous,' and the exact and complete description of these relations follows from the axioms of geometry. '
"These axioms separate certain
pressing
ness " I.
into five groups, each ex-
fundamental facts of our conscious-
:
"
Axioms
Two
i.
of connection (Verkniipfung).
different points, A, B, determine a straight
and we say that AB = a, or BA = a. 1 "2. Any two different points on a straight
line a,
termine that line;
B
is
not C, then
"
BC
if
a plane
a,
Any
"4.
a, 1
Of
AB = a
and
AC = a,
and
a.
Three non-collinear
3.
plane
i.e.,
line de-
points,
and we say that
A, B,
ABC =
C,
determine
.
three non-collinear points, A, B, C, of a
determine
a.
course the symbol "
=
" here
means
"
determines."
THE TEACHING OF ELEMENTARY MATHEMATICS
268 "
two
If
5.
a plane a, " If
A, B, of a straight line a then every point of a lies in a.
lie
points,
in
two planes, /3, have a point A in common, they have at least one other point B in common. " In every straight line there are at least two 7. 6.
points,
in
points,
and
,
'
the concept
"i.
If
at
three
least
non-collinear
in space at least four non-coplanar points.
Axioms
"II.
plane
every
of
arrangement (Anordnung), defining
between.'
C
A, B,
are three collinear points, and
B
lies
A and C, then B also lies between C and A. If A and C are two collinear points, there is
between "
2.
least
one point
D such that "3.
B between
C lies
between
Of any three
at
them, and at least one point
A
and D.
collinear points, there is
one which
uniquely between the other two.
lies
"
C D, can be so definitely arranged that B lies between A and C and also between A and D, and that C lies between A and 4.
four collinear points, A, B,
Any
y
D and also between B and D. "
5. Suppose A, B, C to be three non-collinear points, and a a straight line in the plane ABC, but not con-
taining A, B, or
C;
if
then, the straight line a passes
through a point within the line-segment also
BC 1
pass
AB,
it
through a point within the line-segment
or through a point within the line-segment These
must
five
axioms of Group II were
first
AC. 1
investigated by Pasch
(Vorlesungen uber neuere Geometric, Leipzig, 1882), and the especially due to him.
fifth
is
THE BASES OF GEOMETRY "
Axiom
III.
of
269
the denial
parallelism,
of
which
leads to the non-Euclidean geometry. "
Axioms
IV.
"i.
A B
If
and
A
line
a
1
t
is
a point on the same or another straight
it
is
B
from A' one unique point
ment
AB
A'B'.
.
.
(or
BA)
1
congruent to the line-segment
is
.
If a line-segment
"2.
on a given side of a' such that the line-seg-
possible to find
1
',
of congruence.
are two points on the straight line a t
AB
is
congruent to both
and A"B", then A'B' is also congruent " Let AB and BC be two segments 3.
common of
a\
points;
let
congruent to A'B', and
must follow that 4.
This
is
A"B".
of a, without
AC
is
points
BC
is
if
AB
congruent to B'
congruent
the axiom
then
;
to
A'C
r
C
"6.
>
it
angles corresponding to
for
to
3 for segments. If
for
two
AB=A'B' AC=A'C', 9
then must these also
CBA = angle
=
'
'
and A' B'C' these
for congruence),
angle
BAC = angle B'A'C\
exist,
C'B'A', angle
"V. Axiom of continuity Archimedes.
ABC
triangles,
congruences exist (using
angle
is
"
axiom 2 for segments. This is the axiom for angles corresponding 5. axiom
1
A'B' and B'C' be two segments
common
also without
to
A'B
ACB =
(Stetigkeit)
angle A'CB*. the axiom of
270 THE TEACHING OF ELEMENTARY MATHEMATICS "
A
Let
points that A%,
A
A l
,
and
to the
,
A2 A2
between
,
A
1
and
segments AA V A-^A^ must there be in the
then
equal;
A 2 A 3 A^ between A and A n
a point
,
,
and
also such that the
are
,
series
A
between
lies
etc.,
A ZA B
be any point on a between any given and B ; suppose A z A 8 A, so taken l
The
.
An
such that
B
lies
denial of this axiom leads
non-Archimedean geometry."
Hilbert inserts the necessary definitions for under-
standing these postulates (axioms), and adds numerous corollaries
the
showing
statements
but this
;
it
is
evident
the
of
not the place to enter this
Whether
field.
interesting
is
effect
far-reaching
or not his postulates are
that
or
openly they our elementary rectilinear geometry. Their consideration should convince the teacher that sufficient,
are
assumed
tacitly
in
the question of the postulates
is
by no means the
simple one which the text-books sometimes
make
us
feel.
Thus geometry agree
necessarily
world;
because
that it
is
is
exact, not
with
not of so
postulates
the
because facts
of
much moment.
definitely at
its
postulates
the
external
It is
exact
the outset certain
few statements concerning figures in space, and then applies logic to see what other statements can be deduced therefrom.
CHAPTER
XII
TYPICAL PARTS OF GEOMETRY
The introduction to demonstrative geometry may well be made independent of the text-book, unless the book offers some special preparatory work. If the pupils have not a reasonable knowledge of geometric drawing, a few days may profitably be devoted to this subProfessor Minchin has this to say of
ject exclusively.
the English schools, and the same
our
own
Euclid
" :
is
So
am
far as I
taught
in
is
almost as true of
able to learn
This
of rule, compasses, protractor, or scale. in accordance with the traditional cal
method
nates
which,
English
by
inquiry,
our schools without the aid
all
method
unfortunately,
and
education of
so
is
greatly
quite
quite
the classi-
domi-
conducive
to
the subject.
long-delayed knowledge " Now the use of the above simple instruments for
beginners in geometry is the first change that I advocate, whether we continue to teach from Euclid's all
book or from one proceeding on simpler and better lines. Well-drawn figures possess an enormous teaching power, not merely in geometry, but in
all
of mathematics and mathematical physics." 1
The Teaching
of Geometry,
The School World, 271
Vol.
I,
branches
1
p. 161 (1899).
THE TEACHING OF ELEMENTARY MATHEMATICS
2/2
Before undertaking the ordinary text-book demonstrations the teacher will also find it of great value to
few
a
offer
preliminary
theorems which
pave the
for the usual sequence of propositions, giving a
way
notion of what
is
meant by a
logical proof,
and
creat-
ing a habit of working out independent demonstrations.
The
following, for example,
way:
might be given
(i) All right angles are equal
(if
postulates the demonstrable fact of
in
this
the text-book
the equality of
(2) At a point in a given line not than one more perpendicular can be drawn to that not that one can be drawn, line in the same plane
straight angles);
as so
many
text-books affirm but
complements
of
fail to
tion concerning vertical angles,
to
little
work
first
"
The
book."
of this kind the pupil
is
understand the nature of a logical proof.
dence
(3)
;
and several others of
the simpler ones selected from the
After a
prove
equal angles are equal; the proposi-
to
will
prepared Indepen-
confidence in his
assert
itself, begin handle a proposition without a slavish dependence upon his text-book, while mere memorizing will
ability to
fail
to secure the usual foothold at the start.
These
two points may now be impressed (i) Every statement in a proof must be based upon a postulate, an axiom, a :
definition, or
(2)
No
some proposition previously considered;
statement
is
true
simply because
it
appears
With this preliminary from the figure to be true. treatment of a dozen or more simple propositions, and
TYPICAL PARTS OF GEOMETRY
273
with some instruction concerning geometric drawing,
may be undertaken
the text-book sequence less
of
danger
discouragement, of
much
with
slovenly work, of
groping in the dark, and of mere memorizing. The contest between the opponents of Symbols
all
symbols and the advocates of mathematical shorthand in geometry, as in other branches of the science, is In England Todhunter's Euclid
about over.
place to the Harpur, Hall and Stevens,
make
and others which
giving
McKay, Nixon,
use
extensive
is
of
symbols,
while in America Chauvenet's excellent work has had
more usable
place to less scholarly but
to give
text-
books.
In general one in the
book
is
in the
practically
bound by the symbols class. A few notes
hands of the
upon the subject may, however, be suggestive.
In
place, only generally recognized mathematical in a world-subject like symbols should have place mathematics, provincialism is especially to be con-
the
first
;
We may
would be a better sign of equality than =, but the world does not think so, and we have no right to set up a new sign language.
demned.
In this respect
it
think that
is
||
unfortunate
that
some
of
our
American writers should continue to use the provincial is
symbol for equivalence
difficult
among
to
(=o=),
not only because
make, but because
mathematicians.
it
has no standing
Indeed, the
tween equal and equivalent
is
so
it
distinction
be-
nearly obliterated
2/4 THE TEACHING OF ELEMENTARY MATHEMATICS language that many teachers now use the more exact term "congruent" for what some English writers call "identically equal," even though the textour
in
book
has the word "equal." The symbol for congruence (^), a combination of the symbols for similarity (~, an S laid on its side, from in
classes
their
similis)
and equality (=),
so
by the mathematical world more complete introduction in elementary work
meaning and
so full of
is
generally recognized
that is
is
its
desirable.
It is certainly
of novelty, for
it
not open to the objection
dates from Leibnitz, nor of the provin-
and want of significance which characterize the American symbol for equivalence. cialism
The modern symbols cial stage), identity
for limit (=,
in its provin-
still
(=), and non-equality (=), in addi-
tion to the ordinary algebraic signs, are also convenient.
There
is
much advantage
also
modern method
of
lettering triangles.
in
reading angles and
Among
the ancients,
lines,
80,
it
was a matter
of
and
of
when angles
were always considered as 1
the
following
less
little
than
moment
whether one should read the angle here illustrated AOB or BOA. But A
now number
of degrees, as
we recognize angles of any when we turn a screw through
that
it becomes 80, 270, 360, 450, necessary to two the in The the distinguish conjugate angles figure.
90,
1
,
TYPICAL PARTS OF GEOMETRY obtuse angle
therefore, read
is,
2/5
AOB, and
the reflex
BOA,
counter-clockwise. Pupils brought up to from the beginning have a great advantage in accuracy when they come to speak of figures which
angle
this plan
are at
all
tive
The
complicated.
of positive angles
ones
is
counter-clockwise reading
and the clockwise reading
of nega-
also very helpful in the generalization of
propositions in the earlier books.
great advantage to recognize, before
of
It is also
the pupil has gone too
far,
the distinction between the
segments AB and BA. Negative magnitudes can no longer be kept from elementary geometry, say what line
we may about pure form and ment
of the subject.
magnitudes
of
Pupils understand the negative
algebra
then
to geometry, thus
knowledge and interesting? correlation
is
By
the non-algebraic treat-
so
not
why
opening
doing,
a
apply this both new
fields
mutually
helpful
established between algebra and geom-
a correlation always recognized vanced portions of the science. etry,
The advantage ABC, XYZ, ,
of in
in the
more
ad-
uniformity in lettering triangles counter-clockwise order,
and of
lettering the sides opposite A, B, C, respectively,
a, b, c
(and so for x, y, z, etc.), is apparent to all who have accustomed themselves to the arrangement. There is an interesting line of Reciprocal theorems propositions, early
met by the
pupil, in
which one theo-
276 THE TEACHING OF ELEMENTARY MATHEMATICS
rem may be formed from another by simply replacing the words
point by line
by
line,
point,
angles of a triangle by (opposite) sides of a triangle, sides of a triangle
This
is
seen in the following propositions
two
If
by (opposite) angles of a
triangles
have
two sides and the included
:
triangles
have
two angles and the included side of the one respec-
angle of the one respec-
two sides
tively equal to
two
If
triangle.
tively equal to
two angles
and the included angle of
and the included side of
the other, the triangles are
the other, the triangles are
congruent.
congruent.
two sides of a triangle
If
site
those sides are equal.
Of course
the teacher
as most text-books
ship,
there
is
great advantage
early in the course, for
two angles of a triangle
If
are equal, the sides oppo-
are equal, the angles oppo-
those angles are equal.
site
pass over this relation-
may
do, without
two reasons
to the pupil's interest to see this ject, to
and
But
:
(i) It
adds greatly
symmetry
of the sub-
note that certain propositions have a dual;
(2) It often
investigation ery.
comment.
in recognizing the parallelism
This
a triangle
is
suggests
new
possible theorems
the pupil has the interest of
seen in the following simple exercise
ABC, where
a =
b,
for
discov:
In
the bisector of angle C,
TYPICAL PARTS OF GEOMETRY
produced
to
c,
bisects side
c.
The
2/7
who
pupil
is
led to
discover the reciprocal theorem, and to investigate
its
validity (for reciprocal statements are not always true), has opened before him a field of perpetual interest, a
an independent worker. Converse theorems are often thought uninteresting.
field in
which he
is
Students get the idea that the converses are always true, and that it is a stupid waste of time to prove them. And yet, so necessary are these propositions to the
sequence of geometry, that they have an imporIn arranging to present the subject to a tant place. logical
class,
the teacher
is
met by three questions: (i) What ? (2) Are converses always true ?
are converse theorems (3)
How
Two
are converse theorems best proved
theorems are said
when what
other,
proved
is
in the other,
triangle
and
angle
are converses, and each " In
should read,
then a
what
is
= b"
vice versa.
triangle
is
A=
true
ABC,
;
A=
is
to
be
but
angle B," and,
B
angle
if all
then a
by
"
the second one
if
the angles are equal
the two would not be converses, although
given in the
first
(a
class should
what
is
to
element
is
wanting.
b) is
in the second, for the vice versa
The
what
is
For example, "In
b then angle if
?
be converse, each of the
given in the one
a
ABC, "In triangle ABC, if
to
be proved
be made aware of numerous false
converses, that the necessity for proof
may be
appreci-
For example, "All right angles are equal angles," " If a triangle contains a right angle it is not an equi-
ated.
2/8
THE TEACHING OF ELEMENTARY MATHEMATICS "
lateral
triangle,"
product
two numbers are
If
composite," are
is
all
prime their
true statements, but their
converses are not.
There are so many converses teacher will find
it
be proved that the both as to time and advantageous,
Law
logic, to consider the
At
the course.
to
of Converse rather early in
the expense of one or two lessons
given to the understanding of the law, the time should
be spared, since is
as follows
Whenever
If
it
2. If it
will
be amply repaid
three theorems
must
have the following
If
has been proved that when A>B, then has been proved that when A=B, then
than
it
X>
relations,
be true:
has been proved that when then the converse of each is true. 3. If
The law
later.
:
their converses 1.
it
A
X> Y, and
X
then
F,
and
X< Y,
For
A
can neither be equal to nor less without violating 2 or 3; .*. A>B, which
B
Y, then
proves the converse of i. If Y, then A can neither be greater nor less
X=
than
B
without violating
proves the converse of If
X<
equal to
Y,
B
then
A
i
or
3
. '.
;
A = B,
which
2.
can neither be greater than nor
without violating
proves the converse of
i
or 2
;
.*.
A < B,
which
3.
This law, proved once for such of the converses as
all,
we need
enables us to prove in
elementary geom-
TYPICAL PARTS OF GEOMETRY
279
etry without using the tedious demonstration of Euclid
with every case.
proved if
For example, as soon as
that, in triangle
A>B
a>b
then
ABC,
A =B
if
then a
b
(which, by mere change of
the figure, also proves that
in
this
has been
it
if
A
then a
law shows that the three converses are
and
y
letters
<
#),
true.
Should any teacher feel that this is too difficult for beginners, it should be noticed that the proof is identical
with that usually given, but
set forth for
subsequent use, and
here merely given a name. is
it
is
Until recently elementary geometry seemed afraid to consider a reflex angle, or Generalization of figures
a concave polygon, or an equilateral triangle as a special case of an isosceles triangle, to say nothing of a cross polygon, or a cylinder with a non-circular directrix,
or
But our best
a negative line-segment.
recent works have presented these and other ideas in such a simple fashion
troduction cannot long be
a matter of the text-book;
make much or little of of the work adds more inality, or better
Take the angles
of
lies
an
It
not at
is
all
with the teacher to
and scarcely any feature it, interest, develops more orig-
paves the way for future progress.
familiar theorem that the
n-gon
stated, of course, in less
that their general in-
delayed. it
modern
circumlocution.
equals
n
sum 2
of the interior
straight
various ways and with
After
the simple convex figure,
it
has been
the teacher
angles,
more or
proved for
may
ask
if it
28O THE TEACHING OF ELEMENTARY MATHEMATICS is
case
true in
one
becomes
angle
reflex
he
;
may
then move the vertex back until the angle becomes
and ask the same question. Students have with such questions, and they readily
straight,
no
trouble
follow a
teacher
consideration
the
to
of
the
cross
polygon, a case best presented by moving the vertex of a marked angle through one of the opposite sides.
The
case of the
polygon
sum
of
also a valuable
is
the exterior angles of a
one for beginners.
If the
student will letter the angles for the ordinary convex
polygon, and keep the same lettering when it becomes concave or cross, he will find that the proof is the same for all
When
cases.
the angle
AOB,
for
example
(always read counter-clockwise), becomes BOA, it is to be considered negative, but otherwise the proof is quite Indeed, the one (practically unvarying)
unchanged. principle to
be given the student
ple figure properly, keeping the
formations, and the proof
The
principle
this
same
:
Letter the sim-
letters in all trans-
be the same for
It
all cases.
well illustrated in the case of the
the side opposite an obtuse
square on triangle.
is
will
is
equals the
sum
of
angle of
a
the squares on the
As
other sides plus twice a certain rectangle.
the angle
becomes less obtuse this rectangle becomes smaller; if the angle becomes right, this rectangle vanishes and the theorem becomes the Pythagorean
becomes
;
if
the angle
acute, a certain projection becomes negative,
making the rectangle
negative,
and instead of having
TYPICAL PARTS OF GEOMETRY
28 1
plus twice a certain rectangle we have minus twice that rectangle, the proposition becoming the one con1 cerning the square on the side opposite an acute angle.
This
of
generalization
figures
typical
materially
For example, the geometry. measure the of an inscribed propositions concerning angle, an angle formed by a tangent and a chord, an lessens
the
of
detail
angle formed by two chords, or two secants, or a secant
and a tangent, or two tangents, are all special cases It would be unwise to give this of a single theorem. general theorem
but after considering the cases of an inscribed angle, and the angle formed by a chord first,
and tangent, classes have no trouble in taking the general case and in so transforming the figure as easily to get the special cases from
couple of
it.
The
proof has only a
most general form, and it is a make special theorems for each of the
steps in the
waste of time to
various' simple cases.
The
proposition
concerning the "product" of the
segments of two intersecting chords, or secants, is also one which is often extended through three or four theorems.
It
general case.
requires
only two steps to prove the
If a pencil of lines cuts a circumference,
the rectangle (product) of the two segments from the 1
Upon
this set of theorems,
report of the sub-committee
however, the teacher should read the
on mathematics
in the
Report of the Com-
mittee of Ten, Bulletin No. 205 of the U. S. Bureau of Education, p. 113.,
The
position there taken
is,
however, open to very serious question.
THE TEACHING OF ELEMENTARY MATHEMATICS
282
vertex
is
constant whichever line
theorem four or
five others
is
come
From
taken.
this
as special cases
simply transforming the figure slightly.
by
The time has
" surely passed for fearing so valuable a phrase as pencil
of lines."
These few
illustrations suffice to
tary geometry offers a
field,
show that elemen-
pupils alike, for simple generalizations. lies
on the one side
in
and
interesting to teachers
The danger
always attempting to give the
general before the particular (a fatal error), and on the other in cutting out all of the interest which comes
from generalization, thus falling into the old humdrum of multiplying theorems to fit all special cases. Most of our elementary works devote Loci of points
some space
a few simple loci of
to the treatment of
points, the reciprocal subject of "sets of lines" being
generally regarded as hardly worth considering at this stage little
of
the
student's
progress.
The
subject
is
of
or of great value, depending on the use subse-
quently
made
of
it.
A
few of our recent text-books
have carefully explained the term "locus," and have given
proofs
satisfactory
majority
fail in
two
of
the
particulars,
but
the
to these a
few
theorems,
and as
words may be of value.
To line
say that the locus of points (in a plane) is the containing those points, is entirely inadequate,
for this line
may
may
consist of
contain
other points, or the locus
two or more
lines,
or of a line
and a
TYPICAL PARTS OF GEOMETRY point (as in the
locus
of
a point r distant from a
Perhaps the best plan
circumference).
283
is
to fall
on the etymology of locus (Lat. place) and of
place
all
points
satisfying
a
given
say,
back
The
condition
is
called the locus of points satisfying that condition
giving further explanation by means of illustration. But the most serious error usually found is in the " In proving a theorem concerning the locus of
proof.
necessary and sufficient to prove two things satisfies the (i) That any point on the supposed locus the not on condition (2) That any point supposed locus points
it is
:
;
For
does not satisfy the condition.
if
only the
some other
point were proved, there might be
first
line in
only the second were proved, the suptext-book posed locus might not be the correct one." discarded. be should which fails in these points the locus
;
and
if
A
Methods
of
There
attack
is
a
certain
value
in
turning a pupil into a chemical laboratory, after he has seen some experiments performed, and there telling
him
to discover
something new, or to find the He will fail, but
atomic weight of
some substance.
the attempt
serve to broaden his
some value
also of
ing him salt,
who would attack,
hand him a few
to
ideas. crystals,
It
is
tell-
to prove that they are this or that kind of
leaving
would
may
him do
to his this
own
with
devices.
But the teacher
elementary
students,
who
no general directions as to methods of who would allow a student to wander aim-
offer
284 THE TEACHING OF ELEMENTARY MATHEMATICS groping blindly and wasting his energies attempts, would be looked upon as a failure.
lessly about, in futile
And
yet this
is
about what
we
usually find in a class
geometry; students are turned loose
in
of
and are
exercises,
told
to
invent
among a mass new proofs, to
new theorems, to solve problems and prove theorems entirely new to them. Their only hint is that given find
by the demonstration
some recent proposition their to draw the upon the proof
of
only hope, to stumble
;
prize ticket in the lottery without too great delay.
Mathematicians do not proceed in any such way; they call to their assistance all the general methods possible,
be
a
and
to the teacher of
lesson.
The
least to the pupil
interesting
discovery
and probably
application
"
of
Thus
already mentioned. "
if
geometry
at
to the teacher, is
an
the
law
a student
(If the opposite
mystic hexagram
should
this
theorems, new
of
of
reciprocity
knows
Pascal's
an
sides of
in-
hexagon intersect, they determine three collinear points), it is but a step to rediscover, in the scribed
same way that
it
was
originally
found, Brianchon's
well-known theorem. 1 1
The
teacher will find this theory worked out fully in Henrici and
Treutlein's
Lehrbuch der Elementar-Geometrie, Leipzig, 1881,
3. Aufl.,
one of the most suggestive works on the subject. An excellent little handbook which deserves a place in the library of every teacher of elementary mathematics is Henrici's Elementary Geometry, Congruent 1897,
Figures,
out quite
London, 1879, fully.
a
work
in
which the reciprocity idea
is
brought
TYPICAL PARTS OF GEOMETRY
But
is
it
methods of attack
to
exercises that
is
it
in the
285
treatment of
desired to direct especial attention.
This subject has received much consideration at the hands of Petersen, 1 Rouche" and De Comberousse, 2 and 3 Hadamard, and the following suggestions are largely from their works. 4 1.
In attacking a theorem take the most general
figure
if
E.g.>
possible.
a theorem relates to a
tri-
angle, draw a scalene lateral
triangle; one which is equior isosceles often deceives the eye and leads
away from the demonstration. 2.
Draw
all
An On
figures as accurately as possible.
accurate figure often suggests a demonstration. the other hand, the student
who
relies too
the
the accuracy of the figure in liable to 3.
much upon
demonstration
be deceived.
Be
sure that
what
is
given and what
is
proved are clearly stated with reference to the the figure.
of
is
Neglect
in
this
respect
is
to
be
letters
a fruitful
cause of failure. 4.
Then begin by assuming
the theorem true; see
what follows from that assumption; then see 1
if
this
Methods and Theories of Elementary Geometry, London and Copen-
hagen, 1879. 2
Traite de Geometric, 6 ed., Paris, 1891.
3
Lemons de Geometric elementaire,
Paris, 1898.
4
The immediate
Beman and
source
is,
however,
Smith's
New Plane
Solid Geometry, Boston, 1899, p. 35, 152, to which reference further details.
is
made
and for
286 THE TEACHING OF ELEMENTARY MATHEMATICS can be proved true without the assumption
;
if
so, try
to reverse the process.
Or begin by assuming
5.
the
theorem
show the absurdity
endeavor to
of
false,
the
and
assumption
ad absurdum).
(reductio
To
secure a clearer understanding of the propo-
sition to
be proved it is often well to follow Pascal's and "substitute the definition in place of
6.
advice,
name
the 7.
method
solution
of
a
problem
the
Assume
success.
what
results
the
of analysis suggested in 4, above, will usually
lead to sider
of the thing defined."
In attempting
results follow,
a
until
known
the problem solved, conand continue to trace these
proposition
is
reached
;
then
seek to reverse the process. 8.
One
problems long as its
of is
it is
position
known tion
the most fruitful methods of attacking
by means of the intersection of loci. known merely that a point is on one not definitely determined;
is
that the point
may (and
is
both
if
also on another lines
are
but
if
So line, it
is
line, its posi-
straight
must) be
For example, if it is known uniquely determined. is on a a certain that straight line and a certain point circumference, intersection.
distant
it
may be
Thus,
from two
either of the
two points of
in a plane, to find a point equally
fixed points,
A, B, and also equally
from two fixed intersecting lines, locus of points equidistant from A and B distant
x, is
y;
the
the per-
TYPICAL PARTS OF GEOMETRY
AB
pendicular bisector of
from x and
distant
y
angles xy and yx\
the locus of points equi-
;
the pair of lines bisecting the
is
in
since,
the
general,
cut the other two in two
will
287
line
first
points, both of these
points answer the conditions. Petersen gives numerous other methods, but the above suggestions answer very well for all cases the
student will meet in elementary geometry.
Ratio
chapter there
and
and proportion
In
treatment
the
we have two extremes the
is
Euclidean,
to
logical
the
purely
extreme.
of
method.
of
scientific
geometric,
It
this
First
because of this
is
treatment that English teachers sometimes argue the
more strongly
for
Euclid
although in practice they
The
never use the chapter!
the Association for the
of
Geometrical
fact:
"The
have the same
first
of
the
first
of
and
altogether
Improvement to
ratios
to
read the
be assured of
four magnitudes
is
when any third
first
be
being taken,
less
the multiple of the third
fourth
:
or,
if
and any equi-
is
if
the
than that of the second, also less than that of the
the multiple of the
that of the second, the
said to
equimultiples whatsoever
multiples whatsoever of the second and fourth;
multiple of the
of
the second, which the third
ratio to
has to the fourth,
is
it
One has but
Teaching.
Euclidean definition of equal this
is,
even as modified by the
too difficult for beginners, syllabus
fact
multiple
of
first
be equal to
the third
is
also
288
THE TEACHING OF ELEMENTARY MATHEMATICS
or, if the multiple of equal to that of the fourth the first be greater than that of the second, the ;
multiple of the third fourth."
The
is
also greater than that of the
is
the purely algebraic plan, the
1
other extreme
one adopted by most American text-book plan entirely non-geometric, unscientific,
writers,
a
a break in
the logic of geometry, but so easy that neither teacher
nor pupil need do much serious thinking to master it. Occasionally a writer inserts a proposition at the end of the chapter, intending to bridge the
algebra and geometry, but
chasm between
rarely creates
it
upon the student. Between these extremes, the
any im-
pression
strictly
scientific
and
the strictly unscientific, the too difficult and the too easy,
the
and the
trivial,
there
and usable mean. is
and the
geometric
It
is
at
algebraic,
least
one
the
serious
fairly scientific
consists in proving that there
a one-to-one correspondence between algebra and
geometry, with this relationship
Algebra.
Geometry.
A
A line-segment. The
rectangle
:
number.
The product
of two line-
of two numbers,
segments.
This
having
been
made
a
matter
of
proof,
further postulated that any geometric magnitude 1
Blak clock's Simson's Euclid, London, 1856.
it
is
may
TYPICAL PARTS OF GEOMETRY be represented
and
tions
With these assump-
by a number. laws
proofs, the
289
of
may be as may
proportion
proved either by algebra or by geometry, be the most convenient. The first proposition, stated in dual form,
would then read
four numbers are in
If
proportion, the product of
the
means equals the prod-
uct of the extremes.
The impossible
:
If four lines are in pro-
the
portion,
the
of
rectangle
means equals the
rec-
tangle of the extremes.
in
While
geometry
does
it
not
enter into the province of the teacher to require the
attempt the
to
pupil
the
impossible, at
same time
the questions of the limits of the possible frequently
even in plane geometry.
arise
To
say that nothing
is
pleasant sounding epigram, and possible,
thing,
it
given
bility.
the
true.
is
impossible
infinite
if
It
power,
is
compasses is
impossible.
possible, but
if
one
passes
hyperbola, sible
is
true.
if the u
or
it
is
its
is
possi-
;
To draw
a
with
straight-
To draw
the
circle
a straight line
is
limited to the use of the com-
becomes impossible. cissoid,
that
do any particular
limitations are imposed,
possible
it
it
means
it
to
one has the means to insure
epigram ceases to be
edge only,
if
make a
to
merely asserts that nothing
But the moment that
with the
is
impossible,
conchoid,
To draw an all
necessary instruments
ellipse,
these are
are
pos-
allowed, but
2QO THE TEACHING OF ELEMENTARY MATHEMATICS
and
they are impossible with simply the compasses straight-edge.
From remote
antiquity
men have
tried
to
trisect
an angle, a problem simple enough if the necessary instruments are allowed, but one well known by mathematicians to have been proved to be impossible by the use of compasses and straight-edge alone. It
is
not that the world has not yet solved
cause, like the fact that
might sometimes yield to proof but 1 been proved that it cannot be solved. already
n>2,
it
;
the
Similarly
problem
equal to a given
one
enough been proved if
constructing
"squaring the
circle,
may
of
a
by the
use
has
it
square
circle," is
use a certain curve, but
be impossible
to
be-
it,
n equal z for
xn + yn cannot
it
easy has
of
the
In the same
instruments of elementary geometry. category belong the problems of the duplication of the cube, and the construction of the regular hepta-
gon.
The world and
duplicators,
is
full
of
circle-squarers,
angle-trisectors, simply
and cube-
because these
elementary historic facts are unknown. Euclid paid little attention to solid Solid geometry geometry, with the result that his followers in the English schools have also neglected it. Since the conservative Eastern states have always been influenced 1
Upon
accessible
this
and other problems mentioned
work
for teachers
Geometry, English, by
is
Klein's
Beman and
in this connection, the
Famous Problems
Smith, Boston, 1896.
by
most
of Elementary
TYPICAL PARTS OF GEOMETRY
2QI
the educational traditions of England, solid geometry has
never had the hold in the preparatory schools that it has in the Central and Western states, where tradition counts for
The argument on the one we cannot teach
less.
side
In the time at our disposal
geometry, to say nothing of the solid plane geometry could ever be taught!
on the other side degree
;
is
this
:
The whole
all
as
is
this
:
of plane all
if
of
The argument
question
is
one of
with a year at the teacher's disposal, he would to teach plane geometry about two-thirds of
do better
the time, and solid geometry one-third
;
this
would give
mental training at least equally valuable, which is the first consideration, it would add to the pupil's interest,
and
it
would contribute
to the practical
side through
the added knowledge of mensuration.
The
effort
has several times been made to work out a
geometry along side by The scheme has a number of
feasible plan for carrying solid
side with the plane.
advantages.
1
It is interesting,
for example, to pass a
plane through certain solids (to slice into them, so to speak), and get figures of plane geometry out of them. It is also interesting to
note the one-to-one correspond-
ence between the spherical triangle, the trihedral angle,
and the plane scheme is quite It is
triangle.
But while,
theoretically, this
it
has few followers.
feasible, practically
contrary not only to the historical development of it makes the com-
the science, but also to psychology 1
;
E.g., Paolis, R. de, Elementt di Geometria, Torino, 1884.
2Q2
THE TEACHING OF ELEMENTARY MATHEMATICS
plex contemporary with the simple, the general with the particular,
from the very
ever, to see
how
It is interesting,
first.
skilfully the
how-
Italian writers are han-
dling the matter. Practically,
it
has been found best to take up the
demonstrative solid geometry after a course in plane geometry has been completed. The subject then offers few difficulties to most students a little patience at the ;
outset, a few simple pasteboard models, easily
the class, care in drawing the out the perspective,
first
made by
figures so as to bring
these are the considerations nec-
essary in beginning work in the geometry of three
dimensions.
Models, preferably to be
made by
the
student, are crutches to be used until the beginner can
walk, then to be discarded.
To keep them,
to
have
special ones for every proposition, even to have their photographs, is to take away one of the very things the imagination, the power of we wish to cultivate,
imaging the eral, is
solids,
the power of abstraction.
the appeal to models should be
necessary to return to the crutch
In gen-
made only when the
as
it
pupil
falters.
The same
is
true of the spherical blackboard;
it
is
valuable and should be used in every school, especially in the consideration of polar
but never to depart from
and symmetric triangles
;
it in spherical geometry, or never to take up a theorem without a photograph of the sphere, is wholly unwarranted by necessity or by
TYPICAL PARTS OF GEOMETRY
demands
the
The
of education.
student needs to
abstractions, to get along with a figure
and its
to
293
drawn on a
make plane,
be able to work independent of the sphere or
photograph. teacher will do well to add to the treatment
The
some
usually given
little
which we are indebted able saving
discussion of recent features for to the
A
Germans.
consider-
effected in "producing" lines, planes,
is
and
curved surfaces, in treating prisms, pyramids, cylinders,
and cones, by the introduction of the notion of prismatic, pyramidal, cylindrical, and conical surfaces and spaces.
The concepts
number
of
are simple, and
proofs are
by
their use a
The
considerably shortened.
prismatoid formula, introduced by a German, E. F. August, in 1849, should also have place on account of its
great value in mensuration.
states that in the case of
e
edges, v
vertices,
and
Euler's theorem, which
a convex polyhedron with
f
faces, e
-f-
2
=f+ v,
also
deserves place, both for the reasoning involved and its
interesting
are
additions use,
and
application easily
to
made, whatever text-book
teachers will find
them
of great value.
objection on the score of difficulty
The
These
crystallography.
is
is
in
The
groundless.
correspondence between the polyhedral angle and the spherical polygon should also be noted, a correspondence not always sufficiently promone-to-one
inent in our text-books. as follows
:
This relation
may be
set forth
294 THE TEACHING OF ELEMENTARY MATHEMATICS " Since the dihedral angles of the polyhedral angles
have the same numerical measures as the angles of the spherical polygons, and the face angles of the former have the same numerical measure as the sides of the evident that to each property of a polyhedral
latter, it is
angle corresponds a reciprocal property of a spherical This relation appears by polygon, and vice versa. making the following substitutions:
Polyhedral Angles.
Spherical Polygons.
a. Vertex.
a.
Centre of Sphere.
b.
b.
Vertices of Polygon. Angles of Polygon.
Edges. c. Dihedral Angles. d. Face Angles.
c.
d. Sides.
" In addition to the correspondence
between polyhe-
and spherical polygons, it will be observed that a relation exists between a straight line in a plane dral angles
and a
great-circle arc
on a sphere.
Thus, to a plane
triangle corresponds a spherical triangle, to a straight line perpendicular to a straight line circle arc
corresponds a great-
perpendicular to a great-circle arc, etc."
It
be mentioned, in passing, that the word " arc " may is always understood to mean "great-circle arc/' in the also
geometry of the sphere, unless the contrary
A
further relationship of
interest
is stated.
in the
study of the circle
geometry is that existing between and the sphere, and illustrated in the following ments solid
:
state-
TYPICAL PARTS OF GEOMETRY
"The
A
Circle.
The Sphere.
,
portion of a line cut off by is a chord.
a circumference
The its
295
greater a chord, the less
distance from the centre.
A portion of a plane cut off by a spherical surface is a circle. The greater a #>r/
its
A
A
diameter (great chord) bisects the circle and the circum-
sphere and the spherical sur-
ference.
face.
Two great
Two
diameters (great chords) bisect each other.
Hence may be
line,
4.
',
2.
circumference, 5. diameter.
chord,
The advantage ence
the
each
circles bisect
on the sphere,
by making the following
substi-
:
Circle
i.
3.
circle,
bisects
other.
anticipated a line of theorems
derived from those on the tutions
circle
great
is
evident
\.
3.
Sphere,?., spherical surface,
plane, ^. circle,^, great circle."
in noticing this one-to-one correspond-
if
we
consider
some
of the theorems.
In the following, for example, a single proof suffices for
two propositions
:
If a trihedral angle has
If a spherical triangle
two angles equal
two dihedral angles equal to each other, the opposite
other,
face angles are equal.
are equal.
The
generalization of
the
figures already
has
to
each
opposite
sides
mentioned in
speaking of plane geometry here admits of even more extended use. It is entirely safe to take up the mensuration of the volume or the lateral area of the frus-
tum
of
a right pyramid, and then
let
the upper base
shrink to zero, thus getting the case of the pyramid
296 THE TEACHING OF ELEMENTARY MATHEMATICS as a corollary, or let
it
increase until
base, thus getting the case of
it
equals the lower
the prism;
the prism
would, however, naturally precede the frustum. So for the frustum of the right circular cone, and the cone and
method not only valuable from the considertime, but also for the idea which it gives of the
cylinder, a
ation of
transformation of figures.
Most
of these suggestions can be used to advantage
with any text-book.
Some
by many teachers, and
it
is
are doubtless used already
hoped
all
may be
of value.
CHAPTER
XIII
THE TEACHER'S BOOK-SHELF Although in this work considerable attention has already been paid to the bibliography of the subject, a few suggestions as to forming the nucleus of a library
value.
upon the teaching of mathematics may be of has been the author's privilege, after lecturing
It
before various educational gatherings, to reply to letters
asking
for
advice in this
many
matter, and so he
many among the younger generawho will welcome a few suggestions
feels that there are
tion
of teachers
in this line.
In the
first
place, the accumulation of a large
ber of elementary text-books inspiration
which the
is
teacher
of
little
desires
is
value.
num-
The
not to be
found in such a library such inspiration comes rather from a few masterpieces. Twenty good books are ;
worth far more than ten times that number of text-books.
in
ordi-
a teacher will
Hence, general, nary do well never to buy a book of the grade which he is using with his class; let the book be one which
urge him forward, not one which shall make him satisfied with the mediocre.
shall
297
THE TEACHING OF ELEMENTARY MATHEMATICS
298
Since an increasing number of teachers, especially in our high schools, have some knowledge of German
and would be glad
or French,
that knowledge
The
German,
in
progress
illustrating
of
should be
although
best works, as a whole,
branches,
are
excellent
works
special
Italian.
The
particular
some
are to be found in
it
branches of mathe-
of attacking the various
matics are in French.
lines
do
so, encouraged works which we have upon general
said that the best
methods
make some use
to
to
if
in
in
other Conti-
nental languages offer but little of value that has not been translated into English, French, or German.
Arithmetic
needs to tion
rather than
because for
The
consult
all
some
teacher
of
works on
primary
the
science
those upon the subject
of our special writers
particular
device,
DeGarmo's Essentials
of
educa-
itself,
both
to hold a brief
and because the mathe-
matical phase of the question
the McMurrys' General
seem
arithmetic of
is
exceedingly limited.
Method (Boston, Heath) and their Method of
Method and
the Recitation (Bloomington, Public Sch. Pub. Co.) are
among line,
the best American works.
for
extremes,
teachers
who
will
Along the
special
guard against going
may be recommended
Grube's
to
Leitfaden
(translated by Levi Seeley, New York, Kellogg, and by F. Louis Soldan, Chicago, Interstate Pub. Co.),
Hoose's Pestalozzian Arithmetic (Syracuse, Bardeen), Speer's New Arithmetic (Boston, Ginn), and Phillips's
THE TEACHER'S BOOK-SHELF article in the
299
Pedagogical Seminary for October, 1897.
But the most scholarly work upon this subject that America has produced is McLellan and Dewey's Psychology of Number (New York, Appleton), a work which the author believes to go somewhat to an extreme in its ratio idea, but one which every teacher should place upon his shelves and frequently consult.
Along higher lines, Brooks's Philosophy of Arithmetic (Philadelphia, Sower) deserves a place. Its historical
features,
and
and
it
runs too
and formulae, but
it
has
is
chapter
to cases, rules, it
unreliable,
of
is
much
many good As
recommendation.
worthy showing the views of recent educators as to what matter should be eliminated, what new subjects should be added, and how the leading topics
treated,
author ventures to suggest
Smith's
the
may be Beman and
Higher Arithmetic (Boston, Ginn). In French there is little of value upon primary arithmetic. Upon higher arithmetic, however, numerous works have appeared which cannot fail to inspire Of these the best is Jules Tannery's
the teacher.
Legons d'Arithmetique theorique et pratique (Paris, Humbert's Traite d'Arithmetique Colin), although (Paris,
Nony)
is
also a valuable work.
cares to go into the theory of better
introduction than
bres (Paris, tome
In
German
i,
there
For one who
numbers there
Lucas's
Theorie
des
is
no
Nom-
Gauthier-Villars). is
a veritable cmbarras de richesses.
300 THE TEACHING OF ELEMENTARY MATHEMATICS
The number
of works
upon primary
arithmetic,
and
text-books designed to carry out particular schemes, It
appallingly great.
of is
would be unwise for one begin-
ning a library to attempt to purchase this class of It is better to put upon the shelves a few works. works which weigh these various methods, presenting
their
distinguishing features in
work
best single
brief
The
compass.
Unger's Die Methodik
to purchase is
der praktischen Arithmetik in historischer Entwickel-
ung forth
(Leipzig, Teubner), the latter part of which sets
the
characteristics
Pestalozzi,
Tillich,
is
Methode
suggested by
Diesterweg, second work of
A
al.
Janicke's Geschichte der Methodik des
Rechenunterrichts, which, der
plans
Von Turk,
Stephani,
Grube, Tanck, Knilling, et great value
the
of
in
der
with
Schurig's
Raumlehre,
forms
Geschichte the
third
volume of Kehr's Geschichte der Methodik des Volksschulunterrichtes (Gotha, Thienemann), but which
be purchased separately. those mentioned, however,
A
third work, hardly
is
may
up
to
Sterner's Geschichte der
Rechenkunst (Miinchen, Oldenbourg), the latter part which is devoted to comparative method. For the
of
most
scholarly
treatment
of
arithmetic,
elementary
algebra, and elementary geometry, as of other sub-
by grades, the teacher should own a copy of Rein, Pickel and Scheller's Theorie und Praxis des Volksschulunterrichts nach Herbartischen Grundsatzen jects,
(Leipzig, Bredt), a
work which
also
sets
forth
the
THE TEACHER'S BOOK-SHELF German bibliography
the
of
several
30 1 Al-
subjects.
though advocating a particular method, and therefore outside of the general province of this bibliography,
mention should be made of Knilling's latest work, Die naturgemasse Methode des Rechenunterrichts Volksschule
deutschen
der
in
bourg), on account of
Olden-
(Mimchen,
psychological review of the
its
problem of elementary arithmetic. One of the first works which a teacher Algebra
may
own
profitably
umes,
New
is
Algebra (two
Chrystal's
work which he
York, Macmillan), a
vol-
will not
soon master, but a fountain from which he will get Since this enters but
continual inspiration.
the subject of the equation,
it
lin,
Hodges).
multum
these
in parvo, Fine's
into
should be supplemented
by Burnside and Panton's Theory
To
little
of Equations
be
well
may Number System
(Dub-
added of
that
Algebra
(Boston, Leach).
The most appeared
scholarly
elementary
date and which
matter
which
course
there
French,
is
are
some
of
is
contains a large
usable
many
in
thoroughly up
amount
high-school
other
has
that
Algebre elemen-
in recent years is Bourlet's
taire (Paris, Colin), a work which
to
algebra
excellent
of
work.
new Of
algebras
in
them much more extensive than
Bourlet, but none can be so highly recommended as the first work to be purchased.
From
the
standpoint
of
method, especially as ap-
THE TEACHING OF ELEMENTARY MATHEMATICS
302
plied to the earlier stages, Schiiller's Arithmetik
Algebra (Leipzig, Teubner) deserves a place. practical
book by a
however, run that
becomes
it
difficult
For the teacher who
is
to
Matthiessen's
modernen Algebra der Teubner)
zig, all
will
to
work,
tions
Grundziige litteralen
der
compass,
is
buch
und
Gleichungen (Leipit
is
not at
Quite a remarkable
modern theory
of
equaTheorie der
Petersen's
algebraischen Gleichungen (Kopenhagen,
one cares
literal
antiken
prove a gold mine, but
condensing the
small
in
number.
somewhat master the
of the nature of a text-book.
little
such an extent
a small
taking classes through
equations, and who wishes subject,
to
select
a
German works,
practical teacher.
off into special lines
und
It is
Host).
If
Weber's Lehr-
to look into higher algebra,
der
Algebra (two volumes, Braunschweig Vol. I, French by Griess, Paris, Gauthier-Villars), or Biermann's Elemente der hohere Mathematik ;
(Leipzig, Teubner), are the best of the recent works.
There are
few recent, scholarly, and inexpensive works published in the Sammlung Goschen and the
also a
Sammlung Schubert which
out of
all
proportion to the cost.
Geometry
some good volume on Clarendon
The
will
prove of value
(See
p.
176, note.)
teacher of geometry should have
edition of Euclid.
On
account of
its
second
solid
geometry (Geometry in Space, Oxford, Nixon's may be recommended, Press),
although the Harpur Euclid, Hall and Stevens
(New
THE TEACHER'S BOOK-SHELF
303
As an
York, Macmillan), and others, are excellent.
introduction to the recent development of elementary
geometry, Casey's Sequel to Euclid (Dublin, Hodges) should be
among
the earliest purchases, and to this
also be added, with
profit,
may
three recent manuals by
M'Clelland (Geometry of the Circle, Macmillan), Dupuis (Synthetic Geometry, Macmillan), and Henrici
(Congruent Figures, London, Longmans). In France, where they are not tied to Euclid, nor even to Legendre, there is more flexibility in the course
found in England. Accordingly the modern notions of geometry have more readily found place, and than
is
the reader of
French
will
literature awaiting him.
find
De
Comberousse's
Gauthier-Villars).
Of
inspiring
Probably the best single
for the teacher of geometry, in
and
some very
any language,
Traite"
is
work
Rouch
de Geometric (Paris,
the recent works, Hadamard's
Lemons de Geometric el^mentaire (Paris, Colin)
is
one
of the best.
In Germany
still
more
The making
flexibility is
shown than
in
geometry an exercise in logic, which England carries to an extreme, and which America and France possibly carry too far, is not so noticeFrance.
able in Germany.
of
The
result is a shorter course,
one
divested as far as possible of propositions in the nature of
lemmas, but one in which modern ideas find wel-
come.
To
appreciate this spirit the teacher should
purchase Henrici and Treutlein's Lehrbuch der Ele-
304
THE TEACHING OF ELEMENTARY MATHEMATICS
mentar-Geometrie (Leipzig, Teubner), one of the best books published. As a type of the best of the inexpensive handbooks,
it
would be well
add Mahler's
to
Ebene Geometrie (Sammlung Goschen, costs but twenty cents in
Leipzig,
it
Germany), a bit of concen-
trated inspiration. Italy has
produced some excellent works on element-
ary geometry; indeed, in some features, the leader. (Firenze,
it
Socci and Tolomei's Elementi
Lazzeri
1899),
Geometria (Livorno, (Venezia, Tipog.
and
1898),
d'
Euclide
Elementi di
Bassani's
Faifofer's
has been
various works
Emiliana), and Paolis's Elementi di
Geometria (Torino, Loescher), all have distinguishing features which would entitle them to a place upon the shelves of the reader of Italian.
History and practical first
general
method
works on mathematical history
are Ball's (Macmillan) and
Smith's translation, Chicago, is
the
Probably the most
more popular, the
Open
to
Fink's Court).
latter the
purchase at
(Beman and The former
more mathematical.
Cajori has also written two readable works upon the
general subject (Macmillan).
however, in
The
leading works are,
German, and have been mentioned
in the
foot-notes.
On
general method the pioneer
among prominent
was Duhamel, whose Des Methodes dans Sciences de Raisonnement (Paris, Gauthier-Villars)
writers
five
volumes.
The work
is
les fills
not, however, of greatest
THE TEACHER'S BOOK-SHELF value
practical
to
the
teacher
of
to-day.
Cours de Methodologie mathematique is
Villars)
touch the is
vital points in
work,
interested.
Carre" et is
Dauge's
(Paris, Gauthier-
comparatively recent, but this, too, fails to
especially
(Paris,
305
which the elementary teacher Laisant's
La Mathematique
Naud), frequently mentioned in this it is one of the best efforts
a small volume, but
of its kind,
and
it
may
teacher's book-shelf.
well have a place
Clifford's
Common
upon the
Sense of the
Exact Sciences (Appleton) should also be at hand for consultation.
In the way of periodical liotheca
literature,
Enestrom's Bib-
Mathematica (Leipzig, Teubner)
is
one of the
best publications devoted to the history of the subject.
As
mathematical teaching, Hoffmann's Zeitschrift fur mathematischen und naturwissenschaftlichen to general
Unterrichts (Leipzig,
Teubner), and
L'Enseignement Mathdmatique, Revue Internationale (bi-monthly, Paris,
Carre" et
Naud), are among the
best.
INDEX [Of several foot-note references to the same work, only the
Aahmesu.
See Ahmes. Abacus, 57, 101. ^Eneas Sylvius, 13. Aggregation, signs
Ahmes, n,
given.]
Arithmetic oral, 117.
commercial, 7, 136. year of, 114. applied problems, 136.
of, 182.
first
54, 145.
Alcuin, 16, 60, 61.
ancient divisions, 56.
Algebra
present status, 68. distinguished from algebra, 162.
in arithmetic, 16, 17, 68, 124, 130. ethical value of, 169.
r
first is
Arts, seven liberal, 4.
growth of, 145, kinds of, 155. name, 151.
Aryabhatta, 150.
practical value, 168.
Austrian methods, 122.
Ascham,
what,
and why
when
studied, 170.
Al-Khowarazmi, Allman, 228 n.
Al-Mamun,
taught, 161, 165. 151, 152, 201.
Axioms,
178, 257, 262.
Babylonians, 5, 50, 225. Bachet de Meziriac, 15. Bain, 24^., 28.
of arithmetic, 15.
Ball, 241
Beda,
Angle, 262, 274. Approximations, 142, 159. Arabic numerals, 50, 52, 53. Arabs, 5, 151. Arbitrary value check, 190.
304.
Beman, 148 n., 2iin. Beman and Smith arithmetic, 66 n.
algebra, 159 n. geometry, 285 n. trans, of Fink, 50 n., 304.
Aristotle, 13, 47, 227.
i, 19,
.,
7, 60.
Beetz, 82 n.
Archimedes, 231, 238, 269. Argand, 213. Arithmetic reasons for teaching,
5.
Bagdad, 151.
151.
Al-Mansur, 150.
Amusements
32.
Assyrians,
79, 98.
trans, of Klein, 29072. St., 60.
Benedict,
history of teaching, 71. when to begin, 116.
Bertrand, 214. Bezout, 211.
utilities of, 2, 7, 35.
Biber, 80.
mediaeval, 58.
Bibliography, 297.
crystallizing, 64.
Biermann, 302.
307
INDEX
308 Blockmann, 80
Compound numbers,
.
Comte, 162, 186, Conant, 4472.
Boethius, 10, 59. 10.
Bologna,
Concentric circle plan, 88. Confucius, 33 n. Conrad, 14. Converse theorems, 277.
Bolyai, 265.
Boncompagni, 53 n. St., 60.
Boniface,
Bourlet, 163
n.,
Brahmagupta,
22.
244.
176, 219, 301. 200.
Correlation, 3.
Brautigam, Bin.
Counting, 45.
Bretschneider, 228 n.
Court schools, 59. Cube, duplication
Brianchon, 232, 284. Brocard, 231. Brooks, 67 ., 299. Browning, 12 n. Btirgi, 67 Burnside and Panton, 301. Business arithmetic, 20.
of,
290.
Culture value, 12, 20, 23, 27, 34, 39, 237, 238.
Cycloid, area
of,
244.
.
D'Alembert, 163, 220, 266.
Date
line, 129.
Dauge, 163
Busse, 58, 77.
.,
305.
Davidson, 13 Decimals. See Fractions. .
Cajori, 304. Calculi, 57.
Definitions, 28, 176, 257.
See Easter,
Calendar.
De Garmo, no, inw.,
61.
Cantor, G., 106. Cantor, M.,
Degree, 177, 225.
Capella, 59.
Delbos,
Cardan,
De Morgan,
n.
De Guimp,
14, 153.
80.
4. 44, 148
Denominate numbers,
Cauchy, 169. Charlemagne, 60.
Descartes, 231.
#.,
Desargues, 231.
De De
231, 232.
Stael, 170. 266 n.
Tilly,
Dewey, 45
Chilperic, 59.
Diesterweg,
2, 57.
Chrystal, 163, 164
Chuquet,
Church
n.,
176, 189, 216, 301.
.,
105, 299.
18, 89.
Diophantine equations, 150. Diophantus, 148. Discount, true, 35. Discovery, method Dittes, 6n.
153.
schools, 5, 6, 15, 60, 62.
Cicero, 6. Circle squaring, 290.
Dixon, 257 n. Dodgson, 229 . Drawing, 241, 245, 271.
Clarke, 6 n.
See Church Schools.
Colburn, 117.
Dressier, 120 n.
Comenius, 54. Committee of Ten, 69, 250, 281 n. Committee of Fifteen, 69, 70, 116.
Duhamel,
Compayre, son., 84 n. Complex numbers. See Number tems.
of, 88.
Division, 122.
Clairaut, 240. Cloister.
.,
37, 65.
Denzel, 88.
Checks, 188. Chinese,
177
.,
Carnot, 232. Cassiodorus, 59. Catalan, 41.
Chasles, 228
298.
29^., 304.
Duplication of the cube, 290.
Dupuis, 303. sys-
Easter problem, Ebers, 10.
5, 7, 62.
232.
INDEX Egyptians, 10, n, 12, 50, 145, 226. Elimination, 211. Encyklopadie d. math. Wiss., 29 n.
309
Geometry history of, 224.
non-Euclidean, 233, 265, 269. defined, 234.
Equation in arithmetic, 16, 17, 68, 69, 124, 130.
limits, 236.
of payments, 65.
why
classification of, 152. roots of numerical, 159.
quadratic, 198.
studied, 237. in the grades, 239, 243.
demonstrative, 250, 271. bases of, 257.
impossible
equivalent, 203. radical, 206.
in, 289.
solid, 290.
simultaneous, 208.
inventional, 245.
diophantine, 150.
Erfindungsmethode, 88. Euclid, 229, 235-238.
Examinations, 10, 216. Exchange, 36, 65. Explanations, 140. Factor, 179.
Gergonne, 232. Germain, 208. Gillespie, 162 n. Girard, 6 n.
Girard, Pere, 83.
Goldbach,
41.
Goodwin,
Bp., 171.
Goschen,
176, 302.
n.,
Factoring, 192, 197.
Gow,
Fahrmann, 46
Graffenried, 65 n.
.
227.
Faifofer, 304.
Grammateus,
False position, 124.
Fermat, 41.
Graphs, 208. Grass, 97 n.
Ferrari, 154.
Grassmann,
Ferro, 14, 154. Fibonacci, 53.
Greatest
Fine, i86., 301.
Greenwood, 125 n.
Fingers, 47, 58, 101. Fink, Sow., 304.
Grube,
Greeks,
Fisher and Schwatt, 176.
20.,
24.
Fitzga, 20 n. Formal solutions, 123.
Formal
steps,
Fractions,
n,
in. 23, 54, 119.
decimal, 55, 66, 119.
Francke
Institute, 75.
Frisius, 14,
106.
common
divisor, 39.
6, 12, 50, 51, 55, 150,
89, 118, 298.
Grunert, 202. Guizot, 61 n.
Fiore, 14, 154. Fischer, 202. Fitch,
63.
ico.
Hadamard,
285, 296, 303.
Hall, G. S., 140. Hall and Stevens, 302. Halliwell, 53 n. Hamilton, 95. Hankel, 106, 225 n. Hanseatic League, 8, 62. Hanus, 139 ., 244, 347. Harms, 92 #., 245 n.
Functions, 162, 163.
Harpedonaptae, 226.
Galileo, 244.
Harpur Euclid, 302. Harriot, 156 n. Harris, 125 .
Galley method, 67. Gaultier, 232.
Gauss, 158, 213.
Gemma Frisius,
14,
ioo.
Generalization of figures, 279.
Harun-al-Raschid, 151. Hau computation, 145. Heath, 148 n.
Hebrews,
50.
227.
INDEX Heiberg, 263. Henrici, O., 189, 219, 237, 284 n., 303. Henrici and Treutlein, 284 n., 303. Henry, 53 n. Hentschel, 89, 113, 114^.
Heppel, 190 n.
in.
Herbart, 95,
Kriisi, 81.
Herodotus, 227. Heron, 148. Hilbert, 257
Laboratory methods, 76. Lacroix, 240.
266.
n.,
n,
Laisant,
Hill, 19 n.
Hindu numerals,
I,
42,
145,
224.
Holzmiiller, 173, 174, 251. Homogeneity as a check, 191.
Hoose, 85-*., 298. Horner, 160.
Langley, 245. Laplace, 235. Laurie, 3 n. Lazzeri and Bassani, 304.
Liberal
Hiibsch, 16.
Pisa, 53.
seven, 4.
Loci, 282.
13.
299.
Imaginaries.
arts,
Lobachevsky, 265.
166, 167 n., 170.
Humbert,
Lange, 96.
Lemoine, 231. Leonardo Fibonacci of
Hoiiel, 229, 242.
Humanism,
29^., 39, 49, 104, 140, 156,
240, 305-
50, 52, 53.
History of mathematics,
Hudson,
Kobel, 54. Konigsberger, 257 n. Konnecke, 53 n. Koreans, 57. Korner, 98 n. Kranckes, 88.
See
Number
systems.
India, 3.
Induction, 244.
Locke, 31. Lodge, 125 n. Logarithms, 67. Logic in mathematics,
24, 25, 167, 207,
238, 239. 220.
awakening, simple and compound, 36.
Interest,
Interpretation of solutions, 220. Inventional geometry, 245.
Involution, 31. Isidore, 60.
Logistics, 56.
Longitude and time, Lucas, 299.
Mace, n.
Janicke, 75, 85 Janicke and Schurig, 72 n.
Jews, 5. Journal Royal Asiatic Society, 52 n.
34, 126.
Loria, 229.
32.
Mahaffy, 13 n. Mahler, 304.
Mamun,
151.
Mansur,
150.
Martin, 6n.
Mathews, 239 n. Kallas, 92 n.
Matthiessen, 150 n., 203, 302.
Kant, 95, 265.
McClelland, 303.
Kaselitz, 92.
McCormack, 176 McLellan and Dewey, McMurry, no, 298.
Kawerau,
.
87.
Kehr, 72 n., 300. Kepler, 66 n.
Khayyam,
Mensuration, 137. Mental gymnastic, 79,
201.
Khowarazmi,
151, 152, 201. Klein, 265, 290 n. Klotzsch, 114 n.
Knilling,
2O.,
84,
86 n., 92, 94, 301.
Method,
45^., 105, 299.
84.
rise of, 74.
great question
of, 109.
in geometry, 283.
Metric system, 134.
INDEX Pius
Meziriac, 15.
Middle ages,
II, 13.
Plato, 12, 227, 229, 235.
58.
Minchin, 251, 271.
Pliicker, 232.
Minus and
Plus and minus, 187. Poincare, 257 n.
plus, 187.
Mobius, 232.
Mohammed
ben Musa,
Mohammedans, Miiller,
45
151, 152, 201.
n.
Multiplication
and
Poinsot, 163.
Poncelet, 232. Postulates, 257, 262.
4.
Problems, statement
division, 67, 74.
Murhard, 17 n.
of, 181.
applied, in algebra, 215.
Problem Napier, 67. Negative numbers. tems.
See Number
solvers, 14.
Proklos, 233. sys-
Proportion, 36, 39, 129, 287. Puzzles, 40, 61.
Neuberg, 231. Newcomb, 260 n.
Pythagoras, 13, 228.
Newton,
Quadratic equations, 198. Quadrivium, 60.
48.
Nixon, 302. Non-Euclidean geometry, 233. Notation, 48, 49, 112.
Rashdall,5.
Number
Ratio idea of number, 48, 103. and proportion, 129, 287. Reasons for teaching mathematics,
systems, 157, 184, 213. concept, 99.
pictures, 77.
12,
Object teaching, 71, 100, 102. Obsolete in arithmetic, 68, 69, 70.
Odd numbers, Oliver, Wait,
and Jones,
176.
27,
34,
39,
i,
237,
113,
.
Rechenmeister, 9, 63. Rechenschule, 62. Reciprocal theorems, 275. Recorde, 16, 156 n. Reidt, 32 n. Rein, in, 247.
288, 295.
Oral arithmetic, 117.
Rein, Pickel, and Scheller, 2^n., 300. Remainder theorem, 195. Renaissance, 63.
Oriental algebra, 150.
Oughtred, 156 Oxford, 9.
23,
238.
201.
One-to-one correspondence, 106,
20,
Rebiere, 126
57.
Omar Khayyam,
17,
.
Reviews, 143. Paolis, 304.
Rhyming
Paris, University, 10.
Riese, 14.
Paros,
9.
Rochow,
Roman
Partnership, 65. Pascal, 259 .
Payne's translations, 20 n., 84
Rome, .
arithmetics, 73. 77.
numerals, 50, 51, 54, 55.
6.
Roots, 31. stretchers, 226.
Perception, 78. Pestalozzi, 18, 48, 58, 78, 116.
Rope
Petersen, 285, 302.
Rouche and De Comberousse,
Philanthropin, 76. Phillips, TT,
93 n., 298.
255, 256.
Rosen, 152 n. 303-
Rousseau, 240. Rudolff, 63.
Pincherle, 176, 219.
Ruefli, 86 n.
Pitiscus, 66 n.
Ruhsam,
118.
285,
INDEX
312 Rules, 31, 72, 130, 167.
Subtraction, 121.
Russell, 257
Sully, 31 n. Surd, 180.
.
Saccheri, 233. Safford, 124
Swan
Sylvius, 13.
Sammlung Goschen,
176.
Symbols,
Schubert, 176. Scales of counting, 46. Schafer, 79.
as a check, 191.
Tacitus, 58, 59.
Schiller, 238.
Tanck, 92, 94. Tannery, 162 n., 299.
2. K., 6.
Schmidt, Schmidt, Z., 9. School World, 239 n., 252 n. Schotten, 260 n. Schubert, 176, i860., 265
.,
Tartaglia, 14, 154.
Teachers' failures, 26. Text-books, 70, 139, 173, 254. Thales, 227, 228. Theon of Alexandria, 131 n.
302.
Schiiller, 302.
Tillich, 31, 77, 82, 86. 34, 126.
Time,
245 Schwatt, 176, 239 n. Scratch method, 67. Semites,
66, 148, 155, 182, 222, 273.
Symmetry
Schmid, K. A.,
Schurig, 300. Schuster, 124
pan, 57.
.
.,
.
Tradition, 10.
Trapp, 76. Trigonometry in algebra, 202.
True discount,
5.
Servois, 232.
Turk,
Shaw,
Twelve as a
245.
Short cuts, 137. Signs. See Symbols. Similar figures, 261. Smith, D. E., 50 n., 66 n., 158 285 ., 290 n., 304. Socci and Tolomei, 304. Socrates, 6.
Solon,
6.
Speer, 103 n., 298.
Spencer, W. G., 245. Spencer, H., 27 .
Spiral method, 118.
Square St.
radix, 48.
Tylor, 45 n.
Unger, 7., 300. .,
159
.,
Universities, 8, 9. Utilities of arithmetic, a, 20, 39.
Veronese, 257 n. Vienna, 10.
12.
Spartans,
35.
87.
Vieta, 156, 201. Voltaire, 240.
Von Busse, 58, 77. Von Rochow, 77. Von Staudt, 232.
root, 31.
Benedict, 60.
Wagner,
63.
St. Boniface, 60. Stackel and Engel, 264.
Wallis, 156 n.
Stammer, 20 n. Standard time, 129.
Ward, 42 n. Weber, 302.
Staudt, 232. Stehn, 13, 14
Weierstrass, 106. .
Steiner, 232.
Sterner,
6.,
Stevin, 66
Walker, 39
n., 116.
Wessel, 158, 213.
Wordsworth, 51 300.
.
Young, 174
.
Straight line, 258.
Sturm,
9.
Zahlenbilder, 77.
n.
A HISTORY OF
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