THE TEACHING OF MATHEMATICS
=501.= In mathematics two ends are constantly kept in view:
First, stimulation of the inventive faculty, exercise of
judgment, development of logical reasoning, and the habit of
concise statement; second, the association of the branches of
pure mathematics with each other and with applied science, that
the pupil may see clearly the true relations of principles and
things.
_International Commission on the
Teaching of Mathematics, American
Report; U. S. Bureau of Education,
Bulletin 1912, No. 4, p. 7._
=502.= The ends to be attained [in the teaching of mathematics in
the secondary schools] are the knowledge of a body of geometrical
truths, the power to draw correct inferences from given premises,
the power to use algebraic processes as a means of finding
results in practical problems, and the awakening of interest in
the science of mathematics.
_International Commission on the
Teaching of Mathematics, American
Report; U. S. Bureau of Education,
Bulletin 1912, No. 4, p. 7._
=503.= General preparatory instruction must continue to be the
aim in the instruction at the higher institutions of learning.
Exclusive selection and treatment of subject matter with
reference to specific avocations is disadvantageous.
_Resolution adopted by the German
Association for the Advancement of
Scientific and Mathematical Instruction;
Jahresbericht der Deutschen Mathematiker
Vereinigung (1896), p. 41._
=504.= In the secondary schools mathematics should be a part of
general culture and not contributory to technical training of any
kind; it should cultivate space intuition, logical thinking,
the power to rephrase in clear language thoughts recognized as
correct, and ethical and esthetic effects; so treated, mathematics
is a quite indispensable factor of general education in so far as
the latter shows its traces in the comprehension of the development
of civilization and the ability to participate in the further tasks
of civilization.
_Unterrichtsblätter für Mathematik und
Naturwissenschaft (1904), p. 128._
=505.= Indeed, the aim of teaching [mathematics] should be rather
to strengthen his [the pupil’s] faculties, and to supply a method
of reasoning applicable to other subjects, than to furnish him
with an instrument for solving practical problems.--MAGNUS, PHILIP.
_Perry’s Teaching of Mathematics
(London, 1902), p. 84._
=506.= The participation in the _general development of the
mental powers_ without special reference to his future vocation
must be recognized as the essential aim of mathematical
instruction.--REIDT, F.
_Anleitung zum Mathematischen Unterricht
an höheren Schulen (Berlin, 1906), p.
12._
=507.= I am of the decided opinion, that mathematical instruction
must have for its first aim a deep penetration and complete
command of abstract mathematical theory together with a clear
insight into the structure of the system, and doubt not that the
instruction which accomplishes this is valuable and interesting
even if it neglects practical applications. If the instruction
sharpens the understanding, if it arouses the scientific
interest, whether mathematical or philosophical, if finally it
calls into life an esthetic feeling for the beauty of a
scientific edifice, the instruction will take on an ethical value
as well, provided that with the interest it awakens also the
impulse toward scientific activity. I contend, therefore, that
even without reference to its applications mathematics in the
high schools has a value equal to that of the other subjects of
instruction.--GOETTING, E.
_Ueber das Lehrziel im mathematischen
Unterricht der höheren Realanstalten;
Jahresbericht der Deutschen Mathematiker
Vereinigung, Bd. 2, p. 192._
=508.= Mathematics will not be properly esteemed in wider circles
until more than the _a b c_ of it is taught in the schools, and
until the unfortunate impression is gotten rid of that mathematics
serves no other purpose in instruction than the _formal_ training
of the mind. The aim of mathematics is its _content_, its form
is a secondary consideration and need not necessarily be that
historic form which is due to the circumstance that mathematics
took permanent shape under the influence of Greek logic.--HANKEL, H.
_Die Entwickelung der Mathematik in den
letzten Jahrhunderten (Tübingen, 1884),
p. 6._
=509.= The idea that aptitude for mathematics is rarer than
aptitude for other subjects is merely an illusion which is caused
by belated or neglected beginners.--HERBART, J. F.
_Umriss pädagogischer Vorlesungen; Werke
[Kehrbach] (Langensalza, 1902), Bd. 10,
p. 101._
=510.= I believe that the useful methods of mathematics are easily
to be learned by quite young persons, just as languages are easily
learned in youth. What a wondrous philosophy and history underlie
the use of almost every word in every language--yet the child
learns to use the word unconsciously. No doubt when such a word
was first invented it was studied over and lectured upon, just as
one might lecture now upon the idea of a rate, or the use of
Cartesian co-ordinates, and we may depend upon it that children of
the future will use the idea of the calculus, and use squared
paper as readily as they now cipher.... When Egyptian and Chaldean
philosophers spent years in difficult calculations, which would
now be thought easy by young children, doubtless they had the same
notions of the depth of their knowledge that Sir William Thomson
might now have of his. How is it, then, that Thomson gained his
immense knowledge in the time taken by a Chaldean philosopher to
acquire a simple knowledge of arithmetic? The reason is plain.
Thomson, when a child, was taught in a few years more than all
that was known three thousand years ago of the properties of
numbers. When it is found essential to a boy’s future that
machinery should be given to his brain, it is given to him; he is
taught to use it, and his bright memory makes the use of it a
second nature to him; but it is not till after-life that he makes
a close investigation of what there actually is in his brain which
has enabled him to do so much. It is taken because the child has
much faith. In after years he will accept nothing without careful
consideration. The machinery given to the brain of children is
getting more and more complicated as time goes on; but there is
really no reason why it should not be taken in as early, and used
as readily, as were the axioms of childish education in ancient
Chaldea.--PERRY, JOHN.
_The Teaching of Mathematics (London,
1902), p. 14._
=511.= The ancients devoted a lifetime to the study of
arithmetic; it required days to extract a square root or to
multiply two numbers together. Is there any harm in skipping all
that, in letting the school boy learn multiplication sums, and in
starting his more abstract reasoning at a more advanced point?
Where would be the harm in letting the boy assume the truth of
many propositions of the first four books of Euclid, letting him
assume their truth partly by faith, partly by trial? Giving him
the whole fifth book of Euclid by simple algebra? Letting him
assume the sixth as axiomatic? Letting him, in fact, begin his
severer studies where he is now in the habit of leaving off? We
do much less orthodox things. Every here and there in one’s
mathematical studies one makes exceedingly large assumptions,
because the methodical study would be ridiculous even in the eyes
of the most pedantic of teachers. I can imagine a whole year
devoted to the philosophical study of many things that a student
now takes in his stride without trouble. The present method of
training the mind of a mathematical teacher causes it to strain
at gnats and to swallow camels. Such gnats are most of the
propositions of the sixth book of Euclid; propositions generally
about incommensurables; the use of arithmetic in geometry; the
parallelogram of forces, etc., decimals.--PERRY, JOHN.
_The Teaching of Mathematics (London,
1904), p. 12._
=512.= The teaching of elementary mathematics should be conducted
so that the way should be prepared for the building upon them of
the higher mathematics. The teacher should always bear in mind
and look forward to what is to come after. The pupil should not
be taught what may be sufficient for the time, but will lead to
difficulties in the future.... I think the fault in teaching
arithmetic is that of not attending to general principles and
teaching instead of particular rules.... I am inclined to attack
the teaching of mathematics on the grounds that it does not dwell
sufficiently on a few general axiomatic principles.
--HUDSON, W. H. H.
_Perry’s Teaching of Mathematics
(London, 1904), p. 33._
=513.= “Mathematics in Prussia! Ah, sir, they teach mathematics
in Prussia as you teach your boys rowing in England: they are
trained by men who have been trained by men who have themselves
been trained for generations back.”--LANGLEY, E. M.
_Perry’s Teaching of Mathematics
(London, 1904), p. 43._
=514.= A superficial knowledge of mathematics may lead to the
belief that this subject can be taught incidentally, and that
exercises akin to counting the petals of flowers or the legs of a
grasshopper are mathematical. Such work ignores the fundamental
idea out of which quantitative reasoning grows--the equality of
magnitudes. It leaves the pupil unaware of that relativity which
is the essence of mathematical science. Numerical statements are
frequently required in the study of natural history, but to
repeat these as a drill upon numbers will scarcely lend charm to
these studies, and certainly will not result in mathematical
knowledge.--SPEER, W. W.
_Primary Arithmetic (Boston, 1897), pp.
26-27._
=515.= Mathematics is no more the art of reckoning and
computation than architecture is the art of making bricks or
hewing wood, no more than painting is the art of mixing colors on
a palette, no more than the science of geology is the art of
breaking rocks, or the science of anatomy the art of butchering.
--KEYSER, C. J.
_Lectures on Science, Philosophy and Art
(New York, 1908), p. 29._
=516.= The study of mathematics--from ordinary reckoning up to
the higher processes--must be connected with knowledge of nature,
and at the same time with experience, that it may enter the
pupil’s circle of thought.--HERBART, J. F.
_Letters and Lectures on Education
[Felkin] (London, 1908), p. 117._
=517.= First, as concerns the _success_ of teaching mathematics.
No instruction in the high schools is as difficult as that of
mathematics, since the large majority of students are at first
decidedly disinclined to be harnessed into the rigid framework of
logical conclusions. The interest of young people is won much
more easily, if sense-objects are made the starting point and the
transition to abstract formulation is brought about gradually.
For this reason it is psychologically quite correct to follow
this course.
Not less to be recommended is this course if we inquire into the
essential purpose of mathematical instruction. Formerly it was
too exclusively held that this purpose is to sharpen the
understanding. Surely another important end is to implant in the
student the conviction that _correct thinking based on true
premises secures mastery over the outer world_. To accomplish
this the outer world must receive its share of attention from the
very beginning.
Doubtless this is true but there is a danger which needs
pointing out. It is as in the case of language teaching where the
modern tendency is to secure in addition to grammar also an
understanding of the authors. The danger lies in grammar
being completely set aside leaving the subject without its
indispensable solid basis. Just so in the teaching of mathematics
it is possible to accumulate interesting applications to such an
extent as to stunt the essential logical development. This should
in no wise be permitted, for thus the kernel of the whole matter
is lost. Therefore: We do want throughout a quickening of
mathematical instruction by the introduction of applications, but
we do not want that the pendulum, which in former decades may
have inclined too much toward the abstract side, should now swing
to the other extreme; we would rather pursue the proper middle
course.--KLEIN, FELIX.
_Ueber den Mathematischen Unterricht an
den höheren Schulen; Jahresbericht der
Deutschen Mathematiker Vereinigung, Bd.
11, p. 131._
=518.= It is above all the duty of the methodical text-book to
adapt itself to the pupil’s power of comprehension, only
challenging his higher efforts with the increasing development
of his imagination, his logical power and the ability of
abstraction. This indeed constitutes a test of the art of
teaching, it is here where pedagogic tact becomes manifest. In
reference to the axioms, caution is necessary. It should be
pointed out comparatively early, in how far the mathematical body
differs from the material body. Furthermore, since mathematical
bodies are really portions of space, this space is to be
conceived as mathematical space and to be clearly distinguished
from real or physical space. Gradually the student will become
conscious that the portion of the real space which lies beyond
the visible stellar universe is not cognizable through the
senses, that we know nothing of its properties and consequently
have no basis for judgments concerning it. Mathematical space, on
the other hand, may be subjected to conditions, for instance, we
may condition its properties at infinity, and these conditions
constitute the axioms, say the Euclidean axioms. But every
student will require years before the conviction of the truth of
this last statement will force itself upon him.
--HOLZMÜLLER, GUSTAV.
_Methodisches Lehrbuch der
Elementar-Mathematik (Leipzig, 1904),
Teil 1, Vorwort, pp. 4-5._
=519.= Like almost every subject of human interest, this one
[mathematics] is just as easy or as difficult as we choose to
make it. A lifetime may be spent by a philosopher in discussing
the truth of the simplest axiom. The simplest fact as to our
existence may fill us with such wonder that our minds will remain
overwhelmed with wonder all the time. A Scotch ploughman makes a
working religion out of a system which appalls a mental
philosopher. Some boys of ten years of age study the methods of
the differential calculus; other much cleverer boys working at
mathematics to the age of nineteen have a difficulty in
comprehending the fundamental ideas of the calculus.--PERRY, JOHN.
_The Teaching of Mathematics (London,
1902), pp. 19-20._
=520.= Poor teaching leads to the inevitable idea that the
subject [mathematics] is only adapted to peculiar minds, when it
is the one universal science and the one whose four ground-rules
are taught us almost in infancy and reappear in the motions of
the universe.--SAFFORD, T. H.
_Mathematical Teaching (Boston, 1907),
p. 19._
=521.= The number of mathematical students ... would be much
augmented if those who hold the highest rank in science would
condescend to give more effective assistance in clearing the
elements of the difficulties which they present.--DE MORGAN, A.
_Study and Difficulties of Mathematics
(Chicago, 1902), Preface._
=522.= He that could teach mathematics well, would not be a bad
teacher in any of the rest [physics, chemistry, biology,
psychology] unless by the accident of total inaptitude for
experimental illustration; while the mere experimentalist is
likely to fall into the error of missing the essential condition
of science as reasoned truth; not to speak of the danger of
making the instruction an affair of sensation, glitter, or
pyrotechnic show.--BAIN, ALEXANDER.
_Education as a Science (New York,
1898), p. 298._
=523.= I should like to draw attention to the inexhaustible
variety of the problems and exercises which it [mathematics]
furnishes; these may be graduated to precisely the amount of
attainment which may be possessed, while yet retaining an
interest and value. It seems to me that no other branch of study
at all compares with mathematics in this. When we propose a
deduction to a beginner we give him an exercise in many cases
that would have been admired in the vigorous days of Greek
geometry. Although grammatical exercises are well suited to
insure the great benefits connected with the study of languages,
yet these exercises seem to me stiff and artificial in comparison
with the problems of mathematics. It is not absurd to maintain
that Euclid and Apollonius would have regarded with interest many
of the elegant deductions which are invented for the use of our
students in geometry; but it seems scarcely conceivable that the
great masters in any other line of study could condescend to give
a moment’s attention to the elementary books of the beginner.
--TODHUNTER, ISAAC.
_Conflict of Studies (London, 1873), pp.
10-11._
=524.= The visible figures by which principles are illustrated
should, so far as possible, have no accessories. They should be
magnitudes pure and simple, so that the thought of the pupil may
not be distracted, and that he may know what features of the
thing represented he is to pay attention to.
_Report of the Committee of Ten on
Secondary School Subjects, (New York,
1894), p. 109._
=525.= Geometrical reasoning, and arithmetical process, have each
its own office: to mix the two in elementary instruction, is
injurious to the proper acquisition of both.--DE MORGAN, A.
_Trigonometry and Double Algebra
(London, 1849), p. 92._
=526.= Equations are Expressions of Arithmetical Computation, and
properly have no place in Geometry, except as far as Quantities truly
Geometrical (that is, Lines, Surfaces, Solids, and Proportions) may
be said to be some equal to others. Multiplications, Divisions, and
such sort of Computations, are newly received into Geometry, and that
unwarily, and contrary to the first Design of this Science. For
whosoever considers the Construction of a Problem by a right Line
and a Circle, found out by the first Geometricians, will easily
perceive that Geometry was invented that we might expeditiously
avoid, by drawing Lines, the Tediousness of Computation. Therefore
these two Sciences ought not to be confounded. The Ancients did so
industriously distinguish them from one another, that they never
introduced Arithmetical Terms into Geometry. And the Moderns,
by confounding both, have lost the Simplicity in which all the
Elegance of Geometry consists. Wherefore that is _Arithmetically_
more simple which is determined by the more simple Equation, but
that is _Geometrically_ more simple which is determined by the
more simple drawing of Lines; and in Geometry, that ought to be
reckoned best which is geometrically most simple.--NEWTON.
_On the Linear Construction of
Equations; Universal Arithmetic (London,
1769), Vol. 2, p. 470._
=527.= As long as algebra and geometry proceeded along separate
paths, their advance was slow and their applications limited.
But when these sciences joined company, they drew from each other
fresh vitality and thenceforward marched on at a rapid pace
toward perfection.--LAGRANGE.
_Leçons Élémentaires sur les
Mathematiques, Leçon cinquiéme.
[McCormack]._
=528.= The greatest enemy to true arithmetic work is found in
so-called practical or illustrative problems, which are freely
given to our pupils, of a degree of difficulty and complexity
altogether unsuited to their age and mental development.... I am,
myself, no bad mathematician, and all the reasoning powers with
which nature endowed me have long been as fully developed as they
are ever likely to be; but I have, not infrequently, been
puzzled, and at times foiled, by the subtle logical difficulty
running through one of these problems, given to my own children.
The head-master of one of our Boston high schools confessed to me
that he had sometimes been unable to unravel one of these tangled
skeins, in trying to help his own daughter through her evening’s
work. During this summer, Dr. Fairbairn, the distinguished head
of one of the colleges of Oxford, England, told me that not only
had he himself encountered a similar difficulty, in the case of
his own children, but that, on one occasion, having as his guest
one of the first mathematicians of England, the two together had
been completely puzzled by one of these arithmetical conundrums.
--WALKER, F. A.
_Discussions in Education (New York,
1899), pp. 253-254._
=529.= It is often assumed that because the young child is not
competent to study geometry systematically he need be taught
nothing geometrical; that because it would be foolish to present
to him physics and mechanics as sciences it is useless to present
to him any physical or mechanical principles.
An error of like origin, which has wrought incalculable mischief,
denies to the scholar the use of the symbols and methods of
algebra in connection with his early essays in numbers because,
forsooth, he is not as yet capable of mastering quadratics!...
The whole infant generation, wrestling with arithmetic, seek for
a sign and groan and travail together in pain for the want of it;
but no sign is given them save the sign of the prophet Jonah,
_the withered gourd_, fruitless endeavor, wasted strength.
--WALKER, F. A.
_Industrial Education; Discussions in
Education (New York, 1899), p. 132._
=530.= Particular and contingent inventions in the solution of
problems, which, though many times more concise than a general
method would allow, yet, in my judgment, are less proper to
instruct a learner, as acrostics, and such kind of artificial
poetry, though never so excellent, would be but improper examples
to instruct one that aims at Ovidean poetry.--NEWTON, ISAAC.
_Letter to Collins, 1670; Macclesfield,
Correspondence of Scientific Men
(Oxford, 1841), Vol. 2, p. 307._
=531.= The logic of the subject [algebra], which, both educationally
and scientifically speaking, is the most important part of it, is
wholly neglected. The whole training consists in example grinding.
What should have been merely the help to attain the end has become
the end itself. The result is that algebra, as we teach it, is
neither an art nor a science, but an ill-digested farrago of rules,
whose object is the solution of examination problems.... The
result, so far as problems worked in examinations go, is, after
all, very miserable, as the reiterated complaints of examiners
show; the effect on the examinee is a well-known enervation of
mind, an 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.
--CHRYSTAL, GEORGE.
_Presidential Address British
Association for the Advancement of
Science, 1885; Nature, Vol. 32, pp.
447-448._
=532.= It is a serious question whether America, following
England’s lead, has not gone into problem-solving too extensively.
Certain it is that we are producing no text-books in which the
theory is presented in the delightful style which characterizes
many of the French works ..., or those of the recent Italian
school, or, indeed, those of the continental writers in general.
--SMITH, D. E.
_The Teaching of Elementary Mathematics
(New York, 1902), p. 219._
=533.= The problem for a writer of a text-book has come now, in
fact, to be this--to write a book so neatly trimmed and compacted
that no coach, on looking through it, can mark a single passage
which the candidate for a minimum pass can safely omit. Some of
these text-books I have seen, where the scientific matter has
been, like the lady’s waist in the nursery song, compressed “so
gent and sma’,” that the thickness barely, if at all, surpasses
what is devoted to the publisher’s advertisements. We shall
return, I verily believe, to the Compendium of Martianus Capella.
The result of all this is that science, in the hands of
specialists, soars higher and higher into the light of day, while
educators and the educated are left more and more to wander in
primeval darkness.--CHRYSTAL, GEORGE.
_Presidential Address British
Association for the Advancement of
Science, 1885; Nature, Vol. 32, p. 448._
=534.= Some persons have contended that mathematics ought to be
taught by making the illustrations obvious to the senses. Nothing
can be more absurd or injurious: it ought to be our never-ceasing
effort to make people think, not feel.--COLERIDGE, S. T.
_Lectures on Shakespere (Bohn Library),
p. 52._
=535.= I have come to the conclusion that the exertion, without
which a knowledge of mathematics cannot be acquired, is not
materially increased by logical rigor in the method of instruction.
--PRINGSHEIM, ALFRED.
_Jahresbericht der Deutschen
Mathematiker Vereinigung (1898), p.
143._
=536.= The only way in which to treat the elements of an exact
and rigorous science is to apply to them all the rigor and
exactness possible.--D’ALEMBERT.
_Quoted by De Morgan: Trigonometry and
Double Algebra (London, 1849), Title
page._
=537.= It is an error to believe that rigor in proof is an enemy
of simplicity. On the contrary we find it confirmed by numerous
examples that the rigorous method is at the same time the simpler
and the more easily comprehended. The very effort for rigor
forces us to find out simpler methods of proof.--HILBERT, D.
_Mathematical Problems; Bulletin
American Mathematical Society, Vol. 8,
p. 441._
=538.= Few will deny that even in the first scientific instruction
in mathematics the most rigorous method is to be given preference
over all others. Especially will every teacher prefer a consistent
proof to one which is based on fallacies or proceeds in a vicious
circle, indeed it will be morally impossible for the teacher to
present a proof of the latter kind consciously and thus in a sense
deceive his pupils. Notwithstanding these objectionable so-called
proofs, so far as the foundation and the development of the system
is concerned, predominate in our textbooks to the present time.
Perhaps it will be answered, that rigorous proof is found too
difficult for the pupil’s power of comprehension. Should this be
anywhere the case,--which would only indicate some defect in the
plan or treatment of the whole,--the only remedy would be to
merely state the theorem in a historic way, and forego a proof
with the frank confession that no proof has been found which could
be comprehended by the pupil; a remedy which is ever doubtful and
should only be applied in the case of extreme necessity. But this
remedy is to be preferred to a proof which is no proof, and is
therefore either wholly unintelligible to the pupil, or deceives
him with an appearance of knowledge which opens the door to all
superficiality and lack of scientific method.--GRASSMANN, HERMANN.
_Stücke aus dem Lehrbuche der
Arithmetik; Werke, Bd. 2 (Leipsig,
1904), p. 296._
=539.= The average English author [of mathematical texts] leaves
one under the impression that he has made a bargain with his
reader to put before him the truth, the greater part of the
truth, and nothing but the truth; and that if he has put the
facts of his subject into his book, however difficult it may be
to unearth them, he has fulfilled his contract with his reader.
This is a very much mistaken view, because _effective teaching_
requires a great deal more than a bare recitation of facts, even
if these are duly set forth in logical order--as in English books
they often are not. The probable difficulties which will occur to
the student, the objections which the intelligent student will
naturally and necessarily raise to some statement of fact or
theory--these things our authors seldom or never notice, and yet
a recognition and anticipation of them by the author would be
often of priceless value to the student. Again, a touch of
_humour_ (strange as the contention may seem) in mathematical
works is not only possible with perfect propriety, but very
helpful; and I could give instances of this even from the pure
mathematics of Salmon and the physics of Clerk Maxwell.
--MINCHIN, G. M.
_Perry’s Teaching of Mathematics
(London, 1902), pp. 59-61._
=540.= Remember this, the rule for giving an extempore lecture
is--let the mind rest from the subject entirely for an interval
preceding the lecture, after the notes are prepared; the thoughts
will ferment without your knowing it, and enter into new
combinations; but if you keep the mind active upon the subject up
to the moment, the subject will not ferment but stupefy.
--DE MORGAN, A.
_Letter to Hamilton; Graves: Life of W.
R. Hamilton (New York, 1882-1889), Vol.
3, p. 487._