Percy Arthur Barnett.

Common sense in education and teaching; an introduction to practice online

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!s idle to expect him to use the " best " method of
solving a problem from the first, and it is bad teaching
to show it to him until he has done all he can in succes-
sive efforts to arrive at it by himself.

It is of the highest importance to get pupils to realise
the value of their mathematics in ordinary In enuit .
affairs. From the concerns and details of the process
schoolroom, as we advance up the school, we rather than
:an easily choose such problems or exercises accurac y ln
as are within the scope of the pupils' interest
either commercially, or in production, or as statistics
llustrating some moot point, or as truly scientific com-
mutations pertinent to some school study. So Professor
MEiall and Dr. Wormell would have us set, not long sums
to be worked out mechanically, full of pitfalls and of no
use to any body, but rather easy sums which require thought.
All recent writers agree in urging that we should teach
Dur pupils in performing simple multiplication to multiply
irst by the left-hand figure of the multiplier and to pro-
ved in successive lines in the same way. The reasons
$et forth by Dr. Wormell show that this is imperative as
soon as we consult other processes of arithmetic and
algebra ; and more particularly because it is the basis of
methods of approximation, in which the pupil is positively
breed to think of what he is doing, and cannot woik
mechanically. It is quite legitimate and even desirable
:o forecast roughly what a result is likely to be.

228 Common Sense in Education

Dr. Stanley Hall holds that a good deal of arithmetic
The ossi- should be taught technically that processes
biiity of may often be shown first and examples given,
excessive the reason for the processes being left to " flash
rationaiisa- [ n ^ o the mind at a later ^stage, when reason
is more maturely developed. Without ac-
cepting all that this recommendation implies, it may
perhaps be at once admitted that the strict and unvary-
ing drill of young children during the practice of the
four chief rules or processes in the decimal discrimination
of adjacent figures (units, tens, hundreds, and so on), may
be and often is carried to ridiculous excess. No doubt
they should be frequently reminded that the figures have
a decimal value according to their place, right or left ;
but if we require them invariably to give to figures their
names in the decimal hierarchy, we place an unnecessary
obstacle in the way of rapidity of work in calculation-
just as the " complete sentence/' used inordinately, pro-
duces clumsiness in general ratiocination.

We must not be in a hurry. We can easily disgust
and confuse learners by making calls on capacities not
yet developed. The Ingenious Cocker thought he hac
constructed an " Arithmetick suitable to the Meanest
Capacity for a full Understanding of that Incomparable
Art " ; Professor Bain takes a view somewhat more
modest than that of the earlier sage. " The full bearings
of Arithmetic as a science," he says, " cannot be seer
until the pupil has made some way in the higher branche<|
of Mathematics ; and they are never completely known j
except to the few that attain the conception of the highesl
scientific or logical method. In the lower stage of school
training, ease and accuracy in calculation, extended tc
the ordinary compass of arithmetical problems, must be
chiefly looked to. The persistent practice of years shoulc

Mathematics and Physical Science 229

Dring about this result ; while rapidity is attained by
special drill in Mental Arithmetic." Professor Matthews'
uthority is more recent, but even more weighty. " I
m convinced," he says, " that less harm is done now-a-
ays by teaching by rote than those ill-advised attempts
t rational instruction which are inspired by imperfect
nowledge. Thus, for instance, if you state the binomial
leorem for a fractional exponent, and the conditions for
s validity, you do not educate your pupil, but you give
m a piece of information which he can learn to apply,
nd which may be practically useful to him ; but if you
o on to make him learn one of those unsatisfactory
proofs " of the theorem which still keep their place in
me of the text-books, you are doing positive mischief
nd replacing harmless ignorance by a mere pretence of

The study of Geometry begins properly in the Kinder-
arten, where from the manipulation of the The concrete
rescribed "gifts" and the exercises in draw- beginnings of
g and modelling the children acquire a know- Geometr y
dge of geometrical forms and figures and a sense of the
ality of the subject matter which a more abstract be-
nning certainly lacks. It is not only possible, but it is
so most desirable that we should not begin, as in the
thodox way, by a statement of something that we are
Ding to prove. We should rather, as the highest authori-
es urge, let the class as a first step see the main facts of
ane and solid geometry. Drawings should be made,
odels of cardboard and other materials should be
anipulated and rearranged in such a way as to give
.e learners the opportunity of standing as discoverers
their own right.

This kind of interest can and indeed should be main-
ined all through the study. To quote Mr. Workman,

230 Common Sense in Education

when we are dealing with " that awkward youth who is
always sulkily asking us the wherefore of these

The useful- J J

nessofgeo- triangles, parallelograms, and circles, it is nc
metry must use to tell him that they are whetstones for his
be brought wits. He is not aware that his wits need
home to the sharpening, nor would he greatly relish the
prospect if he were. Indeed, he regards his
discovery of the uselessness of Euclid as a proof of his
already superior sharpness. So we may lawfully use
lower motives with him. We may tell him that there is
a science of trigonometry which is merely the Algebraica'
statement and expansion of Euclid I., 47. That it is
this science which enables ships to sail in straight course
or St. Gothard Tunnels to be pierced so exactly thai
engineers from Switzerland and engineers from Ital>
meet within an inch or two, in the centre of the moun-
tain, after five miles of independent burrowing frorr
opposite sides." The same kind of appeal to living
interest prescribes that we should encourage the youn
learner to prove the validity and usefulness of his mathe
matics by the co-operation of his own hand and eye
He cannot pierce tunnels or sail ships, but he can mak'
plans from his own measurements and approximations
he can model simple geometrical solids in cardboard o
in clay.

Here as everywhere else we must encourage the pup
The making to make his own definitions. It is a gres
of definitions mistake to treat the definitions which we fin
in the books as matter for memory or even for uncor
ditional assent. Passing over the elemental and funds
mental difficulties in getting empirical definitions <
point, line, plane, and so on, it is still easy to get 01
pupils to define for themselves most of the figures thj
form the basis of geometrical reasoning. This, indee

Mathematics and Physical Science 231

constitutes the earliest step in teaching the pupil how to
form unexceptionable definitions, and is a most valuable
service to his intellectual development. The principle
of economy is, of course, of the highest importance in
definition, and it is easier to secure this in plane geo-
metrical definition than in any other, the limitation to
two dimensions, length and breadth, excluding irrelevant
details, as it were, automatically.

There may be some inclination to accept work from
pupils which is slovenly in form, on their pro- Neatness of
ducing proof or even mere indications that work
they understand the point under discussion. But it
should be remembered that geometrical reasoning, in
particular, offers especial opportunity for cultivating the
precision, exactitude, and conciseness which are of such
vast importance in all logical method and statement.
Mere written work, however careful, is not enough ; it
should be largely supplemented by very cautious and
exact oral work. This use of demonstrative geometry
is the best beginning possible of the study of formal or
symbolical logic.

Above all things, it seems to be true that the know-
ledge of the prescribed author, be it Euclid or The need for
any other geometer, is of little consequence "riders"
unless the principles involved can be applied to the
solution of derivative problems or riders. Practised
teachers say that the first step towards cultivating
power to this end is to treat the Euclidian proposition
(if we are working with Euclid) from our own point of
view, to try to worry it out before having recourse to the
book solution, and to provide by all possible devices
that the knowledge of the class shall not be limited to
the figure and letters as they are found in the printed text.
The great purpose to be attained is to free the inferior

232 Common Sense in Education

student " from slavish dependence on his text-book while
the able student ' may ' gain power enough to make his
own geometry". The practice of geometrical drawing
is of course of great service here and elsewhere.

Amongst matters of importance is the question whether
Can we sub- we are * use Euclid's Elements or some one
stitute a else's. Opinions differ. Professor Miall looks
better text to see Euclid disappear altogether from the
book for school. Dr. Wormell and Mr. Workman,
both of them recent writers and teachers of
eminence, would have us retain a text book which is, I
believe, used for the teaching of Geometry by England
alone. The chief reasons for abandoning Euclid may be
compressed under two heads :

(1) Geometry itself has made progress, in its develop-
ment and applications, and in nomenclature. There is
no reason, it is said, except prescription, why the term
" angle " should not be extended to include two right
angles and even an angle greater than two right angles.

(2) Euclid's order is not the order of increasingdifficulty,
and therefore is not the " psychological " order which is
said to be the best for teaching.

On the other hand, it is urged that

(1) We should take some account of the claims of
sacred tradition, which has placed Euclid in such a com-
manding position in England, and

(2) The exigencies of examinations and the need for
uniformity as a guide to the preparation of work and
setting of papers are overwhelming.

The case for the early introduction of Algebra into
Algebra the curriculum seems to be a very strong one,
should come although there are authorities of considerable
early weight on the other side. But it would seem

preferable to treat early training in Mathematics as the

Mathematics and Physical Science 233

[eginning of Algebra as well as of Geometry, for formal
icometry in the school is surely merely the continuation
if the established Kindergarten teaching, with its dis-
inctly geometrical basis ; and Arithmetic very soon
esolves itself into Algebra. On the other hand, it is
ertainly easier to demonstrate geometrical truth from
he first by actual objects which the class can see and
tandle and manipulate for itself, and we may therefore
easonably expect that a class should be able to use the
.bstract language of geometry somewhat earlier than it
ould profitably employ the symbolism of algebra, the
igns of operation, brackets, positive integral indices, etc.
This however should not prevent us from preparing a
lass of young children for the symbolical method by
he occasional translation of numbers and quantities into
etters. It should not be hard to familiarise them with
he notion that a nuts + b nuts = a + b nuts. Nega-
ive numbers, as presenting special difficulties, would be
NDstponed to a later stage. But there seems to be no
eason for postponing the solution of simple equations
.nd the like. Problems should, of course, be exceedingly
asy, and every step made exceedingly clear. But this
Teat advantage will be gained that the pupil will be
ailed upon to use his wits, and the arithmetic will become
ess mechanical than it must otherwise be.

It must be repeated here that there are no strictly
ielimited provinces in the Kingdom of Know- Mathema-
edge ; it is not possible to separate the whole tics, in its
>f arithmetic from the whole of algebra. And various
nore : it is possible, and it may indeed often branches - 1S

desirable, to pass from the elementary parts
:>f the one straight to the other. A certain accurate
<nowledge of arithmetic is presupposed before algebra

attempted, but as soon as a boy can understand

234 Common Sense in Education

symbols, he may, nay he should, according to teachers of
greatest weight, be straightway introduced into problems
involving simple equations.

Again, algebra being in truth inseparable from arith-
metic, being in fact " higher " arithmetic, a teacher who
aspires to teach arithmetic should be well acquainted
with algebra, just as only a teacher with a good know-
ledge of at least general European history can teach
English history profitably. And it is equally necessary
for students to know that the ordinary algebra is only
one of several kinds of algebra.

Moreover, advanced parts of the text-book divisions
have their easier portions which it is possible and desirable
to taste before attacking the whole course of dishes. We
need not, says one who knows, read the whole of a chapter
before passing on to the next, though of course our
pupil must depend on expert direction to save him from
" o'erleaping his selle ".

Professor Matthews notes two points that need special
care. It must always be clear that x denotes a number
and not a concrete quantity ; and the symbol = means
not "is," but " is equal to". The same writer urges,
Even in most reasonably, that "just as practical geo-
aigebra metry may fitly precede the systematic study

"doing" of the science; just as experimental demon-
may precede strat i on o f physical laws helps to the compre-

' ' knowing " . r . , . . . ,

hension of abstract dynamics ; so the practical
application of the laws of algebra, before their logical
necessity is fully realised, is not only harmless but even
helpful towards the complete understanding of the very
abstract considerations upon which their general validity
is based. The proper course would therefore seem to be
to exercise the student as soon as possible in the practice
of the fundamental rules by applying them to rational

Mathematics and Physical Science 235

integral functions of a single variable: the process
'without the theory) of finding the highest common factor
3f two polynomials is particularly valuable for this pur-
)ose." This lends weight to Dr. Stanley Hall's view
quoted above ; the reasons for the processes will "flash"
on the learner when the mechanism of the process has
become familiar.

The conference that reported to the American Com-
mittee of Ten speaks in the same sense. Certain alge-
braical propositions are to be presented to the learner
Defore he can well be expected to understand their
demonstration, as in the case of the rule of signs in mul-
iplication and the binomial formula. The plan here
should be to convince the learner of the propositions by
llustrating them, and to leave the strict demonstration
o a later stage. " That minus multiplied by minus is
blus is proved by never leading us to a wrong conclu-
sion," says Bain.

It is usually laid down that algebra as a special study
can be safely begun in the thirteenth or fourteenth year ;
and it is certainly true that it is better to do a little well
than to spend time "in solving catchy problems and
summing fantastic series ". Accuracy in process is more
mportant than facility and rapidity, for accuracy is atten-
tion, is conscious, and is much more than mechanical.
The chief purpose of the inclusion of algebra in the
school course is that it helps to teach the pupil to reason,
to use his wits.

There seems to be no doubt that very great advantage
is gained in the school teaching of mathematics The common
by the practice of introducing the pupil at the ground of
earliest stage possible to the points where Algebra and
algebra and geometry join hands, and -where ( ometr y
mathematics enters into physical science. We all know

236 Common Sense in Education

that the investigation of mechanical or other physical
phenomena becomes more interesting and the more fruit-
ful in proportion to the power of exact calculation and
prediction. It is this quality which gives to mathematics
its high place as a study of practical importance in the
business of life, quite apart from its value as a gymnastic.
The wise teacher accordingly endeavours to connect
mathematics, in special applications, with the natural
science studies of his form ; never, of course, venturing on
the more abstruse questions, and taking care that pupils
do not delude themselves into thinking they understand
what is still dark to them. We must not, on the other
hand, be in such a hurry as to forgo in school teaching
the preliminary and constant use of practical experiment
and the investigation of concrete phenomena merely
because we can get the same results in a more generalised
fashion, by the deductive or mathematical process. The
penalty paid for this error is that the learner is less
thoroughly interested and less thoroughly convinced ; for
the demonstration is intellectual only, and lacks the
reality that is, we know, associated in young and adolescent
minds with concrete phenomena alone. " We approach
the subject first by the comparatively clumsy experimen-
tal method in order that the boy may acquire the quasi-
instinctive mechanical bias which comes of grappling with
concrete problems," as Professor Miall says.

Out of this subject rises another which has caused con-
siderable discussion, often, I think, the result

Are the

"sciences" of some misunderstanding. Are we to teach
to be taught the " sciences " in any particular order ? And,
severally in in particular, is the teaching of Physics to pre-

rJorto? U " Cede the teachin of Chemistry ? If we remind

ourselves of what has just been said in regard

to the need for making our mathematics as real as pos-

Mathematics and Physical Science 237

-I sible, we ought to be led back to the principle which we
;. have already established, to the effect that in the school
stage of education we must think primarily of connected-
ness in curriculum. The study of the specialised sciences
e should be postponed as long as possible, in order that the
closest connexion may be maintained between the matters
studied. Now there is really no question in dispute as to
the primary claim of the simplest zoology and botany to
a place in a child's education. The animals and plants
about us are so intimate and near, make such com-
paratively slight calls on the powers of abstraction and
lengthened reasoning, that their early claims are gener-
ally conceded. But, at a later stage, the question between
chemistry and physics has been very earnestly contested.
In England it is almost true to say that "all the
speakers are Whigs and all the voters are r

The claims

Tory . Whereas our school science has for of Chemistry
a long time been primarily and often ex- and Physics
clusively chemistry, the experts have mostly respectively
claimed the earlier place for physics. In to pre "
America, on the other hand, the body of ex-
pert teachers of greatest weight has pronounced almost
unanimously in favour of chemistry as the best of the
sciences for early school use.

Now logically physics would naturally precede chemis-
try ; chemists and physicists have inclined to believe
that chemical action, the behaviour of molecules and
atoms, is, at bottom, a series of special cases of mechanical
laws. Some investigators have gone so far as to say that
even nervous changes and adaptations are probably of
this character. That is, there is at least reason to sup-
pose that the laws of mechanics or physics are more
general, cover a wider ground, than the laws of chemistry.
But this does not prove that such laws are apprehended

238 Common Sense in Education

by the learner in that order ; the psychological is not
necessarily the logical order. Indeed, the presumption
is the very contrary.

However, in favour of the precedence of physics is the
fact that a great part of the subject understood by this
term deals with phenomena that can be seen and felt ;
they are distinctly concrete, whereas in chemical action
the greater part of our knowledge is the result of in-
ference. For instance, chemical "affinity" is much
harder to understand than mechanical " cohesion " or
" adhesion ". We infer the affinity ; we see the cohesion.
" To make the study of chemical theory as little artificial
and as much rational as possible, and to secure intel-
ligent conception of its many and close relations to
physical laws, a previous training in the conceptions
and measurements of such fundamental quantities as
mass, density, specific gravity, heat, specific heat, and
others, would seem practically indispensable. ... In
fact it seems not unreasonable to suggest that the whole
subject of elementary physics forms a desirable basis
for the study of the elements of chemistry." So urges
Professor Waggoner.

It seems probable that English teachers are likely to
adopt the view held by Professor Waggoner, and prefer,
as a starting-point for experimental science, the study
of elementary physics, beginning with mechanics, to any
form of chemistry however elementary. It can hardly
be doubted that the daily life of young scholars, the
things they see and handle and talk about, supply
copious material for observation, experiment, and reason-
ing. They notice weight, mass, movement, and the
like, before other phenomena. Chemistry comes next,
if treated very strictly in an experimental way, with a
minimum of lecturing and a maximum of manipulatory

Mathematics and Physical Science 239

work on the part of the class. The elements of physi-
ology or botany may or may not be included, according
to the length of the curriculum ; but with the elements
of physics and mechanics the strictly school course
should be considered sufficient. I cannot say that
either physiology or botany seems to me a good school
subject. However skilful the use which a clever teacher
may make of botany in getting a class to see the adap-
tation of structure to environment, the study makes an
inordinate demand on the mere memorising powers ;
and animal physiology is on many grounds offensive
and unattractive to young people. Botany, however,
has the enormous advantage of sending inquirers out
into the fields and under the trees with their eyes open.
Physiography is a necessary preliminary to geography
and is dealt with under that head ; and geology is too
specialised a study to be suitable for work in school.

It should not be impossible to include in a school
course for pupils up to the age of sixteen or seventeen
a general training in science and scientific method, in-
cluding the things of most importance in physics and
chemistry, and of course physiography as a foundation
for geography. After this stage the studies may become
more special, but there seems to be good ground for
preferring physics to chemistry.

It is generally agreed that the inclusion of science in
the school curriculum is essential. Much harm Science"
has been done by reaction from the violently as discipline
excessive prejudice in favour of the physical and experi-
mental sciences set up by Mr. Herbert Spencer nearly
forty years ago. According to him, " science " is the
sum total of knowledge. The fine arts themselves, he
says, are based on " science " ; "science" is poetry,
" science " affords the best moral discipline, " science "

240 Common Sense in Education

is the great stimulus to religion itself. " Accomplish-
ments, the fine arts, belles lettres^ and all those things
which, as we say, constitute the efflorescence of civilisa-
tion, should be wholly subordinate to that instruction and
discipline in which civilisation rests. As they occupy the
leisure part of life> so they should occupy the leisure part oj

To establish against Mr. Spencer the paramount place
of belles lettres in a liberal education, would be to discuss
again much of what has been already said in this book
about literature generally. It must suffice here to
say that literature is not meant to occupy the leisure
part of life, but rather to pervade the whole of it ; it is

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Online LibraryPercy Arthur BarnettCommon sense in education and teaching; an introduction to practice → online text (page 18 of 25)