Table Of ContentCraftsmanship in the
of
Teaching Elementary
Mathematics
BLACKIE & SON LIMITED
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17 StanhopeStreet,GLASGOW
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Craftsmanship in_the_
Teaching of Elementary
^ - -O - " v
-
Mathematics
BY
F. W. WESTAWAY
FormerlyoneofH.M.InspectorsofSecondarySchools
Authorof"ScientificMethod,itsPhilosophicalBasisanditsModesofA
"ScienceTeaching: WhatitWas WhatitIs WhatitMightE
"TheEndlessQuest: 3000YearsofScience"&c.
BLACKIE & SON LIMITED
LONDON AND GLASGOW
Firstissued1931
Reprinted1934,*937
&
PrintedinGreatBritainbyBlackie Son,Ltd.,Glasgow
" Have some wine" the March Hare said in an
encouraging tone.
Alice looked round the table, but there was nothing
"
on it but tea. / don't see any wine" she remarked.
" There isn't any'' said the March Hare.
"
Then it wasn't very civil of you to offer it"
said Alice angrily.
" You are sad," the White Knight said. " Letme
sing you a song to comfortyou."
" Is it very long?" Alice asked,forshe hadheard a
good deal ofpoetry that day.
" "
It's long" said the Knight, but it's very, very
beautiful. Everybody that hears me sing it either it
"
brings the tears into their eyes, or else
" Or else what?" said Alice, for the Knight had
made apause.
" Or else it doesn't, you know. The song is called
' WAYS AND MEANS ', but that's only what it's called,
you know!"
"
Well, what is the song, then?" saidAlice.
" "
/ was coming to that," the Knight said. The
songreally is 'A-SITTING ONA GATE ': andthe tune's
my own invention"
BY F. W. WESTAWAY
SCIENCE TEACHING
Whatit Was WhatitIs
WhatitMightBe
Second Impression. 10s.6d. net
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PREFACE
When asked to write a book on the teaching of Elementary
Mathematics, I felt doubtful as to the avenue by which the
subject might be best approached. During the present cen-
" " " "
tury, the general policy and attitude to be adopted in
mathematical teaching have been discussed by so many
authorities that there seemed very little new to say. Finally
I decided that class-room craftsmanship might be made a
suitable basis of treatment. Thus the book is not intended for
the experienced teacher who has already acquired skill in his
art, but forthe still struggling beginner. Inthe leading schools,
mathematical craftsmanship probably leaves little to be desired,
but the leaven has yet to work its way into the mass.
From the great variety of topics that come within the
ambit of the various mathematical subjects, I have selected
for treatment those which, in my experience, seem to give
young teachers most difficulty. To treat all topics that come
within the daily practice ofmathematical teachers is impossible;
it would mean writing a dozen books rather than one.
I have sometimes been asked if, as an Inspector pursuing
the same daily round year after year, decade after decade, I
am, when listening to lessons in mathematics, ever amused,
ever really interested, ever inclined to be severely critical,
ever bored.
vii
PREFACE
viii
Amused? Yes; for instance, when a young master tells his
boys that mathematics is by far the most important subject
they learn, inasmuch as it is the only one that leads them into
"
the region of pure thought ".
Really interested? Yes, every day ofmy life. In the crafts-
manship ofeven a beginnerthere is almost always some element
of interest; in the craftsmanship of a really skilful mathe-
matical teacher there is to me always a veritable joy. I never
enter a classroom without hoping to find something which will
make an appeal, and I am not often disappointed. Sometimes
disappointed, of course; unfortunately not all mathematical
teachers have come down from heaven.
Severely critical? Yes, occasionally, more especially at the
rather slavish adoption of certain doubtful forms of traditional
procedure. For instance, a teacher may include in the work
ofthe bottom " Set " ofa Form the Italian method of division,
well knowing that two-thirds of the boys will thenceforth
always get their sums wrong. Another teacher may adopt
"
standard form ", not because he has examined it and found
"
it to be good, but because everybody does it nowadays ".
Instead of saying, " I thought we had to do these things,"
why do not mathematical teachers hold fast to the faith which
is really in them? If their faith, theirfaith, includes the Italian
method, standard form, and the score of other doubtful ex-
pedients that spread like measles from school to school, I
have nothing more to say.
Bored? Yes, though not often. The petrifying stuff often
doled out to Sixth Form specialists, the everlasting Series and
Progressions,theolddodgesanddevicesandbookwork"proofs"
ad nauseam in preparation for scholarship examinations, all
this is virtuallythesame nowas fortyyears ago. True, teachers
PREFACE ix
are not much to blame for this. Boys have to be prepared for
the scholarship examinations, and according to prescription.
But that does not soothe an Inspector who has to listen to the
same thing year after year, and I admit that, with Sixth Form
work, sometimes I am almost bored to tears.
If I had to pick out those topics which in the classroom
make the strongest appeal to me, I should include (i) Arith-
metic to six-, seven-, and eight-year-olds, when well taught;
(ii) Beginners' geometry; (iii) Upper Fourth and Lower Fifth
work when the rather more advanced topics in algebra, geo-
metry, trigonometry,and mechanics are being taken forthe first
time (not the Upper Fifth and its revision work); (iv) Upper
Sixth work when examinations are over and the chief mathe-
matical master really has a chance to show himself as a master
of his craft. Sixth and Upper Fifth Form work often savours
too much of the examination room to be greatly interesting:
everything is excluded that does not pay. But inasmuch as
examination success is a question of bread and cheese to the
boy, the teacher is really on the horns of a dilemma, and very
naturally he prefers to transfix himself on that horn that brings
him the less pain.
Why, of all the subjects taught, is mathematics the least
popular in girls' schools? and whyjs [t_the one subject in
which the man in the street feels no personal interest?
It is not because mathematics is difficult to teach. My own
opinion is that it improbably the easiest ofall subjects to teach.
When it is taught by well-qualified mathematicians, and when
those mathematicians are skilled in their teaching craft, suc-
cess seems always to follow as a matter of course, in girls'
schools equally with boys'. The failure to make any head-
way, even under the best conditions, on the part of a small
x PREFACE
proportion of boys and a rather larger proportion of girls is
probably due to a natural incapacity for the subject. Had
I my own way, I would debar any teacher from teaching even
elementary mathematics who had not taken a strong.-doge^of
the calculus, and covered a fairly extensive field of advanced
work generally. It is idle to expect a mathematical teacher to
handle even elementary mathematics properly unless he has
begjx.through the mathematical mill. And yet I have heard
a Headmaster say, " He can take the Lower Form mathe-
matics all right; he is one of my useful men: he took a Third
in History."
As long as University Scholarships are what they are, so
long will Sixth Form specialists* work proceed on present
lines. But one purpose of the book is to plead for considera-
tion of the many neglected byways in mathematics and for
their inclusion in a course for all Sixth Form boys; suggestions
to this end are made in some of the later chapters. We want
a far greater number of ordinary pupils to become mathe-
matically interested, interested in such a way that the interest
will be permanent; and we want them to learn to think mathe-
matically, if only in a very moderate degree. Why do ordinary
pupils shrivel up when they find a mathematician in their
midst? It is simply that they are afraid ofhis cold logic.
There is, in fact, a curious popular prejudice against mathe-
maticians-a-s.a,class. It probably arises from the fact that we
*~~_-.._.~ - * j
arenotanationofclearthinkers,andwe dislikethefewamongst
us who are. Foreigners at least the French, the Germans,
and the Italians are mathematically much keener than we
are. They seem to become immediately interested in a
topic with mathematical associations, whereas we turn away
from it, disinclined to take part in a discussion demanding
