AMTI Mathetic Muse by Sridhar Narayan Prabhu Maths Olympiad Foundation

40 Pages·2018·13.01 MB·English

by  Sridhar Narayan Prabhu AMTI

(s MUSE

Sridhar Narayan Prabhu
MATHETIC MUSE

(A Humanised Approach to
Mathematics Learning
— A Typical Example)

Author

SRIDHAR NARAYAN PRABU
(INNOVATIVE TEACHER AWARD WINNER)

Edited by
P K SRINIVASAN
‘Academic Secretary

Association of Mathematics Teachers of India

“Truth in closest words may fail
But truth embodied im a tale
‘May enter in at lowly gates’

The Association of Mathematics Teachers of India
MADRAS
1988
PREFACE

At the very outset, I would like to make it quite clear
that ‘Mathetic Muse’ is not at all a mathematics text book.
It is only meant to be a literary piece centring round a
mathematical theme. The text book approach is intended
to make a direct appeal to the head whereas this book
seeks to appeal both to the heart as well as the head.

Mathematical themes are quite different from literary
themes, in that the former are usually treated impersonally
and rigidly while the latter are presented with flexibility
ina human setting. In the former mathematical formalism
and exactitude restrict scope for leisurely thinking
whereas in the latter imagination can have free play and
widen the scope considerably. A literary piece avoids the
arid formalism and injects the warmth of informal treat-
ment attained through mention of origin, stages of
development and interplay of components of the problem,

A word about ‘Mathetic Muse’ will be appropriate. The
word ‘Mathetic’ is an adjective from ‘Mathesis’ meaning
‘learning’. So ‘Mathetic’ means ‘relating to learning’.
“Muse’ means ‘poetic inspiration’ and the phrase, ‘Mathe-
tic Muse’ is therefore taken to mean ‘poetic inspiration in
the domain of learning’ and imagery is the basis for poetic
inspiration.

The theme for the ‘Mathetic Muse’ is the solution of
the following mathematical problem:
A certain sum of money is distributed equally among
a certain number of casual labourers. Were there six
fewer, each would have got a rupee more; were there eight
fewer, each would have got a rupee and a half more. What
is the number of labourers and what is the sum dis
buted?

The problem starts with unknown quantities and ends
with an enquiry about the unknown quantities. The two
hypothetical cases are meant to serve as clues to determine
the unknown quantities.

The problem provides scope for exercise of one’s
imagination about an array of labourers, distribution of
money among them, withdrawal of some labourers and
redistribution of the share of the withdrawing labourers
and so on. Thus the problem presents many an aspect.
To a mind wedded to algebraic processes, it suggests an
algebraic approach leading to formation of equations
and solution thereof. To a calculative mind, it poses com-
parison of magnitudes. To a mind geometrically inclined,
it presents models of rectangles of equal area. For one
imbued with an aesthetic sense, it presents a panorama of
scenes in succession.

Since viewing a situation in different perspectives is a
source of enlightenment as well as enjoyment, an attempt
is made to treat the problem in an interesting and instru-
ctive manner with an apt depiction of human sentiments
that express themselves while dramatising the solution of
the problem viewed in a human situation setting.

I leave it to the readers to see for themselves to what
extent, ‘Shyama’, the heroine of this ‘Mathetic Muse’ suc-
ceeds in humanising the problem on hand by associating
themselves with’ her’ moods ‘and “ecstacies, trials and .
travails. Even a serious student of mathematics can share
her delight in detecting remarkable patterns through
excursions in mathematical thinking and in passing from
the particular to the general to breathe an air of finality.

The desire to humanise mathematics by using
mathematical material for literary purposes arose in me
in 1924 while solving sums in mixtures. In 1932, the
desire became pronounced while considering problems
leading to simultaneous equations which centre round
Tectangles of equal areas with modification of dimen-
sions. In 1945, the desire got crystalised and this book
“Mathetic Muse’ took shape. I wanted to publish it. Some
old students evinced interest in my project but they could
not succeed. In 1970 I revised it thoroughly.

twas strange that Shri P K Srinivasan, whom I came to
know through the fortnightly magazine ‘Junior Scien.
tis, Madras and the then bimonthly Mathematics
Teacher(India), Madras showed interest in my contri.
butions. I sent him 2 copy of ‘Mathetic Muse’ some time
in 1964 and ever since I was in correspondence with him.
When he went to USA on Fulbright Teacher Exchange
programme in 1965, our correspondence discontinued.

After a gap of about twenty years, our correspondence
revived. He even paid me a flying visit on 21.6.1985 and
1 found him still interested in me and my contributions.
When I requested him to edit my book and get it published
in Madras, to my great joy, he consented. My well wishers
and old students among whom I should make special
mention of Shri N C Muzumdar and Shri V P Kamat who
have also been keen on seeing my work published have
magnanimously provided the funds.

This book is published under the auspices of the
Association of Mathematics Teachers of India of which
Iam a life member. I trust that the book would be, to
quote Shri P K Srinivasan, the present Academic Secretary
of AMTI, ‘a pace setter in methodological literature in
Mathematics.’ I thank Shri R Athmaraman, the adminis-
trative secretary of AMTI, for accepting my offer to
AMTI to be the sole distributor of this publication.

Haldipur SN Prabhu
5.3.88
CONTENTS

(1) Exposure at class 1
(2) Visualising spree 5
(8) Flitting of scenes 7
(4) Unfolding the relationships —a glimmer 12
(5) Dé nouement 18
(6) Visions galore a1
(7) Quest for more discoveries 23
(8) Global setting 28

(9) Projects 33
1, EXPOSURE AT CLASS

It was a Friday afternoon in a village high school and
the bell went for the mathematics class. The teacher
entered and the pupils stood up and greeted him. He
signed to them to be seated. It was the last period for the
day. He took the attendance and checked the homework.

He went to the blackboard and wrote a word problem
taken from the topic on simultaneous equations that he
had commenced teaching the previous week. The word
problem read as follows:

‘A certain sum of money is distributed among certain
number of casual labourers. Were there six fewer, each
would have got a rupee more; were there eight fewer,
each would have got a rupee and a half more. What is the
number of labourers and what is the sum distributed?”

‘After asking a few pupils to read out the problem, the
teacher started explaining and working out the problem
on the board. Pupils took down the working in their note
books. The working was as given below:

Let M be the sum of money and X the number of casual
labourers among whom the money is to be distributed.
Now using the conditions given in the problem, we can
write

= Me 1d ——)
Me Miiy
x-8 x
From (1) we get

M _ M+x

x-6 x

=> Mx = Mx + xt — 6M ~ 6x
=> xt = 6M + 6x

x-8 x

=> Mx = Mx + 14x 2 8M — 12x.
=> [Vat = 8M + 12x
Substituting

x? = 6M + 6xin (4), we get

1% (6M + 6x) = 8M + 12x

=> 9M + 9x = 8M + 12x
=>M = 3x

Putting M = 3x in (8), we get x? = 18x + 6x
xt = 2x
x=% ox e0
Putting x = 24 in (5) we get M = 72

So the sum of money is Rs.72/- and the number ‘of
casual labourers 24.

Shyama was one of the bright pupils known for indepen-
dent thinking. The teacher usually asked her if she could
get the solution in a different way, after a particular way
was presented to the class. When Shyama was asked about
the approach to solving the above-mentioned problem,
she went to the black board and worked out as shown
here under.

=

Let the number of labourers be N and the rate at which
the payment is to be made to casual labourers be Rs x per
head. Now, using the conditions given in the problem, we

get
(N-6) (+1) =Nx
(N ~ 8) (x +114) = Nx
From (1) we get
= 6x +N-6=0
N-6x=6
Similarly from (2) we get 1% N — 8x = 12

Now multiplying eqn (3) by 114, we get
1aN-9%=9

Subtracting (5) from (4) x = 3
Putting x = 3 in (3), we get
N-18=6
N=24
The teacher complimented her. A few more problems of
the same type were set from the text book for practice. The
closing bell went and the class was over.

Shyama was in no mood to join her friends in play; soshe
went home, wondering all the time whether the problem
could not be done without building equations, so that she
could share her knowledge with her beloved parents who
had not gone to school. She was happy that it was weekend
when her school did not work.