growing tall

D A V I D KENT AND K E I T H H E D G E R

G R O W I N G T A L L

I N T R O D U C T I O N

Jeff normally teaches English. He is talented, experienced and assured with all

groups of pupils and recently became pressed into service to cover a vacant

mathematics class for a term. It was with some surprise that we witnessed his reaction to the simple yet

elegant solution to an elementary trigonometry problem produced by Diane,

one of his fourteen year old pupils. The door burst open and Jeff yelled

excitedly " I 'm sorry, but you've got to look at this kid's solution. It's great

isn't it." As a teacher of English, Jeff had been known to enthuse before over the

results of the creative activity of juxtaposing words, but seldom had he experi-

enced the act of being creative which in mathematics is sometimes associated

with beauty and form, but also truth. Such we think was the novelty of the experience to Jeff; that the learning

and doing of basic secondary school mathematics can lead to a creative act in

which the creation contains elements of truth unique to mathematics. And we

give respect here to the philosophical arguments on the nature of truth. There can be no doubt that truth is part of the strength and fascination of

mathematics, philosophical objections not withstanding. Yet it is this strange

element of truth which turns so many children away from mathematics.

Despite all the innovations of the past two decades so many children reach the

age of fifteen having made so many mistakes they are convinced of their

inability to cope.

At Belper we believe it is our responsibility to offer pupils situations in which they are able to express themselves to the extent of believing that they

can make some contribution to mathematics, that our subject is not one they

need to fear, but on the contrary it can make a positive contribution to their

lives. Adolescence can be a difficult period, no less for our pupils than any

others. But in between the domestic and personal crises which dot their lives

and confuse their aims, desires and aspirations, we hope we offer some clues

in their search for a sense of identity. Our satisfaction has been to observe those pupils who have developed

Educational Studies in Mathematics 11 (1980) 137-179. 0013-1954/80/0112-0137 $04. Copyright �9 1980 by 1). Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

138 DAVID KENT AND KEITH HEDGER

confidence in themselves through their actions in mathematics, and built on

our attempt to help them 'grow tall' within and through the subject itself.

BELPER HIGH SCHOOL

We thought it advisable to write a little about the school in which we teach. In

a sense it is dangerous to divorce education from its local environment. But at

the same time we feel it necessary to add that we would have operated in a

similar way whatever the circumstances may have been. The educational needs of the children of Belper, a small town in central

Derbyshire, are served by a three tier system of schools. Junior schools cater

for children up to the age of 9; from then to 13 they attend one of the two

middle schools in the town after which they move to the town's only High

School.

The children will normally be at the high school for three years, which we

call lower, middle and upper. In mathematics the teaching is in all ability

groups, a system which we have of late extended into the upper, external

examination year.

Children are in mixed ability groups for almost all of their lower year

classes. There are undoubtedly problems associated with such groupings;

but we share the Headmaster's (Michael Tucker) belief that the benefits out-

weigh the disadvantages. Our main aim is to cultivate and extend the math-

ematical talents of as many children as we can; we believe that the general

philosophy governing the organisation of Belper High School helps us achieve

our aim.

MATHEMATICS AT THE SCHOOL

Belper High School opened in 1973 and for the first five years mathematics

teaching there was dominated by an 'individualised work-card system'. Both

of us find the thought of teaching mathematics almost exclusively through the

written word to be abhorrent. We had worked together before, but only in an unofficial basis in a school and college in outer London. It was a conscious decision to come together at Belper to explore the possibilities of teaching our subject in a personally supportive environment. The inevitable change in methodology was partly responsible for a massive turnover in departmental staff, giving us the opportunity to start with a group of individuals who would

be both committed and supportive. The philosophy which governs our methodology is simple. We believe that

the first task of the mathematics teacher is to cultivate those mental powers

GROWING TALL 139

characteristic in mathematicians, and that it is our professional responsibility

to create situations which allow these powers to develop. We are appreciative of the need to teach elements of basic mathematics

such as simple equations, fractions, graphical work, matrices and the other topics commonly taught to children of our age range. But we offer continual

injections of other work, not normally taught to this age group. For instance,

during a two or three week period around March 1979 we offered about half

of the 'A' level Further Mathematics syllabus in the Theory of Numbers to

our 13-15 year olds. This work (usually intended for 17-18 year olds) includes

Euclid's Algorithm, Linear Diophantine Equations, Euler's function r

Linear Congruences and more. Our criteria for offering anything to the

children are that we should find the work stimulating, that it should be stimu-

lating to them and that we can offer it in such a way that it is accessible to

pupils of all abilities.

Our criteria have been well satisfied not only with Number Theory but also

Eigen Vectors, Integration, Transformation Geometry, Algebra, Loci and other

unusual topics, or standard topics taught in an unusual way.

Part of our task has been to improve the public examination pass rate and to

increase the numbers going on to take mathematics in the sixth form. Both

have been vastly improved in the short term and the indications are that

58% of the 1%18 year olds will be sitting the 'Advanced' (A) level math-

ematics examination in the 1979/80 year and 50% of the 15-16 year olds may well be entered for the 'Ordinary' (O) level examination in the summer of 1980.

(For further information about the British Examinations see Appendix 5.) We are not happy with the present standards of mathematics attained by the

vast majority. The subject is held in awe by too many, the majority of children

fired it difficult and it is the favourite subject for only a few. At Belper we are

trying to change that, we want our pupils to fired mathematics a stimulating

challenge, an enjoyable subject and to overcome some of the strange barriers

which surround it.

In this paper we would like to share with you some of our experiences of

teaching some of our various classroom activities, and in some cases to discuss them through the work of our pupils.

LESSONS ON LOCI

Loci as we teach it is one of our favourite topics and one we regard as vital to

our general methodology. It is a mathematical topic which makes immediate appeals to the children's imagery and hopefully we can encourage them to

re-capture the power of 'thinking in moving pictures'. Taught as we teach it,


Loci offers our pupils the opportunity to engage in some mathematical activity

in which the use of abstract symbolism is minimised.

Lesson 1. The Film

For this session we had put three classes in the hall and showed them the film

'Three points determine a circle'. It is three minutes long, silent and concerns

itself with various circles, which are shown to expand or contract and even-

tually be fully determined by three fixed points on the screen. The film is

shown once in complete silence. We believe what comes after the showing is

different, and would like to describe a typical session.

"Anyone see the cup final on Saturday?" A few eyes screw up and search for others to confirm the weirdness of such a question in such a context.

Eventually a few brave if dilatory hands rise slowly into the air. "Who scored

the first goal?" we ask. "Alan Sunderland", says a lad. "Which foot? Where

was the goalkeeper? Who passed him the ball? How did he control it? Describe

in detail that ,goal". Usually the bravest of lads demonstrates his folly in

volunteering, since he is usually the target of free contradiction by some of his

peer group. We like to pick out the most authoritative corrector and get him to

give a generally accepted version. In doing this session on a number of

occasions we have yet to come across anyone who does not at some stage screw

up his eyes or close them or look to blank space in the room in order to re-live

the moments with images on his own private visual display unit. The immediate

demonstration of this action is always a powerful moment, since it is precisely

the moment when almost all of them realise they have access to experienced

visual images.

Then the switch back. "How did the film start? How many dots? Where

did the circle come in? The left-hand side? Are you sure?" There are times when we feel it is necessary to impose a certain benign

pressure on pupils. For some it is too easy to say "I do not know" and they

need to be chivied, cajoled, encouraged and praised in their faltering efforts.

"What, didn't I notice you? Have you just come in? No, well you must have

seen the film, come on think", we demand. Eventually most respond and most begin to realise they can recall and work

on mental images and the slow process of utilizing the ability begins. Of course, for some like Chris (see Hedger 1976) things just refuse to happen on cue, but

that's another story. "Close your eyes all of you. Now clear your screens. Bring a dot in from

the left and fix it. Now expand a circle from that dot. What happens?" " I 'm losing it, it just disappears."

GROWING TALL 141

The comments flow quickly in the freedom of a darkened and disembodied

universe. "Right, bring it back in, till it disappears into the dot. Now fix a

second dot and let 2 circles expand at the same rate from each. Describe the

locus of the points where they meet."

FIG.2

t There are not many who judge this locus incorrectly and there are some

whose ability to handle these and other harder loci is phenomenal. Helen is one

such girl, whose fluent visual ability sets her apart from others for the first

time. Her demonstrable complete confidence in the truth and sophistication of

her answers acted as a spur to the others who witnessed a previously ordinary,

average mathematician produce extra-ordinary responses provoking our spon-

taneous comment "It seems unfair that one kid should have so much talent."

The session at different times has taken different directions but on this

occasion we examined some more standard loci. And then:

"My left arm is a straight line fixed, and my right forefinger is free to move

142 DAVID KENT AND KEITH H E D G E R

(being careful not to let it do so) so as to be a constant distance away from my

arm", followed by

"My left foref'mger is fixed and the end of my right forefinger is free to

move so that it is always the same distance from it."

"What are the loci?'"

FiO.

GROWING T A L L 143

Even given the success of the session we were not prepared for 'a cylinder' to

the first and 'a sphere' to the second. But then Martin (of whom we hear more

later) is rather special.

At the end of the session, Helen and a few more gave up their break time to

watch another film on harder loci and we had some interesting responses to

the shadows formed by projecting our light source from the projector onto

two cubes on sticks, as in Figure 4.

FIG.4

"Well what shadows can you make? Can you get them all? How do you

know? Can you get this one?"

FIG.5

"Try it."

Helen did.

Lesson 2. The Counters

Two counters, red and green, are placed on the floor as in Figure 6 and the

class, gathered round in a circle, are asked "Can you place this white counter

somewhere so that its distance from the red counter is equal to its distance

from the green." There is usually some initial reluctance to get involved. It


@ FIG.6

means being prepared to put your thoughts and possible mistakes on view. But

the class' attention is drawn to a quotation from the actress Susan George, displayed in the room:

"Such is life and such is growing

and through mistakes we end up knowing."

Fortunately in every class we always have some child prepared to come forward

and try; usually it has been one of the rougher lads for whom mathematics

may for the first time have been freed from the iron grip of symbolism. The problem is changed to "Now place the white counter so that its dis-

tance from the red counter is two times its distance from the green one."

Slowly more and more children become involved, each being asked to place a

white counter in a position satisfying the constraints.

Usually a group of children settle for the locus in Figure 7 and we have witnessed a few heated debates as others realise the folly of their ways.

�9 O@ 0

FIG.7

On occasions we have offered the substance of this session to groups of teachers and been fascinated to observe the difference in response. Beyond the

normal barrier of reservation, mathematics teachers appear to think they have a vested interest in 'not being seen to be wrong'. The act of sharing becomes more stilted and incorrect solutions and strategies abound privately. Until one man became quite animated about his vision of the solution to this problem and finally having convinced himself of the correctness of his solution he strode to the front of the class. "This is another position for your counter,"

GROWING TALL 145

he said as he put the counter in exactly the reverse position of that required,

twice as far from the green as it was from the red. When led to the error of

his venture he slowed, but still offered a strategy. "You draw a large circle

using the red counter as centre, and halve its radius for a circle on the green one. Then look at where they cross." This, as shown in Figure 8, seemed to

have the general agreement of the assembled company.

FIG.8

' \

DAVID KENT AND KEITH HEDGER 146

But the question remained-where were the other points? Indeed were

there any others? Some asserted not. Pebbles in clear water, images of expanding circles do not seem to figure as

prominently in the symbol dominated world of the classroom teacher as they

can in the minds of their pupils. Latent abilities, such as handling the dynamic

involved in the above problem, come easier when they are practised. And lead

to solutions, based on dynamic images such as that shown in Figure 9, which assume the authority of intuitive truth. We offer it to our pupils on the

basis that something cannot be practised until its existence has been recognised.

As the final stage we have usually offered the following Scholarship (S) level

Mathematics problem: "Find the equation of the locus of the points of intersection of conjugate

tangents to the ellipse x2/a 2 +y2/b2= 1, which in purely visual terms b e c o m e s . . , can you tell me something about the shape that is produced

when those tangents are moved around the ellipse so that the angle between

them remains a right angle?" Most fourteen year olds are usually quite uninhibited in their response:

"Oh, course it's a circle."

"How do you know?" "It just is. You can see it is." The confidence borne of lack of awareness of future hazards, to base a

response purely on intuition and the manipulation of dynamic images is

GROWING TALL 147

something which comes quite naturally to most pupils. Again it is their

teachers who suffer the torments of self doubt: "How do you know it is not another ellipse with eccentricity nearly equal

to one?" But then, how do I know that when I've just used a compass to describe a

circle, that I have not produced an ellipse of eccentricity nearly one. How do we know anything that others are not around to confirm for us?

And how do we know that those who do confirm things are not themselves

mistaken and are passing on false information? There comes a time when in a

civilised world we must begin to work in an intuitive way and develop our

individual confidence, our ability to make personal judgements, and our

belief in them once made.

In encouraging some of these qualities in our pupils we are performing the

teaching task at its purest. For them to be interested in, to observe closely,

to hypothesise, test and modify are surely amongst the foremost pre-requisite

qualities through which our pupils will be able to fulfil themselves.

Those like Mark, Heather and Martin, about whom we hear later, who

continue to be creative in a publicly acceptable manner are surely those who

will be the mathematicians of the future.

Lessons Later

A triangle is constructed out of cardboard, as in Figure 11.

FiG11 A couple of counters, or drawing pins if the floor allows, are placed on the

floor, as in Figure 12.

FIG.12


A V

B

The piece of cardboard is placed between them, as in Figure 13 and the

children asked to predict the locus of the vertex (V) as the triangle takes on

all such positions with the two sides VA and VB in contact with the counters,

as in Figure 14.

We are pleased with the response, in terms of the number of children, by

now used to our loci work, who see the circle. This alone allows us to explore

the traditional 'Circle theorems' of the G.C.E. 'O' level syllabus. We are

shocked, but delighted, by the response of the children like Martin, David, Mark and others who see the locus, usually all loci, in three dimensions.

Lesson: Binary

The encouragement of the ability to handle mental images has no meaning unless it can be applied to ordinary problems, such as number bases.

GROWING T A L L 149

G. 15 We ask them to 'clear a screen' with several circles on it, all of equal radii.

"Imagine that seven counters are placed in the right-hand circle - and I give

you a rule - t w o counters are worth o n e in the next circle along to the left.

Keep moving the counters according to this rule and tell me what you get."

FIG.16 Most pupils quickly supply the answer "l l l " . A change of rule to groups

o f 3 being worth 1 in the next circle yields Figure 16 and the response '21"

from a lot of people.

Pupils are told a rule of "change to base k" simply means that groups of

k are worth 1 in the next circle along, and are given chalk and counters to

repeat the mental exercise in front of them on the desk. (If anyone wishes to

try, remember that chalk rubs off and the games o f tiddley winks do not

usually break out until the second lesson, and even then in a well controlled

environment provide an opportunity for a teacher to exercise and re-assert

his or her authority.)

"Pardon? You do not need the counters? Okay. What's a circle with 30 in,

with a rule for base 37" A moment 's pause, the eyes squint, the mental process

activates. "1010", the boy replies.

Using these methods it is quite possible to cover the whole of any G.C.E.

or C.S.E. syllabus.

Psychological research evidence notwithstanding we share a belief that the

ability to think in images, to construct and transform mind pictures is a


fundamental thought process which many children lose. It is a power of the

mind which we feel may be lying dormant within so many children.

SOME CHILDREN

1. Simon

Judged by his previous school in September 1978 as: "C +. Erratic and forget-.

ful. Slow with poor number sense. More effort needed," Simon was considered

to be in need of remedial attention by the High School's remedial specialist.

Organisationally he was given the undoubted benefit of a mixed ability

teaching unit and began by acting out his predicted attitudes and qualities.

He was difficult to motivate and needed to be encouraged, cajoled and press- ured even to pay attention. But merely by being present in the group he had the chance to sample the flavour of parts of the subject which might otherwise have been deemed beyond him, presented sometimes in an unusual manner.

There was never any moment which we could mark as a turning point,

since his growth appeared to be continuous and undramatic. The initial reluc-

tance diminished as his involvement grew and his realisation that whatever

others might have said to him the subject was not beyond him and was there

for the taking.

In a sense Simon encapsulates our thesis. Our 'method' as such is not

easily defined nor obvious in its way of working. We do not base our course on

a particular textbook nor relegate the task of teaching the subject to the

ubiquitous 'workcard' or worse 'workcard system'.

Rather, our common experience with the works of Gattegno, Mary Boole

and at Exeter University has guided our belief that all of our pupils are math-

ematicaUy gifted and that it is our task, and sometimes privilege, to make

them individually aware of it. Certainly this involves us in long and heated

discussions about how to initiate our generative approach with particular

topics, what stimuli to use and how to extend the theme to different levels

in varied directions. In this respect perhaps we are unusual in our use of film

(as described in the loci session) and through our harnessing of Number Theory and other topics; but in all cases we are appealing to our pupils' mental powers and abilities, and trying to loosen the hold of the omnipresent inherited neur- rosis that our subject is only available to a predetermined elite of individuals.

Teaching large groups of pupils in the school hall is never easy, but is made workable by the support and attitudes of our colleagues. Curiously in a school system, like Belper, designed to promote individualised learning our 'method' here is not out of place. The large group session always creates an intense

GROWING TALL 151

group dynamic, engendered perhaps through the tension of pupils possibly

being asked to make a public statement in response to a question, and encourag-

ing our pupils to make a major individualised intellectual effort.

It is our shared belief that each of our pupils who has re-lived an experience as a mental image, or has realised that the order in which things are done can

bring different effects, or who has perceived a sameness or equivalence in two

objects differing in colour, shape and size, or who realises as a small child that

the aeroplane which is but a dot in the sky is in fact far bigger than the window

he is looking at it through, actually possesses mental powers characteristic of

the mathematician. This places upon us the obligation to make pupils aware of

their existence before lack of use leads to the normal adult stage of corrosion

and decay. Certainly we have a conviction that the dynamics of mental imagery

can play an important role in the learning of our subject and try to harness it

in our teaching. But our successes do not occur by magic, rather through

unrelieved pressure, encouragement and cajoling, shared experiences and down-

right nastiness to those who would seek to divert attention in our classroom.

As Simon found to his cost, there is no evidence of a laissez-faire attitude

with liberal undertones of "have a go if you are in the mood" in our class-

rooms. We both believe strongly in the values and benefits of good order,

politeness and common sense, of being in a position of choosing who we work

with at a given moment and restraining by the most effective method any

dissident elements in the class. Simon is one of those who have benefited through our system. By Easter

1979 he was no longer "erratic and forgetful" but was demonstrating a sophisticated facility with and confidence in his ability to manipulate numbers.

Factorising quadratics is not a popular topic with pupils, yet it is one of

those which we expect ours to handle; and certainly it plays a significant role

in the examinations they take. At the moment of writing Simon is one of the most capable pupils in his year group at this topic (and also at trigonometry,

algebraic manipulation and simple equations). His own words give testimony

to the efforts he has put into his work: he had just factorised some expressions of the form x 2 + x -- 20.

"You look at it, Mr. Kent, and want to think you cannot do it. But really

only the first two or three are like that, they are quite hard. If you keep at it

you can see how the numbers go and after a bit of time they get easier. I look

at them now and can do most of them in my head." If Simon's success had been either isolated or accidental we would not have

bothered putting pen to paper to compose this article. Gattegno wrote about the 'blind' law which produces our mathematicians and there will always be some children who become good at mathematics, no matter what we do. But


Simon is not a product of a 'blind' law, but rather a product of a deliberate

plan. We are fortunate to share essential beliefs about mathematics teaching,

although this is not surprising for, as said, we have both been influenced by

Gattegno and Mary Boole, especially their ideas on the 'dynamics of imagery' and 'moving mind pictures'. No one has as yet tapped the mysteries of the sub-

conscious or unconscious workings of the minds of the mathematicians, and perhaps they never will. It is too easy and too glib to say that different people

think in different ways and that children should be encouraged to develop their own ways. We have some very, very bright pupils at Belper who are not men-

ti6ned at all in this paper. There is no need, for they would have been good at our subject no matter what the system. Certainly it would be immoral to

impose any one method on a child, but it would also be professionally

irresponsible not to try to offer any floundering pupil that which we believed

was a respectable and functional way of operating. We believe, for instance, that the ability to move things in the mind will in

no way hinder any mathematical growth, and as such the films produced by

Nicolet and others form a significant part of our curriculum. We do not use these films merely to illustrate a point or show a construction, but offer them

to the children as a way of working and see them as one means of developing

their imagery.

Imagery is usually thought of in geometric terms, so that development of

the image is seen to be related to visual images and thus geometry. Such a view

is too narrow. We would conjecture that the ability to create and move images

in the mind cannot be anything but helpful to the development of mathematics

in children, and that these images are not necessarily related to geometry

p er se.

Simon, and many others in his group, belonged to the group of pupils who

could solve simple equations like:

3x = 12

but for whom:

2 x = 5

would be very difficult, whilst:

7x = 3

would be impossible. And it is not difficult to see why. Even the children who interpret 3x with its usual meaning, and that is not all, often solve the equation

3 x = 12

GROWING TALL 153

by going through the '3 times table' until an answer is obtained. Such a

methodology is obviously far too restrictive. To see that

3x = 12

is equivalent to

12

3

offers a far greater generalisation, untidy until simplified that is true, but never-

theless powerful. But in our experience few pupils can see this method in terms of a 'movement' of tlie symbols and that in solving the equation

7 x = 3

we need to 'move' the 7 from its starting point to being under the 3. We do

not believe it to be too surprising that this facility can be improved by printing

the 7 on a piece of cardboard and making the movement at the blackboard in a

continuous fashion as

[ ~ ] x = 3

so that 3

We accept that the understanding of why the technique works is at a different

level which needs to be worked on. But just being able to go through the

mechanics, with pieces of cardboard if necessary, or better still in the mind, has

to be better titan not offering such equations to some children or having them

sit with glazed looks of "I cannot do this" in their eyes. So we see Gattegno's and Mary Boole's ideas on the development of mind

pictures and images to be about more than the conventional visual properties

associated with geometry. We believe that the films we offer to supplement our

work on loci, the imagining of growing circles and movements of points will have a pay off in other areas of the children's mathematical development.

As one of the products of our belief and methodology Simon was working

through 'O' level problems when he was 14 years old, he was asking for extra work and was very confident, assured and hard working about his mathematics. There is more though, for many of Simon's peer group have observed the transformation in him, they now seek his help in class, recognising him to be one of the most capable mathematicians in the school and a lot of them would like to be like him. They have the chance.


2. Paul and Isobel

Paul and Isobel deserve special mention in that they were the first pupils at the

school to produce some high quality work, and by so doing injected us with an

energy for and belief in what we were doing.

A large comprehensive school such as Belper provides many opportunities

to dissipate energy. Creative teaching is partly about finding ways of generating

energy and the most positive way of doing so comes through the excitement of

a pupil producing a piece of quality work; especially when that pupil is one

who would not normally be expected to produce such quality.

Isobel's story has been told before (Kent, 1979) and there is no doubt that

she has a powerful mathematical talent. She is a complex character for she

tends to make silly errors in adding fractions or with other basics, whilst at the

same time producing some startling awarenesses in unconventional situations.

FIG.17

In the investigation we call 'frogs' three black and three white counters are

laid on a grid as in Figure 17 and the idea is to move the counters according to the rule: 'black moves 1 space to the right, white moves 1 space to the left' unless we reach a situation as in Figure 18 in which the black could 'jump over'

the white rather as in the game of draughts, or vice versa for the white to jump over the black. The game is completed when the counters are as in Figure 19.

FIG. 18

FiG J9 The mathematics of the investigation includes working out and proving the

rule that exists between the number of counters and the number of moves required to reach the end result.

GROWING TALL 155

At the time when most pupils were still shuffling counters around Isobel

had generalised the problem to knowing that if we had "10 black and 10 white

counters it will take 10 times 12 moves." But when asked to express this result

in terms of having n counters of each colour she wrote "n times 2n" moves. It is typically 'Isobel' to confuse 2n with n + 2, yet this complex character had

generalised the result to " I f you have 7 black and 10 white counters it will take you 7 x 10 + 7 + 10 moves," which with some persuasion from her teacher she

could generalise still further to " n m + n + m."

Isobel is a very intuitive sort of girl. She is perceptive and seems to possess

highly developed visual powers (Art is her favourite and best subject). In

situations in which such visual powers are imperative, or particularly advan- tageous, she can usually triviafise the mathematics.

Sitting no more than three yards away from Isobel is Paul, a lad of fourteen who was labelled by his previous school as ~ average, but with adequate

knowledge of his multiplication tables', at about the time of his thirteenth

birthday.

It is not easy to describe the emotional effect Paul had on us, but some of

his work from the weeks around June to July 1978 came close to bringing tears

of joy to our eyes. The full impact of his story carries added weight if we

tell you that he was tested on arrival at the school and scored a mark of 46

out of a possible 140, a mark that placed him well into the lower half of the

ability range.

For two and a half terms this boy sat in class producing average or below

average pieces of work and being no discipline problem to his teacher. Then in

May 1978 there was an explosion which brought shock waves of excitement

across the mathematics department. Quite simply the work in his mathematics

book for the period from late May to July 1978 is of a staggeringly high standard.

The change in Paul started with a problem we call 'Routes on a Cube' in

which the pupils are invited to investigate the relationship between the number of edges of various polyhedra and the number of routes from one vertex to

another. What Paul produced was a masterpiece of clear thinking which is today the centre-piece of the display in the mathematics department. (See Appendix 1 for a copy of part of his work.)

There followed a succession of other pieces of high quality work as Paul put

forward perfect solutions to all that was placed before him. On a visit to the Open University, John Mason, a lecturer there, had posed a problem which our 6th formers had found something between 'very taxing' and 'impossible'. Back

at school we offered it to Paul's class thinking we could make a lesson or so of it. Not so, Paul brought forward a perfect solution in about ten minutes, shrugging his shoulders and commenting that "it was pretty obvious."


It would be nice enough if it ended there, but the story continues into the

next year. We spent a week in October 1978 doing an introduction to trig-

onometry, just very basic right-angled triangle work. This came a week after

some revision of basic mensuration. The class were given some work to do at

home and as an extra Paul was asked if he would like to take a look at an

'O' level mensuration question. He was given an old examination paper with

the words "you might like to look at Question 1, you should be able to do

parts of it."

As if to demonstrate his complete mastery Paul did not only Question 1

but also the rest of the paper. In so doing he dismissed a 3D geometry question

as if with contempt, he tackled a problem on Linear Law with ease and more.

Perhaps the highlights of his weekend endeavour were that he 'invented' the

formula �89 sin C for the area of a triangle and also 'invented', for himself with

no aid, the 'Cosine rule'.

How does one feel when you mark his b o o k ? - shocked, excited, elated, surprised, pleased, thankful, privileged and pensive. When the shock waves of

excitement have died away you begin to realise that in one weekend a boy who

was once described as 'below average' taught himself, at 14, enough to guarantee a top grade 'O' level pass. It made us wonder how many more like him there are, and what we can do about it if we are to discover and develop

them. Our simple belief is that there ire far more children like this, or potentially like this, than most educational theories predict or give credit for.

3. Slow Learners: Like Bridget

There can be no doubt that Paul and Isobel are talented individuals and we

have others like them in every class. But we have others for whom mathematics means virtually nothing at all, just as we have children with such severe emotional problems that one could hardly expect mathematics to be at the forefront of their minds.

It would be quite wrong if readers get the impression that we specialise in

teaching only the most able and emotionally mature children. Far from it,

growing tall is not something restricted to any elite but rather our offering to all pupils. Before we progress any further into this paper we would like to

point out that we try to offer mathematical experiences which will contribute to all our pupils at a level which is personally significant to each individual.

Any sensitive class teacher will feel a thrill when pupils such as Paul pro-

duce extremely high level work. But those like Bridget can and do create the same level of excitement. Such was the case with her attempt at our 'Co- Prime' investigation (see Appendix 2), which starts by defining the term

GROWING TALL 157

Co-Prime and finishes with an examination of Euler's function for a prime

raised to a power, i.e., ~o(pn). Bridget is a girl who finds everything to do with mathematics extremely

difficult (all numbers are random numbers for Bridget). And she evidently

has similar difficulties with other areas of the curriculum. But she is the sort

of girl who is always wilting to try. We could cope by either not having chlidren

like her in the class or by giving them pages of trivial sums. But teaching is

about more than coping. We hope that every lesson offers every pupil the

chance to add to his/her learning and not just keep them occupied. The first part of our Co-Prime investigation asks the boys and girls to decide whether

a pair of numbers are Co-Prime or not. For many such an exercise will be

fairly trivial, taking no more than a few minutes, especially as we have usually

taught Euclid's algorithm. But there are others for whom this will present

enormous difficulties and create exciting possibilities.

After nearly half-an-hour of deciding whether 3 and 7 are Co-Prime Bridget bounced across the floor of the classroom.

"Mr. Kent, Mr. Kent," she chirped with excitement written right across her face, "one goes into three, doesn't it." Trying to catch her breath, she con- tinued, "I've been going 'one times one is one', 'one times two is two', 'one

times three is four', 'one times four is four', . . . . Isn't that daft".

Bridget has really severe problems. Her I.Q. was once measured at 67, a

statistic which might give some indication of her potential at the age of 15.

In many schools we know a topic such as basic algebra, including simple equations, would be out of the question for a girl like Bridget. Yet her develop-

ment in this field has been a source of excitement as rich and satisfying as

anything we have known, especially as the girl's I.Q. was so low, but more

especially because she tries.

Not too long ago the symbol 'x' meant nothing to Bridget other than a

letter in the alphabet. At a later stage she falsely acquired results such as:

x + l = y

X - - I = w

x + 2 = z

x + 3 = "there is no answer."

Although she has perceived the operation wrongly there can be little doubt that Bridget was making some sort of headway; just to get the right answer as she moved through the alphabet was no mean achievement. Twelve months later we were absolutely thrilled that Bridget had reached the stage of correctly solving equations such as

158 DAVID KENT AND KEITH H E D G E R

x + 2 -- 7 x = 5

5 - x = 2 x = 3

and even, in a test,

3 x + 5 = 23 x = 6

There could be no doubt that Bridget would solve such equations by trial and

error, substituting in values for x, but nevertheless, she had made some con-

siderable progress. We, and more importantly she, were pleased at her

achievement.

To share such emotions, such moments, with a girl like Bridget is at least as

thrilling as anything we can experience in mathematics education. So although

some of the other pupils have provided some high quality mathematics we

would wish readers to know that our energies are directed towards all pupils of

all starting points of ability. Our concept of 'growing tall' is about changing

levels o f ability, about transforming attitudes, confidences and performances;

as such it has no restrictions.

4. Dawn

The following question is taken from an 'A' level Further Mathematics paper,

as such it is designed to test the most able 18 year olds in England; it is also

part of our Co-Prime investigation of Appendix 2. The solution that follows

is a neatened form of the one offered by a girl just a few days after her 15th

birthday.

" I f ~ n ) represents the number of positive integers less than n which are

Co-Prime with it and p is a prime number and m a positive integer show that

(i) ~0(p,) --- p -- 1

(ii) ~o(pm) = pm-X(p _ 1)"

~ ( 2 ) - - 1 1

~(3) -- 2 1,2

~o(5) = 4 1 , 2 , 3 , 4

~ 7 ) --- 6 1 , 2 , 3 , 4 , 5 , 6

If p is prime all the numbers less than p won' t go into p or have a common

factor. So

~ ( p ) = p - 1

GROWING TALL 159

Try p = 2 and m = 2.

tp(2 2) = ~ (4) = 2 = 2 x l

r 3) =~o (8) = 4 = 4 x l

~o(2 4) = so(16) = 8 = 8x 1

1,3

1 ,3 ,5 ,7

1 ,3 ,5 ,7 ,9 ,11 ,13 ,15 .

Try p = 3

~o(3 2) = ~9)

= 3 x 2

r 3) = ~(27)

= 9 x 2

= 32x2

= 32 x(3- - 1)

1,2, |174

1,2, O ,4 ,5 , @ , 7 , 8 , ~ ,

10, 11, ( ~ , 13, 14, Q

16, 17, ~ , 1 9 , 2 0 , ( ~ ,

22, 23, ( ~ , 25, 26

Try p = 5

: (55) = •(25)

= 5 lots of 4

= 5 x 4

~ 5 3 ) = ~125)

= 25 lots of 4'

= 52x4

~ p ~ ) =

= p l o t s o f p - - 1

= p ( p - - 1)

~,p~) =

= p,~-I lots o fp -- 1

= p m - l ( p _ 1)

1 , 2 , 3 , 4 , @ , 6 , . . .

1,2, 3,4, (Z) , 6 , . . . ,

(~) , 2 6 , . . . , 124

1 , 2 , 3 , . . . , ( ~ , . . .

1,2,3,| .... , |

. . . . . 24


Imagine the scenario. It is the last ten minutes of your first lesson with a

lively class of fourteen year olds. You have decided to kill these closing

minutes with a quick test of numeracy and mental arithmetic.

Teacher (to the class) "I leave home with a s note, go into Belper on the bus

at a fare of 20p. In town I buy a copy of a record for s then go to the pub

for a pint of beer costing 30p. Then I leave the pub and go back home on the bus, again for 20p. How much change do I have? - give me the answer in base

fifty four.

Dawn (immediately) One one.

And she has not touched base work for six months.

She is one of the most complex characters we have met. An ordinary girl

she hates to be singled out as special, wishing rather to be 'one of the girls'. She

once deliberately scored only half marks in a short test because in the previous

one she beat everyone else and found that to be too embarrassing. She has all

the normal tastes and emotional stresses of a girl of her age, and finds all the usual outlets such as pop music, discos and so on. But at times she is a most precocious talent who will surely become a first class mathematician unless she decides to do something else with her life; she is in that enviable position of

being able to make a positive choice of whether to be a mathematician or not.

She constantly sits in class telling her friends how to do things, telling theft

that they are looking at the problem in the wrong way. "No, no, no, you must not think of it like that," she informs a friend floundering over the

formula for quadratic equations, "you look at this one here and then just see

how the numbers go. It's easy after a bit." Her work on Number Theory easily surpassed that of the 6th former who

studied this as a Further Mathematics option, she found differentiation and integration of polynomials easy, she asked what s in-ix meant because "my

dad wants to know, he has seen it at work." She is so often full of self doubts

and at times difficult to motivate and convince of her abilities. She tends to convince herself that she was "lucky" with something such as her solution to

"Prove that if 11n = 37 (Mod 111) then n is a multiple of 37," she tells you that she just "guessed the answer" or that "it was obvious," and that "Anyone

can do that, it's easy." In her other subjects she shows no special talent, being somewhere between

reasonable and fair, never terribly weak, never outstandingly good. But in

mathematics classes she is one of the most charismatic figures we have met, carrying the respect of all her peers and an authority that is hers alone. She is the girl who at times seems to be "touched with the spirit of Gauss."

GROWING T A L L 161

Dawn is the girl about whom her former teacher wrote at the end of her

lower year:

"Fairly bright, keen, but lacks ability to see round the problem."

Such judgements should not be made in haste.

5. Mark and Heather

If we had been asked to assess Mark's progress and ability after a term and a

half at the school our report would have read something like:

"A likeable lad who tries hard, does neat work and is of average ability."

"But before he had completed two terms at the school we were moved to

write to his parents informing them that we now believed Mark to be:

"Outstanding."

Precisely the same comment can be made about Heather, but Mark came to

our attention first so we have had more time to measure his improvement.

We believe that 13 to 14 year olds who produce work of the calibre these

two produced are 'outstanding'.

In our 'Last Digit' investigation of Appendix 3 the children are told how

one of us first became aware that a number was even if its last digit was a

0, 2, 4, 6 or 8; and this was a lot simpler than dividing the whole number

by 2. They are then told a few things about modulo arithmetic and finally

given some problems. The final problem is a former 'S ' level question (S level

being for the most able 18 year olds in the land), and reads:

"Prove that the sequence

1! + 2 ! , 1! + 2 ! + 3!, l! + 2 ! + 3! + 4 ! . . . .

contains one and only one perfect square."

Separated by three months Mark and Heather offered the following solution:

1 ! + 2 ! = 1 + 2 = 3

1 ! + 2 ! + 3 ! = 1 + 2 + 6 = 9

1 ! + 2 ! + 3 ! + 4 ! = 1 + 2 + 3 + 2 4 = 33

1 ! + 2 ! + 3 ! + 4 ! + 5 ! = 1 + 2 + 6 + 2 4 + 1 2 0 = 153

1 ! + 2 ! + 3 ! + 4 ! + 5 ! + 6 ! = 1 + 2 + 6 + 2 4 + 1 2 0 + 7 2 0

= 873

and after here the last digit will always be a 3, because you are always adding

on a 0 at the end. But there is no perfect square whose last digit is a 3, the


pattern goes 0, 1,4, 9, 6, 5, 6, 9 ,4 , 1,0, t , 4 . . . . . So the only perfect square

is 9~ So the sequence contains only one perfect square.

Whether Heather will experience a similar transformation to Mark remains

to be seen. when he was told that the problem he had just done was designed

for the top 18 year olds in the country and that his solution was both soph-

isticated and elegant Mark assumed a new authority overnight.

Work that was before competent and meticulous became not quite so neat,

riddled with new theories, almost totally correct and nearly always going

beyond the confines of the set problem. He has, quite simply, become one of

the best four or five mathematicians in his year group and is constantly develop-

ing the confidence to try new problems and exercise his belief in himself.

What honest mathematician and teacher, in the privacy of his own conscience

would not admit to some branch of the subject which fills him with awe and

apprehension? How many of us can remember a time when we felt about the

subject as Mark does now?

5. Clive and Richard

There is far more to being a mathematician than having flashes of inspiration,

moments of insight, unusual perceptions and isolated successes. One of the characteristics of the mathematician is a willingness to put in the hard work,

time and concentration when necessary.

Clive's attempt at the 'Co-Prime' investigation, which he produced for one

of our colleagues was a minor classic. Like a fifteen year old Lakatos his twelve pages of script were full of 'Proofs and Refutations' showing how he

had toiled, adapted his theories, changed his mind and finally made his decisions. His work and ultimate solution followed much the same lines as

Dawn's and would be included here except that it is long enough to warrant a separate publication. Not only did it show unusual talent for one so young,

but it also demonstrated his willingness to give of his time to his mathematics.

Talent supported by energy is a complete formula for success. What made it so particularly nice is that Clive is taught by one of our

colleagues, who after showing the work around the department offered the

comment, "I doubt if I 'd be in teaching if I had his talent." Such incidents go a long way to establishing the unity and common spirit so vital to the

organisation and running of a successful mathematics department. The sharing

amongst interested and involved colleagues is surely one positive way of

relieving the pressures of a taxing job. In a similar way having boys like Richard around serves as a continual

source of energy. It was interesting to observe the differences in style when

GROWING TALL 163

Richard and Martin (about whom we tell you next) shared the same lesson on

Diophantine equations. At the end of the lesson the bell rang and Martin

dashed off, perhaps to see his girlfriend, possibly for a cigarette or maybe to

kick a ball. But for ten minutes or more Richard sat quietly copying notes

from the board, making absolutely sure that he had not missed any of the important points. As he left the room we whispered to ourselves 'professional

mathematician', and we would lay a heavy wager now that this fourteen year

old will earn t~is living doing mathematics. Richard experienced an unexpected bonus for his diligence and hard work.

His elder sister was at home from Oxford for the Easter holiday of 1979.

She had just started the University course on elementary Number Theory.

There was a point she was not quite sure about and as she was talking to her

little brother about the work, he was able to sort out her difficulty. He enjoyed

that.

7. A n d then there was Martin

We have put a lot of effort into developing the individual talents of our pupils.

Clive has always been classed as good but Richard is the only one who came to us with extremely high credentials. Paul, Dawn, Mark and the others men-

tioned here are the products of our system. Dotted around the school we have some, and not just those mentioned in this paper, really talented indi-

viduals. But shining above them all, the star at the top of our tree, the jewel in

the crown, is Martin. Allow us to recapture just a few of his comments made at the tender age

of 13.

In his f irs t ever lesson on trigonometry.

Trigonometry is introduced through the idea of the projections of a unit,

rotating vector, as in Figure 20.

COSINE

FiG.20 There follows some class discussion and the children are asked to rotate

some arms and make some measurements.


Martin

Teacher

Martin

Teacher

Martin

Teacher

Martin

Teacher Martin

"You only need to do it for angles up to 45 degrees"

"Why?"

"Because Sine of 70 is the same as Cosine of 20, and its obvious

anyway."

"What is Tan 90?"

"Hasn't got one"

"What do you mean?"

"Infinity"

"Give me another angle whose tangent is infinite"

"270, - 9 0 , there's lots of them"

Whilst we were doing the introductory lesson on trigonometry he sat scribbling

a solution to a three-dimensional geometry problem taken from an 'O' level

examination.

Teacher "How do you know how to work out the volume of that

tetrahedron?" Martin "Is that what you call it. Well three of them make up a box"

Teacher "How do you know it?"

Martin "Well you can see it. It's obvious"

At the start of Permutations. Teacher "I 'm just going out of the room. You saw me put those books on

the shelf. Imagine I had three books: in how many ways could I

arrange them side by side?"

Martin "Six" Teacher "Okay, well whilst the others try it for three you try it for four

books."

Martin "That will be twenty four"

His reply is so immediate that you feel sure he has either guessed it or remem-

bered the result from somewhere else, so you ask him how he knows.

Martin "Well those three can go in six ways. Now your fourth one can go in four ways for each one of your six, so that's four lots of six.

And that's twenty four. It's obvious really."

He came close to ruining our generative lesson on the 'Routes on a Cube'

investigation this year. The children had been introduced to the basic idea,

invited to offer various routes from one vertex to another and then given

the investigation (see Appendix 1).

GROWING T A L L 165

Martin "Mr Kent, there's 16 routes on the cube, and 12 on that one, and

20 on that one, and . . . . "

He shouted out a list of correct results to all the polyhedra we had available

and then went the one stage further.

Martin "Wait a minute. It could be 15, 11 and 19 and so on, that happens

when the corner you end up at is the one directly below the one

you start at."

D.K. "How do you get that?"

Martin "Well say you start at this corner, you have two choices of which

way to start, then you go along that edge until you get to this

corner and then . . . . "

The power of the 'combinatorics' going on in this boy's head was quite aston-

ishing. Like Paul's written contribution it offered a perfect solution, only unlike

Paul's this was merely a mental exercise to Martin. The only thing that stopped

him spoiling the entire lesson was that none of the others in the room seemed

to be able to follow his reasoning.

In the first lesson on Matrix transformations

"We should know now how to multiply matrices and we want to use them to

shift points and shapes in the plane." Figure 21 is projected onto the screen in

the school hall and about 90 children look at it.

i y'

j/ /

/

/

(2,3) / B • / /

/ " A (3) / ,2

/ /

/

X

FIG.21

"What is the matrix which moves the point (3, 2) to its reflection in the line


y -- x , that is to the point . . . . What point, tell me,you . Right, in other words we wish to find the matrix

C ;t(:t=(:) or, if you want to

?(3) 4- ?(2) = 2

and

*.(3) + *.(2) = 3.

Go on, find some simple numbers to make this work, remember your number theory, only now we are using 2 and 3 (see the 3s and 5s investigation that comes next).

David "0 and 1, and 1 and 0"

David gives the matrix (~ ~) and we are happily ready to progress.

Martin "It's not the only one" Teacher "Pardon"

1

Teacher "Eh"

Martin "'(~ - - 2 0 ) t i m e s ( 3 2 ) a l s o g i v e s ( ~ ) . I t does the job for allthe

points on the line." Teacher "What line*."

Martin "The one that goes through (0, 0) and (3, 2).

Teacher "How do you know?" Martin "It's obvious"

And so it goes with Martin. At night he can be seen walking around the town

with his friends or his latest girlfriend, even sometimes sitting on a wall staring into space. At lunchtime he often joins one of the groups of rowdy boys who charge around the school with a football, or disappear for a cigarette. When the bell tings to mark the end of a lesson he is usually first out of the door. In the evenings he will opt to play golf, go to a disco, or visit a public house rather than do his homework. He is a most precocious boy who is so competitive and does not possess an ounce of humility. He chatters in class, flirts with the girls and is in need of a lot of control. But mathematically the only way we can

GROWING TALL 167

K.H.

Some lad

K.H.

Louise

Martin

K.H.

Martin

K.H. Martin

K.H. Somebody K.H.

Sarah

Martin K.H. Martin

K.tl.

Martin

come anywhere near to adequately describing him is by offering a description

o f one of our joint, two classes in the room sessions. We call it the 3s and 5s

investigation: the pupils have to tell us which numbers can and cannot be

made with positive integer multiples of 3 and 5.

"Give me two numbers (less than 6 and not 1,2 or 4)

"3 and 5"

"I am going to make up another number using multiples of 3 and

5, like:

17 = 4 ( 3 ) + 1(5)

Now you make me 12"

"4(3) + 0(5)"

"14(3) -- 6(5)"

"No, now you have made me tell you a rule, you can only add"

"-- 6(5) + 14(3)"

"No negatives at all"

"Oh"

"Now make me 3"

"1(3) + 0(5)- "Now 4"

There is a slight pause as young minds start to work.

"You cannot do it"

"Can"

"How?" "�89 4 �89 "Positive integer multiples only"

"You should have said."

At this stage the class are invited to examine the set of numbers that can be

formed by such integer multiples, and particularly the largest number that

cannot be made. In the life of a mathematics teacher a few moments can stand

out above the rest as being special, Martin was about to provide one of them.

The column of numbers starting at 1 are written on the board and we tick off

when we can make a number:

1

3 4 -- 1(3) + o(5)

4


5 ~/ = 0(3) + 1(5)

6 ~/ = 2(3) + 0(5)

7

8 ~/ = 1 ( 3 ) + 1 ( 5 )

9 ~/ = 3(3) + 0(5)

10 ~/ = 0(3) + 2(5)

We think that for a thirteen year old the spontaneous comment now shouted

out by Martin, as if without thinking, is staggering, to say the least.

Martin "You've finished now Mr. Hedger. You can do all the rest."

K.H. "How do you know?"

Martin "It 's obvious"

We will leave that for you to decide.

They are then invited to examine the same problem for other pairs of

starting numbers such as (4, 7); (2, 11); (7, 9); (6, 9 ) ; . . . and to find a generalisation for the largest number that cannot be made.

Martin "You cannot do it for 6 and 9.

K.H. "Why not?" Martin "Because it's 3. You can only make up the multiples of 3"

K.//. "Okay, try it for 6 and 11" Martin "It 's 49" K.H. "What is?"

Martin "The biggest one you cannot make"

K.H. "Why?"

Martin "It 's 66 - 17"

K.H. "Why?"

Martin "You go 6 x 11 take 6 + 11"

K.H. "How do you know?"

Martin "Does in all the others"

K.H. "How do you get that?"

Martin "I saw it" K.H. "How?"

Martin "It 's just obvious" K.H. "Can you prove it"

Martin "Wait a moment"

The theorem is that given two positive integers which are Co-Prime n and m

GROWING TALL 169

say, then the largest number which cannot be represented in the form

"an + bm", in which a and b are positive integers, is " n m - (n + m)".

8. Sheila's Belief

We would like to offer you a typical 'A' level Further Mathematics problem from the Theory of Numbers option.

"Use Euclid's algorithm to obtain the highest common factor of 143 and 91.

Obtain all the integer solutions to the equations

(i) 143x + 91y = 26

(ii) 143x + 9 ly = 20."

It is nice to be able to report that 15 year old Sheila can handle such problems, and in Appendix 4 we offer her solutions.

But being able to do isolated 'A' level problems is of little value in itself,

unless that pupil can also handle the basic work offered to people of that age. We believe that we can use the advanced work to supplement and

strengthen the basic stuff. We believe that children need to develop a belief in themselves and in their ability to master the mathematics they are required to learn. Sheila's former teacher classed her as 'a little below average'. To us she was rather like Mark, a nice kid who works hard and whom we thought we

could get through 'O' level. But now we see her as a future 'A' level math-

ematician who might even go on to study the subject after that. There is obviously a contradiction between a 15 year old being 'a little below average'

and her doing problems set for the top 18 year olds.

When children produce work of a significantly high standard it is likely

that they will start to believe in themselves. Confidence begets energy, energy

breeds success and success is self-generative.

As practising class teachers we are fully aware of the need to teach the

basic essentials and to obtain good examination results. We could not consider

our time at Belper a success if it became clear that our pupils were enjoying

the course we offer but failing their public examinations. Quadratic graphs is a topic which falls clearly into the category of 'O' level

and C.S.E. essentials which form a part of our basic syllabus; it is a very stan-

dard topic which we teach in a fairly orthodox fashion. Sheila came to us with one such graph, disturbed that hers looked so awful: it was hideous, looking

something like Figure 22. Obviously she had made one of the classic mistakes that many pupils

make with negative numbers. But the conversation which followed was short, very sweet and different, very different.


/

Sheila D.K. Sheila

D.K.

"I can't do it" "Don't be silly"

"I can't do it" "Sheila, there is no way in the world that a young lady of 15 who was doing 'A' level mathematics problems only two weeks

ago can possibly convince me that she cannot find her mistake in

something like this problem."

The strength of the argument seemed to appeal to Sheila. With a charming

little smile she retreated to her desk and set about the task in hand. Needless

to say, once she realised that she should be able to find her mistake, actually Finding it was trivial. Within a few minutes her graph was lovely and smooth.

Growing tall is about cultivating confidences and a belief in one's own

ability. Too many children fail in school mathematics simply because having

failed before they believe they will fail in perpetuity.

CONCLUDING REMARKS

We have written earlier that teaching mathematics, and perhaps especially in a

large comprehensive school such as Belper, is not an easy task. We feel that

there is a dichotomy in existence over mathematics and mathematics teaching. To the people on the outside mathematics is seen to be a very difficult

subject; yet to those on the inside, the mathematicians, it is usually quite easy and seems to get easier the further one gets into it. We think quite the reverse is true about mathematics teaching. To the people on the outside teaching mathematics is seen to be relatively simple - one presents the knowledge, some children grasp it whilst others grasp parts of it and some flounder completely. But to some of us such a vision is not good enough, so we get

GROWING TALL 171

more and more involved. To committed mathematics teachers, the act of

teaching the subject is extremely complex and tends to become even more

complicated the more one gets involved in it.

We believe that too many children seriously under-achieve in mathematics

and that far more are capable of success, no matter how subjectively we define

it, than is customarily accepted.

The words "I cannot do it" flow too easily from the lips of adolescents (and others). In our subject those words are supported by a generally accepted

belief that only a few are capable of success; the idea of a natural gift, aptitude

or brilliance for mathematics pervades society. We continually question that

belief; although accepting the reality that some children demonstrate far

greater mathematical ability than others we still believe that virtually all children are capable of or potentially capable of trapping the elusive butterfly

of success. Seven decades ago Henri Poincar6 put it so provocatively by writing

"A first fact should surprise us, or rather would surprise us if we were not so

used to it. How does it happen that there are people who do not understand

mathematics?" Surprise us it should, but it does not only because it happens so often.

Happen it does though, so we are faced with a tremendous variety of

mathematical talents when the children arrive at Belper High School. But

we do not pay a great deal of attention to that fact, merely accept it as a starting point. It would be wrong for anyone to think that we have written an

article about some of the work of the 'gifted' children at Belper. Of those mentioned, only Richard came to us with outstanding credentials from his

previous school or teacher. The others were all 'average', 'fairly bright', 'slow',

'weak', 'remedial', or given some description which did not single them out as

'gifted'. They are some of the products of what we have tried to offer and of their growing beliefs in themselves.

Martin is perhaps the exception. His mind is so quick, so clear and so alert

that we do not know how much credit we can claim. Certainly we keep him

under strict control and provide him with enough in the way of a stimulus to

keep him busy. He still amazes us: at the time of writing this he was busily working on some of the properties of p(n) - the number of partitions of a given integer - with a copy of Hardy and Wright's 'The Theory of Numbers'

at his side. We provide the stimulus (a problem) and he provides the response

(a solution, a proof), often amazing us in the process. The 'natural' powers of the mind characteristic of the mathematician are so clearly developed in

Martin and seem to be so without too much effort on our part. But we believe that there is a 'little bit of Martin' in all of our pupils and that much of what

he has they have, but their latent powers lie dormant, perhaps seized up


through lack of use. We accept the challenge, carry the responsibility for

making these boys and rids feel the presence of their own potential.

The teaching of mathematics, in schools or elsewhere, is governed by an

interwoven set of complex parameters. We are not so foolish as to claim to

have 'solved' the problem - we have not and we have our failures just as we

have our successes. But we do believe that by working together as we have

at Belper we have helped a number of young people to grow in confidence,

to develop a belief in themselves and to gain access to and a healthy regard

for mathematics and themselves. In short, to grow tall.

APPENDIX 1. ROUTES ON ACUBE

The children are asked to examine the relationship between the number of

edges on the base of certain polyhedra and the number of routes from a 'top'

corner to a 'bottom' one. We throw in the rules (i) you can never move up-

wards, (ii) you can never go along the same edge twice in one route. They

are asked to generalise the result.

What follows is a copy of Paul's work, and we would like to remind you

that in the time it takes you to read through Paul's solution Martin had done

this investigation in his head.

Paul drew Figure 23 and completed the table below.

I I 3

3 4 5

No. of sides on the base No. of Routes

3 5

4 7

5 9

6 11 N 2N--1

FIG.23

i.e.,f: x -+ 2x -- 1.

GROWING TALL 173

7~--8. -9 dO -II---12

5~--8~--7~I 2

L " t 0 .11 -12 3---d0<-9 ~8----7 42

,2----I , g z ,8__vZ_! 12 t - - f ~

1 5

1 2 - - . . . . . , ~ i B d2 7

V .12

Fig. 24.

No. of edges on base and top No. of Routes

3 11

4 15

5 19

6 23 N 4 N - - 1

i . e . , f : x -+ 4 x - - I .

We gave it a mark of 11 out of 10 and asked him if he would make a display

unit for the department. That display is the centre piece of the mathematics

department's suite, hopefully acting as a spur to the other pupils.

.

2.

APPENDIX 2. THE CO-PRIME I N V E S T I G A T I O N

State whether or not these pairs of numbers are Co-Prime

(a) 3 and 7 (b)4 and 6 . . . (j) 136 and 391

Complete this table

Integer N Integers tess than N The Number of

and Co-Prime with it such integers

1 2

5 1 , 2 , 3 , 4 4

6 1,5 2

24

174 DAVID KENT AND K E I T H H E D G E R

3. We shall let ~ n ) denote the number of integers less than n and Co-Prime

with it. (a) State clearly and prove the relationship that exists between n and ~(n)

for n being prime.

(b) Work out ~(3), ~(4), ~(2), ~p(6) and ~o(12).

(c) Does ~p(3) x ~p(4) = r

Does ~(2) x ~(6) = ~o(12)?

(d) Under what circumstances does ~o(n) x ~(m) = ~0(nm)? (e) I f p is prime and a a positive integer work out a formula for ~p(pa).

4. The big, big, big one.

Two positive integers n and m are said to be Co-Prime if their only common

factor is unity. Show that n and ab are co-prime if and only if n and a are

coprime and also n and b are co-prime.

Let ~ n ) be the number of positive integers less than n and co-prime with it,

so that for example ~o(6) = 2 since 6 is co-prime with 1 and 5 only. Verify that

~ 2 4 ) = r and also that ~0(42)= ~ 6 ) x ~0(7). Show that, if p is a prime

number and m is a positive integer

(i) ~ p ) = p - - 1 .

(ii) ~(pm) = pro-1 (p _ 1).

A P P E N D I X 3. THE LAST D I G I T I N V E S T I G A T I O N

1. Copy and complete the following table.

Last Digit of

N N 2 N 3 N 4 N s

1 2 3

12

2. State an obvious theorem. What is the last digit of (387529647) s ?

3. Solve the congruence equations

(a) 3x - 1 Mod (5)

etc. 4. Prove that the sequence 1! + 2!, 1[ + 2! + 3 ! , . . . . contains exactly one

perfect square.

G R O W I N G T A L L 175

A P P E N D I X 4. S H E I L A ' S S O L U T I O N S

To solve 2x + 3y = 5 you guess a solution x = 1 , y = 1 will do.

For the Kernel you solve 2x + 3y = 0.

-3y So 2x = - - 3y and so x -- ~ .

2

Put y = 2k then x = - - 3(2k)_ = _ 3k. 2

So the Kernel is x = -- 3k, y = 2k.

So the Complete general solution is, g u e s s + k e r n e l so x = 1 - 3 k ,

y = 1 + 2k for integer.k.

In the problem 143x + 91y = 26 she was able to reduce it to 1 lx + 7y = 2,

guess a solution x = 4, y = - 6 and then proceed as before. She was also able

to dismiss 143x + 91y = 20 as having no solutions because "20 is not a

multiple of the Highest Common Factor of 143 and 91 ."

A P P E N D I X 5. B R I T I S H E X A M I N A T I O N S

At the end of the fifth year our pupils are entered for either the General

Certificate of Education Ordinary level examination (G.C.E. 'O' level) or for

assessment at the somewhat lower level of the Certificate in Secondary Edu-

cation ( C . S . E . ) - a top, grade 1 pass at C.S.E. is officially recognised as

equivalent to an 'O' level pass. 'O' level is designed for the top 20% of the

abil i ty spectrum, but we would feel disappointed i f less than 30% of our

intake passed this examination, and hopeful of many more. (The only statistic-

ally reliable product o f our joint beliefs, philosophy and methodology has

brought an 'O ' level pass rate o f over 40% for 1978-79.)

For those successful at 'O' level we offer courses in Mathematics and

Fur ther Mathematics at the Advanced (A) level of the G.C.E. These exam-

inations are designed for the most able 18 year olds, with the Further Math-

ematics being for the very top pupils. A few pupils are offered the oppor tuni ty

to sit for the post advanced Special or 'S ' level examination. Officially an 'S '

level Further Mathematics paper is the highest possible public examination a

school mathematician can enter.

A P P E N D I X 6. S P I N - O F F

We set the following homework question to our 4th form - the same kids we

did our number theory with last March:

176 D A V I D KENT AND K E I T H H E D G E R

"A ruler costs twice as much as a pencil. 15 pencils and 6 rulers

cost s Calculate the cost of a ruler and a pencil."

Mark - aged 14 - solved it by:

Let x = cost of pencil

y = cost of ruler

lSx + 6y = 108

guess solution x = 2 y = 1 3 .

Kernel:

1 5 x + 6 y = 0

- 6 y X - -

15

- 2 y 5

put y = 5k

x = -- 2k

So kernel is (-- 2k, 5k).

Therefore general solution is (2 -- 2k, 13 + 5k)

k = 1 (2, lS)

k = o (2, 13)

k = - - 1 (4, 8) Solution.

Therefore Pencil costs 4p

Ruler costs 8p

That shocked us, because Mark has not touched the number theory for

several months. What is more we have been teaching them to solve similar

equations by the more or thodox methods.

NOTES ABOUT:

Caleb Gattegno

During two or three decades of substantial innovations in the teaching of

mathematics Caleb Gettegno's ideas have not reached the general teaching

public to the same extent as those of some of the other contemporary writers

GROWING TALL 177

on mathematics education. He has remained, in this way, as some sort of

under-ground or cult figure whose ideas are appreciated by only a few.

From what we have seen, reading lists for student teachers of mathematics in training usually include the works of such as Piaget, Dienes, Skemp and

others but only rarely give mention to Gattegno. We suspect that something

similar may well be true outside the United Kingdom. We are both grateful to

Dennis Crawforth and Dick Tahta at Exeter University for introducing us to

Gattegno's works and believe that the benefits we so received should be made

available to more people.

Gattegno's ideas are not easy to digest and we feel that it would be unfair

to try to sum them up in a few words. Anyone wishing to find out about Gattegno for themselves could do no better than read from his works on

mathematics teaching which include the books:

The Common Sense of Teaching Mathematics,- Educational Solutions, 1973. What We Owe Children, - Routledge and Kegan Paul Ltd, 1971.

Gattegno Mathematics, Books 1-7 , -Educat ional Solutions 1969 to 1973. For the Teaching of Mathematics, Vols 1 to 3 , - Educational Explorers, 1963.

Mary Everest Boole

The story of the fife, love and tragic early death of the English 19th century

mathematician George Boole is the kind that, were its 'hero' someone other

than a mathematician, would make suitable material for the 'Hollywood script

writers'. His story includes his marriage to the young Mary Everest who was

left at the age of 32 to bring up 5 children; that alone would be enough for many women.

There can be no doubt that marriage to George Boole had a profound effect

on Mary Everest. He himself produced much in the way of new and original

mathematics but wrote very tittle on the act of mathematical creation and mathematics education. As a one-time school teacher of mathematics George

Boole had many thoughts on the teaching of his subject, thoughts which he shared freely with his wife.

With her 5 children grown, and she herself 50 years of age, Mary Boole set

out on a career as a writer. She wrote freely and it would be very difficult,

perhaps impossible, to define her 'field', but certainly her writings covered

much that is to do with mathematics education. Mary Boole died, aged 84, in 1916. In 1931 C. W. Daniel produced four

volumes of her collected works, simply titled - Collected Works, Mary Everest Boole. Those attracted by the titles such as The Forging of Passion into Power, or Philosophy a~ut Fun of Algebra or the psychological interpretation of her


husband's work and how it related to the ideas of the French Oratorian Gratry, The Mathematical Psychology of Gratry and Boole, will surely not be disappointed. But as with Gattegno, Mary Boole's ideas are not always easy to grasp and how they might relate to current classroom practice is something which needs to be worked on and developed.

She is very much a 'lost figure' amongst mathematics educators. Her style of writing hardly appeals to 20th century tastes and it is only of late that a few have been rediscovering her themes. Some of her insights into the nature of mathematical thinking are, so we believe, profound.

The Collected Works are now out of print and not easy to obtain from

libraries. Perhaps the most accessible introduction to her work is the booklet A Boolean Anthology compiled by D. G. Tahta and published by the Association of Teachers of Mathematics (Market Street Chambers, Nelson, Lancashire, England) in 1972.

J. L. Nicolet

It was around 1940 that the Swiss teacher of mathematics J. L. Nicolet first

had the idea of using animated geometrical drawings as an aid to teaching

classes of pupils in the age range 10 to 18 years. Gattegno became enthusiastic about Nicolet's use of film and did much to promote the idea.

We see a strong correlation between Nicolet's films, Gattegno's use of such,

the fact that Mary Boole invented 'curve stitching' and all of their joint beliefs in 'the dynamics of imagery', 'moving mind pictures', and 'mathematical intuition' or 'the act of creating mathematics'. In this article we have written about some of our sessions using the Nicolet films:

Three points determine a circle. The generation of the ellipse. The generation of plane curves.

We obtained our copies of these and other similar films through Educational Explorers Limited of Silver Street, Reading, England.

In Section 4 of Volume 2 of For the Teaching of Mathematics, Gattegno writes fairly extensively about the use of films in the teaching of mathematics. Using films such as those produced by Nicolet in the manner described by Gattegno and ourselves is an essential feature of our mathematics teaching.

ACKNOWLEDGEMENTS

We would like to acknowledge the debt we owe our colleagues at Belper. Expecially we thank Phil Fisher for doing the artwork for us. Our main thanks

G R O W I N G T A L L 179

though must go to all the kids at the school, w i thou t whose help noth ing in

the way o f 'Growing Tall ' would have been possible.

Belper High School, Derbyshire

R E F E R E N C E S

Hedger, K.: 1976, 'Hey Mr. Hedger, what's all this rubbish? What have you been reading?' A.T.M. Supplement 19.

Kent, D.: 1979, 'Isobel', Mathematics Teaching 85, 12-14.

ABSTRACT. If the process of education is about freeing pupils to enable them to take a more positive role in their learning, then the task of teachers becomes concerned with examining the ways in which this can happen.

Over a two-year period at Belper High School we have offered the pupils opportunities to master our subject, to develop their confidence in their ability to cope, to 'grow tall' within and through the subject itself.

Pupils across the full ability range are described in their sometimes faltering attempts at topics varying from solving simple equations to sophisticated number theory and loci problems.

Actual teaching is demanding, tiring and sometimes confused with spurious and krel- evant activities. What we describe are sessions, observations and individual transformations which have generated, rather than dissipated, energy.

This is a descriptive analysis of our methods and observations, offered in the hope that others will accept the challenge, endure the failures, examine the idiosyncracies and eventually experience the joys available to all involved in interactive teaching.

growing tall

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