TAMAS VARGA

ON P R I M A R Y S C H O O L T E A C H E R S ' M A T H E M A T I C S

1. PRE-MATHEMATICS AS A WHOLE

Not very long ago the prevailing opinion was that mathematics proper could not

be assimilated before the age o f 12 -+ 2, but arithmetic could serve as pre-mathe-

matics, laying a solid base to subsequent, more sophisticated ideas. Nearly all teachers now in service have been trained in this philosophy.

The last few decades have shaken this philosophy, but they have not replaced

it by a new one which would be as universally accepted as the earlier. Even so it may not be unjust to call the earlier philosophy obsolete, in the light o f con-

temporary psychology, mathematics, educational research, and classroom ex-

periences. Without denying the existence o f important changes in mathematical thinking during the period of pubescence, pre-pubescence has been found a

particularly susceptible period to a wide range o f mathematical ideas. It seems that some sort o f pre-mathematics as a whole rather than arithmetic can serve as a solid base for further or proper mathematics. But even those who would agree in such broad terms, have rather different views about the details.

2. TEACHER TRAINING IN THE CENTRE. ITS DIMENSIONS

Teacher training at present shows the most motley picture, with solid strong-

holds o f obsolescence, impulsive armoured troops ready to destroy the remnants o f the old frames, and more or less successful attempts at integrating whatever

is found serviceable in old and new. Curriculum planning and actual classroom

practice can be analysed in similar terms, with different proportions of the

main ingredients. But the trend is largely determined by the course teacher training takes, especially if it is meant to include in-service training, the influence o f non-formal meetings, the press etc. No curriculum planner can tehch instead

o f the teachers (not even through the most outstanding programmed materials -

by the way, where are they?), but no matter what the actual classroom practice is like, it can undergo very deep changes in as short a period as a generation: to its improvement or to its deterioration alike; or possibly to one and to the other in different respects.

Education is reputed to be the most conservative component of human culture. Unfortunately? Or rather, fortunately? Were it not so, much harm could be done to humanity.

Educational Studies in Mathematics 7 (1976) 171-177. All Rights Reserved Copyright �9 1976 by D. Reidel Publishing Company, Dordrecht-Holland

172 TAMAS V A R G A

Teacher training is by a long way not the only means of influencing education.

Curriculum planning, educational technology, the advancements of technology in

general (e.g. calculators, computers, the whole range) and many other agents

have their share. Were I asked about the most influential agent, I would vote for

teacher training in the above broad sense - combined, of course, with other

agents, not independently of them. Teacher training itself has various dimensions, both cognitive and affective.

Much research has been done in order to find out what makes someone a success-

ful teacher. What I could distil from the available literature and personal ex- perience is that there are some indispensable traits, which include understanding mathematics and the child (e.g. his ways of thinking), and liking mathematics and the child (e.g. accepting him with his imperfections), none of which can be

replaced by any excess of the rest, though some weaknesses can be partially

compensated. This seems to be equally valid for teachers of teachers.

With all that in mind, let us now restrict our attention to the mathematics

needed by a primary school teacher. One important question is: how much mathematics? Another, still more important: what sort of mathematics?

3. C A N TOO MUCH M A T H E M A T I C S DO H A R M TO A T E A C H E R ?

A non-specialized teacher, for whom mathematics is one of four to eight subjects he has to teach and still more to learn, cannot be expected to devote much time

to mathematics during his initial training. The specialization of teachers in the primary school is a controversial ques-

tion. In the United States and a number of other countries the usual practice is to have all-round teachers in the first s/x grades, i.e. up to the age of 12 or so. In

other countries (one of them is the G.D.R.) this is only the case in the first three grades, up to the age of 9, if not currently, then prospectively. There is also a

trend to blur the line of division between the segment where all or most of the

teaching is done by non-specialized teachers and where the contrary is the case. For instance after the first or second grade partially specialized teachers share the subjects between them. This may or may not be reflected in their initial training, possibly only in their further training, or only their line of interest. However it may be, the mathematical training of a partially specialized teacher can hardly come up to that of a specialized teacher. His less extensive mathema- tical training is, however, not necessarily a serious drawback. The most dangerous trap of a teacher of mathematics, that of seeing his subject as an end in itself, can more easily be avoided by an all-round teacher than by a specalist.

Another point is that of superiority. The highest ambition and source of self- respect for a teacher of mathematics should not be the possibly great difference in mathematical understanding between himself and his pupils, but the extent

O N P R I M A R Y S C H O O L T E A C H E R S ' M A T H E M A T I C S 173

to which he can reduce it. Yet a specialized teacher is more likely to have am-

biguous feelings if the difference is reduced too much, or even reversed. He is

usually less ready to admit he is in the wrong to a pupil. Also, he is less ready to admit his lack o f knowledge when he is in the role o f a student. For a non-

specialized teacher both are easier. One of our most striking experiences during 13 years o f pilot work in Hungary has been the advantage o f non-specialized

over specialized teachers in both situations. As a consequence, the proverb that

every beginning is difficult does not accord well with our reform of mathematical

education. The beginning, with non-specialized primary school teachers, has

been found to present less problems than the continuation, with specialists. Is this a local phenomenon with local causes? Or is there something more general

behind it? Can mathematics immunize someone against further mathematics? Or is it rather pseudo-mathematics and pseudo-education which gives such mis- service? My only reservation about the latter wording is that it suggests too sharp

a dichotomy.

4. P R I M A R Y S C H O O L T E A C H E R S ' M A T H E M A T I C S : W H A T S O R T O F ?

Let me present my description of the dimensions of teacher training. True, it

is rough, vague, contestable. But it is an attempt to find and use a mathematical

model. Maybe only to express an idea, maybe with an eye to drawing inferences,

designing an experiment, an evaluation. I have in mind a vector-scalar function,

the success (a scalar, for simplicity) depending on a number o f independent variables. My spatial imagination cannot go beyond three. Surveying a number o f

possible mathematical models o f the given type (vector-scalar functions), I reject those which are clearly inadequate. The sum of the modules is one o f them. That

would mean: all but one of the component vectors can vanish, provided that the one which does not is sufficiently large. The volume o f the parallelepiped (the

mixed product o f the three vectors I imagine) is a better candidate. This is just one way of looking at an (initially) non-mathematical situation

from a mathematical angle, and not necessarily the best. Moreover, the best

angle is not necessarily a mathematical angle. But those angles add another dimension (again: dimension) to thinking and to changing ideas. Lacking one

dimension is a very sad state. Nobody actually lacks it. Mathematicians and non-mathematicians, con-

sumers o f much or o f very little mathematics, everybody uses mathematical models wherever he looks and whatever he says. They are hidden in every (well, in almost every) sort o f human speech and thinking. But our resources o f mathe- matical models can be richer or poorer. We can be more aware or less aware o f each of them. And we can apply them more adequately or less adequately to a situation. (That last 'And' does not stand here to suggest that there may not be

174 TAMAS V A R G A

further variables - just to avoid the word ' d imens ion ' - , only that we restrict

our attention to these three.) I consider it to be a quality o f high priority in

primary school teachers that they be aware o f a good sample o f mathematical models in their relationship to a great many situations. It is o f higher priority

than skill in solving the sophisticated problems of the usual examination papers.

Those are mostly problems within a mathematical model; or problems of ap- plication with an intended particular mathematical model in mind. Open situa-

tions, in which there is no intended model, are rarely presented in papers which decide on enrolment, or promotion, or a diploma. That would makeevaluation

too awkward. Fitting a mathematical model to reality is no mathematical problem in the

strict sense. Very often it requires personal judgment and decision, which can

be challenged. This sort o f activity is performed on the borderland of mathe- matics and non-mathematics. But that is precisely what is most needed by some- one who is to teach primary school mathematics. The reason is clear: that is what is most needed by the average user o f mathematics. They may like mathe-

matics as an entertainment, or appreciate mathematical models for their

beauty (see below!), but what they mostly need is mathematics as a tool. At an

informal level this means hardly more than seeing the world (both the observable

world and that of ideas) with a mathematically educated eye, being able to understand and properly to express views about it in mathematical terms, and

to distinguish an adequate use o f such terms from pompous misuse. (I say, hardly more. But has the bar already not been set very high? May I recall that

our topic is now what sort of,, rather than how much.) At a more formal level it means solving problems of the environment or of imagination by the use of

mathematical models. This more formal level appears very early; a ~vord

problem' which leads to a subtraction like 16 - 10 is an example. An author

whom I have been discouraged from mentioning in this paper expressed his

appreciation of the wealth of situations suggested by primary school teachers which are related to such simple models.

5. FOLLOWING IDEAS TO THEIR DEPTHS

I come back again to the word dimension. We live in a three dimensional world. Which three? 'Length and width and height', we have learned. Or forward and backward, left and right, up and down. Not a very deep view of the dimension idea, though at a very low level even that may be revealing. Rethinking shows

that it is restricted to bricks (cuboids), or objects packed that way. Those same persons who have the length-width-height idea of what spatial

dimensions are, may understand and use the word dimension in other, more

abstract contexts, without ever realizing that there is something in common. The

ON P R I M A R Y S C H O O L T E A C H E R S ' M A T H E M A T I C S 175

number o f dimensions considered as the number o f data (co-ordinates) needed to individualize an object is not a very difficult concept. It is deep without being

difficult. I f it still remains hidden, mathematical education is at fault. I do not

mean here the topological idea o f dimension, which leads very far, though it

starts, too, at a very intuitive level. But the 'number o f data needed' sense can be

grasped very early through familiar examples. I mean, for instance, the (usually) four dimensions o f the logic blocks; the four words in the name of each, e.g.

small blue thick triangle, testify for them. You can even match them with the

spatial dimensions (in their most simple cuboid sense), if you get rid o f one

A---A , o

2 0

Fig. 1.

i / i I , I 7, .,&--'/--7 /

# ' / I ~ # / I s I : / /

3D

attribute. See Figure 1, with subsets o f two blocks 'along each dimension'. As another example, consider the six dimensions o f six letter words. Or the data o f

persons (of the pupils themselves) as their dimensions. Decide how many are, needed, and what kind, so that everybody can be identified. The idea can be followed in other directions. Take the three dimensions o f the set o f circles in

the plane; or. the six o f the triangles, which is reduced to three if they are only

distinguished up to congruence, and two, if up to similarity. Another direction, especially with finite sets, is the arbitrariness o f the number o f data, its depen- dence on the sort o f data. Even one is always enough, a whole number; the

serial number, after having established an order. But is a two digit number really one datum? Even if we can say it is, don ' t we have a more sensitive measure o f how much we need to know if we consider it to be two data? Do we not obtain a still more sensitive measure if we use binary numerals? Here is a ramification o f the idea of measuring information in bits, at least in a simple special case.

Another possible ramification leads to the idea of distance, again in a deeper and richer sense than usual. The spatial idea can be enriched in such steps:

Distances on surfaces which can be spread (nape o f a cone, walls o f a room)

176 TAMAS V A R G A

sets o f points, a shortest path - is there always one? (Open segments on a line

defined by inequalities.) Distances on surfaces which can be spread (surface o f a cone, walls o f a room)

and which cannot (sphere). These can be looked at as constraints. Other con-

straints: mountains or lakes in between; certain lines have to be followed (roads),

everything else is 'in between'; a special case: distances on grids, on a plane, in

the space. Then here is a junction to the more abstract idea: distance o f two logic blocks

arranged as on Figure 1, along the grid. Distance as the number of differences. This opens up the way to metric spaces, in the simplest sense, o f course. In

what ways are we different? Born when, where, colour o f hair, eyes, skin ... We

can decide only to count the different attributes (born in the same year, no dif- ference, in another year, one difference, just as an example), or to measure the

differences (as if there were three sizes o f blocks and we had to climb two units from the bot tom to the top, if the blocks are o f smallest and largest kind, other- wise not different). We can venture such questions as the distance between

Dutch and German, Dutch and English. Which is greater? How to measure? Or a

less serious example: is London nearer to Dublin or to Lisbon ? In what sense

to one or to the other? It is not difficult to find what is in common in such examples: the definition

of a metric space. And it is not difficult to find counter-examples: situations which do not match the idea o f metric space, though they are relevant to the

idea o f distance. Under the usual conditions o f traffic with oriented streets and roads there is no symmetry o f the distance, whether it is measured in kilo-

meters, or in minutes; sometimes not even in money. For the distances o f sets

o f points the triangle inequality does not hold; look at the map, and see that the

distance between Belgium and France is 0, between France and Switzerland is 0,

but Belgium and Switzerland are hundreds of kilometers apart.

6. T H E E N J O Y M E N T OF M A T H E M A T I C S

Children are more likely to enjoy mathematics if their teachers do so. One important source o f enjoyment is the wealth o f brilliant mathematical jokes, problems for recreation, paradoxes, astonishing questions and answers, pas- times. Many of them can be found scattered in books and journals; many of them belong to folklore and are transmitted from parents to children in fortunate families. The teacher can make even these more fortunate, let alone those who

receive less at home. I cannot imagine successful teacher training that does not avail itself o f this treasury. Historically, much of present day mathematics has risen from such sources. They are excellent starting points even today. They can help to involve children intellectually and emotionally.

O N P R I M A R Y S C H O O L T E A C H E R S ' M A T H E M A T I C S 177

To sum up: the most powerful promotor of mathematical education is

teacher training, and within it the training of primary school teachers (nursery teachers included!). The best we can do is to offer them, in formal or informal

courses and through every available channel, what we would like to be reflected

in the classroom and later in adult society.

National Institute o f Education (OPI), Budapest