On primary school teachers' mathematics

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    Not very long ago the prevailing opinion was that mathematics proper could not be assimilated before the age of 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 of con- temporary psychology, mathematics, educational research, and classroom ex- periences. Without denying the existence of important changes in mathematical thinking during the period of pubescence, pre-pubescence has been found a particularly susceptible period to a wide range of mathematical ideas. It seems that some sort of 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.


    Teacher training at present shows the most motley picture, with solid strong- holds of obsolescence, impulsive armoured troops ready to destroy the remnants of 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 of non-formal meetings, the press etc. No curriculum planner can tehch instead of 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


    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?


    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


    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 of knowledge when he is in the role of a student. For a non- specialized teacher both are easier. One of our most striking experiences during 13 years of pilot work in Hungary has been the advantage of 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.


    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 of independent variables. My spatial imagination cannot go beyond three. Surveying a number of possible mathematical models of the given type (vector-scalar functions), I reject those which are clearly inadequate. The sum of the modules is one of 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 of the parallelepiped (the mixed product of 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 of much or of very little mathematics, everybody uses mathematical models wherever he looks and whatever he says. They are hidden in every (well, in almost every) sort of human speech and thinking. But our resources of mathe- matical models can be richer or poorer. We can be more aware or less aware of 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


    further variables - just to avoid the word 'd imension' - , only that we restrict our attention to these three.) I consider it to be a quality of high priority in primary school teachers that they be aware of a good sample of mathematical models in their relationship to a great many situations. It is of 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 of 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 of 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 of 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.


    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


    number of dimensions considered as the number of data (co-ordinates) needed to individualize an object is not a very difficult concept. It is deep without being difficult. If it still remains hidden, mathematical education is at fault. I do not mean here the topological idea of dimension, which leads very far, though it starts, too, at a very intuitive level. But the 'number of data needed' sense can be grasped very early through familiar examples. I mean, for instance, the (usually) four dimensions of 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 of one

    A---A , o


    Fig. 1.

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

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


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

    the plane; or. the six of 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 of the number of data, its depen- dence on the sort of 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 of 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 of 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 of a cone, walls of a room)


    sets of points, a shortest path - is there always one? (Open segments on a line defined by inequalities.)

    Distances on surfaces which can be spread (surface of a cone, walls of 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 of 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, of course. In what ways are we different? Born when, where, colour of 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 of blocks and we had to climb two units from the bottom to the top, if the blocks are of 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?



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