88 H. Guggenheimer

1221 D. VAN HIELE GELDOF, De didaktiek vun de Meetkunde in de eerste klas van hct

[23] R. DAVIS, Discovery in Mathematics (Teacher’s Edition) (Addison-Wesley 1964). [24] Les modeles dans l’enseignement mathematique, Documentation no 5, Ministere de

I’instruction publique (Bruxelles 1965). C. GATTEGNO et al., Le matbriel pour l’enseignement des math6matiques (Delachaux et Niestl6, NeuchAtel 1958).

V. H . M . 0 (J . M. Meulenhoff, Amsterdam 1957).

A CONSERVATIVE’S V I E W O F T H E R E F O R M O F S C H O O L MATHEMATICS

by H. GUGGENHEIMER, Brooklyn, USA

1. Education is first and foremost a social phenomenon. In the United States, where elementary and secondary education is controlled by local School Boards with, usually, a minimum of interference by State Agencies, it is always a very hot political issue. However, the political problems involved in any instructional reform, (mainly the problem of instruction for the maxi- mum number of children versus high quality instruction for a small number of gifted children) will not be discussed here. I shall be concerned solely with the instruction in College Preparatory Mathematics.

For the orientation of the European reader, I want to point out the main features which characterize the situation in the United States as compared to Central Europe. The United States were not touched by the reform move- ment which between 1910 and 1930 replaced the traditional view of geometry (Euclid teaches logics first and geometry second) and algebra (manipulate formulae) by a more modem and mathematical approach. Until a few years ago, a ‘modem’ book in geometry would show its modernity by using a very great number of axioms in order to make explicit most of the hidden assump- tions in Euclid. Then it would teach the proving of a great number of useless theorems in a painstaking fashion. An algebra book very often would teach the rules by drill before an explanation was attempted.

Another feature characteristic of earlier mathematics texts, and which is variously ascribed either to effects of the Depression, or to the tendency to educate college-bound and terminal students in the same class, or to the influence of JOHN DEWEY,iS the emphasis on the social utility of mathema- tics, Naturally, no such book will teach useful knowledge in Statistics.

A Conservative’s View of the Reform of School Mathematics 89

It will therefore be clear that some reform was long overdue. On the other hand, the absence of any fixed requirements for the curriculum other than a due regard for the different College entrance examinations gives an unlimited field for experimentation with curricula and teaching methods. Similar exper- imentations are quite impossible in countries with government standards for graduation and/or centrally administered final examinations.

2. Two problems must be considered for any reform of mathematical instruction: topical content and methods. It is quite important to keep these two aspects well apart.

As an introduction to a discussion of the topics which should be covered in any mathematics course for collegebound students, I want to quote from Professor BUSEMANN’S introduction to the book ‘Convex Figures’l) :

‘Plane Euclidean Geometry introduces the student to deductive reasoning ; if he is gifted, it awakens his geometric intuition, one of the sources of math- ematical inspiration: and it gives him a sense of historical continuity, because the subject has come down to us from Euclid almost unaltered.

Lately all these merits of plane geometry have been challenged. There are mathematicians who would substitute subjects like elementary group and point set theory for geometry. Are these fields, as far as they can be taught in high school, really as important as the properties of triangles and circles, and are they not exceedingly dull compared with the wealth of ideas accessible to the student in Euclidean geometry ? Geometric intuition as a way to math- ematical creation is now not infrequently held in low esteem because modern mathematics is believed to be too intricate or sophisticated for direct intuition. Further, historical continuity is considered of minor importance in view of the usefulness of such modern concepts as group theory. It seems highly presump- tuous to project our present valuations into the future and to interrupt on this basis a tradition of more than 2000 years. If every, or every second, generation were to take this attitude in the future, the resulting chaos would be inconceivable.’

While I agree fully with these remarks, it seems to me that it is possible to be conservative in the topics treated but modern in the methods chosen. Also, school mathematics should recognize the structure of modern mathe- matics and not hide analysis under the cloak of algebra.

Set theory presents a special problem. I have heard the remark that the invasion of set theory into kindergarten is a sure sign that set theory is on its

1) I. M. YAGLOM and V. G. BOLTYANSKII, Convex figures. Translated by Paul J. Kelly and Lewis F. Walton. Foreword by Herbert Busemann. Holt, Rinehart and Winston, New York 1961.

90 H. Guggenheimer

way out as a foundation of mathematics. The symbols of set theory, E, v, and n , are certainly of great value and should be taught early in the symbol- ism. The spirit of mathematics, on the other side, should not be that of set theory but rather that of the functional approach represented by both cate- gory theory and universal algebra. Similarly, it is possible to teach the clas- sical theorems of Euclidean geometry by group theoretical methods on the high school level. In my opinion, too much emphasis has been placed on the introduction of modern topics and much too little on the introduction of modem methods. For instance, instead of basing plane geometry on the real number system via the Birkhoff axioms, it would be much better to have a simultaneous development of plane geometry and algebra and to base the treatment of the real numbers on both geometry and algebra. This treatment will have several advantages. First, the notion of real number is quite compli- cated. The basic notions of geometry are much simpler in their structure. Also, only a very weak form of completeness is needed in Euclidean geometry. A group theoretical treatment of plane (and, if possible, solid) geometry will provide easily accessible non-commutative groups and therefore give meaning to the different laws introduced in algebra. It is then possible to teach algebra as a general theory of computation and not simply as glorified arithmetic.

There really is no sense in discussing details of a program in a general inquiry like this one. Also, my own interest is not so much in the teaching of high school mathematics than in teaching the high school teachers. Neverthe- less, I hope to be able to direct and/or prepare texts for teacher training and high school which is conservative in its topics and functional in its approach.

As Professor GONSETH has pointed out, a philosophy of the development of science must contain a principle of stability. While it is true that the future development of science will force us to make revisions of our opinions which cannot be predicted, nevertheless these revisions will not destroy the body of acquired knowledge. Any revision of the instruction in any scientific discipline should consciously take this stability into account. Just as the requirement of the preservation of acquired knowledge serves as a test of new theories and, a t the same time, insures that a methodology open to unpredictable changes is not relativism, so the same requirement if applied to the instruction in any science will teach us to avoid the chaos feared by BUSEMANN.

3. The main drawback of modern texts and course curricula in the United States is, in my opinion, that they are developed by committees and not by individuals. It is my experience, and is confirmed by reports of my former students who are now teaching in high schools, that most committee-produced texts are exceedingly tedious. In any case, the development of widely diver- gent experimental texts will go on for a long time and, in the absence of

A Conservative’s View of the Reform of School Mathematics 91

centrally administered standards, it is questionable whether they will converge towards a universally accepted curriculum. In particular the SMSG texts seem farther away from being candidates for a uniform curriculum than ever. On the other hand, centrally administered standards are a political impossi- bility and, as the center of graduate instruction moves away from the East Coast, the influence of the standards of the College Entrance Examinations Board will decrease further. One should also expect that in most states the State Boards of Education will not welcome a specialization of mathematical instruction which will make it more difficult for rural high schools to compete with urban ones. (In the upper Midwest, the average level of instruction seems to be highest in smaller cities.)

4. The dimensions of the problem of training and retraining the teachers of mathematics are quite different in Europe and in America. In Europe, college preparatory education is concentrated in a relatively small number of high level schools. This arrangement is strongly biased in favor of the intelligent urban student. A rural or small town student will often be severely handicapped by the distance from home to the next academic high school. In the United States, every local school board provides for a high school which teaches both college-bound and terminal students. This system is strongly biased in favor of the student of average ability. The attempt to cater for the high ability student by enriched or accelerated classes may create political problems. In many cases, extracurricular mathematics and science clubs will be needed to give the necessary stimulus. In any case, the number of high schools is very great and so is the number of teachers of mathematics. The preparation of most teachers in the higher grades is equivalent only to that required from the teachers of lower grades in European high schools. There- fore, a revision of the curriculum in high schools must start with a revision of the curriculum for the training of high school teachers. Unfortunately, today the revisions of the college curricula are hampered by the general dis- orientation of the revision of high school programs. In my opinion, the creation of modem curricula for teacher training receives much too little attention relative to the effort expended on school teaching.

H. Guggenheimer Polytechnic Institute of Brooklyn Brooklyn, New York 11201