Strategies of Proof in Secondary Mathematics.

IT HAS been the experience of the author that graduate and undergraduate students are not very knowledgeable about direct and indirect strategies of proof used in mathematics. In methods courses for secondary teachers, the author has ob served that even those students who have an undergraduate major in modern mathe matics are not at all conscious as to why books usually check the elements of a solution set to an equation before assert ing that it is indeed the solution set. Neither are they aware, even at an intui tive level, of the "internal" logic of in direct proof. This lack of awareness leads one to question whether it is possible for them to do a passable job of teaching high school students how to prove theorems. The author has on numerous occasions asked secondary school teachers what they did with indirect proof in algebra and geometry. The responses have all been of the variety, "Well, it is so hard to get across to the students that wTe don't really do much with it." Just recently a student teacher told me that her cooperat ing teacher had told her to prove one or two theorems indirectly in geometry so that the "kids" would at least have en countered an indirect proof, but not to expect them to understand the proof or to be able to produce an indirect proof. In view of the emphasis now being placed on mathematical structure in the secondary curriculum, it is surprising that so little has been said about proof. Most certainly, the concept of structure cannot be understood without some idea of the role of proof, since proof is an integral part of the "whole story." Unless teachers understand what we mean by structure and the role of various strategies of proof in building a structure, their teaching efforts will not be in line with the modern movement for mathematics reform. It wilJ be the position in this paper that at least an intuitive knowledge of certain principles of logic is necessary for under standing and teaching proof. These prin ciples may be accepted on an intuitive basis arising out of linguistic conventions, or they may be accepted as a result of a more rigorous treatment that may include the development of truth tables. The posi tion presented here will be that the intui tive acceptance is sufficient and is, in fact, desirable. This position is, of course, debatable; indeed, the whole topic of logic in secondary mathematics is debatable. The author recently heard a mathe matician assert that all logic should be left to the teachers of English. Ignoring the role of logic in teaching secondary mathematics may be the pattern of the past, and it may seem to be inevitable. However, it has not produced creditable results. It will not be the purpose of this paper to debate the alternatives in detail. Suffice it to say, however, that if, in understand ing proof, it is not necessary to recognize the explicit use of certain principles of logic, then why is it necessary to recognize explicitly the properties of rational num bers to understand the concept of a field? From the point of view of the students' acquisition of an understanding of proof, the "leave logic to the English teachers" position is naive and not practical. To understand proof as exhibited in secondary mathematics, it is essential to (1) have a good understanding of what an if-then statement means and (2) recognize the role of selected inference patterns. In