At the very beginning of this chapter, we discussed the importance of the logic and rigor of the school course in geometry. Obviously, however, the level of rigor found in Hilbert is not the same as that found in Euclid. What level of rigor, then, do schools need? This pertains not only to proofs of propositions but also to the rigor of definitions and to the precision of language in general.
Let us begin with the latter. It is not difficult to see that Kiselev’s textbook, which has to this day been considered a model of rigor and deductive logic, contains propositions that turn out to be simply false because of what may be called linguistic sloppiness. For example, it contains the following theorem: “The three altitudes of a triangle intersect at one point” (Kiselev and Rybkin, 1995, p. 108). Meanwhile, generally speaking, an altitude is a perpendicular dropped from a vertex of the triangle to the side opposite it or its extension. Obviously, in an obtuse triangle, the altitudes do not intersect at one point; rather, it is the straight lines that contain the altitudes that intersect at one point.
Moreover, generally speaking, from a certain point of view, virtually all theorems that involve areas and volumes are meaningless. Say, consider the statement “The area of a triangle is equal to one half of the product of its base and height.” How can one multiply a base, i.e. a segment?
One should refer, rather, to the length of the base.
Kolmogorov (1971) wrote: “Traditional geometry textbooks are weighed down by the extreme polysemy of their definitions and notations” (p. 17). It turned out, however, that avoiding such polysemy completely is very difficult, while using symbolic notations overburdens the teaching of the course and, most importantly, alters somewhat its direction. The student in effect has to learn a new language and then to pay attention to subtleties of notation — making sure to distinguish betweenAB,←→AB,AB, and other expressions, instead of focusing on−→
geometry itself. Of course, no one would deny that it would be good if all students acquired a command of precise mathematical symbolic notation, but usually the time that teachers have at their disposal is limited and they must choose what to spend it on. Russian textbooks subsequently simplified symbolic notation, writing simplyAB, verbally indicating what was meant or even expecting students to understand what was meant from the context.
Precise definitions are indispensable in mathematics (as in any other science). Moreover, they are vital in everyday life [recall the example cited by Vygotsky (1986) of a child who said that someone had once been the son of some woman but was not her son any longer: the child had formed his definition of “son” spontaneously and associated it with a certain age — thus, an adult could not be
a son!]. The problem, however, is that to give a rigorous definition of, say, a polyhedron is very difficult (Alexandrov, 1981); meanwhile, students already have an intuitive notion of it, which is sufficient for solving certain problems, including quite substantive ones. This intuitive notion may be made more precise when necessary, and various relevant details may be mentioned explicitly, which can itself be useful, but striving to give a complete and precise definition of a polyhedron is probably not useful (at least attempts to do so in Russian textbooks have not met with success — teachers and students have usually simply skipped over them). As Alexandrov (1984b) emphasized: “The purpose of definitions is not for students to memorize them by rote, but to make students’ understanding more precise. We must try to achieve not empty memorization, but effective learning, i.e. learning that allows students to apply what they have learned” (p. 45).
Consequently, in dealing with any new concept, the authors of textbooks — and teachers as well — must confront the question of whether working toward a precise definition of this concept is justified.
In a very large number of cases, such a definition may be given without difficulty (here, we will not discuss the question of how this should be done, but merely point out that, almost always, the precise definition of a concept must grow out of working with the concept rather than precede it). Nonetheless, it should be borne in mind that even the great mathematicians of the past sometimes worked without having at their disposal definitions that we would consider precise according to today’s standards (for example, of a limit).
Attempts to sustain high standards of deductive logic, approxi- mating the standards of modern science, can hardly be considered successful. Schools have rejected them — theorems that were too difficult were simply not proven in practice, and as a result the level of deductive logic fell rather than rose. The school course in geometry is not a course in the foundations of geometry. The high- est level of deductive logic that is feasible in the classroom is the one that should be aimed at, and this should be done by giving teachers and students difficult problems — difficult but not impos- sible. The balance of mathematical and pedagogical considerations
will be different in each situation and depend on numerous social circumstances.