PROOFS AS A TOOL TO DEVELOP INTUITION
Alexander Khait
Ramat Beit HaKerem,
POB 3566,
e-mail: khait@mail.jce.ac.il
As a result of the computer revolution there is a large increase in high school graduates who continue their professional education in mathematics-related subjects. From the point of view of a mathematics educator there never was so large a population that is not naturally inclined to study mathematics but needs it for professional activities. These students learn mathematics exclusively for practical purposes, which typically mean work with computer-related technologies. So their mathematical needs are closely connected to various computer applications. Two facts should be emphasized in this context:
1. Programmers usually do not prove the correctness of their algorithms.
2. To become a good programmer one has to develop intuition for preparing good samples of inputs to check his programs.
Most methods of advanced mathematics education for non-mathematicians have been developed for physicists and engineers. Relevant mathematics is mostly of the continuous nature: calculus, differential equations, etc. It is based on a limited number of important concepts such as limit, continuity, derivative, etc. These concepts arose from practical experience. The most important contribution of the founding fathers of the modern analysis was the translation of the human intuition into precise terms. Teaching these subjects is typically aimed at creating proper concept images (Tall & Vinner, 1981) that can be adequately applied in relevant situations. Understanding of definitions, while certainly helpful, is not essential. Proofs of theorems are not too important for engineers, neither as the basis to believe in a mathematical fact, nor as an essential part of mathematical discourse. When the extent of the challenge that the proving presents became clear, it was sidelined in the teaching process (Niss, 1999). So for non-mathematicians proofs are seriously taught only to high- level classes as the way to formalize intuition.
De Villiers (1990) gave the following list of roles of proofs play in mathematics:
1. verification (concerned with the truth of the statement);
2. explanation (providing insight into why it is true);
3. systematization (the organization of various results into a deductive system of axioms, major concepts and theorems);
4. discovery (the discovery or invention of new results);
5. communication (the transmission of mathematical knowledge).
Proofs proceed from definitions and involve logical arguments, so all these roles assume an existence of some level of initial abilities of a participant in a proving endeavor (a prover, a listener, a reader) to make sense out of mathematical discourse. This is usually not the case with a typical student I am talking about. The main problem of my students is the lack of understanding the basic structures of standard mathematical linguistic expressions of which definitions are composed (e.g Selden & Selden, 1995).
Discrete mathematics relevant to computer professionals is not a hierarchical theory based on a small number of concepts like calculus, but a broad theme that can be explored in various directions according to specific needs. For each direction suitable definitions are introduced in a matter-of-fact manner. It implies a decrease of importance of any particular contents, facts or methods, while the ability to think and communicate in a proper way becomes crucial. The same may be said about the nature of professional activities of computer experts. The work of a computer professional could be seen as incessant transitions between formal (human-computer) and informal (human-human) communications. Such a specialist has to have working abilities to translate intuition to the formal language creating new definitions (e.g. of functions, structures or classes), to write succinct and comprehensible commentaries and to understand those created by others. Object-oriented programming actually concentrates on definitions. However, one cannot compare a definition of some basic concept like that of the limit with definitions occurring in programming. Definitions of new objects in computer programming have different levels of importance and depth: one does not expect to develop a deep intuitive understanding of each object created for particular needs of an occasionally encountered program; one has, however, to understand these definitions, that is, their meaning in terms of operational consequences. The possibility of being satisfied with intuitive explanations and understandings that could have been proper for past generations is unacceptable now: a computer can be engaged only in a formal talk. So, on the one hand, professionals in computer-related specializations have to have abilities to communicate in a formal language, on the other hand, they have to develop intuition to formulate and discern correct statements (e.g. computer programs), that is, behaving as expected.
All this implies a new role for proofs, namely transitions from formal to informal proofs, striving to distinguish intuitively between true and false propositions. At the first stage the students learn to understand definitions, developing feeling (atomic concept images) for the standard mathematical linguistic structures. Here the formal aspect is stressed. Then, proceeding from this ability a teacher starts to work on intuition development. Intuition is built on the basis of some formal abilities. Initially, to obtain new facts (typically very simple from mathematical point of view) formal proofs are used. Little by little, as intuition develops some formalities are dropped. (Khait, submitted, presents a detailed description of a course based on these ideas.)
In standard mathematical courses there are many theorems proved for historical reasons or because they are important for professional mathematicians. The next example (Khait, 2003) highlights the difference between the two approaches to proofs.
We compare a relative importance of the following two questions
1) Prove the theorem of intermediate value, stating that a continuous function that changes its sign on a closed interval has to pass through zero;
2) Let R and S be two transitive relations on a set A. Prove or refute by an example that: a. RÈS is transitive; b. RÇS is transitive.
There is no question that the former is an important theorem of calculus while the latter is an almost trivial exercise and this is the way most lecturers at college level see them. My experience is that for an average student that belongs to the population we are concerned with, the former statement is obvious, as far as the student is convinced that the definition of continuity does not contradict a common sense intuition. In his future professional life this will suffice to accept the truth of a fact (remember: programmers do not prove correctness of their programs), so our student typically sees a proof of this theorem as a pure nuisance and word playing. Returning to the latter question we note that most students do not have previous intuition concerning transitive relations. Here a proof and a refutation that involve analysis of the definition of transitivity help to understand the concept of transitivity and (what may be even more important) to develop an intuitive understanding of basic logical structures. These structures are the building blocks of both mathematical statements and computer programs.
So a new point should be added to the De Villiers's list: Proofs as a tool to develop intuition.
References
de Villiers, M. (1990) The role and function of proofs in mathematics, Pythagoras, 24, 17-24.
Khait, A. (2003) Goal orientation in mathematics education. International Journal for Mathematical Education in Science and Technology, 34(6), 847 - 858.
Niss, M. (1999) Aspects of the nature and state of research in mathematics education, Educational Studies in Mathematics, 40, 1-24.
Selden, J. and Selden, A. (1995) Unpacking the logic of mathematics statements, Educational Studies in Mathematics, 29, 123-151.
Tall, D. and Vinner, S. (1981) Concept image and concept definition with particular references to limits and continuity, Educational Studies in Mathematics, 12, 151-169.