Another Approach to Proof: Arguments from Physics

In the first part of the paper we will explore the use of arguments from physics in mathematical proof and give some reasons why this approach might be worthwhile. In the second part we will relate this idea to Freudenthal’s concept of local organization. The third part of the paper will present the results of an empirical study conducted in Canada on the classroom use of arguments from physics in mathematical proof. Kurzreferat: Im ersten Teil diskutieren wir den Gebrauch physikalischer Argumente in mathematischen Beweisen und begründen, warum dieser Zugang im Unterricht sinnvoll sein kann. Im zweiten Teil stellen wir eine Verbindung zu Freudenthals Begriff des lokalen Ordnens her. Der dritte Teil enthält Ergebnisse eines in Canada durchgeführten Unterrichtsversuchs zur Anwendung physikalischer Argumente bei mathematischen Beweisen. ZDM-Classification: E50, G40, M10, M50 An old and new approach to the teaching of proof Mathematicians often use arguments from physics in mathematical proofs. Some examples, such as the Dirichlet principle in the calculus of variations or Archimedes’ use of the law of the lever for determining the volumes of solids, have become famous, and have in fact been regarded by the best mathematicians as elegant proofs, if not necessarily rigorous. It is only natural, then, that several authors, notably Polya and Winter (1978), have proposed that arguments from physics could and should be used in teaching school mathematics. Besides the publications of Polya (1981) and Winter (1978) there are a number of other papers and booklets with examples as Tokieda (1998) and Uspenskii (1961). Unfortunately, however, this approach to classroom teaching has not been sufficiently explored. The application of physics under discussion here goes well beyond the simple physical representation of mathematical concepts, which of course has long proven its own usefulness in teaching. What is being explored here is the classroom use of proofs in which a principle of physics, such as the uniqueness of the centre of gravity, plays an integral role in a proof by being treated as if it were an axiom or a theorem of mathematics. This application of physics is also entirely distinct from experimental mathematics, which purports to employ empirical methods to draw valid general mathematical conclusions from the exploration of a large number of instances. Let us look at a typical example. It is a well-known theorem of elementary geometry (the so-called Varignon theorem) that, given an arbitrary quadrangle ABCD, the midpoints of its sides W, X, Y, Z form a parallelogram (see figure 3 below). A purely geometrical proof of this result would divide the quadrangle into two triangles and apply a similarity argument. An argument from mechanics, on the other hand, would consider points A, B, C, D as four weights, each of unit mass, connected by rigid but weightless rods. Of course such a system, with a total mass of 4, has a centre of gravity, and it is this which we need to determine. The two sub-systems AB and CD each have weight 2, and their respective centres of gravity are their midpoints W and Y. From static considerations we may replace AB and CD by W and Y, each having mass 2. But AB and CD make up the whole system ABCD. Its centre of gravity is therefore the midpoint M of WY. In the same way we can consider ABCD as made up of BC and DA. Therefore the centre of gravity of ABCD is must also be midpoint of XZ. Since the centre of gravity is unique, this midpoint must be M. This means that M cuts both WY and XZ into equal parts. Thus WXYZ, whose diagonals are WY and XZ, is a parallelogram. We can learn from this example that an argument from physics may add to our intuitive understanding of the mathematics involved. To an untutored mind it may seem rather surprising that every quadrangle, however irregular, has a property of such high regularity. When one considers the quadrangle as a four-point system of masses, however, it is immediately clear that its centre of gravity must also divide into equal parts the two levers that connect the midpoints of the opposite sides. We have said quite deliberately that this proof adds something to our understanding of the theorem, not that it gives the “real” or the “best” explanation of it. Certainly the more common purely geometric proof also provides insight. We simply want to say that the argument from physics allows us to look at the theorem from a new and intuitively appealing point of view. In many cases, arguments from physics are a way for the mathematician to produce a more elegant proof. This is as true today as it has been in history. Frequently such a proof may be illuminating in different ways. It may reveal the essential features of a complex mathematical structure, or point out more clearly the relevance of a theorem to other areas of mathematics or to other scientific disciplines. In some cases, too, using an argument from physics may also help create a “holistic” version of a proof, one that can be grasped in its entirety, as opposed to a necessarily elaborate and almost inscrutable mathematical argument. Frequently, arguments from physics may help to generalize. Following the lines of our previous physical argument, for example, we can determine the centre of gravity not only for systems with four masses, but also for those with 3, 5, 6 and so forth. It is highly plausible that in these cases, too, one would be able to translate the respective statements about the centre of gravity into purely geometrical theorems. In the case of three masses, for example, such a theorem would say that in any triangle the three medians intersect in a single point and that this point of intersection divides each median in a ratio 2 : 1 (see below). If we look at the educational context from a broader perspective, we can see several reasons why this approach to the teaching of proof should be further developed and tested. First of all, in most Western countries there is a trend away from using proof in the Analyses ZDM 2002 Vol. 34 (1) 2 classroom. In our view, this trend threatens to undercut the educational value of mathematics teaching, since conveying to students the concept of mathematical proof is an important component of this value. One way to counter this unfortunate tendency is to introduce fresh and possibly more attractive approaches to the teaching of proof. Such new approaches, and especially the use of arguments from physics, might well motivate teachers to rethink their attitude to proof. Another reason to pursue the use of arguments from physics is that present-day mathematical practice displays a significant emphasis on experimentation, and it is only right that this be reflected in the classroom by a similar emphasis on experimental mathematics. But it would be dangerous from an educational point of view if experimental mathematics were to be represented in the schools only by “mathematics with computers”. Quite the contrary: under the heading of experimental mathematics, the curriculum should include a strong component devoted to the classical applications of mathematics to the physical world. In cultivating this type of mathematics, students and teachers should be guided by the question of how mathematics helps to explore and understand the world around us. With such an approach, the teaching of proof would be embedded in building models and in inventing arguments to answer the question “why”. Both of these activities force one to think about the derivation of consequences from assumptions, or, in other words, about proof. Working in this way on the borderline between mathematics and physics, it would also become clear that we can not only apply mathematics to physics, but also very often use statements from physics for the derivation of mathematical theorems. For many reasons physics, the discipline nearest to mathematics, has become less and less a required subject in our schools. To maintain meaningful and interdisciplinary mathematics teaching it may therefore become necessary to include some elementary physics in the mathematics curriculum. Of course this will have to be done carefully, bearing in mind the value of the manifold applications from the social sciences which have entered the curriculum in the last few decades. Nevertheless, we think that some adjustments to the curriculum will be necessary if we are to convey to students a more valid and balanced view of mathematics. The educational evolution of proof In our view, introducing concepts and arguments from physics into the teaching of geometry could significantly contribute to the development of students’ understanding of proof. One of the most difficult problems faced by educators when they start doing proofs with their students is the systematic nature of Euclidean geometry. Today, of course, nobody would teach Euclidean geometry in an axiomatic way. Yet a closer analysis of geometry textbooks and the practice of teaching would show that Euclid’s system is always present in the background. It determines to a large extent the sequence of theorems and the arguments students are allowed to use in a proof. Though they bring only a small part of it to the attention of their students, teachers nevertheless have a mental picture of a “grand theory”. For the teacher, the main function of a proof is to incorporate a new theorem into this grand theory. The students know little or nothing of this grand theory, however, and so they must necessarily have a completely different image of the function of proof. As a consequence of this deeply rooted problem, geometry is bound to appear arbitrary and dogmatic to many students. Why are they asked to prove the angle sum theorem, but are allowed to use facts about angles formed by parallel lines intersected by a third line, rather than vice versa? Most educators are aware of this fundamental difficulty, and it would seem to be one of the main reasons why in the last twenty years teachers have limited the role of proof in their teaching more and more. The justification for this reduced role offered by many educators is the mantra that the important thing is not to teach proof, but rather to develop a “culture of reasoning” in the classroom. As is frequently the case with such “soft” slogans, this one is true and wrong at the same time. On the one hand it does take into account the undeniable fact that we cannot really teach systematic geometry in our schools. On the other hand, however, it ignores the reality that one cannot really discuss proof without discussing theories. The very notion of proof is tied to the notion of theory (and here we use “theory,” of course, in the sense of “systematic structure”). Every proof is based upon an understanding, explicit or implicit, of what can be taken as given and what kinds of argument are acceptable, and these questions can be answered only in the context of a theory. Seen this way, it is clear that proof necessarily involves a formal element, and that providing a proof involves much more than setting out some everyday argument. Even more significant in the context of our discussion, perhaps, is that one cannot really understand the importance of proof or its educational value unless one understands that it is intimately tied to the idea of theory. Most people would accept without elaboration the statement that we are having bad weather because of a drop in atmospheric pressure. This is a typical everyday explanation, and in everyday situations it might be sufficient. The difference between educated and uneducated people, however, is that the former, though they too would accept this explanation, are very conscious of the fact that it would need considerable elaboration, involving reference to a number of theories and laws, if we wanted to understand fully why it is raining today. What we are aiming for in the teaching of proof is precisely to develop this sense for the qualitative difference between scientific explanations and everyday arguments. As we develop the notion of proof in the classroom, then, we must also develop the notion of a theory. In this connection we find it very useful to consider Freudenthal’s concept of local organisation in geometry. He states that “... in introductory geometry the student can be lead to learn to organize shapes and phenomena in space by means of geometrical concepts and their properties. At a higher level he should organize these concepts and their properties by means of logical relations. Above this level this relational system can become a subject of investigation” (Freudenthal 1973, 458). ZDM 2002 Vol. 34 (1) Analyses 3 What Freudenthal had in mind is shown by his example of the theorem on the perpendicular bisectors of a triangle. It is not necessary, in his view, to give a complete proof that calls upon the entire (implicit) background of the Euclidean system, starting with the equidistance property of perpendicular bisectors and then progressing to the fact that they meet in one and the same point. Rather, one can concentrate on certain aspects which, for one reason or another, are of interest in the specific teaching situation, while taking other aspects for granted. Thus, local organization aims at the exploration of a certain configuration and not at establishing a deductive truth within a larger system. Of course, the things taken for granted in the “local organization” are consistent with the truths of the larger system and could be proved in it. But since the students do not have an idea of the larger system, we have to take this “local organization” itself as the theory with which we are dealing. Thus, in line with Freudenthal’s concept of “local organisation” we would propose a distinction between “large” and “small” theories. Instead of building a large theory (namely, Euclidean geometry) in the course of the curriculum, it seems to be more appropriate to work in several small theories around stimulating applications. The physical mechanisms described and analysed in elementary statics could provide fruitful examples of such small theories. What does it mean to work in a “small theory”? We are well aware that students will always have a somewhat fragmented knowledge of geometry, because it has to be developed step by step and cannot be conveyed to them all at once by a simple verbal explanation. The idea of a “small theory” recognises that the same is true of their understanding of proof. As educators, we have to accept the fact that in the minds of the students there is no fixed (axiomatic) basis of argumentation, and that we cannot expect them to believe that there is a procedure called proof, superior to their intuition, which establishes the truth of a theorem. But in that case what would a proof accomplish for the students? The answer is that a proof exhibits relations and dependencies between different facts or statements. We will explain this by a classic example. The angles in a circle theorem can be proved by recourse to properties of isosceles triangles. In the context of Euclidean geometry as a system, this means that we can add the angles in a circle theorem to the bag of true (proved) theorems, since we have already proved that the base angles in isosceles triangles are equal, and this, in turn, can be derived from another theorem, and so on. In our teaching, for the reasons we have discussed, we cannot rely on such a long chain of deduction, and thus the proof must be made to mean something different to the students. They are working in the context of a small thematic unit on angles inscribed in circles, a context in which it is natural to consider that isosceles triangles can be inscribed in such circles. By a proof the students might understand establishing a logical connection between the invariance of the inscribed angles and the properties of isosceles triangles. These properties, then, “explain” the invariance of inscribed angles. In regard to truth, of course, the proof wouldn’t change the situation, since both facts, the invariance of the inscribed angles and the equality of the base angles in isosceles triangles, are equally plausible to the students. Nevertheless, the proof would enhance the certainty of the inscribed angle theorem in the sense that a counterexample to it would also be a counterexample to the equality of the base angles in isosceles triangles. This also sheds some light on the relationship between proof and measurement (see Hanna & Jahnke, 1996). Students are often required to measure the angles of triangles and work out their sum. After having found that they always get around 180°, they are told that this has to be proved mathematically in order to show that it is true “for all triangles”. This pedagogically well-meant step is problematic, since it implies that we can arrive at a law valid for the empirical triangles on the paper in front of us by a purely deductive procedure, without any measurement. The fact is, however, that when we are dealing with empirical triangles we must insist that the angle sum property is true only because we have measured it, and not because we have proved it mathematically. This having been said, it is also true that a mathematical proof of the angle sum theorem will enhance its empirical certainty, since it connects this theorem with some other statement (for example, a statement on angles at parallels cut by a straight line) which we can also establish by measurement. Taking this example, the statement on angles at parallel lines functions as an independent test of the angle sum theorem in triangles. If there is a counterexample to one of the theorems it will be relevant to both. For the students, the epistemological situation is similar to that of a physicist. No physicist will believe in a statement simply because it has been proved mathematically. He will test it by measuring, of course. Mathematical proofs are essential in physics nevertheless, because they connect the empirical statements. A theory of physics is a network of measurements and laws connected by proof. The theory as a whole is tested by the system of all measurements taken as a whole, and therefore it has a status more certain than a single measurement could have, irrespective of how often it might have been performed. In educational terms, one could say that this method teaches geometry like a theory of physics. What does the idea of local organization mean for the approach of using arguments from physics in mathematical proof? First of all, since students usually do not have the necessary background in physics, we have to start with building up such a background. This means, for example, that we introduce a unit on statics into geometry teaching. This would be a “small theory” of a mixed nature, since it comprise statements and principles from both geometry and physics. (In the 18 century, in fact, people spoke of “mixed mathematics” in such cases.) The facts of statics which would be taken for granted in this theory are empirical laws about the equilibrium of 1and 2-dimensional “bodies”. As we have said, the empirical character of these laws does no harm to the idea of a mathematical proof, because it is the function of such a proof to establish theoretical connections among statements, whether the statements themselves are empirical