Missing:
Focus on Learning Problems in Mathematics Spring & Summer Edition 1998, Volume 20; Numbers 2 & 3 @Center for Teaching/Learning of Mathematics
Proof as Explanation in Geometry Gila Hanna Ontario Institute for Studies in Education Introduction
The increasing use of computers in mathematics and in mathematics education is strongly reflected in the teaching of geometry, in particular in the use of dynamic graphics software. This development has raised questions about the role of analytic proof in school geometry. The power of such software has suggested to some that one might best abandon analytic proof as an aid to comprehension in favor of visual displays, and even that analytic proof in its role of justification could be replaced by the examination of a large number of cases. This paper views dynamic software as an excellent tool for exploration in the classroom, as it is in mathematical practice, and for helping students see that a theorem is true. It argues, however, that justification without proof would be untrue to the theory and practice of mathematics. It also argues that such software cannot show students why a theorem is true or, more pointedly put, why it is always true, and so cannot replace analytic proof as the source of this key insight. The paper points out, however, that analytic proofs can play this role fully only if chosen for their explanatory value. When they enhance mathematical understanding, analytic proofs both reflect mathematical practice and remain an indispensable instructional activity. Reflections on Proof in School Geometry
Western mathematicians have considered proof an essential part of mathematics ever since Euclid. Because they have seen it as indispensable for establishing the validity of propositions, they have put a great deal of effort into both proving theorems and justifying their methods of proof. This has not been easy. There have always been controversies over the connection between proof and truth in mathematics and over what constitutes an adequate proof. Even today there is no unifonnly accepted theory or practice of mathematical proof (Kitcher, 1984; Davis and Hersh, 1981; Lakatos, 1976; Thurston, 1994). Proof is continuing to evolve, in fact, in response to new problems and new developments, as evidenced by such new tenns as "semi-rigorous proof' (Zeilberger, 1993), "ex-4-
perimental mathematics" (Epstein and Levy, 1995), "zero-knowledge proof' (Blum, 1986), and "holographic proof' (Babai, 1994). Yet proof remains central to mathematics, and for this reason it has been a declared aim of the mathematics curriculum to teach students both the practice and the importance of proof. This aim notwithstanding, proof did not permeate the entire mathematics curriculum, but came to reside almost exclusively in geometry. History bad assigned geometry a privileged position in relation to mathematical proof, probably because it was the first theory to be cast into axiomatic fonn. Rightly or wrongly, geometry came to be seen as the domain in which mathematical reasoning attained its ideal form. Today's curriculum still reflects this historical development. It is still in geometry classes that students first encounter the concept of proof and its related tenns (such as axiom, theorem, deductive method, hypothesis, and two-column proof). It is there they are first told that they are expected to behave as mathematicians, that is, to prove theorems. The importance of proof also finds its expression in the broadly accepted view of school geometry as the study of spatial objects along with the axiomatic systems that purport to represent them (Clements and Battista, 1992). One could even say that geometry is taught, in North American schools at least, primarily as a paradigm for deductive proof. Geometry textbooks differ in the way they explain to students what is expected of them and what proof in geometry is supposed to accomplish, but not on the centrality of deduction. A textbook by Ebos, Tuck and Crowley ( 1970) is representative. After introducing Euclid's Elements as including all the deductive geometry known at the time, the authors illustrate deduction by means of a syllogism, making clear that it is a method of thinking not limited to geometry, and state: ''To develop our work in geometry in an organized way, we will think as Euclid did" (p. 262). They then explain the process by saying that Euclid and his followers began with undefined terms, defined terms and axioms or postulates, and then used logical reasoning (deductive thinking) to expand their knowledge of geometry by proving deductions, the more important of which are known as theorems. Their text goes on to define and illustrate quite carefully all the tenns used in this statement of intent. Further evidence of the importance accorded to proof in school geometry is the benefit which it is expected to bring beyond the borders of that subject. The consensus seems to be that the key goals of geometry instruction are the development of thinking abilities, of spatial intuition about the world, of knowledge necessary to study more mathematics and of the ability to interpret mathematical arguments (Fehr, 1973; National Council ofTeachers ofMathematics [NCIM, 1989]; Suydam, 1985). Thus geometric proof is seen as training for logical reasoning (Suydam, 1985). Students have been expected to acquire not only a degree of competence in understanding and constructing proofs, but also a "general accuracy of thought" (National Council ofTeachers of Mathematics [NCfM], 1989). Unfortunately the few relevant studies show that the majority of geometry students do not attain an adequate mastery of proof itself (Fischbein, 1982; Martin and Harel, 1989; Senk, 1985). In addition, no convincing research has con-5-
finned the supposition that engaging in mathematical proof results in a transfer of learning in the form of an ability to apply reasoning skills in other areas of the cuniculum (Hiebert and Carpenter, 1992). For these reasons geometry is no longer defended on the grounds that its use of deductive proof produces "logical thinkers," and it has had to find its place in the curriculum in its own right
The lnDuence of Computers The Influence on Mathematics in General The use of the computer in mathematics has led some to announce the immi-
nent death of proof itself (Horgan, 1993). Horgan makes this prediction in a thought-provoking article entitled '"The death of proof' that appeared in the October 1993 issue of Scientific American. One of the developments that prompted Horgan's announcement was the important role of the computer in zero-knowledge proofs, holographic proofs and the creation and verification of extremely long proofs such as that of the four-color theorem. Yet even these innovative types of proof are traditional, in the sense that they remain analytic proofs. Of greater import for the present discussion is Horgan's observation that more and more mathematicians, as they tum to the computer to confirm mathematical properties experimentally, appear to be abandoning deductive proof itself. Horgan claims that some mathematicians, rightly impressed by the power of computational experiments and computer graphics in the communication of mathematical concepts, have come to believe that "the validity of certain propositions may be better established by comparing them with experiments run on computers or real-world phenomena" (1993, p. 94). And indeed, the very existence of the quarterly Experimental Mathematics, founded in 1991, would seem to lend support to this claim. This new quarterly differs markedly from traditional journals in that it publishes not theorems and proofs, but rather the results of computer explorations, presenting significant conjectures and describing how they have arisen in the course of exploration. But does this mean that the contributors to Experimental Mathematics or its editors think that proof is dead? On the evidence, the answer is certainly negative. In their paper "Experimentation and Proof in Mathematics," the editors of the quarterly, Epstein and Levy, point out the enhanced potential of experimentation in the age of the computer: "the use of computers gives mathematicians another view of reality and another tool for investigating the correctness of a piece of mathematics through investigating examples" (1995, p. 674). At the same time, they make it very clear how they believe this experimentation fits into the mathematical scheme of things: Note that we do value proofs: experimentally inspired results that can be proved are more desirable than conjectural ones.... The objective of Experimental Mathematics is to play a role in the discovery of formal proofs, not to displace them. (p. 671) -6-
...We believe that, far from undennining rigor, the use of computers in mathematics research will enhance it in several ways. (p. 674) The Potential Influence on the Teaching Geometry It is safe to say that computers will also change the way mathematics is taught, if only to take advantage of the many new tools and activities available in the classroom. New approaches to teaching are now a subject of discussion. But the changes may eventually go beyond teaching approaches to include a reassessment of the weight given to different topics in the classroom. The advent of software with dynamic graphics capabilities has brought a renewed interest in the teaching of geometry, where these new tools are particularly appropriate, and so it may regain a privileged place in the curriculum, though with goals somewhat different from its traditional ones. Geometry software can make it simple and attractive to draw and manipulate figures. The Geometric Supposer (Schwartz and Yerushalmy, 1986), the Geometer's Sketchpad (Jackiw, 1991) and Cabri Geometry (LSD2, 1988), for example, help students understand proofs by allowing them to perform geometric constructions with a high degree of accuracy. This makes it easier for them to see the significance of propositions that form part of the proof, or that are to be proven. With such software students can also easily test conjectures by exploring given properties of the constructions they have produced, or even "discover" new properties. The Sketchpad workbook, for example, has several sections on exploration, organized under seven headings. Most of them not part of the traditional geometry curriculum: Investigation, Exploration, Demonstration, Construction, Problem, Art, and Puzzle (Jackiw, 1991, p. 3). Because computers make it so easy to pose and test conjectures, the curriculum will be likely to encourage more experimentation. Exploration is not inconsistent with the traditional view of mathematics as an analytic science or with the central role of proof. But unfortunately the introduction of this powerful software and its successful use in exploration has lent support to a view among educators that deductive proof in geometry should be abandoned in favor of an entirely experimental approach to mathematical justification. Mason (1993), for example, maintains that with dynamic software one can check a large number of cases and ''by appeal to continuity, an infinite number of cases" and concludes that ''truth will be ascribed to observations made in a huge range of cases explored rapidly on a computer" (p. 87). This view reflects the tendency of some educators and others, as discussed above, to misinterpret the growing use of computers in mathematics as an indication that proof is no longer a central aspect of mathematical theory and practice. Greeno (1994) expressed his concern for the ramifications of this misconception, saying:
Regarding educational practice, I am alarmed by what appears to be a trend toward making proofs disappear from precollege mathematics education, and I believe that this could be remedied by a more adequate theoretical account of the epistemological significance of proof in mathematics (p. 270-271). -7-
One can demonstrate by an example how the dynamic aspects of geometry software could move some to question the need for analytical proof. Suppose the student wants to "prove" the theorem that in any triangle the perpendicular bisectors intersect at one point The student could, on paper, construct a triangle and its three perpendicular bisectors and verify that these intersect at a single point But performing this construction with Cabri Geometry has an important advantage. It allows the student to grab a point on the triangle and pull the triangle over the screen in such a manner that it changes its shape. As this is done, the perpendicular bisectors are continuously redrawn correctly. This procedure shows the student that the three perpendicular bisectors still intersect at one point, no matter what the shape of the triangle. It is at least equivalent to drawing a large number of triangles on paper, or imagining that one had drawn them, and probably more impressive. This powerful feature provides the student with strong evidence that the theorem is true. It certainly helps the student form a mental image (Mason, 1993) and see what Wittgenstein called a "proof-picture" or "self-evident proposition." It would only be natural if the student were to jump to a general conclusion and regard this exploration as entirely sufficient to establish that the perpendicular bisectors always intersect in a single point This shows the value of exploration in giving students confidence in a theorem. Most educators would agree, however, that students need to be taught that exploration, as useful as it may be in making and testing conjectures, does not constitute proof. They would agree, too, that their challenge is using the excitement of exploration to motivate the students to supply a proof, or at least to make an effort to follow a proof presented by the teacher. The first reason is that exploration cannot establish the degree of certainty associated with an analytic proof. But much more importantly in the educational context, even the most powerful dynamic software can only show students that something is the case. True understanding demands that the students see why it is the case, and furthermore why it must always be the case, and this understanding is best engendered by an explanatory proof. It is hard to see why some seem to have viewed exploration and proof as mutually exclusive. In the first place, mathematical exploration, with or without the aid of a computer, makes much use of deductive reasoning itself (Epp, 1994). Polya (1957) has shown that deductive reasoning is central to exploration and problem solving. To solve a problem is to find the connection between the data and the unknown, and to do this one must use what Polya calls a "heuristic syllogism," a kind of reasoning that uses deduction, in addition to circumstantial, inductive and statistical evidence. In the second place, it is a simple fact that exploring and proving are separate activities that reinforce each other. Both are part of the overall problem-solving process and both are needed for success in mathematics. Exploration leads to discovery, while proof is confmnation. Exploration of a problem can lead one to grasp its structure and its ramifications, but cannot yield an explicit understanding of every link. Thus exploration leads to conclusions which, though precisely formulated, must remain tentative. It is the proof -8-
which, by providing a derivation from the premises, finnly establishes the conclusions. Proof as Explanation
In mathematics itself the main function of proof is clearly to validate propositions. A proof provides the greatest justification for a mathematical statement and gives greatest confidence in it. A proof may perform a number of incidental functions as well: it may contribute to the discovery of other mathematical truths, to the systematization of a body of results, and to the communication of mathematical knowledge (de Villiers, 1990). But the most important additional function of proof is that of explanation or clarification. The best proof, even in the eyes of practicing mathematicians, is one that not only establishes the truth of a theorem but also helps understand it Such a proof is also more persuasive and more likely to be accepted. Scholars pointed out the importance of clarification, as distinct from mere justification, as early as the 17th century. Arnauld and Nicole, who studied Euclid in great detail, are an example (Barbin, 1988). Though these two scholars had tremendous admiration for Euclid's work, they thought it appropriate to formulate a number of criticisms in their book La logique ou I'art de penser (1674; republished 1965). They saw it as a significant flaw that the Elements deal more with proofs that ~nvince than with proofs that clarify (eclairent). They used the term "clarify" to mean showing why a theorem is true and "convince" to mean being satisfied with demonstrating that it is true, and thus their distinction is the same as that made above between explanation and justification. Barbin shows that Lamy, another 17th century geometer, also made the distinction between convaincre and eclairer in a work published in 1683. Lamy advises students to make sure the proof of a proposition provides a clarification before they accept it as true. Proofs that clarify, he adds, proceed from definitions through deductions to the conclusion in a clear way. In this they differ from proofs such as reduction ad absurdum, which justify a proposition by deriving an evident absurdity from its negation; such a proof may well convince us that the proposition is true, but can shed no light on why this is so. Lamy also devotes an entire book to the methodology of proof, in which for some propositions he presents two proofs, one that merely convinces and one that also provides clarification (Barbin, 1988). Another view of proof as explanation is that of the ancient Chinese mathematicians. Greek axiomatics had not reached China, and mathematicians proceeded there unconcerned with axiomatics and deductive proofs. For them a proof consisted of "any explanatory note which serves to convince and to enlighten ..." (Siu, 1993, p. 346). These explanatory notes took various forms, consisting of a balanced use of arguments, heuristic reasonings and dissections of diagrams known as proofs without words. A very famous such proof is the use of dissected squares to prove the Pythagorean theorem. Because the Chinese mathematicians were concerned with the explanatory power of their proofs, as op-9-
posed to mere verification, they often devised several different proofs for the same theorem. Here is a proof of the Pythagorean theorem by the dissection method as adapted from the Chou pei suan ching (author unknown, circa B.C. 2001) (Nelsen, 1993).
a
a
b
b
b
a
Figure 1. Another insight into explanatory proof is Steiner's definition ( 1978) that such proof must make " ...reference to a characterizing property of an entity or structure mentioned in the theorem, such that from the proof it is evidence that the results depend on the property" (p. 143). Closely related to Steiner's definition is the concept of "inhaltlich-anschaulicher Beweis. "Witbnann and Muller ( 1990) use this term to characterize a proof in which "[the] method of demonstration calls upon the meaning of the term employed, as distinct from abstract methods, which escape from the interpretation of the terms and employ only the abstract relations between them." Though in mathematics itself the main function of proof is to validate propositions, it is clear that from the earliest days mathematicians have preferred or even insisted upon proofs that also explain. In mathematics education, however, where students are studying propositions that ~ known to be true, the main function of proof is obviously that of explanation. What is important for practicing mathematicians becomes even more important in the classroom: the ability of a proof to help understand the meaning of the theorem being proved and so provide good reason to believe it. Hanna (1990) discusses in more detail the advantages of such explanatory proofs in creating understanding in the classroom. To prove that the three angle bisectors in a triangle meet at a single point, for example, one normally makes use of the defining property that an angle bisector is the locus of all points equidistant from the edges of the angle. Similarly, to prove that the three perpendicular edge bisectors meet at a single point, one makes use of the defining property that a perpendicular bisector is the locus of all points equidistant from the end points of its edge. These proofs are thus explanatory in Steiner's (1978) sense, in that they make "reference to a characterizing property of an entity or structure mentioned in the theorem" (p. 143). In
-10-
the classroom they have been found to be both convincing and illuminating, helping students see why the theorems must be true. Of course one cannot always find an explanatory proof for every theorem one wishes to present in the classroom, much less a proof that comes directly from the properties or the meanings of the terms involved. In many mathematical subjects some theorems need to be proved using contradiction, mathematical induction or other non-explanatory methods. It so happens, however, that geometry enjoys a special position in this regard, in that most of its proofs, are also explanatory ones. In geometry textbooks it is quite rare to see a proof that makes use of non-explanatory arguments such as reductio ad absurdum or mathematical induction. Even in geometry, though, one can sometimes find a proof which is rather more explanatory than the one commonly used. To prove that the area of a rectangle is the same as that of a parallelogram of t11e same length and height, for example, one most often uses the argument that by definition this area is the product of the length and height. This is certainly correct. But one could just as easily apply the more explanatory method of dissection of diagrams discussed above, as illustrated in the following figure:
h
b
Figure 2.
The challenge to educators is to devise ways to use dynamic software in the service of explanatory proofs.
Preparation of this paper was supported in part by the Social Sciences and Humanities Research Council of Canada. -II-
REFERENCES Babai, L. (1994). Probably true theorems, cry wolfl Notices of the American Mathematical Society, 41(5), 453-454. Barbin, E. ( 1988). La demonstration mathematique: significations epistemologiques et questions didactiques. Bulletin de l'A.P.M.E.P., n. 366. Blum, M. ( 1986). How to prove a theorem so no one else can claim it Proceedings ofthe International Congress ofMathematicians, 1444-1451. Clements, D.H., & Battista, M.T. (1992). Geometry and spatial reasoning. In D.A. Grouses (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics. New York: Macmillan. Davis, P., & Hersh, R. (1981 ). The mathematical experience. Boston: Birkhauser. de Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24,17-24.
Ebos, F., Tuck, B., & Crowley, M. ( 1970). Math is I 4 (teacher's edition). Toronto, ON: Thomas Nelson & Sons. Epp, S. ( 1994). The role of proof in problem solving. In A. Schoenfeld (Ed.), Mathematical thinking and problem solving. Hillsdale, NJ: Lawrence Erlbaum Associates. Epstein, D., & Levy, S. (1995). Experimentation and proof in mathematics. Notices of the American Mathematical Society, 42(6), 670-674. Fehr, H.E. ( 1973). Geometry as a secondary school subject In K.B. Henderson (Ed.}, Geometry in the mathematics curriculum: 1973 Yearbook (pp. 369380). Reston, VA: National Council of Teachers of Mathematics. Fischbein, E. (1982). Intuition and proof. For the Learning of Mathematics, 3(2}, 9-18, 24.
Greeno, J. (1994). Comments on Susanna Epp's chapter. InA. Schoenfeld {Ed.), Mathematical thinking and problem solving. Hillsdale, NJ: Lawrence Erlbaum. Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 613.
Hiebert, J. & Carpenter, T.P. ( 1992). Learning and teaching with understanding. In Grouws, D.A. (ED.), Handbook ofresearch on mathematics teaching and learning. New York: Macmillan. Horgan, J. (1993). The death of proof. Scientific American, 269(4), 93-103. Jackiw, N. (1991). The geometers sketchpad. Berkeley, CA: Key Curriculum Press.
Kitcher, P. (1984). The nature of mathematical knowledge. New York: Oxford University Press. Labroratoire de Structures Discretes et de Didactique Institut d'Informatique et de Mathematiques Appliquees de Grenoble {IMAG). (1988). Cabri Geometry [Computer Software]. Lakatos, I. (1976). Proofs and refutations. Cambridge: Cambridge University Press.
-12-
Martin~ W.G., & Harel, G.
(1989). Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 20, 41-51. Mason, J. (1993). Questions about geometry. In D. Pimm & E. Love (Eds.), Teaching and learning mathematics: A reader. London: Holder & Stoughton. National Council of Teachers of Mathematics, Commission on Standards for School Mathematics (1989). Curriculum and evaluation standardsfor school mathematics. Reston, VA: Author. Nelsen, R.B. ( 1993). Proofs without words: Exercises in visual thinking. Washington, DC: The Mathematical Association of America. Poly~ G. (1957). How to solve it: A new aspect ofmathematical method, New York: Doubleday. Schwartz, J., Yerushalmy, M. (1986). The geometric supposer (Computer software). Pleasantville, NY: Sunburst Communications. Senk, S.L. (1985). How well do students write geometry proofs? Mathematics Teacher(6), 448-456. Siu, M-K. (1993). Proof and pedagogy in ancient China: Examples from Liu Hui's commentary on Jiu Zhang Suan Shu. Educational Studies in Mathematics, 24(4), 345-357. Steiner, M. (1978). Mathematical explanation. Philosophical Studies, 34, 135151. Suydam, M.N. (1985). The shape of instruction in geometry: Some highlights from research. Mathematics Teacher, 78, 481-486. Thurston, W.P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161-177. Wittmann, E.C. & Muller, G. ( 1990). When is a proof a proof. BUIL Soc. Math. Belg., /, 15-40.ยท Wittgenstein, L. (1953). Philosophical investigations. Oxford, U.K.: Blackwell. Zeilberger, D. (1993). Theorems for a price: Tomorrow's semi-rigorous mathematical culture. Notices ofthe American Mathematical Society, 40(8), 978981.
-13-
