Theorems in School: From History, Epistemology and Cognition to Classroom Practice

This collection of articles addresses different aspects of teaching mathematical proof in school. As the book’s subtitle suggests, the articles cover different topics from history and epistemology to the nature of argumentation and proof to classroom studies and pedagogy. It is hard to characterize the collection as a whole, other than to say that the various authors, believing that proof deserves a prominent role in K–12 mathematics education, have thought deeply in various ways about the issues involved in helping children learn to prove theorems.

I have recently begun teaching a proof-based geometry course for prospective middle and high school teachers aimed at equipping them to teach the same subject. This book provided me with new perspectives on the nature and teaching of proof, along with promising activities for the classroom. I will briefly survey ideas that I found particularly stimulating in some of the articles.

Gilbert Arsac and Ferdinando Arzarello write about, respectively, the “Origin of Proof” and “The Proof in the 20th Century.”

Arsac analyzes the way that ancient Greek mathematicians and philosophers dealt with the incommensurability problem. He points out that many classic Greek geometry problems were solved via simple inspection of a diagram. Not amenable to such a direct approach are problems in which incommensurability enters. (Examples include considering the length of the diagonal of a square having side length one or the ratio of the side of a regular pentagon to a diagonal of the pentagon.) To handle these problems, one must transcend merely consulting the figure; one is forced to discuss properties of classes of objects. For instance, the classic proof for irrationality of the square’s diagonal depends on properties of even (and odd) numbers. Incommensurability in the regular pentagon example is shown via an infinite process that relies on all regular pentagons sharing certain properties. Thus abstract properties, rather than the figure itself, are now the focus. Geometric objects have become ideal. While discussing the Eleatic School’s perspective on this, Arsac comments, “Breaking away from reality is the condition which allows geometry to rank among real knowledge, perhaps against what a contemporary mind would maintain!”

One lesson Arsac draws from his historical survey concerns the role of proof: is it (primarily) to convince or to explain? Consider the classic arithmetic proof for the irrationality of the square’s diagonal. Not only does it abandon geometry for arithmetic; it also relies on reductio ad absurdum! To moderns, this proof convinces but does not explain. Perhaps for the ancient Greeks, however, the proof explained the infinite process they had encountered with the pentagon and similar problems.

Arzarello defends mathematical proof against claims that it is a useless formal exercise. A proof of the implication A ⇒ B is simply a chain of statements of the form A ⇒ A1 ⇒ A2 ⇒ … ⇒ An ⇒ B satisfying transitivity. The purpose of adding the various statements Ai to the chain is to allow the intended audience to see the logical connection between A and B. Novices require more statements, experts fewer (except those working in foundations). Proof/proving is therefore a dialogue that ends with everyone in the audience in agreement that there are no gaps in the chain. Arzarello even enlists Gödel’s incompleteness theorems in describing proof as an “infinitary game of interpretation.”

Arzarello also discusses the distinction between a proof and a derivation, the latter a formal argument within a formal system in which statements follow along by syntax rather than semantics (e.g., the statements C ⇒ D and C yield the statement D, regardless of what C and D mean). An analogy using computers: proof is to derivation as algorithm is to program.

Around 1980, the mathematics curriculum for schools in England and Wales largely abandoned deductive proof, emphasizing empirical methods instead. In “Curriculum Change and Geometrical Reasoning,” Celia Hoyles and Lulu Healy report on their study from 1996 of the geometrical reasoning of some high-achieving British 14–15 year-olds. The students were presented with six alleged proofs that the angles of a triangle sum to 180 degrees. Among the six were two empirical demonstrations and a correct deductive proof in two-column format. The students were asked which proof they would probably produce (result: one of the empirical demonstrations) and also which proof they thought would receive the best grade from their teacher (result: the two-column deductive proof).

Almost 75% of students claimed that the empirical demonstrations proved the theorem for all triangles, while only about 20% thought that the deductive proof did so (despite its being the proof that teachers supposedly would prefer)! In an interview, one student pointed at the deductive proof’s accompanying fully-labeled diagram (the usual: a triangle with a line drawn through a vertex parallel to the base opposite), claiming that the proof held only for “that triangle.”

The students were also asked to provide a proof to a familiar statement and a proof to an unfamiliar statement. Most gave empirical arguments for the former and failed to make much progress on the latter.

After the British curriculum was revised in 2000 to place more emphasis on deductive proof, the entire test was given to a new group of students in 2002. A comparison of how the two groups of students handled generating their own proofs yielded predictable results. Some members of the new group of students successfully attempted more deduction than the 1996 group, while others, lacking background in experimentation, could make almost no headway at all. Such are the pros and cons of emphasizing empirical versus deductive methods.

Raymond Duval’s “Cognitive Functioning and the Understanding of Mathematical Processes of Proof” is the slowest read in the collection, but worth the effort. He analyzes differences between informal social argumentation and mathematical proof. Perhaps the crux of his analysis is the discussion of how statements gain or change epistemic value (e.g., obvious, absurd, likely, certain) in each setting. In informal argument, the semantics of a statement combine with common beliefs or rules to determine/change epistemic values. In contrast, semantics take a back seat in deductive proof, in which the (strictly logical) relationship of a statement to surrounding statements and the larger mathematical structure determine/change the statement’s epistemic value.

Novices often have trouble attempting a proof of a statement in geometry that appears “clearly true” from a figure. For success, “the theoretical epistemic value must repress the pregnant semantical value!” To aid in this, Duval recommends flowchart proofs. A hermeneutical circle is effected as the student alternately consults the geometric figure and the developing flowchart. Constructing the flowchart promotes recognition of the logical dependence of statements: arrows (representing implications between statements) cannot be drawn, and hence epistemic values of statements cannot be set/changed, unless and until sufficient evidence has been gathered to establish the deduction. (Also, the flowchart’s visual nature makes it as attractive to the eyes and as able to command attention as the geometric figure, so seductive with all of its “obvious” semantics.)

Duval’s viewpoint is disputed by Nadia Douek in “Some Remarks About Argumentation and Proof.” She discusses the nature of the socially-constructed “reference corpus” for mathematical proof, that is, the background assumptions and rules of the game. For example, in both the development and presentation of proofs, shared background knowledge enables professional mathematicians (except those working in foundations) to omit steps that would be essential for a high-school student. Regarding the integrity of a proof, W. P. Thurston points out that reliability rarely comes via “mathematicians formally checking formal arguments.” He remarks on Wiles’s proof of Fermat’s Last Theorem, “The experts quickly came to believe that his proof was basically correct on the basis of high-level ideas, long before details could be checked.” Douek draws the conclusion that “the model of formal proof as described by Duval and based on the ‘operational status’ of propositions rather than on their ‘semantic content’ does not seem to fit the description of the activities performed by many working mathematicians when they check the validity of a statement or a proof.”

In “Construction Problems in Primary School”, Maria Bussi, Mara Boni, and Franca Ferri spotlight the reasoning ability of fifth graders by reporting on a classroom activity involving the construction of a circle. Given two wheels in gear, the students were to determine the size and location of a third wheel so that it would be in gear with the other two. In other words, given two circles of different radii tangent to each other, find a third circle tangent to both of them. Using a compass and guided by their teacher, students began by using trial and error, progressed to recognizing properties of the radii, and finished by developing a precise method to determine the third circle.

 Noteworthy was the change in the role of the compass during this process. Initially a physical instrument that draws circles, the compass was transformed into a mental object that measures and determines distance. During classroom discussion, students often invoked this new theoretical compass by rotating their hands or arms. “When the compass is used to produce a round shape its main goal is communication; when the compass is used to find the points which satisfy a given relationship it becomes a tool of semiotic mediation (Vygotsky, 1978), that can control — from the outside — the pupils’ process of solution of a problem, by producing a strategy that 1) can be used in any situation, 2) can produce and justify the conditions of possibility in the general case, and 3) can be defended by argumentation referring to the accepted theory.” Not bad for fifth grade!

For “Approaching Theorems in Grade VIII,” Paolo Boero, Rossella Garuti, and Enrica Lemut illustrate the “cognitive unity of theorems,” the solidarity of the conjecturing and proving processes. To nudge eighth-graders toward deductive proof, they advocate dynamic exploration leading to a crystallized statement. Their class activity involved two sticks in the ground, one vertical and one oblique. Students were asked, “Can shadows be parallel? At times? When? Always? Never? Formulate your conjecture as a general statement.”

Students were provided sticks and polystyrene platforms but remained inside the classroom, away from the sun. Many students performed dynamic explorations, moving their hands or themselves as a proxy for the sun, moving the oblique stick, moving the platform. Evidence of the cognitive unity between the exploration phase and the resulting proofs: 1) Both the wording of a student’s conjecture and the description of processes (sun moving, stick moving) tended to be preserved in the student’s proof. 2) Students who had at first posited an incorrect conjecture, and had to quickly backtrack, ended up with even more description of processes in their proofs. 3) Students who originally made the correct conjecture, but produced little informal argumentation to back it up, finished with serious gaps in their proofs. The authors also note that for the students, conjecture led naturally to deductive argument: “When they verify their conjecture most of them seem to be aware of the fact that they must get the truth of the statement by reasoning, starting from true facts.”

In “Geometrical Proof,” Maria Mariotti discusses the teaching of proof using Cabri, computer software that allows construction and manipulation of geometric objects. The software promotes the transition from pure intuition to deductive justification, and from informal argumentation to proof. A geometric object that is correctly constructed in Cabri will resist deformation when the student tries to “drag” part of it. Passing the dragging test validates the construction; the student is led to consider why this construction passes and others do not. The focus is shifted from the figure itself to the procedure that created it. “In other words, the problem is shifted from validating by dragging, to explaining the ‘validating by dragging’ itself.” Cabri has many construction commands. To help students avoid confusion between axioms and theorems, Mariotti recommends starting with only a select few Cabri commands (corresponding to axioms) and then building up the Cabri command vocabulary as theorems are proved.

Most of the authors of this collection of articles reside in Europe (several in Italy). North American readers stand to benefit from this exposure to predominantly overseas scholarship, further pointers to which are found in the bibliography at the end of each chapter. That the English is not perfectly fluent in all of the articles is a minor point; the text is always understandable. Mathematics teachers and teacher educators should find this collection valuable, especially those interested in the teaching and learning of proof. From the introductory essay, “The Ongoing Value of Proof,” Gila Hanna pleads, “With today’s stress on making mathematics ‘meaningful,’ teachers are being encouraged to focus on the explanation of mathematical concepts and students are being asked to justify their findings and assertions. This would seem to be precisely the right climate to make use of proof, not only in its role as the ultimate form of mathematical justification, but also as an explanatory tool. But for this to succeed, students must be made familiar with the standards of mathematical argumentation; in other words, they must be taught proof.”

David A. Huckaby is an assistant professor of mathematics at Angelo State University.