Herbst P. G. (1999) On proof, the logic of practice of geometry teaching and the two-column proof format Proof Newsletter Janvier/Fevrier 1999.
© Patricio Herbst

In the practice of mathematics education in the United States of America, proof has been usually associated with a year-long high school course in geometry and with a specific format -- the two-column proof format. In Sekiguchi's (1991) ethnographic study of one of those geometry courses, he describes the two-column-proof format as follows: One draws a long horizontal line and a vertical line downward from the middle to form a letter T, creating two columns under the horizontal line. In the left column, one writes a deductive sequence of statements leading to the statement to prove, numbering each statement. For each step of the deduction one has to write in the right column a reason for the deduction with a corresponding number. (pp. 78-79) The reasons given for each statement usually identify a given (hypothesis), postulate, axiom, theorem, or definition -- one and only one of them per line -- on which the statement is warranted. Figure 1 shows an example quoted from a recent text: A two-column proof (from Clemens et al., 1994, p. 301). Figure 1.    1. Two-Column Proofs and the Logic of Practice  2. Studying the Euclidean Canon to Discipline the    Reasoning Faculties  3. Emerging Discussions on the Curriculum:    Learning to Prove in Geometry  4. On the Defensive: Studying Logic or Dumping    Geometry  5. Some Claims on the Two-Column Proof    Format   A Provisional Conclusion This article has proposed some conjectures regarding what historical circumstances permitted the two-column proof format to emerge and endure during the first half of the century in the United States. The two-column proof format itself did not remain unchanged, but adapted to fit the changing characteristics of the logic of practice that it served. Assuming those historical conjectures are plausible, I have argued that the interaction between (changing forms of) the two-column proof format and (changing conditions on the) study of geometry have contributed significantly to shape a conception of (school) geometry as the descriptive study of figures, a conception of mathematical proof as at most a method for knowledge-certification, and a conception of mathematical reasoning as just logical reasoning about "given" mathematical objects of knowledge.    These plausible effects, and not the physical form of two-column proofs are what is of interest: As one may note, current textbooks sometimes induce the students to write proofs in other formats as well (such as paragraph- or flow-proofs; see Rubinstein et al., 1995, p. 396) but what has been said about the two-column proof format could be said about the explicit teaching of alternative formats as well.    From the perspective of the reader, the finished product looks like an argument that validates the proposition stated. A closer look shows that its production may not have any more meaning than the protocols used by lawyers or notaries, as indeed the division of labor in the practice that produces the proof does not look like the division of labor among a group of mathematicians manufacturing a mathematical argument. Two-column proofs can be called formal, but their formalism has little to do with the productive kind of rigor and formalism that helps mathematicians advance human understanding of mathematics (Thurston, 1995). Those formats play an important role in the logic of practice of geometry teaching, but they do not necessarily involve students in experiences with mathematical rigor and formalism. References   Acknowledgement : I thank Jeremy Kilpatrick and Daniel Chazan for valuable comments to earlier versions of this article.