3.4 Factors influencing functional understanding of proof
3.4.1 Two-column proof ritual
Learners tend to be convinced that a proof is true and correct based on its form or appearance rather than examining its validity themselves (Harel & Sowder, 2007). That, is, they consider a logically correct deductive argument to be a proof if and only if it is in accordance with a specific mathematical convention; two-column format. Hence, learners tend to believe that this format for geometric proofs is at least as important as its content (Schoenfeld, 1989). Shibli (1932) and Herbst (2002) suggest that the 1913 second edition geometry textbook by Arthur Schultze and Frank Sevenoak is the first to express proofs in two parallel columns of statements and reasons divided by a vertical line. That is, the layout is such that the left hand column consists of statements relating to the current proposition, and the right hand column for references to theorems being assumed as
The theoretical frameworks Factors influencing functional understanding of proof
already known and established truths (Bell, 1976). Shibli (1932) argues that the two-column custom has brought improvement in the geometry text. He describes it as not only a work of art and beauty, but also an excellent instrument of instruction in the mathematics classroom.
Knuth, Choppin, and Bieda’s (2009) conducted research with 400 learners about their understanding of the functions of proof. None of them mentioned that proof is a means to explain.
In addition, they found that if some given diagram consists of facts that are visually obvious to them, learners often see no need to go beyond their observations in proving a proposition as true.
Hence, de Villiers (1997) proposes that there need to be less focus on this form of proof in geometry since this can be done with algebra. This presentation of proof (that is, two-column) represents a development in both style and form, from essay type to two-column style. According to Herbst (2002), this format enables both teachers and learners to examine each other’s explanation of their deductive written work and facilitates the marking and correction of learners’
written work and thus bringing stability to the study of geometry in schools. Hence it has remained the default mode of proving in textbooks and a customary tool of engaging learners in proving in school mathematics (Weiss, Herbst, & Chen, 2009). This format is so prevalent that when proofs are written in narrative format – which uses conversational but logical arguments – learners tend to be unsure of their validity (McCrone & Martin, 2009).
However, if emphasis were that classroom activities must be reflective of the practices of mathematicians, two column proofs distort mathematics because no mathematician has ever worked that way (Lockhart, 2002). Wu (1996) concurs and points out that the format is different from how contemporary mathematicians write proof. Most notably, proofs presented in the two- column format have been found to promote understanding of the function of proof merely as a means to verify the truth of an already known proposition (Ersoz, 2009). In addition, Sowder and Harel (2003) argue that this format influences the development of authoritarian proof schemes wherein a proposition is accepted as true solely on the basis of authority, namely, teacher or textbook.
The theoretical frameworks Factors influencing functional understanding of proof
Apart from the fact that the format is foreign to the mathematical practice, it has ‘brought to the fore the logical aspects of a proof at the expense of the substantive function of proof in knowledge construction’ (Herbst, 2002, p. 307). This statement need not be construed as an indictment on logic in proof; while proof is central to mathematics, logic is central to deductive proof. However, the point is that attention given to the logic of a proof may take away conceptual understanding and limit or distort understanding of the construction of knowledge in the mathematics discipline. I concur with Schoenfeld’ (1988) view that advocates for flexibility in the way a deductive proof can be written given that ‘what matters to the mathematical community is the argument's coherence and correctness’ (p. 11).
Further, Herbst and Brach (2006) found that geometry learners were accustomed to tasks that required proving a proposition presented in given-prove format than in proving a general proposition such as ‘a line through the midpoints of two sides of a triangle is parallel to the third side and half its length’ (p. 84). In my extensive exposure to assessment instruments, I can confirm that such tasks are also found in tests and examinations papers in South African high schools (for example, Department of Basic Education [DBE], 2015). In these assessments, the given-prove format is followed by the two-column format that “guides” learners’ proving activity. In addition, from my experience as a Grade 12 Paper 2 (which included geometry proof questions), there is a rigid memorandum in terms of which learners’ answers had to be marked. Hence, McCrone and Martin (2009) are of the view that geometry learners conceive the function of proof to be the application of recently learned theorems rather than a mathematical process for establishing the truth of propositions. Thus, the approach to proof as merely involving the absorption of what the teacher required reflects distorted understanding of the functions of proof in mathematics.
According to Lortie (2002), the reason why two-column proofs are so prevalent is because of what he terms “apprenticeship of observation”. Lortie (1975) coined this phrase to refer to the phenomenon whereby preservice teachers study for the profession after having spent more than twelve years as learners observing and evaluating the practices of their teachers in action.
Specifically, he points out that the average learner spends 13,000 hours in direct contact with classroom teachers by the time he or she finishes high school (Lortie, 2002). Chazan (1993) argues
The theoretical frameworks Factors influencing functional understanding of proof
that simply presenting a two-column proof along with a diagram not only obscures the generality of the proposition but also gives learners no indication that the argument presented is not an argument for a single case. To a limited extent, Wu (1996) provides a reconciliatory argument to the use of two-column proof scheme. He attributes the criticism of two-column proof ritual to its abuse by previous generations. While seen as giving learners a distorted view of the functions of proof, the two-column scheme is an admirable educational tool and he advises that the format is only supposed to be used to introduce proof for at most approximately a month. This, he asserts, allows learners to make a smooth transition to writing a proof in a narrative format as it happens in the mathematical practice. Although I think that this assertion is sensible, it does not seem to have been taken serious by curriculum designers and curriculum delivery monitors; not so even by the CAPS designers. Next, I turn to instruction factor.