3.4 Factors influencing functional understanding of proof
3.4.2 Instruction
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.
The theoretical frameworks Factors influencing functional understanding of proof
at school, they did not even attain 50% on Level 1 and that attainment of Levels 2 and 3 is even rarer in the test results.
Once in the teaching field setting, very few will contest the conclusion that these teachers will either avoid teaching Euclidean proof or encourage memorisation of proofs rather engage learners in what Edwards (1997) refers to as “exploring the territory before proof” (that is, exploration, conjecturing, and argumentation). Giving learners predetermined propositions to prove reinforces the predisposition that the proposition must be true; so they merely need to verify.
Sadly, this is bound to be of little real interest to the learners – the undesirable consequence of such instructional practice is the development of negative attitudes towards Euclidean geometry in particular and mathematics in general.
Generally, presentation of proof to learners as text and diagrams to memorise left them incapacitated to appreciate its value in mathematics. This perspective need not be construed as proposing that memorisation, achieved through drill and practice, is frowned upon in my view of learning. I view memorisation of important concepts of mathematics as a practice that allows learners to acquire basic skills that enabled fast, accurate, and effortless processing of information which frees up working memory for more complex aspects of proof. In this study, in contrast to Western perspectives on learning, memorisation is viewed as the route to understanding. My view of learning draws on perspectives of learning from both Western and Asian cultures. This view rests on the premise that learning is the acquisition of knowledge through primarily the teacher or text and also through learner’s own effort. From this view, learning does not preclude memorisation with a view to gain understanding.
I concur with Tavakol and Dennick (2011) who view memorisation not as an end in itself but as a path to understanding. Thus, the memorisation of propositions—not their proofs—is helpful because it engenders understanding. It is in this light that I take the position that instruction designed in terms of van Hiele’s five sequential phases of learning (inquiry/information, directed orientation, explicitation, free orientation, and integration) promote learners’ acquisition of Level 4 partly through memorisation. The theory suggests that learners have attained Level 4 if they
The theoretical frameworks Factors influencing functional understanding of proof
understood the functions of deduction (why proof is important in mathematics) and the roles of postulates, and theorems such that proofs could be done meaningfully.
Van Hiele proposes characteristics of these levels which, like Usiskin (1982), I labelled:
fixed sequence; adjacency; distinction; separation; and, attainment. The last characteristic emanates from van Hiele’s suggestion that cognitive development in geometry can be accelerated by instruction. They provide detailed explanations of how instruction can move a learner from one level to the next. However, it is not the intent of this study to examine these phases – Hoffer (1981) provides a detailed account. That said, I argue that learners’ understanding of proof must be a feature of the information phase because it is in this phase that learners can be acquainted with the significance or importance of proof by emphasising its functions in mathematics. Instruction designed along such van Hiele lines would not only improve learners' ability to write formal deductive proofs but also help to develop learners’ understanding of the function of proof in mathematics and thus provide them with insight into the activities of mathematicians. For learners to see the functions of proof and to experience the work of mathematicians, they must see how mathematicians use proof as a way of thinking, exploring, and of coming to understand (Schoenfeld, 1994).
A controversial issue in the field of mathematics education is whether classroom instruction should promote more instrumental (traditional or knowledge transmision method) or relational understanding (reform-based methods). It is my view that certain topics in mathematics need to be taught more effectively with one method or another – teaching methods were guided by context. Put another way, teaching should not be exclusively “instrumental” or “relational”.
Consistent with this view is the National Mathematics Advisory Panel’s (2008) instruction that the widely held belief among teachers in high schools that one method is better than another is not supported by research. However, whichever method is used in the classroom, the argument in this project is that learners need to be exposed to experiences that help learners to develop appropriate conceptions of the functions of proof in mathematics. Thus, I am of the view that teaching for functional understanding of proof needs to be an integral part of whichever method of teaching Euclidean geometry.
The theoretical frameworks Factors influencing functional understanding of proof
In the mathematical practice, proving is a process of learning and discovering new mathematics. First, a conjecture would be made based on observation of a number of cases. If available, dynamic geometry software could be an aid in hastening conjecturing. Next, an attempt would be made to explain the conjecture through proof. But, this is not the end of the story in the mathematician’s work on proof. The created proof needs to be communicated to other mathematicians before final acceptance of the conjecture as a mathematical truth. For learners to experience these functions of proof, they must make and test their own conjectures. Of course, the created conjectures and proofs would not necessarily be new mathematics; but, to the learners they would be.