3.3 Learners’ functional understanding of proof
3.3.1 Proof as a means to verify the truth of a proposition
A proof can be viewed as a tool to establish certainty of a conjecture, that is, verifying (making sure) that a conjecture is true for all cases. In validating the correctness of a mathematical proposition or simply verification, all that is required is to logically connect axioms to arrive at a conclusion regardless of its form or aesthetic appeal (Hanna, 2007). Verification denotes the removal of uncertainty by seeking, in the vocabulary of Harel and Sowder (1998), to “convince”
The theoretical frameworks Learners’ functional understanding of proof
or “persuade” someone or oneself about the validity of a conjecture. Harel (2013) takes this idea of certainty further and claims that the ‘need for certainty is the natural human desire to know whether a conjecture is true—whether it is a fact’ (p. 124). Schoenfeld (1994) describes the benefit of reaching certainty so eloquently thus:
One of the glorious things about proof is that it yields certainty: When you have a proof of something you know it has to be true, and why. That feeling of certainty is really powerful, for patterns and trends can be deceptive. All mathematicians have their favorite examples of patterns that look like they ought to hold but fail, or of conjectures that are true for the first N tries but then fail. (p. 26)
This function of proof is most familiar to research mathematicians but regrettably missed by learners as they often complain that it is pointless to prove theorems that “everybody knows" or that have already been proven by other people in the past; a proposition is not a true until it is verified to be so by the construction of a proof (CadwalladerOlsker, 2011). In school mathematics, verification is associated with providing examples as proof that a conjecture is true; nothing more.
However, empiricism is only an important process in merely gaining conviction to seek a proof rather than a proving process itself. Empiricism is defined as making an assertion about the truth of a conjecture after verifying several cases (Balacheff, 1988). Therefore, empiricism is defeasible;
there are historical examples where counterexamples overturned earlier generalisations. This approach reflects an appreciation of the fact that empiricism and quasi-empirical investigations are unsafe; therefore, a proof provides what is refutably absolute guarantee (de Villiers, 1998).
Another traditional approach in mathematics classrooms is to use some examples and then proceed to introduce deductive proof only as a means to verify that the conjecture being tested with examples is true and thus attain conviction. A conjecture is a proposition that is consistent with data and has not been proven to be either true or false (Uploaders, 2013). The main point here is to note that verification of a mathematical proposition can take two forms: empirical or deductive; empirical by selecting a few cases and deductive by logically connecting a set of axioms to produce a new result. Lakatos (1991) argues that even though proof is regarded as the ultimate authority on the truthfulness of a conjecture, its certainty is vulnerable since the axioms on which it is based continue to be open to revision by the mathematics community. The revision may either
The theoretical frameworks Learners’ functional understanding of proof
have been necessitated by a recognition of human error or inconsistencies in an axiomatic system (Umland & Sriraman, 2014).
Mudaly (1999) argues that research has shown that by engaging in appropriate exploratory activities using DGS learners can gain conviction. This, that is conviction, is the predominant justification method used by learners. Notably, this function of proof is noble on its own in that although the proposition is already undisputed mathematical knowledge, there is value in the learner gaining conviction following the same creative path a mathematician would have taken when discovering that knowledge for the first time (Bartlo, 2013).
De Villiers (1990) argues that if learners see proof only as a means “to make sure” through their own experimentation then they will have little incentive to generate any kind of logical proof.
He points out that instead, it is this conviction that propels mathematicians to seek a logical explanation in the form of a formal proof to know why a conjecture must be true. This suggests that it is this role of proof as a means to explain that can motivate learners to seek to generate a proof for a conjecture. Important to consider is that learners need to be aware that the proofs they are learning are new to them but consists of results that are known to be true (Hanna, 1995).
Indeed, given the scientific nature of mathematical knowledge, for each correct conjecture there should be a sequence of logical transformations moving from hypothesis to conclusion (de Villiers, 1990). However, Davis and Hersh (1981) characterise this as a naïve view of mathematics in light of the fact that proof can be fallible. The history of mathematics is littered with instances of “theorems” whose proofs were later found to be false. Hersh’s (1979) position that ‘[w]e do not have absolute certainty in mathematics. We may have virtual certainty, just as in other areas of life. Mathematicians disagree, make mistakes and correct them’ (p. 43) captures the tentative nature of proofs. This problem notwithstanding, formal verifications maintain an important and useful function of proof in mathematics (Stylianou et al., 2015). The next section discusses the function of proof as a means to understand why mathematical propositions are true.
The theoretical frameworks Learners’ functional understanding of proof