3.4 Factors influencing functional understanding of proof
3.4.7 Empirical argument
As mentioned previously, an empirical argument is an argument that purports to show the truth of a mathematical claim by considering a few selected cases. Empirical arguments relied on either evidence from examples (sometimes just a single case) or direct measurements of quantities and numerical computations or perceptions to justify the generality of a proposition (Harel & Sowder, 2007). Appreciating the functions of proof is made especially more difficult ‘when these proofs are of a visually obvious character or can easily be established empirically’ (Gonobolin, 1975, p.
61). However, it needed to be mentioned that this belief persisted despite Popper’s (1988) attempts to demystify it by pointing out that ‘no rule can ever guarantee that a generalisation inferred from true observations however repeated is true’ (p. 25).
I take exception to Harel’s (2013) argument that seems to find fault in his learners when they suggest that conviction of the truth of a mathematical proposition is based on empirical evidence rather than on deductive proof. Specifically, he claims that ‘students viewed their actions of verifying an assertion in a finite number of cases as sufficient for removing their doubts about the truth of the assertion’ (p. 125). Considering this claim in light of his other conclusion that ‘[a]
person is said to have proved an assertion if the person has produced an argument that convinced him or her that the assertion is true. Such an argument is called proof’ (p. 124). I am ultimately convinced that his argument is false and his learners were correct. In support of my argument, de Villiers (1998) concludes that conviction is ‘probably far more frequently a prerequisite for the finding of a proof’ (p. 375). Similarly, Bell (1976) stated that ‘Conviction is normally reached by quite other means than that of following a logical proof’ (p. 24). A similar sentiment is echoed by Schoenfeld (1994) who points out that mathematicians try to produce a proof of a conjecture to show that it works once they suspect that it is true. De Villiers (1998) goes on to clarify why such an observation is flawed:
For what other, weird and obscure reasons, would we then sometimes spend months or years to prove certain conjectures, if we weren’t already convinced of their truth? (p. 18)
The theoretical frameworks Factors influencing functional understanding of proof
Stylianides (2009) argues that learners’ engagement in empirical arguments is likely to reinforce the common conception that an empirical argument can be used to “prove” the generality of a proposition. This notion of proof as something constituted by empirical arguments is further perpetuated by the fact that ‘all real-life proofs are to some degree informal’ (Hersh, 1993, p. 391).
According to CadwalladerOlsker (2011), primary school mathematics also contributes to the treatment of empirical arguments as proof. She points out that in primary mathematics, the weight of several examples might be enough to “prove” that the sum of two even numbers is always even.
He hypothesises that when these same learners engage in high school geometry, they may try to use similar empirical evidence to prove propositions.
Proof is something quite distinct and as such evidence alone may support a conjecture but would not be sufficient to be constitute a proof (Bretscher, 2003). However, learners develop conflicting understanding within the sciences. For instance, whereas deductive proof is the focus in the mathematics classroom, outside Euclidean proof space, including in the physical and life sciences classrooms, learners freely make generalisations based on a limited number of experiences. Thus, the inability to understand the epistemological distinction between proving in mathematics and proving in science contributes to learners’ weak appreciation of the power of deductive proof. Hence, learners often require further empirical evidence even after having proved a proposition in Euclidean geometry (Conner & Kittleson, 2009).
Learners’ empirically-based responses in deductive proof tasks indicate a weakness or lack of understanding of the functions of proof in mathematics (Stanovich, 2005). However, Mariotti (2006) remarks that an experimental investigation or a task that require learners to prove the validity of a given proposition do not seem to be as effective in triggering the production of arguments and justifications when compared to the task requiring the production of a conjecture.
In an attempt to suggest an approach that bridges the gap between empirical arguments and deductive arguments, Stylianides (2009) describes how learners could follow a mathematician’s practice that culminated in a deductive proof. He suggests an activity that involves exploring mathematical relationships to identify and arrange significant facts into meaningful patterns and structure, and using these to formulate conjectures. Then, the conjectures are to be tested against
The theoretical frameworks Factors influencing functional understanding of proof
new evidence leading to their revision to formulate new conjectures that are consistent with the evidence, and providing empirical arguments to verify the viability of the conjectures. However, I think that he overlooked the inherent difficulties that arise from implementation of this approach in the classroom. From my point of view, there is a hard wall between actual classroom practice of mathematics and the practice of mathematicians and to break this wall requires a concerted effort from a variety of stakeholders, most of all from politicians. In addition, to produce a conjecture is a task that does not fit the didactical contract13 in school mathematics wherein propositions are presented and illustrated by the teacher, absorbed and applied afterwards by the learner in tests and examinations (Douek, 2009).
Empirical arguments, frequently the only type of proof comprehensible to learners, may be mathematically valid for establishing refutation by counterexample but invalid if few cases were used for a proof (Hanna, et al., 2009). Hence, “proof” through providing empirical evidence rather than through validation, though prevalent, only limits learners’ understanding of the functions of proof to that of verification or justification. The term “validation” is used to refer to the construction of reasons to accept a specific proposition, within an accepted framework shaped by accepted rules and other previously accepted propositions (Balacheff, 2010).
That empirical explorations provide limited insights into the functions of proof is based on the premise that they provide inconclusive evidence by verifying truth of propositions only for a proper subset of all the cases covered by a deductive proof (Stylianides & Stylianides, 2009). That is, unlike deductive proof which combines logical propositions, there may be an exception or counterexample that negates a conjecture; in actual fact, empirical arguments lead to conjectures because it is virtually impossible to consider every case. Having said that, Harel and Sowder (1998), Healy and Hoyles (2000), Knuth (2002), and Knuth et al. (2009) demonstrated that naïve empiricism is widespread and pervasive way of reasoning among high school learners of mathematics. Balacheff (1988) uses the phrase “naïve empiricism” to describe the practice of
13 Brousseau (1997) refers to the teacher’s routine instructional obligation as a didactical contract.
The theoretical frameworks Factors influencing functional understanding of proof
asserting the truth of a proposition after verifying several cases; that is, using empirical arguments as mathematical proof.
My own personal experience as a Grade 12 examination marker also attest to this argument.
I suggest that naïve empiricism is encouraged in examinations. For instance, in the South African context, the high school question papers is often sequenced such that immediately after proving a theorem, subsequent questions require learners to verify the validity of that theorem empirically by considering specific cases. This instance is exemplified in Figure 3—8. The sample question is an adjusted version of the National Senior Certificate14 (NSC) examination (mathematics paper 2) prepared and written in the February/March supplementary examinations period. The question reflects a practice which is akin to seeking conviction about the truth of a proposition by considering particular cases. This practice predominates mathematics classrooms in high schools.
For instance, Schoenfeld (1989) and Fischbein and Kedem (1982) found that learners tend to seek conviction by empirical means although they had just performed a deductive proof of a conjecture.
This behaviour reflects learners’ failure to appreciate that proof provides a firm intellectual foundation which meant that they did not have to appeal to outside experience.
14 In the context of the South African education system, the National Senior Certificate (NSC), commonly referred to as “matric”, is a national, standardised examination, which represents the final exit qualification at the end of high school (Grade 12).
The theoretical frameworks Factors influencing functional understanding of proof
QUESTION 10
10.1 In the diagram, 0 is the centre of the circle and P is a point on the circumference of the circle. Arc AB subtends ∠AOB at the centre of the circle and ∠APB at the circumference of the circle.
10.2 In the diagram, O is the centre of the circle and P, Q, S and R are points on the circle.
PQ = QS and ∠QRS = y. The tangent at P meets SQ produced at T. OQ intersects PS at A.
Use the diagram to prove the theorem that
states that ∠AOB = 2∠APB. 10.2.1 Give a reason why ∠P2 = y.
10.2.2 Prove that PQ bisects ∠TPS . 10.2.3 Determine ∠POQ in terms of y.
Figure 3—8. An example of examination questions sequenced to written in the February/March supplementary examinations (pp. 12-13)
This phenomenon can be traced back to instances where learners are generally provided with
“riders” as extension of classroom instruction to provide practice and thus, at least unwittingly, consolidate conviction in the truth of theorems. Learners need to understand that an empirical argument merely confirms the validity of a conjecture and serves as a spark to understanding why the conjecture is true. This is not to be construed as an indictment on empiricism; it must be stressed that empirical evidence is a necessary but not an altogether sufficient step towards the
The theoretical frameworks Factors influencing functional understanding of proof
development of a proof. The point is, one other way in which the seeing proof solely as a means to verify develops is through believing that empirical arguments are mathematical proof.