BREAKING THE EQUATION
Andreas Stylianides describes working with students on a journey to
appreciate the difference between being convinced and ‘proof’.
Empirical argument vs. proof (e.g., 213 + 399 = 612), some with the same
odd numbers (e.g., 25 + 25 = 50), and some
Consider the generalisation: ‘the sum of any two with prime odd numbers (e.g., 17 + 31 = 48).
odd numbers is an even number.’ What argument No pair gave me a counterexample – the sum
would your students offer for it? Would that be a was always an even number. So the sum of any
proof? two odd numbers is an even number.
An overwhelming body of research shows that
students of all levels of schooling including high- Even though both arguments are invalid, the
attaining secondary students ‘prove’ mathematical second argument can be considered more advanced
generalisations such as the above by using empirical than the first, because, by seeking possible counter-
arguments (e.g., Coe and Ruthven, 1994). By examples, it communicates a concern that the
empirical arguments I mean those that purport to generalisation may not be true. Balacheff (1998)
show the truth of a generalisation by validating the used the terms naïve empiricism and crucial experiment
generalisation in a proper subset of all possible to describe the special categories of empirical
cases. These arguments are clearly invalid, because arguments represented by the first and second
they cannot exclude the possibility of the existence examples, respectively. The search of possible
of a counterexample to the generalisation. Here are counterexamples in crucial experiment requires a
two examples of empirical arguments for the above strategic selection of cases in contrast to the
generalisation: random (or convenience) sampling of cases in naïve
Empirical argument 1: naïve empiricism empiricism.
I tried many different pairs of odd numbers The fact that a generalisation is true in some
and their sum was always an even number: cases does not guarantee and, thus, does not prove
7 + 9 = 16, 15 + 21 = 36, 25 + 27 = 52, that the generalisation is true for all possible cases.
etc. So the sum of any two odd numbers is an This is the main limitation of any kind of kind of
even number. empirical argument that many students find diffi-
cult to understand. What would be a proof for the
Empirical argument 2: crucial experiment generalisation then? Figure 1 shows three possible
I checked different kinds of pairs of odd proofs for the generalisation on the set of whole
numbers: some with small odd numbers (e.g., numbers.
1 + 9 = 10), some with big odd numbers
Figure 1: Three possible proofs (on the set of whole numbers) for “odd + odd = even.”
MATHEMATICS TEACHING 213 / MARCH 2009 9
Notice the correspondences among the three empiricism as a method for validating patterns, to
arguments: they all seem to be saying the ‘same using crucial experiment, to feeling a need to learn
thing’ using different representations. Notice also about more secure methods for validating patterns
how each argument can be used to help someone (i.e., to learn about proofs). Note that a pattern is
understand why the generalisation is true, but also a kind of generalisation. The teacher and student
convince someone that the generalisation is true names are pseudonyms.
for all cases without requiring that person to make
Activity 1: The squares problem
a leap of faith. A proof ’s potential to promote
understanding and conviction is one of the main Kathy, the teacher, introduced the Squares Problem
reasons for which proof is so important for (Figure 3). The hardest part of the problem was the