Constructivism reconsidered

When I first entered teaching I was very opposed to ideas of constructivist teaching such as delineated in the Yilmaz article. The first reason for this is that my supervising teacher was in the midst of tete-a-tete over the direction of math instruction in her department with her supervisor who was fresh out of grad school and a strong advocate of constructivism. Working with her as a cooperating teacher I was encouraged to look at constructivism with a pessimistic eye from the start. The second is that I entered the teaching profession through the alternate route program, coming from a technological industry with a scientific degree. In my ignorance it seemed at the time that constructivism must be muddle headed nonsense from ivory tower researchers who never saw a day in the 'real world' let alone ever stepped foot in a classroom. Yet as I have progressed through the profession I have seen the absolute necessity of teaching to meaning and that the constructivist perspective is a useful point of view for reaching this goal. I found it interesting that Yilmaz does in fact stress that constructivism is not so much a cogent theoretical structure as it is a group of perspectives with social, psychological, and radical dimensions. As I have progressed through teaching many different high school math classes I have come to find myself very much in agreement with positions such as Piaget's (excerpted from Gillani) that “...an individual encountering a new learning situation draws on prior knowledge to make a new experience understandable.”

The field that most choose to communicate their doubt in the efficacy of a constructivist approach is invariably mathematics. At first blush this seems to be the quintessential example of a discipline which must surely be devoted to the imputation of external unchanging facts. A popular example cited is that no matter what, 2 + 2 = 4. Since this is the example that was also discussed in class I would like to examine it. Far from being a trivial matter whose meaning should not be examined and understood, the concept of the number 2 as a cardinal rather than an ordinal was a crucial development of human thought. Indeed there are some cultures (the Tauade of New Guinea and to some extent in Japanese) where there is no number 2 per se – the term for 2 things depends on the objects being enumerated. Cardinal numbers in general are more properly thought of as the first piece of modern technology and therefore to be examined as a constructed tool whose meaning should be fundamentally understood before being used.

From a practical point of view in the classroom if we do not have an appreciation for the meaning that underlies the concept of number we will not be able to answer some rather natural questions from students first learning this concept in the early grades. This will necessary hobble learning the dependent critical concepts upon which it builds on in later grades. For example why is it that 2 + 2 = 4 and yet for a totally different operation, multiplication, 2 * 2 also = 4? We can use this for a jumping off point for exploration – an activity normally seen as only fruitful for 'exceptional' children. What other numbers work like this? What if we use three numbers in each operation instead of just 2? Is there a pattern where this always works where the numeral itself is the same as the number of operands, and if so why, if not why not?

I have had direct experience with an even simpler example than that above. In conversations with members of the public or even the teaching profession in the math wars on the side of 'good old-fashioned' math teaching one will invariably hear that 'No matter what 1 plus 1 will always equal 2', and thus we should eschew teaching to fundamental meaning in the interest of teaching students basic skills. Yet I have had the opportunity of teaching a computer class where it was decidedly not the case that one plus one equals two - in the binary number system of computers one plus one equals ten! When I reached this unit in my curriculum what I thought would be a straightforward lesson in digital electronic components became a fascinating discussion of why this shocking statement is true in binary, and the discussion quickly extended out to a discussion of number base systems in general. Students took the ball and ran with it: “Can we do math in binary and if so how does it work? Is it easier than the math that I am learning in my math class? If so can we please learn it right now?” This kind of invigorating student directed inquiry is the natural by-product of encouraging students to construct and discover their own meaning in topics that seem to be natural candidates for a non-constructivist approach.