Canons have been around in the humanities for a long time. My definition of a canon would be a list of authoritative works that set the standard or define the essence of what that discipline would be about… like one of those books that are popular at the moment such as “1001 paintings you should see before you die”. Perhaps mathematics could assemble a list of proofs that could act as our discipline’s canon….or 1001 ideas that shape mathematics?
What I have been thinking about for just over 12 months now is assembling some activities that can be used in the classroom for various mathematical concepts that would encapsulate the idea and allow students to engage with the concept, understand it in a way that would enable them to transfer that learning to other contexts, and also have adequate space within its execution to extend students’ thinking. That is, activities that give students the opportunity to connect to previous learning, extend and challenge their thinking and cross a bridge to new learning. These are what I would call the set of canonical activities in mathematics for secondary students….a set of activities that would be the essence of the ideas of secondary mathematics and set the standard.
Last night, I attended, along with various mathematics educators from MonashUniversity and the University of Melbourne, a colloquium at the Education Faculty of Monash University, led by Dr. Anne Watson (Oxford University, UK) and Professor John Mason (Open University, UK). In this session they talked about templates, algorithms, the making of meaning and embedding meaning into the examples we use in mathematics classes so that they are ‘enabling’ for students in the way that they enable students to engage with big mathematical ideas.
Anne started with an example of what they define as a template: she found the distance between two points and showed all steps in the working so that anyone could follow this ‘template’ using co-ordinates of any other two points. Students could substitute values into the appropriate places by following the pattern showed in the template. How much of what we teach is ‘learnt’ by students in this fashion, whether we intend them to do so or not?
Anne next gave us all strips of paper and said we were going to be taught how to fold it into sevenths. She instructed us to fold over the left end of the strip then unfold it again. The part we had folded over was going to act as our first approximation to 3/7. This, of course, meant that the remaining part of the strip was our first approximation of 4/7. We were then told to fold the 4/7 section into quarters and unfold. Next, we took 3 of these parts and made this our second approximation of 3/7. This meant the remaining section on the left was now our second approximation of 4/7 etc. This process was continued until we were satisfied we had a good representation of 1/7.
Now, this activity was interesting in itself BUT it hasn’t actually allowed students to connect, extend and challenge their thinking as yet or enabled them to embed any meaning. In order to do this, a discussion is then needed to consider questions such as
“What other fractions could we divide a strip of paper into using this method?”
“Suppose we wanted to divide a strip of paper into fifths – how could we do it using this method?”
Some realisation is needed that the strip needs to be divided into numbers of parts so that one end is an even number (which will allow us to halve or quarter by folding) and an odd number. This could then lead into the partitioning of integers using powers of 2 and a remainder. The activity started out as a template but had sufficient space in it for extension and to embed mathematical ideas that took the physical manipulation into some other dimension of learning.
John then presented us with a question: If R = number of rows and C = number of columns then there is some quantity defined by 2RC – 2R + C. Is it possible for the value of this quantity to be 37?
After some trial and error we found two numbers that made it possible and we were then asked “Are they the only two values of R and C that would satisfy this equation?”
In further discussion, it was revealed that we could approach this problem algebraically using factorisation:
2RC – 2R + C = (2R + 1)(C – 1) + 1
Putting the expression on the right equal to 37 helped us formulate the problem and consider how many values of R and C would make this true. He then asked “Does the fact that 37 is prime make any difference?”
Again, the problem started out as a simple trial and error but it had many spaces it could expand into… a window into worlds of making meaning and exploring mathematical ideas.
Finally, we were invited to consider the concept of multiplication and asked to write down an example that encapsulated what multiplication meant to us. After thinking for a bit, we were then asked to write down another example that was somehow different to the first. Yet again, after writing this down, we were told to write down a third example that was different again to the first two. Examples were shared then we were instructed to write down a final example that now represented the essence of multiplication for us.
I thought this was a fantastic exercise and could be utilised in many different topic areas as both a pre-instruction (oh how I hate that word ‘instruction’) tool to establish students’ prior knowledge in terms of their actual understanding of the idea and also as a review tool to see if students had actually gained the understandings we wanted them to.
If we could assemble a bank of such activities for all ideas in secondary mathematics, imagine what we could do….