I am noticing an increasingly large number of teachers embracing the theory of variation. I have found myself wondering: what are the benefits and drawbacks of this emphasis on variation?
A simple interpretation of variation as creating an exercise in which something varies whilst the rest remains invariant is attractive. At the very least, such an exercise provides some 'practice', and at best an opportunity to spot some pattern that might give insight into underlying structure.
But it is also possible that a series of minimally different examples, or an example-problem-pair, will enable some learners to switch into a more automatic mode than is usual, mimicking what has gone before. There is also the danger of creating exercises that lead to a game of 'can you see what I see?' [I speak from my own experience!], or of pattern spotting becoming the focus of attention rather than underlying relationships.
What are the features of exercises that use variation in a way that draws attention to structure? To get a sense of what these features might be, try this exercise by Krause (from this paper on variation by Watson and Mason):
In that paper, Watson and Mason describe "the break in pattern that caused many to begin to think about the mathematics behind what they were doing." This was the case for me: A sense of surprise as I went from just doing, to thinking 'oh, ok, a straight line', to realising what was really going on, followed by questions: how is this connected to 'normal' geometry? And then: Oh! Yes... of course...
Watson and Mason describe the challenge in creating exercises that use variation as beautifully as this one:
We do not know how long it took Krause to develop this exercise, but such artistry and precision in helping a learner learn does not come instantly. Constructing tasks that use variation and change optimally is a design project in which reflection about learner responses leads to further refinement and further precision of example choice and sequence... This process... can be done by teachers for themselves.
In this article from MT 252, Anne Watson suggests two questions that designers of exercises may wish to consider:
- What juxtaposition of items, or sequential presentation, or use of materials or layouts can make the underlying relationships discoverable?
- How can I make it likely that all students appreciate the underlying structures intended in the design process?
Another way of bringing variation into mathematical activity is to ask learners to generate examples 'like this'. Inviting learners to create their own examples has a number of additional benefits, including requiring learners to attend closely to the features of a type of problem, to what happens if I...
The apparent simplicity of variation may have led to it becoming another panacea. But designing exercises that use variation in a way that draws attention to underlying relationships is not as simple as it seems. One must research one's own practice, as described here by Caleb Gattegno:
We must study [education] seriously, not go to courses and wait for someone to tell us what we think. I hear that there was one session today about the teacher as researcher. If you read what I wrote in the late 40s, early 50s, this is spelt out there. And it is spelt out with a certain force, in which I said: ’Only you, who are in the classroom, can do this study. If you don’t do it, it will not be done’.... there are years of reflection, experiment and trials, with sometimes a correct answer found in a reasonable time, sometimes not found, ever...The opportunity we have of educating ourselves has been wasted year after year after year. I did not waste it; but I say that to myself because nobody believes it, nobody knows it. I didn’t waste it. I know that every time I was in a classroom, working in a certain way, when I finished I could tell myself: ‘you have found this that you did not find earlier in similar circumstances, and it belongs to the situation, it is part of it’.