Suppose you want students to gain facility in factorising the difference of two squares, and make use of this when expanding a product of the sum and difference of two quantities. A direct approach might be to invite learners to expand products of sums and differences of quantities, varying the complexity and format of those quantities:
Expand: (x + y)(x – y); (a – p)(a + p); (2x + 3p)(2x – 3p); (x²y-2wz³)(x²y+2wz³); ((x+y)²-(x-y)²)((x+y)²-(x-y)²) and 8xy(x²+y²); and so on.
Factorise: ; and so on.
According to the sophistication and maturity of your learners, you might want to develop the variation in smaller steps, being careful not to over-simplify. You would use terms that from experience you know obscure the presence of squares, perhaps garnered from examples constructed by previous students. You can check the depth and complexity of a learner’s awareness (the extent to which they have internalised these actions so that they come to the surface when needed) by inviting them to construct complex examples of their own which show that they appreciate what can be varied and to what extent.
In all of these cases, the recognition of two distinct entities is key. In the expanded form, each entity is a square and the expression is the difference of these squares. In the factorised form, you have a product of the sum and difference of the two entities. In effect, you must ignore the complex form of the entities, identify them as squares, factorise the squares, then transform back to the complex form of each entity; or identify them as the product of a sum and difference, then transform back. You are stressing the form and ignoring the individual components. So factorising a difference of two squares can be seen as a transformation (of attention, that is of what is focused on), followed by a familiar and well-internalised action, and then transforming back. This process of doing, acting, and undoing is known as a by-pass (or as ‘conjugation’ when it appears in group theory). When you can’t do what you want to do where you are, you transfer to somewhere else where you can do it, do it, and then transfer back again.
For example, if you want to differentiate a compound function like y=sin³(x²+3), you use the substitution u=sin(x²+3) so that you can differentiate y=u³ with respect to u; then using the substitution v=x²+3, you can differentiate u=sin(v) with respect to v; and finally, you differentiate v with respect to x and put it all together. Each time, you are transforming ‘where you are’ to an easier place where you are confident and can make use of techniques already secured.
Consider the calculation:
You look first for relationships, and recognise the products in the numerator as products of a sum and a difference, while in the denominator there is a common factor and the same thing is happening again. You ‘park’ the first impulse to ‘calculate’ and ask yourself whether there is an easier way.
A by-pass, or conjugation, is a classic mathematical move and it can really only be developed through experience and intentional reflection upon that experience. The important pedagogic action is to get learners to articulate for themselves (to become aware of) the transformations they are using.
It might be tempting to start by getting learners to construct squares of complex expressions so that they become familiar with recognising squares, but again this would depend entirely on the sophistication of the learners. There is time enough to work on recognising squares while in the midst of something else. If everything is ‘stair-cased’ and presented to learners with little steps, they are likely to be caught up in the flow of ‘doing tasks’ without thinking much about them. By presenting them with complexity, you are calling upon their natural powers to restructure the experience for themselves according to their own needs. By inviting learners to appreciate what is the same and what is different about several instances, you are making use of what ‘variation theory’ considers to be the way human beings actually learn.
John Mason worked for the Open University Mathematics Department, where he designed two of the mathematics summer schools, contributed to numerous courses, and then helped form and run the Centre for Mathematics Education. He has written numerous books and booklets, as well as research articles and book chapters, the best known being ‘Thinking Mathematically’ (1982, 2010).