Here are two accounts from today's Higher maths lesson, followed by some reflection. J and K were working on this exercise, designed to stimulate a deeper understanding of the relationship between a function and its gradient function, as well as introduce points of inflexion. [It would be beneficial to have a look through the exercise before reading the rest of the post]
K created a symmetrical cubic, which resulted in a gradient function of a similar form to one that he had previously had difficulty with, which was good. Here is where he has got to:
The exercise could be said to employ 'variation theory', with elements of surprise, something along the lines of the 'taxi-cab geometry' exercise described in this paper by Anne Watson and John Mason.
Each example varies the amount of freedom and constraint that is afforded - sometimes there are many possible solutions, sometimes only one. There is a constant shifting between doing and undoing, and between the iconic and symbolic.
Some of the examples require students to make choices, to use their initiative. There are opportunities for creativity (see #2 above).
Having a range of examples allows identification of sameness and difference. This task includes prototypical examples, unusual examples and boundary examples (such as f'(x)=3). Some examples brought surprise - a natural means of stimulating learners to check whether something 'makes sense', or whether it fits in with their current schema.
There were connections to other things we have studied. For example, finding the range of values for which a cubic is increasing or decreasing is the same as finding where the (quadratic) gradient function is positive or negative (i.e. connecting to previous work on quadratic inequalities).
I was once again reminded of the importance of being creative when planning mathematical activity, even - or especially - a couple of months before an exam. Sometimes a set of exam problems might be what is required, but often there is something lacking in working through a set of exam problems, even if it is done conscientiously and with reflection. A carefully crafted task often creates more involvement, and reveals more about the mathematical structure you would like to direct learners' attention towards.