This post is about a particular aspect of a session with two Higher maths students (J and K).
We were working on analysing some exam questions that the students had tried for homework, but found difficult. This is the first time we have tackled some of the difficult questions at the end of Higher papers (English A-level teachers might find these papers a source of interesting / challenging problems).
I decided to break the problem down into a set of questions and prompts for the students to work on, rather than through. A loose structure for these prompts started to emerge that might be interesting to develop:
- Some reflection on a first attempt at the problem. What did you try? Why? Why was it difficult?
- Some 'breaking down' of the problem with isolated / simpler / numerical examples, designed to give some insight into structure.
- A return to the original problem in light of this breaking down.
- Then, an invitation to try a similar (exam) problem, and then create a problem 'like this'.
- Reflection on what was learned about the content, the 'type' of problem, and/or solving problems more generally.
Here are some of the responses (click to enlarge):
There were further reflections on this problem that were very interesting - more on this below.
I had also created a similar set of questions related to this problem, along the lines of the structure described above. Here are some responses (click to enlarge):
At this stage, I asked J and K what they had learned from working on these problems. Their responses were generally about the content, logs. Then J had a real insight. He said: "We can change something we don't know into something we know about by re-naming it, and then replacing it later." He gave the example of using A and B in place of log(3,x) and log(3,y) in problem 2 (see axes, and equation):
Re-naming log(3,x) and log(3,y) as A and B allows us to solve the problem as if it was about straight line graphs, before replacing A and B with what they represent - log(3,x) and log(3,y) - in order to find the relationship between x and y.
We then realised that something similar had happened in problem 1! Renaming log(3,x) with P [and log(9,x) with 1/2.P] allowed us to solve the equation in terms of P, before replacing P with log(3,x) to solve the original equation (see circle):
This is of course not new for mathematicians. It is a form of substitution, but is not only substitution. It is rather a type of working as-if, a suspension/simplification of the original problem by re-naming something, thus making it easier to work with, and then re-placing that which was re-named in order to solve the original problem.
This is an exciting realisation / strategy, and now that it has come into our conscious awareness and we have named it, will be something we can come back to.