Last week Andrew Blair (@inquirymaths) tweeted a link to a blog post by James Pearce (@mathspadjames) about deep connected mathematical understanding. It came at just the time I had been reflecting on this activity that I run with my students. Here’s how it goes:
Fireworks is a term that my students quickly got used to me using. When skimming through an exam paper with them (Core 4, I think) I was thinking out loud: demonstrating how I noticed the ‘tell-tale signs’ in a question that mean I recognise it as a certain topic in mathematics. ‘Fireworks’ describes what happens (for me, at least) between the stages of recognising the topic and tackling the question.
There’s a *between* stage?!
Students need to be taught not to dive straight into answering a question. Not told. Taught. It’s taken me all the time I’ve been teaching so far to realise this. “Always read the question carefully.” “Look at what information you are given and what the question is asking for.” “Try and draw a diagram whenever relevant.” All of these wise pieces of guidance come out of my mouth so often that I should probably get t-shirts made. But, I would argue, none of these is especially helpful nor do they encourage the students to pause and think.
The fireworks display
Let’s take an example:
What lights the blue touch paper here? The term ‘binomial expansion’.
What do the fireworks look like? As soon as we’ve identified the topic, I blank the projector and ask the students “What could possibly happen in this question?” I don’t really want the product to be a mind map so I rarely write on the board at this stage. I want the students’ minds to fill with all the mathematical concepts that they connect with the binomial expansion:
- its domain of validity
- how it is found (and does the formula book help us out)
- how many terms might we be looking for
- does it allow us to find an approximate value
- do we need to add/multiply more than one expansion
- does the bracket ‘begin with a 1’
- potential risk of minus signs everywhere
- if we take out a factor, the power comes with it
- might we need to find an approximate value for an integral
The list can almost certainly be continued but, as you can see, it accumulates some basic theory, common causes of mistakes, and ideas about how the question might pan out. As my students are happy to play along in my imaginary world, I tell them to sit back for a minute and ‘enjoy the fireworks display’ that they create mentally before we share these ideas in discussion.
I truly believe that this deliberate pause best sets them up for tackling the questions and, moreover, really emphasises the connections throughout mathematics.
What if the blue touch paper was ‘partial fractions’? We’d be in for a long show: the techniques involved, the ‘risk’ of a repeated factor, the possible lead-in to binomial expansions (accompanied with the above fireworks offshoot), the possible lead-in to integration (with even more fireworks… logarithms? power -2?…) etc
After the fireworks have ended, the students are much more focussed (excited, even) to read through the full question and see if they had better ideas than the examiner. “Oh, it’s all rather boring and we only have to expand and substitute a value. Ah, but they have been sneaky and given us an 8 to factor, a negative coefficient on the second term and a cube root sign that needs representing as a power.” (Often followed by some rather disparaging name for the examiner.)
Not just November 5th
Because the students responded so positively to this activity, it is now quite a routine part of our lessons. Sometimes I will project part of an exam question as the prompt. Other times I will simply write a word or two on the board. (Try ‘range’, ‘tangent’ or ‘a uniform rod’…)
Think, Pair, Share
- Have a go at ‘fireworks’ with one of your exam classes: give them the individual time to think before discussing as a whole class. Do they make good predictions about how an exam question could develop? Are they aware of possible careless mistakes that could creep in? Do they see creative connections between mathematical concepts?
- Do your students typically dive straight in to answering exam questions? Do you have other strategies to slow them down and encourage them to think more carefully?