Recently I’ve noticed something odd. Most maths teachers, when asked, can put forward a relatively strong case in support of why they want pupils to learn about problem solving. However, and this is the odd bit, if you drill down to a particular problem-solving question and ask them what they intend students to learn from that problem, they don’t seem to be able to give such a clear answer. More often than not, all they can say is that they were hoping that the students might get better at problem-solving.
This jars with the way we usually like to do things. If I were to ask the same about any other mathematics question, lesson or topic, I’m quite confident that those teachers would not only be able to describe to me exactly what knowledge they would want students to have by the end of the lesson, but they would also articulate how that knowledge fits into what students have learnt previously and what is planned next.
Leaving it to chance leads to inequity
This feels like (if you pardon the pun) a big problem. It is indeed situations like these – where teachers are unclear about what they are teaching or how it fits into their curriculum – that are liable to widen the gap for disadvantaged learners. Colin Foster puts this well when he says of problem solving that:
"Students with more advantaged socio-economic backgrounds are more likely to have received enhanced opportunities for support... to develop dispositions that allow them to make good use of open-ended opportunities, which other students may simply struggle with and learn very little from."
Explicit Teaching of Problem-Solving Strategies
So, what can we do about this? One potentially fruitful avenue is to consider how we might deliberately plan to teach problem-solving strategies to students.
“If pupils lack a well-rehearsed and readily available method to solve a problem they need to draw on problem solving strategies to make sense of the unfamiliar situation.”
We’ve known now for some time that when we don’t explicitly teach vocabulary to students the gaps between the most disadvantaged students and their peers grow. It seems we’re in a similar situation with teaching problem solving strategies.
Where might you start?
In my next blog I’m going to be exploring how comparing worked solutions can be a great vehicle for focussing your teaching on problem solving strategies and reducing the cognitive load for students. But for now consider taking these steps to help identify the problem-solving strategies that you might want to teach.
Work through some problem-solving tasks with your team. See if you can identify if you use any of these strategies suggested by Polya: (a prominent mathematician renowned for his seminal 1945 work on problem-solving, ‘How to Solve It’).
Guess and check
Make an orderly list
Eliminate possibilities
Use symmetry
Consider special cases
Use direct reasoning
Solve an equation
Look for a pattern
Draw a picture
Solve a simpler problem
Use a model
Work backwards
Use a formula
Be ingenious
Review your curriculum and identify which strategies might be helpful for different topics or year groups.
Read this article from Colin Foster on Domain Specific Tactics.
Consider how the general strategies you’ve identified above might be supported by specific tactics for different topics. e.g. ‘Draw a picture’ in probability might involve a Venn diagram.
