This weekend I had the pleasure of talking at my first #MathsConf. I spoke about how I approach teaching students how to become better problem solvers. This is a breakdown of that talk and includes the PowerPoint that was used at the end for download. This topic definitely warrants further exploration in the future but here is most of what was discussed.
Defining Problem Solving
It’s vital that we define problem solving if we’re going to talk about problem solving strategies.
There are many different types of problems that need overcoming. These vary from solving the climate crisis, how to organise one’s time to produce the perfect Christmas dinner, solving Fermat’s Last Theorem to counting how many squares there are in a diagram.
These are undeniably problems that need solving and may have strategies that are useful to employ when attempting to answer them. They do not, however, represent the sort of problem solving students need to perform well in a KS2-KS5 terminal exam. Luckily, the National Curriculum defines exactly the sorts of skills that are needed.
Unfortunately though, these are about as open to interpretation as sentences go. At KS4 these are referred to as AO3 (Assessment Objective 3). It is possible to search on an exam board’s bank of questions for where this objective has been assessed in the past. Doing so yields examples like these:
Some, like the nested triangles and multi-step circle theorem questions, may look familiar and people are sometimes surprised to know that these even count as “problem solving” questions. That’s good news though. It shows that the types of questions that assess a student’s ability to problem solve can be quite narrow and predictable. Because of that, it makes the skills needed to succeed with them teachable.
There will be students that can access these questions without any extra help. That can make it tempting to assume all students should be able to do this. This is not the case however and, much like it would be our job to fix any gaps in a student’s times tables knowledge for example, it is also our job to equip them with any skills they are missing out on compared to their peers. If we ever want to close the attainment gap in this country we need to ensure that all the implicit skills the more advantaged students have are distilled and taught to the others.
The Goldilocks Zone
So what are these skills? Well, they need to be specific enough to be explicitly taught. They also need to be generic enough to be useful in a variety of situations. They need to be “just right”.
There are a few labelled in the diagram, some may look familiar, others should not (I made them up). I will discuss “Zoom In – Zoom Out” and “Number-Free Problems” below. But first…
Embedding Problem Solving into the Curriculum
These strategies, once distilled, need to be interleaved throughout the curriculum. They need to be introduced at an appropriate stage and then included in any retrieval activities, codified and shared across a department, and referred to throughout a student’s time in their school. They are not to be left until the last 2 weeks of Year 11, nor are they something to be talked about once and then forgotten. That doesn’t work for teaching anything else and it won’t work for this either.
The Strategies
Number-Free Problems
The first idea is to encourage students to ignore the numbers when they first read a question.
I’d argue that most people fluent with the idea of area, proportion, and substitution would confidently be able to say they can solve these questions. There’s obviously an important factor missing here, but the numbers are not a crucial part of the formulation of the problem, just of the almost arbitrary calculating of the solution.
The human brain can hold about 7 things in its working memory at any one time. If you focus on the surface details, in this case the numbers, then you are taking up valuable space. It doesn’t matter so much with the low-complexity level of the questions above but what about this one:
Trying to solve this, whilst caring about the 10 numbers on show is very tricky. Ignore the numbers of this “problem solving” question however and, I think, it becomes easier. My argument is that those students who successfully answer questions like this are already doing this strategy. We just need to make the implicit explicit. It forces students to engage with the deep structure of the problem to produce a plan like below:
They cannot get carried away with the numbers and just randomly try adding or multiplying the first few values they see. Here it is with another problem:
These are KS4 questions but this strategy can be applicable to KS2-KS5 content. Equipping students with the strategy of ignoring the numbers and producing a written plan for those big mark questions feels like a positive step towards closing the gap. Teaching this to students early on and referring to the strategy throughout their schooling could be immensely powerful.
Zoom In – Zoom Out
This strategy is more suitable to geometry questions. Again, I think this is something a lot of people do automatically, and its something we should be explicitly teaching all students.
The idea is to ignore certain parts of a diagram at any one time and focus in on what it is that’s needed. If asked to find x in the diagram below it could be overwhelming.
Once you’ve isolate the 2 lines that make x and the other parallel line though you are left with:
I think this is the sort of mental “zooming in and zooming out” that successful problem solvers are doing. Making this clear to students and giving them time to practice it feels purposeful.
Making it clear with this question:
That it is useful to either mentally or physically have this image in your head:
In Summary
I think there are a set of explicit, teachable, skills we can pass onto students in order for them to better access the AO3 or “problem solving” marks available to them. This involves distilling the things that experts do implicitly, automatically, and turning them into named strategies that are embedded throughout a curriculum.
My thanks to you for reading this, to MathsConf for letting me talk about this, and to the audience for choosing and then engaging with the session. Please find the PPT that was used for the session below.
I also spoke about this for Tip#1 in Craig Barton’s Tips for Teachers podcast if you want to hear parts of it again (along with 4 other topics) https://tipsforteachers.co.uk/craig-latimir/
I’m always interested in what people make of this so please feel free to comment with thoughts, questions or incomplete musings. Follow this or my Twitter account Teach_Solutions for similar content in the future.
Teach Solutions 
One thought on “Teaching Problem Solving (Kind Of)”