Research School Network: The Problem with Problem-solving

As Director of Maths for the South West in the Greenshaw Learning Trust, I recently met with the maths subject leads in our various secondary schools. The focus was on raising standards with a particular lens on Year 11. The hot topic around the table was problem-solving. It was very clear that this remains one of the key challenges within the new 9 – 1 GCSE and with the exams looming, we were all looking for a quick fix, that one magic formula which might solve everything.

Problem solving constitutes a substantial part of the new GCSE, appearing within both AO2 and AO3. These assessment objectives account for 50% of the Foundation paper and 60% of the Higher so clearly, if we wish our students to be successful, we need to optimise their problem solving capabilities. This is reflected in the EEF Guidance Report on Improving Mathematics in Key Stages Two and Three where Recommendation 3 is ​‘Teach pupils strategies for solving problems’. There seems little room for debate: problem-solving needs to be a fundamental and substantial part of students’ mathematical education. But there is a catch: teaching problem solving is very much more difficult than it sounds.

In maths we rely heavily on well-established sequences of steps executed in a particular order to complete a set task – also known as an algorithm. The issue is, as the EEF points outs, that problem-solving occurs in a situation where there is not a readily available method, where the algorithms do not apply – this is almost the defining essence of problem solving. The consequence of this situation is that our normal approach to maths is not relevant so what should we teach them instead that will enable them to solve these mathematical problems? In fact, the issue is not just what but also how? Maths teachers rely heavily on the concept of practice in order to embed mathematical learning but the problems our students are being asked to solve are inherently non-routine and unpredictable (if they were not, there would be no problem to solve) which makes them arguable impossible to practice. How can one practice for something that is random? So although we recognise the criticality of improving our students’ problem solving capabilities, we face a serious challenge in working out how best to achieve this because our traditional habits, both in terms of mathematical knowledge and mathematical pedagogy do not appear to apply to the topic of problem-solving.

In the search for answers, I went to ​‘How I wish I taught maths’ by Craig Barton. I have always found this an excellent starting point for any journey into the research. I quickly found Chapter 9: Problem-solving and independence. The thesis advanced in HIWITM is that those who are good at problem-solving do this through an accumulation of domain-specific knowledge. Since algorithms do not apply and we cannot explore limitless possibilities and permutations, all that we can do is help them to be secure in their domain-specific knowledge so that, as the EEF advises, they are able to draw upon a wide variety of strategies, flexibly, to solve problems. Put another way, they need not to be novices. How many times have we quickly categorised a question only to find upon closer inspection that it is something else? As experts, we see past the surface features and look at the deep structures but this is incredibly difficult for a novice. By providing students with strong domain-specific knowledge we enable them to move along the continuum from novice to expert and thus equip them to diagnose the nature of the problem before them and to select the most appropriate strategy to solve it.

It was with this understanding that my home school, Five Acres High School, approached the problem of teaching problem-solving.

To help avoid the potential for cognitive overload when supporting year 11 students to become better at problem solving and reasoning, we utilised the recommendations from the EEF guidance to show them how to interrogate and use their existing knowledge to solve problems and use worked examples to enable them to analyse the use of different strategies. We did this by designing topic-based problem-solving booklets that were done in reverse of the order usually considered when referring to the QLA (question-level-analysis) from an assessment. Lesson-by-lesson booklets with problem-solving questions collected by topic, where the students started with their strongest topics and then worked downwards first enabled us to build their confidence and flexibility with topics in which they had stronger domain-specific knowledge. As we progressed through the programme, students were beginning to build a range of different strategies whilst simultaneously progressing through more curriculum content which enabled us to do more topic-specific problem-solving. As in the EEF guidance, we would use worked examples and then support them to apply this approach to other similar questions (limiting cognitive overload). A great deal of modelling was used including silent modelling and sharing of other approaches as students became more confident and creative in their strategies (cue the visualiser).

But of course the answer to GCSE success is not about being reactive in year 11 so we applied the same understanding to the design of our new KS3 curriculum. For further reading on this I turn to Jo Morgan’s new book, ​‘A Compendium of Mathematical Methods’ (yes, it was at the very top of my Christmas list!) The initial assumption is that a range of methods could be confusing, but once again it is all about that progression from novice to expert: ​“Once a pupil has developed procedural fluency and an initial comprehension of a chosen method, exploring alternative approaches is likely to deepen their understanding”. As an example of how this has impacted our curriculum, in year 7 we teach students a variety of methods of formal multiplication including column method, grid method, Chinese multiplication and long multiplication. There are many more methods in Jo Morgan’s book and I can’t wait to try them.

It is still too early to see the long term impact of our approach in terms of students getting those qualifications that will open doors to more exciting and successful futures but the short term results are very promising. The results for the original three schools that started developing this curriculum have improved year on year, with P8 scores in 2019 of +0.66, +0.67 and +1.01. However, in education one cannot afford to stand still, there is always more to be done, which is why it is great to be a part of the family of schools in a trust that has in their mission statement:

It is through working together than we can really make a difference and I look forward to reading the EEF KS1 guidance report just published as surely what is done in those first, embryonic, years holds the key to the future success of our young mathematicians.

Rhiannon Rainbow, Five Acres High School, Director of Maths for the South West in the Greenshaw Learning Trust