There is a tension between delivering high-quality problem-solving content focused on exposing students to complex, multi-step challenges, and the current incentive structure of the teaching profession.
Axiom 1: Schools are judged by their results in national exams.
Axiom 2: Teacher time comes at a premium.
Axiom 3: Lesson time is a zero-sum game.
Axiom 4: The GCSE curriculum is too broad and insufficiently deep.
It is in the interaction between these four axioms that problem-solving encounters immovable obstacles:
It is worth noting here the difficulties previous problem-solving initiatives in the UK have run into. For example, in the 1980s, the problem-solving movement gained traction with a focus on teaching general strategies like “working backwards” or “drawing a diagram,” inspired by George Pólya’s How to Solve It. However, this often resulted in students memorising heuristics without truly understanding how to apply them in unfamiliar contexts. More recently, the problem-solving push at GCSE has been met with similar challenges, with criticisms that students are drilled to recognise question types rather than solve problems from first principles.
The Singapore Model, as delivered in Singapore rather than its adapted versions used elsewhere, is widely considered to have been more effective. Why is it regarded as more successful? What differentiates Singapore’s approach? Firstly, the model is implemented with strict consistency, while adaptations in other countries often dilute key aspects, such as the emphasis on bar modelling and structured problem-solving sessions, due to differing curricular demands or teacher training limitations. The significant differences are that in Singapore a particular model is mandated, and that teachers receive high quality training and professional development to deliver these sessions.
The Singaporean model does not clash with axioms 1 or 3 since the sessions are built into schemes of work, so no school can gain a competitive advantage by skipping sessions and focusing on more commonly examined topics. Nor does it clash with axiom 2 since the lessons are centrally prepared to a high standard, and teachers will have received training focused on the delivery of these lessons. From government policy to teacher training, there is coherence.
The UK Model, however, is not truly a structural model, but rather an ideal. En route to mastery learners will have applied their knowledge to a variety of contexts, and will then be offered the opportunity to engage with rich problem-solving tasks.
Suppose now that you are a UK based Teacher of Mathematics. You teach a year 10, set 3 class, and recent GCSE results suggest that both the median and modal mark of your learners will be grade 5. You covered a module on advanced linear equations and graphs ahead of time and now have a lesson to fill that is not mandated in the scheme of work. Which of the following do you choose?
1) Consolidate the current topic.
2) Recap the previous topic, contextual graphs which will also be included in the next internal test.
3) Deliver a broader revision lesson.
4) Undertake a standalone one-lesson problem-solving task, not directly related to their final exam.
Most teachers would probably select one of the first three options, and this is entirely sensible:
Revision sessions on either of the last two topics will help your learners embed knowledge which will likely appear in the final exam, and preparing these will not be a time-consuming exercise:
Options 1) and 2) do not clash with any of the axioms.
Whilst preparing a broader revision lesson may require more time to prepare, it is unlikely to be a significant burden, therefore:
Option 3) does not clash with none of the axioms.
Option 4), however, clashes with all four axioms:
If you do not mandate problem-solving sessions there will always be the incentive to focus on grade metrics (both internal and external). Axioms 1, 3 and 4 work in tandem here, incentivising the teacher to focus on likely exam questions which are predictable given the insufficient depth of problem-solving within GCSE examinations.
Axiom 2, however, is of particular interest to me. I have spent years writing problem-solving questions and can assure the reader that creating high-quality content for some topics can be very difficult. How does one add a problem-solving element to questions on reverse percentages or compound interest? These methods are not adaptable to a wide range of problems as are quadratic equations or Pythagoras’ Theorem, and most authors simply add context devoid of mathematical substance or elegance. As such, it is unreasonable to expect a classroom teacher to create such questions or longer activities for their learners.
Consider again the case of the hypothetical year 10 class above. One needs to recognise the difficulty of providing problem-solving opportunities given the level of the learners. Implicit in the description was that probably each member of this set 3 will sit the higher tier paper. As learners progress through KS4 content, however, problem-solving questions increasingly rely upon strong foundations across a broad range of topics: forming and solving both linear and quadratic equations, proportional reasoning, geometric reasoning etc. As such, the problem-solving tasks undertaken with this group of learners may need to be different to the tasks undertaken by learners in the top set.
Even for a top set teacher, however, preparing an extended problem-solving activity can be a time consuming exercise. True problem-solving is dependent upon less established methods, and so the delivery of this content is fundamentally different to, for example, the teaching of how to calculate the area of a circle. In the case of an area of a circle you are teaching a clearly defined concept, and questions vary in predictable ways. The training undertaken by teachers in Singapore on the delivery of problem-solving content can be focused on established resources and because of precedent delivers more predictable outcomes. If you consider that in the UK there are no mandated resources, and therefore there can be no relevant training available to teachers.
Furthermore, now consider yourself as the year 10 top set teacher with possibly more license to undertake problem-solving activities: where do you go for them? What precisely are you trying to teach? These are far from straightforward questions, and the solution to the latter is surely for staff to be trained in delivering high quality sessions, otherwise significant preparation time will be required for no obvious benefit.
I developed Maths Advance as a platform to support schools in delivering high-quality problem-solving experiences. By offering free to access, structured and accessible problem-solving resources, it contributes to the broader effort of embedding problem-solving into curricula. If the government is serious about integrating problem-solving into the provision across all schools then they need to indicate specific points in mandated curricula where this could take place. In the meantime, you can use a resource such as Maths Advance to provide enriched learning opportunities, and please feel free to reach out to me for advice on how to integrate such a resource.
I worked through the following problem with some enthusiastic year 9/10 learners during Friday lunch time. The session was highly productive and I encourage you to have a go at it yourself:
Given the digits 0, 2, 4, and 5, it is possible to make all of the integers between 1 and 5 inclusive using subtractions which involve only two of the digits, as follows:
1 = 5 – 4
2 = 4 – 2
3 = 5 – 2
4 = 4 – 0
5 = 5 – 0
Q. Is it possible to select a group of 4 digits such that every digit between 1 and 7 inclusive can be arrived at using only two digits and a subtraction?
This session was absolutely not about methods, rather about how to take a step back from performing a set of subtractions, and forming an argument that cut through. True problem-solving.
Another blog post, coming soon.
George Bowman
Founder, Maths Advance
MATHS ADVANCE BLOG 
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