**Amy Dai, Maths Subject Advisor**

Problem solving is an integral part of mathematics, both at GCSE and A Level. For any of the A Level Maths or Further Maths specifications, problem solving makes up a significant portion of the skills that students learn. Teachers and students can agree that problem solving is an important life skill, one that is needed for future education and employment. However, we often hear from teachers that problem solving is difficult to embed in the classroom and that resources for it can be hard to come by.

### Problem solving in the curriculum

When the A Level Maths qualifications were reformed, one of the key areas of change was the increased focus on problem solving. This was shown by an increase of synopticity and requiring students to show adequate mathematical reasoning within their responses.

All A Level Maths qualifications must contain the same proportions of questions that target the three different Assessment Objectives (AOs). For more information about this, Ofqual has published a document that goes into more detail.

- AO1: Use and apply standard techniques (50%)
- AO2: Reason, interpret and communicate mathematically (25%)
- AO3: Solve problems within mathematics and other contexts (25%)

AO3 consists of the problem solving and modelling skills which are broken down further. Learners should be able to:

- Translate problems in mathematical and non-mathematical contexts into mathematical processes.
- Interpret solutions to problems in their original context, and, where appropriate, evaluate their accuracy and limitations
- Translate situations in context into mathematical models. Use mathematical models. Evaluate the outcomes of modelling in context, recognise the limitations of models and, where appropriate, explain how to refine them.

### What makes a maths question a problem?

The generally accepted definition of a maths problem is one which is non-routine. This means the student needs to make some decisions about which techniques need to be applied to find a solution. In the classroom this could be a very open-ended task, with the opportunity to go off track with false starts and potentially multiple re-starts. In the current pen and paper exam system problems could be included by removing any wording that might provide scaffolding for a particular solution route. These often involve applying aspects of different parts of the specification content within the question.

### OCR support for problem solving

We have a wealth of resources available for teachers to use in the classroom. These can all be found on Teach Cambridge.

When considering A Level Maths, there’s the benefit of having two qualifications available – there’s specification A H240, and specification B (MEI) H640. Although there are differences in the assessment model, the content is broadly similar, which means the majority of the resources provided for one specification can be used to support the other. Each year we produce twice the number of problem solving questions that have been sat by students under exam conditions – something that other exam boards can’t offer.

Our test-building platform, ExamBuilder, can be used to source problem-solving questions by filtering on the assessment objective AO3. You can also filter the questions by topics/subtopics. This enables you to produce customised tests and mark schemes containing problem-solving questions on any topic quickly and easily.

### Teaching and learning ideas

One strategy that has proved to be effective in the classroom is the ‘first step’ exercise. Provide students with a worksheet of problems – perhaps look for exam questions that have high mark tariffs. These could be gathered from past exams using ExamBuilder or from other resources such as the textbook mixed exercises. Then have students work together in pairs focussing on just the first step – reading the question and outlining the mathematical steps needed to answer the question. Just as students can practice their essay planning skills in subjects such as history or English, this will give students a similar opportunity to practice their solution planning skills.

When implementing this idea in your classroom, think about changing the lesson format. I’ve had success in the past with formats such as group work, paired relays, and carousels. Introducing a competitive atmosphere can also help provide motivation for teamwork.

Another way of adapting questions to help build problem-solving skills is to look for exam questions which have scaffolding included. Look at removing the part a, part b, part c etc of the question and see if your students can tackle it without the structure of the solution being given to them. A handy benefit of this approach is that you have hints pre-written that you can give out to students who get stuck.

Finally, another approach is to give your students the opportunity to develop their problem solving skills away from the structure of exam questions by using questions from outside sources such as the UKMT Senior Mathematical Challenge or the MEI Riteangle challenge. These allow students to puzzle through questions which test their mathematical knowledge and how they can apply that in a larger variety of questions.

For something even more flexible, take a look at ‘open middle’ questions – these are problems that really test the procedural and conceptual understanding of a student. These questions all have a closed beginning (everyone has the same question) and a closed end (everyone should have the same final answer) but have an open middle (there are many ways to solve the question). More information, along with examples, can be found on the Open Middle website.

### Other resources for problem solving

There are countless support websites out there which provide resources and guidance for maths. I’ve tried to summarise a few of these below and to highlight how they can be used to support problem solving.

This list is by no means exhaustive, and if you have any extra, we would love to hear them – please leave a comment!

- Underground Maths: hosted by Cambridge Mathematics, this website is full of rich resources, from activities to articles and puzzles. It places a lot of emphasis on exploring the connections between different topics and the focus is predominantly on pure mathematics.
- Nrich: this website is similarly full of different articles, activities and puzzles, all with an emphasis of offering rich resources for teachers and students to use. They run a ‘live problems’ programme where new questions are published regularly for students to send their solutions in. The top solutions are then published for everyone else to see.
- RISP and MSV: these are websites that are full of interesting puzzles. They’re sorted by topics, which makes it quick and easy to pick one to use in class. Working through these helps students to develop a greater understanding of standard content.
- MEI: provides teaching and learning resources and professional development events for maths teachers at all levels. Their annual problem-solving competition Ritangle offers a great opportunity to develop the skills needed for A Level Maths.

### Stay connected

Share your thoughts in the comments below. If you have any questions, you can email us at maths@ocr.org.uk, call us on 01223 553998 or tweet us @OCR_Maths. You can also sign up for monthly email updates to receive information about resources and support.

Join us at our teachers’ summer network webinars to discuss, amongst other issues, proposals for new support resources. Dates are 25 April 2024 at 4–5.30pm for A Level Maths, and 2 May 2024 at 4-5:30pm for A Level Further Maths. Also, look out for our full programme of professional development webinars for 2023/24.

### About the author

Amy joined OCR in 2023 after teaching for five years in both state and independent schools. She provides support across all the OCR Maths qualifications, but with a focus on GCSE, A Level Maths and Further Maths. She graduated from the University of York with a degree in Mathematics and Economics before gaining a PGCE in Secondary Mathematics and an MA in Education.