Making problem solving and reasoning accessible to all

Maths Horizons has identified four areas of change that we believe will support the goal of a richer maths curriculum with problem solving at its heart. 

Suggested changes
Curriculum	1	Rebalance content from upper primary to lower secondary, allowing more time for knowledge to be secure and flexible, including opportunities for problem solving and reasoning for all students
	2	Greater specificity and coherence within the programme of study about the mathematical meaning and purpose of what students are taught
	3	Make problem solving and reasoning accessible to all by specifying that students should solve problems using mathematical content from prior years
Assessment	4	Incentivise the teaching of problem solving and reasoning by increasing the proportion of marks allocated to problem solving and reasoning in key stage 2 SATs and GCSEs.

Make problem solving and reasoning accessible to all by specifying that students should solve problems using mathematical content from prior years  

Maths Horizons has argued (see blog 2) that rich mathematical knowledge that allows for problem solving and reasoning must be made explicit within every mathematical sub-domain. Number, algebra, geometry, measures, data and probability should all embed reasoning as a shared entitlement.  

In this way, pupils have opportunities to experience problem solving where the mathematical content is predictable and clearly connected to what they are currently learning. They can build confidence and expertise in reasoning about and applying methods, explaining their thinking, and working systematically towards a solution. But if this is the limit of the opportunities for problem solving that pupils encounter, it will not equip them to succeed when faced with problems in which the relevant mathematics is not signposted in advance.  

A design limitation of the current programme of study is that problem-solving statements are either overly general or tied too closely to specific content statements. This means that problem solving is either overlooked or made predictable. In a typical lesson on fractions, pupils solve fraction problems. In a lesson on algebra, they solve algebra problems. Even when these problems are rich and demanding, the mathematical domain is predictable. Pupils know the sub-domain in which they are operating. This does little to develop a pupil’s ability to recognise relevant mathematical structures or to combine mathematical content from across sub-domains.  

All pupils must also be taught to tackle problems where they cannot rely on the maths being what has most recently been taught.  All pupils need opportunities to solve problems where the mathematical content is not immediately apparent. Here they may be challenged to identify relevant mathematical structures and reason to determine appropriate methods, as well as working systematically to find and justify a solution. To do this successfully, pupils need secure understanding of the mathematical content involved. It is therefore appropriate that this type of problem solving draws on mathematical content first taught in previous years.  

If we are serious about preparing pupils to use mathematics flexibly and powerfully the curriculum must do two things: it must increase specificity within sub-domains, making explicit the richer ways in which pupils should apply content knowledge, and it must also provide opportunities for them to solve problems based on previously secured knowledge. A crucial feature of problem solving – applying it in unpredictable situations - cannot be secured within sub-domain-specific statements alone. 

Example questions 

Richer content questions for Year 5 using content first taught in Year 5 

For the content statement: Identify multiples and factors, including finding all factor pairs of a number, and common factors of two numbers. 

1. A number has at least 3 factor pairs such that:   

1 x ? = 𝑥

3 x ? = 𝑥

? x 9 = 𝑥

What could the number be?  

2. A number has factors 2, 5, 7. What is the smallest number it could be?  

3. A pupil says, “The larger a number is, the more factors it has.” Show that they are incorrect. 

Problem-solving questions for Year 5 using content first taught in previous years 

1. Is the sum of three consecutive numbers always divisible by 3? Explain.  

2. Write a decimal that is between 1/3 and 2/5  

3. Which rectangle with perimeter 40 has the greatest area? Explain your reasoning. 

Maths Horizons is an independent programme drawing on an extensive evidence base to inform and support system change in maths education.