What's the Problem with Problem Solving?

Stage: 1 and 2
Article by Alison Borthwick

Think about the last few mathematics lessons you have taught. How much problem solving did the children do? Despite being one of the three aims of the National Curriculum (DfE, 2013) problem solving often gets forgotten about or added in, often at the end of the lesson or on a Friday. Ofsted continually remind us that ‘problem solving is not emphasised enough in the curriculum’ (Better Mathematics Keynote, Spring 2015).

So, perhaps one of the problems with problem solving is that we don’t do it enough! At NRICH we believe that every lesson can be a problem solving lesson. Our curriculum mapping documents link NRICH activities to different curriculum areas and this article provides further support when considering types of problems and strategies to include in your planning, as well as reminding you about the NRICH four stage process in solving problems.

Planning for Problem Solving
In planning for problem solving the key is to be clear about the type of problem you want to use, the strategies you are going to focus on and teaching the stages of the problem solving process. Combining these different elements allows children to build their confidence, skills and resilience in solving problems.

Remember, children don’t become problem solvers just by doing problems!

1. Types of problems
This section includes five different types of problems (suggested originally by The Primary National Strategy, May 2004), to which we have matched some NRICH activities. It may be useful to consider which type of problem you want to focus on in a lesson, or over a series of lessons. This does not detract from the mathematical content of a lesson, but rather adds to it, in that children can practice the addition of numbers, but through the context of a problem, thus ensuring every lesson can be a problem solving lesson.

Types of problem Examples of NRICH activities
Word problems We have a collection of short problems that would be ideal to fit into this category. While these problems sit within our secondary pages, the short problems on number (11648) would be suitable for upper primary students.
Visual problems Tangrams (1) and Baravelle (6522)
Finding all possibilities Button Up (7227) and Zios and Zepts (1005)
Logic problems Teddy Town (108) and Multiplication Squares (1134)
Rules and patterns Ip Dip (7185) and Tea Cups (32)

2. Problem solving strategies/skills
Alongside the type of problem you choose, it is helpful to consider the strategy or skill you wish the children to develop. Selecting a strategy and then offering opportunities for it to be modelled can help children to realise there are many different ways to solve a problem. Children learn to select a strategy to use as they become more experienced at problems.

Problem solving strategy/skill Examples of NRICH activities
Pattern spotting Domino Patterns (9970) and Consecutive Numbers (31)
Working systematically Same Length Trains (4332) and Beads and Bags (7374)
Using diagrams and pictorial information Fraction Match (6938) and Build It Up (10592)
Working backwards Doing and Undoing (8292) and Andy’s Marbles (2421)
Trial and improvement Find the Difference (6227) and Fifteen Cards (7506)
Visualising Happy Halving (217) and How Would We Count? (8123)
Conjecturing and generalising Break it Up! (2284) and Take Three Numbers (8063)
Reasoning logically I Like … (6962) and Sealed Solution (1177)

3. Four Stages of the Problem Solving Process
Perhaps another problem with problem solving is that children don’t know how to solve problems. This four stage process is a helpful guide to scaffold the skills being developed, from getting started to thinking more deeply about the task.

      Stage 1: Getting started
      Stage 2: Working on the problem
      Stage 3: Digging deeper
      Stage 4: Concluding

Our article on Developing Excellence in Problem Solving with Young Learners offers a more detailed account of this process.

Pulling it all together
Perhaps another problem with problem solving is there is too much choice. It can be confusing thinking about which type of problem to select or which strategy to focus on. In this feature we have offered different NRICH activities for each type and strategy of problem solving. However, one approach is to decide upon the type of problem and strategy you want to focus on, and then find one activity that exemplifies both. Here is an example.

Focus for the lesson: addition and multiplication
Type of problem: Finding all possibilities
Problem solving strategy: Working systematically
NRICH task: Zios and Zepts

Becoming a mathematical problem solver really is the point of doing mathematics.  This will enable children to ‘understand[ing] the world, [have] the ability to reason mathematically, [have] an appreciation of the beauty and power of mathematics and [develop] a sense of enjoyment and curiosity about the subject’ (DfE, 2013).

Department for Education (DfE) (2013) Primary Mathematics Curriculum. London: DfE.
Primary National Strategy (2004) Problem solving. A CPD pack to support the learning and teaching of mathematical problem solving. DfES Publications.
Ofsted. Better Mathematics Keynote Spring 2015. Accessed: 4.07.17 at https://www.slideshare.net/Ofstednews/better-mathematics-keynote-spring-2015