Teaching a structured approach to mathematical problem-solving

A pupil may calculate accurately in familiar exercises but struggle to decide what a problem is asking, select relevant information or choose a mathematical approach. Extra support should make this hidden decision-making visible without reducing every problem to a rigid keyword rule.

Identify where the process breaks down

Use a small set of carefully chosen problems and observe whether the pupil can:

explain the situation in their own words;
identify the question and relevant information;
represent the structure with a diagram, objects, a table or a bar model;
connect the problem to known mathematics;
choose and carry out an approach;
check and explain the result.

Check the underlying calculation separately. Otherwise, weak arithmetic fluency may be mistaken for a problem-solving difficulty.

Teach a repeatable routine

Use a short routine such as:

Understand: What is happening and what must be found?
Represent: What diagram, model or notation shows the structure?
Choose and solve: Which known ideas could help?
Check and explain: Does the answer fit the problem?

Model your thinking aloud with a worked example. Explain why a representation and method were chosen, then compare another valid approach. Move from joint work to prompted practice and then independent use.

Avoid teaching pupils to choose an operation from isolated words such as “altogether” or “left”. The same word can appear in problems with different mathematical structures. For example, “I had 12 sweets and ate 5. How many are left?” and “I ate 5 sweets and have 12 left. How many did I have?” both use “left”; the first needs subtraction and the second addition. A bar model shows the difference.

Build flexible application

Present related problems together: the same structure in different contexts and different structures within a similar context. Ask pupils what stays the same, what changes and when a previous strategy might help.

Use visual representations to reveal structure, not as decoration. The NCETM bar model guidance explains how a bar model can bridge concrete experience and abstract notation when introduced carefully.

Monitor progress

A typical pattern is two or three sessions a week of 15 to 20 minutes for a small group with the same breakdown point. Keep going for as long as the pupils need it, and choose the timetable slot and review point that suit the school (many schools review after six to eight weeks). Track whether the pupil identifies the question, selects relevant information, chooses a useful representation and explains a reasonable solution. Include unfamiliar problems so that progress reflects transfer rather than memorising one task format.

If the pupil still cannot apply learning, revisit prerequisite concepts, vocabulary and calculation. Do not extend problem-solving practice when the barrier lies elsewhere.