Concrete, pictorial, abstract

A widely used approach moves pupils through three stages. First they handle physical objects such as counters, cubes or fraction strips. Then they draw representations such as bar models or number lines. Only then do they work with symbols alone.

The stages are not a ladder to climb once. Older pupils meeting a new idea, such as algebraic expressions, still benefit from a concrete or visual starting point before moving to notation.

Reasoning and talk

Asking pupils to explain an answer, not just give one, shows whether they have understood or remembered a procedure. Sentence starters help: I know this because, I noticed that, This will always work when. Pair discussion before whole-class answers gives more pupils a chance to put their reasoning into words.

Mistakes are useful material. Showing an anonymous wrong answer and asking the class to find the error builds understanding without putting anyone on the spot.

Open questions give more room for reasoning than closed ones. Instead of asking for the answer to 24 divided by 4, a teacher might ask pupils to find as many division facts as they can with an answer of 6, then explain how they know they have found them all. The same content is practised, but pupils have to look for patterns and justify a conclusion.

A sample activity: fraction walls

This suits pupils aged 8 to 11 in a 45-minute lesson. Each pair needs a set of paper strips of equal length, scissors and coloured pencils.

• Keep one strip whole and label it 1.

• Fold and cut other strips into halves, quarters and eighths, then label each piece.

• Stack the strips to build a wall and look for pieces that line up exactly.

• Record every equivalence found, such as two quarters equal one half.

• Draw the wall in books and write one sentence explaining a pattern.

Practice that sticks

Short, spaced practice of earlier topics helps pupils retain methods better than long blocks on a single topic. A few mixed questions at the start of each lesson keep older learning active. Free digital tools such as Desmos and GeoGebra let pupils explore graphs and shapes and see the effect of changing a value immediately.

Parents can help at home by talking about numbers in everyday settings: sharing food, comparing prices, reading timetables. This builds confidence without turning the kitchen table into a second classroom.

Key points

• Introduce new ideas with objects or pictures before symbols.

• Ask for explanations, not just answers.

• Revisit earlier topics with short mixed questions.

• Use everyday situations at home to talk about numbers.

Photos: Vitaly Gariev, Chris Liverani / Unsplash