Key Messages:

• Mathematical thinking is central to deep and sustainable learning of mathematics.

• Taught ideas that are understood deeply are not just ‘received’ passively but worked on by the student.

• They need to be thought about, reasoned with and discussed.

Mathematical thinking involves:

• looking for pattern in order to discern structure;

• looking for relationships and connecting ideas; reasoning logically, explaining, conjecturing and proving.

For example:

There are some important general pedagogical strategies which support students’ mathematical thinking in all lessons:

• Encourage an atmosphere where asking questions, offering explanations, agreeing or disagreeing with others and giving reasons is encouraged and welcomed.

• Ask questions which encourage explanation, reasoning and proof like “why does this work?”; “can you draw a diagram to explain?”, “can you make up an example of your own?”, “what is the same and different about these examples?”, etc.

• Make use of learning journals (in class and for homework) where students are encouraged to make their own notes, create aide memoires for the important points in a topic or series of lessons and create their own worked examples with commentary.

• Move from simplistic teacher/pupil question and answer to a more dialogic teaching where active argument and debate is encouraged and where:

• interactions encourage students to think, and to think in different ways

• questions invite much more than simple recall

• answers are justified, followed up and built upon rather than merely accepted as correct or rejected as incorrect

• exchanges chain together into coherent and deepening lines of enquiry involving everyone

• discussion and argumentation probe and challenge rather than unquestioningly accept

More specific activities that will encourage mathematical thinking include:

•  Classifying mathematical objects - learners devise their own classifications for mathematical objects, and apply classifications devised by others.

• Interpreting multiple representations - learners match cards showing different representations of the same mathematical idea.

• Evaluating mathematical statements - learners decide whether given statements are ‘always true’, ‘sometimes true’ or ‘never true’.

• Creating problems - learners devise their own problems or problem variants for other learners to solve

• Analysing reasoning and solutions - learners compare different methods for doing a problem, organise solutions and/or diagnose the causes of errors in solutions.

For examples of these activity types see section 4 (‘The types of activity’) in ‘Improving learning in mathematics: challenges and strategies, by Malcolm Swan, University of Nottingham -