Key Messages:

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

  • Taught ideas that are understood deeply are not just β€˜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 β€œwhy does this work?”; β€œcan you draw a diagram to explain?”, β€œcan you make up an example of your own?”, β€œwhat 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 β€˜always true’, β€˜sometimes true’ or β€˜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 (β€˜The types of activity’) in β€˜Improving learning in mathematics: challenges and strategies, by Malcolm Swan, University of Nottingham -

Staff meeting - Is there a shared understanding of mathematical reasoning at your school? 

Mathematical reasoning snip.PNG

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