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 