# Teaching for conceptual growth through powerful ideas

Student progress in mathematics depends on drawing together multiple strands of knowledge and understanding that have been developed across and throughout their previous experiences of mathematics.

In some cases, a new kind of reasoning has to be learnt, and this can sometimes run counter to students’ everyday reasoning. Teachers need to know how to support students in new forms of thought. Students who are taught no more than how to adopt rules for action can misapply them and over-generalise their use. They need to understand why particular actions are relevant for particular mathematical situations, and experiencing what does not work, and why, can help them do that.

##### Understanding through experience

Digital technologies, particularly those which offer several mathematical representations (e.g. graphical, numeric, symbolic), can enable rapid exploration of different situations so that students are not so dependent on teachers and textbooks for examples and non-examples but can generate such things for themselves. With the most complex ideas, it is multiple experiences from several points of view, and over time, that are required to achieve a workable understanding.

##### New ways of thinking

Learners can find it difficult to recognise and adopt new ways of thinking, particularly if such ideas are left solely as implicit. Across mathematics at school level, the following are the key powerful ideas:

- variable
- proportionality
- similarity
- symmetry
- linearity
- measure
- dimensionality
- representations
- prediction
- accuracy
- discrete/continuous number
- transformation
- proof

Most of these ideas are either about how quantities relate to each other, or about how such relations can be formalised and represented so they can be transformed, adapted and hence used.