# Powerful aspects of the curriculum

Several mathematical ideas run throughout the curriculum:

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

They gradually become more overt and are often the explicit focus of advanced mathematics. Making these powerful ideas explicit and connected when they arise early on can introduce students to what it means to do mathematics.

Relations between quantities and properties provide a strong connecting theme in mathematics. The curriculum typically introduces quantities first, then their relations and formalisations. To reason deductively about mathematical properties, students need to shift from thinking about measuring or counting quantities to thinking about the formal relations between them, such as functions and geometrical theorems.

The role of formalisation of everyday understandings, and the introduction of new formal ideas, is a key progression in the secondary curriculum. Students at this level have to be introduced to formal ideas that are sometimes counter- intuitive but provide new ways to look at phenomena in and outside mathematics.

A sustainable curriculum from the point of view of value to the learner is one that is powerful for later employment, study, and citizenship where students need to apply their mathematical thinking to novel situations. Curricula that are restricted to testable topics in a traditional order do not adequately prepare students and their contents are often forgotten.