Whole school approaches
To provide multiple experiences over time, teachers need to liaise and construct a coherent, developmental whole school approach to teaching. Guidance from research about learning suggests that any such scheme needs to:
- Build on students’ knowledge about contexts and relative quantities (that cannot be counted), while recognising that these may be limited. Use these to identify variables and learn how to express relations.
- Build on students’ capacity to design suitable units, such as ‘little recipes’, in contexts
- Represent and carry out division in many ways, maintaining the connection between division and ratio by using simplified fraction notations.
- Represent ratio in a variety of ways: a:b, a/b, and as a decimal number. When expressed as a fraction, understand that this is a multiplier and not a part-whole representation.
- Using the simplest version of a mixture, i.e. the simplified ratio/fraction, as a unit to build with, and extending this idea to all relative measures.
- Use models (images, metaphors, diagrams, contexts) which are extendable over a considerable time: avoid models that might encourage over-simplification.
- Provide ways to ‘hold’ information about variables and their relations, e.g. through model use, ratio tables, fraction notation, graphs.
- Develop confidence with multiplication and division of non-integers
- Use a range of types of problem that need proportional reasoning and cannot be done in other ways.
- Spend time on distinguishing linear (proportional) from non-linear situations (including linear functions which have non-zero constants).
- Investigate how changes in variables affect the relation: a/b = c/d.
- Systematically develop all these ideas over time, and give them time wherever they arise in later mathematical topics.
- Do not assume that students understand ratio well enough to apply it fluently in higher levels of mathematics – return to the fundamental ideas before any use of ratio and proportional reasoning.