**Good mathematics teaching**…

should promote good mathematical thinking, which is essential to all stages of learning, particularly for low-attainers.

It is often the case that the mathematical diet for low attaining pupils consists of little more than basic arithmetic, presented in simple step by step learning sequences, and repeated frequently. This is usually because it is felt that they cannot cope with anything at a higher level or with more demanding work. Many teachers have found, however, that this diet does not enable their pupils to succeed. They find their pupils saying that they do not understand a topic after they have met it over and over again. They find them getting bored with meeting work they have been doing, with varying degrees of success, since primary school. Teachers talk of their pupils' inability to concentrate, and their inability to decide which operations to use in different contexts. However, when teachers have presented these pupils with a broader based mathematical challenge, they have found them actually achieving more. The pupils can cope with the frustrations and floundering inherent in such challenges provided it is in a supportive atmosphere environment, where the process of struggle is viewed as successful in itself.

H.M.S.O. (1987) p.14 [http://stem.org.uk/rx7ca]

The guiding principles of the Association of Teachers of Mathematics give similar and further advice.

*The ability to operate mathematically is an aspect of human functioning which is as universal as language itself. Attention needs constantly to be drawn to this fact. Any possibility of intimidating with the mathematical expertise is to be avoided.**The power to learn rests with the learner. Teaching has a subordinate role. The teacher has a duty to seek out ways to engage the power of the learner.**It is important to examine critically approaches to teaching and to explore new possibilities, whether deriving from research, from technological developments or from the imaginative and insightful ideas of others.**Teaching and learning are cooperative activities. Encouraging a questioning approach and giving due attention to the ideas of others are attitudes to be encouraged.*

*Influence is best sought by building networks of contacts in professional circles.*Final report of the Mathematics Matters project also provides a useful list of

* see examples from STEM Centre

*principles for effective teaching and learning*in mathematics. The following is extracted from the mini-site in the NCETM portal (see elaboration there or my note and work of Malcolm Swan).*Teaching is more effective when it ...*

*Builds on the knowledge learners already have**Exposes and discusses common misconceptions* and other surprising phenomena**Uses higher-order questions**Makes appropriate use of whole class interactive teaching, individual work and cooperative small group work**Encourages reasoning rather than 'answer getting'**Uses rich, collaborative tasks**Creates connections between topics both within and beyond mathematics and with the real world**Uses resources, including technology, in creative and appropriate ways**Confronts difficulties rather than seeks to avoid or pre-empt them**Develops mathematical language through communicative activities**Recognises both what has been learned and also how it has been learned*

* see examples from STEM Centre

For more specific advice on individual domains / aspects of mathematics curriculum, here is a good source from a recent Nuffield Foundation project: Key Ideas in Teaching Mathematics (Research-based guidance and classroom activities for teachers of mathematics. You can find in the introductory page following ...

*key domains ('ideas'):*- Relations between quantities and algebraic expressions
- Ratio and proportional reasoning
- Connecting measurement and decimals
- Spatial and geometrical reasoning
- Reasoning about data
- Reasoning about uncertainty
- Functional relations between variables

*recurring themes:*- Powerful aspects of the curriculum
- Conceptual growth
- Teaching approaches
- Teaching for conceptual growth through powerful ideas

**Pedagogic constructs**…

are useful reminders or prompts for reflection on teaching. Think about how learners are engaged in a lesson with these prompts:

- Do; Talk; Record
- Manipulating; Getting a Sense of; Articulating
- See; Experience; Master
- Same and Different
- Creating an Atmosphere for Useful Conjecture
- Learners Constructing Mathematical Objects

- Imagining & Expressing
- Specialising & Generalising
- Conjecturing & Convincing
- Organising & Classifying