1. Key understandings in mathematics: synthesis of research Anne Watson NAMA 2009 Research with Terezinha Nunes and Peter Bryant for the Nuffield Foundation

2. Spatial reasoning Pre-school knowledge of space is relational, not just descriptive Children learn the logic of measurement, comparisons, transitive inferences, & 1-to-1 correspondence early Iteration of standard units has to be understood – and is hard Relations between shapes should be understood conceptually

3. Teaching spatial reasoning Build on spatial relational knowledge from outside school The conceptual basis of measurement relates to children’s knowledge of relations and correspondence Children need lots of experience of iteration with standard and non-standard units

4. Number Pre-school knowledge of quantities and counting develop separately Additive understanding does not precede multiplicative: very young children can reason multiplicatively from everyday experiences Three principles relate to success in mathematics: the inverse relation between addition and subtraction; additive composition; one-to-many correspondence It is difficult to think about relations rather than quantities

5. Teaching number Children have to connect their knowledge of counting with their knowledge of quantities Focus on inverse addition/subtraction, additive composition and one-to-many correspondence Multiplicative reasoning does not follow additive in a linear fashion

6. Relations between quantities Numbers are used to represent quantities and relations. Children tend to interpret statements about relations as if they were about quantities Experts use diagrams, tables and graphs to explore the relations before solving a problem Informal knowledge can obstruct formal learning

7. Teaching about relations Children need to learn to distinguish between quantities and relations Include the study of relations explicitly Link mathematical representations in teaching Focus on the relationships between informal knowledge and formal learning

8. Fractions Knowledge of fractional quantities starts to develop before school Two types of quantity are represented as fractions: measurement and division Transferring understanding between division to measurement is really hard Links between fraction concepts and procedures for comparison and operation are not made Fractions are usually taught as quantities, not as relations

9. Teaching fractions Different types of reasoning in measurement and division need to be linked Fractions need to be taught as relations between quantities, as well as quantities

10. Algebra Success depends on (i) understanding underlying operations and relations and (ii) using symbolism correctly It is important to learn how operations combine and relate ‘Letters stand for numbers’ in a variety of ways: constants, unknowns, variables, parameters Children interpret ‘=’ to mean ‘calculate’; but ‘=’ means either ‘equal to’ or ‘equivalent to’ We all use ‘guess-&-check’ rather than algebra if it’s easy ‘Generalised arithmetic’, e.g. doing substitution, does not give students reasons for learning to use algebra Young students can use letters to express generalities

11. Teaching algebra Read numerical and algebraic expressions relationally Teaching should be based on describing generalisations based in properties (arithmetical rules, logical relations, structures) as well as inductive reasoning from sequences Hands-on ICT provides representations of relations Support moves from expressing patterns to relating properties Understanding takes time, purpose & multiple experiences Explore use of algebraic manipulation software

12. Learning higher mathematics Rules & exploration lead to different kinds of knowledge; neither method necessarily enables learning new concepts Students respond to familiarity, try to apply prior knowledge, and generalise from their experience They need a repertoire of functions, operations, representations, concepts and methods Choice of technology helps problem solving, understanding of relations, identifying variables and functions

13. Teaching higher maths Neither teaching to tests, nor problem-solving activities, lead to new abstract understandings on their own Obvious perceptual reactions need to be recognised Students need time for new meanings and to develop from earlier methods Students need to control switching between representations and comparing to understand functions Authentic, realistic, mathematical exploration uses spreadsheets, graphing tools, etc. and deductive reasoning

14. Developmental themes throughout mathematics Number – not a unitary idea Logical reasoning Correspondence Inversion Inclusion Composition Transitivity Tool use and implicit models Symbol systems Mathematical modes of enquiry Comparisons Properties Representations Action and reflection Direct and inverse relations Formal and informal reasoning Local and global changes

15. Overarching messages for primary and secondary Relating number, quantities, spatial properties, expressions … Interplay of formal and informal understanding Multi-strand approach to number and spatial reasoning Multiple experiences over time, with several representations, of new concepts Rules & exploration do not necessarily enable new concepts Visual impact and recent similarity dominate over analysis of meanings, variables and relations Students rarely use mathematical ICT tools authentically