1. www.beam.co.uk Matching not patching: primary maths and children’s thinking Anne Watson June 2009

2. www.beam.co.uk In this talk: Children’s spatial understanding Children’s understanding of quantity Measure Relations between quantities Roots of algebra

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6. Spatial understanding Pre-school knowledge of space is relational, not just descriptive: size and transitivity distance between corners and edges fitting in and together

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8. Talk about relations between shapes: size, corners, edges, fitting

9. www.beam.co.uk Number as quantity Pre-school knowledge of quantities and counting develop separately: interacting with objects stretching/scaling fitting sharing out pouring

10. www.beam.co.uk Elastic: stretching and scaling Comparing lengths Same shape different size What makes it the same?

11. www.beam.co.uk Success in mathematics is related to understanding: Addition/subtraction as inverses ‘Undoing addition’ feels different to ‘adding on’ Relations as well as quantities, e.g. difference

12. www.beam.co.uk Additive relationship (fitting) a + b = c c = a + b b + a = c c = b + a c – a = b b = c - a c – b = a a = c - b

13. www.beam.co.uk Difference Write down two numbers with a difference of 3 … and two more numbers with a difference of 3 … and another very different pair

14. www.beam.co.uk Sharing

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16. Sharing by counting out

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18. Relations involved in multiplicative reasoning One to one Many to one One to many Stretching and scaling

19. www.beam.co.uk How many …? (fitting and measuring)

20. www.beam.co.uk Actual measurement Iteration of standard units has to be understood – and is difficult

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22. Exact measurement: multiplicative relationship a = bc bc = a a = cb cb = a b = a a = b c c c = a a = c b

23. www.beam.co.uk Fractions 5 is the multiplicative relation between 5 and 3 3 measurement (inexact units) and division (as when sharing one to many) transferring understanding between division to measurement is really hard

24. www.beam.co.uk Inexact measurement: what do children know?

25. www.beam.co.uk Sharing by chopping up

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27. One to many Many to one Fairness Iterative process of dividing and distributing

28. www.beam.co.uk ‘Continuous’ quantities: pouring

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32. Pouring questions are about multiplicative relations How many …. in ….? How many times ….? How much is left over?

33. www.beam.co.uk Relational reasoning 53 + 49 – 49 = ? 2 x 2 + 2 x 4 = 2 ( 2 + 4 )

34. www.beam.co.uk Implications for teaching?

35. www.beam.co.uk Implications about shape and space Use their knowledge of comparisons and relations between 3D shapes and spaces Use their experience of 3D to develop spatial reasoning and ideas about size, and scaling, and multiplication Measuring is about comparing one unit to another – and is hard

36. www.beam.co.uk Implications for teaching number Additive understanding does not precede multiplicative Very young children can reason multiplicatively from everyday experiences of sharing one between many, distributing many to one, comparing quantities, and measuring Multiplication is not only repeated addition; this meaning can get in the way of understanding it fully

37. www.beam.co.uk Implications about relations Understanding relations between quantities, shapes and measures is a strong foundation for later learning ‘=‘ expresses a relation With many quantities it makes more sense to talk about and = at the same time Young students can use letters to express relations between quantities Understanding addition and multiplication as two kinds of relation, rather than knowing four operations, draws on ‘outside’ knowledge and also helps in understanding scaling, ratio, proportion …