Teachers Talking About Teaching Mathematics

1. Teachers Talking About Teaching Mathematics Terezinha Nunes Professor of Educational Studies University of Oxford

2. Are fractions too difficult for primary school children? The ESRC-Teaching and Learning Research programme Team Terezinha Nunes, Peter Bryant, Ursula Pretzlik, Daniel Bell, Deborah Evans The Lauriston Primary School Team Sue Dobbing, Hilary Cook, Heather Rockhold, Aidan O’Kelly

3. Teachers often say that it is difficult to teach fractions • What types of errors do pupils make?

4. Difficulties with fractions • The representation using two numbers • children may attend to only one number (1/4 and 1/5 are considered as the same because they are both 1) • children may not realise that there is an inverse relation between the denominator and the quantity (1/4 is thought to be less than 1/5 because 4<5) • children may attend to both numbers but think that their relation is additive (2/5 and 3/6 are thought to be the same because 2+3 is 5 and 3+3 is 6) • A fraction is part of a whole (children think that 6/5<1)

5. It is not clear what we can infer from students’ difficulties • Experience is crucial in mathematics learning and we need to think more carefully about how children’s everyday experiences relate to knowledge of fractions

6. Different perspectives on number knowledge • Piaget: to understand number is to understand the logic of quantities • Gelman & Gallistel: to understand number is to tag a set with its counting label

7. How many pence do you have? Five. Piaget bought sweets from the children. For each sweet, he gave the child a penny. He made sure that the children knew how to count up to the number of objects he used.

8. How many sweets do I have? Seven. Piaget bought sweets from the children. For each sweet, he gave the child a penny. He made sure that the children knew how to count up to the number of objects he used.

9. Children can learn to use number labels without understanding quantities • This causes a false impression of learning but children later show lack of understanding • They may understand quantities without knowing the number labels • The same could happen with fractions • A different option is to start with children’s understanding of quantities and teach them labels as they make progress in understanding

10. Why do we need fractions? • Some quantities cannot be represented by a single natural number • Addition, subtraction and multiplication do not give rise to the need for fractions • The need for rational numbers emerges when we divide and the dividend is smaller than the divisor

11. Actions schemes in division • Sharing and correspondence: two quantities in a fixed correspondence (the origin of the concept of ratio) – both quantities can be different from 1 • Partition: one whole divided into equal parts • Teaching of fractions traditionally works with partition • We wanted to know what would happen if we were to use sharing and correspondence in the teaching of fractions

12. What are difficult but basic ideas that children need to master? • It is possible to have a dividend that is smaller than the divisor • There are many ways of dividing something: these will result in different fractions but the same quantity • The inverse relation between the denominator and the quantity: the larger the denominator, the smaller the quantity (for the same numerator) • Do children have insights into these ideas?

13. Continuous quantities – dividend smaller than the divisor Same dividend (1), same divisor (3) Same dividend (1), different divisors – 1/3 vs 1/2

14. Percent children above chance (Kornilaki & Nunes) • Equivalence: same dividend, same divisor means they all get the same • 5 years – 62% • 6 years – 84% • 7 years – 100% • Order: the more people sharing, the less each one gets • 5 years – 31% • 6 years – 50% • 7 years – 81%

15. Mamede (2006): different dividends and different divisors but proportional equivalence Each girl eats more than each boy Each boy eats more than each girl Each girl eats as much as each boy

16. Equivalence inference with different dividends and different divisors • 40% of the 6-year-olds and 65% of the 7-year-olds gave more correct responses than expected by chance • Ordering questions with the same numerator • 55% of the 6-year-olds and 71% of the 7-year-olds gave more correct responses than expected by chance (very similar to English children’s rate of success)

17. Conclusions • At about age six or seven children can make inferences about equivalence when the same dividend is divided by the same divisor • They understand the inverse relation between the divisor and the quotient a bit later but most 7-year-olds understand this even when the dividend is smaller than the divisor • The most difficult move is to understand that different dividends and different divisors can still result in equivalent quantities but two thirds of the 7-year-olds give more correct responses than expected by chance

18. A teaching investigation • We wanted to know whether it is possible to use children’s insights in the sharing and correspondence situation in order to teach them about fractions • We carried out two studies • one run by the researchers, outside the classroom • one run by the teachers

19. Nunes, Bryant, Bell, & Evans • 65 children working in 12 groups in two age levels: • Year 4: mean age 8y6m • Year 5: mean age 9y6m • The children’s conversations during their group work were recorded and analysed • The children’s own arguments can be used in designing the instruction for other children

20. The teaching sessions • From Streefland (1994) • A closed packet of biscuits, 6 girls: if each girl receives one biscuit and there is nothing left, how many biscuits in the packet? • If each girl receives a half biscuit and there is nothing left, how many biscuits in the packet? • If some more girls come and they all share fairly, will they now receive more, the same, or less than before? • Show how 3 chocolates can be shared by 4 children; what fraction does each one get?

21. 1. The waiter brings one pizza at a time. How can they share the pizzas? How much does each one get from the first pizza? How much does each one get from the second pizza? How many sixths does each one get? . 2. If the waiter brought both pizzas at the same time, could they share it differently? What fraction would each one get? Is 2/6 the same as 1/3? Second session: 6 children went to a pizzeria and ordered 2 pizzas.

22. K: [her drawings suggest some failed attempts at partition and then a successful solution using correspondences between the children and the pizzas and labelled the fractions for each drawing] If they have 2 pizzas, then they could give the first pizza to 3 girls and then the next one to another three girls. (…) If they all get 1 piece of that each, and they all get the same. Correspondence reasoning

23. How much would they get altogether from the two pizzas? L: I think 1 third because 1 sixth and 1 sixth is actually a different way in fractions and it doubled [the number of pieces] to make it littler and halving [the number of pieces] makes it bigger, so I halved it and it became 1 third. Correspondence reasoning (shown in drawings) and scalar argument (twice the number of parts, each part is half the size)

24. C: [did not present verbal arguments; her drawings show an attempt to solve the problem by partitioning and perceptual comparison, which was not successful] Partitioning and perceptual comparison – usually leading to wrong answers because of the difficulty of partitioning

25. How often were these arguments used? • Correspondence and sharing – 11/12 groups • Scalar arguments (double the number of parts, half the size) – 8/12 groups • Partitioning and perceptual comparisons – 3/12 groups

26. Some pleasant surprises • We thought the problems might become repetitive but some children spontaneously used this repetition to extend their reasoning • They explored the idea of equivalent fractions beyond our expectations

29. Can this approach be used successfully in the classroom? • LauristonSchool • One form entry primary school. • South Hackney, East London. • Open plan building. • Year 3 and 6 children in pilot study • Year 4 and 5 children took part in main study

32. PIZZERIA • 12 children go to a pizzeria. They order 6 pizzas. • There are no tables in the pizzeria big enough for all 12 children to sit at. • Can you think of ways that the children can arrange the seating using smaller tables? • Can you decide how the pizzas should be shared out fairly?

38. ‘I like maths now. The first thing I liked was fractions. I liked doing drawing and working together. It helped me see it.’ ‘Doing the drawing and working as a group made it more fun – you wanted to ‘go’ more to the lessons.’ ‘I used to hate fractions.’ ‘It was nice working together.’ ‘I don’t like maths, but I feel confident about fractions now – they’re the best bit. ‘I liked talking together and seeing other people’s methods.’ ‘People got to see how you worked it out. Then people could use your method, or another person might have an easier way of doing it.’ ‘Fractions are where you share it out.’

39. Conclusions • When they use sharing and the correspondence schema, children can make inferences about order and equivalence of fractions • Teaching that uses this schema of action in the classroom can provoke interesting discussions and help them become more aware of equivalence and of the inverse relation between the divisor and the quotient • It is possible that this different approach to teaching fractions from the beginning of primary school could results in fewer errors BUT TEACHERS MUST INVESTIGATE IT FURTHER