Children’s understanding of probability A review prepared for The Nuffield Foundation

1. Children’s understanding of probabilityA review prepared for The Nuffield Foundation Peter Bryant and Terezinha Nunes

2. Thisgreat book of the universe, which stands continually open to our gaze, cannot be understood unless one first learns to comprehend the language and to read the alphabet in which it is composed: the language of mathematics Galileo -1564/1642

3. Susan and Julie were cycling at the same speed in the Velodrome. Susan started first. When Julie had completed the circuit 3 times, Susan had completed it 9 times. How many times had Susan completed the circuit when Julie completed it 15 times?

4. Using mathematics to understand the world • Quantities • how many times Julie had gone around the circuit • how many times Susan had gone around the circuit • Relations between the quantities • Susan had gone around 3 times the number that Julie had gone around • Susan had gone around 6 times more than Julie

5. Using and testing mathematical models Susan and Julie were cycling at the same speed. Susan started first. • When Julie had completed the circuit 3 times, Susan had completed it 9 times. How many times had Susan completed the circuit when Julie completed it 15 times? • Susan: 3xJulie Susan: 15x3: 45 • Susan: 6 circuits ahead Susan: 15+6=21

6. Quantitative reasoning • Representing the world with numbers • Representing relations with numbers • Operating on the numerical representations • Following assumptions to their logical conclusions • Testing the adequacy of the model

7. Two sorts of quantitative thinking • Modelling non-random events If you have £20 to distribute fairly to 5 people, you know exactly how much each one will get

8. Two sorts of quantitative thinking • Modelling with probability If you toss a coin, we don’t know exactly what will happen each time but we know: • what is possible • roughly what is most likely to happen if you toss the coin a large number of times • if a coin that we tossed lots of times is departing from the expected pattern of probability

9. At first glance, it seems pointless to model random events • But in many probabilistic situations, we want to know whether something that looks like an association could be random • This is crucial to scientific and statistical reasoning • If you get a flu jab, are you less likely to have flu?

10. What is common to both forms of quantitative reasoning • A way of thinking • Use of mathematical concepts • What may be different • Specific concepts used • What can be used in one form of thinking even though it was learned in the other?

11. Relations are crucial in problem solving • A review prepared for the Nuffield Foundation on how children learn mathematics stressed the importance of relations http://www.nuffieldfoundation.org/sites/default/files/P2.pdf http://www.nuffieldfoundation.org/sites/default/files/P4.pdf • A study carried out for the DfE showed how understanding relations predicts KS2 and KS3 mathematics results

12. Cognitive Measures (8 yrs) IQ Working Memory Arithmetic Maths Reasoning 5.5 years School achievement (14 yrs) Maths Science English School achievement (11 yrs) Maths Science English 3 years 8 9 10 11 12 13 14 Age in years

13. Key stage 2: N=2488 Mathematics .60 Key stage 2 .12 .31 .46 Attention and Memory Mathematical Reasoning: Arithmetic .53 .48 Year 4 .64 • Key findings • Mathematical reasoning and arithmetic knowledge make separate contributions to the prediction of KS2 Mathematics results • Reasoning makes a much stronger contribution than arithmetic

14. N=1595 Key stage 3: Mathematics .61 Key stage 3 .11 .33 .47 • Key findings • Mathematical reasoning and arithmetic knowledge make separate contributions to the prediction of KS3 Mathematics results • Reasoning makes a much stronger contribution than arithmetic Mathematical Reasoning Memory and Attention .56 Arithmetic .51 .62 Year 4

15. There are 3 chips in a bag, two red and one blue. You shake the bag and pull out two chips without looking. What is most likely to happen? It is most likely that you would pull out two red chips It is most likely that you would pull out a mixture, one red and one blue Both of these are equally likely

16. Thinking systematically and logically about random events What is most likely to happen? R R B 1st pulled out 2nd pulled out R It is most likely that you would pull out two red chips It is most likely that you would pull out a mixture, one red and one blue Both of these are equally likely R B R R B R B R

17. Thinking about random events • Identifying quantities and relations • Using concepts that are specific to understanding random events • Questions so far?

18. Understanding probability • Randomness produces uncertainty: we can’t make precise predictions about uncertain situations • Nevertheless we can, and often do, think about uncertainty rationally and we can analyze uncertain contexts logically. • If we know what all the possible outcomes are, we can work out the probability of specific outcomes, and this is tremendously useful. • This raises the question: is it possible to show young children how to work out and to understand probability?

19. Three crucial areas in the understanding of probability • The nature of randomness and randomising • The need to work out and organise the sample space • The quantification of probability

20. 1. The nature of randomness : understanding randomising & the independence of separate events

21. Randomness & randomising • Most previous research on children’s understanding of randomness has concentrated on their reactions to various kinds of randomising (usually in quite unfamiliar contexts) e.g. research by Piaget & Inhelder).

22. Piaget & Inhelder worked with 5-13yr-olds on progressive randomisation Older children predicted progressive mixing Younger children predicted continued order

23. Randomness & randomising • Most previous research on children’s understanding of randomness has concentrated on their reactions to various kinds of randomising (usually in quite unfamiliar contexts) e.g. research by Piaget & Inhelder). • But randomising is actually a socially useful activity (e.g. shuffling cards, lotteries), and is familiar to most young children. • It’s possible that teaching children about randomness through useful randomising would be successful.

24. Independence of separate events • The probability of a particular outcome at any point in a random sequence is unaffected by what has happened before in the sequence: if I’ve throw 5 heads in succession, the probability of another head on the next throw is still 0.5. • This independence seems hard for many children, and some adults, to understand

25. Difficulty in understanding the independence of random events • 15greenand 15 blue balls in a bag • Someone has already drawn four balls from the bag (replacing the ball after each draw) and all four were blue. This person is going to make another draw. What is likely to happen on his next draw? 1. The next draw is more likely to be a blue ball than a green one; 2. The next draw is more likely to be a green ball than a blue one; 3. The two colours are equally likely. Positive recency Negative recency Correct answer Chiesi & Primi

26. Percentages for the different answers in Chiesi & Primi’s task Positive Negative Correct recency recency answer 8yrs 0 10yrs 40 College 41 student

27. Percentages for the different answers in Chiesi & Primi’s task Positive Negative Correct recency recency answer 8yrs 66 34 0 10yrs 30 30 40 College 16 43 41 student

28. Randomness and fairness • Randomness has a positive and useful side. It is a valuable way of ensuring fairness in some situations. • This is particularly true in games and lotteries e.g tossing a coin to decide which side bats first • In games, there are some ways of randomising that are better than others: eeny-meeny-miny-mois a more predictable and therefore less random procedure than shuffling cards or having to throw a six in order to start in a game like Ludo.

29. Conclusions and a possible solution : randomness ensures fairness in games • Children have to learn about determined, reversible cause-effect sequences • Initially their view of random, uncertain events is that they will be as predictable and reversible as determined events • Children may learn well about randomising if they are given randomisation tasks in more familiar contexts where randomising is a way of ensuring fairness: like shuffling cards and tossing dice

30. 2. The need to work out and organise (aggregate) the sample space

31. Tree diagram to represent the sample space of four successive tosses of a coin HHHH H T HHHT H H HHTH H T T HHTT H T H H HTHH T HTHT T H HTTH T HTTT THHH H THHT T H H THTH H T THTT T T TTHH T H H T TTHT T H TTTH T TTTT 16 equiprobable possible outcomes

32. Two-dice problem • Fischbein & Gazit asked children about the problem of two dice adding up to particular totals • What is the probability of: (a) 6? (b) 13? (c) a number bigger than 9?

33. The sample space for the two-dice problem 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12

34. What is the probability of 13? 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12

35. What is the probability of 6? 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 5 ways of getting 6 probability of 6 =5/36 p=.139

36. What is the probability of bigger than 9? 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 6 ways of getting >9 p=.167

37. Percent correct in two-dice task 10 years 12 years p of 13 38 78 p of 6 0 51 p of >9 0 28

38. Conclusions and a possible solution : • Teaching about need for working out sample space as first step, again in the context of games • Use of diagrams – particularly tree diagrams • This teaching must involve some categorization

39. 3. The quantification of probability

40. Difficulty with quantification: a PISA question Box A contains 3 marbles of which 1 is white and 2 are black. Box B contains 7 marbles of which 2 are white and 5 black. You have to draw a marble from one of the boxes with your eyes covered. From which box should you draw if you want a white marble?” The proportion of white in A is .33: The proportion of white in B is .29 The white-black ratio is 1:2 in A and 1:2.5 in B • Only 27% of a large group of 15-year olds got the right answer: worse than chance level PISA, 2003

41. The cards are shuffled several times and then put into the box where they belong.

42. Tick box 1 or box 2 or tick the It doesn’t matter which box These are the cards in box 1 These are the cards in box 2 or box 2 box 1 . It doesn’t matter which box

43. This is a problem which does not need a proportional solution • In this quite easy comparison, the child can solve the problem just by directly comparing the number of squares in the two sets

44. Tick box 1 or box 2 or tick the It doesn’t matter which box These are the cards in box 2 These are the cards in box 1 or box 2 box 1 It doesn’t matter which box

45. This is a problem which does need some kind of a proportional solution • Because the number of both kinds of card differed between the two boxes. This makes the problem a genuinely proportional one. • There is overwhelming evidence that this kind of comparison is difficult for children • Piaget & Inhelder report that the children who do solve such problems reach their solution by calculating ratios, not fractions.

46. Conclusions • Children’s understanding of and interest in fairness • seem a good start for working on their learning about • randomness • The importance of the sample space has been badly • underestimated in existing research. We think that it • can be taught with the help of diagrams and concrete material. • Children are more successful at solving proportional • problems using ratios than using fractions. This • gives us an important lead into how to teach them • to quantify and compare probabilities • Our central idea is that we can teach these three elements • separately, but in a cumulative way

47. Children’s understanding of probability.A report prepared forThe Nuffield Foundation http://www.nuffieldfoundation.org/sites/ default/files/files/Nuffield_CuP_FULL_REPORTv_FINAL.pdf