Children’s understanding of probability and certainty: An intervention study

1. Children’s understanding of probability and certainty: An intervention study Peter Bryant Terezinha Nunes Deborah Evans Laura Gottardis Maria-Emmanouela Terlektsi

2. Why probability and certainty? What did we need to consider when designing the intervention? • The cognitive demands of the conceptual field of probability • Concepts that are relevant also in other conceptual fields • The technical skills that children must have to be good problem solvers

3. The design Random assignment of children from the same class to different treatment groups • Baseline: an unseen control group that received extra attention from the class teacher (better than business as usual because the same number of students with their own teacher) • Certainty group: taught relevant technical skills (problem solving, inverse relations between operations, multiplicative reasoning) • Probability group: taught technical and conceptual knowledge about probabilities

4. The probability teaching programme The programme was designed to: • address the cognitive demands of the conceptual field of probability • address the required technical skills • use research in other conceptual fields that make the same cognitive demands

5. The problem solving teaching programme The programme was designed to: • promote technical skills related to problem solving (reading problems and questions carefully) • promote conceptual skills (additive and multiplicative reasoning) • use diagrams and be as interesting as the probability lessons (playing games, throwing dice, recording results)

6. Randomness Sample Space Sample space and quantification of probabilities Association between variables (1 day) Test 1 Test 2 Test 3 Test 4 Test 5 The pre- and post-tests used tasks adapted from the literature Test 1: pre-test measures for sample space and problems involving – provided a reliable measure of initial ability Test 2: provided test-retest reliability Test 3: pre-test measure for quantification of probabilities and more difficult problems about non-probabilistic situations Tests 4 and 5: post-tests used to assess the programmes (problem solving and probability) Summer break

7. Randomness The development of children’s understanding of randomness is related to their understanding of certainty • consider how children use the word “random” • compare random and certain outcomes • consider the possibility of local patterns, over a restricted number of events

8. Randomness • Randomness as a fair way of starting a game • Fair and unfair ways: • shuffling cards: Happy families game • throwing dice: normal and loaded dice (recording outcomes) • Possible, probable, impossible and improbable events • Predictable and unpredictable sequences (recording and analysing sequences) • some things are difficult to predict but not random • local patterns do not allow for prediction over many trials

9. b c a Predict the correct order of the figures cat broccoli apple

10. c p a Predict the correct order of the figures p c a

11. Sample space The definition of an event and of all possible events • an object as the junction of two (or more) dimensions leading to the concept of an event • teaching children about Cartesian product problems • considering how the same outcome may be obtained in different ways and defining a sample space with aggregation of outcomes

12. An object as the junction of two properties Identifying all the possible objects in a matrix: Cartesian product

13. The difficulty of generating all possible combinations The need for a system

14. Aggregation in sample space

15. Quantification of probabilities Understanding ratio and proportions • connecting sample space and quantification of probabilities • quantification using frequencies and ratios is more easily understood • comparing probabilities when the total number of cases is different • evaluating chances when the ratio is presented in different ways

16. When repetition must be eliminated

17. Probabilities and changes in the sample space The difficulty of comparing when the sample space is not the same

18. The need for ratios or proportions when the total is different Analysing ratios with concrete materials Using ratio notation to compare probabilities Noting inverse relations

19. Calculating ratios with a calculator, when you don’t “see” it

20. Results • The groups did not differ at pre-test • There were no school differences at pre-test • Intervention conditions varied considerably between schools: separate spaces in one school, a large room with little interference between groups in the second, a smaller room with noise from one group interfering with the second in the third school • Analyses use repeated measures (T4 and T5), a covariate (T1 for sample space and T3 for quantification)

21. Proportion of correct answers to sample space questions The group effect was significant and the covariate was significant. The school effect was not significant and neither was the interaction between group and school. The probability group differed significantly from each of the other two.

22. R R B What is most likely to happen? The Diagram 1st pulled out 2nd pulled out R It is most likely that I would pull out two red chips It is most likely that I would pull out a mixture, one red and one blue Both of these are equally likely R B R B Lecoutre’s (1996) problem: There are 3 chips in a bag, two red and one blue. You shake the bag and pull out two chips without looking. Complete the diagram to work out the possible combinations and answer the question.

23. Percent correct before and after connecting sample space and quantification of probabilities

24. Mean correct to quantification of probabilities questions The group effect was significant and the covariate was significant. The school effect was not significant and neither was the interaction between group and school. The probability group differed significantly from the unseen group but the other group differences were not significant.

25. Conclusions • Good evidence of programme effectiveness, supporting the notion that children can learn much about probability by the end of primary school • Both groups show significant improvement in the specific domains taught when compared to the unseen group • Some conceptual field specificity (sample space and quantification based on sample space) • Some general conceptual and technical knowledge (proportional reasoning)