1. Strategy flexibility in the number domain up to 100 Joke Torbeyns, Lieven Verschaffel, and Pol Ghesquière University of Geneva 21/01/2006

2. Introduction Procedural or strategy flexibility in the domain of elementary arithmetic -Two phases 1.Phase 1: number domain up to 20 2.Phase 2: number domain 20-100 -Major goal: analysing development of strategy flexibility in children of different mathematical achievement levels

3. What is strategy flexibility? -Flexibly applying multiple strategies, making efficient choices among diverse strategies -Choosing on the basis of …

4. What is strategy flexibility? Item 1.Item difficulty (cf. Geary, 1991) 2.Number characteristics (cf. Blöte et al., 2000, 2001; Selter, 2001)  Objective criteria

5. 1.Item difficulty: solving easy items by heart, solving difficult items with a counting strategy E.g.2 + 3 = 5, I know this by heart 8 + 3 = (8), 9, 10, 11 2.Number characteristics Solving 62 - 29 with compensation strategy (62 - 30 + 1) Solving 62 - 23 with jump strategy (62 - 20 - 3)

6. What is strategy flexibility? Item 1.Item difficulty (cf. Geary, 1991) 2.Number characteristics (cf. Blöte et al., 2000, 2001; Selter, 2001)  Objective criteria Strategy performance 1.Strategy accuracy 2.Strategy speed (cf. Siegler et al., 1995, 1997)  Subjective criteria

7. What is strategy flexibility? Using the strategy that leads fastest to an accurate answer to the item (cf. Siegler et al., 1997)

8. Solving 62 - 29 with compensation strategy and not with jump strategy is flexible / adaptive if and only if compensation strategy leads this child faster to an accurate answer to 62 - 29 than jump strategy

9. How to investigate? -Definition of strategy flexibility on the basis of strategy performance characteristics requires two types of data 1.Data about each child’s strategy choice on each item 2.Data about the accuracy and speed of each strategy on each item -Problems with so-called choice method

10. How to investigate? -Choice method * Only one condition: choice condition * Information about each child’s strategy choice on each item * No information about the accuracy and speed of each strategy on each item -Therefore: choice/no-choice method

11. How to investigate? Choice/no-choice method -Two types of conditions 1. Choice: preferential strategy 2. No-choice: obligatory strategy use -Data 1. Choice: each child’s strategy choice on each item 2. No-choice: strategy accuracy and strategy speed on each item

12. How to investigate? -Strategy flexibility For each item Strategy accuracy and speed in no-choice conditions  Strategy choice in choice condition  Flexible if use of the strategy that leads fastest to an accurate answer to the item

13. Phase 1 Number domain up to 20 Additions over 10, like 8 + 7 =. Subtractions over 10, like 15 - 7 =. Three types of strategies 1.Retrieval 2.Calculation 3.Counting

14. Retrieval strategy 8 + 9 = 17, I know this by heart Calculation strategies Decomposition-to-108 + 9 = 8 + 2 + 7 = 10 + 7 = 17 Tie8 + 9 = 8 + 8 + 1 = 16 + 1 = 17 One-less-than-108 + 9 = 8 + 10 - 1 = 18 - 1 = 17 Counting strategies Count-on8 + 9 = (9), 10, 11, 12, 13, 14, 15, 16, 17

15. Phase 1 First series of studies Developmental changes in flexible use of retrieval, calculation, and counting strategies in children of different mathematical achievement levels Second series of studies Flexible use of diverse calculation strategies  Today’s focus

16. Phase 1 Regular schools, end of first grade Instruction aiming at strategy variety and strategy flexibility Aim: flexible use of decomposition-to-10 and tie strategy in children of different mathematical achievement levels

17. Subjects -83 Flemish first-graders (five classes) -Selection criteria 1.Instruction 2.Choice condition beyond level of counting -Three achievement groups 1.High achievers (n = 31) 2.Average achievers (n = 20) 3.Low achievers (n = 32)

18. Subjects -Instruction in sums over 10 1.Retrieval of tie sums 2.Decomposition-to-10 strategy 3.Tie strategy -Relation between type of sum and strategy 1.Tie sum -Retrieval 2.Near-tie sum -Tie strategy 3.Other-Decomposition-to-10

19. Materials and conditions -Five near-tie sums over 10 -Four conditions 1.Choice condition 2.Decomposition-to-10 condition 3.Tie condition 4.Retrieval condition

20. Choice condition -Choice between decomposition-to-10 and tie strategy  Pictures

21. Choice condition 7 + 6 = … 7 + 7 =...

22. Choice condition -Choice between decomposition-to-10 and tie strategy  Pictures -Five extra buffer items Difference of more than one unit E.g.9 + 5 =.

23. No-choice conditions -Decomposition-to-10 condition Obligatory use of decomposition-to-10 (picture) -Tie condition Obligatory use of tie strategy (picture) -Retrieval condition Obligatory use of retrieval (time limit) 15 extra retrieval items (tie sums over 10, sums up to 10)

24. Analysis 1.Strategy repertoire in the choice condition Written strategy reports 2.Strategy flexibility in the choice condition Strategy performance characteristics: strategy accuracy, strategy speed

25. Strategy repertoire Dec-to-10 + Tie Dec-to-10Tie High771013 Average651025 Low315019 All572518

26. Strategy flexibility -Strategy efficiency in dec-to-10 and tie condition  Strategy use in choice condition For each child and for each sum -Flexible if use of strategy that leads fastest to an accurate answer to the sum E.g.If decomposition-to-10 condition as accurately as but slower than tie condition for this child on this sum Thentie strategy for this child on this sum in choice condition flexible dec-to-10 for this child on this sum in choice condition non-flexible

27. Strategy flexibility

28. Conclusions -Strategy flexibility Not only item characteristics But also strategy performance characteristics Also mathematically low achievers -Two questions 1.Measure of strategy flexibility 2.No group differences in strategy flexibility

29. Conclusions 1.Measure of strategy flexibility -Single observations -Strategy flexibility at individual level -But: unreliable data due to momentary distraction? -Exclusion of unreliable trials (observations, RT, errors)

30. Conclusions 2. No group differences in strategy flexibility -Hypothetical explanation: high achievers fitted strategy choices most frequently to teacher expectations and socio-mathematical classroom norms -Thus: also influence of instructional context on children’s strategy choice process

31. Phase 2 Number domain up to 100 Two-digit additions, like 48+17 Two-digit subtractions, like 53-26 Three types of strategies 1.Split 2.Jump 3.Varying

32. -Split strategy E.g.48 + 19 =. 40 + 10 = 50; 8 + 9 = 17; 50 + 17 = 67 E.g.53 - 41 =. 50 - 40 = 10; 3 - 1 = 2; 10 + 2 = 12 -Jump strategy E.g.48 + 19 = 48 + 10 + 9 = 58 + 9 = 67 53 - 41 = 53 - 40 - 1 = 13 - 1 = 12 -Varying strategies E.g.48 + 19 = 48 + 20 - 1 = 68 - 1 = 67 E.g.53 - 41 =. 41 + 9 = 50; 50 + 3 = 53; so the answer is 9 + 3 = 12

33. Phase 2 Regular schools, medio and end of second grade Instruction aiming at perfect mastery of jump strategy Aim: flexible use of jump and split strategy in children of different mathematical achievement levels

34. Subjects -69 Flemish second-graders (four classes) -Three achievement groups 1.High achievers (n = 30) 2.Average achievers (n = 20) 3.Low achievers (n = 19) -Two measurement times 1.Before instruction in + and - up to 100 2.After instruction in + and - up to 100

35. Instruction -Focus on only the jump strategy -No explicit instruction in 1.Split strategy 2.Varying strategies

36. Materials Four types of + and - up to 100 (n = 20) 1.Two-digit additions without a carry (ANC) E.g. 45 + 23 =. 2.Two-digit additions with a carry (AC) E.g. 45 + 28 =. 3.Two-digit subtractions without a carry (SNC) E.g. 65 - 23 =. 4.Two-digit subtractions with a carry (SC) E.g. 65 - 28 =.

37. Conditions Three conditions 1.Choice condition -Choose between the jump and the split strategy -Pictures

38. Choice condition First I add 45 and 10 … First I add 40 and 10 … 45 + 12 =.

39. Conditions Three conditions 1.Choice condition 2.Jump condition -Obligatory use of the jump strategy -Pictures

40. Jump condition First I add 45 and 10 … 45 + 12 =.

41. Conditions Three conditions 1.Choice condition 2.Jump condition 3.Split condition -Obligatory use of the split strategy -Pictures

42. Split condition First I add 40 and 10 … 45 + 12 =.

43. Analysis 1.Strategy repertoire in the choice condition Written strategy reports 2.Strategy selection in the choice condition -Classroom instruction: jump strategy -Item characteristics: additions, subtractions -Strategy performance characteristics: strategy accuracy, strategy speed

44. Strategy repertoire -Use of jump and split strategy at both measurement times -Consistency in strategy use at both measurement times Time1: 70% only one strategy Time2: 46% only one strategy

45. Strategy repertoire Time1 JumpSplitJ+STotal Time2Jump136423 Split119323 J+S823 54 Total224830100

46. Strategy repertoire -Use of both jump and split strategy at both measurement times -Consistency in strategy use at both measurement times -Consistency in strategy use between two measurement times 55% did not change strategy repertoire between time1 and time2

47. Strategy repertoire Time1 JumpSplitJ+STotal Time2Jump136423 Split119323 J+S823 54 Total224830100

48. Strategy repertoire -Use of both jump and split strategy at both measurement times -Consistency in strategy use at both measurement times -Consistency in strategy use between two measurement times -No group differences

49. Strategy flexibility 1.Classroom instruction -Frequency of jump strategy at time1 and time2 -Increase in frequency of jump strategy in high, average, and low achievers between time1 and time2 Increase in frequency of jump strategy is largest in high achievers

50. Strategy flexibility

51. 2.Item characteristics -Frequency of jump strategy on additions and subtractions -Flexible strategy use Group differences

52. Split strategy -Efficient on additions  Memorised knowledge of sums up to 20  Jump strategy: new knowledge -Subtractions: two difficulties 1. Change of operation E.g. 45 - 22 =.; 40 - 20 = 20; 5 - 2 = 3; 20 + 3 = 23 2. Negative numbers (smaller-from-larger bug) E.g. 45 - 28 =.; 40 - 20 = 20; 5 - 8 = -3; 20 + (-3) = 17  Jump strategy: no special difficulties

53. Strategy flexibility

54. 3. Strategy performance characteristics -Per item: correlation between *Differences in accuracy and speed between jump and split condition *Frequency of jump strategy in choice condition -Flexible strategy use Only high achievers

55. Conclusions -Flexible strategy use 1.Classroom instruction  High achievers 2.Item characteristics  High and average achievers 3.Strategy performance characteristics  High achievers  Relatively low proportion of flexible strategy use in average and low achievers

56. Conclusions -Two questions 1.Measure of strategy flexibility 2.Group differences in strategy flexibility

57. Conclusions 1.Measure of strategy flexibility -Multiple observations -Strategy flexibility at group level -But: individual differences? 2.Group differences in strategy flexibility -Influence of instruction on development of children’s knowledge and inclination

58. Conclusions Instruction: focus on only the jump strategy * Routine expertise in all children * Adaptive expertise in only high achievers -Invention of other strategies -Understanding strengths and weaknesses of diverse strategies -Inclination to apply diverse strategies flexibly

59. Discussion 1.Strategy flexibility -Influence of strategy performance characteristics -But also other variables! * Item characteristics * Socio-cultural and socio-emotional variables: classroom culture, teacher’s expectations, children’s beliefs and motivation, …

60. Discussion -Direct evidence for influence of item and strategy performance characteristics Only indirect evidence for influence of socio- cultural and socio-emotional variables -Therefore: future studies: *More than teacher interviews and textbook analyses *Also children’s ideas about strategy flexibility

61. Discussion 2.Choice/no-choice method -Strategy efficiency data for each child and for each item -Strategy flexibility on the basis of (individual) strategy performance data -But …

62. Discussion * Restricted choice condition Rich diversity of strategies  For each strategy one no-choice condition Completeness? Ecological validity? * Strategy control Written strategy reports Other strategies, like retrieval?

63. Discussion * Analysing strategy flexibility Individual level  Single observations Group level  Ignoring individual differences