1. Using the Competence-Performance Theory as a Tool for Modelling Child Development Michael D. Kickmeier-Rust Cognitive Science Section Department of Psychology University of Graz

2. The development of the understanding of distance, speed, and time concepts and their interrelations What means … - TIME - DISTANCE - SPEED How are these conceptes interrelated? - More time means more distance at constant speed - More speed means more distance at constant speed - More speed means less time at constant distance

3. The development of the understanding of distance, speed, and time concepts and their interrelations PIAGET (1969, 1970) - utilzed a framework of logical operations Importance for everyday‘s tasks - crossing a street safely before an oncoming car - planning and timing a sequence of teaching these concepts Sequence / stages of the development?

4. The development of the understanding of distance, speed, and time concepts and their interrelations Previous Research PIAGET (1969, 1970) - sensorimotor stage - stage of concrete operations - stage of formal operations LEVINE (1979) - direct relations - inverse relations

5. The development of the understanding of distance, speed, and time concepts and their interrelations Previous Research LEVINE (1992) - understanding of distance and speed concepts but not time values - understanding of direct relations in the distance-speed- time triad, while the respective third concept is ignored - understanding of the inverse relationship between time and speed, while the respective third concept is still ignored - understanding of all three concepts; coordination of the three concepts is not fully mature - full integration of the distance-speed-time triad; children can correctly derive one concept from both others

6. The development of the understanding of distance, speed, and time concepts and their interrelations Previous Research MATSUDA (2001) - correctly discriminate between time, distance, and speed - understanding of the direct relations; limited in the ability to verbalise the reasoning processes - understanding of the inverse relations; limited in the ability to verbalise the reasoning processes; limited in the ability to coordinate both kinds of relations - full understanding of the relations between the three concepts; still unstable and based on two-by-two relations - considering the triadic system but not be fully conscious of it - consciously refer to the triadic distance-speed-time system

7. The development of the understanding of distance, speed, and time concepts and their interrelations DISCUSSION So, what‘s true?  One model?  All models? What do these 4 models have in common? What are the advantages / disadvantages? (for planning and timing a sequence of teaching)

8. WHAT CAN KNOWLEDGE SPACE THEORY DO ABOUT THIS? CAN IT HELP TO MODEL DEVELOPMENT MORE PRECISELY?

9. Using CPT to model child development Competence-Performance Theory (Korossy, 1997) Based on KST Distinguishes latent competences and observable performance  Competence Structure  Performance Structure Maps both utilzing interpretation and representation functions Allows conclusion from observable performance to latent underlying competencies

10. Using CPT to model child development EXTRACTING COMPETENCIES FROM PREVIOUS RESEARCH Based on a variety of previous studies we extracted 15 elementary competencies required to understand the DST system - Focusing on physical knowledge ESTABLISHING A SURMISE RELATION Based on a variety of previous studies we established a surmise relation between competencies

11. Using CPT to model child development tUnderstanding of time values dUnderstanding of distance values sUnderstanding of speed values tcDetection time as constant variable dcDetecting distance as constant variable scDetecting speed as constant variable a1Detecting the direct relation between distance and time a2Inference from longer time to longer distance a3Inference from longer distance to longer time b1Detecting the direct relation between speed and distance b2Inference from longer distance to higher speed b3Inference from higher speed to longer distance c1Detecting the inverse relation between speed and time c2Inference from longer time to lower speed c3Inference from higher speed to shorter time 15 elementary competencies

12. Using CPT to model child development Surmise Relation

13. Using CPT to model child development CREATING TASKS Based on a research paradigm by Fumiko Matsuda (1994) 6 task types: DT(1) Inference from longer distance to longer time at constant speed. (2) Inference from shorter distance to shorter time at constant speed.. TD(1) Inference from longer time to longer distance at constant speed. (2) Inference from shorter time to shorter distance at constant speed. SD(1) Inference from higher speed to longer distance at constant time. (2) Inference from lower speed to shorter distance at constant time. DS(1) Inference from more distance to more speed at constant time. (2) Inference from less distance to less speed at constant time. ST(1) Inference from more speed to less time at constant distance. (2) Inference from less speed to more time at constant distance. TS(1) Inference from more time to less speed at constant distance. (2) Inference from less time to more speed at constant distance.

14. Using CPT to model child development ESTABLISHING AN INTERPRETATION FUNCTION SessionRequired Competences DT{s, d, t, sc, a1, a2} TD{s, d, t, sc, a1, a3} SD{s, d, t, tc, b1, b2} DS{s, d, t, tc, b1, b3} ST{s, d, t, sc, dc, tc, a1, a2, a3, b1, b2, b3, c1, c2} TS{s, d, t, sc, dc, tc, a1, a2, a3, b1, b2, b3, c1, c3}

15. Using CPT to model child development CREATING A PERFORMANCE STRUCTURE Based on the interpretation function and the tasks P = {{}, {DT}, {TD}, {SD}, {DS}, {DT, TD}, {DT, SD}, {DT, DS}, {TD, SD}, {TD, DS}, {SD, DS}, {DT, TD, SD}, {DT, TD, DS}, {DT, SD, DS}, {TD, SD, DS}, {DT, TD, SD, DS}, {DT, TD, SD, DS, ST}, {DT, TD, SD, DS, TS}, {DT, TD, SD, DS, ST, TS}}

16. Using CPT to model child development OVERGENERALIZATION “a too frequent application of a rule through which it results in mistakes” 1.Overgeneralization from direct to inverse relations 2.Overgeneralization from inverse to direct relations Frequent misconception in developmental psychology Persisting problem to differentiate between actual capablities and systematic misconceptions

17. Using CPT to model child development DISCUSSION In your opinion, can CPT contribute to this problem? How could we model overgeneralization using CPT?

18. Using CPT to model child development DEFINITIONS OF OVERGENERALIZATION 1.Complete overgeneralization If a child is capable to solve tasks ST and/or TS we would expect/surmise that this child is also capable to solve tasks DT, TD, SD, and DS. In case of overgeneralization we would expect that a child who is capable to solve tasks ST and/or TS fails in tasks DT, TD, SD, and DS. This definition of overgeneralization results in 3 additional performance states P a = P  {{ST}, {TS}, {ST, TS}}

19. Using CPT to model child development DEFINITIONS OF OVERGENERALIZATION 2.Complete overgeneralization by factors Overgeneralization could occur from the factor speed in inverse relations tasks (ST and TS) to the factor speed in the direct relation tasks (SD and DS) and, equivalent, from the factor time in the in the inverse relations tasks (ST and TS) to the factor time in the direct relations tasks (DT and TD). This definition results in 7 additional performance states P b = P  {{ST}, {TS}, {ST, TS}, {DS, SD, TS}, {DT, TD, TS}, {DT, TD, ST}, {DS, SD, ST}}

20. Using CPT to model child development DEFINITIONS OF OVERGENERALIZATION 3.Partial overgeneralization by factors Similar to complete overgeneralization by factors, this definition of overgeneralization states that overgeneralization occurs partially within a specific factor. For instance, if a child overgeneralizes the factor speed from inverse to direct relations, s/he fails in one (SD or DS) or both (SD and DS) tasks. This definition results in 15 additional performance states P c = P  {{ST}, {TS}, {ST, TS}, {DS, SD, TS}, {DT, TD, TS}, {DT, TD, ST}, {DS, SD, ST}}, {DS, TS}, {SD, TS}, {DT, TS}, { TD, TS}, {DT, ST}, {TD, ST}, {DS, ST}, { SD, ST}}

21. Using CPT to model child development DEFINITIONS OF OVERGENERALIZATION 4. Partial overgeneralization Overgeneralization might occur from inverse to direct relations for at least one, two, three, or four tasks. (1) Failure in at least 1 direct relation task P d1 = 2 Q (2) Failure in at least 2 direct relation tasks  18 additional performance states P d2 = P  {{SD, DS, ST, TS}, { TD, DS, ST, TS}, {TD, SD, ST, TS}, {DT, DS, ST, TS}, {DT, SD, ST, TS}, {DT, TD, ST, TS}, {SD, DS, ST }, { TD, DS, ST }, {TD, SD, ST }, {DT, DS, ST }, {DT, SD, ST }, {DT, TD, ST }, {SD, DS, TS}, { TD, DS, TS}, {TD, SD, TS}, {DT, DS, TS}, {DT, SD, TS}, {DT, TD, TS}}

22. Using CPT to model child development DEFINITIONS OF OVERGENERALIZATION 4. Partial overgeneralization Overgeneralization might occur from inverse to direct relations for at least one, two, three, or four tasks. (3) Failure in at least 3 direct relation task  12 additional performance states P d3 = P  {{DS, ST, TS}, {SD, ST, TS},{DT, ST, TS},{ TD, ST, TS},{DS, ST }, {SD, ST},{DT, ST },{ TD, ST }, {DS, TS}, {SD, TS},{DT, TS},{ TD, TS}} (4) Failure in at least 4 direct relation tasks  3 additional performance states P d4 = P a

23. Empirical Investigations QUESTIONS Do the proposed performance structure cover a significant proportion of empirical answer patterns? Are (one ore more of) the proposed definitions of overgeneralization oberserved in empirical data?

24. Empirical Investigations Two cross-cultural investigations Two equivalent studies on Austrian and Japanese children (data of Japanese children were recorded by Fumiko Matsuda and published in 2001) 222 Japanese children / 42 Austrian children Age ranging from 4 to 11 Experimental paradigm accourding Matsuda (1994)

25. Empirical Investigations PROCEDURE

26. Using CPT to model child development TASKS DT(1) Inference from longer distance to longer time at constant speed. (2) Inference from shorter distance to shorter time at constant speed.. TD(1) Inference from longer time to longer distance at constant speed. (2) Inference from shorter time to shorter distance at constant speed. SD(1) Inference from higher speed to longer distance at constant time. (2) Inference from lower speed to shorter distance at constant time. DS(1) Inference from more distance to more speed at constant time. (2) Inference from less distance to less speed at constant time. ST(1) Inference from more speed to less time at constant distance. (2) Inference from less speed to more time at constant distance. TS(1) Inference from more time to less speed at constant distance. (2) Inference from less time to more speed at constant distance.

27. Empirical Investigations RESULTS – Investigation 1 Overgeneralization PS 1 Complete Complete by factors Partial by factors Partial (1) 6 Partial (2) 6 Partial (3) 6 Size 2 19222428643728 Avg. Distance 3 0.40 0.380.340.000.240.30 Percent 4 66.39 68.0771.43100.0075.6369.75 Distance 5 079 81851199083 132 312702936 28877000 30000000 1 Performance structure without overgeneralization 2 Number of patterns 3 Average minimal symmetric distance 4 Percent of answer patterns covered by the theoretical patterns 5 Number of patterns with a distance of 0 to 3 (the maximum distance for six items is 3). 6 The numbers in parentheses denote the minimal number of errors. Please note that partial overgeneralization with at least four error is equivalent to complete overgeneralization.

28. Empirical Investigations RESULTS – Investigation 2 Overgeneralization PS 1 CompleteComplete by factors Partial by factors Partial (1) 6 Partial (2) 6 Partial (3) 6 Size 2 19222428643728 Avg. Distance 3 0.51 0.480.430.000.340.46 Percent 4 54.29 57.1462.86100.0065.7154.29 Distance 5 019 2022352319 114 131101216 22222000 30000000 1 Performance structure without overgeneralization 2 Number of patterns 3 Average minimal symmetric distance 4 Percent of answer patterns covered by the theoretical patterns 5 Number of patterns with a distance of 0 to 3 (the maximum distance for six items is 3). 6 The numbers in parentheses denote the minimal number of errors. Please note that partial overgeneralization with at least four error is equivalent to complete overgeneralization.

29. Empirical Investigations RESULTS – Size-fit trade-off

30. Empirical Investigations CONCLUSION KST / CPT valuable tool for modelling individual develpmental courses Allows to formulate very precise hypotheses and to investigate them empirically It allows to account for a large number of individual knowledge (competence) states and for individual learning paths Not an abstract mathematical construct but rather a tool for psychological modelling and research in a variety of disciplines and fields

31. Using the Competence-Performance Theory as a Tool for Modelling Child Development Michael D. Kickmeier-Rust Cognitive Science Section Department of Psychology University of Graz