using the competence-performance theory as a tool for modelling child development michael d....

Using the Competence-Performance Theory

as a Tool for Modelling Child Development

Michael D. Kickmeier-RustCognitive Science Section

Department of Psychology

University of Graz

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


• 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?


Previous Research

• PIAGET (1969, 1970)

- sensorimotor stage

- stage of concrete operations

- stage of formal operations

• LEVINE (1979)

- direct relations

- inverse relations


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


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


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)

WHAT CAN KNOWLEDGE SPACE THEORY DO ABOUT THIS?

CAN IT HELP TO MODEL DEVELOPMENT MORE PRECISELY?

Using CPT to model child development

Competence-Performance Theory (Korossy, 1997)

• Based on KST

• Distinguishes latent competences andobservable performance Competence Structure Performance Structure

• Maps both utilzing interpretation and representation functions

• Allows conclusion from observable performance to latent underlying competencies


EXTRACTING COMPETENCIES FROM PREVIOUS RESEARCH

• Based on a variety of previous studies we extracted15 elementary competencies required to understandthe DST system

- Focusing on physical knowledge

ESTABLISHING A SURMISE RELATION

• Based on a variety of previous studies we established a surmise relation between competencies


t Understanding of time values

d Understanding of distance values

s Understanding of speed values

tc Detection time as constant variable

dc Detecting distance as constant variable

sc Detecting speed as constant variable

a1 Detecting the direct relation between distance and time

a2 Inference from longer time to longer distance

a3 Inference from longer distance to longer time

b1 Detecting the direct relation between speed and distance

b2 Inference from longer distance to higher speed

b3 Inference from higher speed to longer distance

c1 Detecting the inverse relation between speed and time

c2 Inference from longer time to lower speed

c3 Inference from higher speed to shorter time

15 elementary competencies


Surmise Relation

tcsc

dc

b1a1

a2 a3 b2 b3

c1

c3c2

d ts


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.


ESTABLISHING AN INTERPRETATION FUNCTION

Session Required 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}


CREATING A PERFORMANCE STRUCTURE

• Based on the interpretation function and the tasks

DT TD SD DS

ST TS

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}}


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


DISCUSSION

• In your opinion, can CPT contribute to this problem?

• How could we model overgeneralization using CPT?


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

Pa = P {{ST}, {TS}, {ST, TS}}



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

Pb = P {{ST}, {TS}, {ST, TS}, {DS, SD, TS}, {DT, TD, TS}, {DT, TD, ST}, {DS, SD, ST}}



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

Pc = 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}}



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

Pd1 = 2Q

(2) Failure in at least 2 direct relation tasks 18 additional performance states

Pd2 = 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}}



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

Pd3 = 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

Pd4 = Pa

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?


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)


PROCEDURE


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.


RESULTS – Investigation 1

Overgeneralization

PS1 Complete Complete by factors

Partial by factors

Partial (1)6 Partial (2)6 Partial (3)6

Size2 19 22 24 28 64 37 28

Avg. Distance3 0.40 0.40 0.38 0.34 0.00 0.24 0.30

Percent4 66.39 66.39 68.07 71.43 100.00 75.63 69.75

Distance5 0 79 79 81 85 119 90 83

1 32 32 31 27 0 29 36

2 8 8 7 7 0 0 0

3 0 0 0 0 0 0 0

1 Performance structure without overgeneralization2 Number of patterns3 Average minimal symmetric distance4 Percent of answer patterns covered by the theoretical patterns5 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.


RESULTS – Investigation 2

Overgeneralization

PS1 Complete Complete by factors

Partial by factors

Partial (1)6 Partial (2)6 Partial (3)6

Size2 19 22 24 28 64 37 28

Avg. Distance3 0.51 0.51 0.48 0.43 0.00 0.34 0.46

Percent4 54.29 54.29 57.14 62.86 100.00 65.71 54.29

Distance5 0 19 19 20 22 35 23 19

1 14 14 13 11 0 12 16

2 2 2 2 2 0 0 0

3 0 0 0 0 0 0 0

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.


RESULTS – Size-fit trade-off

.5 0

.4 0

.3 0

.2 0

.1 0

.0 0

Goo

dnes

s-of

-fit

10 20 30 40 50 60

Size

1

1

2

3

4

5

6

w/o OvergeneralizationComplete

Partial (1)Partial by factors

Partial (2)Partial (3)

Complete by factors

2

3

4

5

6

1

2

3

4

5

6

w/o OvergeneralizationComplete

Partial (1)Partial by factors

Partial (2)Partial (3)

Complete by factors

12

3

5

6

Investigation 1

Investigation 2


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

Using the Competence-Performance Theory

as a Tool for Modelling Child Development

Michael D. Kickmeier-RustCognitive Science Section

Department of Psychology

University of Graz

using the competence-performance theory as a tool for modelling child development michael d....

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