IPN Leibniz Institute for Science Educationat the University of Kiel

Reacting to challenges for the research in mathematics education:

case studies of ICT learning environments

Timo Ehmke (Kiel), Martti Pesonen (Joensuu) and Lenni Haapasalo (Joensuu)

Based on the project:

From Visual Animations to Mental Models in Mathematics Concept Formation

(sponsored by DAAD / Academy of Finland)

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Introduction

Starting point: - learning of tertiary mathematics

Problem: - difference between school and university mathematics- School focus on procedural knowledge- University focus on abstract conceptual knowledge- Challenge: Linking procedural and conceptual knowledge

Research interest:- Interactive Graphic Representation (IGR) as tool for learning and assessment

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Features of interactive graphic representations (IGR)

dragging points by mouse automatic animation/movement

dynamic change in the figure tracing of depending points hints and links (text) hints as guiding objects

in the figure response analysis / feedback

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Theoretical background:MODEM-Framework

The 5 phases of Multiple representations

concept formation: of concept attributes:

1. Orientation

2. Definition

3. Identification

4. Production

5. Reinforcement

verbal

symbolicgraphic

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Objectives

1. What kind of connection has the representation form (verbal, symbolic, graphic) of the mathematical problem to the difficulty of the task?

2. Does students’ prior knowledge have impact on the solving of the interactive problems?

3. Which kind of levels can be distinguished in students’ conceptual and procedural knowledge of binary operations?

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Design

First course on Lineare Algebra (N = 92) Four exercises (tests) are computer-based (WebCT) One paper & pencil test (examination) Schema of course and study design:

Test 1

Functions 1

(Web-CT)

Test 2

Functions 2

(Web-CT)

Test 3

Binary Operation 1

(Web-CT)

Test 4

Binary Operation 2

(Web-CT)

Test 5

Examination

(Paper&Pencil)

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Design: Description of items in the two binary operations tests

Concept Type Description VSG Scale No of Items Mathematical domains Concept Graphic DIG 16 R, [-c, c], R², Venn Definition Symbolic DIS 22 R, [0,1], N0, Q+, R² Verbal DIV 16 N, Z, Q+, R, R², R³, vowels Concept G --> S IGS 3 R2 Identification G --> V IGV 3 R2 V --> S IVS 12 R2 Concept G --> S PGS 6 R2 Production G --> V PGV 6 R2 S --> V PSV 6 R, R² V --> S PVS 8 R, R²

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Results: Role of the representation form

Is the representation form of the task (verbal, symbolic, graphic) connected to the difficulty?

0.72 0.73

0.560.63

0.770.86

0.630.67 0.67

0.60

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

DIG DIS DIV IGS IGV IVS PGS PGV PSV PVS

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Results: The role of prior knowledge

Does students’ prior knowledge have impact on the solving of the (interactive) problems?

0.00.10.20.30.40.50.60.70.80.91.0

DIG DIS DIV IGS IGV IVS PGS PGV PSV PVS

Low Prior Knowledge High Prior Knowledge

ns ns

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Results: Different levels of concept understanding

Which kind of levels can be distinguished in students’ conceptual and procedural knowledge of binary operations?

Statistical method: Latent-Class-Analysis

Cases: n = 92 Variables: DIS, DIV, DIG, IGS, IVS, IGV, PGV, PGS

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Three types of learners concerning conceptual-procedural knowledge

0.000.100.200.300.400.500.600.700.800.901.00

DIS DIV DIG IGS IVS IGV PGV PGS

Procedure-bounded (44 %) Procedural-oriented (36 %)

Proceptual (21 %)

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Validation of the classification by a comparison of the examination results

0

1

2

3

4

5

Item 1(Procedural)

Item 2(Procedural)

Item 3(Conceptual)

Item 4(Conceptual)

Procedure-bounded Procedural-oriented Proceptual

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Summary & conclusions

• IGR items could successfully adapted in the MODEM framework for diagnostic purpose.

• Item difficulty of the sub dimensions (IGR) was not crucial.

• Solving items with IGR (eg. DIG and IGV) was less dependend from prior knowledge.

• The class analysis delivered three groups (levels) of concept understanding.

• Challenge for ongoing work: Fostering links between conceptual and procedural knowledge.

• Intervention-study: procedural vs. conceptual training about the mathematical function concept.