1. IPN Leibniz Institute for Science Education at 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)

2. 14.10.2005 Learning and Instruction Symposium - JULIS'05 2 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

3. 14.10.2005 Learning and Instruction Symposium - JULIS'05 3 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

4. 14.10.2005 Learning and Instruction Symposium - JULIS'05 4 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

5. 14.10.2005 Learning and Instruction Symposium - JULIS'05 5 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?

6. 14.10.2005 Learning and Instruction Symposium - JULIS'05 6 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)

7. 14.10.2005 Learning and Instruction Symposium - JULIS'05 7 Design: Description of items in the two binary operations tests

8. 14.10.2005 Learning and Instruction Symposium - JULIS'05 8 Results: Role of the representation form Is the representation form of the task (verbal, symbolic, graphic) connected to the difficulty?

9. 14.10.2005 Learning and Instruction Symposium - JULIS'05 9 Results: The role of prior knowledge Does students’ prior knowledge have impact on the solving of the (interactive) problems? ns

10. 14.10.2005 Learning and Instruction Symposium - JULIS'05 10 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

11. 14.10.2005 Learning and Instruction Symposium - JULIS'05 11 Three types of learners concerning conceptual-procedural knowledge

12. 14.10.2005 Learning and Instruction Symposium - JULIS'05 12 Validation of the classification by a comparison of the examination results

13. 14.10.2005 Learning and Instruction Symposium - JULIS'05 13 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.