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

1. 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. 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 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 Learning and Instruction Symposium - JULIS'05

4. verbal graphic symbolic Theoretical background:MODEM-Framework The 5 phases of Multiple representations concept formation: of concept attributes: • Orientation • Definition • Identification • Production • Reinforcement Learning and Instruction Symposium - JULIS'05

5. Objectives • What kind of connection has the representation form (verbal, symbolic, graphic) of the mathematical problem to the difficulty of the task? • Does students’ prior knowledge have impact on the solving of the interactive problems? • Which kind of levels can be distinguished in students’ conceptual and procedural knowledge of binary operations? Learning and Instruction Symposium - JULIS'05

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

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

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

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

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