Making procedural and conceptual mathematical knowledge and their links alive by Java applets Djordje Kadijevich Megatrend University & Mathematical Institute SANU Serbia & Montenegro Euromath meeting Innsbruck, Austria,7-8 October 2004

Relevant references: Kadijevich, Dj. (2004). Some Aspects of Visualizing Geometric Knowledge: possibilities, findings, further research. In N. Bokan, M. Djorić, A. T. Fomenko, Z. Rakić & J. Wess (Eds.), Contemporary Geometry and Related Topics. Proceedings of the Workshop Belgrade, Yugoslavia May 2002 (pp ). Singapore: World Scientific. Kadijevich, Dj. & Haapasalo, L. (2001). Linking procedural and conceptual mathematical knowledge through CAL. Journal of Computer Assisted Learning, 17, 2, Based on research partly supported by the TEMPUS grant IMG- SCG Euromath meeting Innsbruck, Austria,7-8 October 2004

P-C knowledge distinction Procedural knowledge (P) denotes dynamic and successful utilization of particular rules, algorithms or procedures within relevant representation form(s), which usually requires not only the knowledge of the objects being utilized, but also the knowledge of format and syntax for the representational system(s) expressing them. Conceptual knowledge (C) denotes knowledge of and a skilful “drive” along particular networks, the elements of which can be concepts, rules (algorithms, procedures, etc.), and even problems (a solved problem may introduce a new concept or rule) given in various representation forms.

Euromath meeting Innsbruck, Austria,7-8 October 2004 Procedural knowledge often calls for automated and unconscious steps, whereas conceptual knowledge typically requires conscious thinking. However, the former may also be demonstrated in a reflective mode of thinking when, for example, the learner skillfully combines two rules without knowing why they work. GOAL Acquiring and relating procedural and conceptual mathematical knowledge

Unrealized issues of visualization by Java applets with respect to the P-C issues Euromath meeting Innsbruck, Austria,7-8 October 2004

Three types of Java applets 1. blackbox X applet dealing with X untransparently 2. greybox X applet dealing with X semi-transparently 3. whitebox X applet dealing with X transparently X can be P, C, P&C or P-C from Procedural, Conceptual, Procedural-Conceptual linkage QUESTIONS: Can a whitebox P applet be a blackbox C one? May a whitebox P-C applet be a greybox P one? Can a whitebox P&C applet still be a blackbox P-C one? GOAL: Developing and utilizing whitebox P&C and P-C applets Euromath meeting Innsbruck, Austria,7-8 October 2004

Is this an example of blackbox or greybox C applet? Are basic concepts defined are related themselves? Euromath meeting Innsbruck, Austria,7-8 October 2004

Is this an example of blackbox or greybox P&C applet? Are basic concepts defined and related themselves? Are calculations explained and transparently carried out? Euromath meeting Innsbruck, Austria,7-8 October 2004

Are all calculations transparently carried out? greybox P applet whitebox C applet black or greybox P-C applet ? Euromath meeting Innsbruck, Austria,7-8 October 2004

blackbox C applet or blackbox P-C applet!? What are given explicitly? Euromath meeting Innsbruck, Austria,7-8 October 2004

greybox P& C applet, greybox P-C applet!? Euromath meeting Innsbruck, Austria,7-8 October 2004

SUITABLE TASKS: (1) Analyze some of applets available on the Internet. (2) Design a paper version of a whitebox P&C and P-C applet. TASK: Euromath meeting Innsbruck, Austria,7-8 October 2004

Appendix - Useful collections of Java Applets Manipula Math Applets at Math Online at NCTM Illuminations at WisWeb project at Java View at ON-Math journal published by NCTM at al_id=6. al_id=6 Euromath meeting Innsbruck, Austria,7-8 October 2004