Making procedural and conceptual mathematical knowledge and their links alive by Java applets Djordje Kadijevich Megatrend University & Mathematical Institute

Making procedural and conceptual

mathematical knowledge and their links alive by

Java applets

Djordje KadijevichMegatrend University & Mathematical Institute SANU

Serbia & Montenegro

e-mail: [email protected]

Euromath meetingInnsbruck, 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 15 - 21 May 2002 (pp. 277-283). Singapore: World Scientific.

• Kadijevich, Dj. & Haapasalo, L. (2001). Linking procedural and conceptual mathematical knowledge through CAL. Journal of Computer Assisted Learning, 17, 2, 156-165.

Based on research partly supported by the TEMPUS grant IMG-SCG3018-2003.



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.


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


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-Cfrom 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


Is this an example of blackbox or greybox C applet? Are basic concepts defined are related themselves?


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?


Are all calculations transparently carried out?greybox P applet

whitebox C applet

black or greybox P-C applet?


blackbox C applet or blackbox P-C applet!?What are given explicitly?


greybox P& C applet, greybox P-C applet!?


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:


Appendix - Useful collections of Java Applets

Manipula Math Applets at http://www.ies.co.jp/math/java/

Math Online at http://www.saab.org/moe/start.html

NCTM Illuminations at http://illuminations.nctm.org/

WisWeb project at http://www.fi.uu.nl/wisweb/

Java View at http://www.javaview.de/

ON-Math journal published by NCTM athttp://my.nctm.org/eresources/journal_home.asp?journal_id=6.


http://www.ies.co.jp/math/java/






http://www.saab.org/moe/start.html






http://illuminations.nctm.org/



http://www.fi.uu.nl/wisweb/









http://www.javaview.de/



http://my.nctm.org/eresources/journal_home.asp?journal_id=6





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Making procedural and conceptual mathematical knowledge and their links alive by Java applets Djordje Kadijevich Megatrend University & Mathematical Institute