Euromath meeting
Innsbruck, Austria,7-8 October 2004
Making procedural and conceptual mathematical
knowledge and their links alive by Java applets
Djordje Kadijevich
Megatrend University & Mathematical Institute SANU
Serbia & Montenegro
e-mail: [email protected]
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 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 IMGSCG3018-2003.
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
Euromath meeting
Innsbruck, Austria,7-8 October 2004
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
TASK:
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.
Euromath meeting
Innsbruck, Austria,7-8 October 2004
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 at
http://my.nctm.org/eresources/journal_home.asp?journ
al_id=6.