1. Cognitive strategies and CAS Csaba Sárvári, Mihály Klincsik, Zsolt Lavicza University of Pécs Pollack Mihály Faculty of Engineering Computer Algebra and Dynamic Geometry Systems in Mathematics Education RISC, Castle of Hagenberg, Austria. July 11-13, 2009. University of Cambridge Faculty of Education

2. Cognitive strategies help learners make and strengthen associations between new and already known information; make possible adaptation of attained knowledge in new situation; facilitate the mental restructuring of information.

3. Cognitive strategies guessing analysing, synthesising reasoning inductively and deductively taking systematic notes reorganising information hypothesis testing searching for clues in surrounding material and one’s own background knowledge searching for new rules trying making new structure completion of knowledge elaboration activating of information recalling of the previous knowledge experimenting

4. Metacognition “Metacognition refers to one’s knowledge concerning one’s own cognitive processes or anything related to them, e.g., the learning-relevant properties of information or data.” J. H. Flavell (1976) “thinking about thinking” Most fundamental metacognitive strategies connect the new information with previous knowledge; conscious selection of thinking operations; planning, monitoring, controlling and evaluation of thinking processes.

5. Self-regulated learning The term self-regulated can be used to describe learning that is guided by metacognition strategic action (planning,monitoring, and evaluating personal progress against a standard), and motivation to learn. Self-regulated learners are cognizant of their academic strengths and weaknesses, and they have a repertoire of strategies they appropriately apply to tackle the day-to-day challenges of academic tasks.

6. Cognitive schemes and knowledge-representation network Cognitive schemes (epistemological-, thought- schemes) are such building blocks of our thinking that are meaningful by themselves and having independent meanings. are actively direct one’s cognition and thinking while they are constantly changing in relation to the acquired knowledge. are not independent components of one’s consciousness, but they establish an ever changing relation-system called knowledge-representation network.

7. Knowledge representation network Efficiency of the mathematical knowledge can be approached by evaluating the organization of knowledge elements. A concept is comprehended if the concept is well represented and bounded with other knowledge elements. Consequently, the thickening of the knowledge-representation web is the result of the development and modification of interrelated cognitive schemes.

8. Enlarging of the knowledge representation network The inner representation network Before the learning event During the learning event After the learning event

9. Cognitive strategies and CAS The aim of our presentation to show how we can give - using CAS – additional tools for students to be able to apply efficient (additional) cognitive strategies; to gain a metacognitive knowledge about self, the task and strategies and how to use these strategies; to develop their ability of self-regulated learning.

10. Examples Our examples are the following: using Student Calculus1 Package by integration to demonstrate how the students can efficient learn to integrate applying with different strategies the tools of the package; minmax approximation to show how can be used CAS to combine seemingly distinct areas of mathematics, to help the experimental work investigating solutions of differential equations to present how we can use CAS to gain additional strategies to investigate the different solutions of differential equations.

11. Flow chart of the problem solving process Differential equation General solution Particular solution From local investigation to global picture Vectorfield Links between elements of the mathematical theory Give the domain of existence of the solutions on x-axis Show, there are two solutions go through to the origin (0;0). Determine the limits of the solutions as x tends to ±∞! Problems Determine the asymptotes of the solutions as x tends to ±∞!

12. Aspects of the guided learning tour Experimentations with plotting different solutions Links to the previous knowledge-element (black box, building the problem space) Domain of existence of functions Solutions of differential equation Initial values Limiting value of functions Maple commands: dsolve, solve, subs, plot, odetest, limit, DEplot