Using modularization to thicken the knowledge-representation network in CAS enhanced teaching

1. Using modularization to thicken the knowledge-representation network in CAS enhanced teaching Computer Algebra andDynamic Geometry Systems inMathematics Education University of South Bohemia Pedagogical Faculty 29 June- 1 July, 2010, Hluboká nad Vltavou Csaba Sárvári, Zsolt Lavicza University of Pécs Pollack Mihály Faculty of Engineering University of Cambridge Faculty of Education

2. The main topics of our lecture Knowledge representation network Modularisation, Multiple representation Semiotic approach

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

4. 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 network is the result of the development and modification of interrelated cognitive schemes.

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

6. CAS-module Both the field of mathematics and the mathematics curricula are structured modularly. We can define module as: complex and interconnected elements of knowledge that can be recalled from memory without consciously being aware of it internal structure.

7. Semiotic approach The idea of adopting a semiotic perspective when looking at the nature of mathematics and mathematical activities has its modern roots in the writings of the fathers of contemporary semiotics, Peirce (1839−1914) and Saussure (1857−1913). During the twentieth and current century, a semiotic view has been developed and applied to mathematics or mathematics education by, for example, Rotman (1993), Radford (2005), Presmeg (2006). A Computer Algebra System may be viewed as a semiotic tool: Given the capacity of CAS to transform signs within and between the semiotic registers CAS may facilitate the learning of mathematics.

8. Triadic conceptualisations {Object, Representamen (Sign), Interpretant} (Peirce, 1903) {Object, one of various semiotic systems, Composition of signs} (Duval, 2000)

9. Cognitive activity related to the semiotic registers Three major forms of cognitive activity related to the semiotic registers in mathematics (Duval, 2000): REPRESENTATION TRANSFORMATION PROCESSING CONVERSION Change of register, or semiotic system of the representation, , e.g. the shift from thesymbolic representationto the corresponding Cartesian graph). Change of representation within the sameregister, e.g. manipulationsof an algebraic symbolic expression Mathematical comprehension involves the capacity to change from one register to another The production of mathematical signs in the CAS medium is a possibilitity to achieve higher level of comrehension.

10. Representation and understanding for mathematical knowledge Whenever a semiotic system is changed, the content of the representation changes, while the denoted content of the representation remains the same.

11. The cognitive paradox of access to knowledge objects From an epistemological point of view there is a basic difference between mathematics and the other domains of scientific knowledge. Mathematical objects, in contrast to phenomena of astronomy, physics, chemistry,biology, etc., are never accessible by perception or by instruments (microscopes,telescopes, measurement apparatus). The only way to have access to them and deal with them is using signs and semiotic representations. The crucial problem of mathematics comprehension for learners, at each stage of the curriculum, arises from the cognitive conflict between these two opposite requirements: How can they distinguish the represented object from the semiotic representation used if they cannot get access to the mathematical object apart from the semiotic representations?(Duval, 2006)

12. Modularization-knowledge representation-semiotic approach • The main goals of the CAS-modularisation: • reduction of the complexity, relieve of the attention’s burden, help the experimental work by facilitatig processing and conversion. Lever potencial (Winslow, 2003) : „The idea is for students to operate at a higher conceptual level; in other words, they can concentrate on the operations that are intended to be focus of attention and leave the lower-level oeperations to the computer” (Dreyfus, 1994) • help of transformations within and between registers, to give an answer for cognitive paradox; • effectively develop the cognitive schemes, thicken the knowledge representation network, facilitate the transfer.

13. Constructing of the CAS moduls Modularisation is a dynamic process. By constructing of complex CAS-procedures we apply the following model: • We make the series of the procedures’steps fromalgorithm primitives. (Step by step phase) • We construct the parts of the complete algorithm, asworking succesive procedures.(Semi-automated phase) • We paste the working parts together using themathematical concept of composition (Automated phase).

14. Didactical goals of phases of the algorithm’s building Phase Didactical goal To build the mental map of the algorithm To practise the CAS-notation To study the possible variants Step by step To practise constructing of procedures To look over structure of the algorithms Semi-automated To reduce the complexity To facilitate the computations To supply components for later knowledge-elements Automated

15. Procedure for calculation of local extrema Single steps Single steps Location of critical points Step1 partial derivatives  in CAS diff(f(x,y),x) , diff(f(x,y),y) Step1 Computing second derivatives  in CAS diff (f(x,y),x,y) Step2 Solve system of the equations  in CAS solve( { f ‘x(x,y)=0 , f ‘y(x,y)=0 } , {x,y} ) or fsolve Step2 Building the discriminant  in CAS expression Semi automated algorithms Automated algorithm Epistemic plus value of technique New problems arise Step3 Substitution of the critical points  in CAS subs Test using the second derivatives Experimental work Building element of new algorithms Evaluation of roots: map, allvalues Selection from complex roots: type Step4 Decide from the type of critical points (max, min, saddle)

16. Semiotic view on phases of building CAS procedure Processing Math object Step by step Math-algorithm Conversion, Change of semiotic system CAS-algorithm: Composition of signs (Interpretant) Semi automated phase CAS object Processing . . . . . . . Automated phase . . . . . . .