CAS + PROGRAMMING = MATHEMATICAL CREATIVITY First Central and Eastern European Conference on Computer Algebra and Dynamic Geometry Systems in Mathematics Education CADGAME 2007 Pecs 20th-23th June 2007 G. Aguilera J. L. Galán M. A. Galán A. Gálvez A. J. Jiménez Y. Padilla P. Rodríguez Dpt. Applied Mathematic University of Málaga (Spain)

Index l Reflect on the habitual uses that nowadays are given to CAS in the teaching of Mathematics in Engineering. l Our experience. l Example. l Final conclusions.

Computer Technology and Mathematics Education l It is essential for mathematics education to be as complete as possible: it must help students “learn how to learn”. l Such adaptation to social changes can only be accomplished by teaching students to skilfully use information technologies.

CAS as didactic resources in Mathematics teaching l Using computers makes it enormously easier to do exercises and apply mathematical subject matter to engineering problems. Therefore, the use of computers is especially appropriate in education.

Problems l Even nowadays there are still many teachers who hesitate to use such technology due to technical, personal or even political reasons. l Most teachers were not taught to use CAS when they were studying to be teachers.

Reasons for using CAS in Engineering l CAS helps to make especially difficult abstract concepts more accessible and understandable to students. l CAS helps to increase student motivation and improve students’ attitudes towards Mathematics. l Students are able to attain a higher level of abstraction in mathematical problem-solving.

Reasons for using CAS in Engineering l Using CAS leads to significant quantitative improvements in students’ academic performance, especially significant in cases where there exist deficiencies or a lack of prior mathematical knowledge and skills. l CAS represent extremely useful tools for providing an appropriate attention to diversity in the classroom.

Risks involved l CAS can potentially prevent students from making the proper connections between the techniques used and their mental approach to Mathematics. l Using CAS could theoretically lead to undesirable modifications due to the significant changes made to the teaching process itself.

Nowadays l In most cases, the use of CAS is reduced to using computers as powerful high-performance calculators. l It is therefore necessary to change the way people think about information technologies in order to optimise the opportunities they offer and try to encourage mathematical creativity among students.

Nowadays l Math teachers that use CAS have to change the traditional uses given to these tools. l It is a mistake to use CAS in teaching as simple problem-solving machines. l They should be used in ways that maximize the opportunities that these technologies offer: positively affecting student learning, significantly increasing opportunities for experimentation and allowing students to construct their own mathematical knowledge.

Nowadays l Math teachers must first set out a series of appropriate activities. l The use of CAS in Mathematics has not reached optimum conditions. Mostly are blackbox (showing the result in one step without teaching students how to get there) and should be whitebox (showing intermediate steps).

Combining programming and CAS l When students program, they must read, construct and refine strategies, modify previously written programs and lastly, use the programs to solve problems. This makes them the protagonists of their own learning. l The most appropriate approach involves using programming and CAS together to allow students to create the specific necessary functions that will allow them to solve the problems involved in the subject matter under study.

Our experience Power of CAS Language program Flexibility + Mathematical creativity = Programming with DERIVE: elaboration of programs or specific functions by students

l In the elaboration of programs to calculate triple integrals the student will have to deal with: –The function to be integrated. –The system of coordinates. –The three variables of integration with their corresponding limits of integration. –The order of integration. l The fact that the students themselves are making the programs and including all of their arguments has a positive influence on their ability to subsequently apply them to solve specific exercises. Example: Triple Integrals

Calculate the volume of the solid bounded below by the cone z 2 = x 2 + y 2 (z ≥ 0) and above by the sphere x 2 + y 2 + z 2 = 2 using cartesian, cylindrical and spherical coordinates.

triple command

triplecylindrical and triplespherical commands

Solid

PROJECTION IN PLANE XY

Triple integral in Cartesian coordinates triple(1, z, sqrt(x^2+y^2), sqrt(2-x^2-y^2), y, - sqrt(1-x^2), sqrt(1-x^2), x, -1, 1)

Triple integral in cylindrical coordinates triplecylindrical(1, z, rho, sqrt(2-rho^2), rho, 0, 1, theta, 0, 2pi)

Triple integral in spherical coordinates triplespherical(1, rho, 0, sqrt(2), theta, 0, 2pi, phi, pi/4, pi/2)

Final Conclusions l Our accumulated experience reveals that CAS are computer tools which are easy to use and useful in Mathematics courses for Engineering. l The traditional uses given to CAS in the teaching of Mathematics for Engineering must be changed to maximize the opportunities offered by CAS technologies. Optimal use should be aimed at improving student motivation, autonomy and achieving participatory and student-centred learning.

Final Conclusions l One powerful idea involves combining CAS resources with the flexibility of a programming language. l There exists reasonable evidence to show that making programs with DERIVE facilitates learning and improves student motivation. l It is not necessary to substantially modify the traditional program of studies of Math courses for Engineering to introduce the innovation of having students make their own programs with DERIVE.

CAS + PROGRAMMING = MATHEMATICAL CREATIVITY First Central and Eastern European Conference on Computer Algebra and Dynamic Geometry Systems in Mathematics Education CADGAME 2007 Pecs 20th-23th June 2007 G. Aguilera J. L. Galán M. A. Galán A. Gálvez A. J. Jiménez Y. Padilla P. Rodríguez Dpt. Applied Mathematic University of Málaga (Spain)