COMPUTER ALGEBRA SYSTEMS (CAS) AS RESTRUCTURING TOOLS IN MATHEMATICS EDUCATION Keith Nabb Moraine Valley Community College Illinois Institute of Technology Twenty-second International Conference on Technology in Collegiate Mathematics 2010 Chicago, IL

MAIN QUESTIONS How should CAS be used in undergraduate mathematics? Is there an appropriate time to introduce CAS into the learning experience? To what degree does CAS influence subject matter and pedagogy?

AGENDA Theoretical Considerations Important Research Findings Utility When CAS is implemented Curriculum & Instruction Final Thoughts/Questions

THEORY Instrumental Genesis (Drijvers & Trouche, 2008; Guin & Trouche, 1999; Vérillon & Rabardel, 1995) Cognitive Technologies: Epistemic or Pragmatic? (Pea, 1987; Artigue, 2002; Ruthven, 2002) The Technical/Conceptual Divide (Heid, 2003; Zbiek, 2003) Constraints, Boundaries, & Obstacles (Drijvers, 2000, 2002; Guin & Trouche, 1999; Hoyles, Noss, & Kent, 2004)

CAS USE Black Box (Bossé & Nandakumar, 2004; Buchberger, 1989, 2002) *White Box (Pedagogical Tool) (Buchberger, 1989; Child, 2002; Heid & Edwards, 2001; McCallum, 2003) Amplifier (Heid, 1997; Pea, 1987) Discussion Tool (Guin & Trouche, 1999; 2002; Heid, 1997; Pea, 1987; Pierce & Stacey, 2001) *Catalyst for Reform (Heid, 1988; Judson, 1990; Palmiter, 1991)

WHITE BOX (Heid & Edwards, 2001)

CATALYST FOR REFORM Concepts-first curriculum ( Heid, 1988; Palmiter, 1991) Challenges the assumption that procedural fluency need precede conceptual fluency CAS often treats a wide array of situations uniformly ( Hillel, Lee, Laborde, & Linchevski, 1992) Challenges the pedagogical assumptions of traditional hierarchy in mathematics

WHEN TO IMPLEMENT CAS INTO LEARNING? Before a concept is learned (Cedillo & Kieran, 2003) In conjunction with new learning (Kieran & Drijvers, 2006) After a concept has been mastered (Boyce & Ecker, 1995)

BEFORE... Using CAS to facilitate the genuine learning of algebra (Bruner, 1983) Students examined tables, wrote short programs, reflected on these activities Student conceptions: “variables”—objects that store values “expressions”—computer programs that are written, compiled and executed

IN CONJUNCTION WITH... Knowledge development through practice (Chevallard, 1999) CAS often displays factorizations and expressions that vary with solutions found by hand. This can lead to confusion, reflection, conjecture, etc. Techniques, tasks, and theoretical aspects of student thinking contain elements from CAS and pencil-and- paper environments

AFTER... Students had knowledge of Taylor/Maclaurin polynomials Student: Why is the midpoint chosen and is this the “best” Taylor polynomial given a fixed degree? “This discussion of Taylor polynomials was at a much higher conceptual level than any of our previous lectures on the topic. The interactive and spontaneous nature of our discussion would not have been possible without a computer algebra system. The much deeper understanding of Taylor polynomials gained that day by these students was due, in part, to the use of Maple, which allowed them to see past the details into the conceptual heart of the subject.” (Boyce & Ecker, 1995, p. 49)

CURRICULUM & INSTRUCTION *Curriculum “Freeing” students from drill and practice (Heid, 1988) Rearranging the sequence in traditional curricula (Heid, 1988; Judson, 1990; Palmiter, 1991) Refocusing what we think of as important “math” (Boyce & Ecker, 1995; Lagrange, 2003; Pierce & Stacey, 2002) Instruction More facilitating and less “telling” (Guin & Trouche, 1999; Hoyles et al., 2004) This may put teachers in an unfamiliar/uncomfortable position (Zbiek, 1995) Teachers may feel the need smooth out these challenges so minimizing the effects of CAS (Zbiek, 1995)

CURRICULUM “General” word problems (Lagrange, 2003) Presence of many interconnecting variables spark insights to specific problems Solutions may/may not depend on specific variables Such insights are lost or not even considered in conventional analysis What questions do we pose? What should we look for? Algebraic Insight & Algebraic Expectation (Pierce & Stacey, 2002) Much like we know 113 x 46 is close to 5000, students need to be skilled in anticipating mathematical structure

CAS: THE GOOD AND THE BAD Good Look past routine procedures Open doors to unconventional topics Classroom discussion CAS give immediate, nonjudgmental feedback Nominal effects on one’s ability to perform routine procedures Bad Time is needed for informed use Situated knowledge constrained by CAS Mysteries of the black box Professed use may not align with observable action (in the case of teachers)

THANK YOU! QUESTIONS??