COMPUTER ALGEBRA SYSTEMS, PEDAGOGY, AND EPISTEMOLOGY

Mathematics and Computer Education, Fall 2004 by Boss�, Michael J, Nandakumar, N R

ABSTRACT

The advent of powerful Computer Algebra Systems (CAS) continues to dramatically affect curricula, pedagogy, and epistemology in secondary and college algebra classrooms. However, epistemological and pedagogical research regarding the role and effectiveness of CAS in the learning of algebra lags behind. This paper investigates concerns regarding typing expressions into Texas Instruments TI-92 and TI-89 CAS and offers suggestions for future CAS to be more pedagogically and epistemologically sound.

One of the authors of this paper was in a position to overhear a conversation among two university colleagues. The newer faculty member questioned if it was fair for a student in a college algebra class to be allowed to use a TI-92. The senior faculty member asked, "Does the student know how to use the algebraic functions?" The junior faculty responded, "No." The senior faculty then answered the original question with, "Then let him use it." Unfortunately, for many, the irony within this anecdote connotes the state of debate regarding the use of calculators with Computer Algebra Systems (CAS) in secondary and college mathematics classrooms.

The authors caution readers against inferring that this paper is a polemic against the use of CAS within secondary or college algebra classes. To the contrary, the authors strongly support the judicious advancement of CAS technology within the classroom. CAS should be employed when doing so would be pedagogically and epistemologically sound and when it would enhance student learning.

We now live in an era in which students have ready access to calculators and computers programmed with CAS. At an impressive rate, curricula are being developed to utilize CAS technology. Educators and curriculum developers continuously investigate mathematical content within the curriculum which is most appropriate for applications of CAS. Missing from many of these investigations and curricular developments, however, are scholarly considerations of the role that CAS plays in teaching and learning. While CAS may greatly enhance student learning and the manipulation of algebraic expressions, they are also intrinsically fraught with epistemologically unsound automated functions which may lead to misunderstandings. While this paper tacitly argues in favor of increased adoption of CAS, its focus is to demonstrate that CAS (particularly associated with the TI-92 and TI-89) still carry some programming nuances which should in the future be modified for CAS to be more consistent with student learning. This paper begins to investigate this phenomenon and suggest programming alternatives.

WHAT HAPPENS WHEN WE PRESS ENTER? AUTOMATED PROCEDURES AND OPERATIONS

To "load" an expression into the TI-89 and TI-92, the expression is typed and then the user must press the ENTER key. However, when the ENTER key is struck, two events occur simultaneously: a nicely formatted expression is displayed on the left hand side of the screen and is often redisplayed in modified or simplified form on the right hand side of the screen. While this may seem quite amazing to novice users, it can also lead to questions and even mathematical misconceptions.

The initial question is weather there is a difference between loading an expression and requesting the CAS to perform an operation. Notably, the verbiage within the question implies some of the underlying inquiries within this investigation. This discussion purposely avoids the phrase "entering an expression", due to both the ambiguity of this phrase and the inconsistency of CAS responses to a mathematical expression after typing an expression and pressing ENTER. Rather, the following phrases will be utilized: "typing expressions", "loading expressions", ENTER, and "performing operations on expressions."

With this vocabulary in place, it can now be asked: "What is the primary purpose for ENTER? Is it to load an expression into the CAS's memory to be stored for future operations, to display the expression on the screen, or to perform operations on the expression?" As a pedagogical tool within the classroom, it would probably be advantageous if the function of ENTER were fully understood by students. However, due to the inconsistent, and often seemingly capricious, behavior recognized after pressing ENTER, it is difficult to fully explicate its role.

When arithmetic and algebraic expressions are typed into the TI-89 and TI-92 and ENTER is struck, automated processes often rewrite and replace the expressions with others. Thus automated processes perform tasks and operations regardless of whether or not the user intended the CAS to do so. For instance, in default mode, if "3 � 12 " is typed into the CAS, upon depressing ENTER, the fraction 3/12 is displayed on the left of the screen and the fraction is auromatically rewritten as �, and displayed on the right side of the screen. While this may be both correct and convenient, it potentially leads to a number of mathematical misconceptions: