I moonlight at night in the math tutoring center at the local community college here in the heart of America. One evening a fellow tutor, after helping a student with their homework, returned to the ‘tutor’s table’ on a vehement rampage about how Texas Instruments is the cause of all mathematical illiteracy in high schools today. He went on the say that calculators with computer algebra systems (CASs) such as the TI-89s and TI-92s decrease the abilities of students to perform by-hand symbolic manipulation. If this is true, then why do some high schools and universities continue to use these calculators in their classrooms? The answer is that these technologies are opening new doors for our students.
CASs promote student access to multiple representations
According to the National Council of Teachers of Mathematics ([NCTM], 2000) standards, different representations support different ways of thinking about and manipulating mathematical objects. An object can be better understood when viewed through multiple lenses (p. 360). CASs adequately answer this important aspect of learning mathematics by providing symbolic, graphical, and numerical representations conveniently in one package. Having these different forms of representation, students are able to employ different methods to solve problems nearly instantaneously while allowing their thought processes to remain on the concept they are supposed to be learning and not get mired in the technicalities of performing different methods to solve a problem. It also allows them to verify their work from different viewpoints. Heid, Blume, Hollenbrands, & Piez (2002) claim that The flexibility allowed by the CAS enables students to access multiple approaches to a single problem and use one approach for multiple purposes. Students often use the CAS to scaffold their problem solving, building their solution by moving among various representations (p. 589). In Australia, several studies have been done on the usage of CASs in high school classrooms. One such pilot study reports that students using CASs are able to transfer between these different representations effortlessly when exploring unfamiliar functions (Forgasz, & Griffith, 2006).
CASs allow students more access to meaningful problems
Having the flexibility of a CAS also allows students to extend the range of problems that can be considered in the classroom. In another Australian study, CASs were shown to supplement the student’s by-hand work because the technology allowed students to access messy real world problems (Pierce & Stacey, 2008). Most of the real world mathematical problems are complicated and would be impractical as an in class lesson because of the time that would be wasted on by-hand manipulation. With CASs, however, students can begin to explore these difficult problems focusing on concepts rather than the tedium of long manipulations and correcting minor algebra errors. Heid (2002) explains the importance of this:
To whatever the extent routine symbolic manipulation can be successfully offloaded to the CAS, the time saved can be spent on developing solid conceptual understanding and facility with using mathematics to model and understand mathematical aspects of the real world. … [Students] use a CAS for equation solving, curve fitting, table generating, and graphing so that they can spend their time investigating real-world problemsthat they model using families of functions. Their understandings of such fundamental algebra concepts as function, equation, variable, equivalence, and systems of equations are solidified in the process (p. 665).
While by-hand manipulation is important and should be taught, students should also learn the essence of mathematics. Conceptual understanding, modeling, and generalizing are the essence of mathematics and by-hand manipulations are superficial at best when it comes to learning these topics. CASs allow students to focus their attention on learning mathematics conceptually by freeing them from the time consuming by-hand manipulations that would take place in these real world problems. Martinez-Cruz & Contreras (2002) reflect on their use of CASs in the classroom and conclude that By performing complicated manipulations, technology gave the students time to reflect on their approaches and to pose, explore, and give answers to more questions (p. 597).
What the studies say about CASs
Studies have shown that students who used CASs throughout the school year perform significantly better than their counterparts on questions that measure conceptual understanding such as functions, mathematical modeling of questions, and translating between different representations (Heid, Blume, Hollenbrands & Piez, 2002). Other advantages pointed out by Mahoney (2002) are that CASs can help students learn pre-calculus and calculus even if they don’t have complete mastery of algebra; CASs allow teachers to introduce calculus concepts in lower level math courses; and CASs lets users rely on the accuracy of the system in use while experimenting with algebra, pre-calculus, and calculus.
Does by-hand manipulation really suffer?
Perhaps the most obvious concern to educators is that the CASs ability to solve problems symbolically will lead to decreased abilities of students to perform by-hand symbolic manipulation. Several studies have been done on this issue. Heid, Blume, Hollenbrands & Piez (2002) have reviewed a number of these and have found that 8 out of 9 studies reviewed show that students who used CASs during classroom instruction did just as well or better on achievement tests where technology was not used. It appears from these studies that CASs have actually facilitated better by-hand symbolic manipulation rather than handicapping students in this area.
Certainly negligent usage of the Texas Instrument’s calculators may be detrimental to student learning. However, negligence in any arena is counterproductive to learning and teaching. It is not the abilities of the Texas Instruments calculators that hinder student learning, but rather the way this tool is used in the classroom. When used properly by savvy teachers, this powerful tool can only benefit students and enhance the experience of their mathematical journey.
Forgasz, H.J. & Griffith, S. (2006). Computer algebra system calculators: Gender issues and teachers’ expectations. Australian Senior Mathematics Journal, 20(2), 18-29.
Heid, M.K., Bloom, G.W., Hollenbrands, K. & Piez, C. (2002). Computer algebra systems in mathematics instruction: Implications from research. Mathematics Teacher, 95(8), 586-591.
Heid, M.K. (2002). Computer algebra systems in secondary mathematics classes: The time to act is now! Mathematics Teacher, 95(9), 662-667.
Mahoney, J.F. (2002). Computer algebra systems in our schools: Some axioms and some examples. Mathematics Teacher, 95(8), 598-605.
Martinez-Cruz, A.M. & Contreras, J.N. (2002). Changing the goal: An adventure in problem posing, and symbolic meaning with a TI-92. Mathematics Teacher, 95(8), 592-597.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
Pierce, R. & Stacey, K. (2008). Using pedagogical maps to show the opportunities afforded by CAS for improving the teaching of mathematics. Australian Senior Mathematics Journal, 22(1), 6-12.