Calculus CAS 1
The Effects of CAS Calculators on Student Learning in AP Calculus
Cally Student
Walden University
Kathryn Arnold
EDUC-6653: Introduction to Educational Research
July 24, 2008
Calculus CAS 2
The Effects of CAS Calculators on Student Learning in AP Calculus
After attending an AP conference for teachers of calculus, I was intrigued by a discussion
on the use of computer algebra system (CAS) calculators in the classroom and their merits and
downfalls. I plan to determine the impact of CAS calculators versus non-CAS calculators on
achievement in Advanced Placement Calculus of high school seniors. When given the choice, do
calculus students choose to use CAS calculators, non-CAS calculators or pencil and paper to
solve simple algebra problems? What are student attitudes’ toward the use of CAS calculators
versus non-CAS calculator methods of solving basic algebra problems? Do students using CAS
calculators produce superior work on advanced mathematical concepts in an AP Calculus course
versus those using non-CAS calculators or paper and pencil methods? Does the use of CAS
calculators increase scores on the AP Calculus tests? During my own teaching I often refer to
the “rule of four” when discussing mathematical concepts in calculus: That all concepts or
problems can be viewed algebraically, graphically, numerically and verbally.
In order for teachers to become aware of the usefulness and the advantages of a CAS
calculator they must first understand the mechanisms behind the software and be able to explain
the system to students. Arthur Nunes-Harwitt (2006) explains, “To be a truly sophisticated user,
it is necessary not only to be fluent with a program but also to know how the program works,”
(p. 177). He goes on to explain the way a simple CAS calculator works and outlines a lesson
plan to use with students to enlighten them on the workings of the system. He notices that
students’ attitudes after conducting the lesson vary from frustration to enthusiasm to
indifference. The lesson he conducts places an emphasis on “translation to an alternate form,
arithmetic of representations, and using formulas,” (p. 186). By students and teachers having
Calculus CAS 3
knowledge of how the CAS calculators work they can better apply the “rule of four” using the
system.
One of the first studies to observe student responses as Nunes-Harwitt (2006) has done is
an article by Robyn Pierce and Kaye Stacey (2001) in Australia in 1998. Data was collected
through surveys, observation and assessments for thirty undergraduate students using DERIVE a
CAS program available in computers in the late 1990s. The study asked whether students used
various representations of functions using CAS, whether it promoted discussions with each other
and the teacher and the overall opinion of students using CAS versus pen and paper (p. 2).
According to the study, “Class observations, made by the teacher-researcher, indicated that all
students swapped freely between algebraic and graphical representations of functions but they
seldom use tables unless specifically directed to do so,” (p. 9). In other words, the students were
using CAS to focus on the graphical and algebraic methods in the rule of four, but not in using
numerical representations as frequently. The study also showed that CAS had a positive
influence on verbal representations, noting that students were more likely to discuss the
mathematical problem at hand rather than their social lives (p. 12). The study concludes that the
use of CAS has positive effects on using various representations and encouraging discussions,
while students still preferred to use pen and paper they saw that CAS could add to their learning.
This leads to the use of CAS in written work as discussed in the next study.
Ball and Stacey (2004) focus on the written work of calculus students using CAS and
non-CAS calculators. According to Ball and Stacey the study documents “four features of the
written record provided in a high-stakes university entrance and final school examination, by a
group of students using CAS and comparing them to matched group of student not using CAS,”
(p. 87). The four features focused on wording using standard mathematical notation, non-
Calculus CAS 4
standard mathematically notation, words found in a dictionary and words that cannot be found in
a dictionary (p. 90) while also looking at the length of the written record (p. 91). This paper
focuses on the communication of mathematical knowledge which connects to the verbal
applications in the “rule of four”. The conclusion of the study found that “CAS solutions will
generally be shorter than non-CAS solutions, that they will contain more words and use function
notation more,” (p. 93). This implies that while the written response of students using CAS
calculators were shorter, correct mathematical notation and wording was present more often.
While these studies refer to the mathematical framework necessary for calculus, an early
study by Jeanette Palmiter (1991) focused mostly on impact of CAS calculators the concepts of
calculus specifically. This study examines the difference between students taught using CAS in
calculus versus those using paper and pencil methods and their knowledge of concepts and
procedures and the impact on grades (p. 151). The focus in this study is on grades rather than
written solutions or student attitudes. According to Palmiter, two groups were taught calculus,
one using MACSYMA a computer algebra system and the other using pencil and paper methods,
and then received identical tests (p. 152). The study found that those using the CAS program
“significantly outscore the traditional group on both the conceptual and computational exams,”
(p. 155). This more quantitative study leads to the question of the use of the more advanced
CAS calculators used today on AP exams.
The last report is a case study from Thomas Dick, Gloria Dion and Craig Wright (2003)
which focuses on AP Calculus scores and the use of CAS calculators. This study set out to
evaluate the difference between assessment scores using various types of calculators. A
graphing calculator is required on the AP exam and according to Dick, et. al., “When we
compare the performance of students using calculators that have different levels of sophistication
Calculus CAS 5
(the highest level being those that include symbolic-algebra capabilities), we find a slight but
discernible relationship that associates higher scores with the use of higher-level calculators,” (p.
593). For example, on the 2002 AP Calculus AB exam, the mean AP grade was 3.13 for
students using a high-end calculator such as the TI-83, while the mean AP grade was 3.23 for
students using a CAS calculators such as the TI-89 (p. 595). While this is not a large gain, this
furthers the debate of the use of CAS calculators and implies a benefit for those who use them.
Dick, et. al., conclude, “Taking full advantage of this aid may allow more emphasis on the “rule
of four” – where verbal representations are added to symbolic, graphical, and tabular ones – in
applying calculus to problem solving and making sense of the meaning of calculus concepts in
context,” (p. 595). The study connects the comprehension of the students using CAS calculators
and the “rule of four” to higher scores on the AP exam.
Each of these studies shows the benefit CAS programs can have for students studying
calculus and their understanding of concepts using the “rule of four”. All of these studies shed a
positive light on the use of CAS calculators, but some seem more reliable than others. For
example the study by Arthur Nunes-Harwitt (2006) only focused on a small group of his own
students and he admits that further study would be necessary (p. 185). Also one of the early
studies by Jeanette Palmiter (1991), shows that there may be reasons for her positive results.
Palmiter admits, “The students in the MACSYMA group were actively participating and were
fully aware of their participation in the study,” (p. 155). In other words, the students knew they
were a part of the study and may have performed better due to that knowledge. This is especially
important because this study and the study by Pierce & Stacey (2001) are both early examples
that are often cited in other studies. Since the 1990s and early 2000s, the CAS program has
grown and is much faster in the hands of the students using graphing calculators. Although all
Calculus CAS 6
the studies show positive effects in using the CAS calculators, it is important to note the growing
technology and the factors that influenced student performance.
For my purposes, the case study from Dick, et. al. (2003) gives the best insight on how
CAS calculators can influence the scores for students taking AP Calculus. Knowing that
technology has advanced and the use of CAS calculators can aid in the understanding of using
the “rule of four” I hope to find what effect CAS calculators have when being use in the
classroom and on the AP test influence scores and average grades. The literature on the subject
shows that the CAS calculators can help students to see a calculus idea graphically, numerically,
algebraically and verbally. I hope to see whether using CAS calculators has a positive effect on
student grades and AP exam scores, but to further investigate the usage of the program with the
“rule of four” in the classroom. The literature on this debate weighs the negative and positives of
using the CAS calculators and suggests that with appropriate instruction the calculator can be
beneficial to students and increase their understanding of complex mathematical ideas especially
through the continued focus on the “rule of four”.
Calculus CAS 7
References
Ball, L., & Stacey, K. (2004, July 1). A New Practice Evolving in Learning Mathematics:
Differences in Students' Written Records with CAS. International Group for the
Psychology of Mathematics Education, (ERIC Document Reproduction Service No.
ED489688) Retrieved July 21, 2008, from ERIC database.
Dick, T., Dion, G. & Wright, C (2003, November). Case Study: Advanced Placement Calculus
in the Age of the Computer Algebra System. Mathematics Teacher, Retrieved July 21,
2008, from Academic Search Premier database.
Nunes-Harwitt, A. (2006). Ideas Behind Computer Algebra and their use in the Classroom.
Journal of Educational Technology Systems, 34(2), 177-188. Retrieved July 23, 2008,
from http://www.cs.rit.edu/~anh/CompAlg.pdf.
Palmiter, J. (1991, January 1). Effects of Computer Algebra Systems on Concept and Skill
Acquisition in Calculus. Journal for Research in Mathematics Education, 22(2), 151.
(ERIC Document Reproduction Service No. EJ429210) Retrieved July 21, 2008, from
ERIC database.
Pierce, R. & Stacey, K. (2001b). Observations on students’ responses to learning in a CAS
environment. Mathematical Education Research Journal, 3(1), 28-46. Retrieved July 21,
2008, from http://extranet.edfac.unimelb.edu.au/DSME/CASCAT/publicationsCASCAT/2001Pubspdf/PierceStaceyObsOnStudsLearnCAS.pdf