Wednesday, May 1, 2013

MAA Calculus Study: Graphing Calculators and CAS

This column continues my report on results of the MAA National Study of Calculus I, Characteristics of Successful Programs in College Calculus. This month I am sharing what we learned about the use of graphing calculators (with or without computer algebra systems) and computer software such as Maple or Mathematica. Our results draw on three of the surveys:
• Student survey at start of term: We asked students how calculators and/or computer algebra systems (CAS) were used in their last high school mathematics class and how comfortable they are in using these technologies.
• Student survey at end of term: We asked students how calculators or CAS had been used both in class and for out of class assignments.
• Instructor survey at start of term: We asked instructors what technologies would be allowed on examinations and which would be required on examinations.
Our first question asked students how calculators were used on exams in their last high school mathematics class (see Figure 1). As in previous columns, “research” refers to the responses of students taking Calculus I at research universities (highest degree in mathematics is doctorate), “undergrad” refers to undergraduate colleges (highest degree is bachelor’s), “masters” to masters universities (highest degree is masters), and “two-year” to two-year colleges (highest degree is associate’s). Figure 1. GC = graphing calculator. CAS = graphing calculator with computer algebra system capabilities (e.g. TI-89 or TI-92).
There are several interesting observations to be made from this graph. First, not surprisingly, almost all Calculus I students reported having used graphing calculators on their exams at least some of the time (“always” and “sometimes” were mutually exclusive options). Second, there is a difference by type of institution. Students at undergraduate colleges were most likely to have used graphing calculators on high school exams (94%), then those at research universities (91%), then masters universities (86%), and finally two-year colleges (77%). The differences are small but statistically significant. My best guess is that these are reflections of the economic background of these students. A second observation is that for most students, access to a graphing calculator was not always allowed. However, it is still common practice in high schools (roughly one-third of all students) to always allow students to use graphing calculators on mathematics exams.

Another striking observation from Figure 1 is that the percentage of students who were always allowed to use graphing calculators on exams is almost identical to the percentage of students who were always allowed to use graphing calculators with CAS capabilities on exams. For all categories of students, over half of them were allowed to use graphing calculators with CAS capabilities at least some of the time, which suggests that over half of the students in college Calculus I own or have had access to such calculators.

The next graph (Figure 2) shows how students at the start of the term reported their comfort level with using graphing calculators or computer algebra systems (Maple and Mathematica were provided as examples of what we meant). The most interesting feature of this graph is that students at two-year colleges are much more likely to be comfortable with Maple or Mathematica than those at four-year programs. I suspect that the reason behind this is that most Calculus I students at two-year colleges are sophomores who took pre-calculus at that college the year before. This gave them more opportunity to experience these computer algebra systems. Figure 2. Student attitude toward use of graphing calculator or CAS on a computer such as Maple or Mathematica.
The graphs in Figures 3–5 show what students reported at the end of the term about use of technology. For the graph in Figure 3, students were asked how frequently each of these occurred in class. Percentage shows the fraction of students who responded “about half the class sessions,” “most class sessions,” or “every class session.” We note large differences in instructor use of technology generally (for this question, “technology” was not defined), and especially sharp differences for instructor use of graphing calculators or CAS (with Maple and Mathematica given as examples). It is interesting that students are most likely to encounter computer algebra systems in undergraduate and two-year colleges, much less likely in masters and research universities. Figure 3. End of term student reports on frequency of use of technology (at least once/month). For this question, CAS refers to a computer algebra system on a computer, such as Maple or Mathematica.
The first two sets of bars in Figure 4 show student responses to “Does your calculator find the symbolic derivative of a function?” The first set gives the percentage responding “N/A, I do not use a calculator.” The second set displays the percentage responding “yes.” Looking at the complement of these two responses, we see that across all types of institutions, roughly 50% of students taking Calculus I own a graphing calculator without CAS capabilities. The third set records the percentage responding “yes” to the question, “Were you allowed to use a graphing calculator during your exams?” Note that there are some discrepancies between what students and instructors report about allowing graphing calculators on exams (Figures 4 and 6), but the basic pattern that graphing calculators are allowed far less frequently at research universities than at other types of institutions is consistently demonstrated. Figure 4. End of term student reports on calculator use. No calculator = do not use a calculator. Calculator with CAS = use a calculator with CAS capabilities. Calc allowed on exams = graphing calculators were allowed on exams.
We also asked how often “The assignments completed outside of class time required that I use technology to understand ideas.” Again, we see much less use of technology at research universities, the greatest use at undergraduate and two-year colleges. Figure 5. Frequency with which technology (either graphing calculators or computers) was used for out of class assignments. Almost never = less than once per month (includes never). Sometimes = at least once per month but less than once per week. Often = at least once per week.
The last two graphs (Figures 6 and 7) are taken from the instructor responses at the start of the term: what technology they would allow on their exams and what technology they would require on their exams. Again, we see a clear indication that technology, especially the use of graphing calculators without CAS capabilities, is much less common at research universities than other types of institutions.

It is interesting to observe that there are large numbers of instructors who allow but do not require technology on the exams. At research universities, 26% require the use of some kind of technology, and a further 25% allow but do not require the use of some sort of technology. For undergraduate colleges, 38% of instructors require technology, an additional 42% allow it. At masters universities, 42% require, and a further 33% allow. At two-year colleges, 52% require, and an additional 36% allow. Figure 6. Start of term report by instructor of intended use of technology on exams. GC = graphing calculator. Most of those who checked “other” reported that they allowed graphing calculators on some but not all parts of the exam. Some reported allowing only scientific calculators. Figure 7. Start of term report by instructor of intended use of technology on exams. GC = graphing calculator. Most of those who checked “other” reported that they required graphing calculators on some but not all parts of the exam. Some reported requiring only scientific calculators.
We see a pattern of very heavy use of graphing calculators in high schools, driven, no doubt, by the fact that students are expected to use them for certain sections of the Advanced Placement Calculus exams. They are still the dominant technology at colleges and universities, but there the use is as likely to be voluntary as required. This implies that in many colleges and universities questions and assignments are posed in such a way that graphing calculators confer little or no advantage. The use of graphing calculators at the post-secondary level varies tremendously by type of institution. Yet even at the research universities, over half the instructors allow the use of graphing calculators for at least some portions of their exams.

The MAA national study of calculus, Characteristics of Successful Programs in College Calculus, is funded by NSF grant no. 0910240. The opinions expressed in this column do not necessarily reflect those of the National Science Foundation.