Reflections on the Changing Pedagogical use of Computer Algebra Systems: Assistance for Doing or Learning Mathematics?

STACEY, KAYE

This article documents a change in the use of a Computer Algebra System, (CAS), with a group of first year, undergraduate, mathematics students. CAS was initially used as an assistant for doing mathematics, enabling students to solve difficult problems. During the period of the study it came to be used as an assistant for learning mathematics, as a partner in the teaching and learning process. This article notes the changes required in organisation, teaching materials, and assessment, then reflects on changes in students' attitudes and learning outcomes.

Surveys, interviews and teacher observations suggested that students' attitudes toward the use of CAS for learning mathematics were positive and that they believed that it aided their understanding. Students appreciated the availability of CAS for examinations. There was no demonstrable change in student achievement resulting from the changed pedagogical use of CAS. However changes in learning goals and assessment procedures mean that no simple comparison is possible.

THE IMPETUS FOR CHANGE

CAS are powerful computer packages that do calculations, offer graphical representations, and also perform symbolic manipulation for algebra and calculus. These tools were initially designed by mathematicians to increase their efficiency in solving problems that could be described by algorithms.

GAS have been available for over a decade but so far they have made little impact on the teaching of mathematics. However, as affordable access to technology increases, the use of these packages as powerful tools for students learning mathematics is becoming a reality. In addition, the user interfaces for computer algebra systems are being refined so that novice users are less hampered by the syntax of the programs. The GAS DERIVE was promoted on its user manual as "a mathematical assistant for your personal computer." Its first use was in assisting mathematicians to do mathematics but more recently its role as an assistant for learning mathematics has also been explored (Berry & Monaghan, 1997).

The introduction of new technologies to the teaching and learning of mathematics has been at two levels "functional" and "pedagogical" (Etlinger, 1974). Etlinger reported that when calculators were initially used in primary and secondary classes most teachers saw their major use as allowing students to consider more real life examples and to check answers. The purpose of allowing calculator use was to improve attitudes and increase student motivation to learn mathematics by increasing its relevance through greater use of applied examples. Very quickly the role of the calculator in the mathematics classroom expanded as teachers found ways in which it could be used not only to do calculations but also to assist students as they learned the mathematics. It is instructive to revisit the journals of the mid-1970s and read the excitement and concern of mathematics educators over the role which simple calculators might play in mathematics education.

Etlinger (1974) wrote about calculators but if the word calculator were replaced with GAS the paper could well have been written 20 years later! He wrote that:

Perhaps the most extreme view is that of the calculator as a purely functional classroom device...According to this view, the calculator will allow us to perform calculations much more easily, and will save us the trouble of learning the older, more tedious methods, much as the ballpoint pen saves us the inconvenience of inkwells and blotters...The pure-pedagogical point of view goes basically as follows: the calculator must not be used to replace learning, but rather to facilitate learning. Children must still learn their facts and their algorithms together with the more abstract concepts and ideas of mathematics. (p. 43-44)

Etlinger (1974) commented that the implications of this pedagogical view had not been clarified. He raised a number of questions about the effect of calculators on student attitudes, hand skills, and understanding of concepts. While scientific and graphics calculators have been accepted in school classrooms the use of the next level of mathematical technology, symbolic manipulators, is only now being explored. Similar questions are being asked of GAS today.

In 1996 a review of the literature on the use of GAS in undergraduate mathematics suggested that not only could GAS be used to do mathematics but it might also help students learn quickly and increase their understanding of mathematical concepts. Heid (1988) reported that for an introductory calculus course a group taught using GAS to present and explore concepts produced similar results to a comparison group on a skills test and showed greater understanding of concepts and the ability to use different representations. Palmiter (1991) reported that an experimental GAS group covered the same course as a traditional group with fewer hours teaching and that these students outperformed the traditional group on both conceptual and computational exams. Day (1993) encouraged us that "the power and flexibility of technology can help change the focus of school algebra from students becoming mediocre manipulators to their becoming accomplished analysts" (p. 30).

Previous experience (Heid, 1988, Palmiter, 1991, Mayes, 1993, Llorens-Fuster, 1995, Heid & Zbiek, 1995, Bennett, 1995, Taylor, 1995; Hirschom &Thompson, 1996) suggested that a GAS could facilitate student's learning by:

* exposure to lots of examples and nonexamples in a short space of time;

* encouraging detailed observation and conjectures;

* development of rules by induction;

* exposure to multiple representations, graphical, numeric and algebraic;

* improving attitudes and class attendance; and

* reducing the anxiety of "making mistakes."

The Setting

At the University of Ballarat, GAS have been used since 1990 in undergraduate mathematics courses. "Maple" has been used by engineering students and DERIVE by all other mathematics students. While the number of students undertaking mathematics courses at the University is not large, their backgrounds are diverse. Ballarat is a regional University that has a large number of educationally disadvantaged students. In cross faculty subjects like mathematics, these students study along side academically successful students who choose the specialist courses offered at this University. Providing teaching and learning experiences which cater for such diverse groups and which enable those students with weak mathematics backgrounds to reach an appropriate level of skill and understanding presents a challenge.

Overview

Initially GAS were used for demonstrating ideas in lectures by aiding visualisation and in 1992 laboratory sessions were introduced. These sessions were based on worksheets which helped students learn to use DERIVE and required them to use the CAS in solving extended problems and harder examples (Yearwood & Glover, 1995). Until 1997 students had three hours of lectures and tutorials for traditional mathematics and one hour in a computer laboratory. Assessment was undertaken in separate parts both with and without GAS. In 1997 the use of GAS was broadened. Students were encouraged to see it as a tool to help them learn and understand mathematics not just assist them with difficult problems. To facilitate this, GAS was made available for all classes and all assessment tasks.

This article examines the experience of the transition years, 1995-1998, and looks at the changes required in organisation and teaching. In 1995/96 students learned mathematics "by hand" and learned to use DERIVE for mathematics. In 1997/98 students also used DERIVE to help them learn mathematics. A study was undertaken to monitor student's responses to the changes in teaching and assessment. Data was collected using surveys, interviews, and anecdotes (Table 1). Changes in student's attitudes, participation in learning, and learning outcomes were noted. Special attention was given to the changes in student's answers to examination questions when GAS was made available during the assessment. Examination scripts were carefully analysed question by question and examination results were recorded. Table 1 summarises the total number of students from whom data was collected each year, the nature of the data collected, and the number of responses.

TEACHING AND ORGANISATION 1995-1998

In 1995/96 the use of CAS to help students tackle more difficult and extended problems was essentially an "add-on" to a traditional first year undergraduate mathematics course. Incorporating the use of CAS into the total teaching and learning of the unit in 1997/98 required changes in both teaching and organisation, key aspects are outlined below.

Constants: Curriculum and Staffing

The unit under review was a standard first year, undergraduate, mathematics unit with a focus on calculus, linear algebra, and probability. During the period of the study the course outline was unchanged but four different staff taught it. These staff all demonstrated good professional practice.

Changes in Delivery Mode

In 1995/96 three hours of lecture/tutorials were timetabled for students to be taught mathematics. During a fourth hour--timetabled in a computer laboratory--students were given worksheets to assist them in learning CAS and solving harder problems. Technical assistance was provided by postgraduate students who saw their role as "trouble shooting" DERIVE rather than teaching mathematics.

In 1997/98, to facilitate the use of CAS as a learning tool, the lecturer conducted all classes and used the computer laboratory whenever this enhanced the overall teaching plan, typically two hours per week. In the computer laboratories students were directed by worksheets but worked at their own pace with the lecturer acting as a facilitator. The lecturer kept the focus on mathematics in all classes.

Changes in Worksheets

In 1995/96 worksheets were written to teach students how to use DERIVE to do the mathematics they had covered in lectures. The exercises used complicated examples which illustrated principles that had been taught. They focused on how to use the technology for problems which could prove difficult for students to solve by hand.

For example the exercises on derivatives were based on the function

F(x) = sin x/x.

In an attempt to broaden students learning experience in 1997/98 worksheets were written to enable students to use GAS to explore concepts and patterns of mathematics as well as to use it as a tool to solve more difficult examples. Explorations were used to introduce topics as well as consolidate concepts. The style of worksheets changed. GAS commands, in the new worksheets, were introduced using exercises, which would either revise assumed mathematics knowledge, or provide exploratory inductive activities. Exercises on derivatives were based on many simple polynomials, for example [x.sup.2], [x.sup.3], [x.sup.-2], [x.sup.-5], [x.sup.3/2] [x.sup.4] + [2x.sup.2]+1. Students were expected to inductively develop a rule for the derivative of a polynomial. They were encouraged to make notes on the mathematics that they were observing as they worked with GAS through guided discovery learning exercises. For example, exercises required students to explore multiple representations of rate of change. These linked grap hs of tangents to the curve with gradients and the derivative function. More advanced or complicated problems followed later once students had been introduced to both the mathematics and the required GAS commands.

Changes in Assessment

It has been often stated (Ramsden, 1992) that assessment drives students' learning, and that students will value and concentrate on those aspects of a course that they know will be assessed. Arnold (1995) observed that students who were not allowed to use GAS in their examinations used it as little as possible in their learning exercises. If GAS is to be taken seriously as a mathematics learning tool then students need to have access to it for assessment tasks.

In 1995/96 assessment comprised a midsemester test without GAS and a final examination in two parts one with and one without GAS. In 1997/98 GAS was available for the midsemester test and all the final examination. In 1995/96 the examination section for which GAS was available consisted of questions which students would find difficult to do by hand Figure 1.

In the 1997/98 examinations most of the questions did not look any more complicated than problems solved by hand in previous years Figure 2.

A key change in examination questions was the extension of assessment from students' recall of facts and by-hand skills to questions that require students to interpret and explain their answers. Of 12 final examination questions in 1996, two asked students to interpret the mathematics, whereas in 1997 this was required in six questions. For example a 1996 linear algebra question asked students to "Find the following, if they exist," while the equivalent 1997 question asked student to not only "Find the following if they exist," but also "If they don't exist explain why not." Moving beyond routine mathematical manipulations, or translating conventional mathematical words and symbols into GAS commands, to explaining working, justifying reasoning or interpreting results, required students to verbalise their mathematical thinking thus making these questions more difficult.

RESULTS: STUDENTS' RESPONSES

The results of the study are presented in two sections, first those based on student surveys, interviews, and anecdotal evidence, then second, changes in learning outcomes as measured by the examinations.

Students Viewed CAS as Useful Learning Tool

There was a change in mindset from seeing CAS as being a topic or subject, related to, but separate from, mainstream mathematics, to viewing GAS as a tool which may be used to solve mathematical problems and explore mathematical ideas.

Anecdotal comments from students and lecturers suggested that students who studied this course in 1995/1996 saw the subject as consisting of "maths and DERIVE." The phrase "I need to do some DERIVE" and reference to the "DERIVE exam" indicated that the students saw GAS as an entity separate from mathematics. This also suggested that students believed learning to use the GAS was in itself an achievement. In surveys monitoring student response to the changes in teaching and organisation in 1997/98 students were asked to indicate their reaction to statements, such as those in Figure 3, using a five point Likert scale.

More than 70% of students responded positively to the first four questions. They either agreed or strongly agreed that the use of GAS had helped them understand mathematics, it had given them confidence, and was a helpful tool for producing answers to maths problems. More than 75% disagreed with the statement "When I use DERIVE I don't think about maths." Overall, their responses implied that most students used CAS to help increase their understanding of mathematics by exploring new ideas and looking at variations on set questions.

Classroom Discussion Focused on Mathematics

Anecdotal evidence suggested that in 1995/96 students' questions and discussions in their computer laboratory sessions focused on how to use GAS. In 1997/98, using the new worksheets, the emphasis of these discussions had shifted to mathematics.

The students interviewed, in 1997, agreed that when sharing a computer to work on these exploratory exercises they usually talked about mathematics rather than social events.

Student A: In the labs we get together as a group. Something will happen on one machine and everyone will go and look at the graph or equation.

Student B: In ordinary classes we take down notes but in the labs we discuss what we are learning.

Lecturers in 1997/98 also commented that student's conversations in these laboratory sessions tended to focus on the mathematics problems. They shared conjectures about patterns in algebra and discussed key features of graphs. Lecturers commented that this contrasted with students' behaviour in the more traditional "pen and paper" tutorials where the students either worked in silence or discussed unrelated topics.

Students Liked the Changes in Assessment

Interviews and survey responses in 1997/98 showed these students all agreed that they liked having CAS available in tests and examinations. The students interviewed commented that they believed that, provided they knew what mathematics they wanted to do, using CAS saved time and helped them produce answers which were both mathematically neater and clearer in presentation. The availability of GAS meant they were more confident that they would be able to show their understanding under examination conditions.

Interviewer: One of the things that's been unusual about your situation is having open access to the computer-algebra system for exams. What's your reaction to that?

Student A: It saves a lot of time in the exam. If you've got a fairly hard problem you've got to sit down and do like 2 pages of calculations. It takes up a fair bit of time, whereas with a computer, it's just a matter of knowing what you have to do on the computer and it gives you the answer. If you want to, you can work it out step by step so it can help you either way and saves a lot of time.

Student B: It saves time and also you get tested on a wider range of questions.

Student C: My handwriting's not good so when I'm going through problems and trying to solve them, the problem gets a bit messy and I'm sure the markers struggle solving it. But I can do all the calculating on the computer and just write down the steps rather than all the of little bits in between and I'm sure that would make it much easier for the marker.

Student D: I think it adds a little bit of interest too because when you've just got a paper and you're slaving away hand writing everything, you get bored, sick of doing all the questions, but when you can use the computer every now and again it's a bit different, a bit more enjoyable.

With CAS Who or What Does the Mathematics? Students Believed They Did.

When CAS was introduced more widely into the teaching and assessment of these first year courses, some mathematics staff expressed concerns that the students would not really be "doing" mathematics. This proposition was put to students at group interview session following the release of final results in 1997.

Interviewer: One of the other things that people argue about is whether or not people are really doing mathematics when working with a computer-algebra system. Are you doing it or is the machine doing it?

Who's doing the maths?

Student A: I reckon that we are actually doing it. The computer only spits out an answer to what you type into it

Student B: It's just like with a calculator...it's just going a bit further, we're not just doing multiplication and division quickly, we're doing simple differentiations and stuff quickly.

Student C: Also, you still have to interpret the answer or for that matter interpret the question so you can convert it into what the computer wants...you're still doing a lot of mathematics.

Mixed Views on the Effect of CAS on Recall of Facts and By-Hand Skills

The staff expressed similar concerns about students maintaining their "by-hand" skills. Students interviewed in 1997 expressed mixed views about this.

Interviewer: Do you think it's had any impact one way or the other on your general facts and skills?

Student A: Sort of, it's slipped a bit...not doing everything by hand you lose a bit of the skills. Then on the other hand you've got the computer there, so if you've lost it, you can play around and find out what you've lost.

Student B: I think it actually helps me learn new things because when there are new things that I'm learning, while I'm finding them difficult, I can use DERIVE and go through the steps. With more practice and seeing DERIVE go through it, I pick it up myself and then I can feel confident doing it myself without a package.

Student C: I think it helps me. It's probably psychological as well, just knowing that you've got the computer there if you get stuck. It helps me understand the concepts of different problems. Using the package helps me understand the concepts behind the maths as well as doing the maths.

Student D: I think we've been taught the facts for 13 years now, so I suppose if you used GAS earlier on, you'd probably lose a lot of the facts, but because we've already been taught a lot of skills and details you don't really lose them.

Overview of Student Response

Overall students' responses to the incorporation of the use of GAS in teaching exercises and assessment were positive. Students felt that the worksheets, which made use of multiple representations and a range of examples, helped their understanding of mathematics and the access to CAS in examinations increased their confidence.

RESULTS: EXAMINATION RESPONSES

A key aim of the use of CAS in teaching exercises was to improve learning outcomes that were assessed by written examinations. Examination results from 1995 to 1998 were analysed. In particular, changes that may have been due to the availability of CAS were noted; the student responses for the curve sketching questions in 1996 and 1997 were looked at in detail, and the overall scores for each year were compared.

The overall impression, from comparison and error analysis, of the 1996, 1997, and 1998 examination papers, was that when CAS was available, more students attempted more parts of questions. The 1996 without CAS and 1997 with CAS examination scripts were analysed question by question. In 1996, without CAS, on average, students either did not attempt or scored zero for 42% of questions. In 1997, with CAS this figure dropped to 20%.

There was evidence that in 1997/98 with CAS available, students did not necessarily check answers with CAS. Top students still showed evidence of simple numerical and algebraic calculation errors which would have been detected had they used CAS.

CAS presented some new problems, especially for weaker students. Some students seemed confused by CAS notation especially the use of [x.sup.[conjunction]]a for [x.sup.a] or sin[(x).sup.2] instead of [sin.sup.2](x) Some students had poorly labelled sketch graphs that they had copied from CAS when they had chosen inappropriate windows or not considered the scale.

An increased emphasis in 1997/98 questions, on explaining methods or interpreting results meant a loss of marks for students who had difficulty verbalising their reasoning. Such weaknesses may not have been identified by the earlier examinations.

Comparison of Responses to Curve Sketching Questions: 1996 and 1997

The questions. Curve sketching questions were chosen for comparison since this was a topic for which the multiple representations of CAS had been used extensively in the teaching exercises. Each examination question (Figures 4 to 6) assessed the students' ability to use calculus to correctly sketch and label all salient features of a graph. Despite differences, the questions were considered to be sufficiently comparable in difficulty to enable revealing analysis. Question 1996N (Figure 4) was to be done without DERIVE, whereas Question 1996D (Figure 5) and 1997D (Figure 6) were done with DERIVE available. The 1996D question (Figure 5) presented the student with a function that looked complicated. It was the opinion of the teacher that these students would not have been able to sketch this function without the aide of GAS. This was a GAS dependent question. In Figure 6, the sentence "Use the scale x=2, y=2" from the 1997D question shows the infiltration of GAS notation into the writing of mathematics. GAS coul d be of help for 1997D but the teacher felt that these students would have been able to establish many features of this graph without the aid of technology.

Comparison of responses. Analysis of examination scripts showed changes (Table 2) in the pattern of students' answers to these questions. When compared to 1996N, assessed without GAS, in 1996D more students attempted the question and correctly located the maximum and minimum points. Without DERIVE some students only plotted a series of points. With DERIVE all students sketched smooth curves but many did not label axes and salient points or choose a suitable scale to show key features of the graph. These details could not be copied from the computer screen but required mathematical thinking to interpret the screen image. Comparison of 1996N and 1996D showed that more students were successful at sketching a graph without the aid of GAS. Most students accepted the default scale in 1996D and did their best to read off values of points from the screen. It seemed that these students were used to following directions for the use of GAS rather than thinking for themselves as to how they might best use it as a tool to explore mathematics.

The 1997D scripts indicated that the students were familiar with using GAS to examine the features of functions. Most students in the 1997 group found the turning points by using calculus and checking graphically. Eighty-eight percent used GAS to find or check f(x). Fifty-six percent wrote that they had then used DERIVE's algebra commands to locate turning points while 32% only plotted the graph. The use of calculus focused students' attention on the interval required to show the salient features of the graph. Good choice of scale and labelling of salient points enabled 75% of these students to sketch clear, correct graphs of the function (Table 2).

Comparison of Examination Scores

The aim of the changes in the teaching and organisation of this subject was to improve students' learning outcomes. A key measure of this was their examination scores. A direct comparison of the distribution of scores has been carried out (Figure 7) but the emphasis put on these results must be moderated by an understanding that the availability of CAS forced changes in the style of at least some examination questions. Many questions on previous papers were trivialised by access to a GAS but asking students to explain, justify, or interpret their mathematics required skills not assessed in the past.

A comparison of the distribution of scores for 1995/96 and 1997/98 (Figure 7) did show a slight overall improvement in results. The mean score for the 67 1995/96 examination results was 54.03 while that for the 45 1997/98 results was 60.1. A t-test for independent groups (df = 106.47, t=1.486) indicated that this increase was only statistically significant at the 0.07 level. However there was stronger evidence for the decrease in the variability of the results. This may be seen in the boxplots of the data in Figure 7. In 1995/96 the scores ranged from 4 to 97.5 with a standard deviation of 23.92. In 1997/98 no student scored less than 21, the maximum score was 95 and the standard deviation was 19.21. A Levene's test for homogeneity of variance confirms that there is a statistically significant decrease in variation (test statistic 4.284, p-value 0.04).

The decrease in variance is due to the increase in the minimum score. This is most likely a reflection of an increase in students' confidence to at least attempt examination questions with the aid of CAS and gain a few marks on most questions.

Allowing the use of CAS for all assessment resulted in more students attempting more questions but analysis of examination scripts showed that its use did not seem to mask conceptual misunderstandings or lack of effort in learning mathematical conventions and vocabulary. The required knowledge of syntax and conventions combined with an increased emphasis on students' interpreting results was an extra burden to weaker students. It was still easy to "separate" students' level of understanding as they needed to understand the mathematics in order to use CAS effectively.

CONCLUSIONS

The evolving use of technology in the teaching and learning of mathematics brings new insights and more questions. This study found that students responded positively to the use of GAS as a tool to support guided discovery exercises that encouraged inductive thinking. This reinforced the findings of previous research.

This use of CAS, for the exploration of patterns and concepts, reduced the distinction between the mathematics and the use of CAS. Rather than seeing the ability to use DERIVE as an end in itself, students saw it as a tool for doing and exploring mathematics.

During laboratory sessions the focus of learning shifted from CAS to mathematics and the discussion from social events to mathematics problems. In these classes the teacher interacted with individuals and small groups rather than the whole class and students were actively engaged in the learning process.

The availability of CAS for examinations appeared to reduce the level of test anxiety. Students believed that it gave them a better opportunity to demonstrate their understanding of mathematics. Students were more likely to attempt questions and could validate their ideas by considering alternative representations. Measuring changes in learning outcomes was difficult because the availability of GAS required changes in the wording and style of examination questions. Gomparison of results suggested that either there was some improvement in learning or that the access to GAS gave students a greater opportunity to demonstrate their knowledge. While the non attempt rate was reduced students still made procedural errors and had difficulty explaining their results.

As CAS becomes more accessible in both interface and cost, it inevitably impacts on both what is valued in mathematics and how this is taught. This study provides a piece in the picture that is being built to inform mathematics educators in their choice of when and how to make best use of GAS. There are difficulties in comparing learning outcomes, as the availability of GAS forces changes in questions and goals. What does it mean to be able to do mathematics...understand mathematics when the GAS can do the routines and get the right answer? Such fundamental issues need to be discussed because the evolving pedagogy needs to focus not only on school and undergraduate mathematics as they are now but also to consider what they will become.

References

Arnold, S. (1995). Learning to use new tools: A study of mathematical software use for the learning of algebra, Unpublished doctoral dissertation, University of New South Wales.

Bennett, G. (1995). Calculus for general education in a computer classroom, The International DERIVE Journal, 2(2), 3-11.

Berry, J., & Monaghan, J. (Eds.) (1997). The state of computer algebra in mathematics education. Sweden: Chartwell-Bratt.

Day, R. (1993). Algebra and technology, Journal of Computers in Mathematics and Science Teaching, 12(1), 29-36.

Etlinger, L. (1974). The Electronic calculator: A new trend in school mathematics. Educational Technology, December, 43-45.

Heid, K. (1988). Resequencing skills and concepts in applied calculus using the computer as a tool, Journal for Research in Mathematics Education, 19(1), 3-25.

Heid, M.K., & Zbiek, R.M. (1995). A technology-intensive approach to algebra, The Mathematics Teacher, 88(8), 650-656.

Hirschorn, D.B., & Thompson, D.R. (1996). Technology and reasoning in algebra and geometry, The Mathematics Teacher, 89(2), 138-142.

Lorens-Fuster, J. (1995). A mathematics course with DERIVE at technical colleges, The International DERIVE Journal, 2(2), 33-39.

Mayes, R.L. (1993). Computer use in algebra: And now the rest of the story, The Mathematics Teacher, 86(7), 38-541.

Palmiter, J. (1991). Effects of computer algebra systems on concept and skill acquisition in calculus, Journal for Research in Mathematics Education, 22(2), 51-156.

Ramsden, P. (1992). Learning in higher education. London: Routledge. Taylor, M. (1995). Calculators and computer algebra systems-their use in mathematics examinations, The Mathematics Gazette, 79(484), 68-73.

Yearwood, 1., & Glover, B. (1995). Computer algebra systems in teaching engineering mathematics, Australian Journal of Engineering Education, 6(1), 87-93.

Table 1
Sources of data 1995-1998
Year Data Collected Number of responses
1995 (N=28) Results: overall marks 28
1996 (N=45) Testlexam scripts analysed: 18
 Midsemester-without CAS 16
 Final exam sector: without CAS 15
 Final examination section: with 15
 Results: test, exam, overall marks 45
 Anecdotal verbal comments from
 students and lecturers
1997 (N=16) Background:
 Diagnostic test 16
 Secondary maths experience 16
 Surveys:
 Week3 13
 Week6 7
 Unit evaluation 7
 Test/exam scripts analysed:
 Midsemester-with CAS 16
 Final exam: with CAS 16
 Results: test exam overall 16
 Videoed group interview (post 4
 results)
 Anecdotal verbal comments from
 students and lecturers
1998 (N=29) Background:
 Diagnostic test 24
 Secondary maths experience 24
 Surveys:
 Midsemester review 23
 Unit evaluation 13
 Testlexam scripts analysed:
 Midsemester-with GAS 29
 Final exam: with GAS 29
 Results: test, exam overall marks 29
 Anecdotal verbal comments
 from students and lecturers
Table 2
Percentages of Students Giving Specified Responses to Curve
Sketching Questions
 1996N 1996D 1997D
 n=15 n=15 n=16
Made no attempt 18 12 0
Only plotted points 7 0 0
Correct asymptotes marked 46 N/A 62
Correctly located local maxima
and minima 25 88 88
Clear sketch graph with suitable
scale, labels and salient points
marked. 32 25 75

[Graph omitted]