Towards student instrumentation of computer-based algebra systems ...

International Journal of Mathematical Education in Science and Technology, Vol. 36, No. 7, 2005, 741–749

Towards student instrumentation of computer-based algebra systems in university courses SEPIDEH STEWARTy, MICHAEL O. J. THOMAS*y and JOHN HANNAHz yMathematics Education Unit, Department of Mathematics, The University of Auckland, PB 92019, Auckland, New Zealand zMathematics and Statistics Department, University of Canterbury, Private Bag 4800, Christchurch 1, New Zealand (Received 2 May 2005) There are many perceived benefits of using technology, such as computer algebra systems, in undergraduate mathematics courses. However, attaining these benefits sometimes proves elusive. Some of the key variables are the teaching approach and the student instrumentation of the technology. This paper considers the instrumentation of computer-based algebra systems (CAS) by first- and secondyear university students. The responses to a survey of the classes showed that instrumentation was more advanced for the second year students. The progress of each group is described and a number of student categories are outlined, in terms of their attitude to the computer and their instrumentation level.

1. Background Thomas and Holton [1] give numerous examples to show that many educators have seen benefits from using CAS technology in learning, and have examined areas where they might be used, and how they could be useful. For example, the calculus reform programme, which began in 1986, had as two of its goals the encouragement of problem solving skills and the integration of computer technology. While research suggests that there have been considerable overall benefits from such programmes [2, 3], it is also clear that there is a difference between evaluating programmes and evaluating student learning. It is also true that use of technology outside of calculus has been slower to take hold in universities. While potential is one thing, finding a route to CAS benefits often proves to be quite another. It is easy to identify several key relationships influencing the manner in which the technology is used, and hence the beneficial outcomes for learning. These are the personal perspective of the teacher (lecturer, etc) on the technology, the relationship between the teacher and the students, and the individual student’s instrumentation of the technological tool [4]. The last of these three is important since the students have to be able, not only to see how the technology might be useful in solving mathematical problems, but also have

*Corresponding author. Email: [email protected] International Journal of Mathematical Education in Science and Technology ISSN 0020–739X print/ISSN 1464–5211 online # 2005 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/00207390500271651

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the skill and ability to utilise it to do so. In this paper we are primarily concerned with instrumentation, but recognize the importance of each of the relationships to a successful outcome, and the fact they are closely inter-related. For example, the students’ relationship with the technology is influenced by the decisions teachers make. Kendal and Stacey [5] demonstrated this when they investigated the teaching of differential calculus using CAS. They found that teacher privileging influenced student learning. When teachers gave greater emphasis to certain aspects of learning, evidenced by personal teaching styles and attitudes, it resulted in students acquiring similar preferences. The student’s relationship with CAS technology can be a complex one. For example, Goos et al. [6] describe a hierarchy of student– technology interactions culminating in use of technology as an extension of oneself, integrated into mathematical working. However, they also report [7] some student resistance to movement through this hierarchy. The analysis of the transformation of an artefact into an instrument by Rabardel and others [4, 8] has been applied to tools, such as CAS, employed in learning mathematics. In order to gain the most learning benefits from a tool, the student has to be able to adapt it to the particular mathematical task at hand. It is through the actions and decisions that the student makes that the CAS tool or artefact is transformed into an instrument. In this process of instrumental genesis, the student considers what the tool can do and how it might do it. Such instrumental genesis comprises two phases, instrumentalization and instrumentation. In the first the subject adapts the tool to himself while in the second he adapts himself (through his social environment) to the tool. While the teacher may assist by providing an appropriate learning environment, each student has to work out the role that CAS will play in their learning for themselves. This involves deciding what CAS might be useful for, what will be better done by hand, and how to integrate the two. As part of this process they will need to address a number of factors, such as the differences between mathematical and CAS functioning [9]. Part of the complexity of the process of instrumentation of CAS is the need for explicit attention to both the technical and the conceptual aspects of instrumentation [10], and the nature of student interactions with the algebraic, graphical, and matrix representations, etc. [11]. Here again the role of the teacher is crucial, since to encourage conceptual uses of technology they must see it as more than a mere computational or procedural tool; it must be viewed as a teaching and learning one. In the research described in this paper we have considered student instrumentation of CAS programs during study in university mathematics courses. We were interested in two main research questions: what is the nature of the progress of university students’ instrumentation, and what are some of the factors influencing it?; and to what extent does the instrumentation of experienced computer users differ from that of new users? Preliminary answers to these questions that we have identified are described below.

2. Method The research comprised a case study of the first and second year science and engineering majors studying mathematics at the University of Canterbury, New Zealand. In each of these years students have alternating tutorials and computer laboratory sessions as an integral part of their course, the latter employing Maple in the first year and Matlab in the second. The first year course (MATH 105) covers

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calculus and an introduction to linear algebra (matrix algebra and systems of linear equations: Gaussian elimination, row echelon form, invertibility of matrices, and determinants), and the second year course (MATH 254) is an advanced linear algebra course, adding vector spaces (linear independence, rank and basis, linear transformations), inner product spaces (norms, projections) and eigenvectors to the linear algebra, and is restricted to students with Bþ (or better) grades in the first year. While the computer work was primarily procedural, it did not just aim to solve a particular problem (which could often be done with one command), but also to help students with the steps they should take to solve it. An example of a problem was to use determinants to show that a given a system of linear equations (with some coefficients in terms of k) had only a trivial solution, regardless of what real value k took. The students were given a questionnaire comprising section A, which examined their general attitude to the use of computers in learning mathematics, using a three point Likert response scale. Section B contained seven open questions specific to the use of computers in the course they had taken, and the final sections of the questionnaire gave further opportunity for open responses on the value of the computers for specific content, and other relevant comments (see figure 1 – some formatting has been changed). The vast majority of the students took the questionnaire seriously and responded with good, pertinent comments, and we received a total of 252 first year, and 88 second year responses, from enrolments of 712 and 108, respectively. The ratio of male : female was 3 : 1 for each, and in the first year there were 136 Europeans and 85 Asians, and in the second year 33 Europeans and 49 Asians.

3. Results and discussion In the construction of our questionnaire we had determined to focus on both the student attitudes to the computer, and their instrumentation of the tool. We were aware of the specific categories of use that Thomas and Hong [12] had identified for beginning university students using calculator-based CAS for the first time, namely: performing a direct, straightforward procedure; checking of procedural by-hand work; performing a direct complex procedure, for ease of use, or because the procedure is too difficult by hand; performing a procedure within a more complex process, possibly to reduce cognitive load; and investigating a conceptual idea. Hence our analysis of our data was structured towards identification of similar types of use, and the reasons behind them. 3.1. Attitudes to computer use Table 1 gives a summary of the results of the attitude test for the students from the two years. The first thing that one notices from the data is that there is a striking similarity in the level of responses to many of the questions. In both years the subscale on attitude to computers (1, 2, 8, 9, 10, 14, 18, 20) showed that the students wanted to learn more about computers and wanted to use their knowledge in further courses. They did not believe that the CAS had adversely affected their by-hand skills. In the teaching provision subscale (11, 19, 21, 22), both groups were relatively neutral on the value of the computer labs and the support provided outside them, although both tended to agree that the tutor support was good.

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S. Stewart et al. Section A: For each statement below please circle the number which most closely corresponds to your own view.

1. Computers do not improve my understanding of mathematics problems. 2. I waste a lot of time using the computer. 3. Computers help me to visualise the problems. 4. I would prefer to use the computers to do the calculations for me so I can concentrate on understanding the concepts of the course. 5. I can solve problems using computers even though I don’t understand the theory. 6. My answers are usually different from the answers that the computer gives me. 7. I often check my answers using the computer. 8. I would like to learn more about the computers, so I can use them fully. 9. I believe technology is the way to go to learn mathematics. 10. I hope to use my knowledge in computing in other courses when applicable. 11. My tutors are very supportive and encouraging in using the computer software. 12. I explore the computer by myself to learn more. 13. I find it difficult to decide when to use the computer in doing mathematics problems. 14. Since I have been using the computer, I have forgotten how to do the basic skills. 15. I like to use both computer and pen and paper when working on problems. 16. I only use the computer when I am stuck using pen and paper for solving problems. 17. I find all the commands and instructions too difficult to remember. 18. Computers make mathematics fun. 19. There is not enough support outside lecture time for using the computers. 20. I believe the computer gives me an unfair advantage in learning mathematics. 21. I think our computer labs are very helpful, and I enjoy going to the lab. 22. I would like to see the lecturers use computers in lectures.

A 3 3 3 3

N 2 2 2 2

D 1 1 1 1

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Section C: Specific Questions Section B: Open Questions • Which of the following topics do you find using the • What do you like using the computer software for? (Why?) computers most helpful ? Tick as manythat are • How do you feel about using the computers this applicable (please briefly explain why): year? Linear Equations Matrix Algebra • Should the computers be used in the linear algebra Vector geometry Determinants lectures? If so, how? [Different appropriate list for MATH 254] • How do you decide when to use the computer software? Why were they helpful for this? • Has the computer software helped you to learn any linear algebra? If so, what? Section D: • How much do you feel you rely on the computer? Comments: (Please write anything else you consider For example, could you still do the problems without relevant to the use of computers.) having one? • Do you just try to apply the applications of the computer from what is given in the lecture material or do you explore for yourself?

Figure 1.

Table 1.

The questionnaire.

The means of the questionnaire responses (Maple (Y1) and Matlab (Y2)).

Statement

1

2

3

4

5

6

7

8

9

10

11

Mean (Y1) Mean (Y2) Statement Mean (Y1) Mean (Y2)

1.90 1.90 12 2.10 2.13

2.05 2.00 13 1.85 2.02

2.22 2.51 14 1.53 1.63

2.02 2.43 15 2.27 2.59

2.41 2.28 16 1.97 2.21

1.70 1.69 17 2.12 2.34

1.82 1.78 18 1.89 2.03

2.30 2.54 19 2.14 2.14

2.00 2.10 20 2.18 2.27

2.52 2.67 21 2.17 2.06

2.20 2.34 22 1.74 2.20

The difference between the groups concerned their attitude to the use of computers in lectures. None of the students had been exposed to such use, since all computer work was in a laboratory situation, however, the first year students were significantly less positive towards lecturers using computers (t ¼ 4.72, p < 0.00001).

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This difference was confirmed by the responses to the third section B open question, ‘Should the computers be used in the linear algebra lectures? If so, how?’ Here 113 (52.8%) of the first year students said ‘No’, they should never be used. In contrast 44 (50%) of the second year students said that they should, and a further 11 (12.5%) thought they could sometimes be used. Again there was a significantly higher proportion (2 ¼ 36.5, p < 0.001) of the second year students in favour of computer use in lectures. Some of the reasons given for these contrasting responses may help to identify the reason for these disparate opinions. For example, the MATH 105 students (and one/two MATH 254 ones) commented that: No, it would take too much time. Computers in lectures are prone to malfunction. Linear algebra lectures are fine just using the blackboard and overhead projectors. No, it takes the focus away from the actual mathematics-gets caught up in learning commands etc. No the lectures are for the understanding of the operations, plugging and chugging on a computer can never help my understanding. However, the second year students (and occasionally first year ones too) commented: Yes, because it would make it a lot more interesting and able to see concepts graphically. Yes, because it is always easier to visualise things and helps understanding the topic. Yes, some topics are quite hard to understand if lecture can use computer to draw some pictures or do the modelling, it might be easier. There seems to be a shift here away from understanding as the sole domain of the blackboard explanation, towards the idea that the computer, with its visual capability, can assist with understanding. We do not know what has caused the change, but it may be a factor of the increased exposure to computers of the second year students, the greater difficulty of the mathematics, and especially the need to visualise in linear algebra, or the difference in the software used. Certainly we did get some comments about the difference between Maple and Matlab, but generally it may be that the second year students preferred the former. Of the 22 students who made a meaningful comment, 6 directly stated that they preferred Maple to Matlab since it was simpler – ‘I prefer the more simple Maple to Matlab, which is more complicated to run,’ and easier to use – ‘Matlab is not a very good program because it’s very, very hard to use. (commands are too complicated)’ A further eight students cited problems with Matlab, such as its failure to be ‘user-friendly’. In contrast none of the first year students made a negative comment about Maple. It appears that the steeper learning curve for Matlab, coupled with the increased demands of the content, is causing this direct, negative comparison with Maple, used the previous year. Maple is a reasonably friendly worksheet environment combining text, commands and pictures all in one place, while Matlab is really a programming language where the commands, pictures and numerical output all appear in different places. Students, even these better performing ones, tend to struggle with the discipline of programming, which needs forward thinking and planning, and this makes Maple, with its single input commands, seem easier.

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3.2. Instrumentation of the CAS In the instrumentation subscale (3, 4, 5, 6, 7, 12, 13, 15, 16, 17) of the attitude test, both year groups slightly agreed that they could solve problems with CAS without understanding the theory, but disagreed that their answers differed from the CAS and that they often checked answers with it. They were neutral about using the computer to explore and on the difficulty of deciding when to use the CAS. Regarding their instrumentation, the MATH 254 students were significantly more inclined: to use the computer to visualise (t ¼ 3.34, p < 0.005); to want to use CAS to free-up time for thinking about concepts (t ¼ 4.66, p < 0.00001); to like using both by-hand and CAS on problems (t ¼ 3.89, p < 0.0005); to use CAS only when stuck (t ¼ 2.49, p < 0.05); and to find computer commands, etc. difficult to remember (t ¼ 2.49, p < 0.05). Following their secondary school research project on the use of CAS for understanding functions, Weigand and Weller [13] described a working style with true integration of both methods of working as ‘quite rare’. One possible reason, identified by Artigue [14], is that by-hand techniques often have an elevated status compared with CAS instrumented techniques, which may lack mathematical status in the eyes of students and teachers. However, it seems that the MATH 254 students were making greater movement towards such a style of working than the MATH 105 students. This may be due to greater experience with computers, to their ability level, or to the teaching approach. The first and fourth section B questions in particular, along with section C, were included to examine in detail the types of computer use, the reasons for them, and how the decision was made to use CAS. Table 2 lists the main reasons given by each year group for deciding when to use the CAS. Only two students in each year mentioned using CAS to assist with understanding. The main differences between the two year groups seem to be a decrease in those not using the CAS, or using it when stuck on a problem (2 ¼ 0.34, ns), and a corresponding increase in use when the problem is perceived as being too tedious to do by hand (2 ¼ 32.7, p < 0.001) or too complex (2 ¼ 2.80, ns). Thus the second year students seem to be a little better able to do the problems, but less inclined to work through them by hand, preferring to use the CAS to do long or difficult calculations. Using the categories of Thomas and Hong [12] we are not able to say whether such use is due to a straightforward complex procedure, or one embedded within a more complex problem. Responses to the question of what the students liked using the computer for are summarised in table 3. To the categories of Thomas and Hong [12] we add that of visualisation, which, when it includes the very specific graphing category seen in Table 3, becomes a major form of use.

Table 2. Reason Problem too long/tedious Problem too complex When stuck To visualise

Reasons given for deciding when to use CAS. % Year 1 % Year 2 (N ¼ 252) (N ¼ 88) 2.8 6.0 17.5 6.3

21.6 11.4 14.8 4.5

Reason Told to use To check answers Unsure of method Never use

% Year 1 % Year 2 (N ¼ 252) (N ¼ 88) 13.1 4.4 3.2 6.0

NB where students gave more than one reason all these reasons were counted.

12.5 9.1 5.7 0.0

Student instrumentation of computer-based algebra systems Table 3. Use Graphs Complex questions To save time Checking answers

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What students like to use computer-based CAS for.

% Year 1 (N ¼ 252)

% Year 2 (N ¼ 88)

Use

% Year 1 (N ¼ 252)

% Year 2 (N ¼ 88)

15.5 11.1 7.9 11.9

12.5 18.2 22.7 8.0

When stuck Specific content To visualise Nothing/No use

1.6 2.4 3.6 7.1

0.0 2.3 3.4 1.1

This was confirmed by a number of comments from both years in section C, such as ‘Norms and projection: because they are harder to imagine about (compared to other parts). So there is a need for its visualisation’, ‘Eigenvalues and vectors: computer provides examples of eigenvectors and gives pictures of them. It was quite helpful’, ‘You get an image of the vectors. It helps to understand the geometric meaning’, and ‘It’s easy to see the surfaces, and see what happens to them when you change stuff ’. The only significant difference in usage between the year groups was in saving of time, which became a more important reason for using the CAS for the second year students (2 ¼ 13.7, p < 0.001), possibly due to increased time pressures in that year’s study. There is also a significant drop in the number displaying anti-computer sentiment (2 ¼ 4.46, p < 0.05), although the numbers are too small for the result to be reliable. The sixth section B question addressed the issue of reliance on the technology; did the students think that they could still do the questions without the CAS. In both cases the answer was a ‘yes’, with 65.0% of the first year and 62.5% of the second year group confident that they could, and a further 10.3 and 14.8%, respectively, fairly confident. 3.3. Categories of students We believe that, from our data, we can confirm the identity of some primary types or classes of students regarding the instrumentation of computers in university mathematics (these may generalize to school mathematics), and some of their identifying features, as indicated by the following composite portraits developed from student comments from both years. First we have the student who is openly opposed to computer work, believing strongly in the superiority of by-hand working for doing and understanding mathematics. I’d rather do with my pen . . . I do them faster by pen than computer. Sorry, I don’t think computer software can help me learn . . . I don’t rely on the computer. If you can do it in your head I really believe you should not be using computers. You improve your mathematical skills by attempting them, Not by learning answers from computers. I only use Maple when I have to. Otherwise I’ll use my brain and pen and paper. The second type of student is one who is happy to use the computer, but has limited instrumentation, seeing its value only as a tool for checking answers already found, to avoid careless errors, or to take the tedium out of by-hand work.

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I know I can do them but they get so tedious when doing a lot . . . it’s so easy to make a little mistake. Most of the time using hand to solve the problem . . . [computers were useful] because I did mistakes quite often. I could do the answers on my own . . . [computers were useful] so that I could check the answers that I have with the computer. A third category is the student who tends to rely too much on the computer and would find the mathematics harder without it as a support. Hong et al. [15] found that such students tend to be weaker at mathematics and perform worse without the CAS support. [I use the computer] when stuck using pen and paper . . . [it] helped solve them quickly . . . if I didn’t quite understand the theory I could still get an answer. I often have the computer running when solving problems . . . Rely on [computer] a lot. My maths ability would drop considerably without one. I cannot do any of my assignments without computer . . . Strongly rely. A final group is those that are positive towards computer use and are trying different methods, including applications to conceptual understanding, to integrate it into their by-hand working. It is this group that is making the most progress towards instrumentation of the CAS. I . . . work it out using Maple then try and work out how Maple finds solution. [I use the computer] to find the answer and then see if I can achieve that answer. Helps in learning the basic method . . . without being distracted by the calculations in between. [I use the computer for] understanding concepts . . . helped explain concepts. [It] helped show some gradual steps that needed to be taken. [I use the computer for} . . . shedding light on possible next steps in problems . . . to make the parts they skip understandable. If we consider the application of CAS to the learning of mathematics at university to be desirable, then a difficulty that remains is how we move students toward instrumentation. Goos et al. [7] report that such movement has proven difficult. It may be that Leigh-Lancaster’s [16] idea that true integration of technology necessitates a congruency between curriculum, pedagogy and assessment, where CAS is specified in the curriculum, actively used in the teaching, and expected in assessment, including examinations, is a way forward. Our evidence is that it also takes time for students to make progress with instrumentation, and the categories that we have identified above may assist in clarifying the instrumentation needs of some specific groups of students.

References [1] Thomas, M.O.J. and Holton, D., 2003, Technology as a tool for teaching undergraduate mathematics, In: A.J. Bishop, M.A. Clements, C. Keitel, J. Kilpatrick and F.K.S. Leung (Eds) Second International Handbook of Mathematics Education, Vol. 1, pp. 347–390 (Dordrecht: Kluwer).

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[2] Ganter, S.L., 1999, An evaluation of calculus reform: a preliminary report of a national study, In: B. Gold, S. Keith and W. Marion (Eds) Assessment Practices in Undergraduate Mathematics (Washington, DC: The Math. Assoc. of America), pp. 233–236. [3] West, R., 1999, Evaluating the effects of reform, In: B. Gold, S. Keith and W. Marion (Eds) Assessment Practices in Undergraduate Mathematics (Washington, DC: The Math. Assoc. of America), pp. 219–223. [4] Rabardel, P., 1995, Les hommes et les technologies, approche cognitive des instruments contemporains (Paris: Armand Colin). [5] Kendal, M. and Stacey, K., 1999, Varieties of teacher privileging for teaching calculus with computer algebra systems. The International Journal of Computer Algebra in Mathematics Education, 6(4), 233–247. [6] Goos, M., Galbraith, P., Renshaw, P. and Geiger, V., 2000, Reshaping teacher and student roles in technology-enriched classrooms. Mathematics Education Research Journal, 12, 303–320. [7] Goos, M., Galbraith, P. and Geiger, V. and Renshaw, P., 2001, Integrating technology in mathematics learning: what some students say, Proceedings of the 24th Mathematics Education Research Group of Australasia Conference, Sydney, pp. 225–232. [8] Ve´rillon, P. and Rabardel, P., 1995, Cognition and artifacts: a contribution to the study of thought in relation to instrumented activity. European Journal of Psychology of Education, 10(1), 77–101. [9] Thomas, M.O.J., Monaghan, J. and Pierce, R., 2004, Computer algebra systems and algebra: aurriculum, assessment, teaching, and learning, In: K. Stacey, H. Chick and M. Kendal (Eds) The Teaching and Learning of Algebra: The 12th ICMI study (Norwood, MA: Kluwer Academic Publishers), pp. 155–186. [10] Drijvers, P. and van Herwaarden, O., 2000, Instrumentation of ICT tolls: the case of algebra in a computer algebra environment. The International Journal of Computer Algebra in Mathematics Education, 7, 255–275. [11] Thomas, M.O.J. and Hong, Y.Y., 2001, Representations as conceptual tools: process and structural perspectives, Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, Utrecht, The Netherlands, 4, pp. 257–264. [12] Thomas, M.O.J. and Hong, Y.Y., 2004, Integrating CAS calculators into mathematics learning: issues of partnership, Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, Bergen, Norway, 4, pp. 297–304. [13] Weigand, H.-G. and Weller, H., 2001, Changes of working styles in a computer algebra environment—The case of functions. International Journal of Computers for Mathematics Learning, 6, 87–111. [14] Artigue, M., 2002, Learning mathematics in a CAS environment: the genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematics Learning, 7, 245–274. [15] Hong, Y.Y., Thomas, M.O.J. and Kiernan, C., 2000, Super-calculators and university entrance calculus examinations, In: M. Chinnappan and O.J. Thomas (Eds) Mathematics Education Research Journal Special Issue, 12(3), (MERGA) pp. 321–336. [16] Leigh-Lancaster, D., 2000, Curriculum and assessment congruence-Computer algebra systems (CAS) in Victoria. http://www.math.ohiostate.edu/
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