Colloquium

Do undergraduate students view calculator usage as a proxy for learningwith understanding?_1289 90..92

Samuel King and Carol Robinson

Address for correspondence: Dr Samuel King, Learning Research and Development Center, University of Pittsburgh,3939 O’Hara Street Pittsburgh, PA 15232, USA. Email: soking@pitt.edu

IntroductionThe ideal or intended use of calculators is that by aiding the students in the performance ofrepetitive computational processes, students may be “freed to focus on strategies” (Becta, 2003)and to “analyse and reflect on the relationships between data” (Hennessy, Fung & Scanlon,2001). This means that a judicious use of calculators should encourage a student learningapproach that is predicated on understanding the requisite concepts being studied. However, ourstudy focusing on the type of learning approaches students adopt towards mathematics taskspresents contrary evidence, ie, students use calculators as a way of circumventing the need tounderstand a mathematics problem.

Study descriptionIn our study, focusing on whether the inherent characteristics of the mathematics questionspresented to students facilitate a deep or surface approach to learning, 10 2nd-year undergradu-ate students were asked a series of mathematics questions during structured interview sessions.These students were all enrolled in an engineering programme—the admission requirement wasan “A” in A-level mathematics, and the questions they had to answer had been previously pre-sented to them during a regular classroom instruction session that was facilitated with the use ofelectronic voting systems. The students (interviewees) were allowed to use calculators to answerthe questions.

One of the problems the students were presented with during the interviews is shown in Figure 1,a question that at first sight might appear to have relatively low cognitive demand as its object, thesine wave, was a subject that students learnt prior to university. However, the spread of answers,from a class of over 100 students, would suggest otherwise. What makes the question difficult forthe students is that fact that we are asking about sin (np) or sin (-np). If we asked about sin (2p)or sin (3p), the cognitive demand would be much lower—particularly with calculators to hand. Inthe interviews and in class, the students could answer this question by accurately representingthe sine wave or by using their (non-graphical) calculators to determine the answer. The formershows greater understanding, while the latter is way of obtaining the answer without necessarilyunderstanding the sine wave representation.

Using calculators without subject understandingInterestingly, six students adopted the option of using calculators to solve the problem. Instead ofbeing able to accurately represent or visualise a sine wave, these students adopted a trial-and-errormethod by entering values into a calculator to determine which of the four answer options provided(Figure 1) was the correct one. But it was apparent that the students who adopted this approach hadlimited understanding of the concept. Indicative of this was the fact that none of them used theinformation obtained to sketch the sine wave after they have worked out one or two values.

British Journal of Educational Technology Vol 43 No 3 2012 E90–E92doi:10.1111/j.1467-8535.2012.01289.x

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For example, K9 (this is an interviewee synonym; “I” refers to interviewer) appeared to view thetool as a device for working on problems he or she had difficulties with, hence circumventing theneed to understand the subject:

K9: I wouldn’t know how to do this one unless I had a calculator.

Similarly, K1 seemed to construe calculator use as a way of circumventing the need to understanda problem:

K1: Um, this was probably one thing that I didn’t quite get at first.I: Ok. Why didn’t you get it? What were you struggling with?K1: Um, just remembering which one’s which. So I’d probably be tempted to try and use a calculator for it.

Another example is K4 who, when he or she was requested to answer the question, respondedthat he or she would need a calculator:

I: To get this question, do you have to do something? Or what, what would you have to do?K4: I would probably have to sit and work it out.I: Suppose you wanted to sit and work it out, how would you do it?K4: Ok. Try and remember my rules. Sine . . . [unclear muttering by interviewee] I’d probably need a calculator.I’m sure there’s some easy way to work it out.

Using calculators with subject understandingIn contrast, the students who were able to answer the question without using a calculator seemedto show greater understanding. For instance, K3 symbolised this approach of having an accuraterepresentation of the sine wave to solve the problem (eg, he or she started by, “sine of any p isalways zero”):

K3: Sine of any p is always 0.I: So why do you think, where do you think someone might have a problem with this [question]?K3: I don’t see any actual reason as to why there would be a problem. Um, if you’re told not to use a calculator, youmay not know how to visualise a sine wave.I: Did you visualise the sine wave?K3: Yeah. I think that with sine, cos, even when I’m integrating and differentiating sine and cos, I draw down the sinegraph and then think “the gradient at the first point is 1, so then the gradients 0, so you go to 0.” Then I realise it’sgoing like that and think “that’s a cos.” And for the cos one I just look at the gradient on each point . . . to get my mindof sine, and obviously let it —I: You always do that?K3: Pretty much every single time I’ve got to integrate sine or cos.

1. sin(n π) = sin(–n π) = 02. sin(n π) = 1, sin(–n π) = –13. sin(n π) = sin(–n π) = 1, (n odd), –1 (n even)4. Do not know

Which of the following statements is correct for n=1,2,3,4,.....

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sin(n

π) =

sin

(–n

π) =

0

sin(n

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–1

sin(n

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sin

(–n

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Figure 1: The mathematics question on sine wave presented to students

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© 2012 The Authors. British Journal of Educational Technology © 2012 BERA.

Similarly, K10 commented that he or she mentally visualised the sine wave when he or she sawthe question:

K10: . . . as soon as that question came up I thought of the sine wave in my head.

Further, K3 illustrates an ideal approach to using a calculator—to calculate or verify the outcomeof a computational process about which a student has considerable mathematical knowledge:

I: If you didn’t visualise what was happening with the wave... do you still think you could get this answer?K3: Um, obviously you’ve got to plug the values into a calculator. And that would give you them all. But thatwouldn’t be satisfactory to me because I haven’t... even now I still have to remember it was a repeated function andI would still have to remember. ‘Cause no matter how many values you put into a calculator, you’re never 100% sureif it’s going to be right. You can’t be sure it’s 100% true unless you know the fact that it is a repeated function.

SummaryThe (limited) evidence about the largely procedural use of calculators as a substitute for math-ematical thinking presented indicates that there might be a need to rethink how and whencalculators may be used in mathematics classes. This is imperative so that the habitual use ofcalculators does not replace or erode basic or fundamental mathematical skills. Calculatorsprovide optimum aid for mathematical learning when they “release us from the drudgery of acquir-ing speed and accuracy in doing complicated calculations. They do not release us from the task of knowingwhat are the appropriate calculations to do, or whether the answer makes sense” (Skemp, 1989, p. 169).

However, it appears that some students view the calculator as a way of bypassing the need forunderstanding or automacity development. These students probably see the calculator as a tool touse, ie., they are mathematical users instead of a device to learn mathematics with, ie, as math-ematical learners (Hoyles & Noss, 2006). An alternative view of calculator use is that it promotesefficiency, ie, ability to obtain the right answer. In this case, it is important to point out that only39% of the students obtained the correct answer when the question was initially presentedduring regular instruction (Figure 1).

In conclusion, further research needs to be undertaken regarding the role that calculators play inenhancing student mathematical thinking skills, as opposed to their use for computational pro-cessing (NMAP—National Mathematics Advisory Panel, 2008, p. 24), in mathematics and alliedsubjects, such as engineering, economics, statistics, etc. Moreover, a relevant research investiga-tion would be to evaluate whether there is a correlation between how and why students usecalculators for problem solving and the type(s) of learning approaches that students adopttowards solving related mathematics problems with respect to the deep and surface learningtheoretical framework (Biggs, 1999).

ReferencesBecta (2003). What the research says about using ICT in maths. Coventry: Becta.Biggs, J. (1999). Teaching for quality learning at university. Buckingham: Open University Press.Hennessy, S., Fung, P. & Scanlon, E. (2001). The role of the graphic calculator in mediating graphing

activity. International Journal of Mathematical Education in Science and Technology, 32, 2, 267–290.Hoyles & Noss (2006). What can digital technologies take from and bring to research in mathematics

education? In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick & F. K. S. Leung (Eds). Second Interna-tional Handbook of Research in Mathematics Education (pp. 323–349). Dordrecht: Kluwer.

NMAP—National Mathematics Advisory Panel (2008). Foundations for success: the final report of the NationalMathematics Advisory Panel. Washington, DC: Department of Education.

Skemp, R. R. (1989). Mathematics in the primary school. London: Routledge.

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© 2012 The Authors. British Journal of Educational Technology © 2012 BERA.