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An evaluation of computer‐assistedlearning in mathematicsD.M. Mackie aa Department of Mathematics , Napier University ,Edinburgh EH14 1DJ, ScotlandPublished online: 09 Jul 2006.

To cite this article: D.M. Mackie (1992) An evaluation of computer‐assisted learning inmathematics, International Journal of Mathematical Education in Science and Technology,23:5, 731-737, DOI: 10.1080/0020739920230512

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INT. J. MATH. EDUC. SCI. TECHNOL., 1992, VOL. 23, NO. 5, 731-737

An evaluation of computer-assisted learningin mathematics

by D. M. MACKIEDepartment of Mathematics, Napier University,

Edinburgh EH14 1DJ, Scotland

(Received 10 October 1991)

Mathematics courses for science and engineering undergraduates should aimto develop an enquiring and creative approach to mathematics together with goodcommunication skills. Due to their versatility, computational power and graph-ical capabilities, computers can play a significant role in developing these skills.This paper describes a project in which the use of computer packages wasintegrated into the curriculum of several science and engineering courses. Anevaluative study found that this resulted in more emphasis being placed oninvestigative work, more mathematical discussion and better student under-standing of some concepts.

1. IntroductionCreative thinking, the application of mathematical knowledge to unfamiliar

problems and competence in communication are all skills highlighted in recentreports as important aspects of an engineer's training [1, 2]. Mathematics curriculafor science students, too, must aim to stimulate an analytical and creative approachwhich will enable them to tackle a wide variety of problems with confidence. Howbest can these abilities be fostered in an undergraduate course? The Cockcroft report[3] lists problem solving, investigational work and opportunities for discussionbetween teacher and student and between students themselves, as teachingapproaches which should be included in mathematics education at all levels. Clearlythese approaches will encourage the acquisition of the skills required by science andengineering students.

This paper describes a recent project undertaken at Napier Polytechnic,Edinburgh, in which the use of carefully designed computer packages was integratedinto the curriculum of several science and engineering courses [4]. An evaluativestudy found that this resulted in more emphasis being placed on investigative andexperimental work, more mathematical discussion and better student understandingof some mathematical concepts.

2. Teaching approachesMathematical modelling and open-ended investigations are both activities that

foster creative thinking by promoting an environment in which making mistakes isaccepted. The cognitive skills of analysis, synthesis and evaluation required for thesepursuits necessitate deep understanding of the mathematical concepts involved.Deep or 'relational understanding' [5], in which new concepts are related to previousknowledge structures, enables students to tackle unfamiliar mathematical problems,

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thus improving their investigational skills. Conversely, directed investigative workcan encourage better understanding of concepts by guiding the student through thestages of generalization and abstraction which can lead to relational understanding.

At present, the emphasis of many course is on the development of competence inalgebraic manipulation and the application of routine techniques and algorithms,most of which can readily be accomplished by computer programs. One suchexample is the difficult and tedious matrix calculations which are necessary to solve alinear programming problem with three or more variables, by the Simplex method.To tackle real problems, however, is is important for the user to acquire skills offormulating a problem in mathematical terms, interpreting the solution andanalysing its sensitivity, all of which require a good understanding of the underlyingconcepts of the topic. Previously, many undergraduate courses have concentrated onteaching the difficult computation, which had to be done by hand, leavinginsufficient time to practise the wider problem-solving skills.

Due to their versatility, computational power and graphical capabilities,computers can play a significant role in developing relational understanding,investigational and problem-solving skills. Since the advent of microcomputers, theavailability and quality of mathematical software has steadily improved. Teachersand students can now choose from a wide variety of computer packages to assist thelearning process. In general, however, the teaching approach and objectives of boththe programs and the users has remained the same as in pre-computer days. Mostcomputer-assisted learning has been limited to teaching the same things in much thesame way but with the benefit of a computer to reduce the tedious arithmetic orcontrol the pace of presentation of material. Whilst self-paced learning and theelimination of numerical drudgery are important features of computer use, theopportunities it presents for new approaches to learning have not been fully exploredor reported.

Among the more innovative approaches which have emerged are the Computer-Aided Teaching of Applied Mathematics project at Cambridge University [6] and,more recently, the Mathematics Department at Birmingham University [7], both ofwhich aim to develop an investigative approach towards problem solving. Asupportive software environment enables students to write some of their ownprograms thus acquiring a deeper understanding of the mathematics of the problem.

3. Development of mathematics laboratoriesAt Napier Polytechnic, Edinburgh, the Mathematics Department has been

involved in the use of computers in teaching for many years [8]. With the increasedavailability of microcomputers, teaching staff believed that improved methods ofusing computers to enhance the learning of mathematics were possible. It was feltthat an approach requiring students to write their own programs was too time-consuming and not appropriate for the majority of the science and engineeringcourses being taught at Napier. Some microcomputer-based software was developedand a departmental laboratory was established [9].

The use of computer packages at Napier is integrated into the curriculum toillustrate, explore or extend a topic when it arises. Frequency of use of the laboratoryvaries widely depending on class size, availability of software and the suitability ofthe topics being studied. A few classes have weekly laboratory sessions, comprisingperhaps one-quarter of their total mathematics time, whilst others use computersonly to carry out an investigative assignment as part of their coursework. Much of the

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Evaluation of computer-assisted learning in mathematics 733

software used has been developed within the department in response to needsidentified by lecturers and has been designed in close consultation with them.

The packages developed share a common core of objectives, namely, toencourage investigative work, to facilitate problem-solving and to enhance studentunderstanding of certain algorithms and methods. Unlike most other programsavailable for use in mathematics, the design of the software was determined by thesebasic objectives.

Two of the packages on which much of the research is based are described brieflyhere:

LINPROG is a linear programming package with optional step-by-step pro-gression through the Simplex method. The program features easy input of the dataand menu-driven facilities that enable the original problem to be modified thusfacilitating post-optimal analysis and the solution of integer programming problemsby the branch and bound method.

NODES solves single or systems of ordinary, initial value differential equationsnumerically. Emphasis has been placed on graphical output and flexibility whichallows problem parameters to be easily changed and the resultant effect on thesolution observed. It is suitable for investigating both numerical methods andmathematical models.Both packages are used in conjunction with appropriate worksheets, designed toencourage an investigative approach. More detailed descriptions of the programsand their use at Napier can be found elsewhere [10-12].

4. Evaluation of the use of the computer packagesA summative evaluation of the materials developed and their impact on the

mathematics curriculum has been carried out. In particular, changes in teachingapproaches, learning outcomes, methods of assessment and in student attitudestowards mathematics were studied through observation, questionnaires andinterviews.

Initially, two classes in which the use of computers was extensively integratedinto the mathematics curriculum were selected for close monitoring. One was thefourth year of a BSc in Science with Industrial Studies, an interdisciplinarysandwich degree course with specialization, from the third year onwards, in twoscience subjects, one of which may be mathematics. The other class involved was thefourth year of a B.Eng degree in Communications and Electronic Engineering, also asandwich course. Amongst other topics, both groups study optimization during theirmathematics course.

Questionnaires, supplemented by interviews with individual students, were usedto gather evaluative data from the 28 students in these classes. As the projectprogressed it became apparent that the use of computers in teaching at Napier wassteadily growing, and more classes were using the laboratories for investigative work.More than 100 students from several other classes, who were using the NODES orthe LINPROG package, were surveyed also. Additional data were gathered fromlecturers in the mathematics department over a four year period from annual surveysand from extended, informal interviews with five members of staff who had gainedexperience of using computers to assist their teaching over several years.

Both packages were rated favourably by staff and students for their ease of use,reliability and usefulness of the output. Student comments emphasized

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LINPROG's role as an aid to understanding both the Simplex method for linearprogramming and the branch and bound method for integer programming. Thegraphical output from NODES was highlighted as an important feature, whichassisted student understanding of the behaviour of solutions of differential equ-ations. Acceptance of the packages as effective teaching tools is evident from theincrease in their use over the survey period to a level which has subsequently beenmaintained.

Use of the mathematical sciences laboratories at Napier Polytechnic fortimetabled classes increased by about 70% over the four-year period 1985-88. Themajority of lecturers in the mathematics department now use computer-basedpackages as a tool to support their teaching. Many lecturers, therefore, have had toreconsider their teaching objectives and adapt their teaching approach to accommo-date new styles of learning and the wider opportunities offered by computers.Computer packages have enabled teachers to arrange learning situations which werenot previously possible, for example, using graphics to explore the behaviour of afunction, to analyse the solution of a mathematical model or to investigate thestability of a numerical method.

The teacher's role, however, has not diminished in importance and class contacttime has not been reduced. On the contrary, it was found that class sessions in thelaboratory can be harder work for the lecturer because the students ask morequestions, some of which involve concepts that would not have been encounteredotherwise. The role of the teacher and the computer are complementary and bothcontribute towards the learning process. The computer is used as a tool to removetedious calculation, to present information graphically and to increase a student'sinvolvement in his own learning. The teacher's skills lie in recognizing and seeking torectify a student's particular weaknesses or misunderstandings and exploiting thelearning situations which arise.

5. Learning outcomesThe learning outcomes which have resulted from using computer-based

packages as a tool for mathematics represent a broader range and higher quality ofcognitive and attitudinal outcomes than previously present in the mathematicscurriculum at Napier Polytechnic.

Investigative and experimental work has increased and more emphasis is placed onthis type of work. The use of computer-based packages has been successfullyintegrated into the curriculum of several courses at Napier. In such courses there isnow less time than previously spent teaching techniques and more time spent solvingproblems, formulating and testing mathematical models and doing investigativework.

Almost all the lecturers using the laboratories cited 'To carry out investigativeand/or experimental work' as one of the reasons for choosing to do so. Extensive useis made of graphical output to examine the behaviour of functions, to analyse modelsand to investigate the errors in, or the stability of, a numerical method, using one ofthe packages available in the laboratories. Students' work in the laboratory isdirected by worksheets which have been designed to accompany the computerprograms and which contain some open-ended investigations. Results from studentsurveys show that the majority of students consider investigative work to be bothinteresting and useful.

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Evaluation of computer-assisted learning in mathematics 735

In the first four years of the Communications and Electronic Engineering degree,the students carry out a computer-based assignment as part of the engineeringapplications element of the course. This involves investigating the mathematicalmodel of a physical system using a computer package. NODES was found to be anexcellent tool for this purpose.

Student understanding of some mathematical concepts and algorithms has beenimproved or reinforced as a result of using computer packages. More than 40% of thestudents in the two classes in the initial survey considered that use of LINPROGhelped them to understand the Simplex method, duality, the branch and boundmethod for integer programming problems, or some other aspect of linearprogramming. Use of the NODES package for engineering applications assignmentsresulted in increased understanding of the solutions of differential equations by 68%of respondents in a second-class year. In particular, many students found that beingable to study the solution of equations graphically using NODES improved theirunderstanding of the behaviour of the underlying mathematical model.

Mathematical discussion has increased. The importance of being able to communi-cate mathematical concepts confidently has been noted. The use of computer-basedpackages stimulates mathematical discussion in the classroom both between teacherand student and between the students themselves, confirming findings from aprevious study [13]. Graphical output offers the lecturer many unrivalled opportun-ities to probe the students' understanding by appropriate questions. Evidence fromlecturers also suggests that students' competence in discussing mathematical ideasimproves as a result of using computer packages in the laboratory. Working throughan algorithm step by step on the computer (for example, the Simplex method forlinear programming), not only aids student understanding of the method but alsohelps him to become familiar with the mathematical terms being used. This booststhe student's ability to communicate clearly.

The above outcomes relate to the higher levels of Bloom's taxonomy of cognitivelearning objectives [14]. Learning outcomes in the affective domain are alsoimportant and, in many cases, closely interrelated to cognitive outcomes.

Learning is more student-centred. Computer-based work is usually more self-paced than traditional coursework and allows the student to control the level ofexplanatory detail required. It thus passes more responsibility to the student for hisor her own learning. Explanatory feedback from errors enables a student to learnfrom his own mistakes. Some students return to the laboratory in their own time touse packages to reinforce ideas, explore them further or to assist with coursework forother subjects.

Many students enjoy their mathematics course more as a result of using computerpackages. Attitude surveys concluded that many of the students surveyed felt thatuse of the LINPROG and NODES packages had made the topic being studiedenjoyable and more interesting. Increased motivation probably leads to an improve-ment in educational achievement. There was also strong agreement that similarprograms should be used to enhance the learning of other topics.

Most students consider their computer-based work useful and relevant, both to theircourse and to their future career in industry or commerce.

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6. Implications for the curriculumMethods of assessment at Napier are slowly changing to reflect curriculum

developments. Many courses now include a computer-based assignment as course-work replacing a class test. A majority of students surveyed were in favour of somecomputer-based work being assessed. In some examinations, students are asked tointerpret computer output from a package they have used and they might also beasked questions of a conceptual nature based on work done in the laboratory.

The reported shift in teaching emphasis away from techniques towardsunderstanding, applications and investigations is likely to continue, and, indeed, toaccelerate, as the use of computer algebra systems (CAS) increases. The new,broader range of learning outcomes described in the previous section calls for furtherthought to be given to the most appropriate means of assessing students' work.Tentative steps have been taken at Napier towards laboratory-based examination.As computer facilities improve, these may become commonplace.

The use of CAS has wider implications for the curriculum. There has alreadybeen debate at ICME-6 [15] and elsewhere about possible changes to mathematicssyllabi for engineers and scientists. Is it still necessary for engineers to learn severaldifferent techniques of integration if a CAS is available? It seems likely that, infuture, less time will be spent mastering algebraic skills and techniques. The timethus saved should be spent investigating concepts and methods, solving realisticproblems, formulating models and analysing and interpreting results, usingcomputer packages when appropriate.

Experience at Napier has shown that the effective use of computer packagesenables an investigative approach to be incorporated into the curriculum and,further, that when such an approach is adopted, the quality of the students' learningexperience is enhanced.

The introduction and wider recognition of the power of CAS may prove to be thecatalyst required for course committees to reduce the techniques content of syllabiand to recognize fully mathematics as a laboratory-based subject. Mathematicscurricula should be designed to allow regular time to be devoted to laboratory workin addition to lectures and tutorials. The advantage of using computer packages toenhance the learning of mathematics and to encourage investigative work could thenbe extended into other areas of the curriculum and across more courses than ispresently possible. The improved understanding and higher cognitive skillsresulting from such work will better prepare the students for the current needs ofindustry and commerce.

References[1] FINNISTON, M., 1980, Engineering our Future: Report of Committee of Inquiry into the

Engineering Profession (London: Her Majesty's Stationery Office).[2] GRANT, A., 1985, The Formation of Mechnical Engineers (Institute of Mechnical

Engineers).[3] COCKCROFT, W. H., 1982, Mathematics Counts (London: Her Majesty's Stationery

Office).[4] MACKIE, D. M., 1991, Unpublished PhD Thesis, Napier University.[5] SKEMP, R. R., 1976, Mathematics Teaching, 77, 20-26.[6] HARDING, R. D., 1974, Int. J. Math. Educ. Sci. Technol., 5 (3), 447.[7] BEILBY, M., 1987, in Trends in Computer-Assisted Education, edited by R. Lewis and

E. D. Tagg (Oxford: Blackwell Scientific Publications), 174-179.[8] LEACH, D. F., 1974, Proc. Int. Conf. Frontiers in Education, Council for Educational

Technology, London, 212.

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[9] MACKIE, D., and SCOTT, T. D., 1988, Int. J. Math. Educ. Sci. Technol., 19 (1), 83.[10] MACKIE, D., 1985, Conference Proc. Mathematics Teaching 1985, University of

Edinburgh, Edinburgh.[11] SCOTT, T., 1987, In Aspects of Educational Technology XX, Flexible Learning Systems

edited by F. Percival, D. Craig and D. Buglass (London: Kogan Page).[12] MACKIE, D., 1988, Mathematics and Statistics Curricula in Higher Education for the

1990's, Napier Polytechnic, Edinburgh.[13] KATSIFLI, D., 1986, Int. J. Math. Educ. Sci. Technol., 17 (2), 209-227.[14] BLOOM, B. S. (ed.), 1956, Taxonomy of Educational Objectives: Cognitive Domain (New

York: David McKay).[15] ICME-6, 1988, Proceedings of the Sixth International Congress on Mathematical

Education, edited by K. and A. Hirst, Janos Bolyai Mathematical Society, Budapest,Hungary.

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