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The effectiveness of acomputer‐assisted learningprogramme in engineeringmathematicsD.A. Lawson aa Mathematics Division , School of Mathematical &Information Sciences, Coventry University , Priory Street,Coventry CV1 5FB, EnglandPublished online: 09 Jul 2006.

To cite this article: D.A. Lawson (1995) The effectiveness of a computer‐assisted learningprogramme in engineering mathematics, International Journal of Mathematical Education inScience and Technology, 26:4, 567-574, DOI: 10.1080/0020739950260411

To link to this article: http://dx.doi.org/10.1080/0020739950260411

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INT. J. MATH. EDUC. SCI. TECHNOL., 1995, VOL. 26, NO. 4, 567-574

The effectiveness of a computer-assisted learningprogramme in engineering mathematics

by D. A. LAWSONMathematics Division, School of Mathematical & Information Sciences, Coventry

University, Priory Street, Coventry CV1 5FB, England.

(Received 2 November 1993)

Two groups of students were taught the same engineering mathematicsmaterial in different ways. One group was taught using solely 'traditional' lecturesand tutorials. The other group had some lectures and tutorials but also spent sometime using computer-assisted learning material. The results achieved by thesetwo groups are analysed in an attempt to determine if there was any benefit infollowing the computer-assisted learning programme. The views of the studentswho used computers were surveyed by questionnaire and these findings are alsoreported.

1. IntroductionThe use of computers as an aid in the teaching of mathematics has grown

enormously in recent years. There is a rapidly expanding literature on the subject(see, for example, references [1—5]). In addition, there are now regular internationalconferences devoted to the use of computers in teaching mathematics [6]. However,despite this burgeoning literature there is very little attempt made to quantify thedifference that using computers actually makes to the students' learning.

It is undeniable that the computer provides the learner with an extra resourceof a qualitatively different kind to those that have been used in traditionallecture/tutorial teaching methods. The case as to whether or not the learner actuallyderives significant benefit from using the computer in learning mathematics is stillnot conclusively proved. An early study [7] indicated that computer-augmentedinstruction may have a detrimental effect on mathematical achievement. A differentstudy [8] showed that there was no significant difference in achievement betweencomputer users and a control group.

This paper reports on the effectiveness of a particular programme ofcomputer-assisted learning used with first-year undergraduate engineers atCoventry University. Two main sources of data are used to make this assessment.First, the results in tests and examinations of students who followed thecomputer-assisted learning programme (the CAL group) are compared with thoseof students who were taught using only a conventional lecture/tutorial approach (thecontrol group). The second source of data is the opinions of the learners themselvesas expressed in questionnaires completed at various stages through the course.

In the next section there is an overview of the topics covered within the course.A brief description of the course organisation is included. Details of the softwareused on the computer-assisted learning programme are given. This is followed bya section outlining the way the computer packages were used within the learning

0020-739X/95 $1000 © 1995 Taylor & Francis Ltd.

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568 D. A. Lawson

process. The level of mathematical knowledge and ability in the two groups at thestart of the course was measured by using a diagnostic test. The results from thistest are presented in section 4. These are followed by a discussion of the results fromthe assessments of this course. Section 6 presents the students' attitudes asdetermined from the questionnaires. Finally some conclusions about this particularprogramme of computer-assisted learning are made.

2. Course detailsThe material covered by the course was typical of many engineering mathematics

courses. The topics covered were:

Techniques of differential calculus for functions of one variable (product,quotient and chain rules)Applications of differential calculus (stationary points, points of inflexion,Maclaurin series)Techniques of integral calculus (parts, substitution, partial fractions)Numerical integrationIntroduction to differential calculus for functions of several variablesLinear algebra (matrix notation and multiplication, simultaneous linearequations, eigenvalues, eigenvectors and applications)Complex numbers (polar and exponential forms, de Moivre's theorem)Differential equations (lst-order separable and linear, 2nd-order constantcoefficient)Laplace transforms (standard theorems, solution of differential equations, stepand impulse functions)Fourier series

The course was assessed by means of three phase tests and an end of yearexamination. In the phase tests all questions were compulsory. In the examinationthere was some choice. To pass the course a student had to average 40% over the threetests (although each test individually did not have to be passed) and also achieve amark of 40% in the examination.

The Control group received three hours 'conventional' teaching each week. Thisconsisted of a two-hour block on one day and a further hour on a different day. Thesethree hours were spent in a mixture of lectures and tutorials. The exact allocationof time to each of these activities varied from week to week.

The group following the computer-assisted learning programme had a two-hoursession once a week, which was spent in conventional lecture and tutorial work. Inaddition, each student had a one-hour session in a computer laboratory every otherweek. In these sessions the students completed carefully prepared worksheetscovering material relevant to the lectures. Sometimes the worksheets were designedto give the students the opportunity to investigate, experiment and hypothesize inareas which would subsequently be covered more formally in lectures. Otherworksheets further illustrated material already covered in lectures, giving thestudents the chance to consolidate the understanding already gained.

Two computer software packages were used: A Graphic Introduction to theCalculus (GCAL) [9] and DERIVE [10]. GCAL was specifically designed as ateaching tool. Generically it may be thought of as the PC-equivalent of a graphics

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Effectiveness of computer-assisted learning 569

calculator. Although numerical output can be obtained its overwhelming strengthis its visual impact. GCAL is a menu-driven package which is extremely easy to use.Within this course only a small proportion of its facilities were used, mainly inillustrating and investigating the concepts of differential and integral calculus.DERIVE is a computer algebra package. Originally it was designed as a tool for theprofessional mathematician rather than as a teaching aid, however it is now beingwidely adopted throughout the educational community. This package is also menudriven. However it is not as easy to use as GCAL and the graphics are not asappealing. This point will be returned to later when the students' attitudes to thecomputer-assisted part of the programme are discussed.

3. The nature of the computer-assisted learning materialThe first test covered the topics of differential calculus and complex numbers.

By the time this test took place the CAL group had completed four computerworksheets. These introduced both the software packages. Further GCALworksheets explored, mainly from a graphical point of view, the qualitativeinformation that could be obtained from the graph about the gradient (and vice versa)and investigated the effect on the shape of the graph of a zero gradient (includinginstances where the gradient did not change sign). A DERIVE worksheet hadexplored Maclaurin series, using the symbolic capabilities of the package to performthe differentiation and the graphics facilities to compare the original function withpartial sums of Maclaurin series.

Before taking the second test (which covered integral calculus and differentialequations) the students in the CAL group had completed further computerworksheets, all using the GCAL package. These had enabled the students to explorethe idea of the integral as the limit of a sum and how this related to principles ofnumerical integration. GCAL has a numerical differential equation solver for first-and second-order differential equations. This was used in the differential equationsworksheet to investigate the kinds of solutions that second-order linear constantcoefficient equations may have by examining a range of cases with differentcoefficients. This led into a lecture on the role of the roots of the auxiliary equation.

The computer worksheets before the final test all related to the topic of Fourierseries. The first worksheet, based on GCAL, explored some properties of periodicfunctions, particularly sine waves, primarily from a graphical point of view.Subsequent worksheets all used DERIVE. These investigated much of the standardwork in an introduction to Fourier series: functions with period 2n, functions withgeneral period, Fourier partial sums, odd and even functions, etc. The integrationrequired to determine the Fourier coefficients was all carried out using the symboliccapabilities. The convergence of the partial sums was investigated using the graphicsfacilities.

4. Comparison of the prior mathematical knowledge of the two groupsThe students in the two groups had a wide range of mathematical backgrounds

including A levels, AS levels, BTEC level III, Access and Foundation courses,Scottish Highers, Irish Leaving Certificate and a range of overseas qualifications.The sheer diversity of these qualifications makes impossible a comparison of theinitial knowledge and ability of the two groups on the basis of entry qualification.

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570

Number of RsControl groupCAL groupNumber of SsControl groupCAL groupNumber of EsControl groupCAL group

028250

1080

2422

D.

13128

11414

173

A. Lawson

210172

24252

1722

3063

3439

31425

417114

10114

2417

5735735

106

608600600

773700736

Table 1. Full breakdown of results in the diagnostic test (all figures are percentages).

However, all engineering students at Coventry University sit a mathematicsdiagnostic test during their induction week. The results of this diagnostic test givea basis for a comparison of the entry level of mathematical knowledge of the twogroups.

The diagnostic test is divided into seven sections: basic arithmetic, basic algebra,lines and curves, trigonometry and triangles, further algebra, trigonometricfunctions and calculus. Instead of a mark, for each topic every student is given agrade. The three grades are E (excellent), S (satisfactory) and R (revision is needed).The results of the two groups in the diagnostic test are shown in Tables 1 and 2. Inthese tables CAL group stands for the group following the computer-assistedlearning programme.

The results of the diagnostic test (particularly the group averages shown in Table2) indicate that there was very little difference between the two groups in terms oftheir entry level of mathematical knowledge. In other words, there is nothing tosuggest that one group was significantly better than the other in terms of their abilityand knowledge at the beginning of the course. It seems therefore that any significantdifferences in their achievements in the assessments of the course may be, in partat least, ascribed to the teaching and learning methodology adopted.

5. Comparison of the achievements of the two groupsThe three phase tests and the end of year examination were very much of a

traditional nature being similar in content to tests and examinations set on this coursein previous years when computers were not used at all. The control group and theCAL group both took the same assessments. The average marks of the two groupson each of the three phase tests are shown in Table 3.

It can be seen that the CAL group performed significantly better than the control

Average number ofGroup Rs Ss Es

Control 2-03 2-41 2-55CAL 206 2-39 2-56

Table 2. Average performance in the diagnostic test.

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Effectiveness of computer-assisted learning 571

Group Test 1 Test 2 Test 3

Control 61 40 36CAL 72 51 38

Table 3. Average mark (% ) of each group in the three phase tests.

group on the first two tests. In each test the average mark of the CAL group was11% higher than that of the control group. Remembering that the two groupsperformed so evenly on the diagnostic test, this appears to be strong evidence thatthe use of computers with the CAL group had a very positive effect.

The evidence from the third test is not so clear. In this test the results of the CALgroup were only marginally better than those of the control group. This narrowingof the gap between the two groups may be due to a number of reasons. First, thecomputer worksheets relevant to this test all concentrated on Fourier series.However, the test covered Laplace transforms and eigen-problems as well as Fourierseries. The teaching of these first two topics was of the conventional lecture/tutorialstyle for both groups. Secondly, the majority of the computer work was carried outusing DERIVE. Although the students had used this package before they had notdone so very often. The consensus amongst the students was that DERIVE is notas easy to use as GCAL and consequently it was not as popular. Thirdly, the studentsneeded to average 40% over the three tests. Those who had already scored more than120 marks in the first two tests were guaranteed to achieve this 40% average evenif they scored zero in the final test. Some students decided that it was therefore notworth their while putting in much effort for the final test. Although this phenomenonapplies to both groups it is likely to have been more marked in the CAL group wherethe average total mark over the first two tests was 123 (a guaranteed pass) than inthe control group where the average total mark was only 101 (leaving 19 more marksneeded for a pass).

The results from the end-of-year examination are shown in Table 4. The CALgroup achieved significantly better results than the control group. Their examinationmark was on average 9% better and the pass rate (the pass mark was 40%) was almost30% better. This seems to indicate that the computer-assisted learning programmefollowed by the CAL group was of positive benefit.

6. The students' perceptionThe attitudes of the students towards various aspects of the course were sampledtwice during the year by way of an anonymous questionnaire. There were three

Average Group Exam.Exam. Mark Pass Rate

Control 41 56CAL 50 85

Table 4. Performance in end-of-year examination (all figures are percentages).

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572

1.

2.

3.

Interest

Relevance

More lectures

D.A.

1st2nd

1st2nd

1st2nd

Lawson

A

83

30

Y2862

B

3411

3119N2614

C

4735

4656

DM4624

D

1146

2119

E

05

05

Table 5. Summary of student responses in the two questionnaires (all figures arepercentages).

questions specifically relating to the use of computers within the course. Thesequestions are given below.

1. I have found the computer lab sessions:A. Very InterestingB. Interesting

c.D.E.

2. In relatiorsessions

A.B.C.D.E.

O.K.BoringVery Boring

i to the material covered in lectures, I have found the computer lab

Very UsefulUsefulO.K.IrrelevantTotally Irrelevant

3. I would prefer more ordinary lecture/tutorial time and no computer labsA.B.C.

YesNoDon't mind

The students completed the questionnaire in the week before the first phase test(week 8 of the course) and just before taking the final phase test, close to the end ofthe course. As can be seen from the summary of responses in Table 5, their opinionschanged markedly throughout the year.

The results from the first questionnaire show that the students were quite welldisposed towards the use of computers: 42% of them had found using theminteresting or very interesting as opposed to only 11% who found the computer labsessions boring. 34% could see some immediate relevance of the computer work tolectures compared with 21% who found them irrelevant. Almost half had no definiteopinion as to whether or not computer lab sessions should be abandoned and the timespent instead in lectures and tutorials. The 54% that did have a preference werealmost equally divided.

However, by the time of the second questionnaire the students' attitudes hadchanged appreciably. There was a hardening of their attitude against the use ofcomputers. By this stage only 14% found the computer labs interesting and 51% now

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Effectiveness of computer-assisted learning 573

thought them boring. 19% of students saw the relevance of the computer labsto the material in lectures (a large drop compared with the first questionnaire)while 24% found them irrelevant (only a small increase). The greatest changewas in the answers to the third question. In the second questionnaire 62% ofstudents stated that they would prefer more lecture/tutorial time and no computerlabs.

A number of ideas may be advanced to explain some of these results. First, atthe beginning of the course very few of the students have any experience of usingthe computer as a support tool in mathematics. There is a resulting novelty valueand the students like having a new approach to mathematics. As the courseprogresses they become more and more accustomed to using the computers and sothe novelty factor disappears. With it goes the students' initial enthusiasm for thislearning aid. Secondly, their first experience of computers in this course is withGCAL, which is extremely user friendly. They find it easy to complete theworksheets and so can give attention to the mathematics without being distractedby having to worry about how to make the computer achieve what is required. Bythe time the second questionnaire is completed the most recent computer lab sessionshave dealt with Fourier series using DERIVE. Despite the menu-driven nature ofthis package and its user friendliness in comparison to other computer algebrapackages it is harder to use than GCAL. Students found that they had competinglearning objectives, namely, how to use DERIVE and the mathematical content ofthe worksheets. Their experience with DERIVE in exploring Fourier series (itselfa difficult topic for first years) was uppermost in their minds when they completedthe second questionnaire. Finally, by the time of the second questionnaire theend-of-year examination was quite close. Many students are quite conservative whenit comes to what they expect of teaching and, with the examination so close, theywanted the reassurance of a traditional teaching approach of lectures, tutorials andexamples classes.

7. Conclusions

The results of the CAL group in the first two phase tests and the end-of-yearexamination were significantly better than those of the control group. As therewas no significant difference between the two groups in their performance on thepre-course diagnostic test this suggests that there may be some positive benefitfrom using the computers as a learning aid.

The students must be given enough exposure to the software used to feel atease in using it. Where they are struggling to remember how to achieve aparticular result they will be distracted from the mathematical concepts beingexplored.

There may be a perception amongst the students that computer-assisted studyis not as effective as traditional lectures and tutorials, particularly when theassessment is of a traditional nature. More work needs to be done to convincethem of the value of computer-assisted learning. This may be achieved, in partat least, by establishing in their minds a clearer link between the contents oflectures and the computer-assisted learning material. Another advantage wouldbe to use the computers in the assessment process.

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Technol., 22, 791-798.[4] ADAMS, J. L. and STEPHENS, L. J., 1991, Int. J. Math. Educ. Sci. Technol., 22, 889-893.[5] MACKIE, D., 1992, Int. J. Math. Educ. Sci. Technol., 23, 731-737.[6] TMT-93, Proceedings of the International Conference on Technology in Mathematics

Teaching, at the University of Birmingham, 1993, edited by B. Jaworski.[7] ROBITAILLE, D. E., SHERRIL, J. M. and KAUFMAN, D. M., 1977, J. Res. Math. Educ, 8,

26-32.[8] SAUNDERS, J. and BELL, F. H., 1980, Int. J. Math. Educ. Sci. Technol., 11, 465-473.[9] TALL, D., VAN BLOKLAND, P. and KOK, D., A Graphic Approach to the Calculus,

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