the mathematics content of engineering education

by 1. E. Otung This paper reviews the p r o b l ~ z of declining mathematical skills and tzppetite aniongst university entrants.

This decline necessitates a critical yeview Of the traditiorzal approaclzLfallozwed in engineering education in order to cater adequately for the abilities and p f i r m c e s (zf the type 4' students becoming prevalent

i f i our universities. A niiriimal-mafhematics methodolog]? that does aot irrzpact negatively oiz staiadavds is discussed with examples dyawrijom telecoinmunica fions. Dtjsigned to eizdear rather than deter new students, this uppoach rkhtfully puts ergineevirg f irs t and mathematics second,

and removes what a growing nuniber ojpotential recruits now perceive as afz unfriendly g a t e k e e p at the entrance to the study Of eizgineering.

el-trained eiigineers have a vital role to play in ensuring the economic prosperity arid competitiveness of any W nation in a technology-doi~iiated

world. However, the growing importance of engineer- ing, especially telecommunications and its applications, has coincided with a serious decline in both appetite for and competence in mathematics amongst university entrants, includmg those with good A-level grades. Engineering departments are finding it increasingly difficult to fill available degree places with students who havc the required skills and motivation in mathematics. Consequently a growing nuniber of students who comc into engineering programmes are not only apprehensive about mathematics but are in fact deficient in several key areas. Confi-onting such students with a lot ofmathetnatics before they have had the opportunity to remedy their deficiencies and iinprove their attitudc is likely to lead to withdrawal or failure. On the other hand, engineering modules on offer cannot be purged of mathematical content without seriously diluting the depth of trcatrnent and therefore sipficantly underiiuning the competence of graduate engineers.

This papcr examincs the mathematics probleni and points a way out of the above dilemma. Observations of declining mathematical skills and an ovenvhelming preference for minimal-mathematics are reported. The impact of this development on engineering education is then discussed and a possible solution is offered that eliminates a mathematical high-hurdle without lower- ing standards. This approach treats mathematics as a useful tool rather than an end, and seeks first of all to win new students over to the sheer enjoyment of

studying engineering, allowing a departrncnt morc time to remedy their inatherriatical deficiencies with less risk of failure or withdrawal.

The mathematics handicap

Evidence Acadenlics in engineering departments are unani-

mous in acknowledging that there has been a steady decline in essential matheiliatical skills amongst new intakes to their degree progranunes over the last decade. This problem has been the subject of several receiit studies culniiiating in recommendations mainly aimed at improving the mathematics preparedness of new engineering undergraduates. A recent report' published by the Engineering Council presents a number of objective pieces of evidence, two of which are summarised below.

The performance of new undergraduate physicists at the University of York in the same diagnostic mathematics test administered every year since 1979 is similar until 1990, when there is a sharp drop followed by a steady decline over the past decade. In particular, whereas the average score of the 1986 intake was 7694, that of the 1997 cohort was a mere 50%, and none of the intakes sincc 1995 have registered an average score above 56%. Since 1991 Coventry University has given a standard diagnostic test to new students entering its niathe- matics-based courses. The test covers seven topics, namely basic arithmetic, basic algebra, lines and curves, triangles, further algebra, trigononietry and

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basic calculus. Grouping the students according to their A-level mathematics grade, the results show that:

(i) in each year the performance decreases with A-level grade-an expected and reassuring correlation

(ii) the perfornlance of those students with advanced GNVQ is sipficantly below those with a grade N A-level

(iii) the performance of students with the same A-level grade declined steadily over the years

(iv) the performance of the following groups was roughly the same: 1991 grade N, 1993 grade E, 1995 grade D and 1997 grade C. This signifies a du t ion in grade by approximately one grade every two years. In fact the 1998 grade C cohort scored on average 4.6% below the 1991 grade N students.

The general consensus is that students with a good A-level mathematics grade can no longer be assumed to possess the mathematical slulls required by the tradtional approach to the training of engineers. Engineering academics therefore face a critical problem in the teachmg of their modules. In the short- term at least, this problem can only get worse for several related reasons.

Underlying causes There is a growing dearth of competent and well-

motivated mathenlatics teachers in the pre-19 education sectol. Most of the qualified mathematics teachers are aged over 40, but increasingly the younger mathematicians who would replace them opt for other, more rewardingjobs that abound in a booming and IT- driven economy,

The percentage of sixth-formers taking A-level mathematics has been in decline, currently standing at about 9%, of which only about a third go on to read Mathematics, Science or Engineering at university’.2.

An increasing number of students come to university having never had a positive experience of inathenlatics. These students are understandably very apprehensive ofmathematics-based courses and are at a real risk offailure or withdrawal if c o h n t e d with a lot of inatheinatics.

More and more university entrants have poor study skills and problem-solving discipline, having been raised on a revised mathematics curriculum3 that does not adequately cater for the needs of higher education. These students are easily thrown by simple problems involving multiple solution steps, and rely on the calculator for rather obvious computations without a feel for the correct answer-a skdl born of a good grounding in basic arithmetic.

Impact on engineering education

The needfor mathematics The tool of mathematics is indispensable to engi-

neering. Analysis and design of engineering structures

and systems require mathematical models that encapsulate the important parameters and applicable physical laws. Although some of these laws c m be quahtatively described, a precise quantitative statement requires the language of niatheniatics. Yes, it is possible to use software and machines without necessarily understanding their underlying operations, or to sell and even install a piece of ‘black box’ equipment; but engineers routinely invent, build or analyse processes and systems and interpret experimental iiieasurenients, and these tasks, sometimes critical, require important mathematical SMS. Thus, it may be concluded that it is impossible to raise competent crigineers on a non- mathematical diet.

In.kornomyoris intake In view of the current pre-19 educational handicap,

engineering departments have a greatly reduccd pool of studcnts with adequate nwtheinatical preparedness fi-om which to recruit into their degree places. New intakes in many departments now coinc kom a wide range of vocational, A-level, mature, acccss and foundation backgrounds. This gives rise to a very inhomogeneous cohort. I)iaguostic testing of new undergraduates is now widely uscd to identify the mathematical weakness of individual studerits arid tllat of thc whole cohort. Kemedial measures, based on a variety ofstrategies such as supplementary classes, computer-assisted learning and mathematics support centres, can then be individually prcscribcd to bring each student up to speed in their areas of deficiency. Thcre are, ho\vever, nontrivial problems in this belated attempt at imparting matheniatical sk& that should have beconic ingrained in a student prior to entering university.

Pro blenis The attitude (towards mathematics) of a tnathc-

matically deficient student is likely to be x7ery negative at this late stage, and this fosters self-doubt and seriously hampers the learning of the subject. Moreover, there is usually little or no spare capacity in the departmental curriculum. Thus, whatever remedial action is followed will be an extra burden on a weak student, who may not be able to cope. Furthermore, linlited staff and accon- modation resources and an increasingly diverse cohort make nixed-abdity teaching inescapable. As a result, tlie deficient students will be seriously hampered by the traditional approach employed in cnginecring education. It is worth pointing out that this approach was developed prior to 1986, during \vlut has been described’ as the ‘golden age of mathematics’, and inevitably assumes a wide range of inathematical skills that a growing number of new recruits, raised in a different age, do not possess. Many ofthese students may become discouraged and drop out before remedal measures have had time to yield the intended benefits. Even the more deterrnined ones who decide to stick it out remain at a serious risk of acadenlic failure.

The withdrawal rate in one engineering department in a UK university during the 1999/2000 academic

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Fig. 1 selecting each option in the question: 'Which would you prefer in this module? A mathematics only where unavoidable. B: a lot of mathematics. C: absolutely no mathematics'

Percentage of students (during the academic years 1998/99, 1999/00,2000/01)

Fig. 2 Percentage of students (during academic years 1998/99,1999/00,2000/01) selecting each option in the question: 'What would you do if there's a lot of mathematics? A take on the challenge happily. B: endure. C: withdraw'

data to determine what faction of ths ligh casualty was occasioned by a matliematical high-hurdle, it is hiowti that only 1596 of students who withdrew gave financial d8iculties as the rccason. The loss of students hanslates directly irito a loss of income for the deparhnent involved. But when this \vithdrad involves engineering students, then its inore significant impact is perhaps the undesirable dent made on a crucial area of the nation's skius base.

Minimal mathematics

It \vi11 take nmiy years of coiicerted effort and government iiivestriient to correct the mathematics deficiency in prc-I 9 ediicatioii and provide en@- iieeririg departments with a sufficiently large pool of rnatheinatically coiiipeteiit re- cruits. Und this golden age t-etui-ns, however, urgent action is needed to increase retention of matl~ematicrdly deficient intakes in engineering departments. Adopting a non-mathematical curricdurn is out of the question since this would seriously undermine the conipetence of graduate engineers. A new approach can ho\rever be considered tliat puts engineering first and inathenutics second, and gives bcgitming students the opportunity to become endeared to engineeriiig while gradually rcmedying their deficiencies in nntliematics. The with&xvaI statistics given above suggest that students are most at risk in their first year. Thus, there is a pressing need for a review of the tradtionally mathematical app- rcxlch in the delivery of firs-year enpecring modules.

Observed pr&etzce Since September 1998

successive cohorts taking a first-ycar module dealing with basic teleconiiiiunications in the

year was 29% of fii-st-year students-the highest in tlic School of Electronics at the University of Glamorgan entire university-compared to uidy 8% for non-first-year have been given a questionnaire at thc beginning of the students of the saiiie department. Although there is no first lecture. Individual responses are returned

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aiioiiyniously in the same lecture before the cornniencenient of teaching. Students are informed that the information will be used to determine how bcst to deliver the module for their bcnefit, arid are encouraged to be very honest in their answers. Two of the questions are discussed below-.

In one question the students were asked to choose one of three optioiis to iridicate what they would prefer iii the niodule: (A) matheiiiatics only where unavoidable; (B) a lot of tiiathcniatics; (C) absolutcly no mathematics. Fig. 1 shows thc response in terms of the percentage ofstudents choosing each option during the three academic years. The results consistently show that thc vast majority of these students are averse to much mathematics. In tlie three cohorts questioned, an average of only 5% of the students (with a standard deviation of 2%) wanted a lot of mathematics in the inochle. Worryingly, a n average of 3% of the students \vanted absolutely no mathematics-an impracticable approach in view of the discussion o n the need for rnathcrnatics.

A second question sought to find out what the studclit$ would do if there was a lot of matheiiiatics in the module. Three optioiis were provided: (A) take oil the challenge happily; (U) endure; (C) withdraw Fig. 2 shows the response to the second question over the three-ycar period 1998-2000. An average ofordy 27% of the students would happily accept the challenge of a lot of mathematics in the modulc. Re warned, however, that this particular group of students is not necessarily mathematically slulled-as evidenced by responses to another (unreported) question in the questionnaire. The nuniber ofstudcnts at a serious risk of withdrawal is therefore likely to be significantly larger than the 4% (average) shown in Fig. 2 sirice there will subsequently be sonic migration from the ‘eiidurance’ and even the ‘happy’ camps into the ‘\vi th drawal’ caiiip.

It can be seen that R rnathematical approach would be off-putting to about three quarters of the students qucstioncd. Ignoring their preference can contrihute to avoidable withdrawal or failure. Interestingly, when tlie first question was asked to 13 practising engineel-s-~~roducts of various UK universities, some of them with less than 2 years experience-attending a 5-day short course oil digital telecomiiiuiiicationi networks held in Octobcr 2000, 1 1 o f thcrn (or 85?4)

that is absolutcly necessary. Ei.iiphasis is placed on the underlying engineering considerations, arid lucid graphs and diagrmis are employed to fachtate uriderstaiiciing and assinlilation. Mathematics is giveii its rightful place, which is second to engineering, arid a physical insight into the probleiii at hand. A graphical approach is ireely used where neccssary to siinplify a matheiixitical coinputation or illustrate a theoreni.

Developing siicli a teaching approach requires a thorough lreassessiiieiit of current methods, changes in approach being made where^ necessary to nicct students at their level while eiisuriiig that they have an unclouded insight into the underlying engincering concepts arid a good grounding in the theoretical principles. I t is irisuficient to string together an elegant iiiatlieniatical derivation of a concept for students who do not yet possess the skill or iiiotivation to follow it. Such an apprwach is rather insensitive aid leads to students employing ‘black-box’ foriiiulas whose limitations they cannot fully apprcciatc. They will lack the insight, confidence and competence that could very easily have been imparted through ~ilinirnal- mathematics approach.

The above minimal-rriathematics strategy is used in a new book on corririiuiiication eiigiiieeriiig4, which covers corc tclcconiiiiuriication topics such as signal aidysis and ti-ansinission, atnplitudc, frequency and phase modulations, digital baseband transmission, digital modulation, iiiultiplexiiig strategics, noise effects, etc. The book received the endorsement of reviewel-s as an excellent contribution to its field. The approach has also been tested in short courses on teleconiniuiiications for practising engineers, attracting very positive feedbacks such as the following:

‘Made the subjects enjoyable and easy to follow.’ ‘Excellent arialogics arid examples were used to

‘Explained very abstract subjects in a way that itiiprovcd

‘Fantastic.’

describc complex theorenisi

uiiderstaiiding of inathematical descriptions.’

One inference that can be drawn is that students firid the predominantly nlatheiilatical approach commonly followed in undergraduate textbooks 011 the subject less enjoyable arid more abstruse.

It must be eiiiphasised that a nliiind-mathematics categorically chosc option A, with the reiiiairling engineers going for a lot of matheiixitics (B).

Fcatuues ami ir.riplernentntion Tt can be seen from tlie

forcgoiiig that tlie best interests of the vast majority of students and a good engineering cducation can bc jointly served through A

ininiiiial-matheii7laticr approach where tnathcniatics is eriiployed

approach docs not mean syllabus dilution. Neither is the treattncnt shallom: as would be appropriate for educating a non-technical arid non-mathematical auchencc. On the contrary, the same topics are covered to the same depth, but with mathenmtical rigour subordinated to a practical insight into tlie engineering problem. This may sometimes bc achieved with the aid of lucid graphs and diagrams. For example, the topic of analogue signal sampling in tclecorivriunicatioris can be treated in depth using a prcdoninantly graphical approach4 that gives the student a thorough understanding of dl the important concepts and a proficiencry in anti-&as filter design, aperture effect correction, sampling of lowpass and baridpass signals, etc. A brief demonstration of t h s illuminating graphical approach is presented in the next section of the paper.

At other times, a little thought niay reveal an alternative and sinipler niathematical derivation. A detailed cxarnple of this possibility is presented below on the subject of speciFjlng a receive filter that gives optimum sibpal detection in the presence of white noise. A minimal-mathematics solution of this problem places emphasis on a physical insight into the problem and is sib?7iftcantly toned down in matherriatical rigour-an appealing feature to most students.

Minimal-mathematics examples

i2latc.hed-filtcr: the traditional approadi The specification of a receive filter (called a matched

filter) that gives optiniuin dctcction of a signal in the presence of additive white Gaussian noise is usually derived by invoking SchwarzS inequality. In Fig. 3, the transfer function H ( j ) and hence the iiiipulse response h( t ) of a filter is to be specified so as to maximise the instantaneous output signal power E ( 7 ; ) compared to the average output noise power where 7; is the sampling instant or observation interval. The usual approach is to obtain the ratio

where N 0 / 2 is the power spectral density of the white noise and G(f) is the Fourier transform of the input signal g( t ) . The goal then is to find the form of H(.f‘) that riiaxinuses the right-hand sidc of eyn. 1. To accomplish this, Schwarzi inequality is invoked, which states that given two complex functions x ~ ( T ) and g2(4 in the real variable T satisf$ng the conditions

Fig. 4 Sampling a sinusoid of frequency fm at a rate fs. Note how the samples also fit an infinite array of sinusoids nf,*fm, n=l ,2 , 3, ...

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Time domain Frequency domain

Fig. 5 Frequency domain effect of (instantaneous) sampling of a sinusoid

then

The equality holds in this relation if, and only if,

where K is a constant and the asterisk denotes complex conjugation. Eniploying eqns. 2 arid 3 in eqn. 1 with gl(z) = H(f) and xz(z) = G(f)exp(,j2ref'7;), it follows that the right-hand side of eqn. 1 will be rnaxiniuni when the transfer function H(f) of the receive filter is given by

Taking the inverse Fourier transforni ofeqn. 4 gives the impulse response h(t) of the filter:

In the above, the third line follows from noting that the Fourier transform G(f) of a real signal g(t) satisfies C*(.f) = C(-f), the fourth line from malung the substitution = :f, and the last line by definition ofthe inverse Fourier transform. Thus the impulse response of a matched filter that gives optimum detection of a pulse g(t) in the presence of white noise is simply a

time-reversed arid delayed version of the pulse.

iVIatclzdj3trv: the rninirnal-mathcmati~~ appvnach Now consider an alternative solution of the above

problem following a miniiiial-17iathcmatics approach. HCf) and hence h ( t ) may be obtained by making three increasingly prescriptive observations:

The bandwidth of the filter niust be just enough to pass the incoming signal. Ifit is too wide, noise pom7er is unnecessarily admitted, and if it is too narrow then some signal energy is cut out. Thus, G(f) and H(f) must span exactly the same frequency band. How should they be shaped? The gain response I H ( f ) / of thc filter should not necessarily be flat within its passband. Rather, it should be such that the filter attenuates the whitc noise significantly at those frequencies where G(f) is small-since these frequencies contribute little to the signal energy. And the filter should boost chose frequencies at w-hich G(f) is large in order to niaxinlisc thc output signal energy Therefore the filter should be tailored to the incoming signal, with a gain response that is small where G ( f ) is mall and large where G(f) is large. In other words, the gain response of the filter should bc identical in shape to the aniplitude spectrum of the signal. That is,

where K is a constant. To complete the specification of the filter, its phase response is required. This is accomplished by noting that the riiaxiniurn instantaneous output sigial power occurs at the sampling instant t = '1; if ?very frequency component (i.c. cosine function) in the output sihmal po(t) is delayed by the same

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amount 7; arid has zero initial phase so that

&>(t) = Aocosl27Cf,(t-7;j] + Alcosl2qfl ( t 4 J I + A7cos[2Tcf2(t-?;)] + . . . (7)

resdting in d maxinium instantaneous value at t = given by

&>('l!j = Ao + At + A2 + . . . where Ao, Ai, A2, .. . are the aniplitudcs of the sinusoidal coiiipoiients of go(t) of respective freyuencies.fo,Ji,fz,. . .. Notc that these frequcncies are infinitesiiiially spaced, giving risc to a continuous spectrum G,(f ') . Rewriting eqri. 7 in the form

go(t) = Aocos(27Thfot-2~hT) +Alcos(27Tflt-2Zj-lfi) +A2cos(2?Tfzt- anf'2T-J + . .

shows that the phase spectruni of the output signal yo(t) is rpo(f) = - 2 ~ f 7 ; . Arid since

where rp)~(~'") is the phase response of tlie filter arid qn(f) is the phase spectruiri of the input signal, it f0Uows that

Coiiibining eqns. 6 and 8 gves tlie required filter transfer function:

W ) = KI C ( f ) I ex~[[ iq~(J ' ) I = KI W) I exp[-j~~(.f)lexp(-j2?ff?l) = KC:"(j)exp(j2~jT) (9)

'The impulse response h(t) of the filter is the inverse Fourier trarisform of its transfer function H ( f ) , and foUows from cqn. 9 when it is noted that complex conjugation of C ( j ) corresponds to a tirric revei-sal ofthe real signalX(t), arid that multiplying C*(,f) by die exponential term exp(~j2~fTE) corresponds to delayingg(-t) by 71. Thus,

(1 0) h( t ) = Kg( 21: - t )

Thc two results, eyns. 5 and 10 are clearly identical. But the second approach places eniphasis on a physical insight into the problem and is significantly less matherrxitical.

Savrpling: tht, nziniM.ial-niutherMatirs a p p m h The process of sanipling is pivotal in the digital

transmission of audio and video signals. Traditionally,

Fig. 6 Sampling an analogue signal g(t) = (kfi, f f 3 ) at a rate fs yields samples that also fit an infinite array of signals nf, + (kf,, kf3), n = 1, 2, 3 ,...

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Fig. 7 Frequency domain effect of sampling a band-limited analogue signal

sainpling i s covered using a matlieii~atical apprmcli that employs the Dirac delta function, Fourier transform, integrals and a siiic interpolation fiiiiction to derive condtions for a distortion-free saiiipling and to iiisc~iss the process of reconstructiiig the original signal f i o i r i

thc sariiplcs. But a graphical approach can be employed to introduce such importaut coriccpts as the sampling theorem, Nyquist rate arid aliasing in an intuitively satisfying way without obscuring matters with needless n~athematics. Consider the fotlowing.

Fig. 4 illustrates that the samples of a sinusoid of frequency ji7, taken at a rate .f; sarnpledsecond could originate not only from the original sinusoid, but also from an infinite array of siiiusoids of frequencies Y$ Ff;,,, where n = 1, 2, 3, .. . In other words, the spectrum of the saiiipled signal coiitaiiis the frequeiicies n z +.$, n = 0, 1, 2, 3, _ _ . Thus, in thc Grcqueiicy domain, the effect of instantaneous sampling is to take the baseband frequencies Itf;l, of the sinusoid and replicate them witliout change at regular intervals .f; along the irequency axis. The spectrum of the sanipled sinusoid is therekm as shown in Fig. 5.

Fig. 6 illustrates that when an analogue signal made up of a suni of sinusoids ( ih, kf2, $f?, . . .) is sanipled, then every constituent sinusoid is also replicated as above. Note that in Fig. 0 the analogue signal is

y ( f ) = A isiri(2?F/it) + A3si11(2nj3)

which is iteuoteclg(i) (kfi, kfb). Clearly the samples contaiii coiiiponents at nfi. + (kji, kjj,), n = 0, 1, 2, 3, . . . . Since all information-bearing signals can be

(discrete or continuous) suni ofsinusoids, it follows that in general the effect of saiiipling an arbitrary inf;>riiiatioti signal ofbal-idwidth E: is as shown in Fig. 7. Notc that so long as& 2 2B, the undistorted baseband spectrum ( w = O), arid hence the original sigial, can be recovered froiii the samples by passing them tlirougli a low-pass filter, which passes the spectl-uni at n = 0 but blocks all the spectra replicated at n 2 1. Sampling at the Nyquist rate-h = 2 H requires an unrealisable ideal brickwall filter for distortion-free reconstruction. If however -6 < 2B, there is some overlap between replicated spectra. The rcsiiltant spectrum is thereby distorted such that the origiiial signal cannot be ~-econstructed even with an ideal low- pass filter. This distortion is known as aliaing. The underlying cause of aliasing distortion is illustrated in Fig. 8, where we see that if a sinusoid is sampled at a rate less thaii two saiiiples per period (i.e.f; < 2j&), then the samples contain (i.e. fit) a lower frequency sinusoid of frequency .f; = 1.f; --fiI, I . A low-pass filter will unavoidably pass (i.e. reconstruct) this &as siiiusoid.f, along with J$,, and it is the presence of this new frequency component that causes a distortion of the reconstructed signal.

The above &scussion can be successfully extended4

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Fig. 8 Undersampling a sinusoid of frequency fm at a rate f, c 2fm generates an alias of frequency fa = Ifs- f,J

beyond instantaneous sampling without any need for abstruse matheiiiatics. This minimal-niatheiiiatics approach imparts to the student an unclouded insight into the problem and is especially suitable for newcomers to the subject.

Conclusion

The widely acknowledged declining mathematical skills of university entrants and their penchant for inininid-mathematics necessitate a critical review of the teaching approach employed by engineering acadenlics in order to cater to the abihty, experieiice and preference of a new breed of intakes. Since wastage statistics show students to be most at risk in their first- year, it is particularly iniportant that engineering departments explore approaches to modernising the way firstyear students are introduced to the subject in order to eliminate what a growing number of new and potential recruits perceive as an uneiendly mathematical gatekeeper.

A minimal-mathematics approach has been proposed and illustrated with two detailed exaniples h-om telecommunications. Innovative thinking arid further research can lead to thc extension of this approach to more and more engineering topics that are traditionally wrapped in matheniatics. The approach puts engineer- ing first and mathematics second, giving students an excellent insight into each engincering problem and equipping them to use mathematics wherever necessary as a problem-solving tool rather than an end.

Ths innovativc approach can be combined with computer-integrated lecturing, whereby students first interactively ‘dscover’ key parameters and concepts through guided computer simulations and a lecturer then hangs a more detded knowledge on the pegs that have been thus erected. The result would be the transforination of potentially boring or daunting lectures into a multi-sensory and fun engagement of the student. Arnied with a clear appreciation of the underlying engineering concepts, the student can then venture into a inore niatheinatical approach during subsequent eiicounters with the topic. Engineering education would benefit in terms of a significant improvement in recruitment and retention figures as well as the confidence and competence of the h s h e d products.

References

1 ‘Measuring the mathematics problem’. An Engineering Council Report (Engineering Council, London, 2000)

2 WOLF, A., and TIKLEY, C.: ‘The maths we need now’ (Institute of Education, London, 2000)

3 ‘Tackling the mathematics problem’. A LMS Report (London Mathematical Society (LMS),1995)

4 OTUNG, I. E.: ‘Cornniunication engineering principles’ (Palgrave, Basingstoke, 2001)

0 IEE: 2001

Dr. Otung is with the School of Electronics, University of Glamorgan, Pontypridd, CF37 1DL. UK.

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