Reassessing the mathematics content of engineering ... - faraday

EDUCATION

Reassessing the mathematics content of engineering education by I. E. Otung This paper reviews the problem of declining mathematical skills and appetite amongst university entrants. This decline necessitates a critical review of the traditional approach followed in engineering education in order to cater adequately for the abilities and preferences of the type of students becoming prevalent in our universities. A minimal-mathematics methodology that does not impact negatively on standards is discussed with examples drawn from telecommunications. Designed to endear rather than deter new students, this approach rightfully puts engineering first and mathematics second, and removes what a growing number of potential recruits now perceive as an unfriendly gatekeeper at the entrance to the study of engineering.

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ell-trained engineers have a vital role to play in ensuring the economic prosperity and competitiveness of any nation in a technology-dominated world. However, the growing importance of engineering, especially telecommunications and its applications, has coincided with a serious decline in both appetite for and competence in mathematics amongst university entrants, including those with good A-level grades. Engineering departments are finding it increasingly difficult to fill available degree places with students who have the required skills and motivation in mathematics. Consequently, a growing number of students who come into engineering programmes are not only apprehensive about mathematics but are in fact deficient in several key areas. Confronting such students with a lot of mathematics before they have had the opportunity to remedy their deficiencies and improve their attitude 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 treatment and therefore significantly undermining the competence of graduate engineers. This paper examines the mathematics problem and points a way out of the above dilemma. Observations of declining mathematical skills and an overwhelming 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 lowering 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 department more time to remedy their mathematical deficiencies with less risk of failure or withdrawal.

The mathematics handicap Evidence Academics in engineering departments are unanimous in acknowledging that there has been a steady decline in essential mathematical skills amongst new intakes to their degree programmes over the last decade. This problem has been the subject of several recent studies culminating in recommendations mainly aimed at improving the mathematics preparedness of new engineering undergraduates. A recent report1 published by the Engineering Council presents a number of objective pieces of evidence, two of which are summarised below. 5 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 76%, that of the 1997 cohort was a mere 50%, and none of the intakes since 1995 have registered an average score above 56%. 5 Since 1991 Coventry University has given a standard diagnostic test to new students entering its mathematics-based courses. The test covers seven topics, namely basic arithmetic, basic algebra, lines and curves, triangles, further algebra, trigonometry and

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

and systems require mathematical models that encapsulate the important parameters and applicable physical laws. Although some of these laws can be qualitatively described, a precise quantitative statement requires the language of mathematics. 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 measurements, and these tasks, sometimes critical, require important mathematical skills. Thus, it may be concluded that it is impossible to raise competent engineers on a nonmathematical diet.

(i) in each year the performance decreases with A-level grade—an expected and reassuring correlation (ii) the performance of those students with advanced GNVQ is significantly 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 dilution 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.

Inhomonegeous intake In view of the current pre-19 educational handicap, engineering departments have a greatly reduced pool of students with adequate mathematical preparedness from which to recruit into their degree places. New intakes in many departments now come from a wide range of vocational, A-level, mature, access and foundation backgrounds. This gives rise to a very inhomogeneous cohort. Diagnostic testing of new undergraduates is now widely used to identify the mathematical weakness of individual students and that of the whole cohort. Remedial measures, based on a variety of strategies such as supplementary classes, computer-assisted learning and mathematics support centres, can then be individually prescribed to bring each student up to speed in their areas of deficiency. There are, however, nontrivial problems in this belated attempt at imparting mathematical skills that should have become ingrained in a student prior to entering university.

The general consensus is that students with a good A-level mathematics grade can no longer be assumed to possess the mathematical skills required by the traditional approach to the training of engineers. Engineering academics therefore face a critical problem in the teaching of their modules. In the shortterm at least, this problem can only get worse for several related reasons. Underlying causes There is a growing dearth of competent and wellmotivated mathematics teachers in the pre-19 education sector2. Most of the qualified mathematics teachers are aged over 40, but increasingly the younger mathematicians who would replace them opt for other, more rewarding jobs that abound in a booming and ITdriven 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 university1,2. An increasing number of students come to university having never had a positive experience of mathematics. These students are understandably very apprehensive of mathematics-based courses and are at a real risk of failure or withdrawal if confronted with a lot of mathematics. 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 skill born of a good grounding in basic arithmetic.

Problems The attitude (towards mathematics) of a mathematically deficient student is likely to be very 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, limited staff and accommodation resources and an increasingly diverse cohort make mixed-ability teaching inescapable. As a result, the deficient students will be seriously hampered by the traditional approach employed in engineering education. It is worth pointing out that this approach was developed prior to 1986, during what has been described1 as the ‘golden age of mathematics’, and inevitably assumes a wide range of mathematical skills that a growing number of new recruits, raised in a different age, do not possess. Many of these students may become discouraged and drop out before remedial measures have had time to yield the intended benefits. Even the more determined ones who decide to stick it out remain at a serious risk of academic failure. The withdrawal rate in one engineering department in a UK university during the 1999/2000 academic

Impact on engineering education The need for mathematics The tool of mathematics is indispensable to engineering. Analysis and design of engineering structures


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data to determine what fraction of this high casualty was occasioned by a mathematical high-hurdle, it is known that only 15% of students who withdrew gave financial difficulties as the reason. The loss of students translates directly into a loss of income for the department involved. But when this withdrawal involves engineering students, then its more significant impact is perhaps the undesirable dent made on a crucial area of the nation’s skills base.

percentage of class, %

70 60 67

50

59

40 30 20 41

10

35 0

52 30 A

Minimal mathematics

6 minimal maths

C

3 7

2000/01 1999/00

B 1998/99

lots of maths

Fig. 1 Percentage of students (during the academic years 1998/99, 1999/00, 2000/01) 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 class, %

80 70 60 50

59

82

40 30

65

20 10 35

0 (B) endure

18 6

28 (A) enjoy 7

0 2000/01 1999/00

(C) withdraw 1998/99

It will take many years of concerted effort and government investment to correct the mathematics deficiency in pre-19 education and provide engineering departments with a sufficiently large pool of mathematically competent recruits. Until this golden age returns, however, urgent action is needed to increase retention of mathematically deficient intakes in engineering departments. Adopting a non-mathematical curriculum is out of the question since this would seriously undermine the competence of graduate engineers. A new approach can however be considered that puts engineering first and mathematics second, and gives beginning students the opportunity to become endeared to engineering while gradually remedying their deficiencies in mathematics. The withdrawal 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 traditionally mathematical approach in the delivery of first-year engineering modules.

Observed preference Since September 1998 Fig. 2 Percentage of students (during academic years 1998/99, 1999/00, 2000/01) successive cohorts taking 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’ a first-year module dealing with basic telecommunications in the School of Electronics at the University of Glamorgan year was 29% of first-year students—the highest in the have been given a questionnaire at the beginning of the entire university—compared to only 8% for non-first-year first lecture. Individual responses are returned students of the same department. Although there is no ENGINEERING SCIENCE AND EDUCATION JOURNAL AUGUST 2001

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anonymously in the same lecture before the commencement of teaching. Students are informed that the information will be used to determine how best to deliver the module for their benefit, and 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 options to indicate what they would prefer in the module: (A) mathematics only where unavoidable; (B) a lot of mathematics; (C) absolutely no mathematics. Fig. 1 shows the response in terms of the percentage of students choosing each option during the three academic years. The results consistently show that the vast majority of these students are averse to much mathematics. In the three cohorts questioned, an average of only 5% of the students (with a standard deviation of 2%) wanted a lot of mathematics in the module. Worryingly, an average of 39% of the students wanted absolutely no mathematics—an impracticable approach in view of the discussion on the need for mathematics. A second question sought to find out what the students would do if there was a lot of mathematics in the module. Three options were provided: (A) take on the challenge happily; (B) endure; (C) withdraw. Fig. 2 shows the response to the second question over the three-year period 1998–2000. An average of only 27% of the students would happily accept the challenge of a lot of mathematics in the module. Be warned, however, that this particular group of students is not necessarily mathematically skilled—as evidenced by responses to another (unreported) question in the questionnaire. The number of students at a serious risk of withdrawal is therefore likely to be significantly larger than the 4% (average) shown in Fig. 2 since there will subsequently be some migration from the ‘endurance’ and even the ‘happy’ camps into the ‘withdrawal’ camp. It can be seen that a mathematical approach would be off-putting to about three quarters of the students questioned. Ignoring their preference can contribute to avoidable withdrawal or failure. Interestingly, when the first question was asked to 13 practising engineers—products of various UK universities, some of them with less than 2 years experience—attending a 5-day short course on digital telecommunication networks held in October 2000, 11 of them (or 85%) categorically chose option A, with the remaining engineers going for a lot of mathematics (B). Features and implementation It can be seen from the foregoing that the best interests of the vast majority of students and a good engineering education can be jointly served through a minimal-mathematics approach where mathematics is employed only to the extent and at a level

that is absolutely necessary. Emphasis is placed on the underlying engineering considerations, and lucid graphs and diagrams are employed to facilitate understanding and assimilation. Mathematics is given its rightful place, which is second to engineering, and a physical insight into the problem at hand. A graphical approach is freely used where necessary to simplify a mathematical computation or illustrate a theorem. Developing such a teaching approach requires a thorough reassessment of current methods, changes in approach being made wherever necessary to meet students at their level while ensuring that they have an unclouded insight into the underlying engineering concepts and a good grounding in the theoretical principles. It is insufficient to string together an elegant mathematical derivation of a concept for students who do not yet possess the skill or motivation to follow it. Such an approach is rather insensitive and leads to students employing ‘black-box’ formulas whose limitations they cannot fully appreciate. They will lack the insight, confidence and competence that could very easily have been imparted through a minimalmathematics approach. The above minimal-mathematics strategy is used in a new book on communication engineering4, which covers core telecommunication topics such as signal analysis and transmission, amplitude, frequency and phase modulations, digital baseband transmission, digital modulation, multiplexing strategies, noise effects, etc. The book received the endorsement of reviewers as an excellent contribution to its field. The approach has also been tested in short courses on telecommunications for practising engineers, attracting very positive feedbacks such as the following: 5 ‘Made the subjects enjoyable and easy to follow.’ 5 ‘Excellent analogies and examples were used to describe complex theorems.’ 5 ‘Explained very abstract subjects in a way that improved understanding of mathematical descriptions.’ 5 ‘Fantastic.’ One inference that can be drawn is that students find the predominantly mathematical approach commonly followed in undergraduate textbooks on the subject less enjoyable and more abstruse. It must be emphasised that a minimal-mathematics

matched filter x (t)

Σ

signal g(t)

white noise

h(t)

y (t)

sample at t=Ts

Fig. 3 Signal detection in noise


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Minimal-mathematics examples

approach does not mean syllabus dilution. Neither is the treatment shallow, as would be appropriate for educating a non-technical and non-mathematical audience. On the contrary, the same topics are covered to the same depth, but with mathematical rigour subordinated to a practical insight into the engineering problem. This may sometimes be achieved with the aid of lucid graphs and diagrams. For example, the topic of analogue signal sampling in telecommunications can be treated in depth using a predominantly graphical approach4 that gives the student a thorough understanding of all the important concepts and a proficiency in anti-alias filter design, aperture effect correction, sampling of lowpass and bandpass signals, etc. A brief demonstration of this illuminating graphical approach is presented in the next section of the paper. At other times, a little thought may reveal an alternative and simpler mathematical derivation. A detailed example of this possibility is presented below on the subject of specifying a receive filter that gives optimum signal 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 significantly toned down in mathematical rigour—an appealing feature to most students.

Matched filter: the traditional approach The specification of a receive filter (called a matched filter) that gives optimum detection of a signal in the presence of additive white Gaussian noise is usually derived by invoking Schwarz’s inequality. In Fig. 3, the transfer function H( f ) and hence the impulse response h(t) of a filter is to be specified so as to maximise the instantaneous output signal power Po (Ts ) compared to the average output noise power Pn, where Ts is the sampling instant or observation interval. The usual approach is to obtain the ratio ∞

Po(Ts ) |∫–∞ H( f )G( f )exp( j2π f Ts )df|2 Pn = No ∞ |H( f )|2df 2 –∞

∫

where No /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 maximises the right-hand side of eqn. 1. To accomplish this, Schwarz’s inequality is invoked, which states that given two complex functions g1(τ) and g2(τ) in the real variable τ satisfying the conditions

samples at fs analogue signal at fm

fm=4 kHz; fs=12 kHz fs–fm=8 kHz; fs+fm=16 kHz 2fs–fm=20 kHz; 2fs+fm=28 kHz

0·5 0

t, ms

fs+fm fs – fm 0

t, ms

0·5 0

0·5

t, ms

2fs+fm 2fs – fm

0

(1)

t, ms 0·5

0·5 0

t, ms

Fig. 4 Sampling a sinusoid of frequency fm at a rate fs. Note how the samples also fit an infinite array of sinusoids nfs ± fm, n = 1, 2, 3, …


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An sinusoid of frequency fm

g(t)

f

t –fm

0

fm

Aδn gδ (t)

samples taken at rate fs=3fm

t

f –fs–fm

–fs

–fs+fm –fm

Time domain

0

fm

fs–fm

fs

fs+fm

Frequency domain

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

∞

∫ |g1(τ )|2dτ < ∞,

∫ |g2(τ )|2dτ < ∞

–∞

time-reversed and delayed version of the pulse.

–∞

then ∞

∞

∞

–∞

–∞

–∞

|∫ g1(τ )g2(τ )dτ|2 ≤ ∫ |g1(τ )|2dτ ∫ |g2(τ )|2dτ

Matched filter: the minimal-mathematics approach Now consider an alternative solution of the above problem following a minimal-mathematics approach. H( f ) and hence h(t) may be obtained by making three increasingly prescriptive observations:

(2)

The equality holds in this relation if, and only if, g1(τ) = Kg2✽(τ )

5 The bandwidth of the filter must be just enough to pass the incoming signal. If it is too wide, noise power 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? 5 The gain response |H( f )| of the filter should not necessarily be flat within its passband. Rather, it should be such that the filter attenuates the white noise significantly at those frequencies where G( f ) is small—since these frequencies contribute little to the signal energy. And the filter should boost those frequencies at which G( f ) is large in order to maximise the output signal energy. Therefore the filter should be tailored to the incoming signal, with a gain response that is small where G( f ) is small and large where G( f ) is large. In other words, the gain response of the filter should be identical in shape to the amplitude spectrum of the signal. That is,

(3)

where K is a constant and the asterisk denotes complex conjugation. Employing eqns. 2 and 3 in eqn. 1 with g1(τ ) = H( f ) and g2(τ ) = G( f )exp( j2π f Ts ), it follows that the right-hand side of eqn. 1 will be maximum when the transfer function H( f ) of the receive filter is given by H( f ) = KG ✽( f )exp(–j2πf Ts )

(4)

Taking the inverse Fourier transform of eqn. 4 gives the impulse response h(t) of the filter: ∞

h(t) = ∫ H( f )exp( j2π ft)df –∞ ∞

= ∫ KG ✽( f )exp(–j2π f Ts )exp( j2π ft)df –∞ ∞

= ∫ KG(–f )exp[–j2π f (Ts–t)]df –∞ ∞

= K ∫ G(λ)exp[ j2πλ (Ts –t)]dλ –∞

= Kg(Ts –t)

(5)

|G( f )| = K|H( f )|

(6)

where K is a constant. 5 To complete the specification of the filter, its phase response is required. This is accomplished by noting that the maximum instantaneous output signal power occurs at the sampling instant t = Ts if every frequency component (i.e. cosine function) in the output signal go(t) is delayed by the same

In the above, the third line follows from noting that the Fourier transform G( f ) of a real signal g(t) satisfies G ✽( f ) = G(–f ), the fourth line from making the substitution λ = –f, and the last line by definition of the 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


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ϕ H ( f ) = –ϕ g ( f ) – 2π f Ts

amount Ts and has zero initial phase so that

(8)

go(t) = A0cos[2π f0(t –Ts )] + A1cos[2π f 1(t –Ts )] + A2cos[2π f 2(t –Ts )] + … (7)

Combining eqns. 6 and 8 gives the required filter transfer function:

resulting in a maximum instantaneous value at t = Ts given by

H( f ) = K|G( f )|exp[ jϕ H ( f )] = K|G( f )|exp[-jϕ g ( f )]exp(–j2π f Ts) = KG ✽( f )exp(–j2π f Ts) (9)

go(Ts ) = A0 + A1 + A2 + …

The impulse response h(t) of the filter is the inverse Fourier transform of its transfer function H( f ), and follows from eqn. 9 when it is noted that complex conjugation of G( f ) corresponds to a time reversal of the real signal g(t), and that multiplying G ✽( f ) by the exponential term exp(–j2π f Ts ) corresponds to delaying g(–t) by Ts. Thus,

where A0, A1, A2, … are the amplitudes of the sinusoidal components of go(t) of respective frequencies f0 , f 1, f 2,…. Note that these frequencies are infinitesimally spaced, giving rise to a continuous spectrum Go( f ). Rewriting eqn. 7 in the form go(t) = A0cos(2π f0t – 2π f0Ts ) +A1cos(2π f 1t – 2π f 1Ts ) +A2cos(2π f 2t – 2π f 2Ts ) + …

h(t) = Kg(Ts – t)

shows that the phase spectrum of the output signal go(t) is ϕ o( f ) = –2π f Ts . And since

(10)

The two results, eqns. 5 and 10 are clearly identical. But the second approach places emphasis on a physical insight into the problem and is significantly less mathematical.

ϕ o( f ) = ϕ H ( f ) + ϕ g( f )

where ϕ H( f ) is the phase response of the filter and ϕ g( f ) is the phase spectrum of the input signal, it follows that

Sampling: the minimal-mathematics approach The process of sampling is pivotal in the digital transmission of audio and video signals. Traditionally,

analogue signal g(t)=A1sin(2πf1t)+A 3sin(2πf3t) samples at fs

f1=4 kHz; f3=12 kHz fs=30 kHz A1=3A3

0·5

n=0 0

t, ms

A1sin[2π(fs+f1)t]+A3sin[2π(fs+f3)t ] –A1sin[2π (fs–f1)t]–A3sin[2π (fs–f3)t ]

0

t, ms

0·5

n=1 0

0·5

A1sin[2π(2fs+f1)t ]+A3sin[2π(2fs+f3)t]

–A1sin[2π (2fs–f1)t]–A3sin[2π (2fs–f3)t]

n=2

0

t, ms

t, ms 0·5

0·5 0

t, ms

Fig. 6 Sampling an analogue signal g(t) (±f1, ±f3) at a rate fs yields samples that also fit an infinite array of signals nfs + (±f1, ±f3), n = 1, 2, 3,…


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bandlimited analogue signal t

–B

B

–B

B

f realisable LPF response

sampling at fs>2B t f –fs

fs

ideal brickwall LPF response

sampling at fs=2B t

–fs

–B

B

fs

f distorted spectrum, hence distorted reconstructed signal

sampling at fs