Paper ID #12476

Correlation between engineering students’ performance in mathematics andacademic success

Dr. Gunter Bischof, Joanneum University of Applied SciencesAndreas Zwolfer, University of Applied Sciences Joanneum, Graz

Andreas Zwolfer is currently studying Automotive Engineering at the University of Applied SciencesJoanneum Graz. Prior to this he gained some work experience as a technician, also in the automotivesector. On completion of his studies, he intends to pursue a career in research.

Prof. Domagoj Rubesa, University of Applied Sciences FH JOANNEUM, Graz

Domagoj Rubesa teaches Engineering Mechanics and Mechanics of Materials at the University of AppliedSciences Joanneum in Graz (Austria). He graduated as naval architect from the Faculty of Engineering inRijeka (Croatia) and received his MSc degree from the Faculty of Mechanical Engineering in Ljubljana(Slovenia) and his PhD from the University of Leoben (Austria). He has industrial experience in a Croatianshipyard and in the R&D dept. of an Austrian supplier of racing cars’ motor and drivetrain components.He also was a research fellow at the University of Leoben in the field of engineering ceramics. Hisinterests include mechanical behavior of materials and in particular fracture and damage mechanics andfatigue, as well as engineering education.

c©American Society for Engineering Education, 2015

Page 26.410.1

Correlation between engineering students’ performance in

mathematics and academic success

Abstract

It is quite popular among engineering educators to suppose that the academic performance of

undergraduate engineering students depends on their mathematics skills, but there is only a

little data available that substantiates this assumption. This study aims at shedding some light

on this perceived dependency by comparing examination results in mathematics with those of

the core engineering subjects over a period of more than ten years.

To identify the relationship between the students’ individual mathematical proficiency and

their performance in applied engineering subjects, the examination performance of students in

first, second and third semester mathematics courses has been correlated with their

corresponding performances in mechanics and other mechanical engineering subjects. The

results show that there is a significant positive correlation between the mathematics and

mechanics grades; a correlation between mathematics and other engineering core subjects also

exists but is, in general, less distinct.

The study has been supplemented with a comparison of the dropout rate of students enrolled

in the classes from 2002 until 2009 with their performance on an anonymous pre-course

diagnostic test of mathematical skills, which took place every year in the first week of study.

Prior studies on the relationships between students’ university entry scores and their

performance in terms of grade point averages showed that the expected correlations either do

not exist or are too weak to base educational interventions on. A comparison of pre-college

mathematics skills with degree program drop-out rates, on the other hand, shows a clear trend

that a below-average high school mathematics education entails an elevated risk of drop-out.

Introduction

Retention of students at colleges and universities has long been a concern for educators, and

several univariate and multiple-variable analysis models have been developed for the

prediction of a student’s probability to drop out from university (for an overview see e.g.

Murtaugh, Burns, and Schuster1). Precollege characteristics - like high school grade point

averages - as well as university entrance exams have, in general, turned out to be useful

predictors of student retention.

A prior investigation of the drop-out probability at the engineering department of our

university (Andreeva-Moschen2) clearly showed that the university entry scores can be used

to identify groups of students at higher risk of failure. It also turned out that the probability

distribution for student drop-out depends on the type of high school the students graduated

from, namely secondary colleges of engineering or traditional high schools. Interestingly, the

university entry score distribution does not reflect any differences in this respect, which might

be due to the general character of the entrance exams. The written test, which is the main part

of the entrance exam, focuses mainly on cognitive ability and logical thinking, and to a lesser

Page 26.410.2

degree on mathematics and technical understanding. The study also revealed a positive,

though weak, correlation between the grade point averages of our students and their university

entry score. The weakness of this correlation is in accordance with related studies, e.g. the

investigation of the influence of the university entry score on the students’ performance in

Engineering Mechanics. Thomas, Henderson, and Goldfinch3 found it impossible to reliably

predict student performance in first year Engineering Mechanics based on their overall

performance in entrance exams. However, the introduction of a risk factor (i.e. the number of

students failing the course divided by the number of students with a university entry score in a

specific range) showed a clear trend that students with low university entry scores have an

elevated risk of failure.

Due to the trend that the popularity of engineering degrees among undergraduates temporarily

declined in recent years, we are confronted with an increased number of ‘at risk’ students

attending our engineering department. This situation led to comparably high attrition rates of

over 50 % in the degree program under consideration, which by many faculty members are

mainly attributed to difficulties encountered in the mathematical content of the program.

These difficulties are thought to arise from a lack of understanding as to what engineering

involves and an insufficient mathematical preparedness.

This under-preparedness of first-year university students is not only reflected in their

performance in the mathematics classes; it propagates into mathematically-oriented courses

like Engineering Mechanics, Strength of Materials, Thermodynamics, Fluid Mechanics, and

Control Engineering. In our university’s engineering degree programs, drop-out for academic

reasons primarily takes place in the first year of study, and the major “culprit” is Engineering

Mechanics, followed by Engineering Mathematics (the other courses mentioned before are

taught later in the curriculum). This is in good accordance with a study of Tumen, Shulruf,

and Hattie4 that singled out engineering students as more likely to leave after first year of

study than other students, indicating that engineering studies are a special case with specific

challenges and hurdles. Literature reveals that there is in general a strong correlation between

preparedness for college mathematics and the prospect of earning a university degree (see e.g.

McCormick and Lucas5), and especially in engineering education there is no doubt about the

particular importance of mathematics.

The competence of engineers rests to a large extent on their mathematical training, since

mathematics is not only a set of tools to model and analyze systems; it also provides training

in logical reasoning. Within an online survey involving more than 5700 registered engineers,

Goold and Devitt6 found out that, while almost two thirds of engineers use high level

curriculum mathematics in engineering practice, mathematical thinking has an even greater

relevance to engineers’ work. Nevertheless, there are problems associated with the role of

mathematics in engineering education, in particular related to attracting and retaining students

in engineering degree programmes.

In the present work, the examination performance of students in first and second year

mathematics courses are compared with their corresponding performances in mechanics and

other mechanical engineering subjects. This study was initiated by the finding of an

astonishing conformity of the dropout rates and students’ performances on a pre-course

mathematics diagnostics test over a period of eight years.

Page 26.410.3

Comparison of the dropout rate and the students’ performance on a pre-course

mathematics diagnostic test

Due to the fact that our students come from different schools or even educational systems, we

evaluate the precollege mathematics skills of our freshmen by a written 45-minutes

mathematics diagnostics test, which is anonymous and takes place within the first lecture of

an introductory mathematics course. The main purpose of this test is to give the instructors

information about the level of high school mathematics skills in the class. The instructors can

thus structure their lectures to accommodate to the students’ mathematics preparedness that

might be varying from year to year.

The test comprises about a dozen problems that have to be solved without the help of pocket

calculators. They cover essentially standard high school mathematics problems, supplemented

by a few questions that go beyond the average high school mathematics curricula.

Some typical tasks are:

• Basic algebra: Simplify a compound fraction like

yx

y

yx

1

11

2−

−

• Functions: Given the axes of a Cartesian coordinate system, draw the plots of the

functions y = sin(x), y = exp(x) and y = log(x)

• Exponential and logarithmic equations: Solve 312 23 +−

=xx

We started with these tests in 2002 and have meanwhile gained some insight and statistical

data for an evaluation. The test as well as the pre-test conditions remained the same until 2009

and therefore we now have the test results of eight classes available. Table 1 gives an

overview of the percentage of correct answers to the above mentioned tasks (n = 368).

Table 1: Percentage of correct answers in the anonymous mathematics diagnostics tests

simplify fraction plot sin(x) plot exp(x) plot log(x) exponential eq.

28.1 % ± 7.1 % 29.5 % ± 7.6 % 10.2 % ± 5.5 % 6.4 % ± 3.5 % 6.9 % ± 4.3 %

These results lead to the conclusion that many enrolees either did not master the content or

forgot much of the content they once mastered. Obviously, a considerable number of high

school graduates enrol in engineering programs under-prepared for the academic rigor of

university education. This, however, is not an isolated case; McCormick and Lucas’ study

revealed a serious disconnect between a high school diploma and the student’s preparedness

for post-secondary education5. The authors underlined that secondary and post-secondary

institutions have historically functioned as distinct entities with only modest interaction, and

offered a compilation of recommendations from experts who made notable advances in

mathematics preparedness in the education of high school students.

The 2007 class was asked to sit the precollege maths skills evaluation another two times, both

at the end of the first semester and at the end of the second semester. It came to a marked

improvement of the test results (see Figure 1), although the topics of the exam questions were

just a small part of the curriculum of the introductory mathematics course, and only implicitly

Page 26.410.4

involved in the second semester mathematics course. The number of students in the class

decreased in the meantime from 76 to 66 after the first semester (although no exams took

place in between these two evaluations), and to 55 after one year of study. Obviously, more

than 13 % of the class decided to discontinue their study without failing an exam. This

phenomenon was investigated in a previous paper2 and attributed to weaknesses in the

students’ approach to work and dedication to their study, which would have been necessary to

make up in relatively short time for the missing skills. Another 15 % of the class dropped out

after the second semester because of failing a first semester course. In the remaining years

required to complete their degree another 12 % dropped out (mainly in the second year of

study), which eventually led to a 4-year graduation rate of 60 %.

Figure 1: Test results of the 2007 enrollees at the beginning of the first semester (left, blank,

n = 76), at the end of the 1st semester (middle, gray n = 66) and at the end of the 2

nd semester

(right, dark gray, n = 45).

Experience showed that the attrition rates vary appreciably from year to year, even though the

curriculum, the faculty, and the overall study conditions remained essentially the same. The

circumstance that all the students of the classes 2002 to 2009 who persisted must have

graduated in the meantime (at our university, there is a maximum of three extra semesters

allowed for the completion of a degree program) provoked us to compare the percentages of

the anonymous precollege maths skills evaluation with the graduation rates. Due to the small

percentages accompanied by variances of the same order (see Table 1), only a few results

came into question for such a comparison. In Figure 2 the percentages of correct answers to

the compound fraction problem are compared with graduation rates, while in Figure 3 a

similar comparison with the beginners’ ability to correctly draw a sine function is depicted.

Although, of course, no causal relationship can be inferred from these figures, the conformity

of the trends is nonetheless astonishing. The product-moment correlation coefficient between

the graduation rates and the percentage of correctly solved compound fractions (Figure 2) is

0.844 at a confidence level of 99.5 %, and 0.815 at a confidence level of 99.0 % for the sine

functions (Figure 3). These results inspired us to investigate the degree of correlation between

the mathematics performances of our undergraduate students with their performances in other

subjects, which could shed some more light on the fluctuations in graduation rates.

Page 26.410.5

Figure 2: Percentage of graduates per year of matriculation (left, black) and of correctly

simplified compound fractions in the anonymous mathematics diagnostics test (right, gray).

Figure 3: Percentage of graduates per year of matriculation (left, black) and of correctly

plotted sine functions in the anonymous mathematics diagnostics test (right, gray).

Correlation of student performances in mathematics and other subjects

At the Joanneum University of Applied Sciences, we offer a variety of engineering degree

programs. The faculty considers it especially important to apply modern didactical methods

like project based learning in the degree program as early as possible to increase the

efficiency of knowledge transfer and to fortify the students’ motivation to learn and to

cooperate actively. Students are confronted, complementary to their regular courses, with

problems that are of a multidisciplinary nature and demand a certain degree of mathematical

Page 26.410.6

proficiency7. This leads to a closer cooperation among the faculty and thus to a better

coordination of the courses that take place in the same semester. Students' difficulties in

comprehension or application of knowledge induce instructors to review courses and align

course structures and contents. In addition to those internal reviews and adjustments of the

individual courses, the degree program’s curriculum is reviewed on a routine basis and

proposals for new courses, course deletions, and changes in the sequence of the courses are

reviewed and approved typically every five years. The mathematics and engineering

mechanics curricula have not remained unchanged either in the period of consideration and

investigation for several reasons; all these revisions were mainly guided by the aim to reduce

the drop-out rate.

A larger percentage of total drop-out takes place in early semesters than in later semesters (see

Figure 4). According to R. Stinebrickner and T. Stinebrickner8, the decline in the number of

student drop-out that takes place across semesters is well-established; it is partially caused by

the fact that students tend to learn about their academic performance during early semesters.

Figure 4: Decline in student number across the first four semesters, and the number of

students graduating after eight semesters of study (classes 2007 and 2008).

The courses that are considered by our students as the most difficult in the early semesters are

Engineering Mechanics and Engineering Mathematics. Failing one of these courses appears to

be particularly dispiriting and is most likely the main reason for the early student dropout. The

aim of this study was to find out to which extent the students’ mathematics skills affect their

performances in mechanics, which seems to be the crucial point in student retention.

In the degree program under discussion, the undergraduate mathematics education is covered

by a series of lectures taking place in the first three semesters of study. These Engineering

Mathematics lectures are comprehensive courses combining Algebra and Calculus as well as

Numerical Mathematics with an emphasis placed on the methods used in engineering (see

Tables A1 and A2 in the Appendix). There is, in general, not much creative leeway in the

design of the mathematics curriculum; it has to provide a logically organized sequence of

topics, but still permits a certain degree of freedom. Similar deliberations apply to

Engineering Mechanics. The classical sequence of topics in Engineering Mechanics starts

with statics and continues with kinematics and kinetics. However, statics can be regarded as a

special case of dynamics. The resulting, rather uncommon, arrangement of topics allows an

Page 26.410.7

integrative and more deductive approach, which was implemented in our department

beginning with class 2001. Thus Engineering Mechanics 1 in the 1st semester referred to

mechanics of particles, whereas Engineering Mechanics 2 in the 2nd

semester dealt with

mechanics of rigid bodies. Engineering Mechanics 3 in the 3rd

semester was devoted to

analytical mechanics (in the European sense). In 2008 all Engineering Mechanics courses

were shifted one semester later.

For the investigation of the degree of correspondence between the Engineering Mathematics

and Engineering Mechanics results, the Mechanics scores have been plotted against the

Mathematics scores for each student per semester. In Figure 5 the correlation of the scores for

the first two semesters of the classes 2003, 2004, and 2007 are depicted, which are

representative for the mechanics and mathematics curricula prior to 2008 (see Table A1).

Scores of the classes 2005 and 2006 were not available.

Figure 5: Engineering Mathematics scores versus Engineering Mechanics scores of the

classes 2003, 2004, and 2007 (1st semester in the left, 2

nd semester in the right diagrams).

Page 26.410.8

For all correlations Pearson’s product-moment correlation coefficients R and Spearman’s rank

correlation coefficients ρ have been determined. The Spearman correlation is less sensitive

than the Pearson correlation to outliers and especially suited for grade correlations. The

statistical significance of the obtained R and ρ has been estimated by Student’s t-test and is

represented by the corresponding p-values. Although there is a large scatter in the data sets, a

significant positive correlation can be observed, which is more pronounced in the 1st semester

plots.

The increasing number of dropouts (an exponential decay of the graduation rates with a decay

rate of λ = −0.04 year−1

is fitted to the data, see Figure 6) led to a curriculum reform in the

year 2008. The Engineering Mechanics lecture series, which, due to its characteristic

difficulty and level of abstraction, was regarded to be the main cause for early student drop-

out, should start only in the second semester to give the students the opportunity to improve

their mathematics skills beforehand. In addition, a better conformity of the mathematics

taught and the mathematics needed should be achieved (Table A2).

Figure 6: Percentage of graduates per year of matriculation (circles), and the percentages of

the correctly simplified compound fractions (squares) and correctly plotted sine functions

(triangles) in the corresponding anonymous mathematics diagnostics test. An exponential

function is fitted to each set of data.

Another subsequent consequence of the decreasing graduation rates was the introduction of a

university-wide mathematics bridge course for first-year students in the first two weeks before

the beginning of classes. This course can be attended voluntarily by students who do not feel

sufficiently prepared in mathematics for their future studies. As so often in education, the

question arises how students should catch up on the necessary mathematics skills and

understanding in only two weeks and how many of future students are reflective enough to

realize their need for additional preparation.

A comparison of the Engineering Mechanics scores and the Engineering Mathematics scores

in the second semester of the classes 2008 and 2010 (no scores were available of the class

2009) is illustrated in Figure 7. The plots resemble the first semester correlations in Figure 5

with similar correlation coefficients.

Page 26.410.9

Figure 7: 2nd

semester Engineering Mathematics scores versus Engineering Mechanics scores

of the classes 2008 and 2010.

In 2010, the degree program underwent a significant strategic review resulting in major

revisions in order to fulfil the requirements of the Bologna Accord (see e.g. Sanders and

Dunn9). The four-year undergraduate degree program has been transformed into a three-year

Bachelor’s program complemented by a two-year Master’s program. This reform had little

effect on the first year Mathematics and Mechanics courses, except that Analytical Mechanics

(formerly taught in Engineering Mechanics 3) was moved to the 1st semester of the Master’s

program and complemented with the application of variational principles to deformable

bodies (i.e. strength of materials). The correlation of the scores in these subjects of the classes

2011 and 2012 in the 2nd

semester is depicted in Figure 8.

Figure 8: 2nd

semester Engineering Mathematics scores versus Engineering Mechanics scores

of the classes 2011 and 2012.

Much more data are available if the students’ grades instead of scores are taken into account.

These data could be drawn from the final student grades database of the Registrar’s office.

The disadvantage of grades, compared with scores, is that comparability between different

grading systems is not automatically achieved. There are no letter grades in use in our

country; instead of that, grades from 1 to 5 are used (1 is the best grade, 5 means ‘failed’). A

conversion table is given in Table 2 with typical (but not universal) percentage intervals for

the corresponding grades.

Page 26.410.10

Table 2: Grading system used

Grade Percentage Descriptor US equivalent

1 90 – 100 Excellent 'A'

2 77 – 89 Good 'B'

3 64 – 76 Satisfactory 'C'

4 51 – 63 Sufficient 'D'

5 0 – 50 Insufficient Failing grade 'E/F'

The correlations between Engineering Mathematics grades and the final grades in Engineering

Mechanics and other mathematically-oriented courses are illustrated in Figures 9 to 11. These

data were obtained from more than ten classes of the four-year degree program.

Figure 9: Engineering Mathematics grades versus Engineering Mechanics grades for the

respective semesters

The highest correlation coefficient was obtained for the case when both Engineering

Mechanics 1 and Engineering Mathematics 1 were taught in the first semester. Engineering

Mechanics 1 in the second semester (top right in Figure 9) is slightly less correlated with

Engineering Mathematics 2 but leads to a higher number of better final grades. The relative

number of better grades in both Mechanics and Mathematics increases in higher semesters,

which may partially be caused by the fact that the least motivated students in the class have

then already dropped out.

In Figures 10 and 11 the final grades in Engineering Mathematics are related to the final

grades in Strength of Materials and in Thermodynamics. Here we found similar correlation

Page 26.410.11

coefficients as between Mathematics and Mechanics and, again, an increase in the relative

number of better grades in higher semesters can be observed.

Figure 10: Engineering Mathematics grades versus Strength of Materials grades

Figure 11: Engineering Mathematics grades versus Thermodynamics grades in the 2nd

(left)

and 3rd

(right) semester

All the correlations presented above have in common that virtually no student with an

excellent grade (“1”) in the respective engineering courses has failed (“5”) in Mathematics.

However, an excellent final grade in Mathematics can be accompanied by lower grades in the

engineering courses, but hardly ever by a fail grade (“5”).

In the Figures below the correlations between Engineering Mathematics and Engineering

Mechanics final grades (Figure 12), and between Electrical Engineering as well as

Thermodynamics with Engineering Mathematics grades (Figure 13) are shown. These

correlations are obtained from the student grades database of the three-year Bachelor’s degree

program, which started in 2011. Due to the relatively brief existence of this program, the

database contains a comparatively smaller number of students, leading to more statistical

uncertainty. Figure 12 relates Mechanics and Mathematics grades in the 2nd

and 3rd

semester.

Compared with the data in the four-year degree program, the correlation is comparatively

higher, but the relative number of better grades does not increase with higher semesters. A

similar picture emerges from the correlations between Electrical Engineering and

Thermodynamics with Engineering Mathematics. Lower grades are dominating even in the

third semester. However, the number of students seems to be too small to draw conclusions at

this point.

Page 26.410.12

Figure 12: Engineering Mathematics grades versus Engineering Mechanics grades for the 2

nd

(left) and 3rd

(right) semester

Figure 13: Engineering Mathematics grades versus Electrical Engineering (left) and

Thermodynamics (right) grades in the 2nd

and 3rd

semester respectively

In order to assess the conclusiveness of the correlation factors R and ρ of the grade

correlations, all courses offered in the first three semesters were correlated with each other,

and the respective rank correlation factors ρ are illustrated in Figures 14 and 15.

Figure 14: Spearman’s rank correlation for the grade distribution of all courses offered in the

first three semesters of the four-year degree program (classes 1996 to 2010). ρ increases from

magenta (ρ = 0) to red (ρ = 1).

Page 26.410.13

Figure 15: Spearman’s rank correlation for the grade distribution of all courses offered in the

first three semesters of the three-year degree program (classes 2011 to 2013). ρ increases from

magenta (ρ = 0) to red (ρ = 1).

Figures 14 and 15 reveal that the correlation factor on its own does not suffice to make

credible inferences.

Nevertheless, there is information comprised in the bivariate data displays that can provide

better insight into the interrelation between the performances in Mathematics and other

courses. All the correlations of the final grades in Engineering Mathematics with the final

grades in engineering courses have in common that low mathematics grades seem to bear a

higher risk of failing mathematically-oriented engineering courses. This finding suggests to

study in more detail the students’ risk of failure in Engineering Mechanics in the context of

their Engineering Mathematics performance.

The concept of risk factor has already been used by Thomas, Henderson, and Goldfinch3 for

the investigation of the influence of university entrance scores on students’ performance in

Engineering Mechanics. Risk factors are extensively used in medical statistics. The

Framingham study, a heart study of a population cohort from a town outside Boston starting

in the late 1940s, first proposed the risk factor concept in which physiological and behavioral

characteristics such as blood pressure, blood lipids, and cigarette smoking predicted

subsequent disease events10

. The risk factor concept can be adapted to the present study by

assessing the risk of failing the Engineering Mechanics course by relating the number of

students with a specific grade in Mathematics who failed the course to the total number of

students with the same grade in Mathematics.

In Figure 16 the Engineering Mechanics scores are cumulatively plotted against the

Engineering Mathematics scores of the classes 2003, 2004, 2007, 2008, 2010, 2011, and

2012. The scatter plot clearly shows that when the scores are dichotomized at the midpoint to

“passed” (> 50 %) and “failed” (≤ 50 %), almost none of the students who passed Mechanics

failed in Mathematics. On the other hand, there are a number of students with high

Mathematics scores who failed in Mechanics. Obviously, the application of mathematical

methods to real-world problems brings an extra dimension of difficulty.

Page 26.410.14

Figure 16: Engineering Mathematics scores versus Engineering Mechanics scores

The risk factors for the students in their first year of study were determined from the score

correlation depicted in Figure 16 and are presented in Figure 17. This plot shows the

percentage of students who had to take board exams in the Engineering Mechanics course for

those intervals of Mathematics scores represented by the corresponding letter grades.

Figure 17: Risk factor of failing Engineering Mechanics based on Mathematics grades

Students with a Mathematics score of less than 50 % (i.e., who had to take board exams in

Mathematics) have a risk factor of more than 70 % of failing the written examinations in the

Mechanics course. This risk factor reduces almost linearly as the Mathematics score

Page 26.410.15

increases, with a risk factor of only 4 % for those students with a Mathematics score between

90 % and 100 %.

The investigation of the interdependence of the students’ performances in all the

mathematically-oriented subjects in terms of their corresponding risk factors is currently work

in progress.

Implications and possible interventions

Having identified the decline in the students’ basic mathematical skills and level of

preparation on entry into higher education and its impact on the dropout rate, the question is

what to do about it.

In the last decades a multitude of measures has been developed by universities worldwide.

These include supplementary lectures, computer assisted learning, mathematics support

centers, additional diagnostic tests and streaming. Each has its merits, and brief details are

given below. A comprehensive overview is given, for instance, by Hawkes and Savage in

their report11

emerging from the 1999 workshop “Measuring the Mathematics Problem” at the

Møller Centre Cambridge.

Virtually always on top of the list of recommendations is a diagnostics assessment that allows

new students to reflect on their level of mathematics learning on entry to the university and

thereby provides motivation for improvement in skills shown to be weak. Diagnostic tests

play an important part in identifying students at risk of failing because of their mathematical

deficiencies, in targeting remedial help and, last but not least, in removing unrealistic faculty

expectations. A prompt and effective follow-up support of students whose mathematical

background is found to be poor by the tests is regarded as essential.

Once the results of a diagnostic test are known the question arises as to how provide effective

support for those students with deficiencies in basic knowledge and skills. The follow-up

strategies offered by Hawkes and Savage11

include supplementary lectures, additional

modules, computer assisted learning, mathematics support centers, additional diagnostic tests

and streaming.

Many universities have introduced additional units of study (bridging courses) running prior

to or in parallel with the first year courses. The problem with it is the danger of overloading

students with too much additional work when they are probably struggling with other

modules. And, in addition, it often turns out that bridging courses are attended mainly by the

well prepared and most hard working students, who never miss a chance to learn something

extra (see e.g. Sazhin12

).

Some universities attempt to cope with the different levels of mathematical preparedness by

streaming their students and teaching the weaker group separately, sometimes using the

services of experienced school teachers rather than university lecturers. In order to cover the

same syllabus as the stronger group it may be necessary to increase contact hours for the

weaker group13

.

Amongst the range of strategies for coping with a serious decline in students’ mastery of basic

mathematical skills, a further option is to reduce syllabus content or to replace some of the

Page 26.410.16

harder material with more revision of lower level work. This will definitely help the weaker

students. However, if more advanced material is removed to make space for this revision, it

disadvantages the more able making them less well prepared for the mathematical demands of

the more advanced parts of their studies. In addition, it has a longer-term effect of making

these students less able to compete for jobs13

.

In some institutions, the mathematics diagnostic test forms only a part of the prognosis of the

student. Once the students have obtained their results, further informal assessment in the form

of informal pre-study support talks with mathematics tutors are offered. In their review of

eight years of mathematics study support14

, Patel and Little concluded that individualized

learning programs and face-to-face mathematics study support, targeted through diagnostic

assessment tests, do really offer a solution to the problem of retaining and progressing

undergraduate engineering students who struggle with the mathematics content of their

courses. They suggest that a quiet, relaxed, supported study area together with peer-learning

study groups will help to overcome mathematics anxiety and result in tangible student

benefits.

Several universities have established such mathematics support centers that are tailored to

individual needs and where the students can attend at any time. Such centers provide one-to-

one support for all students with any type of mathematical or statistical problem they may

come across in their course. One of the central features of mathematics support centers seems

to be the provision of non-judgmental support of students outside their teaching departments.

It is a place where students can ask questions without being told ‘you should know that’14

.

Such centers can be very effective in dealing with students having specific isolated

difficulties. However, for the less well-prepared students a drop-in consultation is hardly

sufficient. If they need to gain a coherent body of mathematical knowledge there is no

substitute for an extended and systematic study course11

.

As a first measure against high dropout rates we are considering to “deanonymize” the

mathematics diagnostic tests. These tests will then not only allow identifying overall

deficiencies in the cohort but also the students’ individual strengths and weaknesses. This will

help to identify students who are significantly weaker than the rest of the group and thus be

offered individual help and attention. The establishment of a mathematics support center as an

additional provision to support students who are struggling with the mathematical

components of their undergraduate studies is also envisaged.

Conclusions

Student drop-out in engineering studies is certainly a multi-faceted problem that cannot be

simply explained only by comparably poor mathematics skills at the start of the study.

Resilience, motivation, teamwork competency and dedication are also important for the

successful study of engineering. Nevertheless, the comparison of graduation rates with test

outcomes raised the question whether a simple mathematics diagnostics test could perhaps

provide an early indicator for drop-out rates.

In order to gain some insight into the degree of relation between the students’ performances in

Engineering Mathematics and their success or failure in other courses, especially in

Engineering Mechanics, the students’ final grades and, if available, scores in both courses

were correlated. All comparisons resulted in significant positive correlations, which

Page 26.410.17

corroborate the idea that the engineering students’ mathematics skills are closely connected to

their overall academic success.

But, as is well-known, establishing a correlation between two variables is not a sufficient

condition to establish a causal relationship. The correlation factors obtained from the

comparisons of the final grades in various mathematically-oriented courses do not provide a

clear picture.

However, the introduction of a risk factor, which relates the number of students with a

specific grade in mathematics who failed the mechanics course to the total number of students

with the same grade in mathematics, provides clear information and allows the prediction of

failure probabilities.

Page 26.410.18

Appendix

Table A1: Mathematics and Mechanics course contents prior to Fall 2008

Engineering Mathematics 1 (1st semester) Engineering Mechanics 1 (1

st semester)

• Real-valued functions of one variable:

properties, graphs, and operations

• One-variable calculus: limit of a function,

derivative, differentiation rules, power series,

Riemann integration and integration

techniques

• Algebra of complex numbers

• Vector algebra, analytical geometry

• Linear algebra: matrices, eigenvalues and

eigenvectors, diagonalization of matrices,

numerical methods

• Introduction: general introduction into the

Engineering Mechanics, the task and the

subdivision of mechanics, an outline of

vector algebra;

• Axiomatic foundation of Newtonian

Mechanics

• Kinematics of a particle (incl. relative and

absolute motion)

• Dynamics of particles: axiomatic foundation

of Newtonian Mechanics, equations of

motion in inertial and non-inertial systems of

reference, the principles of linear and angular

impulse and momentum, work and energy,

and the conservation of energy

Engineering Mathematics 2 (2nd

semester) Engineering Mechanics 2 (2nd

semester)

• Ordinary differential equations (ODEs):

classes of ODEs, analytical and numerical

solution of ODEs

• Coupled systems of ODEs: analytical and

numerical solutions

• Integral transforms: Laplace and Fourier

transform

• Curvilinear and oblique coordinate systems

and coordinate transforms

• Vector differential calculus: scalar and vector

fields, gradient, divergence, curl, Laplacian

• Kinetics of a system of particles:

generalisation of the dynamics of a particle to

a system of particles with special

applications to the impact of two bodies and

the motion of a body of a variable mass

• Kinematics of rigid bodies

• Dynamics of rigid bodies: Newton-Euler’s

equations of motion, the principles of linear

and angular impulse and momentum, work

and energy, and the conservation of energy,

general, three-dimensional motion of a rigid

body, rotation about a fixed axis and plane

motion as special cases, statics as a special

case of dynamics of rigid bodies

Engineering Mathematics 3 (3rd

semester) Engineering Mechanics 3 (3rd

semester)

• Integral theorems

• Partial Differential Equations (PDE): classes

of PDEs, analytical solution of linear second

order PDEs, numerical solution of PDEs

• Numerical methods in mathematics: root

finding, systems of equations, interpolation,

integration, ODEs, PDEs (finite difference

and finite element method)

• Introduction to probability and statistics

• Introduction to Analytical Mechanics:

statements of the principle of virtual work,

d’Alembert’s principle and Lagrange’s

equations of the 2nd

kind and their application

to particles, systems of particles, rigid bodies

and systems of rigid bodies

Page 26.410.19

Table A2: Mathematics and Mechanics course contents in the academic years 2008 to 2013

Engineering Mathematics 1 (1st semester)

• Real-valued functions of one variable:

properties, graphs, and operations

• One-variable calculus: limit of a function,

derivative, differentiation rules, power series,

Riemann integration and integration

techniques

• Algebra of complex numbers

• Vector algebra, analytical geometry, linear

systems of equations

Engineering Mathematics 2 (2nd

semester) Engineering Mechanics 1 (2nd

semester)

• Linear algebra: matrices, eigenvalues and

eigenvectors, diagonalization of matrices,

numerical methods

• Ordinary differential equations (ODEs):

classes of ODEs, analytical and numerical

solution of ODEs

• Coupled systems of ODEs: analytical and

numerical solutions

• Integral transforms: Laplace and Fourier

transform

• Introduction: general introduction into the

Engineering Mechanics, the task and the

subdivision of mechanics, an outline of

vector algebra;

• Axiomatic foundation of Newtonian

Mechanics

• Kinematics of a particle (incl. relative and

absolute motion)

• Dynamics of particles: axiomatic foundation

of Newtonian Mechanics, equations of

motion in inertial and non-inertial systems of

reference, the principles of linear and angular

impulse and momentum, work and energy,

and the conservation of energy

Engineering Mathematics 3 (3rd

semester) Engineering Mechanics 2 (3rd

semester)

• Curvilinear and oblique coordinate systems

and coordinate transforms

• Vector differential calculus: scalar and vector

fields, gradient, divergence, curl, Laplacian,

integral theorems

• Partial Differential Equations (PDE): classes

of PDEs, analytical solution of linear second

order PDEs, numerical solution of PDEs

• Kinetics of a system of particles:

generalisation of the dynamics of a particle to

a system of particles with special

applications to the impact of two bodies and

the motion of a body of a variable mass

• Kinematics of rigid bodies

• Dynamics of rigid bodies: Newton-Euler’s

equations of motion, the principles of linear

and angular impulse and momentum, work

and energy, and the conservation of energy,

general, three-dimensional motion of a rigid

body, rotation about a fixed axis and plane

motion as special cases, statics as a special

case of dynamics of rigid bodies

Engineering Mechanics 3 (4th semester)

• Introduction to Analytical Mechanics:

statements of the principle of virtual work,

d’Alembert’s principle and Lagrange’s

equations of the 2nd kind and their

application to particles, systems of particles,

rigid bodies and systems of rigid bodies

Page 26.410.20

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