Paper ID #12476

Correlation between students’ performance in mathematics and academic success

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

Andreas Zwolfer¨ is currently studying at the University of Applied Sciences Joanneum Graz. Prior to this he gained some work experience as a technician, also in the automotive sector. On completion of his studies, he intends to pursue a career in research. Prof. Domagoj Rubesa,ˇ University of Applied Sciences FH JOANNEUM, Graz

Domagoj Rubesaˇ teaches Engineering Mechanics and Mechanics of Materials at the University of Applied Sciences Joanneum in Graz (Austria). He graduated as naval architect from the Faculty of Engineering in Rijeka (Croatia) and received his MSc degree from the Faculty of in Ljubljana (Slovenia) and his PhD from the University of Leoben (Austria). He has industrial experience in a Croatian shipyard 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. His interests include mechanical behavior of materials and in particular fracture and damage mechanics and fatigue, as well as .

c American Society for Engineering Education, 2015

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 to drop out from university (for an overview see e.g. Murtaugh, Burns, and Schuster 1). 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-Moschen 2) 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 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 Goldfinch 3 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, , and . 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 Hattie 4 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 Lucas 5), and especially in engineering education there is no doubt about the particular importance of mathematics.

The competence of 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 Devitt 6 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.

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: 1 − 1 x y • Basic algebra: Simplify a compound fraction like y − 1 x 2 y • 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 32x−1 = 2 x+3

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 education 5. 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 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 paper 2 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 1 st 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.

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 proficiency 7. 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. Stinebrickner 8, 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 integrative and more deductive approach, which was implemented in our department beginning with class 2001. Thus Engineering Mechanics 1 in the 1 st semester referred to mechanics of particles, whereas Engineering Mechanics 2 in the 2 nd semester dealt with mechanics of rigid bodies. Engineering Mechanics 3 in the 3 rd 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 (1 st semester in the left, 2 nd semester in the right diagrams).

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 1 st 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.

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 Dunn 9). 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 1 st 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 2 nd 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.

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 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 2 nd (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 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 2 nd and 3 rd 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.

Figure 12: Engineering Mathematics grades versus Engineering Mechanics grades for the 2 nd (left) and 3 rd (right) semester

Figure 13: Engineering Mathematics grades versus Electrical Engineering (left) and Thermodynamics (right) grades in the 2nd and 3 rd 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).

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 Goldfinch 3 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 events 10 . 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.

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 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 report 11 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 Savage 11 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. Sazhin 12).

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 group 13.

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 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 jobs 13.

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 support 14, 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 course 11 .

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 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 .

Appendix

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

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

• Real-valued functions of one variable: • Introduction: general introduction into the properties, graphs, and Mechanics, the task and the • One-variable calculus: limit of a function, subdivision of mechanics, an outline of derivative, differentiation rules, power series, vector algebra; Riemann integration and integration • Axiomatic foundation of Newtonian techniques Mechanics • Algebra of complex numbers • Kinematics of a particle (incl. relative and • Vector algebra, analytical geometry absolute motion) • : matrices, eigenvalues and • Dynamics of particles: axiomatic foundation eigenvectors, diagonalization of matrices, of Newtonian Mechanics, equations of numerical methods 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 (2 nd semester) Engineering Mechanics 2 (2 nd semester)

• Ordinary differential equations (ODEs): • Kinetics of a system of particles: classes of ODEs, analytical and numerical generalisation of the dynamics of a particle to solution of ODEs a system of particles with special • Coupled systems of ODEs: analytical and applications to the impact of two bodies and numerical solutions the motion of a body of a variable mass • Integral transforms: Laplace and Fourier • Kinematics of rigid bodies transform • Curvilinear and oblique coordinate systems • Dynamics of rigid bodies: Newton-Euler’s and coordinate transforms equations of motion, the principles of linear • Vector differential calculus: scalar and vector and angular impulse and momentum, work fields, gradient, divergence, curl, Laplacian 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 (3 rd semester) Engineering Mechanics 3 (3 rd semester)

• Integral theorems • Introduction to Analytical Mechanics: • Partial Differential Equations (PDE): classes statements of the principle of virtual work, of PDEs, analytical solution of linear second d’Alembert’s principle and Lagrange’s order PDEs, numerical solution of PDEs equations of the 2 nd kind and their application • Numerical methods in mathematics: root to particles, systems of particles, rigid bodies finding, systems of equations, interpolation, and systems of rigid bodies integration, ODEs, PDEs (finite difference and finite element method) • Introduction to probability and statistics

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

Engineering Mathematics 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 systems of equations Engineering Mathematics 2 (2 nd semester) Engineering Mechanics 1 (2 nd semester) • Linear algebra: matrices, eigenvalues and • Introduction: general introduction into the eigenvectors, diagonalization of matrices, Engineering Mechanics, the task and the numerical methods subdivision of mechanics, an outline of • Ordinary differential equations (ODEs): vector algebra; classes of ODEs, analytical and numerical • Axiomatic foundation of Newtonian solution of ODEs Mechanics • Coupled systems of ODEs: analytical and • Kinematics of a particle (incl. relative and numerical solutions absolute motion) • Integral transforms: Laplace and Fourier • Dynamics of particles: axiomatic foundation transform 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 (3 rd semester) Engineering Mechanics 2 (3 rd semester) • Curvilinear and oblique coordinate systems • Kinetics of a system of particles: and coordinate transforms generalisation of the dynamics of a particle to • Vector differential calculus: scalar and vector a system of particles with special fields, gradient, divergence, curl, Laplacian, applications to the impact of two bodies and integral theorems the motion of a body of a variable mass • Partial Differential Equations (PDE): classes • Kinematics of rigid bodies of PDEs, analytical solution of linear second • Dynamics of rigid bodies: Newton-Euler’s order PDEs, numerical solution of PDEs 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 th Engineering Mechanics 3 (4 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

Bibliography

1. Paul A. Murtaugh, Leslie D. Burns, and Jill Schuster, “Predicting the retention of university students”, Research in Higher Education, Vol. 40, No. 3, 1999 2. Emilia Andreeva-Moschen, “The five main reasons behind student enrollment and later drop-out”, ASEE Annual Conference and Exposition, San Antonio, TX, June 10-13, 2012 3. Giles Thomas, Alan Henderson, and Thomas Goldfinch, “The Influence of University Entry Scores on Performance in Engineering Mechanics”, 20 th Australasian Association for Engineering Education Conference, Adelaide, SA, Dec 6-9, 2009 4. Sarah Tumen, Boaz Shulruf, and John Hattie, “Student pathways at the university: patterns and predictors of completion”, Studies in Higher Education, 33(3), 233-252, 2008 5. Nancy J. McCormick and Marva S. Lucas, “Exploring mathematics college readiness in the United States”, Current Issues in Education, 14(1), 2011 6. Eileen Goold and Frank Devitt, “The role of mathematics in engineering practice and in formation of engineers”, SEFI 40th annual conference: Mathematics and Engineering Education, Thessaloniki, 2012 7. Günter Bischof, Emilia Bratschitsch, Annette Casey, and Domagoj Rubeša, “Facilitating engineering mathematics education by multidisciplinary projects”, American Society for Engineering Education, ASEE Annual Conference and Exposition, Honolulu, HI, June 24-27, 2007 8. Ralph Stinebrickner and Todd Stinebrickner, “Academic Performance and College Dropout: Using Longitudinal Expectations Data to Estimate a Learning Model”, The National Bureau of Economic Research NBER, Working Paper No. 18945, Issued in April 2013 9. Ross H. Sanders and John M. Dunn, “The Bologna Accord: A Model of Cooperation and Coordination”, Quest, Volume 62, Issue 1, 2010 10. Thomas R. Dawber, “The Framingham Study: the Epidemiology of Atherosclerotic Disease”, Cambridge, Mass., Harvard University Press, 1980 11. Trevor Hawkes and Mike Savage, (eds.), “Measuring the Mathematics Problem”, London: Engineering Council, 2000. Accessible via www.engc.org.uk/about-us/publications.aspx (12 March 2015) 12. Sergei Sazhin, “Teaching Mathematics to Engineering Students”, Int. J. Engng Ed. Vol. 14, No. 2, p. 145- 152, 1998 13. Duncan Lawson, Tony Croft, and Margaret Halpin, "Good practice in the provision of mathematics support centres", LTSN Maths, Stats & OR Network, 2003 14. Chetna Patel and John Little, “Measuring maths study support”, Teaching Mathematics and its Applications 25 (3), 131-138, 2006