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Mathematics diagnostic testing in engineering: an international comparison between Ireland and Portugal a
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M. Carr , C. Fidalgo , M.E. Bigotte de Almeida , J.R. Branco , V. b
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Santos , E. Murphy & E. Ní Fhloinn a
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Engineering Sciences and General Studies, Faculty of Engineering, Dublin Institute of Technology, Bolton St, Dublin 1, Ireland b
Department of Physics and Mathematics, Coimbra Institute of Engineering, Coimbra, Portugal c
School of Mathematical Sciences, Dublin City University, Dublin, Ireland Published online: 20 Nov 2014.
To cite this article: M. Carr, C. Fidalgo, M.E. Bigotte de Almeida, J.R. Branco, V. Santos, E. Murphy & E. Ní Fhloinn (2014): Mathematics diagnostic testing in engineering: an international comparison between Ireland and Portugal, European Journal of Engineering Education, DOI: 10.1080/03043797.2014.967182 To link to this article: http://dx.doi.org/10.1080/03043797.2014.967182
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European Journal of Engineering Education, 2014 http://dx.doi.org/10.1080/03043797.2014.967182
Mathematics diagnostic testing in engineering: an international comparison between Ireland and Portugal
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M. Carra∗ , C. Fidalgob , M.E. Bigotte de Almeidab , J.R. Brancob , V. Santosb , E. Murphya, and E. Ní Fhloinnc a Engineering Sciences and General Studies, Faculty of Engineering, Dublin Institute of Technology, Bolton St, Dublin 1, Ireland; b Department of Physics and Mathematics, Coimbra Institute of Engineering, Coimbra, Portugal; c School of Mathematical Sciences, Dublin City University, Dublin, Ireland
(Received 10 March 2014; accepted 9 September 2014) Concern has been expressed throughout Europe about the significant deficiencies in the basic mathematical skills of many engineering undergraduates. Mathematics diagnostic tests in the UK, Ireland and Portugal have shown these shortcomings, which provide a challenge to those striving to introduce more innovative educational practices into engineering education, such as projects or real-world problems. Every year, in the Dublin Institute of Technology (Ireland) and the Instituto Superior de Engenharia de Coimbra (Portugal), a diagnostic test is given to incoming first-year students. A comparison showed some potentially interesting differences between these students. In September 2013, a project was undertaken to compare mathematical competencies of incoming engineering students in both countries. A modified diagnostic test was devised and the results were then compared to ascertain if there are common areas of difficulty between students in Ireland and Portugal, or evidence of one group significantly outperforming the other in a particular area. Keywords: mathematics education; engineering education research; university education; entry test; numerical; efficiency
1.
Introduction
As early as 1998, the Mathematics Working Group of the European Society for Engineering Education (SEFI), meeting in Finland, discussed the decline in mathematical competencies of incoming first-year students (SEFI Mathematics Working Group 2002). The results of mathematics diagnostic tests carried out in many higher education institutions, in countries such as Ireland (Cleary 2007; Faulkner, Hannigan, and Gill 2010; Gill and O’Donoghue 2007; Ní Fhloinn 2009), the UK (Edwards 1995; Lawson 1997, 2003; Todd 2001) and, more recently, Portugal (Bigotte de Almeida, Fidalgo, and Rasteiro 2012), have shown an ongoing marked decrease in core mathematical skills. This appears to be largely attributable to two main effects. Firstly, many students are now entering higher education who previously would not have had the opportunity, as students are being recruited from an increasingly diverse student body (Department of Education and Skills 2012). The years since 2008 in particular have seen the return *Corresponding author. Email:
[email protected] © 2014 SEFI
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of a large number of students to full-time education after many years in employment, due to adverse economic conditions. Secondly, students are entering higher education with lower mathematics grades than previously (Faulkner, Hannigan, and Gill 2010), and there has been some evidence of grade inflation in higher education, although this is not universally evident. For example, it has been repeatedly observed in the UK, where, more than a decade ago, Todd (2001) stated ‘A student with an A at A-level mathematics today will on average, obtain a score on our test which would have placed them near the bottom of the cohort fifteen years ago’, while Lawson (2003) found ‘students entering in 2001 with A level mathematics grade B exhibit slightly lower levels of competency in these basic skills than those entering 10 years earlier with a grade N’. In Ireland, however, the picture is not as clear, with some conflicting reports emerging: while there is evidence of grade inflation in the Irish Leaving Certificate (the terminal examination at the end of secondary school education) in most subjects (O’Grady 2009), Faulkner, Hannigan, and Gill (2010) showed that in the case of the University of Limerick, ‘students with the same Leaving Certificate mathematics grade on entry into university have not changed significantly over time’ and they attribute the decline in mathematical competency in Ireland to the lower Leaving Certificate mathematics grades of the students upon entry to higher education. Deficiencies in core mathematical competencies lead to difficulties both in mathematics modules and in related subjects for engineering students. Every year, in the Dublin Institute of Technology (DIT) in Ireland (as in many other higher education institutes), a diagnostic test is given to incoming first-year students, consistently revealing problems in fundamental key skills in mathematics. A similar test has been given in the Instituto Superior de Engenharia de Coimbra (ISEC) in Portugal since 2011. A preliminary analysis of both tests showed some potentially interesting differences between students from Ireland and those from Portugal. As a result, in September 2013, a pilot project was undertaken to compare mathematical competencies of incoming engineering students in both countries. A modified version of their usual diagnostic test was given to both groups, with eight of the questions featured identical in both tests, to allow the core skills involved to be easily compared. These questions were selected so as to address a wide range of key competencies (identified by Niss 2003) while still being appropriate to the students’ prior learning. Topics involved included quadratic equations, algebra, logs, fractions and introductory geometry. The test was given to a large and diverse cohort of students in each institute; however, fortuitously, there exists one group in particular in each institute that is directly comparable in terms of prior experience and ability. This is a group that is of special interest, as it consists of students in a one-year ‘second-chance’ engineering programme. This allows students who have failed to reach the required standard to enter directly into an engineering programme from secondary school the opportunity to improve their academic score and enter into the regular programme the following year.
2.
Background
Although the cohort of students involved in this study are engineering undergraduates in both countries, their experiences of mathematics education to date will, of course, have been considerably different. In this background section, we outline the education systems in Ireland and Portugal, highlighting the time spent on mathematics at each stage, before discussing existing comparison between the two countries in the area of mathematics. We then give an overview of the two higher education institutes involved in the study. Finally, we consider the history of diagnostic testing in mathematics in both institutes prior to their involvement in this project.
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2.1.1.
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Education systems in Ireland and Portugal Irish education system
Irish children usually begin school aged between 4 and 5 years, having spent one year in pre-school prior to starting formal schooling (Department of Education and Skills 2004). They spend 8 years in primary school with approximately 4 hours of mathematics recommended per week (3 hours 25 minutes for the first two years, known as ‘infant’ years, and 4 hours 10 minutes thereafter), although this increase only came into effect in 2011 – prior to that, approximately 3 hours per week of mathematics were recommended (Department of Education and Skills 2011a). Students transfer to secondary school at age 12, where they spend between 5 and 6 years. They sit a state examination (the Junior Certificate) after 3 years, then some students take an optional year known as ‘transition year’ which allows time for less academic pursuits, and then spend a further 2 years studying for the Leaving Certificate (Department of Education and Skills 2004). Mathematics is compulsory for all years of secondary school with a recommendation of at least one mathematics class (usually 35–45 minutes) per day, particularly at Junior Cycle, for all 5 compulsory years (Department of Education and Skills 2011b). This is shown in Table 1.
2.1.2.
Portuguese education system
Portuguese children begin school at age 6, although they may attend pre-school between the ages of 3 and 5 (The Embassy of Portugal in Ireland 2012). The school system is divided into four different ‘cycles’, which we will term first, second, third and secondary. The first 4 years of schooling is spent in first cycle with a minimum of 7 hours of mathematics per week (European Commission 2012). The second school cycle, in which children spend two years, incorporates six 45-minute mathematics classes per week, where during the third school cycle, which lasts for three years, students have 5 hours of mathematics per week (Diário da Republica 2012). At secondary school, which lasts a further three years, there are a range of different options (Ministério da Educação 2013), as shown in more detail in Table 2.
Table 1. Minimum recommended amount of mathematics per week for pupils in the Irish educational system. School stage Infant (2 years) Primary (6 years) Secondary (5–6 years)
Age
Mathematics hours per week
4–6 years 7–12 years 13–17/18 years
3 hours 25 minutes 4 hours 10 minutes 5 × 35–45 minutes
Table 2. Minimum recommended amount of mathematics per week for pupils in the Portuguese educational system. School cycle First (4 years) Second (2 years) Third (3 years) Science and technology (3 years) Technological courses (2–3 years) Professional courses (varies)
Age
Mathematics hours per week
6–9 years 10–11 years 12–14 years 15–17 years 2 × 90 minutes Varies
7 hours 6 × 45 minutes 5 hours 3 × 90 minutes
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A quick comparison between the two tables in this section shows some clear differences in time allocated to mathematics in each system (although it should be noted that these are minimum recommended times in most cases). Considerably, more time is devoted to mathematics in early years in Portugal when compared to the Irish system – however, Irish children begin a year or two earlier in their formal introduction to mathematics. As children progress through school, the amount of time on mathematics is almost equal between the two countries for a couple of years, before Portuguese students again moving ahead with greater time allocation in early teenage years, though this changes again in their final years in secondary school, depending on which page the Portuguese students take, although generally they have more contact hours than the Irish students.
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2.2.
Existing mathematical comparisons between Ireland and Portugal
A comparison between the mathematical competencies of undergraduate engineering students in these two countries is of particular interest at this point in time due to their involvement in the Programme for International Student Assessment (PISA). This assessment ranks the performance of 15-year-old students in each participating country. In 2009, both Ireland and Portugal were ranked equally at 487 points, putting them below the OECD average (OECD 2010). These students would now be in their first or second year in higher education, and so would comprise a similar cohort to those involved in our study. PISA is only undertaken every three years, and in 2012, while there was no improvement in the Portuguese ranking, Ireland improved their mathematics score to 501, moving above the OECD average (OECD 2014). In the coming years, it will be of interest to observe whether this improvement also affects the long-term results of our comparative work.
2.3. 2.3.1.
Higher education institutes involved in the project DIT, Ireland
DIT is the largest higher education institute in Ireland in terms of undergraduate students with 22,500 students, of whom about 3000 study Engineering. DIT is split over several campuses in the centre of Dublin with the faculty of engineering split between the Bolton St and Kevin St campuses. There are two distinct routes to achieving an Honours degree (Level 8) in engineering in DIT. Students who have achieved a C3 (55%) or higher in Higher Level Mathematics in the Irish Leaving Certificate in secondary school are eligible to enter directly onto a four-year Honours degree in engineering. Students who do not have this level of mathematics but have a pass in Ordinary Level Mathematics may enter onto a three-year Ordinary degree (Level 7) in engineering. Upon successful completion of this award, students may progress to the third year of the Honours degree and must complete third and fourth years of this programme to leave with an Honours degree. Entry into higher education in Ireland is generally exclusively based on the number of ‘points’ received in the Leaving Certificate, the final examination in secondary school. Normally, students take seven exam subjects, six of which are included for the purpose of calculating points. Mathematics exams can be taken at Higher, Ordinary or Foundation level. Students who take mathematics at Foundation level are not eligible for direct entry to higher education. Mathematics is a compulsory subject and 96% of Leaving Certificate students take an exam in mathematics (Faulkner, Hannigan, and Gill 2010). A maximum of 100 points can be attained in any one subject, (except mathematics which can be worth up to 125 points). A perfect score is 625, which
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is achieved by about 0.2% of students and the median score is usually around 320–330 points (www.cao.ie).
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2.3.2.
Polytechnic Institute of Coimbra/Coimbra Engineering Institute, Portugal
The Polytechnic Institute of Coimbra (IPC) is a public institute of higher education, composed of six schools (of which Coimbra Institute of Engineering (ISEC) is one) with more than 11,000 students. IPC offers a large variety of studies, including cross-border education in partnership with other European institutions of higher education, for both undergraduate and masters degree levels. ISEC has more than 3000 students, distributed among 8 degrees of 3 years duration and 9 masters degrees. Access to higher education is done through specific examinations required by higher education institutions. Students are ranked for entry into higher education based on marks obtained in national examinations in combination with several other factors (including ratings of secondary education and prerequisites). Students enter ISEC with a huge diversity of academic backgrounds, with only 63% coming from mainstream secondary ‘Science and Technology’ education courses (RAC-SA, Relatório de Actividades-Serviços Académicos, Instituto Superior de Engenharia de Coimbra 2011). The other students come from a combination of more vocational second-level courses (‘Professional’ courses), two-year foundation courses in polytechnics (‘Technological Specialisation’ courses) which cater to students who may not have finished secondary school, and some mature students. This leads to a huge variation in the mathematical ability of incoming students. This large spread creates difficulties in teaching the material and it is increasingly important for the lecturers involved to have an idea of what their students know on entry. 2.4.
History of diagnostic testing in mathematics in each institute
In Ireland, research conducted within DIT showed that a student’s mathematics grade in the Irish Leaving Certificate is a key determinant in that student’s progression through engineering programmes (Russell 2005). Furthermore, a study of progression in Irish Higher Education (Mooney et al. 2010) has shown that ‘the higher the points attained in Mathematics in the Leaving Certificate examination by new entrants to higher education, the more likely it is that they present in the second year of their course’. This result is true not just for students studying engineering but also for those studying all subjects. Although these studies show the importance of students having a good grounding in mathematics before they enter higher education, it does not allow us to pinpoint the particular mathematical areas of weakness that students may have on entry. As a result, a mathematics diagnostic test has been given to first-year students for several years now and a Mathematics Learning Centre has been set up in DIT. Mathematics diagnostic tests have been shown to be one of the best predictors of future performance in an engineering course (Lee and Robinson 2005). The DIT mathematics diagnostic test consists of 20 questions. Students have 90 minutes to complete the test. This test has shown that there are marked deficiencies in core mathematical skills (Ní Fhloinn 2006), in particular, in converting units, indices, dealing with fractions and basic algebra. In 2006, the first year that these marks were analysed, the mean mark obtained by first-year engineering students was 55% across all programmes. More worryingly, this mean dropped as low as 29% in some programmes. Similar marks have been recorded every year since. A large spread is seen within most programmes, with many students scoring significantly lower than the mean mark. However, it should be noted that there is also considerable variation between students registered in different programmes, with those on Honours degree programme
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scoring far higher than their counterparts on Ordinary degree programmes. This is to be expected, given that their mathematical entry requirements are far higher. In fact, the majority of first-year Honours degree students have a mark of over 70% in the diagnostic exercise. Diagnostic testing in ISEC does not have as long a history as that of DIT, and has only been implemented in the past couple of years, but was undertaken for similar reasons as in DIT, namely in order to evaluate the mathematical knowledge of incoming students in ISEC and the relation between this knowledge and the student profile (Bigotte de Almeida, Fidalgo, and Rasteiro 2012). This test was performed in the first week of the first semester of the 2011/2012 academic year by students enrolled in courses of Differential and Integral Calculus of different degrees of Engineering and was taken by 272 students. Similar to the results in DIT, the diagnostic tests show a large spread of abilities both between programmes and within programmes (although there are no equivalent programmes to the ‘Ordinary degrees’ offered by DIT). The results in ISEC also reveal that students on their first registration in higher education have better results than repeating students. In terms of mean marks per degree, Biomedical Engineering students have the highest mean (56.4%), while the lowest (25.8%) was found among the Electromechanical Engineering students.
3.
Design and implementation of diagnostic test
In order to select questions for the joint diagnostic test between DIT and ISEC, an initial comparison of the existing questions was undertaken. It was striking, but perhaps not surprising, that there was a considerable overlap in the topics asked on each test; this is most likely the case as students would require similar mathematics to complete a degree in engineering, regardless of the country where their studies are based. As a result, it was decided to compare eight questions on the test, by ensuring that these eight be identical on both tests. This required some minor alterations of questions with a couple of additional questions created where necessary. The exact details of the overlap and the changes made are outlined in the Appendix below, but the questions cover the following areas: (1) (2) (3) (4) (5) (6) (7) (8)
Quadratic equations Equation of a line Algebraic fractions Fractions Logarithms Trigonometry Area of a circle Applying algebra to real problems
The diagnostic test was given in paper-based, multiple-choice format, with four possible answers given, one of which was correct. The test was negatively marked, with students scoring 3 marks for a correct answer, losing 1 mark for an incorrect answer, and receiving no score if they leave a question blank. Students were informed about this immediately prior to attempting the test, and instructed that they should not guess an answer if they are unsure, but should leave it blank. Students were not aware that they would be taking the test beforehand, so they did not have an opportunity to study for it directly, but instead had to rely on their mathematical knowledge at the time. This was done as the topics involved were basic ones that students would not typically revise for mathematics examinations, but often find themselves unable to do, either as stand-alone questions or as part of longer, more difficult ones. Students in DIT answered the test
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in English, while those in ISEC did so in Portuguese, as their first-year mathematics lectures will be in Portuguese. The test was given to a wide variety of first-year engineering students within both ISEC and DIT. In total, 278 students were tested in ISEC. These students consisted of four groups of first-year Honours degree engineering students, namely Engineering Informatics (143), Mechanical (65), Electromechanical (22) and Biomedical engineering (23). In addition, 25 students from ‘Year Zero’ (explained further below) were included in the sample. A total of 178 students were tested in DIT. These groups consisted of first-year Ordinary degree Building Services engineering students (22), third-year Ordinary degree mechanical engineers (50), first-year access students (70) and 36 students from the General Entry Foundation Programme (also discussed below). Access students are those from low socio-economic backgrounds who enter higher education through non-formal entry procedures. The General Entry Foundation Programme is analogous to the Year Zero course in ISEC. From the above description of the student cohort involved in the project, it is clear that there is a huge variety in the mathematical levels of incoming students who hope to obtain an engineering qualification. However, there is one cohort of students that is of particular interest in each, and this is the group known as ‘General entry foundation engineering’ (GEFE) in DIT and ‘Year Zero’ in ISEC. Students in this grouping represent those who have failed to obtain sufficiently high results in secondary school examinations to allow them direct entry into engineering and who instead opt to undertake a one-year programme within the higher education institute which aims to allow students to improve their knowledge in basic engineering disciplines such as Mathematics, Physics and Chemistry, in order to allow them a second-chance to enter into the usual engineering programme the following year. As a result, although the diagnostic test was given to a large number of students, we will focus especially on comparisons between the GEFE students and the Year Zero students in our analysis.
4.
Results and discussion
Table 3 shows the overall percentage of correct answers given to each question in the diagnostic test by each class-group from each institute. It should be remembered that the ISEC students (with the exception of the Year Zero group) are all enrolled onto Level 8 engineering programmes, while the DIT students (again with the exception of the GEFE group) are enrolled onto Level 7 programmes. However, the third-year mechanical engineers from DIT would be in a position to enter into the final two years of a Level 8 programme provided they perform Table 3.
Results by question and class-group for both ISEC (Portugal) and DIT (Ireland).
Question
Quadratic eqns
Eqn of line
Alg fractions
Fractions Logs Trig
Area of circle
App to real problems
% Correct ISEC (Portugal) Informatics Mechanical Electro mechanical Biomedical Year Zero
63 53 39 78 60
35 43 9 44 −5
64 74 36 78 64
62 67 56 83 73
53 56 39 88 41
40 43 62 77 19
57 55 41 61 28
62 83 71 86 75
% Correct DIT (Ireland) Access Mechanical (third year) Building services GEFE
30 25 1 22
10 0 −4 15
− 10 55 30 30
43 97 71 78
49 76 36 52
43 68 13 63
12 73 4 22
81 81 48 74
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Table 4.
Statistical comparison of results for Year Zero (ISEC, Portugal) and GEFE (DIT, Ireland).
Question
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#correct ISEC (n = 36) #correct DIT (n = 25) 2-proportion test (p-value)
Quadratic eqns
Eqn of line
Alg fractions
15 17 0.034
13 4 0.064
17 18 0.04
Fractions Logs Trig 30 19 0.49
23 13 0.35
26 9 0
Area of circle
App to real problems
15 11 0.86
29 20 0.957
sufficiently well in their terminal examinations at the end of the year. Despite the differences between the groups here, the students universally answered the question on the equation of a line very poorly, as can be seen in Table 3. Equally, the questions involving fractions and applying algebra to real problems were among the best answered questions by each class group. There was greater variation between the students’ performance in the other questions, with trigonometry, algebraic fractions or quadratic equations, for example, among the best-answered in some class-groups, but among the worst-answered in others. We will now look at the two groups that are the most comparable from our overall cohort, namely Year Zero from ISEC and GEFE from DIT. Having conducted a 2-proportion test on their results, some interesting results emerge. As shown in Table 4, on five of the eight questions (fractions, area of a circle, equation of a line, logarithms and applying algebra to real problems), there is no statistically significant difference at 95% confidence between the performance of the two groups. One of the questions (trigonometry) shows a strong statistically significant difference between the groups, with the Irish students performing far better than their Portuguese counterparts, while the remaining two questions (quadratic equations and algebraic fractions) showed a slight statistically significant difference with the Portuguese students outperforming Irish students in these areas. This analysis points to the fact that there is a particularly strong difference in relation to performance in trigonometry and so we considered again the results of the full cohort of students, shown in Table 3, to see if such a difference was also observable there. The ISEC students scored 43.7% on the trigonometry, whereas in DIT, 50.36% got the question correct. This difference is not significant (p = .15), but it is interesting that the Irish cohort who are weaker overall and have performed worse on other areas are equal to their Portuguese counterparts on trigonometry.
5.
Conclusions and future work
Having compared similar cohorts of engineering students in Ireland and Portugal, we have shown that many of the problems with basic mathematical skills are common to both countries, with students performing similarly (sometimes very poorly) on questions on logs, fractions, applying algebra to real problems, area of a circle and equation of a line. However, particular differences have been observed in performance in questions on trigonometry, quadratic equations and algebraic fractions. Given that only one question was asked in each area, and a small cohort of students was directly comparable, this warrants further investigation to ascertain if there is a difference in general. To do so, larger groups of comparable students would need to be tested on a number of questions in each of the areas, and the approaches to teaching these topics at secondary school should also be investigated. Looking at the results of PISA in any similar areas would further add to this research, as it would allow for a comparison between retention of mathematical knowledge in these areas also. All three topics are critical to engineering mathematics and therefore
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furthering our knowledge of the optimal approach to teaching these would be hugely beneficial to the engineering community. Given that diagnostic testing and follow-up assistance have been undertaken in DIT for several years longer than in ISEC, many of the strategies introduced in DIT to assist students who are identified as having deficiencies in their mathematical skill-set may also be beneficial in ISEC. As a result, ISEC is planning to establish a Mathematics Support Centre in the coming year, where students can receive free, additional help with mathematical topics of their choosing. In addition, funding is being sought to translate a number of existing mathematics support resources from English into Portuguese, in order to better assist students in ISEC. These translations will in turn be of benefit to the increasing numbers of Brazilian students who attend DIT on international student exchange programmes. Mutual benefits such as these demonstrate the importance of cross-collaborations among different countries. Introduction of extracurricular mathematics units, similar to those in use in DIT as part of the ‘Core Mathematics Initiative’ (Carr and Ní Fhloinn 2009; Carr, Bowe, and Ní Fhloinn 2010, 2013) is also in preparation in ISEC. In DIT, plans to extend initiatives beyond the first year of engineering programmes are underway, to ensure that students’ core skills remain strong throughout their studies, as recent research has shown that many students continue to struggle with poor core skills for the duration of their programme (CCarr, Bowe, and Ní Fhloinn 2013; Marjoram et al. 2008, 2013). Based on the results generated for the comparison between the GEFE and Year Zero students, we now plan to undertake a further study of students in Honours engineering degree programmes, comparing them at various points throughout their studies on a range of topics, with particular focus on entry and exit mathematical skills.
References Bigotte de Almeida, M. E., C. Fidalgo, and D. M. L. D. Rasteiro. 2012. “The Teaching of Mathematics in Engineering: The ACAM – Assessment of Competencies/Improvement Actions Project.” Proceedings of 16th SEFI MWG seminar of mathematics in engineering education. Salamanca: SEFI. http://sefi.htwaalen.de/Seminars/Salamanca2012/16thSEFIMWGSeminar/ficheros/lecturas/Documents_pdf/Session4/SEFIMWG 12_Almeida.pdf Carr, M., B. Bowe, and E. Ní Fhloinn. 2010. “Improving Core Mathematical Skills in Engineering Undergraduates.” Proceedings of 15th SEFI MWG seminar on mathematics in engineering education. Wismar: SEFI. http://sefi.htwaalen.de/Seminars/Wismar2010/SEFI/papers_pdfs/MWG2010_Carr_C.pdf Carr, M., B. Bowe, and E. Ní Fhloinn. 2013. “Core Skills Assessment to Improve Mathematical Competency.” European Journal of Engineering Education 38 (6): 608–619. doi:10.1080/03043797.2012.755500 Carr, M., E. Murphy, B. Bowe, and E. Ní Fhloinn. 2013. “Addressing Continuing Mathematical Deficiencies with Advanced Mathematical Diagnostic Testing.” Teaching Mathematics and its Applications 32 (2): 66–75. doi:10.1093/teamat/hrt006 Carr, M., and E. Ní Fhloinn. 2009. “Assessment and Development of Core Skills in Engineering Mathematics.” In Proceedings of Centre for Excellence in Teaching and Learning Maths, Stats and OR Network (CETL-MSOR) Conference 2009: Continuing Excellence in Teaching and Learning, edited by D. Green, 19–24. Milton Keynes: The Maths, Stats and OR Network. http://www.heacademy.ac.uk/assets/documents/subjects/msor/Proceedings_2009_ Upload_0.pdf Cleary, J. 2007. “Diagnostic Testing – An Evaluation 1998–2007.” In Proceedings of Second National Conference on Research in Mathematics Education (MEI2), edited by S. Close, D. Corcoran, and T. Dooley, 216–228. Dublin: St. Patrick’s College. Department of Education and Skills. 2004. A Brief Description of the Irish Education System. Dublin: Department of Education and Skills. http://www.education.ie/en/Publications/Education-Reports/A-Brief-Description-of-the-IrishEducation-System.pdf Department of Education and Skills. 2011a. Initial Steps in the Implementation of the National Literacy and Numeracy Strategy. Circular letter 0056/2011. http://www.ncca.ie/en/Curriculum_and_Assessment/Early_Childhood_and_ Primary_Education/Primary_School_Curriculum/Assessment/Standardised_Testing/circular_0056_2011.pdf Department of Education and Skills. 2011b. Revised Syllabuses in Junior Certificate and Leaving Certificate Mathematics for Leaving Certificate Examination in 2013 and Junior Certificate Examination in 2014. Circular letter 0058/2011. http://www.education.ie/en/Circulars-and-Forms/Active-Circulars/cl0058_2011.pdf
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M. Carr et al.
Department of Education and Skills. 2012. Projections of Demand for Full Time Third Level Education, 2011–2026. Dublin: Department of Education and Skills. http://www.education.ie/en/Publications/Statistics/Projections-ofdemand-for-Full-Time-Third-Level-Education-2011–2026.pdf Diário da Republica. 2012. Decreto Lei N.°139/2012 de 5 de julho, Diário da Republica, 1ª Série, 129: 3486–3487. Edwards, P. 1995. “Some Mathematical Misconceptions on Entry to Higher Education.” Teaching Mathematics and its Applications 14 (1): 23–27. doi:10.1093/teamat/14.1.23 European Commission. 2012. “Recommended Annual Taught Time in Full-Time Compulsory Education in Europe 2012/2013.” Eurydice – Facts and Figures. http://eacea.ec.europa.eu/education/eurydice/documents/facts_and_ figures/taught_time_EN.pdf Faulkner, F., A. Hannigan, and O. Gill. 2010. “Trends in the Mathematical Competency of University Entrants in Ireland by Leaving Certificate Mathematics Grade.” Teaching Mathematics and its Applications 29 (2): 76–93. doi:10.1093/teamat/hrq002 Gill, O., and J. O’Donoghue. 2007. “The Mathematical Deficiencies of Students Entering Third Level: An Item by Item Analysis of Student Diagnostic Tests.” In Proceedings of Second National Conference on Research in Mathematics Education (MEI2), edited by S. Close, D. Corcoran, and T. Dooley, 229–240. Dublin: St. Patrick’s College. Lawson, D. 1997. “What Can We Expect from A Level Mathematics Students?” Teaching Mathematics and its Applications 16 (4): 151–156. doi:10.1093/teamat/16.4.151 Lawson, D. 2003. “Changes in Student Entry Competencies 1991–2001.” Teaching Mathematics and its Applications 22 (2): 171–175. doi:10.1093/teamat/22.4.171 Lee, S., and C. L. Robinson. 2005. “Diagnostic Testing in Mathematics: Paired Questions.” Teaching Mathematics and its Applications 24 (4): 154–166. doi:10.1093/teamat/hrh017 Marjoram, M., D. Moore, C. O’Sullivan, and P. Robinson. 2008. “Implementing a Key Skills in Mathematics Initiative.” Proceedings of mathematical education of engineers, 14th SEFI (MWG) conference joint with IMA. Loughborough: SEFI. http://sefi.htw-aalen.de/Seminars/Loughborough2008/mee2008/proceedings/mee2008F_Marjoram.pdf Marjoram, M., P. Robinson, C. O’Sullivan, and M. Carr. 2013. “Improving the Key Skills of Engineering Students in Mathematics.” Proceedings of 41st SEFI conference. Leuven: SEFI. http://www.sefi.be/conference2013/images/148.pdf Ministério da Educação. 2013. Documento Orientador da Revisão Curricular do Ensino Secundário [Guidance document for the Revision of the Secondary School Curriculum], 2–7. Mooney, O., V. Patterson, M. O’ Connor, and A. Chantler. 2010. A Study of Progression in Irish Higher Education. Dublin: Higher Education Authority (HEA). http://www.hea.ie/files/files/file/statistics/2010/Retention%20&%20 Progression/HEA%20Study%20of%20Progression%20in%20Irish%20Higher%20Education%202010.pdf Ní Fhloinn, E. 2006. Maths Diagnostic Report. Dublin: Dublin Institute of Technology. Ní Fhloinn, E. 2009. “Diagnostic Testing in DCU – A Five-Year Review.” In Proceedings of Third National Conference on Research in Mathematics Education (MEI3), edited by D. Corcoran, T. Dooley, S. Close, and R. Ward, 367–378. Dublin: St. Patrick’s College. Niss, M. 2003. “Mathematical Competencies and the Learning of Mathematics: The Danish KOM Project.” In Proceedings of Third Mediterranean Conference on Mathematical Education, edited by A. Gagatsis and S. Papastavridis, 115–124. Athens: Hellenic Mathematical Society and Cyprus Mathematical Society. OECD. 2010. “PISA 2009 Results: Executive Summary.” http://www.oecd.org/pisa/pisaproducts/46619703.pdf OECD. 2014. “PISA 2012 Results: What Students Know and Can Do – Student Performance in Mathematics, Reading and Science (Volume I, Revised Edition).” http://dx.doi.org/10.1787/9789264201118-en O’Grady, M. 2009. “Grade Inflation in the Leaving Certificate Examination 1992–2006.” Paper 7, Network for Irish Educational Standards. http://www.stopgradeinflation.ie RAC-SA, Relatório de Actividades-Serviços Académicos, Instituto Superior de Engenharia de Coimbra. 2011. https:// intranet.isec.pt/Documentos/Relat%C3%B3rio%20de%20Actividades%20e%20Contas/RAC-SA%202010.pdf Russell, M. 2005. “Academic Success, Failure and Withdrawal among First Year Engineering Students: Was Poor Mathematical Knowledge a Critical Factor?” Level 3 (3). http://level3.dit.ie/html/issue3_list.html SEFI Mathematics Working Group. 2002. Mathematics for the European Engineer – A Curriculum for the Twenty-First Century, edited by L. Mustoe and D. Lawson. SEFI HQ. http://sefi.htw-aalen.de/Curriculum/sefimarch2002.pdf The Embassy of Portugal in Ireland. 2012. “Family and School Matters.” http://www.embassyportugal.ie/?p = 342 Todd, K. 2001. “Historical Study of Correlation between A-Level Grades and Subsequent Performance.” In Diagnostic Testing for Mathematics, edited by LTSN Maths Team Project, 16–17. http://mathstore.ac.uk/mathsteam/packs/ diagnostic_test.pdf
About the authors Dr M. Carr is a lecturer in Mathematics and Statistics in the College of Engineering and Built Environment in the Dublin Institute of Technology. His research interests include the first year experience of students and development of core mathematical skills in university students, and mathematics education for engineers. C. Fidalgo is a lecturer in Mathematics in the Coimbra Institute of Engineering, Portugal. Her research interests include the first year experience of students and development of core mathematical skills in students of higher education, and mathematics education for engineers.
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M.E. Bigotte de Almeida is a lecturer in Department of Physics and Mathematics of the Coimbra Institute of Engineering. Her research interests include mathematics education for engineers, in particular, the first year experience of students and development of their core mathematical skills. Dr J.R. Branco is a lecturer in Mathematics in the Coimbra Institute of Engineering in Coimbra Polytechnic Institute. He holds a Ph.D. in Applied Mathematics and graduate on Mathematics and Civil Engineering. His research interests include the first year experience of students on high education, the development of mathematical skills of university students, the mathematics education for engineers and also the study of mathematical models to glioma growth.
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V. Santos, MSc, is an invited assistant in Physics and Mathematics Department in Coimbra Institute of Engineering (ISEC) in the Polytechnic Institute of Coimbra (IPC). Ph.D. student at University of Trás-os-Montes and Alto Douro (UTAD). Her research interests include ICT, geometry, mathematics education and mathematics education for engineers. E. Murphy graduated with an ordinary degree in mechanical engineering in 2002 from Dublin Institute of Technology. Following this he worked on high-end engineering projects at both Bord Gáis and the Electricity Supply Board’s international engineering consultancy (ESBI). He returned to college to complete his honours degree in mechanical engineering, graduating in 2009, and is currently three years into his Ph.D. studies on the modelling of blood flow through stented arteries using computational fluid dynamics. He also lectures part time in thermodynamics, supervises undergraduate projects and helps run a mathematics workshop for secondary school pupils. Dr E. Ní Fhloinn is a lecturer in the School of Mathematical Sciences in Dublin City University and the Director of the Maths Learning Centre in the same university. Her research interests include effective mathematics support, development of core mathematical skills in university students, and mathematics education for engineers.
Appendix The eight overlapping questions in the common diagnostic test are as follows: (1) If x2 − x − 6 = 0, the values of x are (b) 3 and − 2
(a) 6 and 1
(c) − 3 and 2
(d) − 6 and 1
(2) The equation of a straight line is 8x + 4y − 6 = 0. The slope (or gradient) of the line is (a) 8
(c) − 2
(b) 4
(d) 2
(3) Which of the following four statements is correct? 1 2x
(a) (4) Is (a)
+
+
1 2
2 3
2 3x
=
−
4 7
25 42
3 5x
1 2x
+
(c)
12 21
(b)
2 3x
=
3 6x
(c)
1 2x
+
2 3x
=
5 6x
(d)
1 2x
+
2 3x
=
7 6x
= (b)
−1 12
(d)
11 42
(5) log2 16 = (a) 4
(b) 8
(c) 14
(d) 18
(6) In a right-handed triangle ABC, with right angle at A, if AC = 3 cm and BC = 5 cm, then the length of AB is √ (c) 16 cm (d) 4 cm (a) 5/3 cm (b) 34 cm (7) The area of a circle with diameter of 6 cm is (a) 36π cm2
(b) 9π cm2
(c) 6π cm2
(d) 12π cm2
(8) Simon bought 6 concert tickets for himself and his friends. Later, 4 other friends bought additional tickets at the door costing e5 more per ticket than the original price. The total cost for all of them was e180. Write an equation that allows you to calculate the price of Simon’s ticket. (a) 6x + 4(x + 5) = 180 (b) 4x + 6(x + 5) = 180 (c) 6x + 4(x − 5) = 180 (d) 4x + 6(x − 5) = 180
