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<?xml version="1.0" ?> <tei> <teiHeader> <fileDesc xml:id="0"/> </teiHeader> <text xml:lang="en"> <front>Prior knowledge of mechanics amongst <lb/>first year Engineering students <lb/>Dick Clements <lb/>Submitted March 2007; accepted May 2007 <lb/>Abstract <lb/>In the last 25 years, A-level Mathematics syllabi have changed very considerably, <lb/>introducing a broader range of application areas but reducing the previous emphasis <lb/>on classical mechanics. This article describes a baseline survey undertaken to estab-<lb/>lish in detail the entry levels in mechanics for the cohort of students entering <lb/>Engineering courses at Bristol University in October 2005. The survey results con-<lb/>¢rm and quantify existing anecdotal evidence indicating that universities must now <lb/>assume a considerably reduced familiarity with concepts in basic mechanics. These <lb/>changes have strong implications for future course design. <lb/></front> <body>1. Background <lb/>In the 1960s and 1970s A-level Mathematics syllabi, whether for 'single' or 'double' mathematics, <lb/>comprised two strands referred to as 'pure mathematics' and 'applied mathematics'. <lb/>More recently the 'pure mathematics' strand has been described (rather more accurately perhaps) <lb/>as 'core mathematics'. The 'applied mathematics' strand used to comprise exclusively a basic <lb/>study of classical mechanics. Development of the syllabi in the last 25 years has been driven by <lb/>a desire to expose students to a wider range of applications of mathematics and has led to <lb/>the inclusion of additional modules on probability and statistics, and on operations research <lb/>(or 'decision mathematics'). These modules, along with the modules on mechanics, are described <lb/>as 'applications modules'. Obviously students cannot be expected to study all of this broader <lb/>range of applications to the same depth and so a system of optional modules has evolved allowing <lb/>students, or more usually their schools and their teachers, to choose which of the application <lb/>areas to study and to what depth. <lb/>These developments have been of great concern to staff in a number of disciplines at universities <lb/>who previously relied upon students entering university with a good knowledge of basic mechanics. <lb/>In the Engineering Faculty at Bristol University in the 1970s, we knew that students would arrive <lb/>with this sound knowledge of basic mechanics and, by the same token, knew that we had to teach <lb/>probability and statistics from scratch. As the A-level syllabi developed, and a wider range of <lb/>optional application units were introduced, we increasingly found that a proportion of students <lb/>knew some basic probability theory and statistics and that a similar proportion were lacking some <lb/>of the knowledge of mechanics which we had previously presumed. Worse, because of the <lb/>introduction of a range of systems of options, we could not rely on all students having a similar <lb/>level of preparation in mechanics and in probability and statistics. This has led to inefficiencies <lb/></body> <front>TEACHING MATHEMATICS AND ITS APPLICATIONS, Volume 26, No. 3, 2007 119 <lb/>Advance Access Published on 6 August 2007 doi:10.1093/teamat/hrm005 <lb/>ß The Author 2007. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. <lb/>All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org <lb/></front> <body>in our teaching because we have to teach both areas from the level of knowledge of the least <lb/>well-prepared students. It also leads to frustration for those students who find themselves being <lb/>lectured on material which they have already covered during A-level studies. <lb/>The latest (2004) revision of the A-level Mathematics syllabus has reduced still further the <lb/>study of mechanics. The new syllabus for Mathematics A-level contains a core of four modules of <lb/>pure mathematics together with two modules of 'applications' chosen from a range of six <lb/>modules, two in each of statistics, operations research and mechanics. Students may therefore <lb/>study up to two modules of mechanics, but an A-level can be obtained without studying any <lb/>mechanics at all. It is likely that most students will study at most one module of mechanics and <lb/>a significant proportion will study no mechanics at all. Thus, from 2006, students will <lb/>be commencing degree courses in Engineering, Sciences and Mathematics, with little or no <lb/>knowledge and experience in mechanics. <lb/>2. The development of a new mechanics strategy <lb/>In response to the changes in the A-level syllabus, a project has been initiated to develop ways of <lb/>bringing all students entering Engineering courses at Bristol University up to a common level of <lb/>knowledge of basic mechanics. No pre-judgement has been made about the method of achieving <lb/>this objective. Techniques which will be considered include self-study methods based on text-<lb/>books and on computer-based materials, and conventional lectured course units. A mix of such <lb/>methods may well allow the strengths of all approaches to be harnessed to the main objective. <lb/>3. Initial work <lb/>To establish a baseline from which to work a survey of entry qualifications in mathematics was <lb/>undertaken in October 2005. All first year students in the Engineering Faculty were asked to <lb/>complete a computer-based survey form. This was completed during an introductory computer <lb/>session during the pre-sessional orientation week of the Autumn Term. As a result the response <lb/>rate to the survey was virtually 100%; 377 responses being completed. Data was collected <lb/>anonymously and was aggregated into six categories by Department. The four main Engineering <lb/>Departments, Aeronautical, Civil, Electrical and Mechanical, each form categories whilst the <lb/>Engineering Mathematics and Engineering Design students, and the Computer Science and CSE <lb/>students have been combined into two further categories. <lb/>The first section of the questionnaire was designed to establish whether respondents had taken <lb/>Mathematics A-levels or an alternative qualification. A total of 44 students (11.7%) reported that <lb/>they had not taken A-levels in Mathematics; instead 5 had taken the International Baccalaureate, <lb/>2 the European Baccalaureate, 4 had taken Scottish Highers, 3 had taken the French <lb/>Baccalaureate, 3 students had BTEC qualifications, 6 students had attended a foundation <lb/>course and 21 did not report which specific alternative qualification they held. The responses <lb/>of these 44 students were not included in any further analysis. <lb/>Amongst the 333 students who had taken A-levels in Mathematics, the students identified the <lb/>Examination Board as shown in Table 1. <lb/>Notice that the total number of data included in Table 1 (356) exceeds the number of students <lb/>taking mathemtics A-levels (333). This is explained by students who reported taking papers <lb/>from more than one Board. It seems that schools, presumably in response to the pressures <lb/>created by 'league tables', are playing different Boards off against each other by entering <lb/>students for A-levels from more than one Board in order to obtain the best possible grades. <lb/></body> <note place="headnote">TEACHING MATHEMATICS AND ITS APPLICATIONS, Volume 26, No. 3, 2007 <lb/></note> <page>120 <lb/></page> <body>Indeed 5 EngMaths/Design students and 1 Mechanical student claim to have taken A-levels from <lb/>all three main Boards. Table 2 shows the incidence by Department of students taking mathe-<lb/>matics A-levels from more than one Board. Percentages in this table are relative to the number of <lb/>A-level Boards declared by students in each Department, that is the figure in the 'total' column in <lb/>Table 1. It is evident from this table that instances of taking A-levels from two or more Boards <lb/>are significantly higher in the EngMaths/Design category (23%) than in any other (510%). This <lb/>is most likely to be a result of the higher incidence of candidates offering 'Double Maths' A-levels <lb/>for these two courses. <lb/>To determine the level of preparation in mechanics, the data was analysed first by <lb/>Departmental category as shown in Table 3. Overall 89% of students had taken module M1 <lb/>with the percentage in individual Departments varying from 80 to 96%. This does not suggest <lb/>a strong bias by discipline. Overall 72% of students had taken module M2 with a variation <lb/>Table 1 Distribution of A-level Boards by Department categories <lb/>AQA <lb/>EdExcel <lb/>OCR <lb/>Other Boards <lb/>Total <lb/>Aeronautical <lb/>20 <lb/>27 <lb/>3 <lb/>8 <lb/>58 (16%) <lb/>Civil <lb/>9 <lb/>27 <lb/>17 <lb/>2 <lb/>55 (16%) <lb/>Electrical <lb/>8 <lb/>18 <lb/>17 <lb/>5 <lb/>48 (14%) <lb/>Mechanical <lb/>18 <lb/>30 <lb/>24 <lb/>11 <lb/>83 (23%) <lb/>Eng Maths and Design <lb/>15 <lb/>26 <lb/>22 <lb/>2 <lb/>65 (18%) <lb/>Computer Science and CSE <lb/>10 <lb/>18 <lb/>12 <lb/>7 <lb/>47 (13%) <lb/>All <lb/>80 (22%) <lb/>146 (41%) <lb/>95 (27%) <lb/>35 (10%) <lb/>356 (100%) <lb/>Table 2 Incidence of candidates taking A-levels from multiple boards <lb/>3 Boards <lb/>2 Boards <lb/>Total <lb/>Aeronautical <lb/>0 <lb/>6 <lb/>6 (10%) <lb/>Civil <lb/>0 <lb/>1 <lb/>1 (2%) <lb/>Electrical <lb/>0 <lb/>2 <lb/>2 (4%) <lb/>Mechanical <lb/>1 <lb/>3 <lb/>4 (5%) <lb/>Eng Maths and Design <lb/>6 <lb/>9 <lb/>15 (23%) <lb/>Computer Science and CSE <lb/>0 <lb/>1 <lb/>1 (2%) <lb/>All <lb/>7 <lb/>22 <lb/>29 <lb/>Table 3 Percentages of students taking the available mechanics modules <lb/>Total <lb/>candidates <lb/>None <lb/>M1 <lb/>M2 <lb/>M3 <lb/>M4 <lb/>M5 þ <lb/>Aeronautical <lb/>54 <lb/>11 (20%) <lb/>43 (80%) <lb/>34 (63%) <lb/>9 (17%) <lb/>3 (6%) <lb/>1 (2%) <lb/>Civil <lb/>55 <lb/>2 (4%) <lb/>53 (96%) <lb/>36 (65%) <lb/>9 (16%) <lb/>5 (9%) <lb/>1 (2%) <lb/>Electrical <lb/>47 <lb/>7 (15%) <lb/>40 (85%) <lb/>35 (74%) <lb/>10 (21%) <lb/>5 (11%) <lb/>2 (4%) <lb/>Mechanical <lb/>84 <lb/>6 (7%) <lb/>78 (93%) <lb/>63 (75%) <lb/>24 (29%) <lb/>7 (8%) <lb/>3 (4%) <lb/>Eng Maths and Design <lb/>46 <lb/>4 (9%) <lb/>42 (91%) <lb/>34 (74%) <lb/>14 (30%) <lb/>10 (22%) <lb/>3 (7%) <lb/>Computer Science and CSE <lb/>47 <lb/>7 (15%) <lb/>40 (85%) <lb/>37 (79%) <lb/>12 (26%) <lb/>4 (9%) <lb/>1 (2%) <lb/>All <lb/>333 <lb/>37 (11%) <lb/>296 (89%) <lb/>239 (72%) <lb/>78 (23%) <lb/>34 (10%) <lb/>11 (3%) <lb/></body> <note place="headnote">TEACHING MATHEMATICS AND ITS APPLICATIONS, Volume 26, No. 3, 2007 </note> <page>121 <lb/></page> <body>between Departments from 63 to 79%, again not suggesting a strong bias by discipline. Module <lb/>M3, M4 and M5 were taken by only 23, 10 and 3% of students overall. <lb/>The data was also analysed by Examination Board to check for significant differences. <lb/>These results are presented in Table 4. No significant difference between candidates taking <lb/>A-levels from different Boards is apparent. <lb/>A parallel study by Robinson et al. (1) reports figures for the proportion of students taking <lb/>the various mechanics modules at A-level, based in their case on surveys undertaken with first <lb/>year Engineering students at Loughborough, Nottingham and Leicester Universities. <lb/>Unsurprisingly these figures reveal a very similar situation to that uncovered by the Bristol <lb/>survey. Robinson et al. discovered that amongst engineering students across the three universities <lb/>surveyed, 9% had studied no mechanics modules and 23% had studied one module only. <lb/>The equivalent figures for Bristol University, given in Table 3, are 11 and 17%. <lb/>4. Discussion <lb/>The results of this survey reinforce the anecdotal evidence which staff in the various Engineering <lb/>disciplines have been reporting for several years past, of students who have limited or no <lb/>familiarity with concepts which those staff had previously taken for granted. Given the evidence <lb/>in Table 3, it is apparent that it is definitely unsafe to assume a prior level of knowledge of <lb/>concepts in mechanics above that of the module M1. It is, for the time being, probably viable to <lb/>base the main first year courses on an assumption of knowledge of the module M1 and to make <lb/>available remedial/introductory courses for the small percentage of students who have not taken <lb/>M1. Those courses would also serve to bring any overseas students who have an inadequate <lb/>knowledge of mechanics up to the common starting point. <lb/>These findings will be used to ensure that all Engineering Faculty Staff are fully aware of <lb/>the reducing level of preparation in mechanics of the students that they are teaching. Robinson <lb/>et al. (1) reported that over half of the academics they surveyed were not aware of the extent of <lb/>deficiency in knowledge of mechanics amongst their students. This work has not formally <lb/>explored that issue, but there is an evident disquiet amongst many lecturers in the Engineering <lb/>Faculty at Bristol University who find deficiencies of understanding of basic mechanics <lb/>concepts amongst their students and a desire to find out what they can and cannot assume in <lb/>the future. <lb/>Another problem which must be tackled is the variation between syllabi. If we were to assume <lb/>that students had a knowledge of the content of A-level module M1 would that give us an <lb/>adequate baseline? Table 5 below shows a summary of the contents of module M1 in the three <lb/>Table 4 Percentages analysed by Examination Board <lb/>Total <lb/>candidates <lb/>None <lb/>M1 <lb/>M2 <lb/>M3 <lb/>M4 <lb/>M5 þ <lb/>AQA <lb/>80 <lb/>5 (6%) <lb/>75 (94%) <lb/>62 (78%) <lb/>16 (20%) <lb/>10 (13%) <lb/>5 (6%) <lb/>EdExcel <lb/>146 <lb/>6 (4%) <lb/>140 (96%) <lb/>114 (78%) <lb/>34 (23%) <lb/>14 (10%) <lb/>5 (3%) <lb/>OCR <lb/>95 <lb/>8 (8%) <lb/>87 (92%) <lb/>68 (72%) <lb/>28 (29%) <lb/>15 (16%) <lb/>4 (4%) <lb/>Other Boards <lb/>35 <lb/>15 (43%) <lb/>20 (57%) <lb/>13 (37%) <lb/>5 (14%) <lb/>0 (0%) <lb/>0 (0%) <lb/>All <lb/>356 <lb/>34 (10%) <lb/>322 (90%) <lb/>257 (72%) <lb/>83 (23%) <lb/>39 (11%) <lb/>14 (4%) <lb/></body> <note place="headnote">TEACHING MATHEMATICS AND ITS APPLICATIONS, Volume 26, No. 3, 2007 <lb/></note> <page>122 <lb/></page> <body>main A-level Boards. Whilst they are by no means identical, there is a sufficiently large measure <lb/>of overlap for most practical purposes. <lb/>5. Conclusion <lb/>Robinson et al. (1) have highlighted the changes in competence in basic mechanics which have <lb/>become apparent amongst entrants to Engineering and other technical and scientific degree <lb/>courses in recent years. This work provides further evidence that these changes are widespread <lb/>and that universities can no longer assume that entrants have that level of familiarity with <lb/>concepts in basic mechanics which they have taken for granted in the past. The evidence suggests <lb/>that degree courses must be designed or modified to take this into account, and that new course <lb/>units are required to impart the knowledge and skills which the students now lack. <lb/></body> <listBibl>References <lb/>1. Robinson, C., Harrison, M. and Lee, S. (2005) Responding to the Changes in the Teaching and Learning <lb/>of Mechanics in Schools. Loughborough University: Mathematics Learning Support Centre, July 2005. <lb/></listBibl> <front>Dick Clements has an MA in Mathematics and a PhD in Aeronautical Engineering from Cambridge <lb/>University, a PGCE from Leicester University and is a Chartered Engineer. He has enjoyed teaching <lb/>mathematics to engineering students at Bristol University since 1973 and was awarded the title of <lb/>Professorial Teaching Fellow in 2005. <lb/>Address for correspondence: Prof R. R. Clements, Faculty of Engineering, University of Bristol, <lb/>Bristol, BS8 1TR, UK. E-mail: rr-clements@cantab.net <lb/></front> <body>Table 5 Content of Mechanics 1 module of the three largest A-level Boards <lb/>AQA <lb/>Edexel <lb/>OCR <lb/>Mechanics 1 <lb/>Mathematical <lb/>modelling <lb/>Mathematical models <lb/>in mechanics <lb/>Force as a vector <lb/>Kinematics in 1D <lb/>and 2D <lb/>Vectors in mechanics <lb/>Equilibrium of a particle <lb/>Statics and forces <lb/>Kinematics of a particle <lb/>moving in a straight line <lb/>Kinematics of motion <lb/>in a straight line <lb/>Momentum <lb/>Dynamics of a particle <lb/>moving in a straight line <lb/>Newton's laws of motion <lb/>Newton's laws of motion <lb/>Statics of a particle <lb/>Linear momentum <lb/>Connected particles <lb/>Moments <lb/>Projectiles <lb/></body> <note place="headnote">TEACHING MATHEMATICS AND ITS APPLICATIONS, Volume 26, No. 3, 2007 </note> <page>123</page> </text> </tei>