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Engineering Subject Centre Report: Responding to the Changes in the Teaching and Learning of Mechanics in Schools

July 2005

Mechanics Report

Engineering Subject Centre Report: Responding to the Changes in the Teaching and Learning of Mechanics in Schools

July 2005

Authorship

This report was commissioned by the Engineering Subject Centre and was written by:

Carol L. Robinson, Mathematics Education Centre, Loughborough University Martin C. Harrison, Mathematics Education Centre, Loughborough University Stephen Lee, Mathematics Education Centre, Loughborough University

Edited by Engineering Subject Centre staff.

Published by The Higher Education Academy Engineering Subject Centre

ISBN 9781904804369 © 2005 The Higher Education Academy Engineering Subject Centre

Executive Summary Over recent years, there has been awareness in higher education of the declining knowledge in mechanics amongst the engineering student intake. Within the mathematics Alevel, it was traditional for students to study mechanics. However, there had been reports of some schools being unable to offer mechanics modules as part of Alevel mathematics and others advising students to study statistics modules in order to achieve a higher grade. As part of this project, a survey of schools in England was carried out, to ascertain the availability and uptake of mechanics modules within Alevel mathematics. First year engineering students at a number of English universities were also surveyed. Finally, as a result of questionnaires and interviews with academics throughout the United Kingdom, this report provides information about:

• the monitoring of changes in Alevel mathematics, • whether academics are aware of the prior knowledge in mechanics of their intake, • what knowledge they assume when they teach mechanics to first year engineering

students.

Where there is evidence of good practice in teaching mechanics to students with diverse backgrounds, to enable them to cope with the demands of a first year mechanics course, this is reported.

The report commences with background reading on the changes to Alevel structure and content. The commonly held view is that after the implementation of the Alevel changes in September 2004, there is likely to be a considerable decline in the number of candidates studying mechanics. Whatever choice of modules students take, their knowledge and experience of 'applied' mathematics will be considerably less than that of students who have studied mathematics prior to the implementation of the 2004 changes.

Following this, findings from a questionnaire, sent to just under 20% of schools in England which teach Alevel mathematics, are given. In over a quarter of schools, no more than one module of mechanics was offered. Thus there are a significant proportion of school students, some of whom may wish to become engineers, who are unable to study mechanics beyond M1, which is a very basic level. The data for the schools’ questionnaire was collected in January 2004, which was prior to the changes to Alevel mathematics that took place in September 2004. There is the expectation that the availability and uptake of mechanics may well decline further.

The findings from questionnaires completed by first year engineering students are then presented. These questionnaires focused on the students’ prior learning in mathematics and, in particular, whether the students had studied mechanics within Alevel mathematics and, if so, how many mechanics modules they had studied. From the three universities surveyed, it was found that there was a large proportion, 32%, of engineering students who have studied at most one (very basic) module of mechanics within Alevel mathematics. Thus significant

numbers of engineering students are commencing engineering courses having studied little or no mechanics.

The final part of the report addresses the issue of students’ prior knowledge of mechanics from the perspective of the university academic. It was found that few of the respondents were aware of the mechanics modules that their students had studied within Alevel mathematics. This lack of awareness on behalf of academics, with regard to the incoming knowledge of mechanics of their students, is concerning. In fact, 58% of the academics in the sample assumed a knowledge of mechanics that their students would not necessarily have. Students without this assumed knowledge may quickly feel disadvantaged, may struggle with the work and this may result in them giving up the course.

Finally, the report reveals that the mechanics problem is indeed an issue in universities. One academic stated, “We as an engineering faculty are reviewing the whole issue and would be in the market for materials – it’s a real problem and is eating up resources.”

Mechanics Report Engineering Subject Centre

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1 Introduction Over recent years, there has been an increasing awareness in higher education in England of the declining knowledge of mechanics amongst the engineering intake. This is believed to be caused by changes in the Alevel structure and syllabus. Within the mathematics Alevel, it was traditional for students to study mechanics. However, it no longer forms part of the core syllabus and there have been reports of some schools being unable to offer mechanics modules and others advising students to study other applied modules (e.g. statistics, discrete mathematics) in order to achieve a higher grade.

In this report the background to the problem is briefly described. Recent changes to the mathematics Alevel syllabus and structure are then explained and discussed.

Following this, findings from a questionnaire, sent to just under 20% of schools in England which teach Alevel mathematics, are given. This questionnaire was devised to ascertain the current availability and uptake of mechanics modules in schools. In addition, the findings from questionnaires completed by first year Engineering, Mathematics and Physics students are then presented. These questionnaires focused on the students’ prior learning in mathematics and, in particular, whether the students had studied mechanics within Alevel mathematics and, if so, how many mechanics modules they had studied.

Finally, academics were surveyed to find out more about the situation with regard to teaching mechanics at first year university level. Questionnaires to academics throughout the UK were followed up with interviews with some of these academics. Where there was evidence of good practice in the teaching/support of first year engineering students studying mechanics, this is reported on.

The conclusions of the findings are then presented and some recommendations provided. There are also some suggestions of future work which could be undertaken to further inform the sector about the current situation with regard to mechanics.

In addition to the information obtained from the various questionnaires and interviews, the authors, at the request of the Engineering Subject Centre, have also provided a list of topics which cover the material taught in the current Alevel mathematics modules Mechanics 1 (M1) and Mechanics 2 (M2), as many entrants to university engineering programmes now need support in, what for them, are important topics. These can be found in Appendix 7.


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2 Background Universities have been aware, for many years, of the ‘mathematics problem’. This relates to the declining levels of mathematical expertise which engineering students and others possess upon entry to university. Much research has been undertaken to address this issue, e.g. Armstrong & Croft (1). Discussion of the development of appropriate techniques for tackling the problem, including diagnostic testing and followup support, are evident, Hawkes & Savage (2). However, very little has been said about the associated ‘mechanics problem’ which is likely to be exacerbated by recent changes, to the Advanced Subsidiary (AS) and A level mathematics courses, which took effect from the autumn of 2004.

In 1997, Kitchen et al (3), in a paper entitled ‘The Continuing Relevance of Mechanics in A level Mathematics’, expressed concern that no mechanics was now included in the ‘core material’ for Alevel mathematics. (The ‘core material’ is the compulsory material that all students must study for certification in AS/Alevel mathematics.) This was particularly relevant at the time because of the proposed developments for the change in curriculum that was to take place in the year 2000.

In September 2000 changes were made to the structure of Alevels, via the introduction of ‘Curriculum 2000’, in order that all Alevel and ASlevel qualifications would become assessed completely by modules. An ASlevel would consist of three modules and an A level would consist of six modules. In most subjects the six modules for an Alevel would be made up from three AS modules and three A2 modules (where A2 modules are modules that are studied in school year 13). In the structure adopted it was suggested that students would take at least four ASlevel subjects (e.g. Mathematics, Physics, Computing, Physical Education) in year 12, and then continue three (e.g. Mathematics, Physics and Computing) onto full Alevel status in year 13. The introduction of ‘Curriculum 2000’ has been widely commented upon with many concerns raised, in particular the high ASlevel mathematics failure rate, James (4) and Porkess (5), (6).


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3 Alevel Mathematics Syllabi/Specifications

3.1 Curriculum 2000 The structure of Alevels was transformed through the implementation of modular Alevels in ‘Curriculum 2000’. (In fact, for mathematics, modular courses had been widely used for over five years.)

In mathematics, all examination bodies offered an array of modules. Table 3.1 shows a typical framework of the AS/A2 mathematics, with six modules in pure mathematics [P1 (AS level module), P2P6 (A2level modules)], along with a selection of ‘Applied’ modules, which would consist of at least the following:

• Mechanics [M1 (ASlevel module), M2M4 (A2level modules)] • Statistics [S1 (ASlevel module), S2S4 (A2level modules)] • Discrete [D1 (ASlevel module), D2 (A2level module)]

However, a few other modules may have been available from certain examination boards e.g. a coursework module.

AS A2 Pure P1 P2 P6 MechanicsM1 M2 M4 Statistics S1 S2 S4 Discrete D1 D2

Table 3.1 – Curriculum 2000 module classification

As in other Alevels, mathematics modules were classified as AS or A2. However, in all other Alevels, students studied three AS modules in Year 12 and three A2 modules in Year 13. This was not the case in mathematics. With the introduction of ‘Curriculum 2000’, for ASlevel certification in mathematics, ASlevel mathematics candidates were required to study two compulsory ‘common core’ modules (P1, P2) and an applied module and thus were required to incorporate a compulsory A2 module in their programme (as P2 is an A2 level module). This raised the issue as to whether ASlevel mathematics was harder than other subjects.

Porkess (5) discusses several issues, which may indicate that this was indeed the case. For Alevel certification, students were required to study three compulsory ‘common core’ modules (P1, P2, P3) and three other modules (at most two of which are classified as AS). Thus for certification in Alevel mathematics, students either studied three AS and three A2 modules or two AS and four A2 modules (see Figure 3.1).


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Figure 3.1 – Examples of possible module combinations taken by Alevel maths students

Thus Alevel mathematics, unlike any other Alevel, could consist of a combination of two AS and four A2 modules and hence potentially be harder than other Alevels. In particular, if a potential engineering student studied three modules of mechanics, he/she would end up with four A2 modules, whereas taking the first module of statistics (S1) together with one or two mechanics modules and pure modules would result in a combination of 3 AS modules and three A2 modules. One can see why teachers, endeavouring to ensure that their students obtain the highest possible grades, may have been tempted to advise students to study the latter combinations.

The various examination boards can specify the ‘common core’ modules differently but all must include certain topics and therefore are very similar across the boards. However, with reference to the mechanics modules M1M4, there is more scope for differentiation between the boards (due to the fact that these are not ‘core’ modules). An example of the content of the four mechanics modules offered by the examination board OCR (Oxford Cambridge and RSA Examinations) is in Appendix 1. In reality, similar content is found in the early mechanics modules of all examination boards, although in later modules greater differences between the boards becomes evident.

3.2 Revised Specifications (Effective from September 2004) Following the introduction of ‘Curriculum 2000’, the numbers of students taking Alevel mathematics fell dramatically, Porkess (6). This, along with other concerns, including the time pressures schools face in trying to cover all the required material, Porkess (7), have been a factor in causing further changes to the specifications, which were approved in October 2003. The changes were implemented for first teaching in September 2004 and thus the first cohort of Alevel students’ results will be available in summer 2006. The changes involved restructuring module content and consequently certification for mathematics A levels will again be different.


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Table 3.2 provides a summary of the changes. There will be four ‘core’ units (units as opposed to modules) labelled C1, C2 (ASlevel units), C3 and C4 (A2level units). These will replace the old common core of P1P3 and so the content of the three old modules will essentially be spread across the four new units. This implies that the mathematics Alevel may become easier. Students then choose two applied units to add to the four core units to create a mathematics Alevel. This is a fundamental change, as the maximum number of applied units, which a student may study for an Alevel, will reduce from three to two.

AS A2 Pure C1, C2 C3, C4 MechanicsM1 M2 M4 Statistics S1 S2 S4 Discrete D1 D2

Table 3.2 – Revised specification module classification.

Another consequence of these changes means that it will be possible for mathematics A level students to study three AS units and three A2 units, as in other Alevels, (for example C1, C2, C3, C4, M1, M2) or four AS units and two A2 units (for example C1, C2, C3, C4, M1, S1). This is different from the Curriculum 2000 specification, where, as has been noted, students may take three AS and three A2 modules or two AS modules and four A2 modules. Again this may lead to Alevel mathematics becoming easier.

Early feedback from mathematics teachers indicates that many may choose to give students two ‘first level’ applied units e.g. M1+S1, M1+D1, S1+D1. One Head of Department said “It would be ludicrous for us to offer M2 to students when S1 is a much easier option”. Thus the numbers of students studying mechanics beyond M1 may well decrease. Also, with the increasing popularity of statistics and discrete mathematics, and the reduction from three to two applied units in Alevel mathematics, there may be a considerable decline in the number of candidates studying mechanics. Whatever choice of units students take, their knowledge and experience of applied mathematics will be considerably less than that of students who have studied prior to the implementation of the 2004 changes.


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4 Questionnaire to Schools in England One of the authors of this report, Robinson, is responsible for the daytoday running of the Mathematics Learning Support Centre at Loughborough University. Through her work in the centre, anecdotal evidence came to light concerning the fact that some engineering students at Loughborough University had been unable to study mechanics modules at school or were advised against studying them. This prompted the authors to investigate the situation in schools. With the changes to the structure of Alevels in 2000 and the fact that more restructuring was due to be implemented for mathematics specifications in September 2004, the authors produced a questionnaire that was sent to schools in January 2004 in order to gather information on the situation at that time. The authors were interested in the availability and uptake of mechanics modules for students.

4.1 Questionnaire Structure and Administration A selection of 500 ‘schools’ (18%) was taken as a sample of the 2750 educational establishments in England where students study for ALevels (in any subject). There was a proportionate selection of schools from the different types of Local Education Authorities (Administrative Counties, Unitary Authorities and London [Inner and Outer]), ensuring a representative sample was taken.

A short questionnaire that was both easy and quick to complete was produced, in the hope that as many recipients as possible would complete and return it. The questionnaire was two pages in length and consisted of either boxes to be ticked or filled in with numbers. There was also scope for comments and written responses where appropriate. Following guidelines, Cohen et al (8), a pilot scheme was setup, in order to avoid some of the inherent problems associated with mailing questionnaires. The pilot scheme consisted of sending the provisional final questionnaire package (i.e. cover letter, questionnaire and memo) to ten participants that were known to the authors. These were mathematics teachers, heads of mathematics departments or exteachers from various areas (Leicestershire, Staffordshire and County Durham). Feedback from the questionnaires and memos received from the participants (eight of the ten replied) was positive, with only minor alterations recommended. It was then possible to produce a ‘final’ questionnaire that was then sent to the 500 schools. A copy of this can be seen in Appendix 2.

4.2 Results In the twoweek period following the initial mailing the authors received 170 completed questionnaires. Thus the initial response rate was approximately 35%. (Results of this initial response are published in Lee et al (9)). The questionnaire package was then resent to the heads of mathematics in the 330 schools which had not initially replied. A further 72 completed questionnaire were received, making 242 responses in total. Thus the final response rate from the sample was 48%. A comparison of the responses from the initial group with the responses from the subsequent group indicated that there were no significant differences in them and hence the two sets of responses were combined and the overall results follow.


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Within the 242 schools who completed the questionnaire, there were some 13,754 students studying either ASlevel or Alevel mathematics courses. Firstly the results on the availability of ‘applied’ modules are presented.

Table 4.1 shows the percentage of schools which did not offer any modules for each of the three applied strands and the percentage of students in the 13,754 student sample who studied ASlevel or Alevel mathematics in one of these schools. It can be seen that over 5% of schools in the sample do not offer any mechanics. Thus potential engineering students attending one of these schools have no opportunity to study mechanics modules within AS level or Alevel mathematics. Very few schools do not offer any statistics modules (just over 2%). The figures are much greater for discrete modules; this is likely to be because discrete mathematics is still relatively new at Alevel.

No Modules Offered

% of Pupils % of Schools

Mechanics 2.6 5.4

Statistics 1.4 2.1

Discrete 43.4 46.1

Table 4.1 – Availability of modules (No modules offered)

Table 4.2 shows the percentage of schools which did not offer any, or offered at most one module, for each of the three applied strands, and the percentage of students in the 13,754 student sample who studied ASlevel or Alevel mathematics in one of these schools. Thus students at these schools were unable to study two or more modules of a given strand. It can be seen that over a quarter (26.34%) of schools in our sample offer at most one module of mechanics. Over 15% of students in our sample are unable to study mechanics beyond M1. Similar numbers of students are unable to study statistics beyond S1 and again it can be noted that the availability of discrete modules is low.

No, or at most 1, Module Offered

% of Pupils % of Schools

Mechanics 15.8 26.3

Statistics 14.1 21.8

Discrete 79.1 79.0

Table 4.2 – Availability of modules (No modules or at most 1 module offered)

For the purposes of this report, the main interest lies in the availability of mechanics modules. As can be seen from the content of mechanics modules, given in Appendix 1, and from specimen examination papers, website (1), the material presented in M1 is an


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introduction to mechanics and at a very basic level. Not until students study M2 do they start to encounter more demanding material. (Traditionally, material in M1 and M2 would have been studied when mechanics was in the core material for Alevel.) Thus these results on the availability of mechanics modules show that a significant number of students (over 15%) are unable to study mechanics to a level which was once compulsory within the Alevel mathematics syllabus. Moreover, as shall be discussed later in Section 6, some lecturers of mechanics at university assume a prior knowledge of M1 and M2. Students from schools where mechanics is not offered beyond M1 are thus at a distinct disadvantage in these universities.

The results until now have focused upon the availability of applied modules. However, even if mechanics modules are available, students may not choose or may be advised not to study them. Thus attention is now turned to the uptake of applied modules in schools.

In Figure 4.1 the percentage of school students who are studying each of the individual modules is displayed. For example, it can be seen that approximately 43% of pupils are studying M1, compared to 51% studying S1. The percentage of students who studied M2 or S2 are similar to each other, at 18%. Few students study the higher level modules, e.g. M3, M4, S3, S4. Consequently it can be seen that few students study more than a basic mechanics module (M1).

These results demonstrate that a significant proportion of school students still study mechanics, at least until M1 level. However it gives no indication of whether the students who chose to study mechanics are those students who will proceed to study engineering. It is possible that potential engineering students may not be among the 43% of students who study mechanics. This report addresses this issue in Section 5 when findings from questionnaires to first year engineering, mathematics and physics students are presented.


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0

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1 2 3 4 5+ Module number

% of S

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Mechanics Statistics Discrete

Figure 4.1 Percentage of school students studying each applied module

4.3 Summary of Findings from the Questionnaire to Schools in England From the analysis it has been found that mechanics modules are not as widely available as statistics modules within AS/ Alevel mathematics. In over a quarter of schools in the sample, no more than one module of mechanics was offered. Thus there are a significant proportion of school students, some of whom may wish to become engineers, who are unable to study mechanics beyond M1, which is a very basic level. In addition, the number of students studying mechanics modules is less than the number studying statistics modules. 43% of students do study mechanics to M1 level. However potential engineering students may not be among this group. For some of these students, mechanics may not even be available at their school. Others decide, or are advised, not to study mechanics.

This data was collected in January 2004, which was prior to the changes to Alevel mathematics that took place in September 2004. These changes mean that students can now only study two applied modules as opposed to previously where they could study three applied modules. Thus there is the expectation that the availability and uptake of mechanics may well decline further.


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5. Questionnaire to First Year University Students The situation in schools with respect to the availability and uptake of applied modules has been considered. As mentioned in Section 4, it was found that 43% of students study some mechanics (M1) within Alevel mathematics. However it was not possible to conclude, from the questionnaire to schools, whether potential engineering students were among this group of students. Thus it was decided to write a second questionnaire and target engineering, physics and mathematics students, in order to ascertain the number of Alevel mechanics modules these students have studied. Initially students at Loughborough University were targeted. (At Loughborough University, approximately 900 students each year register on engineering, mathematics and physics courses.) Subsequently, questionnaires were distributed to students at Leicester and Nottingham Universities. In this section, details of the administration of the questionnaire and the findings from it will be reported.

5.1 Questionnaire Structure and Administration Initially data was collected via administering a simple questionnaire, at Loughborough University, in autumn 2003. A copy of this can be seen in Appendix 3. This questionnaire was a single page of questions obtaining information on what modules students had studied in Alevel mathematics and what grades they had achieved. If students had not studied A levels, there was a question for them to complete to provide information on the type of prior qualification/s they had. The questionnaire was administered in lecture and tutorial periods in weeks 67 of the first semester of the academic year 20032004. There was a response rate of 53% (i.e. completed by 53% of the 855 students registered on the courses) with 457 completed replies, of which 389 students had studied Alevel mathematics.

The questionnaire was readministered to Loughborough University students commencing study in the academic year 20042005. However, some changes were made to the previous year’s questionnaire, both in terms of content and administration. (A copy of this second version can be seen in Appendix 4.) Firstly, the request for detailed information on grades was dropped. (This was to reduce the time required for completion. Also it had been found in the 20034 questionnaire, that some students did not answer this section, perhaps because they could not remember their grades, or because they did not wish to divulge their grades.) In terms of the administration, the timing of distribution of the questionnaire was changed. The questionnaire was distributed much earlier in the academic year, in weeks 12, during lecture and tutorial periods. Moreover, for some students the questionnaire was included at the beginning of a mechanics diagnostic test, which had been developed by the authors. The findings from this test are reported in Lee et al (10). As a result of the change in administration of the questionnaire, there was a substantial increase in the response rate, from 53% to over 90% of the 823 registered students. In this year, of the 755 completed replies, 703 had studied Alevel mathematics.

For the academic year 20045, the questionnaire was also administered to students in engineering departments at Leicester and Nottingham Universities. The numbers studying first year engineering at the two universities differed by a large amount. At Leicester


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University there were approximately 80 first year students studying engineering, compared to over 400 students at Nottingham University (and approximately 650 at Loughborough University). The number of completed replies from students at the Universities of Leicester and Nottingham were 41 and 255 respectively, and of these 39 and 232 respectively had studied Alevel mathematics. The response rate equates to over 50% of students in each case. One reason for the lower response rate than at Loughborough University (over 90%) was that in both universities the questionnaire was administered in a lecture in the final few weeks of semester 2 in the 20042005 academic year, as opposed to week 1 of semester 1 at Loughborough University.

5.2 Results (Loughborough University) Comparative results for the two intakes at Loughborough University can be seen in Figure 5.1. Here it can be seen that the 703 replies in Autumn 2004 exhibit similar results to those of the 389 students in Autumn 2003. In both years, approximately 13% of students had not studied any mechanics modules and a further 26% had only studied only one module of mechanics. Consequently, around 39% of the (Engineering, Mathematics and Physics) students had studied at most one module of mechanics prior to entry to university. The comparative figures, for engineers only, were 10% (no modules), 23% (one module) and hence 33% (at most one module). (There were 318 engineers in 20034 and 498 engineers in 20045 in the sample who had studied Alevel mathematics.).

This means that one in three students commencing an engineering degree, that will require them to use and apply mechanics concepts and theory, will have very little prior knowledge. Subsequently, many of these students may find themselves struggling with their first university module of mechanics, unless no prior knowledge is assumed.


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Figure 5.1 – Comparison of number of Alevel mechanics modules studied by Engineering, Mathematics and Physics students at Loughborough University in 20034 and 20045

5.3 Results (Loughborough, Leicester and Nottingham Universities) Results from Nottingham University and Leicester University were collated with those of Loughborough University (20045). Figure 5.2 shows the percentage of engineering students, in each university, who had studied a particular number of mechanics modules.

Figure 5.2 Comparison of number of Alevel mechanics modules studied by engineering students in three universities

0

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Number of mechanics modules studied

% of S

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nts

20034 20045

2003 04 389 Students 2004 05 703 Students

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Nottingham Loughborough Leicester

Nottingham 232 students

Loughborough 498 students Leicester 39 students


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It can be seen, from Figure 5.2, that the results from Loughborough University and the University of Nottingham are very similar, particularly for the students who have not studied much mechanics. For example, 10% of engineering students at Loughborough University and 7% of engineering students at the University of Nottingham have studied no mechanics modules. The comparable figures for those who have studied one module of mechanics are 24% and 19% respectively. The results from the University of Leicester are distributed rather differently. There is a higher percentage of students, when compared to the other two universities, who have studied either no mechanics modules or one module of mechanics. Also, there is a lower percentage of students who have studied two or more modules of mechanics. Thus the sample of engineering students at the University of Leicester has less prior knowledge of mechanics than those at Loughborough and Nottingham Universities. It should be noted that the sample size from the University of Leicester is much smaller than either the sample size from Loughborough University or from the University of Nottingham. Consequently, the results from Nottingham and Loughborough Universities may be a more accurate reflection of the situation.

Overall three universities and including data from 20034 and 20045, 32% of the sample of engineering students had not studied mechanics beyond M1. Thus many students, commencing an engineering degree in these universities, will have very little prior knowledge of mechanics.

It should be noted that all three of these universities have similar entry requirements for engineering students and thus the findings presented here may not be an accurate reflection of the situation in the country as a whole where entry requirements vary widely.

5.4 Summary of Findings from the Questionnaire to First Year University Students First year engineering, mathematics and physics students at Loughborough University and first year engineering students at the Universities of Leicester and Nottingham completed a questionnaire to provide information on the number of mechanics modules, within Alevel mathematics, which they had studied. Loughborough students were surveyed over two consecutive years (20034 and 20045), Leicester and Nottingham students were surveyed in 20045. In total 1508 students completed the questionnaire. Of these, 1363 [90%] had studied Alevel mathematics and correctly completed the questionnaire. The comparative numbers for engineers only are, 1139 students completed the questionnaire and 1087 [95%] had studied Alevel mathematics and correctly completed it. In all three universities, it was found that there are a significant number of engineering students, 9% on average, who have studied no mechanics modules, 23% on average who have studied one module of mechanics and hence a large proportion, 32% on average, of engineering students who have studied at most one (very basic) module of mechanics within Alevel mathematics. Thus significant numbers of engineering students are commencing engineering courses having studied little or no mechanics.


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In addition to the 32% of Alevel engineering students who have studied at most one module of mechanics, there may also be a concern about the number of students who have come to university via a nonAlevel route. In the sample approximately 5% of the engineering students were from a nonAlevel background – these may include students from a foundation year or who have studied via the BTEC route. Results of a mechanics diagnostic test, Lee et al (9), taken by some of the sample of Loughborough University engineering students revealed that the nonAlevel students performed poorly in comparison with the A level students. Further work to investigate the mechanics background of these students would be very helpful in completing the picture of the prior mechanics knowledge of all engineering students upon entry to university.

As commented earlier, the three universities involved in the survey have similar entry requirements for engineering students and thus the findings presented here may not be an accurate reflection of the situation in the country as a whole, where entry requirements vary widely. Further work gaining information on the mechanics background of students from a wide range of universities may better inform one’s understanding of the situation.


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6. Questionnaire to University Academics Information about the uptake and availability of mechanics in schools and results of the survey of the number of mechanics modules studied by first year engineering students prior to commencing study at university were presented in Sections 4 and 5. This section and those following now address the issue of students’ prior knowledge of mechanics from the perspective of the university academic. With the changes to the Alevel syllabus and structure, outlined in Section 2, it is clearly important for relevant university academics to be aware of the changes and to ensure that universities respond appropriately to them. This section reports on a survey of such staff which was carried out to ascertain:

• whether staff have monitored changes in Alevel mathematics, • whether they are aware of the prior knowledge in mechanics of their intake, • what knowledge they assume when they teach mechanics to first year engineering

students.

Where there is evidence of good practice in the teaching of mechanics it is described.

The methods used to obtain the required information included an online questionnaire and followup interviews. In this section, details of the administration of the questionnaire and the main findings from it will be reported. The structure, format and findings from the follow up interviews will be presented in Section 7.

6.1 Online Questionnaire Structure and Administration Advice was sought from the HEA Engineering Subject Centre on the best method to distribute questionnaires to academics involved in the teaching of mechanics in universities. Dr Sarah Williamson informed the authors that the Engineering Subject Centre has a mailing list of academics and that the best way to contact relevant academics may be via this route. Although the authors were not able to obtain a copy of the list (for data protection reasons) she suggested that an online questionnaire be prepared and then staff in the Engineering Subject Centre would email those on their mailing list with the link to the questionnaire. (The mailing list had over 600 names of engineering academics, support staff and other related staff. There were contacts in almost every engineering department in the UK.) Once the questions were prepared, Paul Newman, a web designer in the Mathematics Education Centre, constructed an online version of the questionnaire.

The questionnaire contained questions in five sections and was longer than the other questionnaires described in this report (approximately 6 pages when printed, although the online version was one page which scrolled down). It was estimated that it would take approximately 10 minutes to complete, as most of the questions had options where the respondent selected an answer by clicking a radio button. It was also made a feature that personal data (namely Title, Forename, Surname, Email Address, University, Department) was made compulsory so that a questionnaire was not submitted without knowing its author. However respondents were assured that all replies would be treated in confidence.


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The online questionnaire was trialled by 6 relevant engineering academics at Loughborough University. The replies to this trial were included with the final data, as no major problems were raised by those who completed it. The final online questionnaire entitled “Students’ Exposure to Mechanics” can be seen in Appendix 5 and on website (2).

6.2 Results of the Online Questionnaire to University Academics

6.2.1 Information on response rates and the departments and universities represented by the respondents

In the oneweek period following the initial emailing (end of June 2004), the authors received 21 completed questionnaires, in addition to those of the 6 completed during trials. The email was then resent (at the beginning of July) to encourage those that had not initially replied to do so. An additional 6 replies were received following the second email. Therefore a total of 33 replies were received and these were from academics in 19 different universities (2 of the respondents indicated that mechanics was not important to their students and therefore these respondents were not required to complete anything other than the introductory section of the questionnaire). In 5 of the universities more than one reply was received back. A list of the respondents’ universities, the number of replies from each university and the type of department represented is in Appendix 6. The respondents represented a wide cross section of universities, including both pre and post1992 institutions and a Scottish university. The number of first year students in each department ranged from small (e.g. 15 students), to large (e.g. 250 students). In total there were 4271 students in the departments of the respondents. There were 4050 students in the 23 departments represented by the respondents who indicated that knowledge of mechanics was important for their students.

Thus in summary it can be seen that, the respondents represented 20% of HEIs in the UK that deliver engineering programmes, and therefore a large number of engineering students for whom a knowledge of mechanics was important.

Comments on the responses to the remaining sections in the online questionnaire now follow.

6.2.2 Section 2 of the Questionnaire:

Gaining an understanding of students’ prior knowledge in mechanics 31 of the 33 respondents completed this section the other 2 were not required to complete it as mechanics was not important to their students. Only 5 of the 31 respondents monitored the type of modules within Alevel mathematics, which their students had studied. 24 did not monitor these and for 2 it was not applicable, as their students studied for Scottish Highers. Thus 17% (5 out of 29) of respondents, whose students studied Alevel mathematics, were aware of the mechanics modules their students had studied. The 5 respondents represented


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380 students. The number of students who had studied a given number of mechanics modules can be seen in Table 6.1.

No of Students

No. of students studied Alevels

% of Students

Studied 0 Mechanics modules 87 380 23 Studied 1 Mechanics modules 138 380 36 Studied 2 Mechanics modules 145 380 38 Studied 3 Mechanics modules 10 380 3

Table 6.1 – Number of mechanics modules studied

It can be seen from Table 6.1 that the percentage of engineering students in the survey who had studied at most one module of mechanics was 59%. This figure is much higher than the 32% obtained in the survey of engineering students at Loughborough, Nottingham and Leicester Universities. This finding reinforces the statement at the end of section five that the results from the three universities involved in the earlier survey may not accurately reflect the situation in the country as a whole. The situation may be worse than predicted in the earlier survey and thus further work gaining information on the mechanics background of students from a wide range of universities is highly recommended.

In addition to the information provided about numbers of mechanics modules that had been studied by current students, 6 academics reported that, over recent years, there had been a decline in the number of mechanics modules that their students had studied.

Another method employed to ascertain prior knowledge in mechanics was that of the diagnostic test. 10 academics stated that they used a diagnostic test with their students. Some of these academics were from Loughborough University, where the authors had administered a mechanics diagnostic test to engineering students. Of the others, it was only confirmed in two cases that the diagnostic test was a mechanics test as opposed to a mathematics test.

In summary, from the responses in this section, it can be seen that few academics (17%) are aware of the mechanics modules which their students have studied within their Alevel mathematics and only two departments in the survey, independently of the authors, used a diagnostic test in mechanics. Thus many academics are unaware of the detail of their students’ prior study in mechanics.

This section of the online questionnaire also drew attention to the Scottish dimension with regard to mechanics. For two of the academics in the survey, Alevel qualifications were irrelevant as the majority of their students studied Scottish Highers and/or Advanced Highers. The respondents also indicated that there were issues arising due to the varied knowledge that their intake had upon arrival at university. These were not the same as the English situation, but nevertheless were important.


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Teaching of mechanics Academics were asked to state the level of prior knowledge, in terms of the number of mechanics modules in Alevel mathematics, they assume their students have. Table 6.2 summarises their responses.

Assumed knowledge, in terms of number of mechanics modules

Number of academics assuming knowledge of the given number of mechanics modules

Zero Knowledge 11 M1 Knowledge 10 M1 + M2 Knowledge 5

M1 + M2 + M3 or more 0

Don't Know or N/A 5

Table 6.2 – Number of mechanics modules assumed to have been studied

From Table 6.2, it can be seen that 11 academics assumed zero prior mechanics knowledge, with a further 10 citing that they expected knowledge comparable to M1 and 5 academics expecting knowledge comparable to two modules of mechanics. 5 stated that they did not know or that it was not applicable. Thus 15 out of 26 academics assume a knowledge of mechanics which their students will not necessarily have.

Three of the academics streamed students in order to teach them mechanics. The criteria for streaming were different in each case. One streamed according to the students’ prior knowledge of mechanics, i.e. two groups one for those who had studied at most one module of mechanics and one for those that had studied more than one module. The other two academics streamed according to whether students were on chartered engineering or incorporated engineering courses, or depending on whether the students had Alevel mathematics/science or not. In the first case the pass rates increased with streaming, however this was not the case of chartered/ incorporated status streaming.

The third question in Section 3 of the Questionnaire asked if any support in mechanics was available above the normal help. More than half, 17 academics said that there was extra help in mechanics available:

• Eight offered extra tutorials in mechanics. • Seven cited that a dropin centre was available. • Four that supplementary material was available.


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The number of hours of extra support that was available was in general between 1 and 5 hours per week, although in one case it was considerably more, at 12 hours per week. Four academics cited the pass rates as increasing after the provision of extra support, and six cited that they had stayed the same. There were a further six academics who selected the 'Didn't Know' option. Several comments supporting these results were made, including:

“The students who use the extra support say they have found it invaluable.”

“We have had great difficulty in persuading those students who really need the help to drop in to the Maths Learning Centre. It is generally the brighter students who use the facility.”


Supporting material for mechanics

In this section, academics were asked whether they were interested in using help sheets covering basic topics in mechanics. This referred to material that was being developed by the Higher Education Academy Engineering Subject Centre and the mathcentre project, website (3). 27 academics said that they would welcome the material, 4 said that they would not want the material and the other two did not comment. In addition 22 of the 27 said that they would be happy to be contacted to review such material as it was being developed. Thus there is a clear interest, from most of the academics in this survey, in material to help lay good foundations in the knowledge of mechanics.


Monitoring Alevels

Of the 29 academics from nonScottish universities, 12 stated that they had a member of staff who monitored the developments in Alevel mathematics, 10 said that they did not, with 7 saying that they did not know. Therefore, in at least a third of the departments represented by the academics in this survey, there is not a member of staff who monitors developments in Alevel mathematics. Clearly this is a worrying statistic and, with the recent changes in A level mathematics, will have implications for the students studying in these departments. Approximately half of those, whose departments monitor the developments in Alevel mathematics, review the content in their first year undergraduate course each time the mathematics AS/Alevels change.

In this section there were also several interesting comments made on this issue including:

“We probably only make major changes to our embedded teaching of mathematics every three years or so. So far we have been trying to increase the backup support to cope with changing intake.”

“This is an uphill struggle since the mixture of modules taken by students also varies from year to year. We can only really assume P1, P2 & P3 yet we MUST assume M1 to make any progress in the first year.”


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6.2.6 Other Comments In addition to the five sections of questions there was the opportunity for any other comments to be made. There were many relevant and interesting comments made by the majority of academics who had completed the questionnaire. These included:

“The mathematical ability of undergraduates is a handicap in learning mechanics.”

“According to my experience students no longer find mechanics interesting. Also, we do not seem to get the same calibre of students as we did a few years ago. We must accept that good quality students do not study engineering nowadays.”

“I no longer teach Mechanics. The general level of preparation is not as good as it used to be.”

“We as an engineering faculty are reviewing the whole issue and would be in the market for materials – it’s a real problem and is eating up resources.”

“We have to assume minimal knowledge quite often because of the different backgrounds of our students. Even in the case of students who have studied Mechanics, some revision has always proved useful.”

“It is important that our students are motivated by mechanics and see the purpose of it. Working on the basis that a good student has the ability to learn whatever he or she wants, it is the area that we are concentrating on when we are introducing engineering.”

6.3 Summary of Findings from the Online Questionnaire to University Academics A total of 33 responses to the online questionnaire were received and these were from academics in 19 different universities. The respondents represented a wide cross section of universities, engineering departments and a large number (over 4000) of engineering students for whom a knowledge of mechanics was important.

It was found that that few (17%) of the respondents were aware of the mechanics modules that their students had studied within Alevel mathematics. This lack of awareness on behalf of academics, with regards the incoming knowledge of mechanics of their students, is concerning. If they assume no prior knowledge of mechanics, as 11 of them did, then the only problem may be that those who had studied mechanics become bored. However, if they do assume some prior knowledge of mechanics, there may be many of their students who do not have this knowledge. In fact, 15 out of 26 academics (58%) assumed a knowledge of mechanics that their students would not necessarily have. Students without this assumed knowledge may quickly feel disadvantaged, may struggle with the work and this may result in them giving up the course.


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Streaming of classes and extra support were cited as two ways that were used to try to overcome the problem of lack of prior knowledge of mechanics amongst some of the students. More than half of the respondents stated that there was extra support in mechanics available. This took the form of extra tutorials in mechanics, dropin centres and supplementary material. There was a definite interest, from most of the academics in this survey, in help sheets in basic mechanics, which the Higher Education Academy Engineering Subject Centre had suggested might be useful in helping to lay good foundations in the knowledge of mechanics.

As well as very little knowledge of the mechanics modules that their students had studied within Alevel mathematics, there was also lack of knowledge of developments in Alevels. In at least a third of the departments represented by the academics in this survey, there was not a member of staff who monitored developments in Alevel mathematics. Clearly this is a worrying statistic and, with the recent changes in Alevel mathematics, will have implications for the students studying in these departments. This is an area where university staff development units or Higher Education Academy Subject Centres should perhaps work to inform the sector about changes taking place which are directly affecting the incoming knowledge of their student intake.

The online questionnaire also drew attention to the Scottish dimension with regard to mechanics. The respondents from Scotland indicated that there were also issues arising due to the varied knowledge in mechanics that their intake had upon arrival at university. These were not the same as the English situation, but nevertheless were important. Clearly it would be advantageous to conduct a survey in Scottish schools, similar to that carried out in England and to survey first year engineering students in Scottish Universities to ascertain the amount of mechanics they have studied prior to university. However, these surveys are outwith the scope of this project.

Finally, the online questionnaire revealed that the mechanics problem was indeed an issue.:



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7. Follow up Interviews with University Academics To obtain more detailed information, from an academic’s perspective, on the issue of students’ prior knowledge of mechanics and how universities respond to this, interviews were undertaken. Anecdotal evidence and evidence from the responses to the online questionnaire had indicated that some academics continue to teach mechanics assuming that all students have a firm grounding in the subject. Others assume no prior knowledge whatsoever. Some assume some prior knowledge and put on extra tutorials for those who struggle to cope. Most of those that were selected for interview were chosen as there was an indication, from their questionnaire responses, of good practice in the teaching and learning of mechanics within their department. This good practice is reported here.

7.1 Followup Interviews Structure and Administration

From the 33 replies to the questionnaire, six academics were selected for interview. In addition, another two academics, who did not complete the questionnaire, were interviewed. These latter two were known to have particular expertise in the area of teaching mechanics. The eight academics worked at five universities, namely, the Universities of Birmingham, Leeds and Strathclyde and Loughborough and Coventry Universities. Two taught in the same department. The size of their departments, in terms of student numbers, ranged from approximately 45 (1st year) students to 150 (1st year) students. The students they taught were studying Aeronautical and Automotive Engineering, Civil Engineering, Electrical and Systems Engineering, Mechanical Engineering, Manufacturing Engineering or Physics. Thus there was a range of different universities, a range in the size of departments involved and a range of engineering and other relevant courses.

The interviews were generally 3045 minutes in length and in most instances were carried out in the academic’s own institution. There were two exceptions to this, where telephone interviews were carried out in place of facetoface interviews. Where participants gave consent, in 2 of the 6 cases, the interviews were recorded. (It was not practical to record telephone interviews.) The interviews were recorded to avoid taking detailed notes while conducting the interview and to allow for clarification of any point at a later date. An interview schedule was produced. It included sections on knowledge of student intake, the teaching and learning of mechanics and academic support.

This part of the report is concerned mainly with good practice and hence only this aspect of the findings from the interviews will be presented.


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7.2 Followup Interviews – summary of findings on good practice

7.2.1 Good practice in gaining an understanding of students' prior knowledge The academics that were interviewed all used some method/s to gather information on their students' prior knowledge. These methods included:

• Interviews (used by 2 of the 7 departments), • Questionnaires (used by 1 department) [4 departments if Loughborough University

(LU) included], • Diagnostic Testing (Maths tests used by 6 departments, Mechanics test by 1

department. [4 departments used mechanics tests if LU included]), • Feedback from students via informal interaction (used by 3 departments).

The questionnaires and diagnostic testing at Loughborough University, which are referred to in the above list, were administered by the authors to the Loughborough University students.

7.2.2 Good practice in the teaching and learning of mechanics Some of the academics interviewed taught the subject essentially assuming zero prior knowledge. This then ensured that students who had not studied mechanics prior to university were not disadvantaged. Others expected students to have studied one or even two Alevel modules of mechanics. In these latter cases, several staff mentioned that they spent time at the beginning of the course revising topics from Alevel mechanics modules. It should be pointed out that for some students, who have not studied mechanics at Alevel, this would be new material. Nevertheless spending some time at the beginning of the module reviewing what in essence is taken to be prerequisite information is clearly much more beneficial than not revising at all.

To ensure the best learning experience for their students, the following methods were employed by at least some of the interviewees;

• Streaming, • Using experienced staff to lecture the material, • Reinforcing learning by using laboratory work, • Using two lecturers at one time to teach the students to encourage discussion of

concepts, • Regularly assessing the students’ knowledge, • Using group work, with staff input, to aid understanding of some of the concepts.

7.2.3 Good practice in the provision of support

With the large variation in the amount of prior knowledge of mechanics, it is not surprising that the interviewees discussed student support with mechanics. Evidence of varying


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degrees of support was found. Some universities had a support centre. One such centre was relatively small but offered mathematics support to students from specific engineering departments and employed final year students to work in the centre to provide help explicitly with mechanics. In most instances, there was an open door policy from the lecturer/ lecturers who taught the module. This meant that appropriate onetoone support was available, although the time lecturers were available was obviously limited. As well as this, regular tutorial sessions were found to be a useful means of supporting the learning. It was stressed that it is important that attendance is monitored so that students do actually attend the sessions and subsequently benefit from them.

In looking at what support and help was available to students having problems with mechanics, it was noted by the interviewees that there is currently little provision in terms of good paperbased resources dealing with specific topics in basic mechanics. Several academics made reference to good resources in mathematics and stated that they would welcome similar resources on mechanics.


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8. Conclusions

In this report the background to the ‘mechanics problem’ was briefly described.

Recent changes to the mathematics Alevel syllabus and structure were then explained. It was noted that, with the introduction of curriculum 2000, Alevel mathematics, unlike any other Alevel, could consist of a combination of two AS and four A2 modules and hence potentially be harder than other Alevels. Following the introduction of ‘Curriculum 2000’, the numbers of students taking Alevel mathematics fell dramatically. Changes were then implemented, which will first be taught in September 2004. A result of these changes is likely to be a considerable decline in the number of candidates studying mechanics. Whatever choice of units students take, their knowledge and experience of 'applied' mathematics will be considerably less than that of students who have studied prior to the implementation of the 2004 changes.

Following this, findings from a questionnaire, sent to just under 20% of schools in England which teach Alevel mathematics, were given. This questionnaire was devised to ascertain the availability and uptake of mechanics modules in schools. From the analysis it was found that mechanics modules are not as widely available as statistics modules within AS/ Alevel mathematics. In over a quarter of schools in the sample, no more than one module of mechanics was offered. Thus there are a significant proportion of school students, some of whom may wish to become engineers, who are unable to study mechanics beyond M1, which is a very basic level. In addition, the number of students studying mechanics modules is less than the number studying statistics modules. Although 43% of students do study mechanics to M1 level, it should be noted that potential engineering students may not be among this group. For some of the potential engineering students, mechanics may not even be available at their school. Others decide, or are advised, not to study mechanics.

The data for the schools’ questionnaire was collected in January 2004, which was prior to the changes to Alevel mathematics that took place in September 2004. These changes mean that students can now only study two applied modules as opposed to previously where they could study three applied modules. Thus there is the expectation that the availability and uptake of mechanics may well decline further. The authors plan to conduct a second survey of schools during the academic year 20056, to investigate whether or not there has been a decline.

The findings from questionnaires completed by first year engineering, mathematics and physics students were then presented. These questionnaires focused on the students’ prior learning in mathematics and, in particular, whether the students had studied mechanics within Alevel mathematics and, if so, how many mechanics modules they had studied. Loughborough University students were surveyed over two consecutive years (20034 and 20045) and the Universities of Leicester and Nottingham students were surveyed in 20045.


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1139 engineering students responded to the questionnaire, of whom 95% had studied A level mathematics. In all three universities, it was found that there are a significant number of engineering students, 9% on average, who have studied no mechanics modules, 23% on average who have studied one module of mechanics and hence a large proportion, 32% on average, of engineering students who have studied at most one (very basic) module of mechanics within Alevel mathematics. Thus significant numbers of engineering students are commencing engineering courses having studied little or no mechanics.

In addition to the 32% of Alevel engineering students who have studied at most one module of mechanics, concern was expressed about the background in mechanics for the students who have come to university via a nonAlevel route. Results of a mechanics diagnostic test written and administered by the authors, revealed that nonAlevel students performed poorly in comparison with the Alevel students. Further work to investigate the mechanics background of these students would be very helpful in completing the picture of the prior mechanics knowledge of all engineering students upon entry to university.

As commented earlier, the three universities involved in the survey have similar entry requirements for engineering students and thus the findings presented here may not be an accurate reflection of the situation in the country as a whole, where entry requirements vary widely. Further work gaining information on the mechanics background of students from a wide range of universities would be very helpful in establishing a clear picture of the situation.

Both questionnaires have proven to be effective methods in gaining an understanding of students’ prior knowledge of mechanics. Thus it is recommended, for any university with a high proportion of Alevel students, that they administer the ‘Questionnaire to First Year University Students’, to make it possible to better organise and produce a first year mechanics curriculum which is accessible to all their students.

The final part of the report addressed the issue of students’ prior knowledge of mechanics from the perspective of the university academic. With the changes to the Alevel syllabus and structure, it is clearly important for relevant university academics to be aware of the changes and to ensure that universities respond appropriately to them. An online questionnaire and followup interviews were used to ascertain whether staff have monitored changes in Alevel mathematics, whether they are aware of the prior knowledge in mechanics of their intake and what knowledge they assume when they teach mechanics to first year engineering students. Where there was evidence of good practice in the teaching of mechanics it was described.

It was found that few of the respondents were aware of the mechanics modules that their students had studied within Alevel mathematics. This lack of awareness on behalf of academics, with regard to the incoming knowledge of mechanics of their students, is concerning. In fact, 58% of the academics in the sample assumed a knowledge of mechanics that their students would not necessarily have. Students without this assumed


27

knowledge may quickly feel disadvantaged, may struggle with the work and this may result in them giving up the course.

As well as very little knowledge of the mechanics modules that their students had studied within Alevel mathematics, there was also lack of knowledge of developments in Alevels. In at least a third of the departments represented by the academics in this survey, there was not a member of staff who monitored developments in Alevel mathematics. Clearly this is a worrying statistic and, with the recent changes in Alevel mathematics, will have implications for the students studying in these departments. This is an area where university staff development units or Higher Education Academy Subject Centres should perhaps work to inform the sector about changes taking place which are directly affecting the incoming knowledge of their student intake.

The online questionnaire also drew attention to the Scottish dimension with regard to mechanics. The respondents from Scotland indicated that there were also issues arising due to the varied knowledge in mechanics that their intake had upon arrival at university. These were not the same as the English situation, but nevertheless were important. Clearly it would be advantageous to conduct a survey in Scottish schools, similar to that carried out in England and to survey first year engineering students in Scottish Universities to ascertain the amount of mechanics they have studied prior to university. However, these surveys are outwith the scope of this project.

During the followup interviews, good practice in the teaching of mechanics was discussed. Streaming of classes and extra support were cited as two ways that were used to try to overcome the problem of lack of prior knowledge of mechanics amongst some of the students. More than half of the respondents stated that there was extra support in mechanics available. This took the form of extra tutorials in mechanics, dropin centres and supplementary material. There was a definite interest, from most of the academics in this survey, in help sheets in basic mechanics, which the Higher Education Academy Engineering Subject Centre had suggested might be useful in helping to lay good foundations in the knowledge of mechanics.

Finally, the online questionnaire revealed that the mechanics problem was indeed an issue.


Future work by the authors includes exploring the relationship between students' prior knowledge of mechanics and their performance in their first year university courses. Other areas of interest include investigating the prior knowledge of students who have not studied Alevels, including Scottish students, and finding out about the prior mechanics knowledge of Alevel engineering students at universities where the entry requirements are different to those at Loughborough, Nottingham and Leicester Universities.


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9. Acknowledgements

The authors are grateful to the following for their assistance in the work involved in this project: Mr Godfrey Pell for advice regarding questionnaire methodology and the statistics involved in analysis of the questionnaire to schools; Ms Rosie Shier for helpful discussions and suggestions regarding the statistical analysis associated with the data; Mr Paul Newman for technical assistance in the production of the online questionnaire; Dr Sarah Williamson for advice regarding the distribution of questionnaires to academics; Dr Tony Croft, Director of the Mathematics Education Centre, for his support of the work; Mrs Clare Wright for administrative assistance.

10. References

[1] Armstrong, P.K. & Croft A.C., "Identifying the Learning Needs in Mathematics of Entrants to Undergraduate Engineering Programmes in an English University", European Journal of Engineering Education, Vol 24, No. 1, 1999, pp5971.

[2] Hawkes, T. & Savage, M., (eds.), “Measuring the Mathematics Problem”, Engineering Council Report, 2000.

[3] Kitchen, A., Savage, M. and Williams, J., "The Continuing Relevance of Mechanics in Alevel Mathematics", Teaching Maths & Its Applications, Vol 16, No. 4, 1997, pp 165170.

[4] James, G., "Mathematics in schools: Implications for undergraduate courses in engineering and other numerate disciplines", Mathematics TODAY, Vol 146, 2002, pp 140 146.

[5] Porkess, R., "AS/ A Level Mathematics modules and the UCAS Tariff", MSOR Connections, Vol 1, No. 4, 2000, pp 1820.

[6] Porkess, R., "The new AS and A Levels in Mathematics", MSOR Connections, Vol 3, No. 4, 2002, pp 1216

[7] Porkess, R., "Mathematics in Curriculum 2000: What will students know? What will students not know?", MSOR Connections, Vol 1, No. 3, 2000, pp 3539.

[8] Cohen, L., Manion, L., and Morrison, K., “Research Methods in Education – 5 th

Edition” Routledge, Falmer, London.


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[9] Lee, S., Harrison, M.C. and Robinson, C.L., "Mechanics Teaching in Schools: Implications for Undergraduate Engineering Courses" In International Conference in Innovation, Good Practice and Research in Engineering Education, Halstead, A., and Lister, P., (eds) University of Wolverhampton, Vol 2, 2004, pp 9398.

[10] Lee, S., Harrison, M.C. and Robinson, C.L., “UK engineering students’ knowledge of mechanics on entry: Has it all gone?”, Accepted for publication in The Proceedings of the International Conference on Engineering Education (ICEE 2005) July 2005, Gliwice, Poland

11. Websites [1] http://www.mathsrevision.net/alevel/specimen.htm (accessed 22/6/5)

[2] http://www.mathcentre.ac.uk/stephen/questionnaire.html (accessed 22/6/5)

[3] http://www.mathcentre.ac.uk/ (accessed 26/6/5)

http://www.mathsrevision.net/alevel/specimen.htm

http://www.mathcentre.ac.uk/stephen/questionnaire.html

http://www.mathcentre.ac.uk/


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Appendix 1 – Content of Alevel Mechanics Modules

An example of the content of the four mechanics modules offered by the examination board OCR (Oxford Cambridge and RSA Examinations) is given below.

M1

• Force as a vector • Equilibrium of a particle • Newton’s Laws of motion • Linear momentum • Kinematics of motion in a straight line

M2

• Centre of mass • Equilibrium of a rigid body • Motion of a projectile • Uniform motion in a circle • Coefficient of restitution and impulse • Energy, work and power

M3

• Equilibrium of rigid bodies in contact • Elastic strings and springs • Impulse and momentum in two dimensions • Motion in a vertical circle • Linear motion under a variable force • Simple Harmonic Motion

M4

• Relative motion • Centre of mass • Moment of inertia • Rotation of a rigid body • Stability and oscillations


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Appendix 2 Questionnaire to Schools in England


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Appendix 2 – continued


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Appendix 3 Questionnaire to First Year University Students – 2003 4


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Appendix 4 Questionnaire to First Year University Students – 2004 5


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Appendix 5 Online Questionnaire to University Academics


36



37



38



39



40



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Appendix 6 Universities Represented by Responses to the Questionnaire to University Academics

University Number of Responses

Departments Involved in the Survey (alphabetical order)

University of Brighton 1 Coventry University 3 University of Huddersfield 1 Napier University 1 University of Ulster 1 Aston University 1 University of Essex 1 University of Exeter 1 Loughborough University 9 University of Strathclyde 2 University of Wales Bangor

1

University of Plymouth 2 UMIST (University of Manchester Institute of Science and Technology)

1

University Of Nottingham 1 University of Newcastle upon Tyne

1

University of Birmingham 1 University of Bristol 3 Cardiff University 1 Imperial College 1

Aeronautical and Automotive Engineering Chemical Engineering Civil Engineering Electrical Engineering Engineering Mathematics Engineering and Technology Informatics Manufacturing Engineering Material Engineering Mechanical Engineering Systems Engineering


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Appendix 7 Possible Topics for Mechanics Leaflets

This appendix contains a list of possible topics for mechanics leaflets. These topics are taught in the Alevel mathematics modules Mechanics 1 (M1) and Mechanics 2 (M2)

Sections 13 relate mainly to M1. Sections 410 relate mainly to M2. Leaflets on the topics shown in bold are already available on the mathcentre website (www.mathcentre.ac.uk/).

Section 1: Introduction

• Introduction to Mechanics • Units & Prefixes • Conversion of Units • Simplifying Mechanics • Forces as Vectors • Force Diagrams • Resolving Forces – i, j, k notation • Equilibrium of a Particle • Velocity & Acceleration (include acceleration due to gravity) • General motion in a straight line (using calculus)

Section 2: Newton’s Laws of Motion

• 1st Law • 2nd Law • Types of force: driving force/pull/thrust of engine, tension • Forces acting together • Motion due to gravity without/with constant air resistance • Forces acting at an angle, e.g. tow rope at an angle • Motion on a slope (including resolving along & perpendicular to slope) • Friction: • limiting friction, angle of friction, coefficient of friction, normal contact force • 2.5, 2.6 & 2.7 with friction • 3rd Law (cars & trailers, lifts) • Pulleys

Section 3: Linear Momentum

• Impulse & Momentum: definition & units • Impulsemomentum equation (impulse = change in momentum): ball hitting wall • Conservation of Momentum: collision of 2 billiard balls


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Section 4: Work, Energy & Power (motion with constant acceleration) • Work & Kinetic Energy: definitions, units & work energy equation • Cases with force at an angle (including perpendicular force) • Case with resistive forces • Power: definition & units • Relationship between Power & Work

Section 5: Potential Energy

• Potential Energy: definition & units, conservative & nonconservative forces • Conservation of Energy • Connected Particles (pulleys) with and without resistance • Motion round curved path, e.g. switchbacks

Section 6: Projectiles

• Independence of horizontal & vertical motion • Greatest Height, Horizontal Range • Trajectory

Section 7: Uniform motion in a circle

• Angular/tangential speed & acceleration: definitions • Horizontal circles • (Examples with forces, e.g. ball in circular groove, bicycle on banked track) • Conical pendulum

Section 8: Coefficient of restitution and impulse (nonoblique impacts)

• Coefficient of restitution: definition • Ball hitting wall • Collisions with up to 3 objects • Collisions involving energy

Section 9: Moments (Rigid Bodies not particles)

• Moment: definition (Turning effect of a force) • Moments & Equilibrium • Examples with nonparallel forces

Section 10: Centre of Mass & Equilibrium of a Rigid body

• Centre of mass: definition


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• Composite Bodies • Hanging & Balancing: simple bodies • Toppling and/or Sliding: simple bodies • Special Shapes; laminas, viz. triangles, sectors of circle, etc • Uniform Solid Shapes: hemisphere, cone & composites • Uniform Shell Shapes: hemispherical bowl, hollow cone

Section 11: Miscellaneous

• Lami’s theorem

Alternative Formats This publication can be downloaded from the Engineering Subject Centre website

www.engsc.ac.uk. Please call 01509 227170 for alternative format versions.

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