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Teaching Mathematics and Its Applications: International Journal of the IMA
, Volume Advance Article – May 29, 2017

19 pages

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- Publisher
- Institute of Mathematics and its Applications
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- © The Author 2017. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. All rights reserved. For permissions, please email: journals.permissions@oup.com
- ISSN
- 0268-3679
- eISSN
- 1471-6976
- D.O.I.
- 10.1093/teamat/hrx009
- Publisher site
- See Article on Publisher Site

Abstract As an increasing number of British universities are now requiring/encouraging mathematics applicants to have taken ‘extension papers’ such as the Mathematics Admissions Test (MAT), Advanced Extension Award (AEA) and Sixth Term Examination Paper (STEP), current students were asked how useful they were in preparation for their degree. The MAT was most commonly described as good preparation for undergraduate mathematics, whilst most participants who had taken the AEA were indifferent regarding its usefulness. Participants were positive about STEP, commending its similarity to undergraduate-style assessment and its challenging questions. The students’ views suggested that those wishing to be well prepared for tertiary mathematics should take one of these papers, preferably STEP. However, whilst universities may not necessarily wish to require applicants to pass extension papers, it may be beneficial for universities to recommend students to take them in order to improve their mathematical thinking and expectations of undergraduate mathematics study. I. Introduction 1.1 The problematic secondary–tertiary mathematics transition The preparedness of British undergraduate mathematicians for the demands of university study has been of concern since the 1990s. This manifests itself in two ways: concerns that undergraduate mathematics students are insufficiently prepared for the demands of their course; and concerns about the number of students taking any post-compulsory mathematics qualifications (in the UK, this equates to the upper secondary stage of schooling). This is commonly referred to as the ‘Mathematics Problem’, both nationally and internationally. Lecturers’ concerns about students’ preparation for undergraduate mathematics are reflected in data that suggests students have negative experiences at university. Low pass rates in mathematical subjects are common in the first year of study (London Mathematical Society, 1995), and the mathematical sciences had the highest drop-out rate of all disciplines in 2012/13 (Higher Education Statistics Agency, 2014). Tackling the Mathematics Problem (London Mathematical Society, 1995) reported that this phenomenon occurs despite many mathematics students achieving good grades in pre-university examinations. According to the London Mathematical Society (1995), incoming students were lacking in three areas: They were unable to fluently and consistently perform algebraic manipulations and simplifications. Their analytical powers were weak in instances where they are required to solve multi-step problems. They were ignorant of the nature of mathematics and, more specifically, undergraduate mathematics. Concerns regarding students’ mathematical preparedness for not just mathematics degrees but degree courses in the sciences which require mathematical competency are clearly long-standing, with similar concerns raised more recently by the likes of ACME (2011), the Institute of Physics (2011), the British Academy (2012), the Royal Society for the Encouragement of Arts, Manufactures and Commerce (Norris, 2012) and the Higher Education Academy (Hodgen et al., 2014). Furthermore, recent work by Darlington & Bowyer (2016) found that many mathematics, science and social science undergraduates did not feel well prepared for the mathematical components of their degrees by A-level Mathematics. The skills taught at school are often considered by universities to be an insufficient basis for further study in mathematics, leading to the perception that there is a ‘gap’ between secondary and tertiary mathematics (Thomas, 2008). This gap is described by Tall (1991) as a shift ‘from describing to defining, from convincing to proving in a logical manner based on those definitions’ (p. 20). Lawson (1997) argues that the apparent discrepancy between what students actually know upon leaving secondary school and what lecturers expect them to know when they begin university study ‘will, at the very least, impair the quality of their education and, at worst, may prove too difficult for them to bridge’ (p. 151). Furthermore, the situation may be worsening. For example, Smith (2004) found that scores on a diagnostic test for first-year undergraduate mathematics students had decreased with each new cohort. Of particular concern are undergraduates’ difficulties with mathematical proof. Selden (2012) calls the new emphasis on proof at the undergraduate level a ‘major hurdle’ for newcomers, with much of it centred on mathematical analysis. This is, however, not a new concern, with lecturers commenting on this since the 1970s (e.g. Bell, 1976; Selden & Selden, 2003; Weber, 2001). This may be a result of inadequate handling of proof in pre-university mathematics qualifications in the UK. These qualifications (the A-level—see Section 1.2) generally only deal with inductive proofs for sums of series, and Anderson (1996) argues that proof is not assessed at this level by questions ‘which demand any depth of understanding or which require any creativity in the process of justification’ (p. 129). Consequently, there is significant concern that the A-level Mathematics curriculum does not sufficiently meet universities’ needs (e.g. Cox, 2000; Hawkes & Savage, 2000; Lawson, 1997; Porkess, 2006). University mathematics departments are reacting in different ways. For example, diagnostic testing is now used in many mathematics departments across the UK (LTSN MathsTEAM, 2003; Williams et al., 2010). Additionally, the 4-year undergraduate ‘MMath’ in mathematics was introduced on the recommendation of the Neumann report, which claimed that changes were necessary in order to respond to: changes to the secondary mathematics curriculum; the continuing growth of mathematics; and to ensure that undergraduate qualifications in the UK could remain comparable with those in other countries. (Neumann, 1992, p. 186) 1.2 Pre-university mathematics assessment in England and Wales In England and Wales, students intending to progress to university commonly take Advanced ‘A’-level qualifications at age 18 in their final year of secondary education. Studied over the course of 2 years,1 students typically take three or four subjects of their choosing to this level. Universities usually require students to achieve certain grades in A-level examinations in order to be offered a place to study there. Two mathematics A-levels are commonly taken: Mathematics and Further Mathematics. A-level Mathematics is the most popular of the A-level subjects (Joint Council for Qualifications, 2016), and Further Mathematics is one of the fastest growing (although numbers are small in absolute terms). A-levels are graded from A*–E (see Table 1). Table 1. Grading in A-level Mathematics and Further Mathematics Grade Mark† required to achieve grade (%) Proportion of A-level candidates achieving the grade in 2016 (%) Mathematics Further Mathematics All subjects A* 80 overall + 90 in ‘A2’ modules 17.5 28.7 8.1 A 80 24.3 27.5 17.7 B 70 22.3 20.6 27.1 C 60 16.1 11.3 24.7 D 50 10.8 6.5 14.6 E 40 6.1 3.5 5.9 Ungraded (fail) <40 2.9 1.9 1.9 Grade Mark† required to achieve grade (%) Proportion of A-level candidates achieving the grade in 2016 (%) Mathematics Further Mathematics All subjects A* 80 overall + 90 in ‘A2’ modules 17.5 28.7 8.1 A 80 24.3 27.5 17.7 B 70 22.3 20.6 27.1 C 60 16.1 11.3 24.7 D 50 10.8 6.5 14.6 E 40 6.1 3.5 5.9 Ungraded (fail) <40 2.9 1.9 1.9 Source: Joint Council for Qualifications (2016). † The mark is calculated as a percentage of the ‘unform mark scale’ (UMS), a standardized means of marking examinations, according to candidates’ performance in the examinations. Table 1. Grading in A-level Mathematics and Further Mathematics Grade Mark† required to achieve grade (%) Proportion of A-level candidates achieving the grade in 2016 (%) Mathematics Further Mathematics All subjects A* 80 overall + 90 in ‘A2’ modules 17.5 28.7 8.1 A 80 24.3 27.5 17.7 B 70 22.3 20.6 27.1 C 60 16.1 11.3 24.7 D 50 10.8 6.5 14.6 E 40 6.1 3.5 5.9 Ungraded (fail) <40 2.9 1.9 1.9 Grade Mark† required to achieve grade (%) Proportion of A-level candidates achieving the grade in 2016 (%) Mathematics Further Mathematics All subjects A* 80 overall + 90 in ‘A2’ modules 17.5 28.7 8.1 A 80 24.3 27.5 17.7 B 70 22.3 20.6 27.1 C 60 16.1 11.3 24.7 D 50 10.8 6.5 14.6 E 40 6.1 3.5 5.9 Ungraded (fail) <40 2.9 1.9 1.9 Source: Joint Council for Qualifications (2016). † The mark is calculated as a percentage of the ‘unform mark scale’ (UMS), a standardized means of marking examinations, according to candidates’ performance in the examinations. Both qualifications consist of a mixture of compulsory and optional modules. A-level Mathematics has four compulsory ‘Core Pure Mathematics’ modules, and optional applied mathematics modules in Mechanics, Statistics and Decision Mathematics. A-level Further Mathematics has three compulsory ‘Further Pure Mathematics’ modules, with an additional three modules available from a mixture of pure and two applied options. However, these A-levels are currently undergoing reform, and substantial changes to their content and structure will take effect in September 2017. The regulator of qualifications in England and the government’s Department for Education (2013) have made changes such that A-level Mathematics will consist of 100% prescribed content (a core of pure mathematics in addition to topics in statistics and mechanics), and 50% prescribed content for A-level Further Mathematics, in a bid to ensure a consistent base of content knowledge amongst students who have taken these qualifications (ALCAB, 2014). Both qualifications prepare students for the workplace and for undergraduate study in a wide range of science and social science subjects, in addition to tertiary mathematics. Consequently the reforms will have implications for some prospective students’ readiness for undergraduate study in mathematically demanding subjects. 1.3 University entry requirements Owing to the increasing number of candidates taking A-level Mathematics and Further Mathematics, and the large proportion of those candidates who achieve top grades, university admissions tutors in the mathematical sciences have a large pool of high-quality candidates to choose from. Students of the mathematical sciences are more likely to achieve top grades at A-level than students in other subject areas, with 8.1% of them achieving three or more A* grades at A-level (Vidal Rodeiro & Zanini, 2015), a figure much higher than in other degree subjects. Furthermore, the number of mathematics undergraduates has been steadily increasing over recent years, from 13,188 in 1996/7 to 30,340 in 2014/15 (Higher Education Statistics Agency, 1998, 2016). Consequently, there is a need for admissions tutors to find additional measures of differentiating between well-qualified candidates, given the A* reflects a high degree of accuracy as well as mathematical competency. Many good mathematicians could therefore miss out on achieving an A* through minor mistakes. A growing number of universities either explicitly require, or recommend, that mathematics applicants take certain ‘extension papers’. By ‘extension paper’, we refer to additional qualifications or examinations which students may take alongside A-level Mathematics and/or Further Mathematics. These extension papers are also used to counter perceived problems with A-level because, although they do not require students to learn any additional mathematics content, they provide a more challenging form of assessment and thus require students to engage with mathematics (especially formal mathematics and proof) in a way that may be more reminiscent of university mathematics. In the UK, the papers which might be required of students are: the Mathematics Admissions Test (MAT); the Advanced Extension Award (AEA); and a Sixth Term Examination Paper (STEP). Some universities specifically require students to take one or more of these papers to be admitted to mathematics courses, others simply encourage it, whilst some universities make lower A-level grade requirements for students who pass extension papers. Sample questions from each extension paper may be found on their respective websites (University of Oxford Mathematical Institute (2017a) for the MAT; Pearson (2017) for the AEA; and Cambridge Assessment Admissions Testing (2017) for the STEP). 1.4 MAT Evolving from admissions tests used by the University of Oxford, the MAT is now used by Oxford and Imperial College London as a means of shortlisting undergraduate mathematics candidates. The MAT is based on AS-level Mathematics material only, as it is examined during the beginning of applicants’ second year of A-level Mathematics study, and aims to assess students’ mathematical understanding, as opposed to knowledge (University of Oxford Mathematical Institute, 2017a). It has no grading system or pass mark—students’ results are interpreted in conjunction with the rest of their application to shortlist for interviews. For illustration, however, in 2016 the mean mark of the 1,973 Oxford Mathematics, Mathematics and Statistics, and Mathematics and Philosophy applicants2 was 50.3%, with the mean amongst those who were offered a place to study being 73.1% (University of Oxford Mathematical Institute, 2017b). Research by Darlington (2015) found that MAT examinations primarily tested students’ skills in the areas of proof, justification and interpretation, conjecturing and comparing, and evaluation. Such skills were found to form the majority of first year undergraduate mathematics examination questions in pure mathematics. 1.5 AEA Unlike the MAT, taking the AEA is not a specific entry requirement for any university mathematics courses; however, many universities describe it as advantageous to students’ applications, and some require it when students are not able to take A-level Further Mathematics.3 Introduced in 2002, the AEA is based on A-level Mathematics knowledge, and candidates find it difficult: in 2016, 56.6% of candidates’ papers were ungraded (Joint Council for Qualifications, 2016). Questions in the AEA typically involve the use of familiar, routine procedures, perhaps in new situations that candidates are unfamiliar with (Darlington, 2015). The AEA is taken in the summer at the same time as A-level examinations. AEAs were originally introduced by the government in order to provide a means of differentiating between high-achieving students, though uptake is very small: in 2016, there were only 742 AEA Mathematics candidates (10.4% fewer than in 2015), compared to 92,163 A-level Mathematics and 15,257 A-level Further Mathematics candidates (Joint Council for Qualifications, 2016). 1.6 STEP Originally created over 30 years ago to form part of the University of Cambridge’s admissions requirements, STEPs evolved from the ‘special paper’ examinations set by Cambridge colleges for students to do in the sixth form of schooling (aged 16–18 years). They have grown in popularity over recent years, and are available for any student to take up, regardless of which university they apply to. Students who apply to the University of Cambridge are required to achieve certain pass grades in one or more of the STEPs. Other universities, such as the University of Warwick, either encourage applicants to take one, or will stipulate lower A-level grade requirements to students who have done so. Three STEPs are available: STEP I and STEP II draw upon A-level Mathematics knowledge, and STEP III draws upon A-level Further Mathematics knowledge. Students usually take two STEPs, depending on the A-levels that they are studying and the university that they are applying to. STEPs are difficult—in 2016, 24.3% of candidates failed STEP I (Cambridge Assessment Admissions Testing, 2016a)—and are aimed at the top 5% of A-level Mathematics candidates (University of Cambridge Faculty of Mathematics, 2015). In 2016, there were 2,050 entries for STEP I, 1,346 for STEP II and 903 for STEP III (personal communication). The STEPs are used to test candidates’ mathematical thinking and the questions are similar in style to university mathematics questions. In fact, in a study analysing and comparing the MAT, AEA and STEP, Darlington (2015) found that the skills assessed in STEPs were the most similar in style to undergraduate mathematics skills of the three extension papers. STEPs assess a combination of students’ proof and deduction skills, along with their ability to perform familiar calculations in novel or more challenging contexts (Darlington, 2015). Furthermore, STEPs have been found to be very good predictors of Cambridge mathematics students’ final degree classifications (University of Cambridge, 2011). 1.7 Aims In response to ongoing A-level reform (see Section 1.2), we undertook a large-scale study investigating students’ experiences of A-level Mathematics and Further Mathematics as preparation for degrees requiring mathematical competency (Darlington & Bowyer, 2016). The wider study surveyed current science and social science undergraduates who had taken A-level Mathematics and/or Further Mathematics regarding their views and perceptions of their mathematical preparation by those qualifications. In the case of undergraduate mathematics, the experiences of those students who had taken extension papers were also investigated as the skills assessed in those papers have been found to be different to A-level Mathematics and A-level Further Mathematics (Darlington, 2014), and share commonalities with the skills required to succeed in undergraduate mathematics (Darlington, 2015). Over 4,000 students took part in the wider study, including 928 undergraduate mathematicians. This article focuses on the responses of the 430 mathematics students who reported to have taken one or more of the MAT, AEA or STEP. 2. Method 2.1 Approach Using the contact details available online, mathematics departments at all universities in Britain4 were contacted via email to invite them to participate in this study. Departments were asked to pass on to their students details of an online questionnaire aimed at those in their second year or above who had taken AS- or A-level Further Mathematics. A response rate is incalculable as it is not possible to guarantee which universities did pass on information about the questionnaire—though many replied to confirm that they had done so. Mathematics students from a total of 42 universities (54.5% of those contacted) took part in the overarching study. 2.2 Instrument An online questionnaire was developed by the research team in conjunction with two mathematics assessment experts. Consisting of 21 questions, the questionnaire was piloted with three recent graduates from mathematically demanding degrees in order to ascertain whether the questions were clear, appropriate and effective. Minor changes were made in response to feedbacks. The questionnaire comprised of three sections. The questions posed relevant to the data reported in this article related to: About you: Gender, university, degree course, year of study, academic performance and the entry requirements for their degree. A-levels: Year taken, grades achieved in Mathematics and Further Mathematics and optional modules studied. Experiences of post-compulsory mathematics: Participants were asked to describe A-level Mathematics, AS- or A-level Further Mathematics, STEP, AEA Mathematics and the MAT (if applicable) as either ‘good preparation’, ‘bad preparation’ or ‘neither good nor bad preparation’ for undergraduate mathematics. They were asked to explain their answers. Participation was voluntary, anonymous and responses were accepted between September and December 2014. 2.3 Sample In common with the undergraduate mathematics cohort in the UK5 where 63% of whom were male in 2015 (UCAS, 2015), majority of the 430 participants were male (73.4%). Participants studied at a total of 27 different universities, with an average of 15.7 participants per university (median = 32.0). As an indication of the quality of the university, according to the Complete University Guide (2016), 7 feature in the top 10, 15 in the top 20 and 20 in the top 30 of the 70 listed. Having taken AS- or A-level Further Mathematics was a condition for participation in the study—87.2% of participants had taken A-level Further Mathematics, and 12.8% AS-level Further Mathematics. The majority of participants (70.2%) were required to have taken A-level Further Mathematics in order to be accepted onto their current university course. Nearly half (49.1%) of respondents were in their second year of study, with 32.2% in their third and 18.8% in their fourth year. The majority of participants (52.3%) were studying for single or joint honours undergraduate 3-year Bachelor’s degrees in mathematics, with 47.7% taking integrated 4-year Master’s degrees such as the ‘MMath’ (see Table 2). Table 2. Degrees studied by participants (N = 430) Degree No. of participants Proportion of participants (%) Mathematics 361 84.0 Joint honours Statistics and Economics 17 4.0 Statistics 13 3.0 Computer Science 12 2.8 Philosophy 9 2.1 Physics 9 2.1 Economics 5 1.2 Business and Finance-related 2 0.5 Humanities subjects 2 0.5 Degree No. of participants Proportion of participants (%) Mathematics 361 84.0 Joint honours Statistics and Economics 17 4.0 Statistics 13 3.0 Computer Science 12 2.8 Philosophy 9 2.1 Physics 9 2.1 Economics 5 1.2 Business and Finance-related 2 0.5 Humanities subjects 2 0.5 Table 2. Degrees studied by participants (N = 430) Degree No. of participants Proportion of participants (%) Mathematics 361 84.0 Joint honours Statistics and Economics 17 4.0 Statistics 13 3.0 Computer Science 12 2.8 Philosophy 9 2.1 Physics 9 2.1 Economics 5 1.2 Business and Finance-related 2 0.5 Humanities subjects 2 0.5 Degree No. of participants Proportion of participants (%) Mathematics 361 84.0 Joint honours Statistics and Economics 17 4.0 Statistics 13 3.0 Computer Science 12 2.8 Philosophy 9 2.1 Physics 9 2.1 Economics 5 1.2 Business and Finance-related 2 0.5 Humanities subjects 2 0.5 2.4 Analysis Descriptive statistics were collected for the multiple choice items on the online survey, and statistical testing conducted to ascertain whether there were any significant differences between the responses of certain groups (e.g. gender, whether Further Mathematics was required by the university the participant was at, and whether the participant had taken AS- or A-level Further Mathematics). Students’ responses to the open-ended questions were coded and subjected to thematic analysis. Thematic analysis infers a description of the students’ ‘truth space’ (Onwuegbuzie, 2003, p. 400); that is, their feelings, experiences and opinions. Guidelines by Braun & Clarke (2006) were used to conduct the analysis in order to ensure a consistent, reliable framework of analysis. This method was used to identify emerging patterns in the participants’ explanations of their experiences of each extension paper. Participants’ responses were coded as themes and sub-themes emerged. The themes are shown in Table 3. Table 3. Themes and sub-themes identified via qualitative analysis Theme Sub-theme Enjoyment Expectations of university mathematics Accuracy Perceived difficulty Preparation for university mathematics Usefulness Question style Introductory insight Mathematical skills Application of learning/methods Thought and understanding Problem-solving Proof and rigour Relationship to A-level Content Question style Interest Relationship to other extension papers Difficulty Theme Sub-theme Enjoyment Expectations of university mathematics Accuracy Perceived difficulty Preparation for university mathematics Usefulness Question style Introductory insight Mathematical skills Application of learning/methods Thought and understanding Problem-solving Proof and rigour Relationship to A-level Content Question style Interest Relationship to other extension papers Difficulty Table 3. Themes and sub-themes identified via qualitative analysis Theme Sub-theme Enjoyment Expectations of university mathematics Accuracy Perceived difficulty Preparation for university mathematics Usefulness Question style Introductory insight Mathematical skills Application of learning/methods Thought and understanding Problem-solving Proof and rigour Relationship to A-level Content Question style Interest Relationship to other extension papers Difficulty Theme Sub-theme Enjoyment Expectations of university mathematics Accuracy Perceived difficulty Preparation for university mathematics Usefulness Question style Introductory insight Mathematical skills Application of learning/methods Thought and understanding Problem-solving Proof and rigour Relationship to A-level Content Question style Interest Relationship to other extension papers Difficulty 3. Results Throughout Section 3, the number of respondents to each question is given, because not all participants answered all questions. Some participants did not complete the entire questionnaire, whilst others accidentally missed some questions. Participants were asked whether any of MAT, AEA or STEP were entry requirements for their current course. Of those who had taken an extension paper, most had taken a STEP (see Table 4). Table 4. Extension papers taken by participants Paper(s) taken No. of participants Proportion of participants (%) MAT (n=139) AEA (n=103) STEP (n=322) • 217 50.7 • 77 17.9 • • 50 11.4 • • 33 7.7 • 24 5.6 • • • 22 5.1 • • 7 1.6 Total 430 100.0 Paper(s) taken No. of participants Proportion of participants (%) MAT (n=139) AEA (n=103) STEP (n=322) • 217 50.7 • 77 17.9 • • 50 11.4 • • 33 7.7 • 24 5.6 • • • 22 5.1 • • 7 1.6 Total 430 100.0 Table 4. Extension papers taken by participants Paper(s) taken No. of participants Proportion of participants (%) MAT (n=139) AEA (n=103) STEP (n=322) • 217 50.7 • 77 17.9 • • 50 11.4 • • 33 7.7 • 24 5.6 • • • 22 5.1 • • 7 1.6 Total 430 100.0 Paper(s) taken No. of participants Proportion of participants (%) MAT (n=139) AEA (n=103) STEP (n=322) • 217 50.7 • 77 17.9 • • 50 11.4 • • 33 7.7 • 24 5.6 • • • 22 5.1 • • 7 1.6 Total 430 100.0 It is not necessarily the case that those who had taken extension papers were required to have done so by their university. Indeed, only 198 were required to have taken a STEP (60.2% of those who had taken one), 15 to have taken the AEA (14.7%) and 69 the MAT (52.3%). The remaining participants were not required to have taken any extension paper. These students may have taken an extension paper in order to receive a lower offer for admission, to assist their overall application, or because they had a general interest in mathematics and wanted the experience of taking the examination. Additionally, not all participants who remarked on their experience of STEPs were current students at the University of Cambridge, and not all participants who discussed the MAT were current students at the University of Oxford or Imperial College London. These students may have been required to take the MAT during the application process, but were not offered a place at either institution or chose not to take up a place at either institution. Similarly, some participants may have been given an offer from Cambridge which was dependent on their performance in a STEP, but did not meet the terms of this offer. The majority of participants (70.2%) were required to have taken A-level Further Mathematics in order to be accepted onto their current course of study. However, it should be noted that this is certainly not the case for the majority of undergraduate mathematics courses in the UK, suggesting that our sample is skewed towards more selective universities. The attainment of the participants is also not representative of the general cohort of students who took A-level Mathematics or Further Mathematics (see Table 5). Table 5. Proportions of students achieving at least an A grade A-level Mathematics (%) A-level Further Mathematics (%) Participants 99.5 94.9 Mathematics undergraduates† 63.8 No data available A-level candidates (2016) 41.8 56.2 A-level Mathematics (%) A-level Further Mathematics (%) Participants 99.5 94.9 Mathematics undergraduates† 63.8 No data available A-level candidates (2016) 41.8 56.2 †This is the most-recently available data, relating to undergraduates in 2011 (Vidal Rodeiro, 2012). Table 5. Proportions of students achieving at least an A grade A-level Mathematics (%) A-level Further Mathematics (%) Participants 99.5 94.9 Mathematics undergraduates† 63.8 No data available A-level candidates (2016) 41.8 56.2 A-level Mathematics (%) A-level Further Mathematics (%) Participants 99.5 94.9 Mathematics undergraduates† 63.8 No data available A-level candidates (2016) 41.8 56.2 †This is the most-recently available data, relating to undergraduates in 2011 (Vidal Rodeiro, 2012). Participants were asked how well they thought that A-level Mathematics and Further Mathematics had prepared them for university mathematics, as well as whether they considered extension papers to have been helpful preparation for university mathematics. No formal definition of ‘preparation’ was given to the participants for them to answer this question. A distinction was not made between the role of studying qualification content, preparing for those examinations and actually sitting the examinations, and thus perceptions of participants’ overall experiences was sought. It might have been difficult for participants to have differentiated between these three experiences, given it would have been at least 2 years since they took any of the A-level or extension paper examinations. Consequently, participants were asked to decide whether studying for and taking the qualifications had been ‘good preparation’, ‘bad preparation’ or ‘neither good nor bad preparation’. Participants had the opportunity to explain their responses regarding extension papers after choosing from the three options. They were asked to consider the following in their responses: ‘Did this test change the way that you learned maths at school? Did it impact on the way you saw maths? Did it affect your expectations of university maths? And, if so, how?’ 3.1 Perceived usefulness of A-levels Overall, the extension papers were generally viewed positively by students who had taken them (see Fig. 3). In the case of both A-level Mathematics and A-level Further Mathematics, most students believed that their content and assessment were good preparation for undergraduate mathematics. However, the proportions differed between students who had taken each extension paper. Figure 1 shows that students who had taken none of the extension papers were more likely to describe A-level Mathematics as good preparation for their degree than those who had taken any of the extension papers. Fig. 1. View largeDownload slide Students’ perceptions of A-level Mathematics as preparation for undergraduate mathematics, by extension paper taken. Fig. 1. View largeDownload slide Students’ perceptions of A-level Mathematics as preparation for undergraduate mathematics, by extension paper taken. Furthermore, Fig. 2 shows that students who had taken a STEP were slightly more likely to describe A-level Further Mathematics as bad preparation for their degree than other participants. Fig. 2. View largeDownload slide Students’ perceptions of A-level Further Mathematics as preparation for undergraduate mathematics, by extension paper taken (students of the full A-level only). Fig. 2. View largeDownload slide Students’ perceptions of A-level Further Mathematics as preparation for undergraduate mathematics, by extension paper taken (students of the full A-level only). Fig. 3. View largeDownload slide Extension papers as preparation for undergraduate mathematics. Fig. 3. View largeDownload slide Extension papers as preparation for undergraduate mathematics. 3.2 Perceived usefulness of extension papers AEA was described by most of those who took it as being ‘neither good nor bad preparation’, with 42.2% describing it as ‘good preparation’ for undergraduate mathematics. The MAT was described by just over half (50.4%) of those who had taken it as ‘good preparation’. Of all of the extension papers for which participants were asked to describe in terms of their usefulness as preparation for university, the STEPs received the most positive outcome, with 81.4% of participants reporting that they believed it was ‘good preparation’. However, Table 6 shows that participants differed in their opinion of STEPs depending on whether they were required to take one for admission to their course (Fisher’s exact test statistic = 30.697, p = 0.000). Specifically, those who were required to have taken a STEP were more likely to describe it as being good preparation for their degree than those who were not. Table 6. Perceptions of STEPs by students who were and were not required to have taken one Perception of the STEP No. of participants (%) STEP required STEP not required Total Good preparation 176 (91.2%) 82 (66.1%) 258 (81.4%) Neither good nor bad preparation 14 (7.3%) 36 (29.0%) 50 (15.8%) Bad preparation 3 (1.6%) 6 (4.8%) 9 (2.8%) Total 193 (100.0%) 124 (100.0%) 317 (100.0%) Perception of the STEP No. of participants (%) STEP required STEP not required Total Good preparation 176 (91.2%) 82 (66.1%) 258 (81.4%) Neither good nor bad preparation 14 (7.3%) 36 (29.0%) 50 (15.8%) Bad preparation 3 (1.6%) 6 (4.8%) 9 (2.8%) Total 193 (100.0%) 124 (100.0%) 317 (100.0%) Table 6. Perceptions of STEPs by students who were and were not required to have taken one Perception of the STEP No. of participants (%) STEP required STEP not required Total Good preparation 176 (91.2%) 82 (66.1%) 258 (81.4%) Neither good nor bad preparation 14 (7.3%) 36 (29.0%) 50 (15.8%) Bad preparation 3 (1.6%) 6 (4.8%) 9 (2.8%) Total 193 (100.0%) 124 (100.0%) 317 (100.0%) Perception of the STEP No. of participants (%) STEP required STEP not required Total Good preparation 176 (91.2%) 82 (66.1%) 258 (81.4%) Neither good nor bad preparation 14 (7.3%) 36 (29.0%) 50 (15.8%) Bad preparation 3 (1.6%) 6 (4.8%) 9 (2.8%) Total 193 (100.0%) 124 (100.0%) 317 (100.0%) 3.3 Explanations of usefulness In addition to describing each of the qualifications as either ‘good’, ‘bad’ or ‘neither good nor bad’ preparation for university mathematics, participants were invited to explain their responses. A number of themes emerged in participants’ responses to the open-ended question. Themes and sub-themes are displayed in Table 3. 3.3.1 MAT Those who indicated that the MAT had not been useful preparation suggested this was because the test focused on logical thinking and puzzles, rather than the type of mathematics studied at university. Others reflected on a lack of available support and believed that, consequently, they were unable to effectively prepare for the test, which they felt made the MAT of very limited use for university study. Conversely, those participants who suggested that the MAT was useful preparation for university reported that this was because the style of questions was more similar to university examinations than A-level. Some of these participants also suggested that the difficulty of the MAT was welcome, as it promoted the idea that it is not necessarily possible to achieve 100% in university examinations, and provided an additional challenge. I found it to be more oriented towards logical thinking or problem solving as opposed to maths, and although I found it enjoyable it certainly has played no role in assisting my university studies. In terms of mathematical skills, most participants suggested that the level of challenge promoted better understanding of the concepts covered in AS-level Mathematics. This was because the answers and methods required were not immediately obvious, and therefore a deeper level of thought was required to find a solution. A minority of participants also perceived the MAT to require a deeper problem-solving ability than A-level. The Oxford exam took the knowledge I had accrued and forced me to think more deeply about the intuition behind the subject. Participants’ comments regarding enjoyment (or otherwise) of the MAT reflect that it is an aptitude test and used to select candidates for interview. Very few participants reported that it had been an enjoyable experience. More students reported that they had found the MAT stressful, and this had been compounded by a lack of support from their school and the university to which they had applied. However, it should be noted that any benefit of the MAT is intended to be for admissions tutors and interviewers, rather than students themselves. Nonetheless, there is a clear difference between the opinions expressed about STEPs and the MAT: whilst participants appreciated the challenge of STEPs because these papers also prepared them for the demands of the university study, the difficulty of the MAT was not counterbalanced by many perceived benefits for the students themselves. The negative opinions expressed about the MAT by certain participants may warrant attention from the universities which require candidates to take it. The MAT was too early in the year and too far from what I had already studied, plus there was very little support for it. This, I feel led to me getting a poor result. There was no preparation for the admissions test available, and the information covered on the test was not a part of either A-level course. I therefore was not engaged by anything that came up on the test, and left me with a bad experience of material covered by these institutions. 3.3.2 AEA Participants who indicated that the AEA was useful preparation for university reported that this was because its questions were less structured, varied and therefore more similar to university-style questions. Conversely, those who suggested that AEA was bad preparation indicated that its similarity to A-level was a negative factor. The majority of these participants suggested that AEA simply consisted of ‘hard’ A-level questions, which whilst being more challenging than a standard A-level examination, did not require a deeper understanding of mathematics or more advanced problem-solving ability. AEA maths gave me a much greater confidence that I was good at mathematics and I feel that it prepared me very well for the types of questions in university (more unstructured or only loosely structured). This was particularly useful. AEA maths also gave me a big challenge and it made me feel that I was able to advance to university maths with somewhat of an advantage over some of the other students. I felt AEA was just a slight advancement on A-level Mathematics, I didn't feel I had to change my way of learning to study for this exam, I just had to understand all areas of A-level very well in order to succeed. AEA is still a long way from the level of maths at university, so it didn't really help prepare me a great deal. Participants were also divided in their opinions regarding whether the AEA developed useful mathematical skills, such as mathematical thought and understanding. For some participants, the unfamiliarity of AEA-style questions required a deeper level of thought about which methods to use to solve the question, and a smaller minority perceived these questions to encourage creative problem-solving. However, two participants perceived the AEA to promote the use of useful ‘tricks’, rather than true conceptual understanding. A-level questions are very predictable and you can recognise immediately what topic it's asking about and what method to use, but AEA questions require more thought and creativity with how to approach it. Getting used to not instantly being able to see how to answer a question was very useful preparation for degree maths. You learn some good ‘tricks’ and lots of practice makes you quite quick at doing dense calculations well, but I don't remember learning much that was really new for this exam. The small proportion of responses that discussed AEA in relation to other extension papers suggested that AEA is ‘between’ STEPs and A-level in terms of perceived difficulty. Additionally, these responses suggest that the AEA is not perceived to be as important as STEPs, presumably because it is not currently a compulsory entry requirement for admission to undergraduate mathematics at any university (although it is often recommended or students might be permitted to achieve lower A-level grades if they perform well in the AEA). A minority of participants (n = 10) reported that they did not prepare for the AEA, and that they only took it because they were offered the opportunity by their secondary school. These students reported that this lack of preparation was either due to their focus on A-levels, or due to their focus on STEPs. 3.3.3 STEP The majority of participants reported that taking a STEP had given them a more accurate expectation of university mathematics, as they felt that the questions emphasized the importance of proof and formal mathematics. Participants suggested that STEPs were good preparation for university mathematics because STEP questions were more similar to university examination questions, and were particularly useful in that the questions required synoptic understanding of different areas of mathematics. Participants also reported that STEPs helped develop certain mathematical skills that they considered to be useful preparation. In particular, participants believed that STEP questions developed their problem-solving ability, which they considered to be good preparation for encountering difficult problems during their undergraduate course. A smaller proportion of participants also reported that the emphasis on problem-solving had encouraged independent self-study, which they considered to be good preparation for the different way of learning at university. The questions were not of any standard form, therefore you were required to draw on all areas of maths you have studied, whereas with A-level I mainly found that each question stuck to one topic of the course. It also required you to consider completely new problems, which was probably too challenging for A-level but which was very useful for university. As well as this it allowed me to see a glimpse of what university maths would be like after being told a lot it wasn't like the maths studied at school. Participants also reported that STEPs were challenging. Nevertheless, the majority of responses indicated that this was a positive thing, and was closely related to enjoyment. Most participants reported that, because STEP questions were so challenging, they required a higher level of mathematical thinking and were therefore perceived to be more interesting. A smaller proportion of participants reported that STEP questions also required creative approaches which showed them how exciting mathematics could be, and considered them more engaging than the examination of A-level Mathematics and Further Mathematics. Most participants also reported that the difficulty of STEPs deepened their mathematical understanding. They believed that this was due to the solutions not being immediately obvious and therefore the participants needed a deeper knowledge of the content they had been taught at A-level in order to understand where to apply particular methods. You actually had to think rather than regurgitate, which made the questions a lot more interesting and enjoyable. It made me realise how much I would enjoy maths at university. Whilst most participants were positive about their experiences of STEPs, a very small minority reported negative experiences. For a very small minority (five participants), the difficulty of STEPs in comparison to A-level had made them worried about the demands of university study. These comments reflect the importance of appropriate support for students who choose or are required to take extension papers. Four participants reported that they had found taking a STEP so stressful that it had nearly put them off studying mathematics at university. Furthermore, five participants suggested that tuition or support is essential for students to have positive results and therefore useful experiences of STEPs. In my experience, this definitely tends to favour people who've had a lot of good tuition in mathematical problem solving, while those who have had fewer such opportunities are largely left to fend for themselves to learn necessary skills. 4. Limitations There are a number of limitations with this study which should be considered when interpreting the results. The self-selecting nature of participation in this study was two-fold: students were self-selecting in their decision to complete the questionnaire, and the opportunity to take part was itself reliant on the self-selection of their university departments. Additionally, it could be that students who felt particularly strong (either negatively or positively) about their experience of A-level and extension papers, as well as their transition to university study may have felt more compelled to take part. Students’ experiences will have been influenced by the entry requirements they had to meet to be admitted to their course. Universities will tailor the first year of their courses to meet expectations of students’ mathematical backgrounds depending on the minimum criteria they must have met. For example, universities which require students to have succeeded in a STEP will expect first-year undergraduates to be able to cope with STEP-style questions and have high expectations of students’ abilities regarding proof and formal mathematics. On the other hand, universities which only require A-level Mathematics will tailor their course so as not to expect any further mathematical knowledge beyond what is covered in this qualification. Some of the questions posed to participants relied on their own interpretation of what constitutes ‘useful’ preparation. It is possible that some students will have interpreted it in different ways. For example, to one person an A-level or extension paper may have been useful if it covered contents that they would go on to learn or use at university. To another, something might have been useful if it assessed a particular style of question or required certain skills which were different or, in their opinion, more akin to that at university. We cannot know what their interpretations were. A larger number of participants in this study took a STEP compared to those who took the AEA or MAT. This was seen to affect the nature of students’ responses to certain questions, hence some of the results reported in Section 3 being divided by extension paper. It should also be remembered that only a minority of those students who go on to study mathematics at university take any of these extension papers. However, these numbers still represent an important proportion of all mathematics undergraduates and therefore we believe that students’ experiences of these extension papers have important implications for future undergraduates’ preparation for undergraduate study. For illustration, in 2014, 10,250 undergraduate students began their first year in the mathematical sciences (Higher Education Statistics Agency, 2016), whilst there were only 899 AEA candidates that summer (Joint Council for Qualifications, 2014), 1,971 STEP candidates (personal communication) and 1,399 candidates were required to have taken the MAT to apply for courses at Oxford that year (University of Oxford, 2015). Students at certain schools may receive coaching or teaching geared towards them taking an extension paper. This could give certain students advantages over those who are not afforded this opportunity, and may therefore have affected the students’ perceptions of their mathematical preparedness. However, some online resources and additional support is available. For example, Cambridge offers a 4-day intensive residential course for students who have received offers to study mathematics but whose schools do not offer any STEP support (University of Cambridge Faculty of Mathematics, 2016). A final limitation is the time elapsed between taking either an A-level or an extension paper and completing the questionnaire. Whilst this means that they will have experienced at least 1 year of university study and therefore be able to reflect upon this, they may not have a fresh memory of what it was like to prepare for and take the extension papers. This is why students were not asked to distinguish between their experiences of preparing for taking the extension papers and their experience of taking the extension papers. Students’ responses outlined in Section 3.3 clearly demonstrate a focus on their preparation for the examinations. 5. Discussion The participants’ responses regarding their experiences of any of the three extension papers were generally positive. Very few described any of the MAT, AEA or STEP as bad preparation for their degree, although STEPs were described most favourably. The limitations of the MAT in preparing prospective mathematics undergraduates for further study appear to lie in the fact that it is not intended to act as such. The MAT is an aptitude test, and used to shortlist candidates for interview at an early stage in their A-level Mathematics studies. Consequently, it relies on some of the more basic mathematics taught as part of that qualification, and probes students’ abilities to be effective problem-solvers and independent thinkers. The AEA and STEPs require the use of more sophisticated and advanced mathematics. Moreover, students’ responses support Darlington (2014), suggest that each extension paper provides a different means of challenging and preparing students. The MAT challenges students by requiring them to solve unfamiliar problems which do not have an obvious method towards a solution. AEA Mathematics challenges students by requiring in-depth use of the content of A-level Mathematics in order to solve more complex versions of A-level Mathematics style questions. STEPs, on the other hand, combine the use of more advanced mathematics with some questions more familiar to A-level Mathematics students, as well as questions which require deeper mathematical thinking. In doing so, STEPs give students an insight into what undergraduate mathematics assessment might be like. It appears to be successful in doing so. The fact that participants who were required to have taken the STEP were more likely to describe it as useful preparation for their degree suggests that the universities which require applicants take it tailor their undergraduate courses to their cohort well. For that reason, STEPs seem to function as a means of managing students’ expectations of further mathematics study, as well as preparing them for demanding assessment at university. Conversely, participants were divided in their opinions regarding whether the AEA and MAT gave them a more accurate conception of undergraduate mathematics study, though they agreed that they were challenging. 6. Conclusion The three extension papers described in this article serve two stakeholders: universities wishing to select the best and most suitable candidates for mathematics study, and prospective mathematics applicants wishing to ascertain their suitability for the course and gain an insight into further study. Though the number of candidates for these extension papers is rather low compared to the number of undergraduate mathematicians or A-level Mathematics candidates, the experiences of those who take them appear to be largely positive. Given these papers serve to prepare students for what undergraduate mathematics might be like through challenging them and presenting them with unfamiliar mathematics questions, more could perhaps be done to promote them. Students who are better prepared for university study and who have accurate conceptions of what its study will be like will probably be more likely to be successful and happy with their studies (Crawford et al., 1994). One means of tackling the high drop-out rates for mathematics would be more effective pre-university preparation. The number (and calibre) of universities which require candidates to be successful in one of these papers suggest that there is a growing awareness amongst admissions tutors that these papers can serve their purpose effectively. Indeed, a new MAT was piloted in the autumn of 2016 in the UK by Cambridge Assessment Admissions Testing after a number of British universities expressed interest in there being a more accessible MAT which could be used by a range of universities. The ‘Test of Mathematics for University Admission’ (TMUA) is a 2-hour multiple choice test comprising two papers—one on mathematical thinking and one on mathematical reasoning. Students applying to study mathematics degrees at Durham University and Lancaster University (both high-ranking universities) were encouraged to take the test as part of their application, and a good performance resulted in lower A-level grades being permitted from some applicants (Cambridge Assessment Admissions Testing, 2016b). It is expected that a number of other high-ranking British universities will begin to use the TMUA next year to help with their admissions process. However, as there are issues regarding some students’ access to support for preparing for these papers, universities often choose not to incorporate them into their admissions requirements in order to ensure equitable access to students from all educational backgrounds. Furthermore, not all universities require students to have achieved the best-possible grades in A-level Mathematics and hence pitch their teaching at a different level. It may therefore not be the case that such universities would want to require prospective applicants to have studied for, let alone passed, extension papers. Nonetheless, the usefulness of studying for and taking these papers from the perspective of the applicants suggests that universities may be wise to recommend that prospective applicants take them or emphasize that they might be permitted to achieve lower A-level grades for admission if they perform well in extension papers. Secondary mathematics teachers might also wish to use some past questions from some of the extension papers to challenge more able students, and perhaps give them an insight into the challenge and nature of university study and assessment. Ellie Darlington is a research officer in the research division at Cambridge Assessment, a not-for-profit non-teaching department of the University of Cambridge. She has a doctorate in mathematics education, and consequently specializes in this area, particularly relating to the secondary–tertiary mathematics transition. Jessica Bowyer is a doctoral student at the University of Exeter, having formerly been a research assistant in the research division at Cambridge Assessment. Her thesis will focus on the differences in student attainment between different geographical areas. 1 It is also possible for students to study the first half of the A-level, and to take examinations at the end of this to earn an Advanced Subsidiary ‘AS’ level. The second year of the A-level, and its associated examinations, is referred to as the ‘A2’. 2 Figures for the total number of MAT candidates is not made publicly available. Imperial College London does not release MAT performance data. 3 Not all schools are able to teach Further Mathematics courses. 4 A total of 77 at the time of data collection. 5 In total, 63% of Mathematical Sciences students in 2015 were male. However, note that ‘Mathematical Sciences’ includes degrees such as Statistics and Finance which are traditionally less male-dominated than single honours Mathematics degrees. In 2016, 72.5% of A-level Further Mathematics candidates were male (Joint Council for Qualifications, 2016). 6 This is the most-recently available data, relating to undergraduates in 2011 (Vidal Rodeiro, 2012). References ACME . ( 2011 ). Mathematical Needs: Mathematics in the Workplace & in Higher Education . London : Advisory Committee on Mathematics Education . 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Teaching Mathematics and Its Applications: International Journal of the IMA – Oxford University Press

**Published: ** May 29, 2017

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