Should I take Further Mathematics? Physics undergraduates' experiences of post-compulsory Mathematics

Kneubil and Robilotta (2015) assert that 'the relationship between physics and mathematics is definitely a rather involved one' (p 2). The mathematics involved in physics degrees stems from a range of areas, including pure mathematics and statistics in addition to mechanics. For example, the Advisory Committee on Mathematics Education (2011) describe the mathematical content of a physics degree course at one British university, which they examined during a study investigating mathematics in Higher Education:

• Mathematics I (Year 1): calculus, complex numbers, vectors, linear algebra and statistics.
• Mathematics II (Year 2): vector calculus, scalar and vector fields, electric, magnetic and gravitational fields, partial differential equations, Fourier series and transforms.
• Mathematics III (Year 2): differential equations (ordinary and partial) and real and complex matrices.

(p 13)

However, research suggests that physics undergraduates are surprised by the volume of mathematics in their course. A survey by the Institute of Physics (2011) found that only 47% of physics students felt that their expectations of the mathematics in their degree were met. Those whose expectations were not met reported that they encountered more mathematics at degree-level than they had expected and that the mathematics content was more difficult than they anticipated. Furthermore, physics lecturers have reported concern about their students' mathematical backgrounds and understanding. The repeated use of the same diagnostic test at the University of Bristol since 1975 has seen average marks decrease from approximately 75% pre-1990 to less than 50% after 2000 (Barham 2012). Similar tests have been used to identify and correct physics students' misconceptions at other universities, such as the University of Edinburgh (Archer and Bates 2009). As a consequence of this perceived fall in standards, many physics departments now offer specialised mathematics support to their students (Institute of Physics 2011).

The volume of mathematical content within undergraduate physics means that it is important that students are prepared for the demands of their course. Additionally, prior attainment in mathematics has been found to correlate with undergraduate performance in physics since at least the 1970s (Hudson and Mcintire 1977, Cohen et al 1978, Chadwick 1985, Meltzer 2002, Buick 2007). Consequently, UK universities usually require that applicants have attained high grades in A-level Mathematics (see table 1). A-level Mathematics predominantly covers pure mathematics content, but students take two units (out of six) from the three applied mathematics strands: statistics, mechanics or decision mathematics (see figure 1). Furthermore, students can take an additional AS- or A-level in Further Mathematics, which can only be taken in addition to A-level Mathematics. Up to four applied mathematics units can be taken as part of A-level Further Mathematics. Depending on the availability of applied units (e.g. according to teaching expertise and resources), students can either specialise in one particular applied strand, or take a mixture of units. For example, a prospective undergraduate physicist may specialise in mechanics.

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No UK university currently requires AS or A-level Further Mathematics for admission to study physics. Nevertheless, the Institute of Physics (2011) found that physics students who had taken Further Mathematics perceived themselves to cope better with the mathematical demands of their degree. Consequently, both lecturers and students believed that Further Mathematics should become a requirement to study physics.

The utility of A-levels in Mathematics and Further Mathematics as preparation for undergraduate physics is especially pertinent as both qualifications are currently undergoing reform for first teaching in 2017. The content in A-level Mathematics will be centrally prescribed by the Department for Education, ensuring that all students will receive the same background in statistics and mechanics, as there will be no optionality. This content will include both mechanics and statistics content. However, there will be no decision mathematics content in either the reformed AS or A-level, due to the A-level Content Advisory Board's (ALCAB) recommendation that this content be removed from A-level Mathematics due to the perception that these units are 'soft' options and are not universally valued by universities (The A Level Content Advisory Board 2014). The introduction of prescribed content is likely to be welcomed by universities, as there is currently significant variability in the applied mathematics content that new undergraduates have studied.

Additionally, 50% of the content in Further Mathematics will be prescribed, with the awarding bodies free to design the remaining content. It is likely that this will be a mixture of applied and pure mathematics. If additional mechanics content is incorporated into Further Mathematics, prospective physics students may find that Further Mathematics is a more attractive A-level option than currently as they will no longer have the option to specialise in mechanics in A-level Mathematics.

This article reports on data from a large-scale study investigating undergraduates' perceptions of AS/A-level Mathematics and Further Mathematics as preparation for the mathematical demands of their course (see Darlington and Bowyer 2016). Over 4000 undergraduates responded to an online questionnaire investigating their perceptions of post-compulsory mathematics qualifications. The data presented here relate only to the 494 undergraduate physics students who participated.

An online questionnaire was created for current undergraduates investigating their perceptions of taking AS-/A-level Mathematics and Further Mathematics. Universities were asked to disseminate the questionnaire to their students. Participants were in their second year of study or above, as it was felt that they would be more able to fully reflect on the utility of A-levels as preparation for their degree than new undergraduates.

The questionnaire included a mixture of multiple choice and open response questions, and covered four areas:

• 1.  
  Mathematical background: post-compulsory mathematics qualifications taken, grades achieved, optional units studied;
• 2.  
  Undergraduate study: course title, year of study, result in previous year's examinations;
• 3.  
  Perceptions of post-compulsory mathematics as preparation: perceived utility of the applied units, perceived overall utility of both AS/A-levels;
• 4.  
  Experiences of Further Mathematics: how influential certain factors were in the decision to study Further Mathematics, experiences of studying Further Mathematics.

The number of responses, N, to each question is given throughout because not all questions were applicable to all participants.

Throughout our analysis, participants' universities were categorised according to their ranking by the 2015 Complete University Guide for physics and astronomy. This guide was used rather than university mission groups because many institutions are no longer aligned with a mission group and it was considered necessary to give an illustration of the different types of universities involved in the study. The Complete University Guide's rankings are considered to be independent. Nonetheless, there are other rankings available and our findings may have differed had one of these been utilised instead. Consequently, any conclusions arising from these rankings should be treated with caution.

3.1. Sample

3.2. Post-compulsory Mathematics

3.3. Experiences of A-level Mathematics units

Participants were asked which applied units they had studied, and how many Further Pure Mathematics units they had taken as part of Further Mathematics (see figure 2). Participants were similarly likely to have studied at least one mechanics or statistics, although they were more likely to have taken multiple mechanics units. This suggests that where possible, participants had chosen to, and had been able to, specialise in mechanics.

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Participants were asked to rate the applied and Further Pure Mathematics units they had studied in terms of utility for the mathematical components of their degree (see figure 3). Mechanics received a highly positive reception, which is to be expected, given the relevance of this area of mathematics to physics. However, Further Pure Mathematics units were reported to be 'very useful' by the same proportion of participants, indicating that participants valued the additional exposure to pure mathematics content, such as matrices and calculus, that these units offer.

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3.4. Motivations for studying Further Mathematics

Given that Further Mathematics is not currently an entry requirement for undergraduate physics, we wanted to understand students' motivations for choosing to study this qualification. Participants were asked to rate how influential certain factors were in their decision-making (see figure 4).

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Perhaps unsurprisingly, prior success with, and enjoyment of, mathematics were strong influences on students' decisions. However, the findings also suggest that, whilst Further Mathematics is not a formal entry requirement, participants were aware of its utility in terms of preparation for their degrees and careers. The factor with the strongest influence was that participants had been considering studying a mathematics or mathematics-related degree. Additionally, most participants reported that they had believed that Further Mathematics would be a useful qualification to have, and that they needed it for their future career. Interestingly, participants from higher ranking universities were significantly more likely to be have been strongly influenced by the latter: 76% of participants from the top 25% of universities for physics agreed that this had influenced them a lot, compared to 66% of other participants (${{\chi}^{2}}(2)=7.271,p~=.025$).

3.5. Experiences of Further Mathematics

Participants were also asked a series of Likert-scale questions regarding their experiences of studying Further Mathematics (see figure 5).

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Responses suggest that participants' experiences were highly positive: they enjoyed the qualification and were glad they had taken it. Indeed, only one participant disagreed that they were glad they had taken Further Mathematics. Despite its perceived difficulty, Further Mathematics thus appears to be both an enjoyable and useful qualification.

One explanation for Further Mathematics' positive reception is the strong overlap between its content and that of undergraduate physics. 96.2% of participants agreed or strongly agreed with the statement, 'In my first year at university, we were taught material that I had learnt in Further Maths'. This suggests that its similarity with undergraduate content makes it a worthwhile qualification to take and may ease the transition into university study. Although some students who had already covered this content at A-level may find the repeated content boring, many students may find the familiarity useful, especially as this content is covered far more quickly and in a more independent manner at university than at A-level. This is corroborated by the high proportion who agreed that studying it alongside A-level Mathematics was sufficient preparation for the mathematical content of their degree. Students from lower-ranking universities were significantly more likely to strongly agree with this statement than those at higher-ranking universities: 31.3% of those from the top 25% of universities strongly agreed, compared to 46.4% of other participants ($\text{Fisher'}\text{s} ~\text{Exact} ~\text{Test};~p~=.003$). This is particularly interesting as the same students were less likely to take Further Mathematics.

3.6. Overall utility of A-level Mathematics and Further Mathematics

Overall, A-level Mathematics and Further Mathematics were considered by most participants to be beneficial in terms of their preparation for undergraduate Physics (see figure 6). Participants were slightly more enthusiastic about Further Mathematics.

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Participants studying at attending lower-ranking universities were significantly more positive about A-level Mathematics as preparation for their degree: 85.4%, compared to 72.1% of those attending universities ranked in the top 25% (${{\chi}^{2}}(2)=12.887,~p=\,.001$)

A self-selecting study of this nature suffers from a number of limitations:

• Participation was self-selecting both in terms of students choosing to complete the questionnaire, and the cooperation of the universities contacted. Data has therefore been correlated with national data where possible, in order to give an indication as to whether this sample might be skewed.
• It could be that students who felt particularly strongly about their mathematical preparedness and its impact on their transition to undergraduate study (either positively or negatively) may have felt more compelled to take part.
• This study only incorporates the views of students who had taken post-compulsory mathematics qualifications. We cannot contrast their responses with students who did not take A-levels in either Mathematics and/or Further Mathematics yet went onto study undergraduate Physics.

This article reports on the views of undergraduate physicists, but is not contrasted with the views of lecturers. The overall study also included interviews with 30 lecturers (including three physics lecturers) regarding their perceptions of their students' mathematical preparedness, and this will be published in due course.

The data collated in this study suggest that both A-level Mathematics and Further Mathematics are very good preparation for undergraduate physics. In particular, participants reported that Further Pure Mathematics and mechanics units were the most useful optional units at A-level. This indicates that prospective undergraduate physicists would benefit from studying Further Mathematics where possible, and specialising in mechanics during the applied strands of the course.

The benefit of post-compulsory mathematics as preparation for undergraduate physics is to be expected when considering the strong relationship between the two disciplines, as is the perceived utility of mechanics units. Nevertheless, what may be more surprising is participants' enthusiasm for Further Mathematics. The positive opinions about Further Mathematics appear to stem from the pure mathematics content that Further Pure units offer. Further Pure Mathematics units cover topics such as imaginary and complex numbers, matrices and advanced calculus: prior experience with these concepts may therefore ease the transition to undergraduate physics. Moreover, 97% of participants who studied Further Mathematics reported that intending to study a mathematics-related degree was an influential factor in their decision to study it. This, coupled with participants' perceptions that Further Mathematics would be useful for their future career, suggests that students are reasonably well-informed about the utility of Further Mathematics as preparation for physics and choose their A-levels accordingly. This is despite concerns that making Further Mathematics an entry requirement would discourage students from progressing to undergraduate physics.

Additionally, although both mathematics A-levels are undergoing reform, it is likely that Further Mathematics will remain a useful qualification for physics applicants. The introduction of prescribed content in A-level Mathematics means that all students will be familiar with basic mechanics concepts such as kinematics, Newton's laws of motion, and moments. Nevertheless, this content will form a small proportion of the overall A-level, although additional mechanics content may be incorporated into Further Mathematics. Furthermore, key concepts such as matrices and imaginary numbers will remain in Further Mathematics. Consequently, prospective physicists should still be encouraged to take Further Mathematics where possible.

A-level reform provides a unique opportunity for universities to reconsider their existing admissions requirements. Despite the positive and enthusiastic reception that Further Mathematics has received in this study, no university explicitly requires students to offer this qualification at either AS- or A-level. Universities may understandably be worried about any potential negative effects on widening participation if they require applicants to have taken Further Mathematics, as not all schools have the resources to offer it. However, support for students in this position has increased over the past few years, through programmes such as the Further Maths Support Network. Consequently, universities should consider requiring that physics applicants study at least AS-level Further Mathematics. This would benefit both universities and prospective students, by ensuring that students are given realistic expectations about both the type and volume of mathematics involved in undergraduate physics. Additionally, a uniform entry requirement would ensure that all students have the same mathematical background, subsequently easing the difficulties of teaching mixed cohorts. Any negative effects of a strict requirement could be counteracted by making differentiated offers to students whose schools do not offer the qualification.