MATHEMATICS WITHIN A-LEVEL SCIENCE 2010 EXAMINATIONS … maths.pdf · score maths in science report 2 score maths in science report 3 ExEcutivE summary m athematics enables students

score maths in science report 2score maths in science report3

ExEcutivE summary

mathematics enables students to understand and describe many scientific phenomena yet there is concern that science assessments at a-level are not reflecting the subject’s analytical nature. to explore whether there was any evidence for this concern, score investigated the type, extent and difficulty of mathematical questions within science a-levels. the findings show that a large number of mathematical requirements listed in the biology, chemistry and physics specifications are not assessed. those that are assessed are covered repeatedly and often at a lower level than required. this is likely to have an impact on the way that the subjects are taught and therefore on students’ ability to have the necessary skills to progress effectively to stem higher education and employment. in addition, the findings show a disparity in the way mathematics is assessed across the different awarding organisations. score recommends that there is a review of the mathematical requirements for each of the sciences at a-level and that a framework is developed to regulate the way mathematics is assessed within the sciences to ensure parity across the system.

Background

there has been growing concern across the science community about the mathematical demand of science qualifications, specifically that Gcse and a-level science qualifications are not meeting the needs of students in the way they assess the analytical nature of science.

in 2009 score published evidence on Gcse science examination papers which reported a wide variation in the amount of mathematics assessed across awarding organisations and confirmed that the use of mathematics within the context of science was examined in a very limited way. score organisations felt that this was unacceptable. mathematics is integral to the teaching and learning of the sciences, and offers a valuable aid in understanding and describing scientific phenomena; as such it should be appropriately represented in the biology, chemistry and physics curricula and their assessments.

to provide further evidence to support these concerns, score set up this project to investigate the mathematics found in the 2010 science assessments at a-level across the unitary awarding organisations in england, Wales and northern ireland.

aims

score’s overall objective for this project was to gather evidence on the type, extent and difficulty of mathematics required to access the sciences in current a-level specifications and to establish whether this was being appropriately met by the assessments. the work did not compare the mathematical requirements between physics, chemistry and biology as it is accepted these will differ between the disciplines.

the findings aim to provide score with evidence to inform the development of policy on the type, extent and difficulty of the mathematics in the criteria and assessments for a-levels in biology, chemistry and physics. the project also supports score’s work on how the examinations system should operate to ensure science qualifications are fit for purpose and also its work on improving the coherence between the sciences and mathematics.

in the project, we looked across all assessments at a-level for a given year, including both experimental and practical examination papers.

mEthodology

the project was designed in three phases. the first was to establish the nature of the mathematics assessed within the biology, chemistry and physics a-level examinations in 2010. the full suite of examinations papers from aQa, ccea, edexcel, ocr and WJec were analysed using the four measures that follow:

1. the type of mathematics. the mathematical areas assessed were categorised against the stated mathematical requirements for biology, chemistry and physics respectively1.

2. the extent of the mathematics. the proportion of the question parts within a paper that included mathematics was measured as was the proportion of the marks within these questions that required mathematics.

3. the difficulty of mathematics. this was measured against 3 criteria: the number of steps in a calculation, the familiarity of the context and the complexity of the question. each category had varying levels of difficulty and each was measured as a proportion of the total number of question parts containing mathematics. it was not measured against the number of marks.

4. the appropriateness of mathematics. We looked at whether the answer required scientific comprehension in addition to mathematical skill. this was measured as a proportion of the total number of question parts containing mathematics.

a subject expert group was established for each of the three sciences. each group analysed the full suite of 2010 examinations papers of aQa, ccea, edexcel, ocr and WJec for their respective subjects at a two-day workshop. examination papers included all the theory papers (Units 1, 2, 4 and 5) and the experimental and practical papers (Units 3 and 6). calculations were based on the assumption that theory papers make up 80% of the complete a-level and the experimental and practical papers the remaining 20%2.

the groups comprised practising a-level teachers, teachers with experience in curriculum research and development and individuals working for awarding organisations as markers, question writers or examiners. standardisation exercises were employed throughout the analysis to verify the reliability of judgements within and across the subject expert groups.

the second phase aimed to measure the coherence between the teaching and learning of mathematics and the sciences. there is an assumption that the mathematical concepts used to access the sciences are first taught within a mathematical context, i.e. within the mathematics curriculum. the project compared the mathematical requirements for the sciences at a-level with the mathematics curriculum prior to Key stage 5 using the current national curriculum level descriptions and a 2012 mathematics Gcse specification3. this work was carried out by a researcher and by a mathematics teacher.

the aim of the third phase was to determine the nature of mathematics that the community would like to see in a-level science assessments. this was achieved through an online survey for stakeholders in the science community. Depending on their expertise, participants answered the survey for biology, chemistry or physics a-level assessment. the participants were chosen in three groups; teaching profession; higher education; and professional bodies. an online survey was completed by 97 participants across the three groups (27 for biology; 38 for chemistry; and 32 for physics). participants from industry were also consulted more generally but as most science-related industries employ at a graduate level their comments tended to focus more on the outcomes at graduate level rather than directly referring to a-level.

1 the five awarding organisations use the mathematical requirements defined by ofqual in developing their specifications for biology, chemistry and physics. the mathematical requirements are available in the full report.

2 a complete science a-level is made up of 6 units. the marks from the theory papers (Unit 1, 2, 4 and 5) make up 80% of the complete a-level assessment and the marks from the practical and experimental papers (Unit 3 and 6) make up the remaining 20%. in the analysis, question parts rather than marks were used in the calculation but we maintained the 80:20 weighting.

3 the edexcel 2012 a specification was used for the purpose of the analysis as it was considered representative of typical content found in the revised mathematics Gcses.

MATHEMATICS WITHIN A-LEVEL SCIENCE 2010 EXAMINATIONS

FULL REPORT FROM SCORE

score maths in science report2 score maths in science report 3

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SCORE Maths in science report2

CONTENTS

CHAPTER 1: INTRODUCTION 4

1.1 Background 4

1.2 Research aims 4

1.3 Overview of findings 4

1.4 Related literature and research 4

CHAPTER 2: 7

2.1 Research design – overview 7

2.2 Research design – Phase 1 7



CHAPTER 3: 13

3.1 Physics 13

3.1.1 Phase 1: A-level papers 13

3.1.1.1 Extent 13

3.1.1.2 Type 15

3.1.1.3 Difficulty 16

3.1.1.4 Appropriateness 23

3.1.2 Phase 2 – Physics A-level in comparison with GCSE mathematics and National Curriculum Level Descriptors 25

3.1.3 Phase 3 – Survey findings 31

3.2 Chemistry 33


3.2.1.1 Extent 33

3.1.1.2 Type 36



3.2.2 Phase 2 – Chemistry A-level in comparison with GCSE mathematics and National Curriculum Level Descriptors 45



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3.3 Biology 52


3.3.1.1 Extent 52

3.3.1.2 Type 54



3.3.2 Phase 2 – Biology A-level in comparison with GCSE mathematics and National Curriculum Level Descriptors 63


3.4 Views from industry 70

APPENDICES 72

Appendix 1: Summer 2010 A-level papers analysed 72

Appendix 2: Mathematical requirements for physics A-level 71

Appendix 3: Mathematical requirements for chemistry A-level 72

Appendix 4: Mathematical requirements for biology A-level 75

Appendix 5a: Examples of mathematical questions within a science A-level of a single step calculation, a multiple step calculation and an extended step calculation 76

Appendix 5b: Examples of mathematical questions within a science A-level of a Level 1, Level 2, Level 3 and Level 4 complexity 81

Appendix 5c: Examples of mathematical questions within a science A-level of a Level 1, Level 2 and Level 3 context 83

Appendix 5d: Examples of mathematical questions within a science A-level where all marks, some of the marks, none of marks require scientific comprehension in addition to mathematical skill 85

Appendix 6: Framework for analysing A-level theory and practical papers 88

Appendix 7: Acknowledgements 90


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1.1 BACKGROUND

There has been growing concern across the science community about the use of mathematical assessments in science qualifications.

In 2009 SCORE published research on GCSE science examination papers. The report showed a wide variation in the amount of mathematics assessed across awarding organisations and confirmed that the use of mathematics within the context of science was examined in a very limited way. SCORE organisations felt that this was unacceptable. Mathematics is integral to the teaching and learning of the sciences, and offers a valuable aid in understanding and describing scientific phenomena; as such it should be appropriately represented in the biology, chemistry and physics curricula and their assessments.

To provide further evidence to support these concerns, SCORE set up this project to investigate the mathematics found in the summer 2010 science assessments for the biology, chemistry and physics A-levels across the unitary awarding organisations in England, Wales and Northern Ireland.

1.2 RESEARCH AIMS

SCORE’s overall objective for this project was to gather evidence on the type, extent and difficulty of mathematics assessed in science A-levels and to establish whether the current assessments reflected the mathematical requirements of the sciences. The work did not compare the mathematical requirements between biology, chemistry and physics as it is accepted these will differ between the disciplines.

The findings aim to provide SCORE with evidence to inform the development of policy on the type, extent and difficulty of the mathematics in the specifications and assessments for A-levels in biology, chemistry and physics. The project also supports SCORE’s work on how the examinations system should operate to ensure science qualifications are fit for purpose and also its work on improving the coherence between the sciences and mathematics.

In the project, we looked across all assessments at A-level for 2010, including theory and practical examination papers.

1.3 OVERVIEW OF FINDINGS

The main findings of the research were as follows:

• A large number of the mathematical requirements listed in the 2010 biology, chemistry and physics AS and A2 specifications were assessed in a limited way or not at all within the examination papers.

• There is a measurable variation between awarding organisations in terms of the amount and difficulty of the mathematics that is assessed in biology, chemistry and physics AS and A2 examination papers. Participants in our survey felt that in some cases the amount of mathematics assessed in A-level science examinations was too low.

• The examination questions that did require mathematics were felt to be of insufficient difficulty; too many involved only single step questions, require only simple recall, and were set only in familiar contexts.

• There were many mathematical requirements identified in biology, chemistry and physics A-levels that go beyond the current GCSE mathematics.

1.4 RELATED LITERATURE AND RESEARCH

Relevant literature and previous research findings are presented below.

SCORE: GCSE science 2008 examinations This project explored the fitness for purpose of GCSE science examination papers, focusing on how the examination papers assessed particular aspects of GCSE science. One aspect was the extent and type of mathematics required by the questions and whether this was the same across the awarding organisations. Findings showed that the demand and type of the mathematics within the GCSE science papers was limited. The assessed mathematical content, in some cases, did not correspond with the mathematics found in the science specifications. This was particularly true for the more advanced mathematics. The mathematics found was judged to be limited in terms of both the type and extent of mathematics required by students. The project considered only the examinations themselves and not the internal assessment/coursework; it was accepted that this may account for discrepancies across awarding organisations.

CHAPTER 1: INTRODUCTION


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SCORE: GCSE science examinations 2008, 2009 and 2010 The SCORE research above was extended to include similar analyses of the 2009 and 2010 science GCSEs. While there had been modest increases in the amount of mathematics required in the GCSEs over the three cycles of assessments, much of the mathematics was found to be at Key Stage 2 level of difficulty with the demand and type of the mathematics still found to be limited when compared to the lists of mathematical requirements provided by some awarding organisations in their specifications.

IOP: Mind the Gap, July 2011 The Institute of Physics had concerns, borne out of anecdotal evidence, that the current physics and mathematics A-levels were not preparing students sufficiently for studying physics or engineering as an undergraduate. This concern was two-fold: that first-year undergraduates were not proficient in the mathematical skills needed, and that the lack of mathematical content in the physics A-levels was not motivating students who enjoyed solving mathematical problems from to take further related study. 55% of the academics surveyed in the project felt that the students were not very, or not at all well prepared to cope with the mathematical content of undergraduate study. 92% felt that a lack of fluency in mathematics was a barrier to students achieving their potential.

In the IOP research, integration, identifying particular equations and techniques to deal with problems, and vectors and scalars were identified by academics and students as being more difficult content areas. The IOP research also showed how academics were concerned that in the physics A-level there was not enough promotion of contextual understanding of the topic.

Royal Statistical Society: The Future of Statistics in our Schools and Colleges, January 2012 This report analysed the current provision in statistics in schools and colleges. There are three recommendations from the report that are relevant to the current discussion: 1. Recommendation 3: national education policy

should ensure that all students are equipped with a working knowledge of basic statistics, including the necessary associated mathematical competence

2. Recommendation 8: the curriculum should be designed so that, wherever possible, students have met statistical techniques in mathematics before they need to use them in other subjects

3. Recommendation 15: the statistics content within mathematics, up to GCSE, should include some topics that are either not currently covered or are only treated lightly


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Dr. Jenny Koenig for the UK Centre for Bioscience and the HEA: A survey of the mathematics landscape within bioscience undergraduate and postgraduate UK higher education, June 2011 This report was prompted in part by the understanding that biology was becoming a more quantitative science with greater of levels of mathematics and so the mathematics skills of biologists needed to increase to meet this demand. The report explained how amongst bioscience undergraduates, there was a wide variety in their mathematics qualifications. There were specific concerns about undergraduate students’ abilities to rearrange simple equations and reliably use ratio and proportion.

Advisory Committee on Mathematics Education (ACME): Mathematical Needs, Mathematics in the workplace and in Higher Education, June 2011 The report outlines the mathematical needs from the perspective of higher education and employers. ACME’s first recommendation in the report is that a large majority of young people should continue with some form of mathematics post-16. ACME’s third recommendation refers to how, in a revised National Curriculum, there should be greater emphasis on essential mathematics techniques and the application of mathematics. ACME’s recommendation 10 refers to using mathematics in a range of familiar and unfamiliar contexts. ACME’s recommendation 14 is that universities should make clear the level and extent of mathematics within their degrees.

The Nuffield Foundation: Mathematics in other subjects at A-level, 2012 The Nuffield Foundation has completed a project which analysed the mathematical content of other A-levels: business studies, computer science, economics, geography, psychology and sociology. The same methodology was used as in the first phase of the SCORE research, that is, the A-level assessments were analysed against a framework of measures of type, extent and difficulty of the mathematical content. The same method of analysis of data was also utilised. The results from the Nuffield research (due April 2012) should determine whether there are common areas among a wider spread of subjects at A-level that would benefit from changes to the mathematics GCSE or future mathematics for post-16.


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2.1 RESEARCH DESIGN – OVERVIEW

The project was designed in three stages. The first was to establish the nature of the mathematics currently assessed within science A-level papers. The second compared the current mathematical requirements for biology, chemistry and physics A-level with GCSE mathematics in order to identify overlap or gaps in mathematical content. The third reviewed the findings with members from the science community and focused on the nature of mathematics that the community would like to see in A-level science assessments. Views were gathered from teachers, professional bodies, higher education and industry.

A working group was established to oversee the research project and to inform the methodology and policy implications. The group comprised representatives from: • The Institute of Physics • Society of Biology • Royal Society of Chemistry • The Royal Society • AQA • ACME • Institute of Education • Institute of Mathematics and its Applications • Sixth Form College • Industry

2.2 RESEARCH DESIGN – PHASE 1

Phase 1 aimed to establish the nature of the mathematics assessed within the biology, chemistry and physics A-level summer 2010 examinations1. Papers from the five awarding organisations that offered single science A-levels in England, Wales and Northern Ireland were analysed: AQA, CCEA, Edexcel, OCR and WJEC2.

The analysis did not compare the mathematical content across the sciences as it is accepted there are different mathematical requirements for biology, chemistry and physics.

A framework was developed in order to make judgements about the type, extent, difficulty and appropriateness of the mathematics within each of the examination papers for biology, chemistry and physics A-level. The framework incorporated measures used in previous research and suggestions from the working group. A pilot exercise was carried out to test the validity of the framework and the effectiveness of the analysis process3. Figure 1 outlines the measures of the framework used in the analysis of science A-level examination papers.

CHAPTER 2: METHODOLOGY

1 Summer 2010 papers were analysed to represent an A-level rather than AS 2009 and A2 2010 papers. This was because awarding organisations usually commission a group of examiners who are responsible for all the assessments for a single examining session; it is unlikely that any comparison would be made with papers for other sessions during the course of that process.

2 Where awarding organisations offered more than one specification for a science A-level, the one with the highest uptake was chosen. Where there were options questions, or sections within an A-level, the awarding organisations were contacted and asked which option had the highest uptake and that option was used in the final analysis, for example, for AQA Physics A, paper 5, option C was used in the analysis. Appendix 1 details the summer 2010 A-level papers analysed in the research.

3 The report from the pilot exercise testing the validity of the framework is available from the SCORE Secretariat.


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4 Mathematical requirements form part of the Criteria set by Ofqual for biology, chemistry and physics A-level.

5 Drake, P., Wake, G. and Noyes, A. Assessing ‘functionality’ in school mathematics examinations: what does being human have to do with it? Research in Mathematics Education, 2012

Figure 1: Measures used to analyse the mathematics assessed within science A-level examinations

EXTENT 1. Question number and part: This allowed for

the number of questions and question parts with mathematical content to be worked out as a proportion of the total number of questions parts, giving a measure of the extent of the mathematical content. A question part was the smallest division that a question was divided into which may be a whole question, as in most multiple choice question or it may be, for example, Q3aii.

2. The number of marks that required mathematics: The percentage of marks awarded for mathematical comprehension was calculated as a proportion of overall marks available in the paper. This provided a second measure of the extent of the mathematical content. Whole marks were counted rather than parts of marks.

TYPE 3. The type of mathematics: The mathematical

requirements for biology, chemistry and physics, as set by Ofqual4, were used to identify the type of mathematics assessed and how frequently they occur in assessments. The mathematical requirements for physics are stated in Appendix 2, for chemistry in Appendix 3 and for biology in Appendix 4. All of the requirements needed for each question part were recorded. The mathematical requirements for the three subjects are different so comparisons across subjects are not straightforward, for example, ratios, fractions and percentages is listed under 1c) in the physics requirements and under 1b) in the chemistry requirements.

DIFFICULTY 4. The number of steps in a calculation: This

aspect of difficulty discriminated between mathematical questions where only one step was needed to gain the solution to a problem (single step), where more than one step was needed in one calculation to gain the solution to a problem (multiple step) and where a value, for example x, had to be found and that value, x, used in a subsequent calculation in order to find the solution to the problem, y (extended calculation). Appendix 5a gives an example at A-level of single, multiple and extended type questions.

5. The complexity of the task: This aspect of difficulty established the complexity of the mathematical question as defined by Geoff Wake’s5 four descriptions of increasing difficulty. Level 1 complexity is defined as straightforward or routine, which requires recall of procedures and relatively straightforward application. Level 2 requires application and understanding of mathematics in one domain. Level 3 requires understanding and use of mathematics across domains and necessitates a decision about the direction in which to proceed. Level 4 involves complex activity requiring synthesis and application across a number of domains with structuring or decision making being necessary. Appendix 5b gives an example at A-level of the four categories.

6. Familiarity of context: It is generally accepted that if a context is more familiar it is easier to apply mathematics than if the context is unfamiliar. Three categories were used to judge the familiarity of the mathematical question, in relation to recognising which

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A subject expert group of 6-8 participants was established for each of the three sciences. The groups comprised practising A-level teachers, teachers with experience in curriculum research and development and individuals working for awarding organisations as markers, question writers or examiners. Each group took part in a dedicated workshop, and analysed the full suite of 2010 examinations papers for their respective subjects, using the framework above. A different framework was used for A-level theory papers and A-level practical papers (see Appendix 6 for the framework templates).

In the analysis of the science A-levels, all six units which make up a complete A-level were analysed. Units 1, 2, 4 and 5 are the theory papers. Units 3 and 6 are referred to in this research as the practical papers6. Subject expert groups were given the A-level papers and associated mark schemes, the A-level specifications, data sheets and any other necessary materials from the awarding organisations needed to fully comprehend the demand and scope of the complete A-level. The data was analysed for single papers and an average was calculated for the theory papers (Units 1, 2, 4 and 5) and the practical papers (Units 3 and 6). These averages were then recalibrated to provide an average for the complete A-level where theory papers make 80% contribution of the complete A-level and practical papers make up 20% contribution7.

Figure 1: continued

6 The nature of units 3 and 6 varied across the awarding organisations and across the subjects. They are referred to in the specifications as experimental and practical papers, experimental tasks, laboratory tasks, practical skills, practical tasks and investigations, project work, controlled assessments or coursework. Following a pilot exercise, it was accepted that in these units we were not comparing like with like, but that judgements about the mathematics expected to be met and used by an average student could be made. The analysts considered the most challenging judgement in these cases to be identifying the number of marks that would require mathematics, however, in their groups they did feel that reliable judgements could be made.

mathematics to apply, as defined by Geoff Wake’s work referenced above. These levels were: Level 1 – typically met through the learning programme; Level 2 – some novel aspects; and Level 3 – situation unlikely to have been met before. Appendix 5c gives an example at A-level of the three levels of familiarity.

APPROPRIATENESS 7. Application: A judgement was made on

whether the content of the question part reflected how mathematics is used in the real world in a scientific context. This was identified in the SCORE GCSE review as a particular issue for biology and one that merited investigation at A-level.

8. Relationship to question: A judgement was

made as to whether the mathematics was a structural part of the question or whether it was just tagged on. This was identified in the SCORE GCSE review as an issue across the sciences at GCSE.

9. Mathematics skill or scientific comprehension:

This measure differentiates between marks within a mathematical question part that require scientific comprehension and marks which are given purely based on mathematics skills. Analysts made a judgement between three categories: whether all the marks in a question part required mathematics skill only, whether the marks included a mixture of scientific comprehension and mathematical skill or whether all of the marks required scientific comprehension. Appendix 5d provides an example at A-level of this measure.


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Standardisation exercises were employed throughout the analysis to verify the reliability of judgements within and across the subject expert groups. These included:

• An individual sample analysis prior to the two-day group workshop. Each analyst on the subject expert groups was sent a sample of A-level questions, along with instructions for analysing the questions. The results from the exercise were compared at the workshop in order to reach agreement on the meaning of the measures.

• Changing the make-up of the subject expert groups throughout the analysis to verify the reliability of judgements across the groups.

• Whole group discussions to clarify issues as they arose to ensure all groups were in agreement with existing measures.

• Groups checking their final judgements against those made at the start of the analysis session to check that they had not slipped throughout the process.

Students’ scripts were not analysed in this research; if they had been there may have been instances of students using higher level mathematics in their responses to Units 3 and 6. For the purpose of this project, however, analysts considered more typical responses expected by teachers and markers, both those given in the mark schemes and what they would expect from their own professional experience. It was accepted that mathematics of higher complexities and a greater number of extended calculations may be present in such scripts, although they may not be rewarded with a greater number of marks than were possible to be given in the mark scheme.


In Phase 2 we measured the extent to which the mathematical requirements of biology, chemistry and physics A-level had been previously taught in the mathematics curriculum. This provided an indication of coherence between the sciences and mathematics. Two comparisons were used to compare the coherence of science A-levels and the mathematics accessed up to Key Stage 4: comparison with National Curriculum level descriptors and comparison with the 2012 mathematics GCSE specification.

The National Curriculum level descriptors go from Level 1, the easiest level, up to Level 8. Mathematics beyond Level 8 is classified as EP, exceptional performance. The content areas from the mathematical requirements were levelled against the National Curriculum level descriptors by two mathematicians with experience in teaching and research. Their responses were collated and considered. As the mathematical requirements were not set in a context of expectations, some of the areas were open to interpretation. In these cases the mathematics experts gave a range of levels that the mathematics could correspond to depending on the actual expectations of an assessment item.

The collated levels were then further analysed by the lead researcher and subject specialists who had been involved with the Phase 1 analysis of the papers. This allowed the levels to be considered in terms of the contexts that the items were set in and the levels could be refined.

7 The 80:20 ratio of theory and practical papers refers to the weighting of marks in a complete A-level. In the analysis the averages were taken as a proportion of question parts containing mathematics rather than as a proportion of marks awarded for mathematics but we maintained the 80:20 weighting. The results are not dissimilar but a more accurate measure would have been obtained with averages based on proportion of marks awarded for mathematics.


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Secondly, the mathematical requirements found in the A-level specifications were analysed against the GCSE mathematics specification to see whether or not the mathematical requirements that featured in the science A-level specifications were covered in the GCSE mathematics specification. This comparison indicates the mathematics that students who are not studying mathematics beyond GCSE will need to be taught within the science curriculum in order to access all content areas of the science A-level. The results are shown for students studying GCSE mathematics at both Foundation and Higher level. The Edexcel 2012 ‘A’ specification was used for the purpose of the analysis as it was considered representative of typical content found in the revised mathematics GCSEs.

The analysis was carried out by the lead researcher and a mathematics teacher who had taught GCSE and A-level mathematics and had taught mathematics sessions for students taking all three of the A-level sciences. His experience over time meant that further comments could be added, for example, whether the content area had been removed from GCSE specifications.


The aim of the third phase of the research was to gather the opinions of the science community on the findings from Phase 1 and 2. In order to do so, a survey was sent to members of the science community and focused on the nature of the mathematics that they would like to see in the A-level science assessments. Four participant groups were identified: • A-level science teachers • Higher education representatives • Representatives from professional bodies • Industry representatives.

The first three of these groups responded to an online survey in order to give their opinions8. The industry representatives responded to a short questionnaire either over the telephone or in writing.

The online survey was taken in two parts to avoid prejudicing the results. Part A asked respondents for general comments about the mathematics in the current science A-levels. These comments were based on previous experience of science A-levels, on viewing a complete A-level9 and on reviewing a selection of questions with mathematical content10. In Part B respondents were asked for their views on the way mathematics is assessed within A-level Sciences having been given the findings from the first phase of the research.

8 The online survey and responses are available from the SCORE Secretariat.

9 A complete A-level theory paper for their subject. The papers that were chosen sat in the middle of the data range from the first phase of the research, i.e. they sat in mid-range for the extent of the paper involving mathematics, the mid-range in terms of difficulty and included a typical range of mathematical content.

10 A set of questions with mathematical content. The questions were chosen to cover the range of mathematical requirements that were assessed in the A-level papers and were chosen so that there was an even spread from across all of the awarding organisations and from the AS and A2 papers.


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The online survey was completed by nearly 100 participants, with the breakdown for each subject in Table 1. This represents a very small proportion of the science community and, in some participant groups, numbers were so small that findings can only offer guidance on how mathematics should be assessed within the sciences at A-level and should not be regarded as strong evidence.

Data was analysed separately for the different participant groups so that similarities and differences of opinion could be identified. The results were reported for the whole group apart from where there were significant differences of opinion between the groups. A variety of question types were used in the survey to elicit data relevant to the research questions. From the

online survey data, basic statistics were generated. Where questions were open-ended, responses were coded and categorised. Similar codes were used to report the responses from the industry questionnaire.

For industry, the most effective means of gathering opinions on the mathematical element of the science A-levels was through written or telephone questionnaire. Six representatives in total took part in the questionnaire.

Table 1: Number of participants completing the online survey for Phase 3

SUBJECT

Teachers Higher education Professional Bodies Total

Biology 21 3 3 27

Chemistry 20 11 7 38

Physics 21 7 4 32

NUMBER OF PARTICIPANTS


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The detailed findings from Phases 1, 2 and 3 are separated into biology, chemistry and physics. The awarding organisations are not named and are represented by A-E. These representations are the same throughout.

3.1 PHYSICS

3.1.1 PHASE 1: A-LEVEL PAPERS

3.1.1.1 EXTENT

This measure seeks to capture how much of the A-level physics assessment is mathematical (independent of the type, appropriateness or difficulty). It is quantified by the proportion of questions or question parts within a complete A-level that require mathematics and the proportion of marks requiring mathematics. Table 2a shows the percentage of question parts containing mathematics within each unit and the percentage of question parts containing mathematics for theory only and practical only papers. Table 2b takes into account the weighting of theory papers (80%) and practical papers (20%) to calculate the

average percentage of question parts containing mathematics in a complete A-level. Figure 2 illustrates the percentage of question parts in a complete A-level containing mathematics for the five awarding organisations.

Figure 2: Percentage of question parts containing mathematics in a complete physics A-level for the five awarding organisations

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CHAPTER 3: DETAILED FINDINGS

Table 2a: Percentage of question parts containing mathematics within each unit and within the set of theory and practical papers across the five awarding organisations

A B C D E

AS units 1 and 2 49 57 68 42 51

AS unit 3 56 15 33 38 76

A2 units 4 and 5 62 48 63 46 63

A2 unit 6 58 31 33 82 58

Theory papers only 56 53 66 44 57

Practical papers only 57 23 33 60 67

Table 2b: Percentage of question parts weighted to take account of the theory component (80%) and practical component (20%) of the A-level assessments

A B C D E

Theory contribution (80%) 45 42 53 35 46

Practical contribution (20%) 11 5 7 12 13

Total A-level 56 47 60 47 59


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Table 3a shows the percentage of marks requiring mathematics for each unit and for theory only and practical only papers. Table 3b takes into account the weighting of theory papers (80%) and practical papers (20%) to calculate the average percentage of marks requiring mathematics in a complete A-level. Figure 3 illustrates this as a graph.

Figure 3: Percentage of marks requiring mathematics in a complete physics A-level for the five awarding organisations

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Table 3a: Percentage of marks11 that require mathematics within each unit and within the set of theory and practical papers across the five awarding organisations

A B C D E

AS units 1 and 2 45 60 66 40 49

AS unit 3 44 15 33 55 74

A2 units 4 and 5 53 49 62 45 66

A2 unit 6 50 30 27 70 25



Table 3b: Percentage of marks requiring mathematics within a total A-level, weighted to take account of the theory component (80%) and practical component (20%) of the A-level assessments

A B C D E



Total A-level 48 49 57 47 56

11 A mark was judged to require mathematics if part or all of the mark could not be achieved without mathematics.


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3.1.1.2 TYPE

The number of occurrences of each mathematical requirement listed in Physics A-level was measured to identify the type of mathematics assessed and the frequency of each type of mathematics assessed. The results are displayed in Figure 4.

1. Arithmetic and Computation(a) use a calculator for addition, subtraction, multiplication

and division;(b) recognise and use expressions in decimal form;

(‘standard from’ deleted from this requirement and recorded separately as 1(h) to illustrate how commonly each occurred.)

(c) use ratios, fractions and percentages;(d) use calculators to find and use xn, 1/x, x2, √x, logl0x , e x,

log ex;(e) use calculators to handle sinѲ, cosѲ, tanѲ, sin-1Ѳ, cos-

1Ѳ, tan-1Ѳ when Ѳ is expressed in degrees or radians.(f) recognise and use SI prefixes 10-12, 10-9, 10-6, 10-3, 103,

106 and 109

(g) handle calculations so that significant figures are neither lost or carried beyond what is justified;

(h) standard form.

2. Handling data(a) show an awareness of the order of magnitude of physical

quantities and make order of magnitude calculations;(b) use an appropriate number of significant figures;(c) find arithmetic means and medians;(d) express changes as percentages and vice versa;(e) understand and use logarithmic scales in relation to

quantities which range over several orders of magnitude.

3. Algebra (a) change the subject of an equation by manipulation of

the terms, including positive and negative, integer and fractional indices and square roots

(b) substitute numerical values into algebraic equations using appropriate units for physical quantities

(c) check the dimensional consistency of physical equations and substitute numerical values into such equations using appropriate units for physical quantities;

(d) solve simple algebraic equations including y=k/x, y=k/x2

(e) formulate and use simple algebraic equations as mathematical models of physical situations, and identify the inadequacy of such models

(f) understand and use the symbols: <, <<, >>, >, ~, , ∑, ∆x, x, dx/dt

4. Geometry and Trigonometry(a) calculate areas of triangles, circumferences and areas of

circles, surface areas and volumes of rectangular blocks, cylinders and spheres;

(b) use Pythagoras’ theorem, similarity of triangles and the angle sum of a triangle;

(c) use sines, cosines and tangents in physical problems;(d) use sinѲ ≈ tanѲ ≈ Ѳ and cos Ѳ ≈ 1 for small Ѳ;(e) understand the relationship between degrees and radians

and translate from one to the other.

5. Graphs(a) translate information between graphical, numerical and

algebraic forms;(b) plot two variables from experimental or other data using

appropriate scales for graph plotting;(c) plot data on a log-linear graph and determine whether

they change exponentially and determine the exponent;(d) plot data on a log-log graph and decide whether data

obey a power law and determine the exponent;(e) select appropriate variables for graph plotting;(f) understand that y = mx + c represents a linear

relationship and rearrange relationships into this form where appropriate;

(g) determine the slope / gradient and intercept of a linear graph in the appropriate physical units;

(h) determine the gradient of a tangent to a non-linear graph by drawing and use the slope of the tangent as a measure of rate of change;

(i) choose by inspection a straight line or curved line which will serve as the best straight line through a set of data points presented graphically;

(j) understand the possible physical significance of the area between a curve and the x axis and be able to calculate it or measure it by counting squares as appropriate;

(k) understand and use the slope of a tangent to a curve as a means to obtain the gradient. Understand and use the notation d/dt for a rate of change;

(l) understand and use multiplicative scales (1, 10, 100...);(m) use logarithmic plots to test exponential and power law

variations;(n) sketch simple functions including y = k/x, y = kx2 y = k/x2,

y = sinѲ, y = cosѲ, y = e-kx.(o) understand or recognise the physical significance of a

straight line passing or not passing through the origin.

Mathematical requirements listed in x-axis on Figure 4


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3.1.1.3 DIFFICULTY

For each of the following measures, percentages are stated as a proportion of the number of question parts. The percentages do not relate in any way to the number of marks awarded for mathematical understanding.

NUMBER OF STEPSThe number of steps involved in a calculation was used as one measure of difficulty, based on the assumption that questions containing mathematics that required multiple step or extended calculation (e.g. value x had to be found and used in a subsequent calculation in order to find the solution to the problem, y) were more difficult than single step calculations, as they require students to use higher order skills and extended reasoning. Appendix 5a shows an example of each type of calculation.

Table 4a shows the percentage of mathematical question parts classified as containing single step (S), multiple step (M) or extended step (E) calculations within each A-level unit and as an average for theory only and practical only papers. These figures are calculated as a percentage of the question parts identified as containing mathematics. Table 4b shows these percentages of the total number of question parts and Table 4c takes into account the weighting of theory papers (80%) and practical papers (20%) to calculate the average percentage of single step, multiple step and extended step calculations in a complete A-level. Figure 5a illustrates the percentage of single, multiple and extended step calculations in a complete A-level for each of the five awarding organisations. Figures 5b and 5c illustrate these percentages for theory only and practical only papers respectively.

Figure 4: The number of occurrences for each mathematical requirement in a full suite of examination papers for a complete physics A-level in each of the awarding organisations. It has no relation to the number of marks awarded for each mathematical requirement.

1a 1b 1c 1d 1e 1f 1g 1h 2a 2b 2c 2d 2e 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 5a 5b 5c 5d 5e 5f 5g 5h 5i 5j 5k 5l 5m 5n

Arithmetic and Computation

Handling Data Algebra Geometry andTrigonometry

Graphs

Mathematical Requirements for Physics A-Level


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Table 4a: Percentage of single, multiple and extended step calculations in physics A-level, calculated as a percentage of question parts identified as containing mathematics

Table 4c: Percentage of single, multiple and extended step calculations in physics A-level, weighted to take account of the theory component (80%) and practical component (20%) of the A-level assessments

A B C D E

Number of steps E M S E M S E M S E M S E M S

AS units 1 and 2 16 63 22 5 65 31 19 55 27 9 48 44 13 74 14

AS unit 3 0 21 79 0 17 83 0 31 69 0 50 50 2 13 85

A2 units 4 and 5 30 37 34 5 63 33 0 60 41 5 38 57 28 69 4

A2 unit 6 0 20 80 0 50 50 0 20 80 0 39 61 0 50 50

Theory papers only 23 50 28 5 64 32 10 58 34 7 43 51 21 72 9

Practical papers only 0 21 80 0 34 67 0 26 75 0 45 56 1 32 68

Table 4b: Percentage of single, multiple and extended step calculations in physics A-level, calculated as a percentage of the total number of question parts in a complete A-level

A B C D E


AS units 1 and 2 8 31 11 3 37 18 13 37 18 4 20 18 7 38 7

AS unit 3 0 12 44 0 3 12 0 10 23 0 19 19 2 10 65

A2 units 4 and 5 19 23 21 2 30 16 0 38 26 2 17 26 18 43 3

A2 unit 6 0 12 46 0 16 16 0 7 26 0 32 50 0 29 29



A B C D E


Theory papers only (80%)

11 22 13 2 27 14 6 30 18 2 15 18 10 33 4

Practical papers only (20%)

0 2 9 0 2 3 0 2 5 0 5 7 0 4 9

A-level total (weighted)

11 24 22 2 29 17 6 32 23 2 20 25 10 37 13


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Figure 5b: Percentage of mathematical question parts identified as containing single, multiple and extended step calculations in theory examination papers for physics A-level

Extended SingleMultiple

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Figure 5c: Percentage of mathematical question parts identified as containing single, multiple and extended step calculations in practical examination papers for physics A-level

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PRACTICALExtended SingleMultiple

Figure 5a: Percentage of mathematical question parts identified as containing single, multiple and extended step calculations in a complete physics A-level for each of the five awarding organisations

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COMPLEXITY OF TASKQuestion parts that contained mathematics within a full suite of A-level examinations were measured against four levels of complexity, with Level 4 being considered the most difficult. These levels included Level 1 (straight forward/routine), Level 2 (requires understanding and application of mathematics within one domain), Level 3 (requires understanding and application of mathematics across domains) and Level 4 (requires synthesis and application of mathematics across a number of domains). Appendix 5b shows an example of each level of complexity.

Table 5a shows the percentage of mathematical question parts classified as Level 1, 2 or 3

complexity12 within each A-level unit and as an average for theory only and practical only papers. These figures are calculated as a percentage of the question parts identified as containing mathematics. Table 5b shows these percentages of the total number of question parts and Table 5c takes into account the weighting of theory papers (80%) and practical papers (20%) to calculate the average percentage of Level 1, 2 and 3 complexity in a complete A-level. Figure 6a illustrates the percentage of Level 1, 2 and 3 complexity type calculations in a complete A-level for each of the five awarding organisations. Figures 6b and 6c illustrate these percentages for theory only and practical only papers respectively.

Table 5a: Percentage of Level 1, 2 and 3 complexity type calculations in physics A-level, calculated as a percentage of question parts identified as containing mathematics

A B C D E

Complexity 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1

AS units 1 and 2 12 61 28 5 67 29 18 66 14 0 14 87 4 85 12

AS unit 3 2 16 82 0 0 100 0 75 25 0 20 80 0 5 95

A2 units 4 and 5 17 63 21 2 63 35 5 45 50 9 24 68 11 86 4

A2 unit 6 0 20 80 0 58 42 0 25 75 0 22 78 0 50 50



Table 5b: Percentage of Level 1, 2 and 3 complexity type calculations in physics A-level, calculated as a percentage of the total number of question parts in a complete A-level

A B C D E

Complexity 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1

AS units 1 and 2 6 30 14 3 38 17 12 45 10 0 6 37 2 43 6

AS unit 3 1 9 46 0 0 15 0 25 8 0 8 30 0 4 72

A2 units 4 and 5 11 39 13 1 30 17 3 28 32 4 11 31 7 54 3

A2 unit 6 0 12 46 0 18 13 0 8 25 0 18 64 0 29 29



12 Level 4 was omitted from the findings as very few examination papers included questions of this complexity.


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Table 5c: Percentage of Level 1, 2 and 3 complexity type calculations in physics A-level, weighted to take account of the theory component (80%) and practical component (20%) of the A-level assessments

A B C D E

Complexity 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1


7 28 11 2 27 14 6 30 17 2 7 27 4 39 4


0 2 9 0 2 3 0 3 3 0 3 9 0 3 10


7 30 20 2 29 17 6 33 20 2 10 36 4 42 14

Figure 6a: Percentage of mathematical question parts identified as containing Level 1, Level 2 and Level 3 complexity-type calculations in a complete A-level for each of the five awarding organisations

Figure 6b: Percentage of mathematical question parts identified as contanining Level 1, Level 2 and Level 3 complexity-type calculations in theory examination papers for physics A-level

Figure 6c: Percentage of mathematical question parts identified as contanining Level 1, Level 2 and Level 3 complexity-type calculations in practical examination papers for physics A-level

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Level 3 Level 1Level 2

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THEORYLevel 3 Level 1Level 2

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PRACTICALLevel 3 Level 1Level 2


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CONTEXTThe question parts that contained mathematics within a full suite of A-level examinations were classified as Level 1, Level 2 or Level 3, with Level 1 the most familiar context. Level 1 is a context typically met through the learning programme, Level 2 is a context that contains some novel aspects and Level 3 is an unfamiliar context unlikely to have been met before. Appendix 5c shows an example of levels of familiarity.

Table 6a shows the percentage of mathematical question parts classified as Level 1, 2 or 3 context within each A-level unit and as an average for

theory only and practical only papers. These figures are calculated as a percentage of the question parts identified as containing mathematics. Table 6b shows these percentages of the total number of question parts and Table 6c takes into account the weighting of theory papers (80%) and practical papers (20%) to calculate the average percentage of Level 1, 2 and 3 context in a complete A-level. Figure 7a illustrates the percentage of Level 1, 2 and 3 context calculations in a complete A-level for each of the five awarding organisations. Figures 7b and 7c illustrate these percentages for theory only and practical only papers respectively.

Table 6a: Percentage of Level 1, 2 and 3 context-type calculations in physics A-level, calculated as a percentage of question parts identified as containing mathematics

A B C D E

Context 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1

AS units 1 and 2 0 27 69 0 61 40 14 26 61 0 12 89 0 0 100

AS unit 3 0 21 79 0 100 0 0 6 94 0 60 40 0 0 100

A2 units 4 and 5 7 40 54 0 46 54 4 30 66 2 11 87 0 0 100

A2 unit 6 6 21 74 0 75 25 33 13 53 0 0 100 0 44 56



Table 6b: Percentage of Level 1, 2 and 3 context-type calculations in physics A-level, calculated as a percentage of the total number of question parts in a complete A-level

A B C D E

Context 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1

AS units 1 and 2 0 13 34 0 35 23 10 18 41 0 5 37 0 0 51

AS unit 3 0 12 44 0 15 0 0 2 31 0 23 15 0 0 76

A2 units 4 and 5 4 25 33 0 22 26 3 19 42 1 5 40 0 0 63

A2 unit 6 3 12 43 0 23 8 11 4 17 0 0 82 0 26 32




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Table 6c: Percentage of Level 1, 2 and 3 context-type calculations in physics A-level, weighted to take account of the theory component (80%) and practical component (20%) of the A-level assessments

A B C D E

Context 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1


2 15 27 0 23 20 6 15 34 0 4 31 0 0 46


0 2 9 0 4 1 1 1 5 0 2 10 0 3 11


2 17 36 0 27 21 7 16 39 0 6 41 0 3 57

Figure 7a: Percentage of mathematical question parts identified as containing calculations set in a Familiar, with Some Novel Aspects, and Unfamiliar context in a complete A-level for each of the five awarding organisations

Figure 7b: Percentage of mathematical question parts identified as containing calculations set in a Familiar, with Some Novel Aspects, and Unfamiliar context in theory examination papers for physics A-level

Figure 7c: Percentage of mathematical question parts identified as containing calculations set in a Familiar, with Some Novel Aspects, and Unfamiliar context in practical examination papers for physics A-level

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TOTALFamiliarSome NovelUnfamiliar

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3.1.1.4 APPROPRIATENESS

APPLICATION Judgements were made as to whether the content of the question part reflected how mathematics is used in the real world in the scientific context. All of the question parts with mathematics were judged to reflect an appropriate way in which the mathematics could be used in a real scientific context.

STRUCTURAL OR TAGGED ON One of the measures was to ascertain whether the mathematics was a structural part of the question or whether the mathematics was tagged on to the question. In all of the question parts with mathematical content the mathematics was judged to be a structural part of the question.

MATHEMATICS SKILLS OR SCIENTIFIC COMPREHENSIONMathematical question parts within a full suite of A-level examinations were measured against the extent to which scientific comprehension was required to achieve the full marks. If a question part required no scientific comprehension to acquire the full marks it was classified as no scientific comprehension (mathematical skill only), if part of the marks required scientific comprehension in

addition to mathematical skill it was classified as some scientific comprehension and question parts where all marks required scientific comprehension were classified as scientific comprehension. Appendix 5d shows an example of each category.

Table 7a shows the percentage of mathematical question parts classified as all marks (S), some marks (B) or no marks (M) requiring scientific comprehension within each A-level unit and as an average for theory only and practical only papers. These figures are calculated as a percentage of the question parts identified as containing mathematics. Table 7b shows these percentages of the total number of question parts and Table 7c takes into account the weighting of theory papers (80%) and practical papers (20%) to calculate the average percentage of mathematical question parts where all marks, some marks or no marks require scientific comprehension in a complete A-level. Figure 8a illustrates the percentage of mathematical question parts where all, some and no marks require scientific comprehension in a complete A-level for each of the five awarding organisations. Figures 8b and 8c illustrate these percentages for theory only and practical only papers respectively.

Table 7a: Percentage of mathematical question parts classified as all mark (S), some marks (B) and no marks (M) requiring scientific comprehension in physics A-level, calculated as a percentage of question parts identified as containing mathematics

A B C D E

Scientific comprehension

S B M S B M S B M S B M S B M

AS units 1 and 2 83 11 7 98 0 2 76 4 20 100 0 0 97 0 4

AS unit 3 16 0 84 100 0 0 69 0 31 100 0 0 48 3 49

A2 units 4 and 5 99 0 2 97 0 4 99 2 0 75 2 24 100 0 0

A2 unit 6 30 0 70 83 0 17 100 0 0 72 0 28 89 0 11




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Table 7b: Percentage of mathematical type question parts where all marks (S), some marks (B) or no marks (M) require scientific comprehension in physics A-level, calculated as a percentage of the total number of question parts in a complete A-level

A B C D E



AS units 1 and 2 41 5 3 56 0 1 52 3 14 42 0 0 49 0 2

AS unit 3 9 0 47 15 0 0 23 0 10 38 0 0 36 2 37

A2 units 4 and 5 61 0 1 47 0 2 62 1 0 35 1 11 63 0 0

A2 unit 6 17 0 41 26 0 5 33 0 0 59 0 23 52 0 6



Table 7c: Percentage of mathematical type question parts where all marks (S), some marks (B) and no marks (M) require scientific comprehension in physics A-level, weighted to take account of the theory component (80%) and practical component (20%) of the A-level assessments

A B C D E




41 2 2 41 0 2 46 2 6 30 0 5 45 0 1


3 0 9 4 0 1 6 0 1 10 0 2 9 0 4


44 2 11 45 0 3 52 2 7 40 0 7 54 0 5


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3.1.2 PHASE 2 – PHYSICS A-LEVEL IN COMPARISON WITH GCSE MATHEMATICS AND NATIONAL CURRICULUM LEVEL DESCRIPTORS

Two comparisons were used to establish the coherence of physics A-levels and the mathematics accessed up to Key Stage 4: comparison with National Curriculum level descriptors and comparison with the 2012 mathematics GCSE specification. These comparisons are displayed in Table 8.

Figure 8a: Percentage of mathematical question parts requiring scientific comprehension in addition to mathematical skill to achieve all marks, some of the marks or no marks in a complete physics A-level for each of the five awarding organisations

Figure 8b: Percentage of mathematical question parts requiring scientific comprehension in addition to mathematical skill to achieve all marks, some of the marks or no marks in theory examination papers for physics A-level

Figure 8c: Percentage of mathematical question parts requiring scientific comprehension in addition to mathematical skill to achieve all marks, some of the marks or no marks in practical examination papers for physics A-level

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TOTALNo ScientificComprehension

Some ScientificComprehension

ScientificComprehension

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Mathematical requirements as listed in the physics A-level specification

Corresponding NC level(s) for mathematics

Comment

1. Arithmetic and Computation F H

(a) use a calculator for addition, subtraction, multiplication and division;

L4 to L5 (L4 + and – decimals, to L5 x and ÷)

ü ü

(b) recognise and use expressions in decimal; (standard form deleted from this requirement and recorded separately as 1(h))

L5 ü ü

(c) use ratios, fractions and percentages;

L5 to L8 (L5 for calculations and fractions to L8 for proportion)

ü ü Percentages: reverse percentages are exclusive to higher level

ü ü Ratios

ü ü Fractions

(d) use calculators to find and use xn, 1/x, x2, √x, logl0x , e x, log ex;

L6 to EP (L6 for xn, 1/x, x2, √x, logl0x , e x, log ex)

ü ü xn, 1/x, x2, √x

û û logl0x , e x, log ex

(e) use calculators to handle sinѲ, cosѲ, tanѲ, sin-1Ѳ, cos-1Ѳ, tan-1Ѳ when Ѳ is expressed in degrees or radians.

EP û ü Degrees

û û Radians appear in AS level mathematics specification

(f) recognise and use SI prefixes 10-12, 10-9, 10-6, 10-3, 103, 106 and 109

EP û ü


L7 to EP ü ü Round to the nearest integer or to any number of significant figures

û ü Extended usage of significant figures

(h) standard form. L8 û ü

2. Handling data NC levels F H Comments

(a) show an awareness of the order of magnitude of physical quantities and make order of magnitude calculations;

L5 to EP (depending on the context e.g. mass of humans or mass of electrons)

û û Not a concept taught in mathematics

Found in the Edexcel GCSE specification Foundation (F) and Higher (H)

Table 8: Comparison of mathematical requirements for physics A-level with mathematics found in the National Curriculum Level Descriptors and GCSE mathematics specification


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(b) use an appropriate number of significant figures;

L7 ü ü

(c) find arithmetic means and medians; L5 to L7 (depending on whether the data is discreet, continuous or grouped)

ü ü

(d) express changes as percentages and vice versa;

L7 to L8 ü ü Changes as a percentage

û ü Percentages as changes

(e) understand and use logarithmic scales in relation to quantities which range over several orders of magnitude.

EP û û This appears in the A-level mathematics specification

3. Algebra NC levels F H Comments

(a) change the subject of an equation by manipulation of the terms, including positive and negative, integer and fractional indices and square roots

L8 to EP ü ü In Foundation Level students are required to change the subject of a formula.

In Higher level this includes cases where the subject is on both sides of the original formula or where the power of a subject appears.


L5 to L8 ü ü


EP û û Not a concept taught in mathematics


L6 to EP û ü Inverse relationships are not in the specification at Foundation level

Table 8: continued


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Table 8: continued


L5 to EP ü ü Formulate and use simple algebraic equations as mathematical models of physical situations: only simple equations would appear at Foundation level mathematics

û û Identifying the adequacies of such models: this is not a concept taught in mathematics


L2 to EP ü ü =, <,>,~

û û <<, >> These symbols are not used in mathematics GCSE or A-level

û ü

û û ∑, ∆x, x, dx/dt appear in the AS mathematics specification

4. Geometry and Trigonometry NC levels F H Comments

(a) calculate areas of triangles, circumferences and areas of circles, surface areas and volumes of rectangular blocks, cylinders and spheres;

L6 to EP ü ü Areas of triangles, circumferences and areas of circles, surface areas and volumes of rectangular blocks

û ü Cylinders and spheres


L5 to L8 (L5 angle sum of triangle, L7 Pythagoras, L8 similarities of triangles)

ü ü 3D Pythagoras is only found in Higher level specification

(c) use sines, cosines and tangents in physical problems;

L8 û ü

(d) use sinѲ ≈ tanѲ ≈ Ѳ and cos Ѳ ≈ 1 for small Ѳ;

EP û û This is only found in the A-level mathematics specification as it requires the use of radians

(e) understand the relationship between degrees and radians and translate from one to the other.

EP û û


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5. Graphs NC levels F H Comments

(a) translate information between graphical, numerical and algebraic forms;

L4 to L7 ü ü

(b) plot two variables from experimental or other data using appropriate scales for graph plotting;

L6 ü ü

(c) plot data on a log-linear graph and determine whether they change exponentially and determine the exponent;

EP û û Not found in mathematics A-level specification

(d) plot data on a log-log graph and decide whether data obey a power law and determine the exponent;

EP û û Not found in mathematics A-level specification

(e) select appropriate variables for graph plotting;

EP û û In mathematics, students would not be expected to identify an independent variable or select two variables for graph plotting from a set ofmore than 2 variables

(f) understand that y = mx + c represents a linear relationship and rearrange relationships into this form where appropriate;

L8 to EP ü ü Recognise that y = mx + c represents a straight line graph

û û Rearrange relationships into this form: taken to mean reducing a more complex function to a linear function

(g) determine the slope / gradient and intercept of a linear graph in the appropriate physical units;

EP ü ü Slope

û ü Intercept

(h) determine the gradient of a tangent to a non-linear graph by drawing and use the slope of the tangent as a measure of rate of change;

EP û û Taken out of GCSE mathematics specification

(i) choose by inspection a straight line which will serve as the best straight line through a set of data points presented graphically; include curved lines

L7 ü ü Curved lines are not considered in mathematics specification in relation to best fit

Table 8: continued


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Table 8: continued


EP û û Counting squares taken out of GCSE mathematics specification


EP û û

(l) understand and use multiplicative scales (1, 10, 100 ...);

EP û û

(m) use logarithmic plots to test exponential and power law variations;

EP û û

(n) sketch simple functions including y = k/x, y = kx2 y = k/x2, y = sinѲ, y = cosѲ, y = e-kx.

L8 to EP û ü y = k/x, y = kx2 y = k/x2, y = sinѲ, y = cosѲ

û û y = e-kx


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3.1.3 PHASE 3 – SURVEY FINDINGS

3.1.3.1 TYPE OF MATHEMATICS ASSESSED AT A-LEVEL PHYSICS

In Part A of the survey, respondents were asked to consider the spread of mathematical content areas assessed within physics A-level. Figure 9 illustrates the percentages of respondents that considered the spread to be good, acceptable (key mathematical areas were assessed), average (limited variation in type of mathematics) and poor (restricted amount of mathematics assessed many times).

Participants were also asked to comment on the areas of mathematics they would like to feature highly in assessments, a little in assessments or not at all. The results are shown in Figure 10. Participants were also asked if there were any other areas of mathematics, not listed in the requirements, that they felt should be included in the assessments. The only areas that were mentioned repeatedly were calculus, differentiation and integration.

Overall, there was concern that the mathematics within physics A-level was not difficult enough, which meant that students lacked fluency in mathematics and problem-solving skills. Some were also concerned that students did not fully understand mathematical concepts, which would limit their ability to apply what they knew.

Prior to finding out results of the analysis, 63% of respondents felt that awarding organisations should use a framework to ensure that a broad spread of mathematical requirements are assessed and 47% felt that all requirements should be assessed over a two- or three-year cycle of the A-levels to ensure that they are taught.

Figure 10: Mathematical requirement areas that physics survey respondents would like to feature highly in assessment, a little in assessment or not at all

1a 1b 1c 1d 1e 1f 1g 1h 2a 2b 2c 2d 2e 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 5a 5b 5c 5d 5e 5f 5g 5h 5i 5j 5k 5l 5m 5n

Mathematical Requirements for Physics A-Level

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Feature Not at AllFeature a LittleFeature Highly

Figure 9: Opinions from the physics online survey respondents on the spread of mathematical content areas within a physics A-level

There was a good spread of different mathematical content areas being assessed

A restricted amount of mathematical content areas seemed to be assessed many times

The Key mathematical content areas were assessed

There was not enough variation in the type of mathematical content being assessed


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In Part B, after receiving the findings from the analysis, more than 75% of respondents felt it was unacceptable that mathematical content areas they considered essential were hardly or not at all assessed. When comparing Figure 4 and Figure 10 it is clear there is a mismatch between mathematical content areas that are assessed and those that the science community would like to be assessed (for example selecting appropriate variables for graph plotting; understand that y = mx + c represents a linear relationship and rearrange relationships into this form where appropriate; and determining the slope / gradient and intercept of a linear graph in the appropriate physical units).

Furthermore nearly three quarters felt it was inappropriate that a few mathematical requirements were assessed repeatedly, both throughout the question papers within a qualification and across awarding organisations, rather than a spread of skills being assessed. Many acknowledged the importance of testing some of the key mathematical skills more frequently, for example, graphs and using formulae, in order to improve mathematical fluency. However, a broader spread was felt to be important in order to understand the breadth of physics that would be required for progression and to test deeper thinking, as opposed to substitution in formulae. The same concern was raised again that if topics were not assessed they would not be taught.

EXTENT OF MATHEMATICS In Part A, 59% of respondents felt that the amount of mathematics in the paper was not enough to adequately prepare for progression to higher education in a physics or related subject. After viewing the findings of Phase 1, the majority thought that the percentage of question parts containing mathematics was appropriate. However, 40% of respondents thought the lower limit of 47% of marks requiring mathematics was too low.

No consensus was reached in Part A or B on whether the mathematics in the theory papers and the mathematics in the practical papers should be different. Those who felt that it should be the same felt the A-level examination papers should be seen as a unified subject, the two being inter-dependent.

Those who felt it should be different explained that in the practical units there would be more interpretation of graphs, data handling, and calculation of errors, derivations, and ‘show’ questions. Some explained that the theory papers should test rigour and the understanding of concepts whereas the practical units should be testing experimental planning, data collection, statistics and analysis so the mathematical element would naturally be different.

DIFFICULTY OF MATHEMATICS Part A of the survey showed that 100% of the representatives from higher education and professional bodies were not concerned if the level of difficulty was perceived to go up due to the A-levels containing more mathematics. However, only 62% of teachers agreed, largely because they felt that increasing the level of demand of the mathematics in physics A-level would result in the overall A-level being perceived as more demanding.

In Part A, when asked about difficulty in terms of the number of steps in the calculations on the paper, 44% of respondents felt that there should be more multiple-step calculations, with a further 15% wishing to see more extended questions. In Part B, nearly two-thirds of the respondents felt that there should be an even spread of all three types of calculation and nearly half of the respondents felt that there should be more multiple-step calculations and more extended calculations13.

When considering context as a measure of difficulty in Part A of the survey, two thirds of the respondents felt that the number of mathematical questions set within a familiar context was appropriate. In Part B respondents were informed that the analysis had found that the two awarding organisations had over 70% of the mathematical calculations set in a familiar context. Results were evenly split between whether this was appropriate, with half of the respondents expressing the view that there should be less mathematics set in familiar contexts so that students have more experience at applying mathematics in unfamiliar situations. However, overall 65% of the respondents felt that the high level of mathematical assessment set within a familiar context impacted adversely on students’ ability to apply mathematics in a novel situation.

13 Some respondents supported both statements so these percentages do not necessarily represent two different findings.


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Respondents were asked to make a judgement on the mathematical difficulty in the paper in terms of complexity. In Part A no respondents thought that the complexity was too difficult and there was an even split between those who thought it was too easy and those who thought that it was appropriate. In Part B, when respondents were told that the vast majority of questions across awarding organisations required Level 1 and Level 2 complexity (i.e. use of straightforward and familiar concepts or required application of one domain of mathematics), 88% of respondents felt that the recall of common mathematical procedures should be assessed alongside procedures that involve application in one or more content areas, that is, that the difficulty should be increased.

COMPARABILITY ACROSS AWARDING ORGANISATIONSIn Part A 88% of respondents agreed that it was important that the A-levels from all of the awarding organisations had the same level of difficulty in terms of the mathematical content. Only 59% of respondents agreed that it was important that all of the awarding organisations assessed the same mathematical content areas. The remainder considered it only important for the same key areas to be assessed by all of the awarding organisations. The vast majority (91%) also thought that the proportion of questions with mathematical content should be similar across the awarding organisations. 63% of respondents felt that awarding organisations should use a framework to ensure a broad spread of mathematical requirements is assessed.

In Part B most respondents (59%) felt that differences across awarding organisations in the proportion of the marks in an A-level with mathematical content were not acceptable.

COHERENCE BETWEEN MATHEMATICS AND THE SCIENCES The appropriateness of the mathematics was deemed by 97% of the respondents to be the most important aspect of mathematical assessment over demand and extent of mathematics.

All respondents felt that physics A-levels should contain mathematics beyond that found in the current mathematics GCSE. There were mixed views on whether the mathematics GCSE should include these extra requirements to ensure GCSE is adequate preparation for the mathematics in a science A-level, with some agreeing that a new mathematics qualification for use alongside physics A-levels would support the students.

3.2 CHEMISTRY


3.2.1.1 EXTENT

This measure seeks to capture how much of the A-level chemistry assessment is mathematical (independent of the type, appropriateness or difficulty). It is quantified by the proportion of questions or question parts within a complete A-level that require mathematics and the proportion of marks requiring mathematics. Table 9a shows the percentage of question parts containing mathematics within each unit and the percentage of question parts containing mathematics for theory only and practical only papers. Table 9b takes into account the weighting of theory papers (80%) and practical papers (20%) to calculate the average percentage of question parts containing mathematics in a complete A-level. Figure 11 illustrates the percentage of question parts in a complete A-level containing mathematics for the five awarding organisations.


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Figure 11: Percentage of question parts containing mathematics in a complete chemistry A-level for the five awarding organisations

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AS units 1 and 2 41 60 41 35 50

AS unit 3 49 27 35 40 53

A2 units 4 and 5 51 55 66 42 55

A2 unit 6 51 34 21 55 41



Table 9b: Percentage of questions parts weighted to take account of the theory component (80%) and practical component (20%) of the A-level assessments

A B C D E



Total A-level 47 52 49 41 51


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Table 10a shows the percentage of marks requiring mathematics for each unit and for theory only and practical only papers. Table 10b takes into account the weighting of theory papers (80%) and practical

papers (20%) to calculate the average percentage of marks requiring mathematics in a complete A-level. Figure 12 illustrates this as a graph.


Figure 12: Percentage of marks requiring mathematics in a complete A-level for the five awarding organisations

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AS units 1 and 2 38 46 33 23 38

AS unit 3 49 10 44 36 22

A2 units 4 and 5 44 45 49 20 42

A2 unit 6 46 29 32 26 22


Practical papers 48 20 38 31 22


A B C D E



Total A-level 43 41 41 24 36


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3.2.1.2 TYPE

The number of occurrences of each mathematical requirement listed in Chemistry A-level was measured to identify the type of mathematics assessed and the frequency of each type of mathematics assessed. The results are displayed in Figure 13.

1 Arithmetic and numerical computation:(a) recognise and use expressions in decimal and

standard form;(b) use ratios, fractions and percentages;(c) make estimates of the results of calculations

(without using a calculator);(d) use calculators to find and use power, exponential

and logarithmic functions (xn, 1/x, √x, logl0x , e x, log ex);(e) +, -, x, ÷.

2 Handling data:(a) use an appropriate number of significant figures;(b) find arithmetic means;(c) construct and interpret frequency tables and diagrams,

bar charts and histograms;(d) use an appropriate number of decimal places.

3 Algebra:(a) understand and use the symbols: =, <, <<, >>, >, ,

~, μ;(b) change the subject of an equation;(c) substitute numerical values into algebraic equations

using appropriate units for physical quantities;

Figure 13: The number of occurrences for each mathematical requirement in a full suite of examination papers for a complete chemistry A-level in each of the awarding organisations.

It has no relation to the number of marks awarded for each mathematical requirement.

Arithmetic andNumerical

Computation

HandlingData

Algebra Geometry andTrigonometry

Graphs

Mathematical Requirements for Chemistry A-Level

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1a 1b 1c 1d 1e 2a 2b 2c 2d 3a 3b 3c 3d 3e 4a 4b 4c 4d 4e 5a 5b 5c 5d4f 4g 4h

70

80


(d) solve simple algebraic equations;(e) use logarithms in relation to quantities which range

over several orders of magnitude.

4 Graphs:(a) translate and interpret information between graphical,

numerical and algebraic forms;(b) plot two variables from experimental or other data;(c) understand that y = mx + c represents a linear

relationship;(d) determine the slope and intercept of a linear graph;(e) calculate rate of change from a graph showing a linear

relationship;(f) draw and use the slope of a tangent to a curve as a

measure of rate of change;(g) interpret a spectrum.

5 Geometry and trigonometry:a) appreciate angles and shapes in regular 2-D and 3-D

structures;b) visualise and represent 2-D and 3-D forms including

two-dimensional representations of 3-D objects;c) understand the symmetry of 2-D and 3-D shapes.



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3.2.1.3 DIFFICULTY

For each of the following measures, percentages are stated as the number of question parts containing mathematics. The percentages do not relate in any way to the number of marks awarded for mathematical understanding.

NUMBER OF STEPSThe number of steps involved in a calculation was used as one measure of difficulty based on the assumption that questions containing mathematics that required multiple step or extended calculation (e.g. value x had to be found and used in a subsequent calculation in order to find the solution to the problem, y) were more difficult than single step calculations, as they require students to use higher order skills and extended reasoning. Appendix 5a shows an example of each type of calculation.


Table 11a: Percentage of single, multiple and extended step calculations in chemistry A-level, calculated as a percentage of question parts identified as containing mathematics

A B C D E


AS units 1 and 2 33 30 39 19 2 80 16 13 72 15 58 28 28 7 66

AS unit 3 19 3 78 0 33 67 0 15 85 53 0 48 0 56 44

A2 units 4 and 5 14 12 75 32 0 69 18 4 79 8 12 81 42 2 57

A2 unit 6 14 7 79 31 6 64 17 17 67 83 0 17 0 33 67



Table 11b: Percentage of single, multiple and extended step calculations in chemistry A-level, calculated as a percentage of the total number of question parts in a complete A-level

A B C D E


AS units 1 and 2 14 12 16 11 1 48 7 5 30 5 20 10 14 4 33

AS unit 3 9 1 38 0 9 18 0 5 30 21 0 19 0 30 23

A2 units 4 and 5 7 6 38 18 0 38 12 3 52 3 5 34 23 1 31

A2 unit 6 7 4 40 11 2 22 4 4 14 46 0 9 0 14 27




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Table 11c: Percentage of single, multiple and extended step calculations in chemistry A-level, weighted to take account of the theory component (80%) and practical component (20%) of the A-level assessments

A B C D E



8 7 22 12 1 34 7 3 33 3 10 18 15 2 26


2 1 8 1 1 4 0 1 4 7 0 3 0 4 5


10 8 30 13 2 38 7 4 37 10 10 21 15 6 31

Figure 14a: Percentage of mathematical question parts identified as containing single, multiple and extended step calculations in a complete A-level for each of the five awarding organisations

Figure 14b: Percentage of mathematical question parts identified as containing single, multiple and extended step calculations in theory examination papers for Chemistry A-level

Figure 14c: Percentage of mathematical question parts identified as containing single, multiple and extended step calculations in practical examination papers for chemistry A-level

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TOTALExtended SingleMultiple

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THEORYExtended SingleMultiple

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PRACTICALExtended SingleMultiple


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Table 12a shows the percentage of mathematical question parts classified as Level 1, 2 or 3

complexity15 within each A-level unit and as an average for theory only and practical only papers. These figures are calculated as a percentage of the question parts identified as containing mathematics. Table 12b shows these percentages of the total number of question parts and Table 12c takes into account the weighting of theory papers (80%) and practical papers (20%) to calculate the average percentage of Level 1, 2 and 3 complexity in a complete A-level. Figure 15a illustrates the percentage of Level 1, 2 and 3 complexity type calculations in a complete A-level for each of the five awarding organisations. Figures 15b and 15c illustrate these percentages for theory only and practical only papers respectively.


Table 12a: Percentage of Level 1, 2 and 3 complexity type calculations in chemistry A-level, calculated as a percentage of question parts identified as containing mathematics

A B C D E

Complexity 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1

AS units 1 and 2 0 48 53 0 12 89 17 46 38 0 35 66 0 25 75

AS unit 3 19 43 38 0 0 100 0 42 58 0 60 40 0 59 41

A2 units 4 and 5 0 23 78 2 24 75 11 12 76 7 72 22 50 50 0

A2 unit 6 11 76 14 0 38 62 0 38 62 0 71 29 0 74 26



Table 12b: Percentage of Level 1, 2 and 3 complexity type calculations in chemistry A-level, calculated as a percentage of the total number of question parts in a complete A-level

A B C D E

Complexity 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1

AS units 1 and 2 0 20 22 0 7 53 7 19 16 0 12 23 0 13 38

AS unit 3 9 21 19 0 0 27 0 15 20 0 24 16 0 31 22

A2 units 4 and 5 0 12 40 1 13 41 7 8 50 3 30 9 28 28 0

A2 unit 6 6 39 7 0 13 21 0 8 13 0 39 16 0 30 11




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Table 12c: Percentage of Level 1, 2 and 3 complexity type calculations in chemistry A-level, weighted to take account of the theory component (80%) and practical component (20%) of the A-level assessments

A B C D E

Complexity 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1


0 13 25 0 8 38 6 11 26 1 17 13 11 16 15


1 6 3 0 1 5 0 2 3 0 6 3 0 6 3


1 19 28 0 9 43 6 13 29 1 23 16 11 22 18

Figure 15a: Percentage of mathematical question parts identified as containing Level 1, Level 2 and Level 3 complexity type calculations in a complete A-level for each of the five awarding organisations

Figure 15b: Percentage of mathematical question parts identified as containing Level 1, Level 2 and Level 3 complexity type calculations in theory examination papers for chemistry A-level

Figure 15c: Percentage of mathematical question parts identified as containing Level 1, Level 2 and Level 3 complexity type calculations in practical examination papers for chemistry A-level

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TOTALLevel 3 Level 1Level 2

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THEORYLevel 3 Level 1Level 2

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PRACTICALLevel 3 Level 1Level 2


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Table 13a shows the percentage of mathematical question parts classified as Level 1, 2 or 3 context within each A-level unit and as an

average for theory only and practical only papers. These figures are calculated as a percentage of the question parts identified as containing mathematics. Table 13b shows these percentages of the total number of question parts and Table 13c takes into account the weighting of theory papers (80%) and practical papers (20%) to calculate the average percentage of Level 1, 2 and 3 context in a complete A-level. Figure 16a illustrates the percentage of Level 1, 2 and 3 context calculations in a complete A-level for each of the five awarding organisations. Figures 16b and 16c illustrate these percentages for theory only and practical only papers respectively.

Table 13a: Percentage of Level 1, 2 and 3 context-type calculations in chemistry A-level, calculated as a percentage of question parts identified as containing mathematics

A B C D E

Context 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1

AS units 1 and 2 0 0 100 0 2 99 0 0 100 0 0 100 0 0 100

AS unit 3 0 0 100 0 0 100 0 0 100 0 0 100 0 11 89

A2 units 4 and 5 0 7 94 0 0 100 0 0 100 0 0 100 0 0 100

A2 unit 6 0 0 100 0 17 84 0 33 66 0 0 100 0 0 100



Table 13b: Percentage of Level 1, 2 and 3 context-type calculations in chemistry A-level, calculated as a percentage of the total number of question parts in a complete A-level

A B C D E

Context 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1

AS units 1 and 2 0 0 41 0 1 59 0 0 41 0 0 35 0 0 50

AS unit 3 0 0 49 0 0 27 0 0 35 0 0 40 0 6 47

A2 units 4 and 5 0 4 48 0 0 55 0 0 66 0 0 42 0 0 55

A2 unit 6 0 0 51 0 6 29 0 7 14 0 0 55 0 0 41




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Table 13c: Percentage of Level 1, 2 and 3 context-type calculations in chemistry A-level, weighted to take account of the theory component (80%) and practical component (20%) of the A-level assessments

A B C D E

Context 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1

Theory papers only (80%))

0 1 36 0 0 46 0 0 43 0 0 31 0 0 42


0 0 10 0 1 6 0 1 5 0 0 10 0 1 9


0 1 46 0 1 52 0 1 48 0 0 41 0 1 51


Figure 16b: Percentage of mathematical question parts identified as containing calculations set in a Familiar, with Some Novel Aspects, and Unfamiliar context in theory examination papers for chemistry A-level

Figure 16c: Percentage of mathematical question parts identified as containing calculations set in a Familiar, with Some Novel Aspects, and Unfamiliar context in practical examination papers for chemistry A-level

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TOTALFamiliarSome NovelUnfamiliar

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STRUCTURAL OR TAGGED ON One of the measures was to ascertain whether the mathematics was a structural part of the question or whether the mathematics was purely tagged on to the question. In all of the question parts with mathematical content the mathematics was judged to be a structural part of the question.

MATHEMATICS SKILLS OR SCIENTIFIC COMPREHENSIONMathematical question parts within a full suite of A-level examinations were measured against the extent to which scientific comprehension was required to achieve the full marks. If a question part required no scientific comprehension to acquire the full marks it was classified as no scientific comprehension (mathematical skill only), if some of the marks required scientific comprehension in

addition to mathematical skill it was classified as some scientific comprehension and question parts where all marks required scientific comprehension were classified as scientific comprehension. Appendix 5d shows an example of each category.


Table 14a: Percentage of mathematical question parts classified as all marks (S), some marks (B) and no marks (M) requiring scientific comprehension in chemistry A-level, calculated as a percentage of question parts identified as containing mathematics

A B C D E

Context S B M S B M S B M S B M S B M

AS units 1 and 2 97 3 0 100 0 0 100 0 0 74 19 7 100 0 0

AS unit 3 0 66 34 0 67 33 0 90 10 0 88 13 0 67 33

A2 units 4 and 5 100 0 0 99 2 0 100 0 0 100 0 0 100 0 0

A2 unit 6 0 69 31 0 78 27 0 83 17 0 100 0 0 66 44




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Table 14b: Percentage of mathematical type question parts where all marks (S), some marks (B) or no marks (M) require scientific comprehension in chemistry A-level, calculated as a percentage of the total number of question parts in a complete A-level

A B C D E


AS units 1 and 2 40 1 0 60 0 0 41 0 0 26 7 2 50 0 0

AS unit 3 0 32 17 0 18 9 0 32 4 0 35 5 0 36 17

A2 units 4 and 5 51 0 0 54 1 0 66 0 0 42 0 0 55 0 0

A2 unit 6 0 35 16 0 27 9 0 17 4 0 55 0 0 27 18



Table 14c: Percentage of mathematical type question parts where all marks (S), some marks (B) and no marks (M) require scientific comprehension in chemistry A-level, weighted to take account of the theory component (80%) and practical component (20%) of the A-level assessments

A B C D E



36 0 0 46 0 0 43 0 0 27 3 1 42 0 0


0 7 3 0 4 2 0 5 1 0 9 1 0 6 4

A-level total 36 7 3 46 4 2 43 5 1 27 12 2 42 6 4

Figure 17a: Percentage of mathematical question parts requiring scientific comprehension in addition to mathematical skill to achieve all marks, some of the marks or no marks in a complete A-level for each of the five awarding organisations

Figure 17b: Percentage of mathematical question parts requiring scientific comprehension in addition to mathematical skill to achieve all marks, some of the marks or no marks in theory examination papers for chemistry A-level

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3.2.2 PHASE 2 – CHEMISTRY A-LEVEL IN COMPARISON WITH GCSE MATHEMATICS AND NATIONAL CURRICULUM LEVEL DESCRIPTORS

Two comparisons were used to establish the coherence of chemistry A-level and the mathematics accessed up to Key Stage 4: comparison with National Curriculum level descriptors and comparison with the 2012 mathematics GCSE specification. These comparisons are displayed in Table 15.

Figure 17c: Percentage of mathematical question parts requiring scientific comprehension in addition to mathematical skill to achieve all marks, some of the marks or no marks in practical examination papers for chemistry A-level

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Mathematical requirements as listed in the Chemistry A-level specification


Comment

1 Arithmetic and numerical computation:

F H

(a) recognise and use expressions in decimal and standard form;

L5 (decimals) to L8 (standard form)

ü ü Decimals

û ü Standard Form: this will be taken off the new 2012 Foundation level specification

(b) use ratios, fractions and percentages;

L5 (AS) to L6 (A2) ü ü Percentages: reverse percentages are exclusive to Higher level

ü ü Ratios

ü ü Fractions

(c) make estimates of the results of calculations (without using a calculator);

L5 to 6 (NB always have a calculator)

ü ü

Table 15: Comparison of mathematical requirements for chemistry A-level with mathematics found in the National Curriculum Level Descriptors and GCSE mathematics specification



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(d) use calculators to find and use power, exponential and logarithmic functions ( xn, 1/x, √x, logl0x , e x, log ex );

û û Power functions

û ü Degrees

ü ü Radians appear in AS level mathematics specification

ü ü Exponential functions

û û Logarithmic functions

(e) +, -, x, ÷. L1 to L7 (due to number of decimal places and significant figures)

ü ü

2 Handling data: NC levels F H Comment

(a) use an appropriate number of significant figures;

L8 ü ü

(b) find arithmetic means; L7 to EP (involves % calculation and inverse means)

ü ü

(c) construct and interpret frequency tables and diagrams, bar charts and histograms;

L4 to L6 (L4 collect and record discrete data, L6 for continuous data)

ü ü Tables, frequency tables, diagrams, bar charts

û ü Histograms

(d) use an appropriate number of decimal places.

L5 ü ü Students are expected to understand how to round off to an appropriate number of decimal places

Table 15: continued

L7 to EP (power and exponentials unlikely to be found).Theory of logs is EP. In context pupils calculate pH using calculator and minus log button or shift log button, so there are some mathematical decisions to be made.


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3 Algebra: NC levels F H Comment

(a) understand and use the symbols: =, <, <<, >>, >, , ~, μ;

L5 to L7 ü ü =, <,>,~

û û <<, >> These symbols are not used in mathematics GCSE or A-level

û ü

û û μ appears in A-level mathematics specification

(b) change the subject of an equation; L8 ü ü In Foundation level students are required to change the subject of a formula.In Higher level this includes cases where the subject is on both sides of the original formula or where the power of a subject appears.

(c) substitute numerical values into algebraic equations using appropriate units for physical quantities;

L8 to EP ü ü

(d) solve simple algebraic equations; L7 to L8 ü ü

(e) use logarithms in relation to quantities which range over several orders of magnitude.

EP (see 1(d) above)

û û

4 Graphs: NC level F H Comment

(a) translate and interpret information between graphical, numerical and algebraic forms;

L5 to L6 ü ü

(b) plot two variables from experimental or other data;

L6 ü ü

(c) understand that y = mx + c represents a linear relationship;

L8 ü ü

(d) determine the slope and intercept of a linear graph;

EP ü ü slope

û ü intercept

(e) calculate rate of change from a graph showing a linear relationship;

EP ü ü Taken to mean working out a gradient

Table 15: continued


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Table 15: continued

(f) draw and use the slope of a tangent to a curve as a measure of rate of change;

EP û û Taken out of the GCSE mathematics specification

(g) interpret a spectrum. L6 to L8 (involves identifying peaks, reading off scales, interpreting data and using tables. L6 compares to reading scatter diagrams and L8 compares to interpreting data from it)

ü ü While a spectrum is not in the mathematics specification, the skills of interpreting a spectrum are

5 Geometry and Trigonometry: NB shapes of molecules relates to drawing 2D and 3D shapes although some different notations are used in chemistry and mathematics. Mathematics can help understanding, but these questions can be done without mathematics.

F H Comment

a) appreciate angles and shapes in regular 2-D and 3-D structures;

L6 ü ü

b) visualise and represent 2-D and 3-D forms including two-dimensional representations of 3-D objects;

L6 ü ü

c) understand the symmetry of 2-D and 3-D shapes

L6 ü ü


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3.2.3.1 TYPE OF MATHEMATICS ASSESSED AT A-LEVEL CHEMISTRY

In Part A respondents were asked to consider the spread of mathematical content areas assessed within chemistry A-level. Figure 18 illustrates the percentages of respondents that considered the spread to be good, acceptable (key mathematical areas were assessed), average (limited variation in type of mathematics) and poor (restricted amount of mathematics assessed many times).

Figure 18: Opinions from the chemistry online survey respondents on the spread of mathematical content areas within a chemistry A-level





Participants were also asked to comment on the areas of mathematics they would like to feature highly in assessment, a little or not at all. These results are displayed in Figure 19. Participants were asked if there were any other areas of mathematics, not listed in the requirements that they felt should be included in the assessments. Content areas suggested by more than one respondent were: • calculus (mentioned by 20% of respondents) • logarithms • statistics • first order, second order equations • quadratic equations and • powers / manipulation of indices and probability.

Overall, participants were concerned about the levels of mathematical content in chemistry A-levels, with many feeling that students were being misled about the mathematical requirements of chemistry as a subject. Concern was also expressed that downgrading the mathematical requirements would also lead to a restriction in the chemistry content that could be assessed. Algebra, problem solving, calculus, data manipulation and units were all mentioned as areas with which many chemistry students struggled.

63% of repondents felt that awarding organisations should use a framework to ensure a broad spread of mathematical requirements are assessed and 58% felt that all requirements should be assessed over a two- or three-year cycle of the A-levels to ensure they are taught. 63% also felt that if areas were not assessed then it would mean they would not be taught. Very few (16%) felt that only key requirements should be assessed as opposed to all requirements.

In Part B, after receiving the findings from the analysis, three-quarters of respondents felt it was unacceptable that essential mathematical content areas were hardly or not at all assessed. The quarter that found it acceptable was almost exclusively made up of teachers. When comparing Figure 13 and Figure 19 it is clear there is a mismatch between mathematical content areas that are assessed and those that the science community would like to be assessed (for example, recognise and use expressions in decimal and standard form). More positively, there were some areas that the science community thought should feature highly and did in fact feature highly across all awarding organisations (for example, substituting numerical vales into algebraic equations using appropriate physical quantities). Most of the comments related to this section of the survey indicated that the missing topics from the assessment were central to chemistry.


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Furthermore, over two-thirds felt it was inappropriate that a few mathematical requirements were assessed repeatedly, rather than a spread of skills being assessed. While there was broad agreement that many key skills would and should be assessed repeatedly as they are linked to different content areas of chemistry, they felt that there was an absence of some other important skills and there was a danger that the repetition encourages more rote learning rather than a deeper understanding of the material. It was felt that a broader range of mathematical skills needed to be assessed to check more in-depth understanding, to give a more meaningful assessment of students’ skills and to show the skills of the more able students. Many felt that a chemistry examination should only be concerned with chemistry content and that the mathematics should be assessed in the context of chemistry.

EXTENT OF MATHEMATICS In Part A 56% of respondents felt that the amount of mathematics in the paper was not enough to adequately prepare for progression to higher education in chemistry or a related subject. However, after viewing the findings of the analysis, the majority (88%) thought that the percentage of question parts containing mathematics was appropriate.

Respondents were told that in the analysis it was found that the percentage of marks that required mathematics varied across awarding organisations between 24% and 43%. 60% of the respondents thought the lower limit of marks was too low and there was no consensus reached on the higher limit.

No consensus was found on whether the mathematics in the theory papers and the mathematics in the practical papers should be different. However, there was some consensus when asked their reasons for their opinions. It was widely agreed that some types of mathematics were more suitable for use in the practical papers, for example, data handling, processing experimental results, measurement and in-depth analysis and that the assessments should draw on appropriate mathematics to support the chemistry content. It was thought to be important across the whole course to use and apply appropriate mathematics skills to appreciate the chemistry. It was felt that mathematical content should not be stipulated for inclusion in either theory or practical papers as that would lead to convoluted questions. Others felt that linking certain mathematical concepts to only one type of paper could lead to compartmentalisation of the mathematics and others felt the mathematics should be found in both types of paper so that students could apply their knowledge in a variety of settings.

Figure 19: Mathematical requirement areas that chemistry survey respondents would like to feature highly in assessment, a little in assessment or not at all

1a 1b 1c 1d 1e 2a 2b 2c 2d 3a 3b 3c 3d 3e 4a 4b 4c 4d 4e 5a 5b 5c

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DIFFICULTY OF MATHEMATICS In Part A 100% of the representatives from higher education felt that it did not matter if the level of difficulty was perceived to go up due to the A-levels containing more mathematics, while only 71% of respondents overall agreed with this.

In Part A, when asked about difficulty in terms of the number of steps in the calculations on the paper, most respondents felt that the balance on the sample paper seemed appropriate, with quite a number of respondents (mostly from HE) feeling that it was not an important issue. In Part B the participants were told that the analysis found that the majority of calculations were single step. 42% felt that there should be an even spread of all three types of calculation and nearly half of the respondents felt that there should be more multiple step calculations (42%) and more extended calculations (34%)16.

When considering context as a measure of difficulty, three-quarters of the respondents in Part A felt that the number of questions with mathematical content set in a familiar context was appropriate. In Part B respondents were told that the vast majority of the mathematics in the A-levels was set in contexts typically met through the learning programme, that is, it was set in a familiar context (92%-100%). Two-thirds of respondents felt that there should be less mathematics set in familiar contexts so that students would have more experience of applying mathematics in unfamiliar situations; thus the majority wanted an increase in difficulty in terms of familiarity of context. Respondents were asked whether or not the familiarity of the context in the majority of the assessments was creating a problem with progression to higher education or industry and students’ ability to apply mathematics in a novel situation. Two-thirds did feel that this was causing a problem.

Respondents were asked to make a judgement on the mathematical difficulty in terms of complexity in the paper. In Part A no respondents thought that the complexity was too difficult and there was an even split between those who thought it was too easy and those who thought that it was

appropriate. When considering a set of questions, two thirds felt that the questions were appropriate in terms of complexity while the remaining third felt it was too easy. In Part B, when respondents were told that the vast majority of questions across awarding organisations required Level 1 and Level 2 complexity (i.e. use of straightforward and familiar concepts or required application of one domain of mathematics), 72% of the participants felt that the recall of common mathematical procedures should be assessed alongside procedures that involve application in one or more content areas, that is that the difficulty should be increased in terms of complexity.

COMPARABILITY ACROSS AWARDING ORGANISATIONSIn Part A 100% of respondents agreed that it was important that the A-levels from all of the awarding organisations had the same level of difficulty in terms of the mathematical content. 37% of respondents also agreed that it was important that all of the awarding organisations assessed the same mathematical content areas, with 61% thinking only that it was important that the same key areas were assessed by all of the awarding organisations. The vast majority (97%) thought that the proportion of questions with mathematical content should be similar across the awarding organisations. 63% of respondents felt that awarding organisations should use a framework to ensure that a broad spread of mathematical requirements is assessed. Lastly, there were requests that the mathematical requirements across all of the awarding organisations should be the same.

In Part B most respondents (89%) felt that the differences across awarding organisations in the proportion of the marks at A-level that are for mathematical content were not acceptable.

COHERENCE BETWEEN MATHEMATICS AND THE SCIENCES Respondents were asked which was the most important feature of the mathematics in chemistry A-levels: proportion, appropriateness or difficulty of mathematical content. 83% of the respondents felt that the appropriateness of the mathematical content was the most significant, with 14%

16 Some respondents supported both statements so these percentages do not necessarily represent two different findings.


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of respondents feeling that the difficulty of the mathematics was the most important aspect.

All respondents felt that chemistry A-levels should contain mathematics that is not found in current mathematics GCSE. A third of respondents agreed that mathematics GCSE should be adequate preparation for the mathematics in a science A-level and two-thirds agreed that the introduction of a new mathematics qualification for use alongside chemistry A-levels would support the students.

3.3 BIOLOGY


3.3.1.1 EXTENT

This measure seeks to capture ‘how much’ mathematics is in the biology A-level assessments (independent of the type, appropriateness or difficulty). It is quantified by the number of questions and question parts within a complete A-level that require mathematics and the number of marks within those questions requiring mathematics. Table 16a shows the percentage of question parts containing mathematics within each unit and the percentage of question parts containing mathematics for theory only and practical only papers. Table 16b takes into account the weighting of theory papers (80%) and practical papers (20%) to calculate the average percentage of question parts containing mathematics in a complete A-level. Figure 20 illustrates the percentage of question parts in a complete A-level containing mathematics for the five awarding organisations.


A B C D E

AS units 1 and 2 19 23 11 18 4

AS unit 3 59 0 55 100 53

A2 units 4 and 5 27 7 3 22 3

A2 unit 6 58 50 50 88 71



Table 16b: Percentage of question parts weighted to take account of the theory component (80%) and practical component (20%) of the A-level assessments

A B C D E



Total A-level 30 17 17 35 15


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Table 17a shows the percentage of marks requiring mathematics for each unit and for theory only and practical only papers. Table 17b takes into account the weighting of theory papers (80%) and practical

papers (20%) to calculate the average percentage of marks requiring mathematics in a complete A-level. Figure 21 illustrates this as a graph.

Figure 20: Percentage of question parts in a complete A-level containing mathematics for the five awarding organisations

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Figure 21: Percentage of marks requiring mathematics in a complete A-level for the five awarding organisations

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AS units 1 and 2 16 22 10 15 3

AS unit 3 46 0 38 56 41

A2 units 4 and 5 21 5 4 12 3

A2 unit 6 48 48 40 45 68




A B C D E



Total A-level 24 16 14 21 13


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3.3.1.2 TYPE

The number of occurrences of each mathematical requirement listed in Biology A-level was measured to identify the type of mathematics assessed and the frequency of each type of mathematics assessed. The results are displayed in Figure 22.

1 Arithmetic and numerical computation:(a recognise and use expressions in decimal and

standard form; (b) calculate or use ratios, fractions and percentages; (c) make estimates of the results of calculations (without

using a calculator); (d) use calculators to find and use mean, standard

deviations, power, exponential and logarithmic functions;(e) use calculations involving simple arithmetic and algebraic

transformations:(f) understand and use correlations;(g) +, -, x, ÷.

2 Handling data: (a) use an appropriate number of significant figures; (b) find arithmetic means; (c) construct or interpret tables, frequency tables and

diagrams, bar charts and histograms; (d) understand simple probability; (e) understand the principles of sampling as applied to

scientific data; (f) understand the terms mean, median and mode and

standard deviation; (g) use a scatter diagram to identify positive and negative

correlation between two variables; (h) select and use a simple statistical test; (i) make order of magnitude calculations;(j) determine and interpret population variance, standard

deviation and standard deviation (error) of the mean;

Figure 22: The number of occurrences for each mathematical requirement in a full suite of examination papers for a complete biology A-level in each of the awarding organisations. It has no relation to the number of marks awarded for each mathematical requirement.

1a 1b 1c 1d 1e 1f 1g 2a 2b 2c 2d 2e 3a 3b 3c 3d 3e 4a 4b 4c 4d 4e 5a 5b

Arithmetic and Computation

Handling Data Algebra

Geometry andTrigonometry

Graphs

Mathematical Requirements for Biology A-Level

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(k) understand probability in order to understand how genetic ratios arise;

(l) frame null hypothesis.

3 Algebra: (a) change the subject of an equation; (b) substitute numerical values into algebraic equations

using appropriate units for physical quantities; (c) understand the use of logarithms in relation to quantities

that range over several orders of magnitude;(d) derive an equation;(e) =, <, >.

4 Graphs: (a) translate information between graphical, numerical and

algebraic forms; (b) plot two variables from experimental or other data; (c) calculate rate of change from a graph showing a linear

relationship;(d) draw and use the slope of a tangent to a curve as a

measure of rate of change;(e) construct and / or interpret line graphs.

5 Geometry:(a) visualise three dimensional forms from two dimensional

representations of three dimensional objects;(b) calculate circumferences and areas of circles, surface

areas and volumes of regular blocks and cylinders when provided with appropriate formulae.



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3.3.1.3 DIFFICULTY

For each of the following measures, percentages are stated as a proportion of the number of question parts containing mathematics. The percentages do not relate in any way to the number of marks awarded for mathematical understanding.

NUMBER OF STEPSThe number of steps involved in a calculation was used as one measure of difficulty, based on the assumption that questions containing mathematics that required multiple step or extended calculation (e.g. value x had to be found and used in a subsequent calculation in order to find the solution to the problem, y) were more difficult than single step calculations, as they require students to use higher order skills and extended reasoning. Appendix 5a shows an example of each type of calculation.


Table 18a: Percentage of single, multiple and extended step calculations in biology A-level, calculated as a percentage of question parts identified as containing mathematics

A B C D E


AS units 1 and 2 0 13 88 0 31 69 17 29 54 6 35 60 0 0 100

AS unit 3 2 12 86 0 0 0 17 17 67 0 40 60 0 18 82

A2 units 4 and 5 0 9 92 0 88 13 0 50 50 7 62 32 50 50 0

A2 unit 6 21 12 67 20 53 28 8 46 46 13 43 43 7 15 78



Table 18b: Percentage of single, multiple and extended step calculations in biology A-level, calculated as a percentage of the total number of question parts in a complete A-level

A B C D E


AS units 1 and 2 0 2 17 0 7 16 2 3 6 1 6 11 0 0 4

AS unit 3 1 7 51 0 0 0 9 9 37 0 40 60 0 10 43

A2 units 4 and 5 0 2 25 0 6 1 0 2 2 2 14 7 2 2 0

A2 unit 6 12 7 39 10 27 14 4 23 23 11 38 38 5 11 55




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Table 18c: Percentage of single, multiple and extended step calculations in biology A-level, weighted to take account of the theory component (80%) and practical component (20%) of the A-level assessments

A B C D E



0 2 17 0 5 7 1 2 3 1 8 7 1 1 2


1 1 9 1 3 1 1 3 6 1 8 10 0 2 10


1 3 26 1 8 8 2 5 9 2 16 17 1 3 12

Figure 23a: Percentage of mathematical question parts identified as containing single, multiple and extended step calculations in a complete A-level for each of the five awarding organisations

Figure 23b: Percentage of mathematical question parts identified as containing single, multiple and extended step calculations in theory examination papers for biology A-level

Figure 23c: Percentage of mathematical question parts identified as containing single, multiple and extended step calculations in practical examination papers for biology A-level

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Table 19a shows the percentage of mathematical question parts classified as Level 1, 2 or 3 complexity18 within each A-level unit and as an

average for theory only and practical only papers. These figures are calculated as a percentage of the question parts identified as containing mathematics. Table 19b shows these percentages of the total number of question parts and Table 19c takes into account the weighting of theory papers (80%) and practical papers (20%) to calculate the average percentage of Level 1, 2 and 3 complexity in a complete A-level. Figure 24a illustrates the percentage of Level 1, 2 and 3 complexity type calculations in a complete A-level for each of the five awarding organisations. Figures 24b and 24c illustrate these percentages for theory only and practical only papers respectively.


Table 19a: Percentage of Level 1, 2 and 3 complexity type calculations in biology A-level, calculated as a percentage of question parts identified as containing mathematics

A B C D E

Complexity 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1

AS units 1 and 2 0 0 100 0 2 99 0 0 100 0 0 100 0 0 100

AS unit 3 0 0 100 0 0 0 0 0 100 0 0 100 0 11 89

A2 units 4 and 5 0 7 94 0 0 100 0 0 100 0 0 100 0 0 100

A2 unit 6 0 0 100 0 17 84 0 33 66 0 0 100 0 0 100



Table 19b: Percentage of Level 1, 2 and 3 complexity type calculations in biology A-level, calculated as a percentage of the total number of question parts in a complete A-level

A B C D E

Number of steps 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1

AS units 1 and 2 0 0 19 0 0 23 0 0 11 0 0 18 0 0 4

AS unit 3 0 0 59 0 0 0 0 0 55 0 0 100 0 6 47

A2 units 4 and 5 0 2 25 0 0 7 0 0 3 0 0 22 0 0 3

A2 unit 6 0 0 58 0 9 42 0 17 33 0 0 88 0 0 71




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Table 19c: Percentage of Level 1, 2 and 3 complexity type calculations in biology A-level, weighted to take account of the theory component (80%) and practical component (20%) of the A-level assessments

A B C D E

Number of steps 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1


0 1 18 0 0 12 0 0 6 0 0 16 0 0 3


0 0 12 0 1 4 0 2 9 0 0 19 0 1 12


0 1 30 0 1 16 0 2 15 0 0 35 0 1 15

Figure 24a: Percentage of mathematical question parts identified as containing Level 1, Level 2 and Level 3 complexity type calculations in a complete A-level for each of the five awarding organisations

Figure 24b: Percentage of mathematical question parts identified as containing Level 1, Level 2 and Level 3 complexity type calculations in theory examination papers for biology A-level

Figure 24c: Percentage of mathematical question parts identified as containing Level 1, Level 2 and Level 3 complexity type calculations in practical examination papers for biology A-level

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Table 20a shows the percentage of mathematical question parts classified as Level 1, 2 or 3 context within each A-level unit and as an average for theory only and practical only papers.

These figures are calculated as a percentage of the question parts identified as containing mathematics. Table 20b shows these percentages of the total number of question parts and Table 20c takes into account the weighting of theory papers (80%) and practical papers (20%) to calculate the average percentage of Level 1, 2 and 3 context in a complete A-level. Figure 25a illustrates the percentage of Level 1, 2 and 3 context calculations in a complete A-level for each of the five awarding organisations. Figures 25b and 25c illustrate these percentages for theory only and practical only papers respectively.

Table 20a: Percentage of Level 1, 2 and 3 context type calculations in biology A-level, calculated as a percentage of question parts identified as containing mathematics

A B C D E

Context 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1

AS units 1 and 2 0 0 100 0 2 99 0 0 100 0 0 100 0 0 100

AS unit 3 66 0 34 0 0 0 90 0 10 88 0 12 67 0 33

A2 units 4 and 5 0 7 94 0 0 100 0 0 100 0 0 100 0 0 100

A2 unit 6 69 0 31 78 0 23 83 0 17 100 0 0 66 0 44



Table 20b: Percentage of Level 1, 2 and 3 context type calculations in biology A-level, calculated as a percentage of the total number of question parts in a complete A-level

A B C D E

Number of steps 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1

AS units 1 and 2 0 0 19 0 0 23 0 0 11 0 0 18 0 0 4

AS unit 3 39 0 20 0 0 0 50 0 6 88 0 12 36 0 17

A2 units 4 and 5 0 2 25 0 0 7 0 0 3 0 0 22 0 0 3

A2 unit 6 40 0 18 39 0 12 42 0 9 88 0 0 47 0 31




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Table 20c: Percentage of Level 1, 2 and 3 context type calculations in biology A-level, weighted to take account of the theory component (80%) and practical component (20%) of the A-level assessments

A B C D E

Number of steps 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1


0 1 18 0 0 12 0 0 6 0 0 16 0 0 3


8 0 4 4 0 1 9 0 1 18 0 1 8 0 5


8 1 22 4 0 13 9 0 7 18 0 17 8 0 8


Figure 25b: Percentage of mathematical question parts identified as containing calculations set in a Familiar, with Some Novel Aspects, and Unfamiliar context calculations in theory examination papers for biology A-level

Figure 25c: Percentage of mathematical question parts identified as containing calculations set in a Familiar, with Some Novel Aspects, and Unfamiliar context in practical examination papers for biology A-level

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STRUCTURAL OR TAGGED ON One of the measures was to ascertain whether the mathematics was a structural part of the question or whether the mathematics was tagged on to the question. All of the question parts with mathematical content were judged to be a structural part of the question.

MATHEMATICS SKILLS OR SCIENTIFIC COMPREHENSIONMathematical question parts within a full suite of A-level examinations were measured against the extent to which scientific comprehension was required to achieve the full marks. If a question part required no scientific comprehension to acquire the full marks it was classified as no scientific comprehension (mathematical skill only), if part of the marks required scientific comprehension in addition to

mathematical skill it was classified as some scientific comprehension and question parts where all marks required scientific comprehension were classified as scientific comprehension. Appendix 5d shows an example of each category.


Table 21a: Percentage of mathematical question parts classified as all mark (S), some marks (B) and no marks (M) requiring scientific comprehension in biology A-level, calculated as a percentage of question parts identified as containing mathematics

A B C D E



AS units 1 and 2 40 7 54 0 25 76 0 75 25 0 53 47 0 75 25

AS unit 3 0 41 59 0 0 0 0 25 75 0 100 0 6 41 53

A2 units 4 and 5 0 71 29 0 13 88 0 100 0 0 76 24 0 0 100

A2 unit 6 0 43 57 0 81 20 0 23 77 0 100 0 52 0 48




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Table 21b: Percentage of mathematical type question parts where all marks (S), some marks (B) or no marks (M) require scientific comprehension in biology A-level, calculated as a percentage of the total number of question parts in a complete A-level

A B C D E

Number of steps 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1

AS units 1 and 2 8 1 10 0 6 17 0 8 3 0 10 8 0 3 1

AS unit 3 0 24 35 0 0 0 0 14 41 0 100 0 3 22 28

A2 units 4 and 5 0 19 8 0 1 6 0 3 0 0 17 5 0 0 3

A2 unit 6 0 25 33 0 41 10 0 12 39 0 88 0 37 0 34



Table 21c: Percentage of mathematical type question parts where all marks (S), some marks (B) and no marks (M) require scientific comprehension in biology A-level, weighted to take account of the theory component (80%) and practical component (20%) of the A-level assessments

A B C D E

Number of steps 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1




3 13 14 0 7 10 0 7 8 0 30 5 4 3 8

Figure 26a: Percentage of mathematical question parts requiring scientific comprehension in addition to mathematical skill to achieve all marks, some of the marks or no marks in a complete A-level for each of the five awarding organisations

Figure 26b: Percentage of mathematical question parts requiring scientific comprehension in addition to mathematical skill to achieve all marks, some of the marks or no marks in theory examination papers for biology A-level

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3.3.2 PHASE 2 – BIOLOGY A-LEVEL IN COMPARISON WITH GCSE MATHEMATICS AND NATIONAL CURRICULUM LEVEL DESCRIPTORS

Two comparisons were used to establish the coherence of biology A-levels and the mathematics accessed up to Key Stage 4: comparison with National Curriculum level descriptors and comparison with the 2012 mathematics GCSE specification. These comparisons are displayed in Table 22.

Figure 26c: Percentage of mathematical question parts requiring scientific comprehension in addition to mathematical skill to achieve all marks, some of the marks or no marks in practical examination papers for biology A-level

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Mathematical requirements as listed in the Biology A-level specification


Comment

1 Arithmetic and numerical computation:

F H

(a) recognise and use expressions in decimal and Standard Form;

L4 to L8 (L4 decimals to L8 standard form)

ü ü Decimals

û ü Standard Form: this will be taken off the new 2012 foundation level specification

(b) calculate or use ratios, fractions and percentages;

L5 to L7 ü ü Percentages: reverse percentages are exclusive to higher level

ü ü Ratios

ü ü Fractions

(c) make estimates of the results of calculations (without using a calculator);

L5 to 6 (NB students will always have a calculator)

ü ü

Table 22: Comparison of mathematical requirements for biology A-level with mathematics found in the National Curriculum Level Descriptors and GCSE mathematics specification



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Table 22: continued

(d) use calculators to find and use mean, standard deviations, power, exponential and logarithmic functions;

L4 to EP (L4 mode, L5 mean, EP deviations. Exponential and logarithmic functions not found in biology)

ü ü Mean

û û Standard deviation: recently removed from GCSEs. Now on AS level mathematics specification

û ü Power functions: students are required to draw graphs of these functions

û ü Exponential functions: students are required to draw graphs of these functions

û û Logarithmic functions: On AS mathematics specification

(e) use calculations involving simple arithmetic and algebraic transformations:

L3 to L4 (depending on the size of the number. Algebraic transformations not found in biology.)

ü ü

(g) +, -, x, ÷. L1 to L7 (depending on the nature of the numbers)

ü ü

2 Handling data: NC levels F H Comments

(a) use an appropriate number of significant figures;

L7 to L8 ü ü

(b) find arithmetic means; L5 to L7 (L5 for discrete data to L7 for grouped data)

ü ü


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(c) construct or interpret tables, frequency tables and diagrams, bar charts and histograms;

L4 to EP (L4 for bar charts, L4 for frequency tables and EP for histograms.)

ü ü Tables, frequency tables, diagrams, bar charts

û ü Histograms

(d) understand simple probability; L5 to L7 (L5 in theory paper, L7 in practicals and coursework)

ü ü Taken to mean only 1 event taking place

(e) understand the principles of sampling as applied to scientific data;

EP û ü Random and stratified

(f) understand the terms mean, median and mode and standard deviation;

L5 to EP ü ü Mean

ü ü Median

ü ü Mode

û û Standard Deviation

(g) use a scatter diagram to identify positive and negative correlation between two variables;

L6 ü ü

(h) select and use a simple statistical test;

EP û ü Taken to mean choosing a sampling technique

(i) make order of magnitude calculations;

L5 to EP (depending on context)

û û Not a concept taught in mathematics. Compare with 1c) estimation techniques

(j) determine and interpret population variance, standard deviation and standard deviation (error) of the mean;

EP û û Appears in A-level mathematics specification

(k) understand probability in order to understand how genetic ratios arise;

L6 to EP ü ü Single events

û ü Joint (i.e. 2 or 3) events

(l) frame null hypothesis. EP û û Appears in A-level mathematics specification

Table 22: continued


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3 Algebra: NC levels F H Comments

(a) change the subject of an equation; L8 ü ü In Foundation level students are required to change the subject of a formula. In Higher level this includes cases where the subject is on both sides of the original formula or where the power of a subject appears.

(b) substitute numerical values into algebraic equations using appropriate units for physical quantities;

L4 to L5 (L5 for numbers in the thousands.)

ü ü

(c) understand the use of logarithms in relation to quantities that range over several orders of magnitude;

EP û û This appears in AS level mathematics

(d) derive an equation. L5 to L6 ü ü

(e) =, <, > L4 ü ü

4 Graphs: NC level F H Comments

(a) translate information between graphical, numerical and algebraic forms;

L3 to L6 (L3 for bar graphs to L6 for line graphs.)

ü ü

(b) plot two variables from experimental or other data;

L3 to L6 (As above in 4(a), but will usually be a line graph in biology.)

ü ü

(c) calculate rate of change from a graph showing a linear relationship;

EP ü ü Taken to mean working out a gradient

(d) draw and use the slope of a tangent to a curve as a measure of rate of change;

EP û û This was previously on the GCSE specification

(e) construct and / or interpret line graphs (straight or curved).

L6 ü ü

Table 22: continued


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3.3.3.1 TYPE OF MATHEMATICS ASSESSED AT A-LEVEL BIOLOGY

In Part A respondents were asked to consider the spread of mathematical content areas assessed within biology A-level. Figure 27 illustrates the percentages of respondents that considered the spread to be good, acceptable (key mathematical areas were assessed), average (limited variation in type of mathematics) and poor (restricted amount of mathematics assessed many times).

Participants were also asked to comment on the areas of mathematics they would like to feature highly in assessment, a little or not at all. These results are displayed in Figure 28. Participants were also asked if there were any other areas of mathematics not listed in the requirements that they felt should be included in the assessments. The only area that was mentioned more than once was converting between different units.

The main concern voiced by participants was the lack of assessment of statistics, with others expressing concern that mathematics should always be taught within a biology context. The lack of alignment with mathematics courses also worried some respondents, since this would mean some students would not have the mathematical skills to access some of the biology content, and this was also expressed as a query about whether there should be a minimum requirement set for biology A-level in terms of grade achieved at GCSE mathematics.

In Part A, prior to the findings of the analysis, 74% of respondents felt that awarding organisations should use a framework to ensure that a broad spread of mathematical requirements are assessed and 52% felt that all requirements should be assessed over a 2- year or 3- year cycle of the A-levels to ensure they are taught. A third of respondents felt that if areas are not assessed then it would mean that area would not be taught at all. No respondents felt that only key requirements should definitely be assessed as opposed to all requirements.

In Part B participants were told that some of the mathematical requirements were well-covered in the A-levels, for example, interpreting frequency

5 Geometry: NC levels F H Comment

(a) visualise three dimensional forms from two dimensional representations of three dimensional objects;

L6 ü ü

(b) calculate length, circumferences and areas of circles, surface areas and volumes of regular blocks and cylinders when provided with appropriate formulae

L4 to EP (L4 for length, L6 for circumference to EP surface area of cylinders.)

ü ü Circles

ü ü Surface area for cuboids

û ü Surface area for cylinders

Table 22: continued

Figure 27: Opinions from the biology online survey respondents on the spread of mathematical content areas within a biology A-level






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tables, diagrams, histograms and bar charts. Other areas were hardly assessed or not assessed at all. There was little difference between those who felt that this was appropriate and those that did not. Some respondents thought that greater breadth in assessments promoted greater breadth in teaching and there were concerns that if content areas were not assessed they would not be taught. When comparing Figure 22 and Figure 28 it is clear there is a mismatch between mathematical content areas that are assessed and those that the science community would like to be assessed (for example, plotting two variables from data and using logarithms in relation to quantities which range over several orders of magnitude).

Some felt that it was not necessary to assess all mathematical content areas in all papers or that it was not necessary to assess all areas every year while others felt that if it was in the specification then it should be assessed.

Still considering the type of content areas that were assessed, respondents were told that a few mathematical requirements were assessed repeatedly, rather than a spread of skills being assessed. 81% of the respondents did not think that this was appropriate. It was reported that students needed a broad mathematical capability and therefore a broad range should be assessed. It was felt that, while the key requirements were the most important skills to be assessed, the majority

of requirements were relevant so the majority of requirements should be assessed. It was also thought that a range of mathematical requirements were necessary for future study or jobs.

3.3.3.2 EXTENT OF MATHEMATICS

In Part A half of the respondents felt that the amount of mathematics in the paper was not enough to adequately prepare for progression to higher education in biology or a related subject. However, after viewing the findings of the analysis, 70% of respondents thought that the percentage of question parts containing mathematics was appropriate (39-54% of question parts across awarding organisations). Respondents were told that the percentage of marks that required mathematics varied across awarding organisations from 13% to 25%. The most common opinion on these results was that differences like this across awarding organisations were not acceptable (77%). 48% of respondents felt that 13% was too low and no census was reached on whether the upper limit was too low.

In Part A, when asked whether the mathematical content in the theory papers and in the practical papers should be different, nearly two-thirds of respondents felt that it should be different. It was felt that mathematical requirements were appropriate to the topic and that the papers lent themselves to different skills. Many cited examples

Figure 28: Mathematical requirement areas that biology survey respondents would like to feature highly in assessment, a little in assessment or not at all

1a 1b 1c 1d 1e 1f 1g 2a 2b 2c 2d 2e 2f 2g 2h 2i 2j 2k 2l 3a 3b 3c 3d 3e 4a 4b 4c 4d 4e 5a 5b

Mathematical Requirements for Biology A-Level

0

5

10

15

20

25

30

Num

ber o

f Res

pond

ents

Feature Not At AllFeature a LittleFeature Highly


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of how practical papers would lead to more analysis and manipulation of data rather than interpretation, which would be assessed in the theory papers. They felt that the practical work was an opportunity for more in-depth mathematics. They also reported that if it was different that it would allow for a greater range of mathematical skills to be assessed and it would give rise to less repetition. Those who felt that the mathematics should be the same in the practical papers felt most strongly that the key mathematical skills should be tested in both types of assessment. In Part B the majority also did not think it was important if the marks were awarded in the theory or practical papers.

3.3.3.3 DIFFICULTY OF MATHEMATICS

In Part A of the survey nearly 60% of respondents felt that it did not matter if the level of difficulty of biology A-levels was perceived to go up due to them containing more mathematics. There was a split in opinions from all three participant groups.

When asked about difficulty in terms of the number of steps in the calculations on the paper, the most popular opinion (41%) was that the balance on the sample paper was appropriate. In Part B the participants were told that the majority of calculations across the examinations papers were single step, followed by multiple step, with extended calculations the least likely type to be found. About a third of respondents felt that there should be more multiple step and extended calculations19 and in total 48% of respondents felt there should be an even spread of the three types of calculation.

In Part A when considering context as a measure of difficulty, around two-thirds of the teachers felt that the number of questions with mathematical content set in a familiar context was appropriate. There was no agreement between the higher education and professional body representatives. In Part B respondents were told that in the analysis

of the papers it was found that between 53% and 69%20 of the calculations were found to be set in a familiar context21. Nearly half of the respondents thought that this was about the right amount of mathematics to be set in a familiar context, with the other half agreeing that there should be less mathematics set in familiar contexts so that students had more experience of applying mathematics in unfamiliar situations. However, fewer than half of the respondents felt that this familiarity of context was creating a problem with progression to higher education or industry and students’ ability to apply mathematics in novel situations.

Respondents were asked to make a judgement on the mathematical difficulty in terms of complexity in the paper. In Part A just over half (59%) thought that the complexity was appropriate, with the remainder considering the complexity to be insufficient. In Part B, nearly three-quarters of participants thought that the recall of common mathematical procedures should be assessed alongside procedures that involved application in one or more mathematical content areas; however, the analysis of the assessments showed that the vast majority of the mathematics required the use of only recall of procedures and relatively straightforward application.

3.3.3.4 COMPARABILITY ACROSS AWARDING ORGANISATIONS

In Part A all of the respondents agreed that it was important that the A-levels from all of the awarding organisations had the same level of difficulty in terms of the mathematical content. When asked about awarding organisations assessing the same mathematical content areas, 63% felt that only the same key areas needed to be assessed, with 30% thinking that all of the mathematical content areas should be assessed in the same way. The vast majority (93%) thought that the proportion of questions with mathematical content should be similar across the awarding organisations. 74%

19 Some respondents supported both statements so these percentages do not represent two different findings.

20 These figures are the average of AS and A2 papers and do not take into account the weighting of the theory and practical papers.

21 There was no comment on the finding that almost 100% of the theory papers contained mathematics set in familiar contexts.


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of respondents felt that awarding organisations should use a framework to ensure a broad spread of mathematical requirements are assessed.

In Part B, 78% of respondents felt that the difference across awarding organisations in the proportion of marks at A-level for mathematical content was not acceptable.

3.3.3.5 COHERENCE BETWEEN MATHEMATICS AND THE SCIENCES

Respondents were asked which was the most important feature of the mathematics in biology A-levels: the proportion of the paper that contained mathematical content, the appropriateness of the mathematical content covered in the assessments or the difficulty of the mathematics in the assessments. Out of these choices, 89% agreed that the appropriateness of the mathematical content covered in the assessments was the most important.

Two-thirds of participants agreed that mathematics GCSE should be adequate preparation for the mathematics in a science A-level rather than students needing the support of a separate mathematics qualification. It was felt that the mathematics GCSE should be adequate preparation to build on, although the mathematics in the biology A-level would go beyond it in terms of the application of the mathematics in novel situations.

3.4 VIEWS FROM INDUSTRY

3.4.1 TYPE OF MATHEMATICS ASSESSED WITHIN SCIENCE A-LEVEL

There were concerns that the mathematical elements did not properly prepare students for higher education, including the fact that the mathematics needed for engineering was only found within the further mathematics A-level. There was also a feeling that there was grade inflation in the current science A-levels, that there was low quality in the mathematical content and that there was a problem distinguishing between the (mathematical) ability of an A and an A* graded science student.

The industry representatives were asked which mathematical content areas from a given list they felt were important in the A-level science assessments. All areas listed (and shown below) were felt to be important, with the subsequent comments clarifying their opinions: • Arithmetic and computation – an important

foundation skill

• Handling data – a vital skill for extracting information from data

• Algebra – a crucial area, in particular rearranging equations and using generic formulae

• Geometry – less important and only necessary at a basic level

• Trigonometry – a lot of scientific work cannot be completed without trigonometry

• Graphs – important to communicate data and trends

• Application of mathematics – very important, if not vital, needs lots of practice.

Calculus was mentioned repeatedly as an important aspect of mathematics, as it was needed in so many careers. It was felt to be important for students to be able to complete calculations without a calculator so that they developed a feel for numbers and could spot mistakes. Complex numbers, vectors, risk assessment and probability were other aspects of mathematics not mentioned above that the industry representatives felt were important to A-level science students. Statistics was mentioned by half of the representatives because it is commonly used in biology, in process, design and manufacturing engineering and in the optimisation of experiments.

There was general agreement that the mathematics in the science A-levels needed to go beyond that found in mathematics GCSE and that this was not a problem. It was felt that the mathematics only needed to be there when the science demanded it, but that it was necessary to go beyond mathematics GCSE if it was to prepare students for higher education.


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3.4.2 EXTENT OF MATHEMATICS

All of the industry representatives agreed that it would not matter if the amount or difficulty of the mathematics in the science A-levels increased, even if this resulted in the A-levels being perceived to be more difficult. Reasons given included that A-levels needed to be more competitive internationally, that if the difficulty was not there that the assessments were of less use to higher education and industry and that if it deterred a student, then perhaps they should not have been taking the A-level anyway. It was recognised that, if changes were made, that more mathematics specialist support may be needed.

3.4.3 DEMAND OF MATHEMATICS

Industry representatives’ main concern regarding the mathematical element of the science A-levels was the perceived lack of fluency and lack of ability of the A-level students to use and apply their mathematical knowledge in a new situation; this was problematic as this was a skill they would be expected to use in work.

There were concerns that the mathematical elements did not properly prepare students for higher education, including the fact that the mathematics needed for engineering sat only within the further mathematics A-level. Lastly, there was a feeling that there was grade inflation in the current science A-levels, that there was low quality in the mathematical content and that there was a problem distinguishing between the (mathematical) ability of an A and an A* graded science student. Industry representatives were asked to comment on aspects of mathematical difficulty used in the Phase 1 analysis of the A-level papers. Most felt that students should be tackling mathematics in familiar and unfamiliar contexts; work contexts would be unfamiliar and unfamiliar contexts differentiated the difficulty in the assessments. In terms of complexity, most of the industry representatives reported that

they would like to see a mixture of calculations that involved straightforward recall of routine classroom procedures, those that required application of mathematics within one area of mathematics and those that required application in more than one area of mathematics. While calculations involving straightforward recall were felt to underpin some science, there was a perceived need to differentiate by including the more complex calculations and to give a greater challenge to keep students engaged. It was reiterated that the mathematics should be what was necessary to support the science, but that the science should not be ‘dumbed down’ in order to avoid the inclusion of more complex mathematics. All of the industry representatives wanted to see more multiple step and extended calculations than single step calculations.

3.4.4 PARITY ACROSS AWARDING ORGANISATIONS

Most of the industry representatives felt that there should not be differences across the awarding organisations in terms of the difficulty of the mathematics assessed, the assessed content or the amount of mathematics assessed in the science A-levels. It was felt that variety in the landscape of qualifications was not understood by industry and that employers did not want to ask which awarding organisations awarded the qualification when assessing candidates. They felt that aligning the mathematics element would remove the temptation for awarding organisations to lower standards to attract more candidates and for schools to opt for an ‘easier’ examination. One respondent felt that admissions tutors were aware of the differences across the awarding organisations and another that, if there were transparent differences across the awarding organisations in terms of mathematical content, that it may lead to more diversity when it came to recruitment.


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APPENDICES

Subject Awarding Organisation A level

Physics AQA Physics A

CCEA Physics

Edexcel Physics

OCR Physics B

WJEC Physics

Chemistry AQA Chemistry

CCEA Chemistry

Edexcel Chemistry

OCR Chemistry A

WJEC Chemistry

Biology AQA Biology

CCEA Biology

Edexcel Biology

OCR Biology (H421)

WJEC Biology

APPENDIX 1: SUMMER 2010 A-LEVEL PAPERS ANALYSED


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4. Geometry and Trigonometry(a) calculate areas of triangles, circumferences and areas of

circles, surface areas and volumes of rectangular blocks, cylinders and spheres;


(c) use sines, cosines and tangents in physical problems;(d) use sinѲ ≈ tanѲ ≈ Ѳ and cos Ѳ ≈ 1 for small Ѳ;(e) understand the relationship between degrees and radians

and translate from one to the other.

5. Graphs(a) translate information between graphical, numerical and

algebraic forms;(b) plot two variables from experimental or other data using

appropriate scales for graph plotting;(c) plot data on a log-linear graph and determine whether they

change exponentially and determine the exponent;(d) plot data on a log-log graph and decide whether data obey

a power law and determine the exponent;(e) select appropriate variables for graph plotting;(f) understand that y = mx + c represents a linear relationship

and rearrange relationships into this form where appropriate;(g) determine the slope / gradient and intercept of a linear

graph in the appropriate physical units; (h) determine the gradient of a tangent to a non-linear graph by

drawing and use the slope of the tangent as a measure of rate of change;

(i) choose by inspection a straight line or curved line which will serve as the best straight line through a set of data points presented graphically;



(l) understand and use multiplicative scales (1, 10, 100...);(m) use logarithmic plots to test exponential and power law

variations;(n) sketch simple functions including y = k/x, y = kx2 y = k/x2, y

= sinѲ, y = cosѲ, y = e-kx.(o) understand or recognise the physical significance of a

straight line passing or not passing through the origin.

1. Arithmetic and ComputationCandidates should be able to:(a) use a calculator for addition, subtraction, multiplication and

division;(b) recognise and use expressions in decimal form; (‘standard

from’ deleted from this requirement and recorded separately as 1(h) to illustrate how commonly each occurred.)

(c) use ratios, fractions and percentages;(d) use calculators to find and use xn, 1/x, x2, √x, logl0x ,

e x, log ex;(e) use calculators to handle sinѲ, cosѲ, tanѲ, sin-1Ѳ, cos-1Ѳ,

tan-1Ѳ when Ѳ is expressed in degrees or radians.(f) recognise and use SI prefixes 10-12, 10-9, 10-6, 10-3, 103,

106 and 109


(h) standard form.

2. Handling data(a) show an awareness of the order of magnitude of physical

quantities and make order of magnitude calculations;(b) use an appropriate number of significant figures;(c) find arithmetic means and medians;(d) express changes as percentages and vice versa;(e) understand and use logarithmic scales in relation to

quantities which range over several orders of magnitude.

3. Algebra (a) change the subject of an equation by manipulation of the

terms, including positive and negative, integer and fractional indices and square roots






APPENDIX 2: MATHEMATICAL REQUIREMENTS FOR PHYSICS A-LEVEL22

22 During the course of the research it was established that the awarding organisations adapted the mathematical requirements set by Ofqual. This list represent an amalgamation of the mathematical requirements used by the awarding organisations, linked closely to that set by Ofqual. The text in red represents additions and changes made to the list during the analysis process in order to more accurately capture the mathematical content.


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APPENDIX 3: MATHEMATICAL REQUIREMENTS FOR CHEMISTRY A-LEVEL22

1 Arithmetic and numerical computation:(a) recognise and use expressions in decimal and

standard form;(b) use ratios, fractions and percentages;(c) make estimates of the results of calculations

(without using a calculator);(d) use calculators to find and use power, exponential

and logarithmic functions (xn, 1/x, √x, logl0x , e x, log ex);(e) +, -, x, ÷.

2 Handling data:(a) use an appropriate number of significant figures;(b) find arithmetic means;(c) construct and interpret frequency tables and diagrams,

bar charts and histograms;(d) use an appropriate number of decimal places.

3 Algebra:(a) understand and use the symbols: =, <, <<, >>, >, ,

~, μ;(b) change the subject of an equation;(c) substitute numerical values into algebraic equations

using appropriate units for physical quantities;(d) solve simple algebraic equations;(e) use logarithms in relation to quantities which range

over several orders of magnitude.

4 Graphs:(a) translate and interpret information between graphical,

numerical and algebraic forms;(b) plot two variables from experimental or other data;(c) understand that y = mx + c represents a linear relationship;(d) determine the slope and intercept of a linear graph;(e) calculate rate of change from a graph showing a linear

relationship;(f) draw and use the slope of a tangent to a curve as a

measure of rate of change;(g) interpret a spectrum.

5 Geometry and trigonometry:a) appreciate angles and shapes in regular 2-D and 3-D

structures;b) visualise and represent 2-D and 3-D forms including

two-dimensional representations of 3-D objects;c) understand the symmetry of 2-D and 3-D shapes.


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APPENDIX 4: MATHEMATICAL REQUIREMENTS FOR BIOLOGY A-LEVEL22

1 Arithmetic and numerical computation:(a recognise and use expressions in decimal and

standard form; (b) calculate or use ratios, fractions and percentages; (c) make estimates of the results of calculations

(without using a calculator); (d) use calculators to find and use mean, standard

deviations, power, exponential and logarithmic functions;(e) use calculations involving simple arithmetic and

algebraic transformations:(f) understand and use correlations;(g) +, -, x, ÷.

2 Handling data: (a) use an appropriate number of significant figures; (b) find arithmetic means; (c) construct or interpret tables, frequency tables and

diagrams, bar charts and histograms; (d) understand simple probability; (e) understand the principles of sampling as applied

to scientific data; (f) understand the terms mean, median and mode and

standard deviation; (g) use a scatter diagram to identify positive and negative

correlation between two variables; (h) select and use a simple statistical test; (i) make order of magnitude calculations;(j) determine and interpret population variance, standard

deviation and standard deviation (error) of the mean;(k) understand probability in order to understand how

genetic ratios arise;(l) frame null hypothesis.

3 Algebra: (a) change the subject of an equation; (b) substitute numerical values into algebraic equations

using appropriate units for physical quantities; (c) understand the use of logarithms in relation to quantities

that range over several orders of magnitude;(d) derive an equation;(e) =, <, >.

4 Graphs: (a) translate information between graphical, numerical and

algebraic forms; (b) plot two variables from experimental or other data; (c) calculate rate of change from a graph showing a linear

relationship;(d) draw and use the slope of a tangent to a curve as a

measure of rate of change;(e) construct and / or interpret line graphs.

5 Geometry:(a) visualise three dimensional forms from two dimensional

representations of three dimensional objects;(b) calculate circumferences and areas of circles, surface

areas and volumes of regular blocks and cylinders when provided with appropriate formulae.


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APPENDIX 5a: EXAMPLES OF MATHEMATICAL QUESTIONS WITHIN A SCIENCE A-LEVEL OF A SINGLE STEP CALCULATION, A MULTIPLE STEP CALCULATION AND AN EXTENDED STEP CALCULATION23

SINGLE STEP CALCULATION

Taken from: CCEA AY111 (AS/1) 21 Jun 11 7(b)(iii) Straightforward one-line calculation using definition of e.m.f.

23 These examples are for physics only; chemistry and biology will be added at a later date


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MULTIPLE STEP CALCULATION

Taken from: CCEA AY111 (AS/1) 21 Jun 11 5(b) Multi-step because (i) Equation must be constructed; (ii) solved algebraically for square root to give speed


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EXTENDED STEP CALCULATION

Taken from: CCEA AY111 (AS/1) 21 Jun 11 9(b)(iii) Extended calculation because the value of cross-sectional area found in an earlier part of the question has to be combined with value of gradient from the graph to give a value of the resistivity


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EXTENDED STEP CALCULATION continued


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EXTENDED STEP CALCULATION continued


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APPENDIX 5b: EXAMPLES OF MATHEMATICAL QUESTIONS WITHIN A SCIENCE A-LEVEL OF A LEVEL 1, LEVEL 2 AND LEVEL 3

LEVEL 1

Taken from: CCEA AY121 (AS/1) 27 Jun 11 2(a) This is a straightforward application of a standard formula


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LEVEL 2

Taken from: CCEA AY211 (A2/1) 24 May 2011 6(b) This requires understanding of algebra and careful application to follow through the calculation correctly

LEVEL 3

Taken from: CCEA AY221 (A2/2) 6 Jun 2011 6(b) This requires understanding of algebra and trigonometry and decision on how to proceed with the solution


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APPENDIX 5c: EXAMPLES OF MATHEMATICAL QUESTIONS WITHIN A SCIENCE A-LEVEL OF A LEVEL 1, LEVEL 2 AND LEVEL 3 CONTEXT

LEVEL 1

Taken from: CCEA AY221 (A2/2) 6 Jun 2011 6(a)(i) On this specification, candidates should have calculated V/d as part of the learning programme


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LEVEL 2

Taken from: CCEA AY111 (AS/1) 21 Jun 11 5(a)(i) Efficiency calculations and ratios are not that routine in the context (of bouncing balls)

LEVEL 3

Taken from: WJEC 1324/01 (PH4) 21 Jun 11 5(b)(ii)(I) It is highly unlikely that the graph and the specific analysis will have been met before


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APPENDIX 5d: EXAMPLES OF MATHEMATICAL QUESTIONS WITHIN A SCIENCE A-LEVEL WHERE ALL MARKS, SOME OF THE MARKS, NONE OF MARKS REQUIRE SCIENTIFIC COMPREHENSION IN ADDITION TO MATHEMATICAL SKILL

ALL MARKS REQUIRE SCIENTIFIC COMPREHENSION

Taken from: 2010 OCR Physics B (G491) Unit 1 2a This requires both recall of the scientific definition of stress and calculation of the quotient


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SOME OF THE MARKS REQUIRE SCIENTIFIC COMPREHENSION

Taken from: CCEA AY211 (A2/1) 24 May 2011 9(a)(i) Some of this part just required pressing the ‘lg’ button, but the third (first blank) column requires some understanding of the physics, derived from the question stem, and the unit has got to be checked as correct


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NONE OF THE MARKS REQUIRE SCIENTIFIC COMPREHENSION

Taken from: WJEC 1321/01 (PH1) 24 May 2011 5(b)(i) Calculation of the area of a circle - no physics required

SOME OF THE MARKS REQUIRE SCIENTIFIC COMPREHENSION continued


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SCORE Maths in science report88 SCORE Maths in science report 89

THEORY PAPERS FRAMEWORK

PRACTICAL PAPERS FRAMEWORK

APPENDIX 6: FRAMEWORK FOR ANALYSING A-LEVEL THEORY AND PRACTICAL PAPERS

Awarding Body: Unit: Specification: Date: AS or A2:

Question number and part

Appropriateness: does it reflect how mathematics in science is used in the real world?

Relationship of the mathematics to the question

Number of steps

Type of mathematics

Complexity of task

Number of marks that require mathematics

Are the associated marks purely for mathematics skills or do they require scientific comprehension

Familiarity of context

Yes No Structural part Tagged on single multi Extended calculation

1 to 4 Mathematics skills

Scientific comp

both 1 2 3

Awarding Body: Unit: Specification: Date: AS or A2:

Question number and part / section

Number of steps

Type of mathematics

Complexity of task

Number of marks that require mathematics

Familiarity of context

Associated marks purely for mathematics or do they require scientific comprehension

Comments

single multi Extended calculation

1 to 4 1 2 3 Mathematics skills

Scientific comp

both


APPENDIX 7: ACKNOWLEDGEMENTS

Geoffrey Wake

Jerry McCarthy

Tandi Clausen May

Peter Hall

Stu Lloyd

David Peet

Laurie Mansfield

Tony Tooth

Ginny Hales

David James

Stu Billington

Clare Green Fiona Miller Tamsin Barton

Charles Tracy

Clare Thomson

Rachel Lambert-Forsyth

John Bentham

Mary Ratcliff

Martin Smith

Alice Rogers

Michael Reiss

Erica Tyson

David Swinscoe

Kay Stephenson

Rosalind Mist

Mario Moustras

Ellen Weavers

Niall MacKay

OTHER ACKNOWLEDGEMENTS AND THANKS

All those involved in analysing the A-level papers, with particular thanks to:

All those involved in completing the survey or questionnaire from teachers, HE representatives, representatives from Professional Bodies and industry representatives

PROJECT TEAM

WORKING GROUP

Documents

MATHEMATICS WITHIN A-LEVEL SCIENCE 2010 EXAMINATIONS … maths.pdf · score maths in science report 2 score maths in science report 3 ExEcutivE summary m athematics enables students