Mathematics within A-level science 2010 examinations - policy report

REPORTMatheMatics within a-level

science 2010 exaMinations

Mathematics within a-level science 2010 examinations2

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 analytical nature of the sciences. to explore whether there was any evidence for this concern, scoRe investigated the type, extent and difficulty of mathematical questions within science a-levels, using the 2010 as and a2 examination papers. the findings show that a large number of mathematical requirements listed in the biology, chemistry and physics specifications are assessed in a limited way or not at all within these papers. the mathematical requirements that are assessed are covered repeatedly and often at a lower level of difficulty than required. this is likely to have an impact on the way that the subjects are taught and therefore on students’ ability 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 all awarding organisations.

BackgROund

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

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 can offer 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 assessments for the biology, chemistry and physics a-levels 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 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 this project, we looked across all assessments at a-level for 2010, including theory and practical examination papers.

Mathematics within a-level science 2010 examinations 3

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 summer 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 mathematics. the proportion of the question parts and marks within a paper that included mathematics was measured.

3. the difficulty of mathematics. this was measured against three criteria: the number of steps in a calculation, the complexity of the question and the familiarity of the context. each category had varying levels of difficulty and each was measured as a proportion of the total number of question parts.

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.

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

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. the project compared the mathematical requirements for the sciences at a-level with the mathematics curriculum prior to post-16 studies using the current national curriculum level descriptors and a 2012 mathematics Gcse specification4. 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 science 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 split into three groups: the 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 through written or telephone questionnaires. six representatives in total took part.

1 the five awarding organisations use the mathematical requirements currently defined by ofqual in developing their specifications for biology, chemistry and physics. the mathematical requirements are available in the full scoRe report Mathematics within A-level science 2010 examinations which can be downloaded from the scoRe website.

2 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. in this research we refer to Unit 3 and 6 as practical examination papers.

3 a complete science a-level is made up of six 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.

4 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.


kEy findings Of This sTudy

the sections that follow pull together the results of all three phases of the research under each of the measures listed on page 3, starting with the type of mathematics.

Detailed findings are available on the scoRe website in the full report, Mathematics within A-level science 2010 examinations.

1. Type of mathematics

this measure separates mathematics into different categories – such as ‘manipulation of equations’, ‘arithmetic’ and ‘graph work’ – in order to see which parts of the mathematical requirements should be included in science a-level specifications and how frequently they occur in assessments.

1.1. Type of mathematics within current specifications

in the current examination system, ofqual sets criteria for biology, chemistry and physics a-level from which specifications are developed. this process is to ensure specifications give an authentic representation of the subject, include the appropriate amount of content and that there is consistency across the different awarding organisations. the biology, chemistry and physics criteria include a list of mathematical requirements which underpin the scientific concepts in the three disciplines. the mathematical requirements differ for biology, chemistry and physics, and the specifications and their assessments are expected to reflect these mathematical requirements.

survey respondents were asked to consider the current mathematical requirements listed for each of the science a-levels and identify any missing areas. For biology, chemistry and physics, it was felt there were underpinning areas of mathematics missing from the requirements and that their exclusion meant students were not adequately prepared for progression in that subject. For example, for physics many of the respondents highlighted the absence of calculus, differentiation and integration, in chemistry the absence of calculus and in biology, converting between different units.

1.2. Type of mathematics within current assessments

For biology, chemistry and physics, the analysis showed that the mathematical requirements that were assessed concentrated on a small number of areas (e.g. numerical manipulation) while many other areas were assessed in a limited way, or not at all. this is illustrated very clearly for biology, chemistry and physics in Figure 1, which shows the coverage of the different mathematical requirements at a-level for the five awarding organisations. the requirements have been grouped into five areas (arithmetic & numerical computations, handling data, algebra, graphs and geometry & trigonometry). although the range of coverage might change year on year, the fact that the pattern is repeated across the awarding organisations suggests that the distributions are typical.

a perceived consequence, raised repeatedly by the science community in the online survey, is that if mathematical content areas are frequently not assessed then these areas will not be taught or practised in depth. if areas within the mathematical requirements are not taught or practised, it will limit students’ access to the subject, their ability to understand scientific concepts and reduce their mathematical fluency. instead, there should be a broader spread of mathematical skills assessed every year to accurately reflect the mathematical requirements and encourage teachers and students to practise them.

survey respondents were asked to identify content areas from the mathematical requirements that should feature highly in assessments. in most cases, the biology, chemistry and physics respondents identified mathematical content areas that were hardly or not at all assessed by the awarding organisations.


Figure 1: number of occurrences in science a-level examinations of each mathematical requirement listed in the specification1 (please note the mathematical requirements are different for each subject and are assigned different labels, for example 4a in physics does not equate to 4a in biology).

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

1A 1B 1C 1D 1E 2a 2b 2c 2d 3a 3b 3c 3d 3e 4a 4b 4c 4d 4e 4f 4g 4h 5a 5b 5c 5d

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

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2. Extent of mathematics

this measure seeks to capture how much of the a-level science 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.

currently, there are no assessment guidelines on the acceptable range of the number of questions and associated marks that should require mathematical understanding and proficiency within the biology, chemistry and physics a-level assessments. there is concern among the science community that competition between awarding organisations discourages them from setting examinations or assessment tasks that might appear more difficult, for example by including both more challenging and more mathematical content. table 1 shows the variation in the amount of mathematics assessed in the different a-level specifications for each of the sciences. the analysis does not explicitly compare between the three sciences as it is accepted that the mathematical requirements will be different.

Table 1: The variation across awarding organisations in the amount of mathematics assessed within a biology, chemistry or physics a-level.

The range in the amount of mathematical content assessed across awarding organisations in a complete A-level assessment

Question parts marks

Biology a-level 15% - 35% 13% - 24%

chemistry a-level 41% - 52% 24% - 43%

Physics a-level 47% - 60% 47% - 57%

the survey findings indicate that the extent of mathematical content in a-level assessment should increase, providing that the mathematics underpins the principles of biology, chemistry and physics. table 1 also demonstrates the degree of disparity in the amount of mathematics assessed within a given subject by the different awarding organisations.

3. difficulty of mathematics

the level of difficulty was measured in three ways: the number of steps involved in the calculation; the complexity of the calculation; and the familiarity of the context. we assumed the difficulty increased with the greater number of calculation steps; as the complexity increased; and as the context of the question became less familiar.

3.1. number of steps in a calculation

the 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 or extended step calculations (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 reasoning5.

For biology and chemistry, the majority of calculations across all specifications were found to be single step whereas physics had a more even spread. Figure 2 shows the range across specifications of mathematical question parts involving single, multiple or extended step calculations. these data are to be considered within the context of the subject and as such comparisons should not be made between the three sciences.

there is variation between the awarding organisations in the amount of mathematical questions using single, multiple and extended step type questions for each of the sciences, suggesting the level of difficulty may vary between awarding organisations. survey respondents for all three subjects felt that there should be a more even spread of single, multiple and extended step calculations, with the inclusion of more multiple and extended step questions than were found in the examination papers analysed. the argument was also made that the inclusion of more in-depth problem solving would allow students to apply


their knowledge and understanding in unstructured problems and would increase their fluency in mathematics within a science context.

Figure 2: The spread of occurrences across awarding organisations of single, multiple and extended step mathematical questions parts within biology, chemistry and physics a-level assessment.

3.2. complexity of the calculation

the questions containing mathematical content were measured against four levels of complexity6, with level 4 being considered the most difficult5.

• level 1 includes straightforward and routine questions which require simple recall of procedures.

• level 2 requires application and understanding of mathematics within one domain (one mathematical content area e.g. algebraic equations).

• 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 and decision making being necessary to answer the question.

Figure 3 shows the spread of mathematical complexity within science a-level assessment across the awarding organisations. level 4 was omitted from the findings as very few examination papers included questions of this complexity.

the mathematical assessments for each of the sciences focused mainly on level 1 and level 2. the respondents to the online survey did not think that this was appropriate. they felt that calculations of simple and higher complexity should be included in the assessment, including those requiring understanding and the use of mathematics across domains which necessitate a decision about the direction in which to proceed.

the complexity of the questions varied significantly across awarding organisations in both chemistry (9% to 33% being more than just recall) and physics (11% to 44% being more than recall).

5 examples of each of the measures can be found in appendix 5 in the full scoRe report Mathematics within A-level science 2010 examinations which can be downloaded from the scoRe website.

6 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

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Figure 3: The spread of occurrences across awarding organisations of complexity levels in mathematical question parts within biology, chemistry and physics a-level assessment.

3.3. familiarity of the context

existing research7 has raised concerns about the context in which mathematics is assessed, concluding that the assessment of mathematical concepts is often set in familiar contexts which weakens students’ abilities to apply mathematics in novel situations. such concerns have been echoed by higher education and industry.

Question parts containing mathematics were measured against three types of contexts5:

• familiar context – context is typically met through the learning programme.

• some novel aspects – context has some novel aspects.

• unfamiliar context – context is unlikely to have been met before.

Figure 4 shows the spread of context levels for the mathematical assessment within the science a-levels across the awarding organisations.

7 For example, Mathematical Needs (acMe, 2011)


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Figure 4: The spread of occurrences across awarding organisations of context levels in mathematical question parts within biology, chemistry and physics a-level assessment.

results showed that respondents would like to see more mathematics set in unfamiliar contexts so that students have more experience at applying mathematics in unfamiliar scientific situations.

4. appropriateness of mathematics

the mathematics required for biology, chemistry and physics a-level is there to support and develop scientific understanding and should only feature in assessment if it underpins the science. we considered a mathematical question part to be appropriate if it required scientific comprehension in addition to mathematical skill to achieve the marks. Mathematical question parts were measured against the following categories for appropriateness:

• no scientific comprehension – marks only required mathematical skill.

• some scientific comprehension – some of the marks required scientific comprehension in addition to mathematical skill.

• scientific comprehension – all marks required scientific comprehension in addition to mathematical skill.

Figure 5 shows the number of question parts containing mathematics that required scientific comprehension to answer them correctly rather than mathematical skill only. For physics and chemistry, the vast majority of question parts with mathematics did require scientific comprehension. however, in biology almost all mathematical question parts within the a-levels included a proportion of marks that could be gained through mathematical skill only. a more detailed study is needed to provide a clearer picture of why this is happening and whether the biology community considers this a problem.

Biology a-level across the awarding organisations appears to show a good spread of questions set in familiar and unfamiliar contexts. however, when considering theory papers only, which make up 80% of the biology a-level, almost all questions are set in a familiar context. in chemistry, the theory and practical papers are almost entirely made up of questions set in a familiar context. the survey


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Figure 5: The spread of occurrences across awarding organisations of mathematical question parts requiring scientific comprehension within biology, chemistry and physics a-level assessment.

5. coherence between the sciences and mathematics

as a measure of coherence, the project compared the mathematical requirements in biology, chemistry and physics a-level with a typical Gcse mathematics specification and the national curriculum level descriptors. table 2 illustrates the mathematical areas identified in the science a-levels that go beyond a typical Gcse mathematics specification and the mathematics national curriculum.

survey participants for all of the sciences agreed that it was necessary for the mathematical requirements for science a-levels to go beyond those in the current Gcse mathematics, which was considered insufficient preparation to access all content areas of the sciences at a-level. in some cases, different terminology was used to describe the same areas of mathematics in mathematics Gcse and the a-level sciences. the mathematical requirements for the sciences also included content areas which were not considered mathematical concepts, which further inhibits coherence across the disciplines.

Most of the mathematical areas identified in table 2 would be taught in as and a2 mathematics. this could affect the students’ ability to access the sciences at a-level in two ways. the first concerns those students who do not take mathematics beyond Gcse. currently these students would need to be taught these areas of mathematics within the science curriculum in order to access all contents of the science a-level. it is not always the case that science teachers will have the confidence and experience to teach this mathematics first-hand.

the second concerns those students who take mathematics as or a-level alongside an a-level in the sciences. improved coherence between these a-levels would support students in applying mathematical concepts in a number of different contexts, including scientific contexts. it is not straightforward to apply mathematics learnt in one context to another and students need many opportunities to do this fluently and confidently.




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Table 2: The mathematical content areas in biology, chemistry and physics a-level requirements that go beyond a typical gcsE mathematics specification and the mathematics national curriculum

issuEs and REcOmmEndaTiOns

the research in this project has provided evidence that the current mathematical assessments in science a-levels do not accurately reflect the mathematical requirements of the sciences. the findings show that a large number of mathematical requirements listed in the biology, chemistry and physics specifications are assessed in a limited way or not at all within these papers. the mathematical requirements that are assessed are covered repeatedly and often at a lower level of difficulty than required for progression into higher education and employment. it has also highlighted a disparity between awarding organisations in their assessment of the use of mathematics within biology, chemistry and physics a-level. this is unacceptable and the examination system, regardless of the number of awarding organisations, must ensure the assessments provide an authentic representation of the subject and equip all students with the necessary skills to progress in the sciences.

scoRe recommends the following to address these issues:

Biology ChemisTry PhysiCs

arithmetic and computations

standard deviation; logarithmic functions

logl0x , ex, logex logl0x; ex; logex; radians

handling data standard deviation; order of magnitude calculations; standard deviation of the mean; Frame the null hypothesis

order of magnitude; calculations; Understand and use logarithmic scales

algebra Use logarithms in relation to quantities which range over several orders of magnitude

<< >>; μ; Use logarithms in relation to quantities which range over several orders of magnitude

Dimensional consistency; identify the inadequacies in simple algebraic equations used as mathematical models;

<< >>; ∑, ∆x, x, dx/dt

graphs Draw and use the slope of a tangent to a curve as a measure of rate of change

Draw and use the slope of a tangent to a curve as a measure of rate of change

Determine gradient of a tangent to a non-linear graph by drawing and use the tangent as a rate of change; Using the slope of a tangent to obtain gradient using d/dt; log-linear graphs; log-log graphs; log plots; select appropriate variables for graph plotting; Rearrange relationship into y = mx + c form; counting squares under a curve; Multiplicative scales

geometry and Trigonometry

Use sin ≈ tan ≈ and cos ≈1 for small ; relationship between degrees and radians

mathematical requirement of science a-levels

1. there should be a review of the current mathematical requirements for each of the sciences at a-level to ensure the inclusion of underpinning areas of mathematics within that science. the science community, working closely with the professional bodies and informed by the results of this research, should advise on and approve the mathematical requirements.

2. a framework should be developed to regulate the way mathematics is assessed within science a-levels. this should include guidance on: the frequency with which each mathematical requirement is assessed; the extent to which each mathematical requirement is assessed; and the number of marks that should be available for each level of difficulty.

3. the mathematical assessments for science a-level should include a more even spread of single, multiple and extended step calculations across the different mathematical requirements. there should also be more in-depth problem solving and more opportunities for students to apply their knowledge and understanding in unfamiliar contexts.

scoRe - science community Representing education 6-9 carlton house terrace london sw1Y 5aG

tel: +44 (0)20 7451 2245 email: [email protected] web: www.score-education.org

Publication date: april 2012

supported by the department for Education and the gatsby charitable foundation.

scoRe, a collaboration of organisations working together on science education policy:

association for science Education www.ase.org.uk

institute of Physics www.iop.org

Royal society www.royalsociety.org

Royal society of chemistry www.rsc.org

society of Biology www.societyofbiology.org

Ensuring parity across awarding organisations

4. awarding organisations should use the revised mathematical requirements for biology, chemistry and physics a-levels to ensure that a broad spread of mathematical requirements is assessed and is appropriately weighted across all theoretical and practical assessments.

5. all mathematical requirements should be assessed over a 2-3 year cycle to ensure they are all taught. this should be monitored by ofqual to ensure assessment across awarding organisations matches the requirements of the specifications.

coherence across the sciences and mathematics

6. the mathematical content of science qualifications must draw on the same terminology (and meanings) used in mathematics to ensure coherence across the disciplines. where there are accepted differences, science and mathematics teachers must work together to make it clear to students that there is a difference and when each usage should be applied.

7. there should be consideration in the next review of Gcse mathematics of whether some or all of the mathematical content required for science a-levels should be included within Key stage 4 mathematical qualifications.

8. the mathematical areas within science a-levels that go beyond Gcse mathematics should be more clearly labelled to inform teachers’ pedagogy in the sciences. these areas should also be made more explicit to students so that they are better informed when making a-level choices. For example, it may be necessary for some students to support their science a-level with a post-16 mathematics qualification.

9. a-level combinations of one or more science a-levels with mathematics are well regarded to facilitate progression to higher education in steM subjects. to promote and support this combination, university admissions for steM-related courses should stipulate clearly the appropriate mathematical requirement for course entry.

10. the scientific community, including scoRe organisations, should work with the advisory committee on Mathematics education (acMe) to consider appropriate and realistic post-16 mathematical options to support the teaching and learning of the sciences at a-level.

http://www.score-education.org

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http://www.ase.org.uk

http://www.iop.org

http://www.royalsociety.org

http://www.rsc.org

http://www.societyofbiology.org

http://www.score-education.org

Documents

Mathematics within A-level science 2010 examinations - policy report