16. Programme Aims s3

Programme Specification A statement of the knowledge, understanding and skills that underpin a taught programme of study leading to an award from The University of Sheffield

This programme specification covers the curriculum in Mathematics and Statistics for the following dual honours degree programmes and should be read in conjunction with the relevant programme specification for the second subject. 1. Programme Titles 2. Programme Codes 3. JACS Codes Mathematics with Study in Europe MASU10 / MASU11 G100 Mathematics with Spanish Language MASU12 G100 Mathematics with French Language MASU13 G100 Mathematics with German Language MASU14 G100 4 Level of Study Undergraduate Bachelor of Science with Honours (BSc Hons) (MASU11) 5a Final Qualification Master of Mathematics with Honours (MMath Hons) (MASU10, MASU12, MASU13, MASU14) Position in the QAA Framework for 5b Honours (BSc), Masters (MMath) Higher Education Qualifications 6a Intermediate Qualification(s) BSc (MMath) Position in the QAA Framework for 6b Honours Higher Education Qualifications 7 Teaching Institution (if not Sheffield) Not applicable 8 Faculty Science 9 Home Department School of Mathematics and Statistics Other Department(s) involved in 10 Modern Languages Teaching Centre (MLTC) teaching the programme 11 Mode(s) of Attendance Full-time 12 Duration of the Programme 4 years Accrediting Professional or Statutory 13 Not applicable Body January 2008, updated January 2009, March 2012, February 14 Date of production/revision 2014, March 2016.

Dual Degrees The University of Sheffield defines a dual degree as the independent study of two parallel subjects. Dual degrees offer students the flexibility to choose a programme of study that reflects their interests and gives the opportunity to develop detailed knowledge and key skills in two major subjects. Whilst the two subjects may be taught independently, they will complement, inform and illuminate one another. Where there are two programme specifications for dual degrees, one for each half of the programme, and students should refer to both documents for a full description of the whole programme. Where there are clear links between the two subjects, details will be included in Sections 15 and 20 of the programme specifications. However, there are some single programme specifications for dual degree combinations where there is a substantial degree of integration between the two subjects.

0164cbd4078e6f46db640855a79b8eb2.doc – ver16-17 1 15. Background to the programme and subject area

Mathematics involves the study of intangible objects (such as numbers, functions, equations and spaces) which necessarily arise in our attempts to describe and analyse the world about us. It is a fascinating subject of great beauty and power. Its abstraction and universality lie behind its huge range of applications, to physical and biological sciences, engineering, finance, economics, secure internet transactions, reliable data transmission, medical imaging and pharmaceutical trials, to name a few. Mathematicians were responsible for the invention of modern computers, which in turn have had a great impact on mathematics and its applications. Teaching in the School of Mathematics and Statistics (SoMaS) is in the three areas of Pure Mathematics, Applied Mathematics, and Probability and Statistics. Pure mathematics is a subject rich in patterns and one in which the development of a theory may begin with identification of behaviour common to various simple situations and proceed, through precise analysis, to the point where rigorous general results are obtained. Solutions of particular problems may involve standard analytical techniques, for example from calculus, or the application of an abstract general theory to a particular concrete example. In applied mathematics and in probability and statistics, a common approach to practical problems, from a wide variety of contexts, is first to model or interpret them mathematically and then apply mathematical or statistical methods to find a solution. In all three subjects it is vital that work should be presented in a clear, precise and logical way so that it can be understood by others. For these reasons, graduates from programmes involving mathematics are highly regarded, by a wide range of employers, for their analytical, problem-solving and communication skills as much as for their knowledge of mathematics. These programmes offer the opportunity to spend the third year studying mathematics at a University in another European country. Several universities offer such opportunities but the Sheffield programme is distinctive for the amount and quality of the language teaching provided by the Modern Languages Teaching Centre. We believe that this is reflected in the quality of the graduates from the programmes. The list of partner institutions is extensive but the most common destinations are in France, Germany and Spain. In their first two years, students normally spend a third of their time on language studies and the rest on mathematics. The primary factor in determining whether a student takes the BSc or the MMath is their Level Two examination result but a student who qualifies for MMath has a choice between the specialist MMath programme, including a substantial project, and the broader BSc programme, augmenting the courses taken abroad. On all programmes the third year is spent in Europe. On MASU12, MASU13 and MASU14, students are required to spend one third of their fourth year studying their chosen language and this component of the programme carries Foreign Languages at Work accreditation (FLAW). On MASU10 and MASU11, students at Level 4 may concentrate entirely on mathematics or may take up to 20 credits of language modules. The single honours programmes in Sheffield have a common SoMaS core, of 40 credits at Level 1 and 20 at Level 2, together with substantial components from each of the three areas, with a wide choice of modules at each of Levels 3 and 4. Students on these programmes normally take 40 language credits at Level 1 though, on MASU10 and MASU11, this may be reduced if the student has a strong background in language. Students study two of the three Mathematical disciplines. Of these Pure Mathematics and Probability & Statistics are easiest to accommodate, with 20 credits of each at Level 1, whereas Applied Mathematics will only be available to students with a strong background in either language or mechanics. Some SoMaS modules concentrate on applicability while others are more theoretical. Some deal with contemporary developments, such as error-correcting codes, signal processing and financial mathematics, and others treat long-established topics of continuing importance. Several, such as the module on History of Mathematics and the module on Galois theory, put the subject in its historical perspective. All are informed by the research interests and scholarship of the staff. Staff in all three areas have international reputations in research, with 89% of research activities being rated as world leading or internationally excellent in the 2014 Research Excellence Framework exercise. Many modules are taught by leading experts in the area in which the module is based. In Pure Mathematics there are particular research strengths in topology, algebra and algebraic geometry, number theory and differential geometry, and there are modules available in all these areas. The main strengths within Probability and Statistics are in Bayesian statistics, statistical modelling and applied statistics, and probability and, again, all these are prominent in the undergraduate curriculum. Several members of the School belong to the Sheffield Centre for Bayesian Statistics in Health Economics. Applied Mathematics research is strong not only in traditional areas of the subject, such as fluid mechanics, but in interdisciplinary areas such as solar physics, particle astrophysics, environmental dynamics and mathematical biology. The School was instrumental, with other departments in the University, in setting up the Sheffield-based NERC Earth Observation Centre of Excellence for Terrestrial Carbon Dynamics. Further information is available from the school web site: http://www.shef.ac.uk/maths

0164cbd4078e6f46db640855a79b8eb2.doc – ver16-17 2 16. Programme aims

Through its programmes, the School of Mathematics and Statistics aims: to provide Mathematics degree programmes with internal choice to accommodate the diversity of students’ interests and abilities; to provide an intellectual environment conducive to learning; to prepare students for careers which use their mathematical and/or statistical training; to provide teaching which is informed and inspired by the research and scholarship of the staff; to provide students with assessments of their achievements over a range of mathematical and statistical skills, and to identify and support academic excellence. In its single honours programmes with Study in Europe and in its dual programmes with French Language, German Language or Spanish Language, it aims: to provide a degree programme in which students may choose either to specialise in one mathematical discipline (Pure Mathematics, Applied Mathematics, Probability and Statistics) or to choose a more balanced programme incorporating two of these disciplines; to offer students the opportunity to study mathematics and statistics in another European country. In its MMath programmes, it aims: to prepare students for progression to a research degree in one of the three mathematical disciplines or for careers in which the use of mathematics is central. The MMath with language programmes ( MASU12, MASU13, MASU14) also aim: to provide language instruction beyond that needed to study mathematics and statistics abroad, giving students the opportunity to acquire all-round fluency in the language.

17. Programme learning outcomes

Knowledge and understanding: a graduate should: K1 have acquired a working knowledge and understanding of the methods of linear mathematics; K2 have acquired a working knowledge and understanding of the methods of advanced calculus; K3 have acquired a broad knowledge and understanding of at least two of Pure Mathematics, Applied Mathematics and Probability & Statistics; K4 have acquired a working knowledge and understanding of specialist mathematical or statistical topics; K5 have acquired a working knowledge of their chosen language.

Knowledge-based skills: a graduate should: SK1 be able to apply core concepts and principles in well-defined contexts; SK2 show judgement in the selection and application of mathematical tools and techniques; SK3 demonstrate skill in comprehending problems and abstracting the essentials of problems; SK4 be able to formulate problems mathematically; SK5 be able to obtain solutions of mathematical problems by appropriate methods; SK6 have experienced first-hand, through a substantial period of study of mathematics at a European University outside the UK, the life, language and culture of a different European country; SK7 be able to converse with native speakers of their chosen language; SK8 be able to interpret mathematical text written in their chosen language. In addition those graduates whose programmes have included a substantial component of Pure Mathematics should: SK9 understand the need for proof and logical precision; SK10 have developed an understanding of various methods of proof.

0164cbd4078e6f46db640855a79b8eb2.doc – ver16-17 3 Skills and other attributes: a graduate should: S1 have skill in calculation and manipulation; S2 understand logical arguments, identifying the assumptions and conclusions made; S3 be able to develop and evaluate logical arguments; S4 be able to present arguments and conclusions effectively and accurately; S5 demonstrate the ability to work with relatively little guidance; S6 have developed the skills to acquire further mathematical or statistical knowledge; S7 have developed the skills to model and analyse physical or practical problems; S8 appreciate the development of a general theory and its application to specific instances; S9 have acquired skills in the use of computer algebra packages.

In addition, MMath graduates should: K6 have enhanced and extended their specialist knowledge in at least one of the three disciplines Pure Mathematics, Applied Mathematics, Probability & Statistics; S10 have enhanced and extended the necessary mathematical skills to consider careers as practising mathematicians or statisticians; S11 have shown the ability to complete an extended individual study of a mathematical or statistical topic and to present an account of the topic.

In addition, graduates on the MMath programmes with named languages (MASU12, MASU13, MASU14) should: K5+ have acquired a near-native level of proficiency in their chosen language; SK7+ demonstrate a near-native command of their chosen language in both professional and social contexts; S4+ be able to present arguments and conclusions effectively and accurately, both in English and their chosen language; S12 have acquired a profound cultural understanding of the country where they have studied. + indicates an enhancement of a previously listed learning outcome. 18. Teaching, learning and assessment

Development of the programme learning outcomes is promoted through the following teaching and learning methods: Lectures A 10-credit lecture SoMaS module (or half-module) at Level 1 or 2 generally comprises 22 lectures supported by a weekly or fortnightly problems class. At levels 3 and 4, a typical 10-credit module has around 20 lectures. The lecturing methods used vary. Effective use is made of IT facilities, for example through computer demonstrations using data projectors. Students also learn mathematical techniques and theories through seeing problems being solved and results proved in lectures. Theory is developed and presented in a clear and logical way and is enhanced by the use of illustrative examples. In many modules, supporting written material is circulated. Some Level 3 modules include an element of project work for which guidance is provided in lectures. Learning outcomes supported by lectures: K 1-4,6 SK 1-5, 9-10 S 1-4, 6-8, 10 Problems classes At Levels 1 and 2, lecture groups are divided into smaller groups for problems classes lasting fifty minutes. Ample opportunity is provided for students to obtain individual help. Coursework, usually in the form of sets of problems, is regularly set and marked and feedback is given. This is usually administered through the problems classes. For a 20-credit ``core’’ module at Level 1, students meet fortnightly in small groups with their personal tutor, and may be required to present their solutions and participate in group discussions. Setting of coursework continues into Levels 3 and 4, together with the associated feedback, but, due to the expected increasing maturity of students and in support of learning outcome S5, the formal mechanism provided by problems classes is replaced by informal contact with the module lecturer. Learning outcomes supported by problems classes:

0164cbd4078e6f46db640855a79b8eb2.doc – ver16-17 4 K 1-4,6 SK 1-5, 9-10 S 1-8 Project work At Level 4 all MMath students are required to take 40 credits of project modules with a view to achieving learning outcome S11. They are given training in presentational skills, including the use of mathematical typesetting packages, and individual guidance in one-to-one sessions with a nominated supervisor. Learning outcomes supported by project work: K 3,4,6, (1,2)* SK (1-5,9,10)* S 11, (1-10)* * dependent on the nature of the individual project Computing and Practical Sessions At Level 1 students taking Probability and Statistics are provided with training on the software package R. This training contributes to SK2, SK5, S1 and S9. Learning outcomes supported by computing and practical sessions: SK 2,5 S 1,9 Language Teaching Language skills are taught using seminars supplemented by more informal group and pair work. Students may enrol to use the MLT Centre’s up-to-date Self-Access Centre, comprising a 27 position Computer-Aided Language Learning (CALL) laboratory, an Audio Visual room, with access to European television channels, and an extensive library of print material, audio and video cassettes. Learning outcomes supported by language teaching: K 5, 5+ SK 6-8, 7+ S 4+, 12

Opportunities to demonstrate achievement of the programme learning outcomes are provided through the following assessment methods: Most SoMaS modules are assessed by formal examinations, augmented in some cases by a component of assessed coursework; several modules now include an element of the latter. The most common format involves the regular setting of assignments, each consisting of a number of problems based on material recently covered in lectures. Some of the Level 3/4 modules include a project and/or poster presentation. Examinations are normally of 1.5, 2 or 2.5 hours’ duration. Where a module is assessed by both examinations and coursework, the latter contributes between 10% and 30% of the final mark. All assessment by examination or coursework contributes to demonstrating the achievement of K1-4,6, SK1- 5,9,10 and S1. However as students progress through the programmes, less explicit guidance on selection of techniques is given and more is expected in terms of formulation of problems and in solving problems requiring several techniques or ideas. The learning outcomes S2-8 feature in some assessment at Level 1, but again more is expected later in the programmes, as the students mature mathematically. For example, as students progress through the programmes, the Pure Mathematics examinations, typically, require more in the way of rigorous proof. Aspects of S9, the use of computer packages, are assessed in the appropriate modules. The learning outcomes K1 and K2 are assessed not only in the core Level 2 modules but also in various approved modules. Each student is required to submit a report on the year abroad to their Erasmus/Socrates tutor. This should include a record of courses taken, examination papers, a sample of coursework, their assessment record and, with SK6 in mind, a discussion of their experiences as a student abroad. The tutor recommends a mark for the year abroad to the Director of Teaching and the agreed mark is presented to external examiners and the final SoMaS Examination Board. If, due to practice at the host university, there is insufficient information available on the assessment of the student’s work in the year abroad, the tutor may recommend that the year abroad mark should be replaced by a weighted average of the student’s marks in Years 2 and 4. The year abroad mark accounts for 100, 110, or 120 credits and is augmented by the grades awarded on any MLTC Year Abroad modules taken in the third year. Learning outcomes K5, 5+, SK7, 7+, S4+ and S12 are assessed through the language modules by methods including tests, records of independent study and for MLTC Year Abroad modules at Level 3, a year abroad diary, a project and a presentation in the target language on return to the UK. There is no direct assessment of SK8 though in many cases it will feature in assessment by the host university. The additional learning outcome, S10, for the M programmes MASU10, MASU12, MASU13 and MASU14, is assessed in Level 4 examinations while S11 is assessed through a submitted project report, examined by at least two members of staff and subject to moderation.

0164cbd4078e6f46db640855a79b8eb2.doc – ver16-17 5 Learning outcomes assessed by mathematics examinations: K 1-4,6 SK 1-5, 9-10 S 1-8, 10 Learning outcomes assessed by mathematics coursework: K 1-4,6 SK 1-5, 9-10 S 1-9 Learning outcomes assessed by mathematics project work (MMath): K 4,6 SK 1-5, (9,10)* S 2, 4, 5, 6,10,11, (1,3,7,8,9)* *dependent on the nature of the individual project Learning outcomes assessed by report on year abroad: K all SK 1-6, 8-10 S 1-9 Learning outcomes assessed by MLTC modules: K 5, 5+ SK 7,7+ S 4+, 12 19. Reference points

The learning outcomes have been developed to reflect the following points of reference: Subject Benchmark Statements http://www.qaa.ac.uk/AssuringStandardsAndQuality/subject-guidance/Pages/Subject-benchmark- statements.aspx University Strategic Plan http://www.sheffield.ac.uk/strategicplan Learning and Teaching Strategy (2011-16) http://www.shef.ac.uk/lets/staff/lts The QAA Mathematics, Statistics and Operational Research benchmark document at: http://www.qaa.ac.uk/en/Publications/Documents/SBS-Mathematics-15.pdf The European Mathematical Society Mathematics Tuning Group report “Towards a common framework for Mathematics degrees in Europe” at www.maths.soton.ac.uk/EMIS/newsletter/newsletter45.pdf pages 26-28. The University of Sheffield Students’ Charter at http://www.shef.ac. uk/ssid/ourcommitment/charter/. The University’s coat of arms, containing the inscriptions Disce Doce (Learn and Teach) and Rerum Cognoscere Causas (To Discover the Causes of Things; from Virgil's Georgics II, 490), at http://www.sheffield.ac.uk/about/arms The research interests and scholarship of the staff.

20. Programme structure and regulations

The teaching year is divided into two semesters each of fifteen weeks, the final three weeks of each being devoted to examinations. The programmes are fully modular, being delivered mainly in 10-credit modules, taught and examined during a single semester, and in 20 credit modules, often examined at the end of the year. Each year of study represents 120 credits. The two Study in Europe programmes differ mainly in the fourth year when BSc students take modules at Level 3 whereas MMath students take modules at Level 4. On the three dual programmes, MASU12, MASU13 and MASU14, students must take forty credits in the appropriate language whereas on MASU10 or MASU11 they may take twenty such credits. The differences in the regulations at Levels 1 and 2 are that MASU12, MASU13 and MASU14 have a named set of four ten credit language modules, and that on MASU10 or MASU11 Applied Mathematics can be taken. On all five programmes the third year is spent at the European University. At Level 1 students take two core 20-credit modules. The material in these modules comprises mathematical methods. The choice of topics is influenced by the potential for application. Depending on their background in language and mechanics, students take a further 40 credits of Mathematics consisting of two of the following: 20 credits of Pure Mathematics (PM), 20 credits of Applied Mathematics (AM), 20 credits of Probability & Statistics (PS). In addition, they take 40 credits in their target language. At Level 2, students usually take 20 SoMaS core credits, a further 60 mathematics credits, and 40 credits of language. The third year is spent at the European host university and should include a component of language, provided either by the host or by the Modern

0164cbd4078e6f46db640855a79b8eb2.doc – ver16-17 6 Language Teaching Centre’s Study Abroad modules. The SoMaS modules offered at Level 4 are in specialist topics and there are no lecture-based core modules. At Level 4, MMath students must take 40 credits of project- based modules of which 10 credits are core, and 40 further credits of mathematics. The Level 3 and Level 4 SoMaS modules are consistent with the guidelines in the Council for the Mathematical Sciences briefing document and the European Mathematical Society document cited at 19. Students on the Study in Europe programmes are normally registered for the MMath programme. At the end of their second year, these students choose between the BSc programme and the MMath programme, the essential difference being the level of courses in the final year. Students averaging less than 59.5% on the mathematics modulesin their Level 1 or Level 2 assessment are required to transfer to the BSc. Students on MASU12, MASU13 or MASU14 who fail to achieve an average of 59.5% or above at Level 1 or Level 2 will also be required to transfer to the corresponding one of the BSc programmes MASU**, MASU** or MASU**, or to the BSc programme MASU11. It is possible for a student who is qualified for the MMath to transfer to the BSc. For example, a student whose host university has a less significant provision than Sheffield in one of the three disciplines could choose between a broad BSc programme including that discipline or a more specialised MMath programme. Students averaging less than 55% on the language modules at Level 1 or Level 2 will be required to transfer to the MMath or BSc Mathematics programme. Masters graduates do not also obtain a BSc but an MMath candidate failing to achieve II(ii) standard or higher may be awarded a BSc. Classification of the final degree is subject to the University of Sheffield General Regulations. Level 1 serves as a qualifying year and does not contribute to degree classification. The weighting for Level 2, the year abroad and Level 4 on the MMath is 1:1:2; for the BSc it is 1:0:2, but students are required to pass their year abroad.

Detailed information about the structure of programmes, regulations concerning assessment and progression and descriptions of individual modules are published in the University Calendar available on-line at http://www.shef.ac.uk/calendar/

21. Student development over the course of study

Mathematics is essentially a linear subject with key skills and core knowledge taught at Level 1 or Level 2 often required at subsequent levels. Level 1 consolidates key technical skills for use throughout the programmes. Ideas of proof and abstraction, illustrated by concrete examples, are introduced in the Pure Mathematics modules and applications are developed in Applied Mathematics and Probability & Statistics. Level 2 introduces more advanced technical methods, in particular those of linear mathematics and advanced calculus. The Pure Mathematics modules put some topics introduced at Level 1 on a sounder theoretical basis than before or treats them at a more sophisticated level of abstraction. There is further development of theory and applications in Applied Mathematics, including differential equations, and Probability & Statistics, including modelling. Modules at Level 3 and Level 4 offer a range of specialist options consistent with the principles outlined in reference points (1), (3) and (4). Some of these build on knowledge acquired in earlier years and others, though requiring skills already acquired and the corresponding degree of mathematical maturity, introduce topics that are essentially developed from scratch. Several of those offered at Level 4 have Level 3 prerequisites and the selection of such modules by an MMath student is monitored to ensure that the student has acquired appropriate background knowledge during their year abroad.

22. Criteria for admission to the programme

Detailed information regarding admission to the programme is available at http://www.shef.ac.uk/prospective/

0164cbd4078e6f46db640855a79b8eb2.doc – ver16-17 7 23. Additional information

A list of possible host universities for the year abroad may be viewed at http://www.shef.ac.uk/ieu/. SoMaS has an active Staff-Student Forum and there is a lively Student Maths Society. Personal Tutorials Each of the three SoMaS subject areas has an Erasmus/Socrates tutor and every student on these programmes is assigned to one of these tutors who oversees the choice of courses in the year abroad and on return to Sheffield. The tutor also maintains contact with the student throughout the year abroad. The School of Mathematics and Statistics runs a personal tutorial system conforming to the guidelines in the University’s Students’ Charter. All students are allocated a personal tutor, who may also be the Erasmus tutor, at the outset of their University career. It is hoped that the association will remain during the whole of each student’s course. However, a system is in place to allow a student to transfer to another tutor if they wish. Personal tutors provide personal support and academic guidance, acting as a point of contact and gateway for University support services, such as Careers and the Counselling Service. Students are expected to see their tutor at scheduled sessions, the frequency of which is highest at Level 1, and may contact their tutor at other times. Many other staff members also have particular responsibility for student support. In particular, the Senior Tutor is responsible for all day-to-day issues for individual students, although the Erasmus co-ordinator will often deal with these for students on MASU10, MASU11, MASU12, MASU13 and MASU14. The web page for SoMaS is at http://www.shef.ac.uk/maths

This specification represents a concise statement about the main features of the programme and should be considered alongside other sources of information provided by the teaching department(s) and the University. In addition to programme specific information, further information about studying at The University of Sheffield can be accessed via our Student Services web site at www.shef.ac.uk/ssid /

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