examiners’ report: honour moderations in mathematics ...examiners’ report: honour moderations in...

Examiners’ Report: Honour Moderations in Mathematics Trinity

Term 2005

Part I of the Report

A. STATISTICS

• Numbers and percentages in each class

Number Percentages %2005 2004 2003 2002 2005 2004 2003 2002

I 64 59 58 53 30 29.5 30.1 29.0II 111 103 115 107 53 51.5 59.6 58.5III 24 18 12 16 12 9.0 6.2 8.7Pass 6 14 2 0 3 7.0 1.0 0.0Fail 4 6 6 7 2 3.0 3.1 3.8

• Number of vivas and effect of vivas on classes or result

As in previous years there were no vivas.

• Numbers of scripts double or triple marked

As in previous years, no scripts were multiply marked by the Examiners. However ateam of graduate students, under the supervision of Jessica Hu, sorted out and then wentthrough all of the exam scripts at the Examination Schools to cross-check against themark sheets and to spot any un-marked questions. A number of omissions were spottedthis way.

• Numbers taking each optional subject:

No subjects were optional.

1

B. EXAMINING METHODS AND PROCEDURES

The format of the examination was the same as last year.

Each candidate was required to display her/his University Card on the desk throughout eachof the examinations.

The marks for written papers and marks for Maple projects (MM) were averaged according tothe following two formulae:

Av1 = [PMI+PMII+AMI+AMII+ 25/40*MM)]/4.25

Av2 = [PMI+PMII+AMI+AMII]/4

rounded up to a whole number.

Procedure for re-calibration of marks and classification of candidates:

• In order to ensure equal weighting across subjects, papers were standardised to the samemean and standard deviation. In practice this provided a good first approximation, thoughsome further minor adjustments were then required. This standardisation was used todetermine approximate borderlines on each paper.

• Using these borderlines provisional USMs for each paper were then allocated to all can-didates using linear interpolation.

• For each candidate, an average USM score was calculated both with Maple (Av1) andwithout Maple (Av2).

• Ordered lists were prepared by the database in both Av1 and Av2 order using raw marksand also USMs.

• Only one classification meeting was held.

• At the meeting the Examiners had before them the list described above. The Examinersconsidered candidates near the provisional borderlines individually, to decide whether theirperformance was accurately reflected by their provisional USMs and USMs were adjustedaccordingly. At the bottom end of the list, it was felt that candidates’ performance wasreflected better by their raw marks, than their re-scaled marks. A second list using theseadjusted marks was then considered.

• The final classification was carried out as described in the Undergraduate Handbook.

As last year, no calculators were allowed.

The Examiners felt that the examination provided a fair measure of the achievement of candi-dates.

The part (a) and part (b) format was again found to be restrictive for the setting of questionsin some subject areas.

This was the third year of the new examination format. As last year the four papers had to besplit for both setting and marking. Any diminution in marking effort was again felt to be smallwith some Examiners marking and recording marks right up to the deadline for marks to besubmitted for entering on the database.

This year the Examiners marked a total of 966 scripts, comprising 844 Mathematics and Mathe-matics & Statistics, and 122 Mathematics & Philosophy, Mathematics & Computer Science andComputer Science. As last year, the Examiners were asked to set the paper M1 and M1(CS)for the final two categories of candidates. As in previous years, the load was not evenly spread.

The innovation of last year that there should be independent checking of the addition of markson the scripts and of the recording of marks on the mark lists and in the data base was continuedwith great success.

Candidates were asked to begin each question in a new answer booklet, and to write the numbersof all the questions to be marked on the front answer booklet. This was highlighted more clearlyin the rubric on the papers and candidates were reminded verbally at the beginning and endof each exam. Even so, many failed to write the numbers of all the questions to be markedon the front answer booklet. Because of the different rubric for Honour Moderations, Part A,Part I and Part II, a separate coversheet that catered for all candidates was decided against forHonour Moderations. Only a few failed to begin each answer in a new answer booklet.

Submission of Maple projects: When the students submit a project they are given a confirmationnumber as part of the submission procedure. This number is also stored in the log files of thesubmission system.

C. Please list any changes in examining methods, procedures and conventionswhich the examiners would wish the faculty/department and the divisionalboard to consider.

In the light of their experience this year, the Examiners ask the Sub-faculty to re-consider thefollowing two points:

Formalizing the criteria for a pass:

The examiners suggest that consideration be given to introducing a criteria for a pass, of thistype: ”To pass Mods, a candidate must have a pass mark in at least 2 papers.”

The ‘One Bad Paper’ Rule:

All of the examiners were again concerned about the “one bad paper” rule and recommend itsabolition. It is unfair in several ways.

The Mathematics & Computer Science Moderators are not constrained this way, neither arethe Mathematics & Philosophy Moderators, classes being decided on total marks.

D. Please describe how candidates are made aware of the examination con-ventions to be followed by the examiners (Please attach any relevant docu-mentation to the report.)

This information was given in the Maths Undergraduate Handbook and re-iterated in the Noticeto candidates and e-mails circulated.

Part II of the Report

Moderators: D. R. Stirzaker (Chair), A. D. Lunn, W. B. Stewart, U. Tillmann, S. T. Tsou.

Production of papers: CRC in the Mathematical Institute by Sarah Hood.

Processing of marks: Carried out by Dr. Lunn, using a data base written by Alan Dyson.

Number of candidates: 2005: 211 (2004: 199) (2003: 193) (2002: 183).

A. GENERAL COMMENTS ON THE EXAMINATION

1. The Moderators would like to record their gratitude to Maria Moreno, Sarah Hood andJessica Hu in the Mathematical Institute for their support. They would also like to thankthe staff at Ewert House Examination Centre.

2. The Moderators are very grateful to Dr C.A. Wilkins and Dr Brian Stewart for theiradministration of the Maple projects.

3. The vast majority of candidates submitted five questions per paper, and no-one submittedmore than five answers.

4. The Moderators anticipate that it may be necessary to appoint an Assessor in 2005-2006,to run the database and implement the recalibration yielding USM’s.

0 20 40 60 80 1000.00

0.01

0.02

0.03

B

C

D

Raw marks

A

The density traces of the raw marks for each paper showed the papers to be of comparablestandard, though Papers B and C were found to be a little harder than Papers A and D. There isno doubt that the examiners achieved consistent values for mean and standard deviation, therebymaking the papers very good for discriminatory assessments: only minor transformations wereneeded for initial calibration.

B. EQUAL OPPORTUNITIES ISSUES AND BREAKDOWN OF THE RE-SULTS BY GENDER

The breakdown of the final classification by gender is as follows:-

I II III P F TotalMale 47(22.3%) 57(27%) 15(7.1%) 4(1.8%) 2(0.9%) 125 (59.2%)Female 17(8%) 56(26.5%) 9(4.3%) 2(0.9%) 2(0.9%) 86 (40.8%)Total 64(30.3%) 113(53.5%) 24(11.4%) 6(2.7%) 4(1.8%) 211 (100%)

C. DETAILED NUMBERS ON CANDIDATES’ PERFORMANCE IN EACHPART OF THE EXAMINATION- Mathematics candidates only

Raw marksA B C D

Mean 58.3 54.7 56.4 58.7Standard Deviation 14.1 16.3 17.1 13.5

Paper A

question Q1a Q1b Q2a Q2b Q3a Q3b Q4a Q4battempts 182 182 202 202 56 56 194 194mean 7.25 1.39 9.77 3.53 6.98 0.98 10.09 4.30


Paper B



Paper C



Paper D



D. COMMENTS ON PAPERS AND INDIVIDUAL QUESTIONS

Report on Paper A: Pure Mathematics I

General

Generally, the paper went reasonably well with raw marks producing roughly the right results(possibly too many thirds and passes). Once again this year, very few attempted the geometryquestion even though at least the first part was terribly easy. But more remarkable, the “hidden”geometry in the other questions was often treated very badly. In Q1 many candidates did not“see” that a plane through the origin is a 2 dimensional vector space. Quite a few thought thatthe dimension of a plane is 3 and that of a line is 2!. Similarly, only few candidates of thosewho attempted Q5 recognised the equation of a plane and only a couple could recognise anddescribe the intersection of two planes.

Question 1: This was quite a popular question though not a very high scoring one. Mostcandidates had trouble arguing why U ∩ W and U + W are finite dimensional. Surprisinglymany could not derive that two distinct planes through the origin in R3 intersected in a line.In part (b) about half of all candidates did not really know what was asked. Only a couplecandidates gave a sound argument.

Question 2: Nearly everyone attempted this question which was generally well done. Possiblyslightly on the easy side, this question showed that most candidates had a reasonable grip onthe rank-nullity theorem.

Question 3: This was not a very popular question. Some unfortunate candidates thought thatpart (ii) is correct, i.e. that the identity and the zero transformation are the only projections.Part (b) saw only a few good solutions.

Question 4: A very popular and straight forward question with many candidates achieving12 marks in part (a). Quite a few candidates tried to short cut calculations for the secondmatrix when they realized that it only had two eigen-values without determining the dimensionof the eigen-spaces. Otherwise, points were lost mainly due to calculational mistakes (includingproducing zero vectors as eigen-vectors). The first part of part (b) posed no great difficultywhile the second part was done often in a very confused and confusing way. Too many did notrealize that a diagonal matrix is diagonisable, and only a handful were able to argue that ifthere are two linearly independent vectors with the same eigen-value then the 2 by 2 matrixmust be diagonal.

Question 5: This question had a standard score of 9: Most candidates could show that the firstand last relations were equivalence relations and saw that the second is not reflexive. About a

third realised that the third might not be transitive but only about half of these were able toargue it convincingly. The answers to part (b) were shocking. I am convinced that the samequestion in an entrance paper would produce many more sensible geometric interpretations thenit did here.

Question 6: This question would have been quite hard if it hadn’t been for part (b) appearingmore or less verbatim on the lecturer’s last problem sheet. Many candidates were able toreproduce the main arguments satisfactory. The first part however proved to be more trickywith some candidates being confused by the notation.

Question 7: This quite straight forward question on the isomorphism theorem proved popular,even with some who could not quote the theorem but nevertheless managed to collect a fairnumber of points anyway by computing the inverse and showing that T is a group in part (a),and proving “by hand” that the Heisenberg group is a normal subgroup of T in part (b). I waspleased by the fair number of candidates who used the isomorphism theorem to solve the firstpart of (b) efficiently.

Question 8: This was really a very easy question, possibly too easy. Most who attempted itdid part (a) rather well. But marks were lost because the arguments were sloppy or no actualcounter example for the last part was produced. Surprisingly few spotted the answer to part(b). I suspect this is at least in part a function of not being prepared to tackle the geometryquestion and not quite believing that it could be this easy.

Epilogue

As last year the questions quite deliberately mixed aspects from different parts of the course.The geometry question Q8 was arguably more an algebra question; question Q7 on the iso-morphism theorem expected candidates to use row operations to find an inverse for a matrix.Elements of geometry were found in Q1 and Q5. As a result, not many perfect score for anyone question were produced.

Mathematics and Philosophy candidates produced half of about a dozen (raw) marks above 80,including the top mark of 91.

It seems to me that this year the Computer Science candidates attained better marks on thelinear algebra than last year.

Report on Paper B: Pure Mathematics II

General

It is clear that students write off—at least for examination purposes—the Trinity Term work,and the Geometry.

The candidates write far far too much; I wish there were 10 marks as in AC1 for presentation(including brevity). Or perhaps next year’s Moderators could set some questions ‘in less thanfour lines define sup(S)’.

Pictures may prove nothing, but they can give clues to the argument; I have been astonishedat the lack of sketches in the answers.

Many candidates, including some strong ones, thought they were doing Honour Moderations inMathematics & Statistics; I marked their scripts anyway.

Impressionistically I thought that there were some very strong candidates amongst the Maths& Philosophers; the statistics bear this out. I have no particular comment on the Maths &Computer Science candidates.

Question 1: (Algebra of Limits etc) The first part was done, and largely done well, byalmost every candidate. The second part needs care, and although the result was not new tothe candidates only the best were able to provide absolutely convincing proofs.

Question 2: (Monotonic sequences) Very popular, but not very well done. A surprisingnumber are prepared to admit sup ∅ to polite society. I did not think it sufficient in part(a)(iii) to use the ‘Approximation Property’ without a word of explanation. The real surprisewas (a)(iv): about a third of the candidates could not do this, producing spurious Cauchyconvergence arguments by considering |b2n+1 − b2n|.The second part was not intended as an exercise in algebra.

Question 3: (Geometric Series and Comparison Test) The first part was reasonablywell done. I attempted to make things easy by the hint in (a)(i), but few took it: they eitherasserted that inf{tn} = 0 or proved it from the Archimdean Property. I rejected all attemptsto use the Ratio Test for part (a)(iii) believing the argument to be circular; I likewise rejecteda surprising number of uses of the Comparison Test which compared

∑tn with

∑αn. One

candidate insisted that Cauchy sequences were not convergent: a non-trivial number requiredan extra condition (boundedness) for convergence. The majority stuck with what they knewwas safe and did part (iv) in two stages. Some merely asserted that the result would followfrom the Comparison Test.

The second half was badly done: almost no one made any linkage with part (a). I suppose thatthe first hint just got in the way, and wish I had set it as a trivial task in part (a).

Question 4: (Continuous functions) Reasonably popular, and part (a) was on the whole welldone. Part (a)(iii) was done almost universally by proving the contrapositive (by contradiction,

of course). I expected candidates to use in their proofs the definition of continuity they hadgiven and not another.

For part (b) it is not necessary to prove that a continuous function on [a, b] actually attainsits bounds. As there’s no mention of differentiability it ought to have been clear that Rolle’sTheorem was a red herring. One extremely good candidate explained elegantly why we couldwithout loss suppose that φ(−1) = φ(1), with φ(x) positive in the interval and negative outside.Many others would have made their ‘proofs’ more convincing (and picked up the cases they’dmissed) if they’d provided some sketches to guide the reader.

Question 5: (The Mean Value Theorem) Reasonably popular, but not well done; perhapsa mismatch between setter’s intentions and candidates’ understanding of the code.

In (a)(i) I expected a proof of the fact that the derivative vanishes at a local maximum. The proofof the MVT consists of trivial algebraic juggling, and two hard facts from analysis: existence ofmaxima for continuous functions (given in the question) and this one. In fact it is clear fromreading the answers to this and to the previous question that many candidates haven’t madethe distinction between ‘c is a local maximum’ and ‘f ′(c) = 0’.

In (a)(iii) I expected something more than the incantation ‘Inverse Function Theorem’; I wishI had added to the end of the sentence ‘(the Inverse Function Theorem)’.

Again, few made any linkage with part (a); tan(x) was increasing ‘from the graph’ even bymany who knew or were able to establish that its derivative is a square.

Question 6: (Power Series) This was not a successful question, and too many attempted it.

The problems in part (a) are these. Part (a)(iii) is too tough for candidates to do unseen, and ishighly dependent on the choice of definition of ‘radius’. Nevertheless, a non-trivial number didget it out, either by a comparison argument, or arguing via integration of uniformly convergentseries. More worryingly, only a minority who did the question realised in (a)(iii) and (iv) thatone series is the derivative (or indefinite integral) of the other. However as a result of thedifficulty of (a)(iii) I marked all the rest of the question generously.

I also realise that by setting the question in this way I may have reinforced the very commonbelief that the radius is always given by the ratio test. It is not; see question 3(b)(ii).

In part (b) almost no candidate thought it worthwhile to differentiate p(x) as well. I knowcandidates have been told how to prove identities like these; I was at one of the Mods lectureswhen it was done.

Question 7: (Basics of Integration and FTC) Not many takers for this question, althoughthose who tried it made good progress.

In (a)(iii) I expected some comment on the relevance of condition λ > 0; ‘linearity of sup’ isn’tthe answer.

I expected in part (b) to have the steps justified by reference to parts of (a). Too manycandidates just wrote down calculations.

Question 8: (Stereographic projection)

Only a handful of candidates offered this question. Without exception (and in old-fashionedgeometric style) they ignored the ‘if’ of part (a)(ii).

Report on Paper C: Applied Mathematics I

Question 1: An easy question intended to get them off to a good start: they were asked to solvea straightforward first-order differential equation in part (a)(i) and an even more straightforwardsecond-order, non-homogeneous equation in part (a)(ii). Part (b) asked them to solve twocoupled first order equations with a hint telling them how to proceed; for the most part thehint was ignored. This was by far the most popular question (202 attempts) and was not aswell answered (average mark 11.61) as one might have expected.

Question 2: This was the second most popular question (181 attempts) and was well answeredby the good candidates. Part (a) was straightforward (average mark 9.98), but part (b) wasquite testing (average mark 3.36). It was well answered by the majority of candidates.

Question 3: The great majority of those who attempted this were successful on part [a]. Butthen dealing with the pair of simple simultaneous equations for A and B in part [b] proved tobe surprisingly tricky for many.

Question 4: This was the least popular question. A good proportion of those who tried itfound it to be very easy, but quite a few had difficulty in simply integrating the equations ofmotion; even though their solution is easier than what typically appears in the second half ofquestion 1 of this paper. [As was the case this year.]

Question 5: This is a slightly non-standard problem, which the candidates are most unlikelyto have seen before. It is encouraging that it nevertheless proved to be a very popular andwell-done question, that yielded high marks and a great many alphas.

Question 6: It was anticipated that this would be an easy problem dealing with standardtechniques; unfortunately a very high proportion of solutions tried to get the answer thus:- ”byusing the independence of X and 1 - X .” This led to the average mark being rather lower thanhad been expected.

Question 7: Also a very popular question, that was really very well done by a great manycandidates. The use and understanding of generating functions was much better this year thanhas been seen at times in the past.

Question 8: The statistics question which, surprisingly, proved to be the third most popular(173 attempts); however, it was fairly clear that for many it was the last question attemptedand they ran out of time. The average mark was 12.14.

Report on Paper D: Applied Mathematics II

General

1. Most candidates do questions 1,2,3,4,5. What is the rationale behind this?

2. Q1 in its present format is too easy and does not provide enough of a test.

3. The gravitational question remains unpopular. Is is because it is lectured in Trinity Term?

Question 1: This standard bookwork question on Fourier series is extremely popular (all butone candidate attempted it). Most did very well on both parts; many gain full marks, althoughthere were almost no elegant answers. Most answers are too long. In part (b) quite a numberof candidates did not make use of the obvious parity of the given functions F (x) and G(x).The format should be changed in future, as this does not provide enough test on candiates’mathematical ability.

Question 2: The question was popular (191 attempts) and was generally answered well. Itproduced the second highest average mark (12.56).

Question 3: The waves question with the focus on being able to deduce boundary conditions.It was a relatively popular question (158 attempts) withan average mark of 10.52, the fifthhighest average. It was, perhaps, a little too long.

Question 4: Disappointing answers for what is really a straightforward piece of bookwork.Many candidates failed to discuss the three cases of the separation constant: positive, zero,negative. Also very few explained why there should be the integer n in the formula. Justdifferentiating the given expression for T (r, θ) to obtain Laplace’s equation does not constitutean answer to this question.

Question 5: Part (a), except the last bit, was well attempted. It is surprising that for the lastbit, many candidates did not know how to “change the origin” properly. Some just calculatedthe second derivatives at the origin, which is not a critical point. Many attempts on (b) aredisastrous, particularly for the following two reasons. First, some candidates memorized themethod of Lagrange multipliers by the formula ∇f = λ∇g, then called the two constaints fand g, and proceeded with the formula without bothering to introduce the distance funtionto be minimized. Second, the notation ∇, misleading in itself, hypnotized some candidatesinto evaluating just ∂/∂x and ∂/∂y, while there are 4 (or 3, depending on the method chosen)variables, thus not producing enough equations to solve the problem.However, there are a small number of very elegant solutions.

Question 6: Very few attempts, and among these no serious attempts for part (b). Theattempts on part (a) are in general successful.

Question 7: Part (a) gets many good answers, although the question is not so popular. Forpart (b), there was no attempt at even a sketch of how to define χ with the requisite properties.

So I decided that merely invoking the word “conservative” would get some marks.

Question 8: Only a handful of attempts, none serious. Once again, a gravitational force/potentialproblem proves extremely unpopular.

APPENDIX

Department of Mathematics: A consolidated list of Examiners’ Conventions:Moderations and Preliminary Examinations in Mathematics

Version of 29th November 2004

This document consolidates and confirms the examining the conventions concerning the settingand marking of the mathematical papers for Moderations examinations. It is subsidiary in allways to the current Examination Regulations and Notes for the Guidance of Examiners andChairmen of Examiners.

Examiners are reminded that any substantive change requires the consent of the TeachingCommittee of the Department.

1 Chairman of Examiners

Examining Boards are urged to make use of the provisions of ‘Regulations for the conduct ofexaminations, Part 6’, in the Examination Regulations and to choose a Chairman early in theacademic year.

2 Paperwork

2.1 Examiners

The Examiners should ensure that they are equipped with:

• The Examination Regulations.

• The Notes for the Guidance of Examiners and Chairmen of Examiners.

• The Educational Policy and Standards Committee Notes of Guidance on Examinationsand Assessment

• The Aims and Objectives of the mathematics courses, as agreed by the Teaching Com-mittee.

• The Course Handbook, including the Syllabuses and Lecture Synopses.

• The examination papers from the preceding two years.

• The Examiners’ Reports on these examinations.

• Any responses to these agreed by the Teaching Committee on behalf of the Subfaculty,and any additional decisions of the Teaching Committee.

• The published tables of Class Percentage Figures for the last two years.

3 The Examination Papers

3.1 The form of the questions

Examiners must be guided by previous papers (together with the Examiners’ Reports) whererelevant; and any Specimen Papers issued by the Sub-faculty.

3.2 Target Marks

Examiners should follow any guidance in the Course Handbook on the profile of marks theyare aiming for the candidates to achieve. Examination marks will be reported to candidates inUniversity Standardised Form. Examiners may recalibrate raw marks, but in setting the papersshould aim to minimise the need for recalibration.

3.3 Number and Form of Questions

Normally the number of questions on each paper is prescribed in the Examination Regulationsor Course Handbook (including the Lecture Synopses). There will be 8 questions on each paper.Each question will be divided into two parts: part(a) of a straight forward nature, and part(b)requiring more advanced understanding or an unseen application of techniques or theory on thesyllabus. Part(a) will attract 12 (raw)marks and part(b) 8 (raw)marks.

4 Setting and checking

4.1 Checklist for Setters and Checkers

The Examiners should provide those who are asked to supply draft questions with a checklistof important considerations.

1. Is the question on the syllabus (as in the Course Handbook)?

2. Is the mathematics correct?

3. Is the notation and terminology standard/obvious/defined? Is it unambiguous?

4. Is it clear what may be assumed, what detail is required, and what would constitute acomplete answer?

5. Is the question of a straightforward character? (This is no longer explicitly mentioned inthe Examination Regulations but setters should avoid unnecessary complexity.)

6. Is the form of presentation familiar/inviting/readable?

7. Has an easy start been provided?

8. Could a second-class or third-class candidate complete, or partially complete part(a)?

9. Can the question be done in half-an-hour under exam conditions?

10. Are the questions as a whole fairly spread across the syllabus?

11. Are the questions as a whole, and as far as it is relevant, of comparable standard toother questions this year and last year (taking into account comments in the Examiners’reports)?

12. Are the questions as a whole of a similar general nature, as far as it is relevant, to questionsin previous years (taking into account comments in the Examiners’ reports)?

4.2 Protocols

For Moderations and Prelims each paper (or part-paper) should be set by an Examiner (orAssessor) and checked by another, the whole paper being reviewed and approved by the wholeexamining board.

4.3 Marking Schemes

4.3.1 Model Solutions

Those setting questions must be asked to provide complete model solutions, annotated so as toindicate what is considered bookwork, and with a draft Marking Scheme for the approval of theexaminers; the solution should also make clear how much of the question is accessible to lessstrong candidates.

4.3.2 Aims of Marking Schemes

Marking schemes for the questions should aim to ensure that the following qualitative criteriahold:

For part (a) of the questions:

10-12 marks a completely, or almost completely, correct answer to this basic partof the question; only minor slips or omissions.

7-10 marks very elementary material substantially correct plus some demonstrationof understanding of standard bookwork or examples in this part of the question.

0-6 marks for the very elementary accessible material.

For part (b) of the questions:

6-8 marks a completely, or almost completely, correct answer to this more advancedpart of the question involving an unseen application or more advanced bookwork.

3-5 marks a worthy though incomplete answer to this more advanced part of thequestion.

0-2 marks answer shows some merit.

4.3.3 Approval of Marking Schemes

The Marking Schemes should be approved by the examining board alongside the papers.

5 Invigilation

The Examiners should inform a candidate’s college if an incident occurs during the sitting ofthe papers which is recorded in the log sheet, so that, for example, a medical certificate can besent to the Chairman of Examiners if appropriate.

6 Marking and Checking

6.1 Marking

Marking Schemes The Examiners have seen and approved the Marks Schemes, and Markersmust use these consistently. However, it may become clear while marking that the alloca-tion of marks should be changed. If you make such a change, then please make sure thatyou do so consistently, and that you tell the Examiners you have done so.

Mark Ranges in Mods In all Mods papers questions are to be marked out of 20, with part(a)attracting 12 (raw)marks and part(b) attracting 8 (raw)marks. Setters should aim to makeat least 6 marks of part(a) accessible to candidates with some basic knowledge of the topicexamined in that question.

Marking The Examiners will want to review at least some of the scripts during the clas-sification process. They will not (normally) want to re-mark (since they cannot do soconsistently across all candidates). They will want to be able to see quickly where markshave been gained. They will also want to be sure that all a candidate’s work has beentaken into consideration. Markers are therefore asked:

i. to indicate the marks given for each part of a question, by writing, e.g., 35 ;

ii. to show the total mark in some distinctive way, e.g., 18 ;

iii. to leave some trace that each page has been marked; pages on which no individualmarks have been shown should not be ticked, but marked \;

iv. to copy the total marks for both part(a) and part(b), plus the total mark for eachquestion on to the cover sheet;

v. to use some colour of ink not used by the candidates.

vi. to make remarks on the quality or otherwise of the answers if they wish; and if anargument is flawed to direct the Examiners to the defect.

Mark Sheets Pre-printed marks sheets will be supplied.

In entering into the pre-printed mark sheets the numerical mark for each part(a) andpart(b) of a question, care must be taken to distinguish between 0 marks for an attemptand − for a non-attempt.

Detailed instruction on the required check-sums will be given by the Examiners.

Before sending in the marks sheets markers should take a photocopy.

Reports The Examiners must have, for their use in the classification process, a brief reporton the performance of the candidates on each paper (or part-paper).

6.2 Checking the Marks

The Examiners should ensure that their procedures allow for:

• an additional arithmetic check of the correctness of the addition of the partial marks foreach question as recorded on the front of the script;

• an appropriate check of the marks entered into the marks database for each candidate;for example, by the use of a checksum.

The annexed note is commended as good practice.

6.3 Logging Scripts

The Examiners should ensure that a central log is kept of the whereabouts of all scripts; andshould instruct all Markers to return ‘sporadic’ scripts or answers to the central contact with anote of explanation.

7 Practical work

7.1 MAPLE in Mods

Marks for MAPLE will be communicated to the Moderators early in Trinity Term. The rawMaple marks will consist of two marks out of 20 but will incorporated into the average as amark out of 25, that is, as the equivalent of a quarter of a paper.

8 Classification of Candidates

Examiners are reminded of the criteria for various classes agreed by the Teaching Committeeand published in the Course Handbook(see 7.1). Examiners are also reminded that they mayexercise individual consideration in assigning USMs for candidates whose marks lie outside thestandard pattern, in order to ensure fair treatment.

8.1 Extract from Course Handbook 2004

All Mathematics candidates take four papers, viz.

1. Pure Mathematics I (PMI)

2. Pure Mathemtics II (PMII)

3. Applied Mathematics I (AMI)

4. Applied Mathematics II (AMII)

and submit two Maple projects.

The first two papers are also taken by candidates in Mathematics & Philosophy

Each paper has eight questions and candidates may submit answers to five questions. Eachquestion is marked out of 20 marks and will consist of two parts: part (a) of a straight forwardnature, part (b) requiring more advanced understanding or an unseen application of techniquesor theory on the syllabus. For complete answers to Part (a) on five questions a candidate willbe rewarded by a sound second class level USM, but further evidence of depth of understandingand the ability to solve problems demonstrated by answering some part (b) questions will beneeded to earn a first class (≥ 70) USM. The paper Applied Mathematics I will be dividedinto two sections: (i) Calculus and Dynamics, and (ii) Probability and Statistics; four questionswill be set on each section and candidates instructed that they should not submit answers tomore than five questions in all and not more than three questions from either section. Markswill be reported in university standardised form: 70+ a first class mark, 50-69 a second classmark, 40-49 a third class mark, 30-39 a pass mark, and below 30 a fail mark. Examiners mayrecalibrate the raw marks to arrive at university standardised marks reported to candidates.

The standardised marks for written papers and marks for Maple projects (MM) will be averagedaccording to the following two formula:

Av1 =PMI + PMII + AMI + AMII + 1

410040 MM

414

,

Av2 =PMI + PMII + AMI + AMII

4,

rounded up to a whole number.

Classes will be awarded according to the following conventions:

First : Av1 ≥ 70 and no standardised mark on any written paper < 50;

Second : Av1 ≥ 70 with at least one standardised mark on a written paper < 50

or

50 ≤ Av1 < 70 and no standardised mark on any written paper < 35;

Third : 50 ≤ Av1 < 70 with at least one standardised mark on any written papers < 35

or

40 ≤ Av1 < 50 and Av2 ≥ 40

or

40 ≤ Av2 < 50;

Pass : 30 ≤ Av2 < 40;

Fail : Av2 < 30.

A ‘Preliminary Examination’ is set for candidates who fail moderations or who, for some goodreason, are unable to sit Moderations.

The Preliminary Examination consists of two papers; one in Pure Mathematics and one inApplied Mathematics. This is an unclassified examination. To pass the examination a studentmust achieve a USM of at least 40 on each of the two papers and demonstrate understandingof sufficient breadth to satisfy the Examiners.

8.2 Qualitative description of examination performance for the various classes

First Class: the candidate shows excellent problem-solving skills and excellent knowl-edge of the material, and is able to use that knowledge in unfamiliar contexts.

Second Class: the candidate shows adequate/basic to good problem-solving skills and(good) knowledge of much of the material.

Third Class: the candidate shows reasonable understanding of at least part of the basicmaterial and some problem solving skills. Threshold level.

Pass: the candidate shows some limited grasp of basic material demonstrated by a mean-ingful attempt of at least one question.

Fail: little evidence of competence in the topics examined; the work is likely to showmajor misunderstanding and confusion, coupled with inaccurate calculations; theanswers to the questions attempted are likely to be fragmentary only.

9 Post-Examination

It will be helpful if Examiners will ensure that:

• Full Marking Schemes are deposited (after the examination is complete) in the Examiners’files in the Mathematics Department Academic Office.

• LATEX source files for the papers (incorporating any corrections) are supplied to the Math-ematical Institute for the electronic archive.

Chairmanon behalf of the Teaching Committee.

examiners’ report: honour moderations in mathematics ...examiners’ report: honour moderations in...

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