Examiners Report: Honour Moderations in Mathematics and Philosophy 2005

Transcription

1 Examiners Report: Honour Moderations in Mathematics and Philosophy Part I A. STATISTICS (1) Numbers and percentages in each class/category Number Percentages % 2004/5 2003/4 2002/3 2004/5 2003/4 2002/3 I II III Pass Fail (2) Marking of scripts No scripts were double marked. B. NEW EXAMINING METHODS AND PROCEDURES There were no changes from last year. C. CHANGES TO BE CONSIDERED We have no proposals. D. NOTIFICATION OF EXAMINATION CONVENTIONS Candidates are made aware of the conventions through the course handbook and through a notice sent to them in Trinity Term. Part II A. GENERAL COMMENTS ON THE EXAMINATION The moderators would particularly like to thank Sarah Hood and other members of the Maths Institute admin team for their support. They are grateful to Dan Lunn for operating the database 1

2 system. They also thank the invigilation team at Ewert House for their notable efficiency and helpfulness. The overall standard was very high, and the performance by the candidates in this examination compared favourably with those in the single school in the shared mathematics papers, as can be seen from the tables below. As a result there was a high proportion of Firsts. The moderators were faced with no difficult decisions. The Firsts were clear-cut; and there were no hard cases to test the rule on excellence on one side, with adequate knowledge on the other. USM Mean USM Standard Deviation Pure Mathematics I Pure Mathematics II Elements of Deductive Logic Introduction to Philosophy For comparison, the corresponding figures for the single school were: USM Mean USM Standard Deviation Pure Mathematics I Pure Mathematics II B. GENDER ANALYSIS I II III P F Total Male 9(60%) 6(40.0%) (57.7%) Female 4 (36.4%) 6(54.6%) 1(9.1%) (42.3%) Total 13(50%) 12(42.3%) 1(7.7%) Moderators: W. B. Stewart (Chairman), A. D. Lunn, D. R. Isaacson, O. E. E. Pooley. 2

3 REPORTS ON INDIVIDUAL PAPERS (The reports on Papers 1 and 2 cover candidates from all schools taking these papers.) Report on Paper 1: Pure Mathematics I 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 geometry question 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 that the dimension of a plane is 3 and that of a line is 2!. Similarly, only few candidates of those who attempted Q5 recognised the equation of a plane and only a couple could recognise and describe the intersection of two planes. Question 1, Paper A, Paper M1, Paper M1(CS) This was quite a popular question though not a very high scoring one. Most candidates had trouble arguing why the U W and U + W are finite dimensional. Surprisingly many could not derive that two distinct planes through the origin in R 3 intersected in a line. In part (b) about half of all candidates did not really know what was asked. Only a couple candidates gave a sound argument. Question 2, Paper A, Paper M1, Paper M1(CS) Nearly everyone attempted this question which was generally well done. Possibly slightly on the easy side, this question showed that most candidates had a reasonable grip on the rank-nullity theorem. Question 3, Paper A, Paper M1(CS) This was not a very popular question. Some unfortunate candidates thought that part (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, Paper A, Paper M1(CS), Question 3, Paper M1 A very popular and straight forward question with many candidates achieving 12 marks in part (a). Quite a few candidates tried to short cut calculations for the second matrix when they realized that it only had two eigen-values without determining the dimension of the eigen-spaces. Otherwise, points were lost mainly due to calculational mistakes (including producing zero vectors as eigenvectors). The first part of part (b) posed no great difficulty while the second part was done often in a very confused and confusing way. Too many did not realize that a diagonal matrix is diagonisable, and only a handful were able to argue that if there are two linearly independent vectors with the same eigen-value then the 2 by 2 matrix must be diagonal. Question 5, Paper A This question had a standard score of 9: Most candidates could show that the first and 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 to argue it convincingly. The answers to part (b) were shocking. I am convinced 3

4 that the same question in an entrance paper would produce many more sensible geometric interpretations then it did here. Question 6, Paper A, Question 4, Paper M1 This question would have been quite hard if it hadn t been for part (b) appearing more or less verbatim on the lecturer s last problem sheet. Many candidates were able to reproduce the main arguments satisfactory. The first part however proved to be more tricky with some candidates being confused by the notation. Question 7, Paper A, Question 5, Paper M1 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 fair number 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 was pleased by the fair number of candidates who used the isomorphism theorem to solve the first part of (b) efficiently. Question 8, Paper A This was really a very easy question, possibly too easy. Most who attempted it did part (a) rather well. But marks were lost because the arguments were sloppy or no actual counter 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 geometry question and not quite believing that it could be this easy. Question 5, Paper M1 (CS) This was an easy question attempted by all but four candidates and which went quite well in general. There was some misunderstanding in that some candidates produced an inverse in part (b) for only one value of β which they picked conveniently. 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 isomorphism 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 any one 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 the linear algebra than last year. 4

5 Report on Paper 2: 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 than four lines define sup(s). Pictures may prove nothing, but they can give clues to the argument; I have been astonished at the lack of sketches in the answers. Many candidates, including some strong ones, thought they were doing Honour Moderations in Mathematics & 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, by almost every candidate. The second part needs care, and although the result was not new to the candidates only the best were able to provide absolutely convincing proofs. Question 2: (Monotonic sequences) Very popular, but not very well done. A surprising number 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 surprise was (a)(iv): about a third of the candidates could not do this, producing spurious Cauchy convergence arguments by considering b 2n+1 b 2n. The second part was not intended as an exercise in algebra. Question 3: (Geometric Series and Comparison Test) The first part was reasonably well done. I attempted to make things easy by the hint in (a)(i), but few took it: they either asserted that inf{t n } = 0 or proved it from the Archimdean Property. I rejected all attempts to use the Ratio Test for part (a)(iii) believing the argument to be circular; I likewise rejected a surprising number of uses of the Comparison Test which compared t n with α n. One candidate insisted that Cauchy sequences were not convergent: a non-trivial number required an extra condition (boundedness) for convergence. The majority stuck with what they knew was safe and did part (iv) in two stages. Some merely asserted that the result would follow from the Comparison Test. The second half was badly done: almost no one made any linkage with part (a). I suppose that the 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 well done. 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 had given and not another. For part (b) it is not necessary to prove that a continuous function on [a, b] actually attains its bounds. As there s no mention of differentiability it ought to have been clear that Rolle s Theorem was a red herring. One extremely good candidate explained elegantly why we could 5

6 without 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 d missed) if they d provided some sketches to guide the reader. Question 5: (The Mean Value Theorem) Reasonably popular, but not well done; perhaps a 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 proof of the MVT consists of trivial algebraic juggling, and two hard facts from analysis: existence of maxima for continuous functions (given in the question) and this one. In fact it is clear from reading the answers to this and to the previous question that many candidates haven t made the 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 wish I 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 by many 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 is highly dependent on the choice of definition of radius. Nevertheless, a non-trivial number did get it out, either by a comparison argument, or arguing via integration of uniformly convergent series. More worryingly, only a minority who did the question realised in (a)(iii) and (iv) that one series is the derivative (or indefinite integral) of the other. However as a result of the difficulty 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 common belief 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 know candidates have been told how to prove identities like these; I was at one of the Mods lectures when it was done. Question 7: (Basics of Integration and FTC) Not many takers for this question, although those who tried it made good progress. In (a)(iii) I expected some comment on the relevance of condition λ > 0; linearity of sup isn t the answer. I expected in part (b) to have the steps justified by reference to parts of (a). Too many candidates just wrote down calculations. Question 8: (Stereographic projection) Only a handful of candidates offered this question. Without exception (and in old-fashioned geometric style) they ignored the if of part (a)(ii). 6

7 Report on Paper 3: Elements of Deductive Logic Questions were marked out of of the scripts were first class, 3 were third class. There were no failures. 1. (3 answers) Two very good answers, and one very poor (the candidate did not read that the question required a symbolization into propositional logic). 2. (21 answers) 12 answers attained 20 or more marks, 1 of which attained 25 marks. This questions was all book-work, which was known pretty well by most candidates who answered this question. Given the familiar nature of the material it was perhaps a little disappointing that so many answers were rather scrappy. 3. (9 answers) Only one candidate who answered this question got a first class mark, although a few others were not too far behind. Only one candidate recognized that two types of case needed to be considered in the inductive step of the proof in part (c). Very few candidates gained marks on part (d), and those who did only proved that the dual of an expressively adequate connective was also expressively adequate; they did not, as was required, prove that no self-dual 2-place connective was expressively adequate. 4. (4 answers) One candidate only answered part (a), the best answer received 20 marks. 5. (17 answers) 9 candidates attained 20 or more marks and most scored very highly. Candidates were very clear about how Hodges restriction on the tableau rule for the universal quantifier allows for the empty domain. They were clear about the truth-values of (namefree) universally and existentially quantified formulas in the empty domain, but those who chose to justify their claims in formal terms tended to be rather scrappy in their handling of the formal semantics of quantified formulas. 6. (23 answers) Only 9 answers were first class and, although relatively few candidates did very badly on this question, it was not done very well in general. Most candidates opted to answer section (b) by providing appropriate tableaux, a rather time-consuming option that they might have been primed to avoid if they had read the comments in last year s Examiners Report on question 8 of the 2004 EDL paper. A surprising number of candidates did not know the definition of a connected binary relation, either omitting the requirement of non-identity from their symbolization, or giving a symbolization of a relation s being serial. More than half the candidates failed to prove that the relation S was symmetric, and only about a third of candidates provided a model in which S was not transitive. 7. (10 answers) Most answers were first class. Part (d) was the least well-answered part of this question, with candidates failing to set out their reasoning clearly and rigorously. Very few candidates really explained why the soundness of rule (ii) alone did not give any information about the central rows of the truth-table. 8. (17 answers) Only 5 answers were clearly first class, and there were a number of surprisingly weak answers. In part (a) very few candidates dealt with the definiteness of the description the philosophy fellow at Wadham. In part (b) very few candidates offered a 7

8 symbolization that took account of the absence of the adjective piano in the conclusion of the argument. In part (c) a significant number of candidates failed to symbolize the premise adequately. Many of those whose symbolizations required the use of the tableau rules for identity in order to close the tableau failed to employ these properly, closing branches, for example, if they contained a = c and c = a. 8

9 Report on Paper 4: Introduction to Philosophy Each question had at least one answer, with an uneven distribution. Two questions (1 and 22) received just one answer and seven questions (2, 5, 8, 18, 19, 20, 21) out of the fifteen received 74% of the answers. There were 50 answers to questions on General Philosophy and 54 answers to questions on Frege (twelve candidates answered two questions on both subjects, six answered three question from General Philosophy and one question on Frege, eight answered one question on General Philosophy and three on Frege). With the exception of a mark of 91, the range of marks was from 53 to (1 answer) The one answer was reasonably good. 2. (12 answers) The most popular question from General Philosophy. The weaker answers ignored the obvious and prevalent cases where the future most certainly is not always and in the long run never is like the past (e.g. surviving to one s next birthday) and stuck to the most hackneyed example of the topic. Some strong answers made good use of the literature, particularly Goodman s New riddle of induction. 3. (5 answers) Most of the answers gave a reasonable exposition of the view that colour is a secondary quality on Locke s distinction, and then went on, with varying success, to considerations in favour of counting colour as a primary quality. 4. (3 answers) Two good answers and one rather weak one. 5. (11 answers) Second most popular answer from General Philosophy. They were all, either explicitly or tacitly, expounding, with varying degree of accuracy and clarity, Frankfurt s arguments in his article Alternate possibilities and moral responsibility. Only one candidate took it upon him or herself to consider what is meant by moral responsibility. 6. (5 answers) All five answers were either explicitly or tacitly based on Strawson s arguments in Self, mind and body. 7. (4 answers) Only four answers, but each one first class and two of them outstandingly good, with very careful and perceptively critical consideration of Putnam s arguments in his Brains in a vat, including exposition of Putnam s account of reference. 8. (9 answers) Some good answers, others fairly unfocused. Candidates tended to treat this question as if they had been asked to give criterion for personal identity, relevant but not by itself an answer to the question asked. 16. (4 answers) One strong answer, two fairly strong answers, and one weak answer. The weak answer was unclear about the relationship between Frege s definitions of number and of natural number. 17. (4 answers) One good answer, the others fairly weak, not clear about the role played by Hume s principle in establishing the infinity of the natural numbers, the weakest ones claiming that the infinity of the natural numbers is equivalent to the existence of a successor for each number (necessary but not sufficient). 9

10 18. (16 answers) The most answered question on the exam, and there were some very good answers. One superb answer included an argument that Hume s principle can t show that Julius Caesar isn t a number by noting that Hume s principle is compatible with taking the natural numbers as the sequence of finite von Neumann ordinals starting with Julius Caesar rather than the empty set. Candidates differed in their awareness of the various contexts in which the Julius Caesar problem presents itself in Frege s project. A considerable number rightly noted, as part of a negative answer to the question, that the Julius Caesar problem arises equally for abstract objects, so the problem cannot be dismissed in the way suggested. 19. (10 answers) Everyone who answered this question answered it in the negative and also, with one exception, took the view, correctly, that Frege would have answered it in the negative himself. 20. (11 answers) All answers made the point that Frege s definition of analyticity differs from Kant s. Relatively few focused on their differing understanding of logic. Most candidates, though not all, noted that despite the differing notion of analyticity, Frege s is in direct disagreement with Kant over Kant s appeal to pure intuition as the basis for arithmetic. One terrifically good answer traced the evolution of Kant s notion of analyticity from the Critique of Pure Reason to the Prolegomena and claimed on this basis that Frege s notion of analyticity was in the spirit of Kant s and so a basis for disagreement. Some candidates usefully brought in Frege s reasons for agreeing with Kant that geometry is synthetic while holding that arithmetic is analytic. 21. (8 answers) Only two first class answers among the eight who answered this question. Pretty much everyone who answered this question discussed the importance for Frege of holding that attribution of number is to concepts and not objects but had relatively little to say as to how Frege analyzed this distinction. 22. (1 answer) The one answer made some basic points about second-order logic in comparison with first-order logic but had nothing to say about Frege s use of second-order logic to define and prove basic properties of the relation of following in the φ-series (ancestral of a relation), arguably his most important reason for developing second-order logic. 10