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Cambridge Mathematical Tripos

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The Mathematical Tripos is the taught mathematics course at the University of Cambridge.

1 Origin of the Mathematical Tripos
2 1909 Tripos reforms
3 Today's Mathematical Tripos
4 CATAM (Computer-Aided Teaching of All Mathematics)
5 Specific Structure of the Mathematical Tripos
- 5.1 Part 1A
6 References
7 See also
8 External links

Origin of the Mathematical Tripos

In its classical nineteenth-century form, the tripos was a distinctive written examination of undergraduate students of the University of Cambridge. From about 1780 to 1909, the "Old Tripos" was distinguished by a number of features, including the publication of an order of merit of successful candidates, and the difficulty of the mathematical problems set for solution.

Influence

According to the study Masters of Theory: Cambridge and the Rise of Mathematical Physics by Andrew Warwick, during this period the style of teaching and study required for the successful preparation of students had a wide influence: on the development of 'mixed mathematics' (a precursor of later applied mathematics and mathematical physics, with emphasis on algebraic manipulative mastery); on mathematical education; as vocational training for fields such as astronomy; and in the reception of new physical theories, particularly in electromagnetism as expounded by James Clerk Maxwell.

Early history

The early history is of the gradual replacement during the middle of the eighteenth century of a traditional method of oral examination by written papers, with a simultaneous switch in emphasis from Latin disputation to mathematical questions. That is, all degree candidates were expected to show at least competence in mathematics. A long process of development of coaching – tuition usually outside the official University and college courses – went hand-in-hand with a gradual increase in the difficulty of the most testing questions asked. The standard examination pattern of bookwork (mostly memorised theorems) plus rider (problems to solve, testing comprehension of the bookwork) was introduced.

Wranglers and their coaches

The list of wranglers, that is, the candidates awarded a first-class degree, became in time the subject of a great deal of public attention. The coaches, of whom Edward Routh was the most outstanding, assumed a para-academic status. The level of technique required of the candidates was high, and the time pressure in the examinations acute. It became common for those with a first degree in mathematics elsewhere to come to Cambridge to take part in the Tripos, as a second degree.

1909 Tripos reforms

The reforms implemented in 1909 did much to dismantle the old Mathematical Tripos system. It continued as an examination (and a course). The influence persists. In Cambridge terms, it has done much to support the particular kind of mathematical approach of the University's Faculty of Mathematics. G. H. Hardy, one of those most responsible for the changes, was concerned in particular to assert the importance of pure mathematics. The undergraduate course of mathematics at Cambridge still reflects a historically-broad approach; and problem-solving skills are tested in examinations, though the setting of excessively taxing questions has been discouraged for many years.

Today's Mathematical Tripos

Today, the Mathematical Tripos course comprises three undergraduate years (Parts IA, IB and II) which qualify a student for a BA degree, and an optional one year graduate course (Part III) which qualifies a student for a Certificate of Advanced Study in Mathematics. Assessment is by written examination at the end of each academic year.^[1] During the undergraduate part of the course, students are expected to attend around 12 one-hour lectures per week on average, together with two supervisions.^[1] Supervisions are informal sessions in which a small group of students work through problem sets, under the guidance of a faculty member, college fellow or graduate student. During the first two years (Parts IA and IB) the schedule of courses is quite rigid, and students have relatively little choice. Courses in these years cover pure mathematics (algebra and analysis); applied mathematics (electromagnetism, special relativity, quantum mechanics and fluid dynamics); and statistics. ^[2] Some students only take Part IA of the Mathematical Tripos, and switch to a related subject (typically Computer Science or Physics) after their first year. During the third year (Part II), a wider choice of courses is available and a student will typically begin to specialise in either pure mathematics or applied mathematics.

CATAM (Computer-Aided Teaching of All Mathematics)

An optional component of the mathematical Tripos is CATAM, a series of computational projects that students can undertake in an attempt to gain extra credit. CATAM projects are completed over the first two terms of the academic year, with the first set of projects due in soon after the first term (these are usually sent into the University just before Christmas).

Specific Structure of the Mathematical Tripos

Part 1A

There are three options; Pure and Applied Mathematics, Mathematics with Physics or Mathematics with Computer Science. In the latter two cases, either the Physics course from the Natural Sciences Tripos or some of the courses from the Computer Science Tripos are substituted for one of the Mathematics courses.

Michaelmas Term

There are four 24 hour lecture courses (all lectured in Cockroft Lecture Theatre). These are 'Groups', 'Vectors and Matrices', 'Numbers and Sets' (Mathematics with Physics students do not take this course), and 'Differential Equations'.

Lent Term

There are four 24 hour lecture courses; 'Analysis I', 'Probability', 'Vector Calculus' and 'Dynamics'. The year is examined over four 3 hour papers in early June - these are the Mathematical Tripos Examination Papers.

Part IA Mathematics Tripos Exam Structure

The Part IA Mathematics Tripos Exam Structure is summarised below. It should be noted that Mathematical Tripos examination papers usually contain two sections. The first section usually consists of short and relatively simple exam paper questions (providing a limited number of 'beta' marks for good attempts) whereas the second section consists of longer and more demanding exam paper questions (which provide opportunities to gain more valuable 'alpha' marks for essentially complete answers or additional 'beta' marks for less complete answers). Paper 1 : Vectors and Matrices , Analysis I Paper 2 : Differential Equations, Probability Paper 3 : Groups, Vector Calculus Paper 4 : Numbers and Sets, Dynamics The papers are not examined in quite this order though. In fact paper 3 follows paper 4. Each course is examined in exactly one paper and there are two section I questions marked out of 10, and three section II questions marked out of 20. Any section I question scoring 8 or more is further awarded a beta quality mark. Any section II question scoring at least 15 marks is awarded an alpha quality mark. Moreover a beta quality mark is given to any question from section II that scores between 10 and 14 inclusive. For ranking of candidates and thus awarding of degree classes, the most significant indicator used is not the raw mark but the merit mark calculated as follows. Merit Mark = m + 10α + 3β − 24 : whenever alpha > 8 Merit Mark = m + 7α + 3β : otherwise Where m denotes the raw mark obtained on examinations before being processed via the use of the statistical weighting formulae, α denotes the number of 'alpha' marks and β denotes the number of 'beta' marks. 30% of candidates are awarded first class honours. 45% are awarded upper seconds and up to 25% of candidates are awarded lower seconds. The number of candidates to be awarded thirds or below is not to exceed 6%.

Part IA Mathematics Tripos Exam Method

The number of marks assigned to a paper is determined by a certain examination process. The allocation of question marks can make a substantial difference when the mark allocated to a student. For section II questions, obtaining 14 marks instead of 15 marks for such questions in actuality results in a 'marks lost' weighting of not merely 1 mark, but (possibly) 8 marks due to the significant weighting given to 'alpha' marks within the Tripos paper. There is substantial emphasis placed upon solving more demanding and complex section II questions than their section I counterparts!

References

Masters of Theory: Cambridge and the Rise of Mathematical Physics (2003) Andrew Warwick, ISBN 0-226-87375-7.
Leonard Roth (1971) Old Cambridge Days, American Mathematical Monthly, 78, 223–236.

The Tripos was an important institution in nineteenth century England and many notable figures were involved with it. It has attracted broad attention from scholars. See for example:

John Gascoigne (1984) Mathematics and Meritocracy: The Emergence of the Cambridge Mathematical Tripos, Social Studies of Science, 14, 547–584.
Nicholas Griffin; Albert C. Lewis (1990) Bertrand Russell's Mathematical Education, Notes and Records of the Royal Society of London, 44, 51–71.
Christopher Stray (2001) The Shift from Oral to Written Examination: Cambridge and Oxford 1700–1900, Assessment in Education: Principles, Policy & Practice, 8, 33–50.

In old age two undergraduates of the 1870s wrote sharply contrasting accounts of the Old Tripos — one negative, one positive. Andrew Forsyth, Senior Wrangler 1881, stayed in Cambridge and was one of the reformers responsible for the New Tripos. Karl Pearson Third Wrangler in 1879 made his career outside Cambridge.

A. R. Forsyth (1935) Old Tripos Days in Cambridge, Mathematical Gazette, 19, 162-179.
Karl Pearson (1936) Old Tripos Days at Cambridge, as Seen from Another Viewpoint, Mathematical Gazette, 20, 27-36.

J. J. Thomson, a Second Wrangler in 1880, wrote about his experience in:

J. J. Thomson Recollections and Reflections London: G. Bell, 1936.

J. E. Littlewood, a Senior Wrangler in the last years of the old Tripos, recalled the experience in:

J. E. Littlewood A Mathematician's Miscellany (2nd edition published in 1986), Cambridge University Press.
G. H. Hardy, A Mathematician's Apology, Cambridge University Press (1940). 153 pages. ISBN 0-521-42706-1.