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The Mathematical Tripos is the mathematics course that is taught in the
Faculty of Mathematics In contemporary education, mathematics education, known in Europe as the didactics or pedagogy of mathematics – is the practice of teaching, learning and carrying out scholarly research into the transfer of mathematical knowledge. Although rese ...
at the
University of Cambridge , mottoeng = Literal: From here, light and sacred draughts. Non literal: From this place, we gain enlightenment and precious knowledge. , established = , other_name = The Chancellor, Masters and Schola ...
. It is the oldest
Tripos At the University of Cambridge, a Tripos (, plural 'Triposes') is any of the examinations that qualify an undergraduate for a bachelor's degree or the courses taken by a student to prepare for these. For example, an undergraduate studying mathe ...
examined at the University.


Origin

In its classical nineteenth-century form, the tripos was a distinctive written examination of undergraduate students of the
University of Cambridge , mottoeng = Literal: From here, light and sacred draughts. Non literal: From this place, we gain enlightenment and precious knowledge. , established = , other_name = The Chancellor, Masters and Schola ...
. Prior to 1824, the Mathematical Tripos was formally known as the "Senate House Examination". 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. By way of example, in 1854, the Tripos consisted of 16 papers spread over 8 days, totaling 44.5 hours. The total number of questions was 211. The actual marks for the exams were never published, but there is reference to an exam in the 1860s where, out of a total possible mark of 17,000, the senior wrangler achieved 7634, the second wrangler 4123, the lowest wrangler around 1500 and the lowest scoring candidate obtaining honours (the
wooden spoon Wooden Spoon may refer to: * Wooden spoon, implement * Wooden spoon (award) A wooden spoon is an award that is given to an individual or team that has come last in a competition. Examples range from the academic to sporting and more frivolous e ...
) 237; about 100 candidates were awarded honours. The 300-odd candidates below that level did not earn honours and were known as ''poll men''. The questions for the 1841 examination may be found within ''Cambridge University Magazine'' (pages 191-208).


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 Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
and
mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
, with emphasis on algebraic manipulative mastery) * on mathematical education * as vocational training for fields such as
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
* in the reception of new physical theories, particularly in
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
as expounded by
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
Since Cambridge students did a lot of
rote learning Rote learning is a memorization technique based on repetition. The method rests on the premise that the recall of repeated material becomes faster the more one repeats it. Some of the alternatives to rote learning include meaningful learning, as ...
called "bookwork", it was noted by Augustus De Morgan and repeated by Andrew Warwick that authors of Cambridge textbooks skipped known material. In consequence, "non-Cambridge readers ... found the arguments impossible to follow."


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 Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
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
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
s) plus ''rider'' (problems to solve, testing comprehension of the bookwork) was introduced.


Wranglers and their coaches

The list of wranglers (the candidates awarded a first-class degree) became in time the subject of a great deal of public attention. According to
Alexander Macfarlane Alexander Macfarlane FRSE LLD (21 April 1851 – 28 August 1913) was a Scottish logician, physicist, and mathematician. Life Macfarlane was born in Blairgowrie, Scotland, to Daniel MacFarlane (Shoemaker, Blairgowire) and Ann Small. He s ...
:To obtain high honours in the Mathematical Tripos, a student must put himself in special training under a mathematician, technically called a coach, who is not one of the regular college instructors, nor one of the University professors, but simply makes a private business of training men to pass that particular examination. Skill consists in the rate at which one can solve and more especially write out the solution of problems. It is excellent training of a kind, but there is not time for studying fundamental principles, still less for making any philosophical investigations. Mathematical insight is something higher than skill in solving problems; consequently the
senior wrangler The Senior Frog Wrangler is the top mathematics undergraduate at the University of Cambridge in England, a position which has been described as "the greatest intellectual achievement attainable in Britain." Specifically, it is the person who a ...
has not always turned out the most distinguished mathematician in after life.
William Hopkins William Hopkins FRS (2 February 179313 October 1866) was an English mathematician and geologist. He is famous as a private tutor of aspiring undergraduate Cambridge mathematicians, earning him the ''sobriquet'' the " senior-wrangler maker." ...
was the first coach distinguished by his students' performances. When he retired in 1849, one of his students,
Edward Routh Edward John Routh (; 20 January 18317 June 1907), was an English mathematician, noted as the outstanding coach of students preparing for the Mathematical Tripos examination of the University of Cambridge in its heyday in the middle of the ninet ...
became the dominant coach. Another coach,
William Henry Besant William Henry Besant (1 November 1828 – 2 June 1917) was a British mathematician, brother of novelist Walter Besant. Another brother, Frank, was the husband of Annie Besant. Parentage William was born in Portsea, Portsmouth on 1 November 18 ...
published a textbook, ''Elementary Hydrostatics'', containing mathematical exercises and solutions such as would benefit students preparing for Tripos. After Routh retired in 1888,
Robert Rumsey Webb Robert Rumsey Webb (9 July 1850 – 29 July 1936), known as R. R. Webb, was a successful coach for the Cambridge Mathematical Tripos. Webb coached 100 students to place in the top ten wranglers from 1865 to 1909, a record second only to Edwar ...
coached many of the top wranglers. Warwick notes that college teaching improved toward the end of the 19th century: :The expansion of intercollegiate and university lectures at all levels through the 1880s and 1890s meant that, by 1900, it had become unnecessary for coaches either to lecture students or even to provide them with manuscripts covering the mathematical methods they were required to master. The prime job to the coach now was to ensure that students were attending an appropriate range of courses and that they understood what they were being taught. … This curtailment of responsibility made it virtually impossible for a private tutor to dominate undergraduate training the way that Hopkins, Routh, and Webb had done. A fellow of Trinity College,
Robert Alfred Herman Robert Alfred Herman (1861–1927) was a fellow of Trinity College, Cambridge, who coached many students to a high wrangler rank in the Cambridge Mathematical Tripos. Herman was senior wrangler in 1882. In the early days of Tripos, coaches wer ...
then was associated with several of the top wranglers as their coach; evidently the University was finally providing their students with education. When A. R. Forsyth wrote his retrospective in 1935, he recalled Webb,
Percival Frost Percival Frost (1817–1898), was an English mathematician. Life Percival Frost was born at Kingston upon Hull on 1 September 1817, the second son of Charles Frost. He was educated at Beverley and Oakham, and entered St. John's College, Cambrid ...
, Herman, and Besant as the best coaches. Other coaches that produced top wranglers include E. W. Hobson, John Hilton Grace,
H. F. Baker Henry Frederick Baker FRS FRSE (3 July 1866 – 17 March 1956) was a British mathematician, working mainly in algebraic geometry, but also remembered for contributions to partial differential equations (related to what would become known as ...
,
Thomas John I'Anson Bromwich Thomas John I'Anson Bromwich (8 February 1875 – 24 August 1929) was an English mathematician, and a Fellow of the Royal Society. Life Thomas John I'Anson Bromwich was born on 8 February 1875, in Wolverhampton, England. He was descended from ...
, and A. E. H. Love.


Athletics

Apart from intellectual preparation, the challenge of Tripos was its duration: "The examinations themselves were intended partly as tests of endurance, taking place on consecutive mornings and afternoons for four and five days together." Brisk walking was taken up by many candidates to build up their stamina. As the nineteenth century progressed walking turned to
athletics Athletics may refer to: Sports * Sport of athletics, a collection of sporting events that involve competitive running, jumping, throwing, and walking ** Track and field, a sub-category of the above sport * Athletics (physical culture), competi ...
and other competitive
sports Sport pertains to any form of competitive physical activity or game that aims to use, maintain, or improve physical ability and skills while providing enjoyment to participants and, in some cases, entertainment to spectators. Sports can, th ...
including
rowing Rowing is the act of propelling a human-powered watercraft using the sweeping motions of oars to displace water and generate reactional propulsion. Rowing is functionally similar to paddling, but rowing requires oars to be mechanically atta ...
and
swimming Swimming is the self-propulsion of a person through water, or other liquid, usually for recreation, sport, exercise, or survival. Locomotion is achieved through coordinated movement of the limbs and the body to achieve hydrodynamic thrust that r ...
. The coaches set the example: Routh had a two-hour constitutional walk daily, while "Besant was a mountaineer, Webb a walker, and Frost was extremely proficient in cricket, tennis, running and swimming." By 1900, there were twenty-three recognized sports contested at Cambridge.


Women

In 1873,
Sarah Woodhead Sarah Woodhead (1851–1912) was the first woman to take and pass a Tripos examination. In particular, she was the first woman to take, and to pass, the Mathematical Tripos exam, which she did in 1873. Education Woodhead’s family had long belon ...
became the first woman to take, and to pass, the Mathematical Tripos. In 1880,
Charlotte Angas Scott Charlotte Angas Scott (8 June 1858 – 10 November 1931) was a British mathematician who made her career in the United States and was influential in the development of American mathematics, including the mathematical education of women. Scott ...
obtained special permission to take the Mathematical Tripos, as women were not normally allowed to sit for that exam. She came eighth on the Tripos of all students taking them, but due to her sex, the title of "eighth wrangler," a high honour, went officially to a male student. At the ceremony, however, after the seventh wrangler had been announced, all the students in the audience shouted her name. Because she could not attend the award ceremony, Scott celebrated her accomplishment at Girton College where there were cheers and clapping at dinner, a special evening ceremony where the students sang "See the Conquering Hero Comes", received an ode written by a staff member, and was crowned with laurels. After this incident women were allowed to formally take the exam and their exam scores listed, although separately from the men's and thus not included in the rankings. Women obtaining the necessary score also received a special certificate instead of the BA degree with honours. In 1890, Philippa Fawcett became the first woman to obtain the top score in the Mathematical Tripos.


1909 reforms

Reforms were implemented in 1909. 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.


The modern tripos

, 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 masters course (
Part III ''Part III'' is the third studio album by American R&B group 112. It was released by Bad Boy Records on March 20, 2001 in the United States. Unlike the previous releases, the album is described as having edgier, techno-flavored jams, resulting ...
) which qualifies a student for a
Master of Mathematics A Master of Mathematics (or MMath) degree is a specific advanced integrated Master's degree for courses in the field of mathematics. United Kingdom In the United Kingdom, the MMath is the internationally recognized standard qualification after a f ...
(MMath) degree (with BA) if they are a Cambridge fourth year student or a Master of Advanced Study (MASt) degree if they come from outside just to do
Part III ''Part III'' is the third studio album by American R&B group 112. It was released by Bad Boy Records on March 20, 2001 in the United States. Unlike the previous releases, the album is described as having edgier, techno-flavored jams, resulting ...
. Assessment is mostly by written examination at the end of each academic year, with some coursework elements in the second, third and fourth years. 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. Supervisions are informal sessions in which a small group of students—normally a pair—goes through previously completed example sheets under the guidance of a faculty member, college fellow or graduate student. During the first year, Part IA, the schedule of courses is quite rigid, providing much of the basic knowledge requisite for mathematics, including
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
,
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
, methods in
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, and
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
. The second year, Part IB, contains no mandatory content but it is recommended that students do particular courses as they are essential prerequisites for further courses. A range of pure courses, such as
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
,
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
and a course studying
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
,
rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
and
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
are on offer as well as applied courses on
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
,
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
and
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
. In Part II, students are free to choose from a large number of courses over a wide range of mathematical topics, these are separated into more accessible C courses and D courses which are more involved. Some students choose to exchange 25% of the first year mathematics options in exchange for the Physics option of first year Natural Sciences Tripos with the possibility of changing to Natural Sciences at the end of the first year.


References


Further reading

* Rouse Ball, A History of the Study of Mathematics at Cambridge * Leonard Roth (1971) "Old Cambridge Days",
American Mathematical Monthly ''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an e ...
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: * * 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 Karl Pearson (; born Carl Pearson; 27 March 1857 – 27 April 1936) was an English mathematician and biostatistician. He has been credited with establishing the discipline of mathematical statistics. He founded the world's first university st ...
Third Wrangler in 1879 made his career outside Cambridge. * *
J. J. Thomson Sir Joseph John Thomson (18 December 1856 – 30 August 1940) was a British physicist and Nobel Laureate in Physics, credited with the discovery of the electron, the first subatomic particle to be discovered. In 1897, Thomson showed that ...
, a Second Wrangler in 1880, wrote about his experience in: * J. J. Thomson ''Recollections and Reflections'' London: G. Bell, 1936.
J. E. Littlewood John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to mathematical analysis, analysis, number theory, and differential equations, and had lengthy collaborations with G. H. H ...
, 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 Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...
. *
G. H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
,
A Mathematician's Apology ''A Mathematician's Apology'' is a 1940 essay by British mathematician G. H. Hardy, which offers a defence of the pursuit of mathematics. Central to Hardy's " apology" — in the sense of a formal justification or defence (as in Plato's '' Ap ...
, Cambridge University Press (1940). 153 pages. . * Kathryn M. Olesko (2004
Review of ''Masters of Theory''
from
American Scientist __NOTOC__ ''American Scientist'' (informally abbreviated ''AmSci'') is an American bimonthly science and technology magazine published since 1913 by Sigma Xi, The Scientific Research Society. In the beginning of 2000s the headquarters was in New ...
magazine. * Theodore M. Porter (2003
Review of ''Masters of Theory''
from
Science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence for ...
. On the importance of the Tripos in the history of mathematics in Britain: search on "tripos" in
The MacTutor History of Mathematics archive
For statistics on the number of graduates (men and women) between 1882 and 1940 see:

For the present-day Tripos see:
Cambridge University: Guide to the Mathematical Tripos
(pdf)
Actual examination papers from 2001 onwardsThe Cambridge Maths faculty's site explaining Part III
* Nelson, Graham
"Miss Warren’s Profession"
''Eureka 51'', 1992. Critique of Part III. {{Authority control
Mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
Mathematics Tripos The Mathematical Tripos is the mathematics course that is taught in the Faculty of Mathematics at the University of Cambridge. It is the oldest Tripos examined at the University. Origin In its classical nineteenth-century form, the tripos was a ...
Mathematics education in the United Kingdom University folklore