Masters of Theory: Cambridge and the Rise of Mathematical Physics by Andrew Warwick
What does it mean to excel at mathematics? And how should you teach towards this kind of excellence?
A common way of telling it is that for as long as people have been learning math, there has been a “traditional” way of teaching it. The tradition of traditional math teaching involves, but is not limited to: textbooks, nightly homework, in class problem solving performances, corrections and grades.
When did it start? Where? One of the things that really stunned me about Masters of Theory is that there is an entirely concise and known answer to this question: it started in Cambridge University, and coalesced in the 1840s and 1850s. And as to the question of “why?,” the answer is largely: because of competition and testing.
More on that development in a moment. First, some trivia.
- Did you know that the word “coach,” as in “mathematics coach” or “basketball coach,” came into practice in the 1830s in England to describe private mathematics tutors? The word was borrowed from “coach” as a mode of transport — the most efficient, fastest, classiest way to travel. When Cambridge undergrads were reaching for a word to describe the work of someone you could hire (if you had money) to help you smoothly move through the mathematical terrain of the mathematics curriculum, they turned to analogy: coaches.
- For a top student, James Clerk Maxwell had a reputation as a sloppy student of math at Cambridge, and his attempt at capturing his thoughts on electromagnetism in a text was basically incomprehensible.
- Before the 19th century, paper had very little to do with learning mathematics.
These pieces of trivia fit together nicely in Warwick’s detailed narrative. There are clear stages marking the shift. Here is a summary.
Step 0: What was learning university math like before the shift?
The goal in the medieval university wasn’t technical facility with a problem solving apparatus. You weren’t supposed to be able to solve problems. The goal was knowledge of works of mathematics.
A central text was Euclid:
“The main form of undergraduate teaching was the thrice weekly college lectures in which students were taught the basics required for a pass or mediocre honors degree…The main job of the mathematics lecturer was to go through required sections of such important texts as Euclid’s Elements, ensuring that the majority of students had at least a minimal grasp of geometry, arithmetic, algebra, mechanics, hydrostatics, optics and astronomy. Lectures were run at the pace of the average student and appear…to have consisted mainly in the lecturer asking students in turn to state proofs and theorems or to solve simple problems orally.”
If you were really great at mathematics, that meant that you were the absolute best at oral discourse in mathematics. You could recite proofs, split hairs, debate and give convincing rhetorical presentations of various proofs or arguments, etc.
Step 1: Newton, Leibniz and the emergence of an extremely powerful technical apparatus for solving problems in math and physics
Then, Calculus happened:
“In the early seventeenth century, most of what was known at the time as ‘mixed mathematics’ was not overly demanding from a technical perspective. However, it was considered by the majority of scholars to be of only secondary importance to the study of ‘physics,’ the search for the true causes of natural phenomena…It was with the gradual translation of Newton’s mechanics into this new [algebraic] language in the early years of the eighteenth century that the fundamental techniques of mathematical physics, indeed the discipline itself, began slowly to come into existence.”
And this was completely different than the mathematics that was previously seen as valuable for undergraduates to learn:
“The aspect of these developments of most immediate relevance to our present concerns is that the increasingly technical nature of physico-mathematics from the mid-seventeenth century made it ever more alien to undergraduates students.”
The new technical apparatus was useful in solving a variety of problems. Yet university mathematics education at Cambridge didn’t significantly change. It continued teaching more or less as it always had:
“Even at Trinity the primary function of a college lecture was still to inform students what they should read and to test their recall and understanding of that reading by catechetical inquisition…In order to find out whether students had learned the definitions, proofs, and theorems they were required to know, the lecturer would go round the class asking them in turn to enunciate propositions and even to solve simple problem orally. the paper and pens provided in the lecture room were not therefore central to the teaching process, but enabled students to take notes as they saw fit during the oral exchanges between the lecturer and individual members of the class…The only visual aid employed by the mathematical lecturer was ‘a cardboard, on which diagrams were drawn relating to the mathematical subject before us’ (Prichard, 1897, 36). This cardboard was handed from student to student as the lecturer went round the class and seems to have functioned as a kind of primitive blackboard.”
This is 100 years after Euler, 150 after calculus, and university math class still didn’t involve learning how to solve problems.
(Going off script here for a moment to note that you were still supposed to be able to solve arithmetic problems. The point is that if you wanted to assess this, you’d do so in an oral recitation. Written mathematics wasn’t expected, and so anything beyond basic mental math wasn’t oriented towards solving problems.)
Step 2: The new mathematics is perfect for ranking students and assigning scholarships
As this is going on, Cambridge University began using the Senate House examination (a.k.a. the Mathematical Tripos) to award valuable college fellowships. If you were one of the top 2 or 3 “wranglers” (as they are apparently called in Cambridge) you were guaranteed “one of the few recognized routes by which a young man from a relatively poor background, but with academic ability, could make his way up the social scale in Georgian Britain.”
At first, it was an oral test and not focused on mathematics. But the new technical mathematics was perfect for clearly ranking students (you either solve the problem accurately, or you don’t) and the exams changed in three ways that Warwick identifies:
- the exam became focused more on math and natural science
- if you wanted to rank in the top few, you had to do a written exam, not just an oral one
- the exam became increasingly competitive
So, the exam, its content, and the incentives students encountered were changing rapidly. But the teaching in the university classes didn’t change at all. Which meant that before long…
Step 3: An extensive system of private tutors (“coaches”) emerged to prepare students for the examination
“The main point to take from this brief survey of Cambridge pedagogy is that professorial and college lectures, tutorial sessions, public disputations and private study were all forms of learning based in the first instance on reading or oral debate. With the gradual introduction of written examinations the preferred form of teaching began to change to suit the new form of assessment. Success in the Senate House examination depended increasingly on the ability to write out proofs and theorems and to solve difficult problems on paper. Ambitious students accordingly turned to private tutors.”
Step 4: These tutors, competing with each other for top students, soon land on a paper-based pedagogy that is distinct from what the university offers, and closely resembles what we think of today as “traditional” math teaching
There was a whole industry of tutors surrounding Cambridge University, but some tutors stood out and gained reputations for reliably producing top scorers on the examinations. The first of these elite tutors was William Hopkins:
“Hopkins’ success derived from his teaching methods, his own ability and enthusiasm for mathematics, and his reputation as a tutor. Unlike the majority of private tutors, he taught students in small classes–between ten and fifteen pupils–composed of men of roughly equal ability. This meant that the class could move ahead at the fast possible pace, the students learning from and competing against each other. Hopkins considered it an ‘immense advantage in Class Teaching when there is a sufficient equality in the ability and acquirements of each member of the class’ that ever student would: ‘hear the explanations which the difficulties of others might require, and thus be led to view every part of the subjects of his studies, through the medium of other minds, and under a far greater variety of aspects than those under which they would probably present themselves to his own mind, or would be presented by any Tutor teaching a single pupil (1854, 19-20).”
“Hopkins’s teaching methods were thus designed to optimize the benefits of intensive, progressive, and competitive learning. He also developed and exploited an avuncular intimacy with his pupils which would have been quite alien to most college lecturers and university professors…In the friendly atmosphere of his teaching room, Hopkins combined the admiration of his students with his own infectious enthusiasm for mathematics to promote the competitive ethos and a dedication to hard work.”
An entire chapter is dedicated to a later tutor, Edward Routh, whose students dominated the exams over many years. Here’s what Warwick has to say about Routh’s teaching:
“The primary method of teaching, around which Routh’s whole system was built, was the one-hour lecture to a class of not more than ten pupils using blackboard and chalk. The blackboard was a fairly recent pedagogical innovation in Cambridge at this time, private tutors having previously worked on paper with their pupils sitting next to them…Routh would begin with a ‘swift examination of exercise work’ set for the class at the end of the previous meeting. These exercises generally required students to reproduce proofs and theorems, and to solve related problems, as they would have to in the examination, and Routh would quickly discuss any errors common to several members of the class and those of an individual from which he felt the class might learn.”
Going over homework on a blackboard — this was state of the art!
“Despite the public nature of these corrections there was ‘no jesting, no frivolous word over a blunder,’ and Routh would neither give any ‘tips’ on which exercises he thought likely to appear in forthcoming examinations nor express an opinion on the relative abilities of members of the class. Having corrected the work of the previous lecture he would at once launch into a ‘continuous exposition’ of the material to be covered that day,each member of the class taking notes as fast as he was able. Routh generally led students through which he considered the best textbook accounts of each subject…”
“Revision sessions apart, Routh devoted little time to solving difficult examination problems in his lectures. At the end of each hour, rather, he would hand out about six problems ‘cognate to the subject’ of the lecture’ which were to be solved and brought to the next class.”
“Once a week Routh gave a common problem sheet to all his students, regardless of year or group. In one week the students were allowed as much time as they required to puzzle out the solutions, but, every other week, the problems had to be solved ina timed three hours under examination conditions. Each students was required to leave the problems in the pupil-room on Friday or Saturday in order that Routh could mark them over the weekend. The following Monday the marked scripts would reappear in the pupil room together with Routh’s model solutions (to save him having to waste precious minutes in the lecture) and a publicly displayed mark sheet ranking all students according to the marks they had scores. These biweekly ‘fights’ gradually accustomed students to working at the pace required in the Senate House, incited and preserved an atmosphere of fierce competition, and provided that objective measure of relative merit upon which Routh himself took care never to comment.”
He gave feedback to everyone, and everyone had room to improve:
“Routh was extraordinarily scrupulous in marking student scripts to the extent that ‘it was one of his peculiarities that he was never wholly satisfied with any work shown up to him’ (Moulton 1923). On one memorable occasion when a brilliant student, Fletcher Moulton, mischievously prepared a problem paper ‘on which no criticism could be offered,’ Routh still found room for improvement by urging Moulton to ‘Fold neatly.'”
In contrast, less successful tutors tended to group students in two and threes, didn’t recreate exam conditions, and individualized instruction more (rather than following a systematic and carefully constructed syllabus). Make of that what you will.
Step Five: All this time, university lecturers still weren’t teaching any of the new technical math that calculus and mathematical physics produced! The only way to learn any of this stuff was through the tutors. This went on for way longer than you might expect.
Step Six: Eventually, both the methods and curriculum of this system of private tutors were mainstreamed and brought into the colleges. From there, it spread to top prep schools and eventually to public schools.
Honestly, none of this is really the main point of the book. The book’s major argument is far more ambitious than just telling this story. It aims to detail how mathematical training at Cambridge involved an extensive local, distinct, technical manner of doing math. If you wanted to learn this way of doing math, you had to be socialized into it at Cambridge. And this manner of doing math, Warwick argues, can help explain the distinctive research that into mathematical physics that happened at Cambridge.
(Historians of science had already argued that the technical apparatus of science — the tools, machines, etc. — involved a local, distinctive area of knowledge that grounds whatever theorems or ideas they shared. Warwick is making a similar argument about paper, examinations, the classes that coaches offered. All of these things are like the microscopes of Cambridge research into mathematical physics. It sounds better when Warwick says it.)
All of this is pretty heady; extracting just the pedagogical story from it is sort of cheap.
(Full disclosure: I only read the first half of the book! That’s where the pedagogical history mostly is.)
At the same time, all of this info deserves to be more widely known in math education. I think it’s particularly relevant to a number of perennial discussions:
- Has there only ever been one mathematics, or has mathematics changed drastically depending on culture and time?
- What is “traditional” mathematics teaching? When did it arise, and why did it become widespread?
- What’s the relationship between the current way we teach mathematics and past ways of teaching it?
I’m eager to read more books like Masters of Theory. Reading recommendations are seriously appreciated.