# How did students manage to study the required Mathematics for Physics

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## Summary:

My question is about the history of education in Physics and Maths.

## Main Question or Discussion Point

I will basically focus on 18th and 19th century, I got to know from the biographies of Max Planck and few other that there were no organized syllabus in Universities for studying. Students had to take classes that they could understand and it was less like a lecture and more like a private tutoring by some professor to a small number of students.

I got this
It is noteworthy that in the 1850s the Stokes' Theorem appeared in the examination for the math Tripos at Cambridge, as a question to generalize Green's Theorem to 3 dimension, from what we surmise that it was not explicitly taught.
information from cesaruliana (a user from another forum). And as I'm reading Arnold Sommerfled's Lecture on Theoretical Physics (on the guidance of @vanhees71) I came across this

Throughout this volume we shall make continual use of vector analysis, that is, calculus applied to vector quantities. Thus, while familiarity with vector algebra and with basic concepts of vector analysis is assumed on the part of reader...

So, I have given two examples where it was assumed by the instructors that the students know higher mathematics. If we focus on first quote, we find that up to 1850s textbooks (textbook in the sense that we use it today, an organized sequence of chapter with explanations and exercises) were not in general public use, only original publications were there (original works of Stokes, Cauchy, Green, Hamilton). My point is the textbooks that we use today contains the explanation of some topic with examples and real life situations but the original papers doesn't intend to teach anything, their only purpose is to establish a result indisputably. So, how those students lear higher mathematics (like Vector Calculus) without the availability of textbooks?

In my second example Sommerfeld is assuming his students to be familiar with vector calculus, although the year was around 1945 - 1951 (and his lectures were meant for undergraduate courses) but it was Germany not the U.K. where universities by this time had adopted an organized course. Was it like that instructors used to teach students all by themselves without means of any textbooks? Do we have any record of lectures where an instructor doing something like this?

Any help will be much appreciated.

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## Answers and Replies

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jedishrfu
Mentor
I think you're reading too much into this. As a student, if you wanted to take a course and talked with other students or the prof you would discover what math they use and either get prepped to learn it on the fly or take the relevant course. Basically you would've mapped your studies out in order to do well in each course.

It's only more recently, used loosely like since WW2 where colleges got more formal and setup dept academic advisors who would tell you what courses you need in order to take other courses. I suspect it came about simply because of student struggles and the profs collectively agreed this needed to be done for the students benefit.

I think you're reading too much into this. As a student, if you wanted to take a course and talked with other students or the prof you would discover what math they use and either get prepped to learn it on the fly or take the relevant course. Basically you would've mapped your studies out in order to do well in each course.

It's only more recently, used loosely like since WW2 where colleges got more formal and setup dept academic advisors who would tell you what courses you need in order to take other courses. I suspect it came about simply because of student struggles and the profs collectively agreed this needed to be done for the students benefit.
I asked it this time only for educational purpose (no personal problem :) ).

In Sommerfeld's time and long beyond, German professors did not use textbooks. No such thing. Lectures were not intended for the benefit of students, but were a forum for the professor's research. Students were on their own to learn what they needed to know to follow a professor's lectures. The closest thing to a "textbook" would be the professor's published theses. All one will ever find would be someone's notes taken in class. This persisted at least through the last decades of the 20th century when I was a student in Germany. More recently the American model where professors are mere teachers has had great influence, but professorial independence is likely still the norm in Germany. If you took a course on philosophy you would, of course, read the philosophers, but the professor was there to give his own interpretations. Lecterns were more like pulpits, I say "his" most decidedly; they were 100% male.

Lectures were not intended for the benefit of students, but were a forum for the professor's research.
What a great line is this !

If possible can you please elaborate some real life examples of
Students were on their own to learn what they needed to know to follow a professor's lectures

[on your own] A bit off topic, but might provide some perspective. I certainly cannot speak from experience about the 19th century, but judging from what we know, as suggested, about the world altering collaboration between the likes of Rutherford, Sommerfeld, Bohr, Heisenberg, et al, it makes no sense to try to make sense of all this through the lens of today's mass universities. Students were not kids trying to get "an education" to prosper in the world. [Seems to me that Sommerfeld doesn't get the credit he deserves for what was the Sommerfeld/Bohr model of the atom; and, of course, Rutherford was behind all of it.] The private tutorial to a small group of students might best be analogized to the seminar, where, again, "students" were participants in group endeavors. We can see the differences on the ground today, where the original sandstone KGi (Kollegien Gebäude) houses the smaller classrooms next to the newer KGii with its Audi Max (Auditorium Maximum); and the Freie Universität has overtaken Humboldt in Berlin. Most everything changed after the war, the population explosion and the opening of German higher education, as reparations, to students all over the world at only token tuition. The Audi Max gets most of the action today.
There was always tension between professors and administrators, but, primarily for financial reasons and the sheer numbers of students today, the postwar world has seen the tug-of-war lean toward administration.
A professor's syllabus would help, but it rarely directed one to specific works, only generally to topics. At least in the 20th century, one had merely to grouse local bookstores which usually had the works on the shelves that were in demand at the nearby university.
Occasionally one was required to pass an exam to get into a class, as opposed to the American model where exams are given afterwards to pile up credits, which don't exist in German higher education, except for the occasional Schein, which is more in the way of an entrance ticket. One attended classes to prepare for final oral and written exams years hence, so it was not imperative as it is in the modern American model to follow lectures immediately. Sooner or later you would figure it out. ON YOUR OWN.

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epenguin
Homework Helper
Gold Member
Provisionally, here is a stab at a few considerations based on my limited knowledge.

For the 18th century it seems to me the question hardly arises. There was essentially no organised formal teaching of physics. In that century universities were little concerned with physics or any other science. Research was pursued in or associated with Royal or national academies, or voluntary associations like the Edinburgh Society, the Birmingham Lunar Society, the Accademia dei fisiocritici, Accademia Dei Lincei, etc. Most of which still survive and exercise some scientific as well as ceremonial function. They concerned themselves to an extent with the propagation of science and public education, but to talk of any educational system in the modern sense is anachronistic, at least as concerns physics.

As concerns the 19th century I have the impression that for England and France at least the, boot is somewhat on the other foot than the one you put it - that mathematical education was quite flourishing, but it was quite hard to get systematic education in physics started. I read that even before then some knowledge of mathematics was expected of all students at Cambridge. 'Grammar Schools' taught arithmetic and Euclid at least. But at Cambridge University maths study was intense, a positive hothouse. You might have heard of "The Old Tripos". This was a system of gruelling (8 day!) mathematical examinations for which you did not so much study as go into training for. Professors did give some lectures, but then the main thing was you went into training under a private coach if you wanted to compete for a top place ("Wrangler").* The competition was followed nationally almost like sporting events today. The subject was just as prestigious as Classics, if not more so. It qualified you to go into almost anything, not necessarily anything related to maths or science. One so qualified fictional character I must mention is Christopher Tietjens, principal character of the Tietjens tetralogy, that major 20th century novelistic achievement, a Smiths Prize man. It seems to me that behind this prestige was something snob, traceable back to Plato, that like classics maths was not a hands-dirty subject and, as such, was suitable for the education of a gentleman. Despite that, or perhaps behind the protection of that, applied mathematics was far from snobbed, on the contrary it turned out to be British math's main strength:

"this system was the nursery for the great flowering of British physics in the 19th century. Its products included Maxwell (2nd Wrangler), Kelvin (2nd Wrangler), Stokes (Senior Wrangler) and Rayleigh (Senior Wrangler). On the pure side it produced Sylvester (2nd Wrangler) and Cayley (Senior Wrangler). Pearson, the father of modern statistics, was a 3rd Wrangler." The above were mostly theoretical physicists, but another Tripos product was JJ Thompson, discoverer of the electron. On the other hand, except for the two above mentioned, British pure mathematics is said to have rather fallen out of the continental mainstream in the 19th century. However, that would not have been a problem for physics, for the question that has been raised. The rather counter-productive swot and mathematically provincial nature of the Tripos was reformed in the 20th century by Hardy and others.

Physics teaching on the other hand seems only to have got really going in English universities about 3/4 of the way through the 19th century, which is the time of the foundation of the Clarendon and Cavendish laboratories (Maxwell played an important part in this). Courses in physics started rather earlier at University College London, but it seems to have been with some difficulty. Interestingly, the pure mathematician JJ Sylvester was a teacher of physics there, but it seems he was something of a misfit, but then for that matter so was he almost everywhere else too. (Note that the above trio were nearly all the Universities in England for most of the nineteenth century).

So in summary it does not seem that English students of physics were starved of mathematics in the 19 century. Nor, I think, were they in France. Here the really great teaching institutions were not universities but the Grandes Ecoles, e.g. the Polytechnique, the Normale etc.These were founded (early 19th century ) and their curriculums designed by mathematicians! Meant for the formation of engineers and other cadres. There was at least the same forcing-house competition (also in entry exams) and concentration on maths as in the Cambridge Tripos. To this day Polytechniciens are known as 'les $x$'. The difference with Cambridge is that the system survived unmodified up till this day - or at least up to when I last heard.

But indeed in Germany from the mathematical history I remember reading (Bell!) arrangements had looked a bit chaotic. Strange indeed - is being systematic too alien to German temperament?

This is what I have rapidly thrown together from what I know - I would not go to the stake for any statement I have made here.

* I took a look at some of the Tripos exam questions, and rapidly closed the book.

Some references for above:

https://en.wikipedia.org/wiki/Mathematical_Tripos

https://www.phy.cam.ac.uk/history

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Students were not kids trying to get "an education" to prosper in the world
Your words are very very powerful.

@epenguin I don’t know why but I believe that solving problems is not the only way or even the best way to learn mathematics. Do you got some references which favours my belief? :)

epenguin
Homework Helper
Gold Member
@epenguin I don’t know why but I believe that solving problems is not the only way or even the best way to learn mathematics. Do you got some references which favours my belief? :)
Can't think of any offhand, and I don't think I agree, nor do I think would most teachers or homework helpers here.
Most would say that if you cannot apply the maths, which is tested in exercises and problems, you cannot be said to have learnt it. (Similarly for any other subject.) Why do students come to Homework Help? Presumedly they have followed the lesson, read the appropriate chapter in the book, okay not always, and many may have thought they understood. But being unable to do the exercises, reveals that in some sense they haven't yet.

Can't think of any offhand, and I don't think I agree, nor do I think would most teachers or homework helpers here.
Most would say that if you cannot apply the maths, which is tested in exercises and problems, you cannot be said to have learnt it. (Similarly for any other subject.) Why do students come to Homework Help? Presumedly they have followed the lesson, read the appropriate chapter in the book, okay not always, and many may have thought they understood. But being unable to do the exercises, reveals that in some sense they haven't yet.
Well, that’s right (or may be right) but there’s a dilemma that surrounds me everytime, how much problem-solving skills do we need so that we can say “we have learnt this topic”. Because as you know it is always possible to put up a question that cannot be solved. For example, it is always possible to give an indefinite integral and there is no certainty that we can solve it (although it is solvable).

epenguin
Homework Helper
Gold Member
Well, that’s right (or may be right) but there’s a dilemma that surrounds me everytime, how much problem-solving skills do we need so that we can say “we have learnt this topic”. Because as you know it is always possible to put up a question that cannot be solved. For example, it is always possible to give an indefinite integral and there is no certainty that we can solve it (although it is solvable).
That's a good question.Not every student can honestly say they have your problem of doing too much!

I would say be reasonable and rational, the exercises are meant really to help you; if they are allowed to get on top of you, if you make it a question of self esteem, there is a point where they become counter-productive.Make reasonable efforts without making it that you have absolutely to prove yourself! 'Reasonable' takes into consideration your time, and the time you need for other things in study and the rest of your life. And whether the problem to be solved is a big issue or a small one. Take into account that once you have put a real effort into something, but not before, giving it a few day's rest often results in you more easily finding the solution when you come back to it. One university teacher told us (we used to work each week's problems in a group) if you're just not seeing the way to an answer, it's no use insisting. But the aim of the course was not to train mathematicians.

Another point is that education has to be general, and generally enabling. That is, your question may come in the exercises at the end of section 3 of chapter 7, but what you have to use to solve it comes also from what you should have learnt earlier. The problem does not tell you that, and some students can't solve it because it doesn't occur to them to use anything else but what is in that section.This is like a real life problem that presents itself in physics (but maybe we could illustrate better in other fields that haven't had such symbiosis with maths, like economics or biology) a problem that presents itself in that field (or even within maths) comes without a visiting card announcing what book, chapter and verse it needs to find solution, or anything useful.

For your particular example, integrals, I do not think one should overinvest.It is worth having some familiarity with the eight or a dozen main types or tricks. But they are so bitty there is a good chance you will forget some of it; it seems best to return to it from time to time especially when you meet them in physics or other application. By the way I think we did not quite finish enough the one you asked about a few weeks ago and I will come back to that with some tips when I have time.

Finally, there is no completely general rule. I say on the one hand don't overdo anything. On the other hand on there is probably no scientist worth his salt who hasn't at some point got obsessed with one particular resistant problem and spent a lot of time on it. And probably doesn't regret it. This often leads to something, but better if the problem is worthwhile.

Another point is that education has to be general, and generally enabling
Aha! Seems like we share some common thoughts

I can recommend a superb book that addresses some of the points raised in this thread: "Masters of Theory: Cambridge and the Rise of Mathematical Physics" by Andrew Warwick, U. Chicago Press, 2003.

The now-widespread practice of teaching physics and math by drill in increasingly difficult problem sets originated in the early decades of the nineteenth century at Cambridge, and, as noted by others posting in this thread, was intimately connected with the institution of the Math Tripos. It came about as a consequence of a movement to reform the practice of mathematics in Britain in light of recent advances on the continent, most notably France.

As part of this movement, the Tripos examination, which had been held at Cambridge since roughly the time of Newton, was completely overhauled, and a class of specialized tutors called coaches, prepared students for that ordeal. I believe how the modern use of "coach" for an instructor came to be.

@Palomides Means the sole purpose of teaching Physics and Maths through drill was to increase the number of well problem-solver students? And it was only and only for competition with other nations.

According to Warwick, the purpose of university training in nineteenth century Oxbridge was to produce scholars imbued with Victorian virtues (among them a passive attitude towards politics) and, through teaching in schools or in the pulpit, to propagate them in the wider population. Nationalist competition was an important part of the package but not the whole of it.

At Cambridge, mathematical instruction was not primarily intended as preparation for a career in math or science, or even for a university career: Apart from the univerity path, nineteenth century Britain had no such career paths, and a university professor was not ordinarily expected to conduct original research (still true at Oxbridge well into the twentieth century.) If, however, a Tripos candidate were not annointed First Wrangler, he* had essentially no chance of a career in any endeavor involving mathematical skill, much less a university career.

* Women were only allowed to sit the Tripos late in the Victorian period, and their Tripos ranking was separate from that for the men, which once led to the University announcing that a woman had outranked the First Wrangler!

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