I read an article the other day about how physics professors have to teach their introductory classes in a somewhat old-fashioned manner. This is due to pressure from the other departments (other sciences, engineering, computer science, etc).

This has me wondering what compromises math professors make when teaching things like calculus, linear algebra, and ODEs -- all of which are required courses for science and engineering majors.

This raises two questions for me:

How are introductory math courses designed so as to prepare not only math students, but also science and engineering students for their later works?
How might introductory courses be changed if science and engineering students were no longer part of the equation (so to speak)?

Additionally, because there is no Physics Educators SE, if someone can answer the same questions for introductory physics courses, I'd be interested in that as well.

  • $\begingroup$ As someone who used to be an engineering student before I switched into mathematics, I am very curious to see a good answer for this. Personally, I have enjoyed my math courses far more than the handwavy engineering courses because you actually get to see how some (previously seemingly magical) mathematics works. So, my guess would be that the introductory classes would have more time to focus on foundations that are usually skipped over, but which would eventually lead to better student understanding in advanced courses? Maybe that's just what I wished I had :( $\endgroup$
    – bzm3r
    Commented Apr 19, 2015 at 21:35
  • $\begingroup$ It's not unlikely that there already are universities where only math majors take math classes. For example, here at Jyväskylä, the department of physics teaches the necessary (from their perspective) mathematics to their students without any help from the department of mathematics and there is a similar thing with economics. The full story is more complicated here, but this general phenomenon is certainly not unique to this university. $\endgroup$ Commented Apr 19, 2015 at 21:35
  • $\begingroup$ This is basically the idea behind the "Honors" track in Notre Dame's math program. It's more theoretically oriented with the goal of preparing students for grad school in mathematics. $\endgroup$
    – Livius
    Commented Apr 21, 2015 at 1:48
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    $\begingroup$ In the UK courses are usually only taken by students on a mathematics degree (that is, spending nearly all their class time on maths classes). The programs differ according to what standard the students are when they start, and what the overall picture of the degree is (very applied, very pure, etc). $\endgroup$
    – Jessica B
    Commented Apr 21, 2015 at 10:20

3 Answers 3


In a world where the audience was pure of heart:

  1. proof methods course is the first course
  2. matrix linear algebra is taught along-side early calculus
  3. calculus is taught with methods of analysis, never just with a primarily computational focus
  4. all courses are about proofs, at least 50 percent
  5. real analysis as an advanced calculus course ceases to exist because we already covered that in the "calculus sequence" (which is not the calculus you typically find, rather, the sort my Chinese colleague learned... replete with analysis from the outset)
  6. higher analysis replaces the traditional real analysis course
  7. required second course on theoretical linear algebra
  8. generally the level of courses is higher since the expectation of math from math majors ought to exceed that we have for other folks who have not declared math as their life's work.

Main Point: because the major math courses are taken only by math students the audience of the major math courses would share common prerequisites. These prerequisites could be used to bring greater cohesion to the entirety of the curriculum. This cohesion would produce a student with greater sophistication at the end of the course of study (say over 4 years). Students would be challenged in ways we simply cannot hope to attain in the current mixed-audience format.

Admittedly, I have in mind more or less pure mathematics students. My focus would be a bit different if I was to consider the applied math. I don't think this program is realistic for a large major. Mostly, this sort of thing would really only be reasonable with a small group of highly motivated pure students.

Also, to be clear, I speak merely from the viewpoint of curriculum design. Even in this specialized course of study I sketch here, the individual would still need to take the initiative to study beyond the rudiments. I do think it is time we (as in the society of mathematicians, or, at least people who care deeply about math as a professional activity) made more of modern mathematics an expectation for the undergraduate in math. In particular, I have in mind the subject GRE, tricky calculus and linear is great, but, I'd love to see more of an expectation in abstract courses. There is a wide gulf between the standard course load most places and the necessary scholarship for doctorate work. So, I'd rather see the undergraduate be a little less leisurely in the interest that the graduate study can be more meaningful. This is why I push undergraduates. But, only the ones who want a push. Of course, the battle I wish to fight is a losing one from a statistical perspective, the entry point of most students invariably declines (USA-based comment).

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    $\begingroup$ Even in an idealized situation, how/why are all math students "on the same track"? No one takes any initiative? Or are they simply forbidden to read anything that's not explicitly required? :) $\endgroup$ Commented Apr 19, 2015 at 21:57
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    $\begingroup$ @paul garrett What I was trying to get at is half the reason I can't cover things I'd like to cover is that some majors never take, for example, calculus III. That influences what I can assume as a common experience in linear algebra. I think it's safe to say, In many schools, the lack of proof type experience of non-majors necessarily dilutes the linear algebra course. Perhaps I'll think of a better way to phrase it... $\endgroup$ Commented Apr 20, 2015 at 0:37
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    $\begingroup$ The stricture of homework/exam/grade, and accidental adversarial relation of teacher/student, also creates situations in which the only motivation to try to understand is to avoid getting a bad grade, and "fairness" doesn't allow us (=teachers) to create hazards for disadvantaged students... That is, it is essentially impossible to talk coherently about genuine mathematics in a school setting, where so many corruptive influences degrade the context. That is, "Math is not a school subject!". $\endgroup$ Commented Apr 20, 2015 at 1:10
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    $\begingroup$ @paulgarrett there is much truth to your point. Grades in my independent study courses with one or two students mean very little. So, you're observation is pretty much on target in my experience, the place where I have students do the best thinking is where I offer virtually no cajoling for them to do homework... yet, they do the homework because they want to. For the masses, somehow I'd need people who just wanted to do math for the fun of it as opposed to finish a degree. But, I am happy the degree seeking folks tolerate my mathematical exuberance. I am grateful for their existence. $\endgroup$ Commented Apr 20, 2015 at 1:34
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    $\begingroup$ If none of the historical and logistical baggage was in place, and negotiations were starting from scratch, I suspect many or most mathematics departments would say to the physics/engineering/etc people, "OK, if you really want us to, we'll teach calculus without analysis or proofs to freshmen. However, we'll recommend those planning to major in mathematics not to take it, and we'll let them take a proper elementary analysis course instead". Or just make the engineers teach their own "Abusing integral signs for monetary profit I" course ;-) $\endgroup$ Commented Apr 20, 2015 at 8:33

You may want to take a look into the German education system, where undergraduate studies happen at universities and are mostly separate. Right now, at my (German) university, there are the following courses in the first semester:

  • linear algebra (for mathematicians)
  • calculus (for mathematicians)
  • math for physicists
  • math for computer scientists
  • math for chemists
  • math for agronomists
  • mechanics and thermodynamics (for physicists)
  • physics for chemists, mathematicians and computer scientists
  • physics for biologists, pharmacists, medics and similar

I started studying math and physics in the last year when physicists did not have their own math course but had to attend the first two linear algebra and the first four calculus courses for mathematicians (though they had to pass the exams in two of them). These courses were already very much as described in James S. Cook’s answer and primarily focussed on theory and proofs. For example, I heard about modules, rings and groups in the first semester. Keep in mind though that the students had better prerequisites, e.g., I was suprised to learn that some of the other students had not already done proofs in school.

I later tutored the math-for-physicists series several times and the respective courses were still mainly theory- and proof-focussed¹. Mostly some aspects of linear algebra were missing (such as modules and rings) that are not that relevant to physicists. After what I heard, the first two courses from the list did not change much after physicists got their own courses.

However, I cannot really compare the contents of the courses to a system where everybody attends the same courses, as I have no experience with the latter.

¹ which I as a physicist consider a good thing: We need people to understand the mathematics, being able to apply it is secondary. You cannot really understand quantum mechanics without linear algebra, for example.

  • $\begingroup$ As a physics major myself, I don't know that I agree that "we need people to understand the mathematics, being able to apply it is secondary". Personally I love to learn the math, but math is neither taught rigorously to physics major at my university, nor do I think it should be. Rigorous math takes time and energy that could be better spent on learning physics. As long as physicists are taught the basic ideas behind a theory and the tools to compute when necessary, that's the minimum that's required. Any bit extra couldn't hurt, but I don't see it necessarily helping either. $\endgroup$
    – user5083
    Commented Apr 20, 2015 at 14:41
  • $\begingroup$ @Bye_World: I am not certain where you draw the line between “rigourous mathematics” and the “basic ideas behind a theory”, so let me give an example: Physicists need central concepts from linear algebra so often and in such a theoretical manner that only understanding vector spaces in a manner of geometric visualisation and being able to solve linear equation systems (which is much less relevant today than, say, thirty years ago) does not suffice and even wastes a lot of time. They really need to understand what a basis, an eigenvector and so on are. $\endgroup$
    – Wrzlprmft
    Commented Apr 20, 2015 at 15:35
  • $\begingroup$ Of course the line depends on the subject we're discussing, but let's consider real analysis. Obviously every physicist should know how to compute derivatives and integrals and have some idea about what those operations mean, how they relate to each other, and some of the most important theorems regarding them. But do they really need to know the definition of a limit point or of open and closed sets or the Heine-Borel theorem or any of countless other definitions/ lemmas/ theorems that mathematicians must know about real analysis? $\endgroup$
    – user5083
    Commented Apr 20, 2015 at 15:42
  • $\begingroup$ As for the definitions: You do not know definitions for their own sake but as a side product of understanding the underlying concept. I never memorised the definitions of a limit or an open set, but I can produce them from scratch with some thinking. Definitions are inevitable on the way to a good understanding of most mathematical concepts, because the alternative is a lot of handwaving which does not make things easier. As for Heine–Borel: No, they do not need that and it’s not part of my university’s math-for-physicists curriculum, if I recall correctly. $\endgroup$
    – Wrzlprmft
    Commented Apr 20, 2015 at 16:14

I applaud James Cook for professing his opinion with some eloquence. As I disagree with him on tone and some points, I (with respect for his opinion) suggest the following:

  1. Thinking is taught in the first, and every course.

  2. Learning how to apply in some situations is taught in the first (and every) course.

  3. Learning how to make a proof and how to apply a result is a skill/talent/goal that develops over time with practice; both should be considered extensions of thinking.

  4. Methods of analysis do not take a back seat to computations, nor do computations take a back seat to analysis.

  5. Real analysis is covered when an idea of how it is to be used is presented first.

  6. Linear algebra is taught as needed: a course heavy on applications with some theory (and where to find more theory and proofs when needed) for those who need to use old tools and develop new tools for application, and a course heavy on theory and light on applications (and where to find more problems and possible room for application) for those who need to advance the theory for the sake of both the theory and the applications.

  7. Generally the level of courses is better (not higher) because we give people the information to decide what education will be best for their intended growth and that of the society they live in. Even if we don't know the proper direction for society, we should make several directions available by the courses we teach and the actions we do.

Main point: who needs a main point?

Edit: 2015.04.20

Rather than respond to the comments point by point, I choose to explain the rationale and the nature of my disagreement with the post of James Cook. Understand first that I don't think James Cook is wrong. From his perspective and experience, his answer may be as apt as any.
I see his answer as producing people who are adept at checking and generating proofs, and I take his "higher level" as meaning "more facility with handling activities centered around understanding proofs". Whether he intended this slant on it or not, in my view this slant is wrong, and I attempt to correct it with the list above.

Mathematics is not just proofs and understanding proofs. Mathematics is about pattern, observing, creating, convincing ourselves, convincing others, and building something that is both beautiful and useful.

I want a student coming out of a mathematics course to think "I could do that." That is not just doing computations, or solving problems, or writing up a proof of an assertion, or checking one's logic, or trying to find a flaw in someone else's reasoning. It is not a game of axioms and propositions.

Mathematics is an activity which both has the purity and potential beauty of intellectual play admired and pursued by the Greeks (and others) of old, and has the power of being applied by someone who would work in the physical world and help society through such application. It may be a pedagogical convenience to keep these two things separate, but it is a mistake socially and perhaps totally to do so.

I want a student coming out of a mathematics course to think that they could imagine or create structures and realize or apply them, check to see that they withstand the brutal force of reason and reality, fix or modify these ethereal visions and either apply them or give them to someone else to use. Most importantly, I want them to be able to tell other people about them in a way that helps.

So I prefer that education involve teaching the student to think. Not "do I have to remember to write the definition this way" or "is it pemdas or pedmas or samdep" or "what chapters do I have to read the night before the test". I want them to do mental simulations: "I have to take a notion of continuity in one dimension and generalize it to two dimensions. Can I think of counterexamples I want to avoid so that I can frame the definition to avoid them?" or "This notion of a group acting on itself is a little too self-referential. Can I use group of symmetries to simplify how to write up this case-by-case combinatorial proof of this other thing that interests me?" or "My uncle's going to ask me to help tomorrow in the backyard: can I use any of this to make things better/easier/cooler for us tomorrow?"

Even if the student will not be a theoretician, she or he needs to explain and interact with others, and may need a fragment of the theoreticians perspective. Similarly, the student may not be an engineer, or customer support worker, or salesman, or actuary, or one of a number of other intended destinations, but they will need to know that others have to apply math, and they would do well to prepare explanations for those who aren't as adept at theory. In short, all of the students need to be teachers to some extent, for themselves as for others. I do not see this as a potential result from James Cook's list. I do see it in mine.

I will stop here.

End Edit: 2015.04.20

Gerhard "Not A Professional Mathematics Educator" Paseman, 2015.04.19

  • $\begingroup$ I appreciate that you (digitally) signed and dated your work. :P $\endgroup$
    – user5081
    Commented Apr 19, 2015 at 22:42
  • $\begingroup$ Thank you. I often go months without a sign of such appreciation. Gerhard "Thanks For Shortening The MTBA" Paseman, 2015.04.19 $\endgroup$ Commented Apr 19, 2015 at 23:13
  • $\begingroup$ Thank you for this, my opinions are not strongly held on this answer. Mostly a result of my particular teaching load etc. I think, thinking is a good suggestion. Most of my answer's tone is based on my colleagues' experience with a more specialized-education where the idea of leisurely selecting one's educational future is not an option. The question was: to succeed professionally in education, or, to work in the fields. So, perhaps some of that tone has found its way to my answer. $\endgroup$ Commented Apr 20, 2015 at 0:52
  • $\begingroup$ I downvoted the answer, and was prompted to provide a comment on why, so I obliged. Please consider it openly! What do you mean by "thinking" (i.e. too general)? James' answer is clearer, because by focusing on teaching proofs right from the outset, his scheme implicitly gets students to think about how "mathematics is a game, where we play with the rules (axioms), and their consequences (proofs)". What do you mean by learning how to apply? Existing curricula tend to teach students how to apply by presenting mathematics as a set of monolithic rules that must simply be followed -- not nice! $\endgroup$
    – bzm3r
    Commented Apr 20, 2015 at 2:09
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    $\begingroup$ @user89, I think you have misunderstood, but that's OK. James has his version of part of Utopia, my version is different. Perhaps the edit to my post will help. If not, feel free to post in an answer your version with a brief comparison to others. Gerhard "Help Make A Better Forum" Paseman, 2015.04.20 $\endgroup$ Commented Apr 20, 2015 at 17:49

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