7

I’m currently getting my Master’s in Math at Portland State University so that I can teach community college mathematics. I’m specifically hoping to teach calculus, statistics, and linear algebra, so I’m trying to pick classes that will give me deeper knowledge of these subjects. In particular, I’m hoping the degree will give me deep enough knowledge to be able to teach multivariable and vector calculus. Since we are required to take at least two 600 level sequences, next year I’m thinking of taking the Mth 614-616 sequence called “Modern Analysis”. I’m currently taking Mth 512, which is part of the 500 level Real Analysis sequence, and is the prerequisite for Modern Analysis. I’m getting 100% in the class this term so I expect to be ready for Modern Analysis next year. As far as I’m aware, Modern Analysis is the only 600 level analysis sequence that will be offered next year. Here is the course description (copied from here) for all three courses in the sequence:

Topics from nonlinear analysis, harmonic analysis, analytic functions, ordered vector spaces, analysis on Lie groups, and operator theory.

It seems like the class will obviously cover very high-level material, but I’m concerned it might not relate enough to multivariable/vector calculus. However, I’m not that familiar with these topics, so maybe there’s more of a connection than I currently realize. Would a sequence like this give me deep enough knowledge to teach a multivariable/vector calculus course at a community college? Thank you for your help!

blakedylanmusic
  • 261
  • 1
  • 9
  • 1
    I am not sure what they mean by "nonlinear analysis". Harmonic Analysis will deal with things like the Fourier Transform which could be single-variable, multivariable, or done on topological groups. Analytic functions of one complex variable require understanding multivariable calculus of 2 real variables. Not sure about ordered vector spaces. Analysis on Lie groups will require a lot of multivariable calculus on differentiable manifolds. Operator theory is essentially infinite dimensional linear algebra. Seems like a lot of different topics in one course! – Steven Gubkin Feb 23 '23 at 14:39
  • @StevenGubkin This will cover an entire school year so it makes sense that there are a lot of topics. That being said, usually for these 600 level courses the topics that are actually taught are up to the teacher (from what I’ve heard). – blakedylanmusic Feb 23 '23 at 15:11
  • 2
    I realize that you're probably constrained in what you can take, but my feeling is that things like nonlinear analysis (@Steven Gubkin: probably means "nonlinear functional analysis"), ordered vector spaces, etc. will not help much for subject matter (not a good fit for what you actually need to know) or mathematical maturity (huge overkill to the point of diminishing returns). Best would be some kind of stiff 2-semester advanced calculus course such as would use Advanced Calculus of Several Variables by C. H. Edwards (1973/1994) (continued) – Dave L Renfro Feb 23 '23 at 16:05
  • 2
    OR Advanced Calculus: A Differential Forms Approach by H. M. Edwards (1994) OR Functions of Several Variables by Wendell Fleming (1977) OR Advanced Calculus by Loomis/Sternberg (1968/1990) OR Advanced Calculus by Nickerson/Spencer/Steenrod (1959/2011) OR others at what I'd call an "honors advanced calculus" level. – Dave L Renfro Feb 23 '23 at 16:05
  • 1
    (returning two hours later) Of the books I listed, probably the most helpful for your later teaching is C. H. Edwards or Fleming, as the others are in my opinion more theoretically oriented than is optimally helpful for your needs. Of these two, I'd go with C. H. Edwards, since Fleming gets into Leb. integration and is less computationally oriented than C. H. Edwards. Incidentally, you'll want to be sufficiently "computationally oriented" so that you don't fall for the paradox I describe in this 20 May 2001 sci.math post. – Dave L Renfro Feb 23 '23 at 18:15
  • @DaveLRenfro Thank you, I’ll be sure to read the C.H. Edwards! I wonder also if finding opportunities to tutor people taking that course would help me get a refresher and get practice teaching it. It’s certainly helped me get a refresher on other topics when I tutored for them too. – blakedylanmusic Feb 23 '23 at 20:10
  • 1
    If you have the opportunity to tutor people for the course, then this is excellent additional preparation, maybe more than trying to master C. H. Edwards (let alone more advanced stuff). Indeed, if you're able to do a lot of tutoring, you probably only need to work though something like Vector Calculus by Marsden/Tromba (any edition), saving something like C. H. Edwards for when you're actually teaching the course (not necessarily to help with teaching the course, but for your own elucidation of the material you're teaching). – Dave L Renfro Feb 23 '23 at 20:34
  • 3
    I can't know for sure of course, but it seems likely that the course description is just some boilerplate list of suggestions, and the individual instructor decides what actually ends up in the course. Each of the topics in the list would be sufficient to fill half a year on its own just for an introduction. I also agree with other comments that the topics on the list are very diverse (but if you happen to really meet someone who works both with ordered vector spaces and Lie groups, please send me their name - I would be very interested to check out their work). – Jochen Glueck Feb 24 '23 at 10:46
  • 3
    @Jochen Glueck: Now that you mention it, this does seem to be way, way too broad of a list of topics for a single course. Moreover, given the size of the graduate program there, it's even more likely (in my view) that this is, as you say, simply a smorgasbord list of topics in analysis (likely not exhaustive either) among which one could be selected, according to the interests of the instructor and expected students. – Dave L Renfro Feb 24 '23 at 11:01
  • @JochenGlueck yes, usually that’s how the 600 level sequences go, a few of the topics are picked out of a big list of topics based on the instructor’s discretion. So it makes sense that covering all of this would be impossible. – blakedylanmusic Feb 24 '23 at 15:16
  • 3
    An efficient method to see how to teach multivariate calculus is to watch others do just that. I've learned much from videos on You Tube from various folks, I think Edward Frenkel's videos from multivariate calculus on Lagrange multipliers were very nice. Of course, keep in mind the target audience for the video your watching. Fwiw I have complete lecture series of Calculus III and Advanced Calculus (with a differential forms/ calculus on normed linear spaces... and more) posted on my You Tube channel. Lot's of folks have lots posted these days. – James S. Cook Feb 25 '23 at 04:41
  • 2
    Since you're asking how knowing more advanced analysis will help you teaching a calculus course, the answers on the page https://matheducators.stackexchange.com/questions/15237/how-does-knowing-more-about-mathematics-help-ones-teaching-of-lower-level-cours might interest you. – KCd Feb 26 '23 at 22:47

4 Answers4

18

It is impossible to say whether you will be ready to teach multivariable calculus after taking this course. Certainly many of the topics you list will employ some multivariable calculus, but using bits and pieces of an undergrad course as a tool in a higher level course will probably not be the best preparation for teaching.

If you are going to be teaching multivariable calculus, I would recommend getting a few really good textbooks and making sure you can do all of the exercises, motivate all of the definitions and theorems, and feel comfortable enough with the proofs that you can both present the main ideas without any prep work and present a detailed proof with a little bit of prep work. You should also work on having some kind of overarching narrative for the course which makes sense to you.

The textbooks I would recommend are "Multivariable Mathematics" by Shifrin and "Vector Calculus" by Marsden and Tromba.

If you want a "higher level perspective" on multivariable calculus then you should learn some differential geometry. I would recommend John M. Lee's introductions to topological manifolds, smooth manifolds, and riemannian manifolds in that order.

I want to say that this answer is not meant to discourage you from taking the course you mention. It sounds interesting, and it is often true that we really learn something when we are applying it in different situations. I am sure that taking the course will strengthen your understanding of multivariable calculus, just probably not in the most systematic way.

J W
  • 4,654
  • 2
  • 22
  • 44
Steven Gubkin
  • 25,127
  • 4
  • 61
  • 110
  • That makes a lot of sense, thank you! It should be a good class and I’ll still learn a lot. I won’t have any more room to take more classes next year, so I’ll probably have to self study differential geometry and the textbooks you mentioned in that case. After getting exposure to 600 level topics it shouldn’t be too hard to self study those topics. – blakedylanmusic Feb 23 '23 at 15:14
  • 8
    @blakedylanmusic I find it a bit strange that you are assessing your ability according to which classes you have taken. At some point you need to develop the self-confidence in your mathematical and learning skills that you feel like you can tackle any subject! You shouldn't let the fact that you have not taken an 800 level class to stop you from self-studying an "800 level topic" (whatever that means!). These class numbers are arbitrary. I have certainly had undergrads who have only taken "300 level" courses who are more capable than grad students who have taken "800 level" courses. – Steven Gubkin Feb 23 '23 at 15:26
  • 1
    That’s a good point. I probably do have the skills and have successfully self studied both mathematical and non mathematical subjects. Maybe I need to remember that. Thank you :) – blakedylanmusic Feb 23 '23 at 16:51
  • I think part of the issue for me, too, is I’m paying thousands of dollars for them to prepare me for my career goals. I want to feel like the classes I’m paying a lot of money to take will help me with my goals. – blakedylanmusic Feb 23 '23 at 17:03
  • 4
    @blakedylanmusic Understandable. Hopefully you do feel like learning math for the sake of math is also worthwhile, especially if you are going to be a math professor. I think that knowing this higher level content is not really essential to being a good instructor at the community college level except that it ensures the correct level of "mathematical maturity". Knowing Fourier analysis is neither necessary nor sufficient to teach Calculus, but the experience of learning Fourier analysis can definitely make you a more competent mathematician (which is only part of what is needed to teach). – Steven Gubkin Feb 23 '23 at 17:28
  • 1
    +1 for recommending Marsden and Tromba. Going through that text from cover to cover doing all exercises would be a great prep. – MPW Feb 24 '23 at 16:02
16

Assuming you already have a good understanding of the material in a standard course in multivariable calculus and vector calculus, I would recommend that you take a class in electricity and magnetism from the physics department. I would guess that a sizable fraction of your students are going to major in engineering and physics, and their courses in electricity and magnetism will likely be where they use vector calculus most heavily. So as their teacher, you ought to understand that material yourself. If you don't already have a lot of physics under your belt, then taking a physics class will broaden your horizons more than studying more analysis will.

Timothy Chow
  • 493
  • 3
  • 9
  • 2
    Maybe take fluid dynamics too. – J W Feb 24 '23 at 06:37
  • 2
    Great suggestion! – Steven Gubkin Feb 24 '23 at 10:43
  • 5
    Excellent suggestion. I'm inclined to add that taking a course in electricity and magnetism will not only be useful to better understand one's future students' needs, but might also lead to a much better understanding of the mathematics itself (maybe not so much at the proof level, but rather at an intuitive and geometric level). – Jochen Glueck Feb 24 '23 at 10:56
  • I love this idea! I think this will also have to go on my self study list. I won’t have room on my schedule to take another course, and I do need this 600 level sequence to graduate. Do you have any books you recommend on this topic? – blakedylanmusic Feb 24 '23 at 15:18
  • 4
    @blakedylanmusic I enjoyed Div, Grad, Curl, and All That by H. M. Schey when I was a student, as well as Electricity and Magnetism by Edward Purcell. – Timothy Chow Feb 24 '23 at 19:12
  • 3
    @blakedylanmusic: I'll second Timothy Chow's textbook suggestions. Note that "Purcell" is now "Purcell and Morin"; the latter was responsible for updating to the third edition after Purcell's death. After those, if you've got an appetite for more, try Griffiths's Introduction to Electrodynamics. – Michael Seifert Feb 24 '23 at 22:00
5

I'm going to throw in a frame challenge: you stated your goal is to be prepared to teach at a collegiate level.

All the other answers talk about whether you have the correct material, whether various courses will help you understand what you are teaching, etc.

If you want to be prepared to teach, then take workshops/courses on university-level teaching (I have).

There is a lot of evidence-based teaching methods out there, and most of this focuses on how you deliver the material, not what you teach. Practically none of the teaching training I had was topic-specific, possibly also due to lack of attendance.

Knowing the material is necessary, but nowhere near sufficient to be a competent teacher.

obscurans
  • 167
  • 2
  • 1
    Yes, that makes a lot of sense, and I'm thinking about getting my PhD in Math Education because I'm very interested in that kind of thing. Great suggestion, thank you! – blakedylanmusic Mar 03 '23 at 21:51
3

I echo Steven's comments. Honestly, independently was writing the same and then dropped it when I saw his response...then came back like a dog.

It's pretty tenuous the connection. You could be a great teacher without this class and visa versa. It's just not as high bang for the buck as taking a few texts and working them through (every problem, especially the hard ones at the end of sections).

I would only maybe add that you might consider a PDE course also if you want something multivariable to take now (Steve's list is great, just an addition). It will still not be as applicable as the material itself.

I'd also consider some stats courses (maybe a DOE class based on Box, Hunter, Hunter or similar) as that is both "multivariable" in tone (multiple regression) and also fits your other desire for stats coverage.

guest troll
  • 41
  • 2
  • Yes, I’m taking a stats class now and will take a grad level stats sequence next year, so I think I’m all set on stats. There were PDE courses offered this year as well but I didn’t have room in my schedule, so maybe that’ll also have to go in the self study list. I’m sure that would be a huge help – blakedylanmusic Feb 24 '23 at 15:21