College Math Teaching

January 15, 2020

Applying for an academic job: what I look for in an application

Filed under: academia, editorial, research — Tags: , — collegemathteaching @ 4:48 pm

Disclaimer: let me be clear: these are MY thoughts. Not everyone is like me.

I have served on several search committees and have chaired several of them as well. My university: an undergraduate university where we have a very modest, but mandatory research requirement (much less than what an R1 has). Teaching loads: mostly service classes; maybe an upper division class; typically 10-12 hours per semester. Service loads are heavy, especially for senior faculty.

These thoughts are for the applicant who wants to be competitive for OUR specific job; they do not apply to someone who just wants to make a blanket application.

General suggestion: Proofread what you submit. You’d be surprised at how many applications contain grammatical mistakes, spelling errors and typos.

Specific suggestions:
Take note of the job you are applying for. We have two types of tenure track positions: assistant professor positions and tenure track lecturer positions.
One can obtain promotion from the assistant professor position; one cannot be promoted from our lecturer position.

Assistant professor position: has a research and service requirement and involves teaching across the curriculum (lower and upper division classes)
Lecturer position: has a service requirement and involves teaching freshman courses..occasionally calculus but mostly pre-calculus mathematics (college algebra, precalculus, perhaps business math, non-calculus based statistics).

Teaching: What I look for is:
1. Relevant experience: how will you do when you walk into the classes YOU are teaching? Can we reasonably predict success from your application?

2. References: if you are applying for the lecturer position, you should have a teaching reference from someone who has observed your teaching in a pre calculus class (algebra, trig, pre calculus, etc.). Letters that say “X is a great teacher” and is followed by the current buzzwords really don’t stand out. Letters that say “I observed X teaching a precalculus class and saw that…” get my attention.

If you are applying for the assistant professor position, I’d like to see the observation from, say, a calculus class.

Observations of how well you lead a graduate student review for the Ph. D. exam on the topology of manifolds really isn’t helpful to us.

3. Teaching statement: I don’t care about all of the hot buzzwords or how you want to make the world a better place. I’d like to see that you thought about how to teach and, even better, how your initial experiences lead to adjustments. Things like “I saw students could not do this type of problem because they did not know X..” catch my eye as do “I tried this..it didn’t work as well as I had hoped so I tried that and it worked better ..” also catch my eye. Also, “students have trouble with these concepts and I have found that they really haven’t mastered…” are also great.

4. Please be realistic: if you ware applying for the lecturer position, it doesn’t help to state your heart is set on directing student research our teaching our complex variables class. If your heart IS really set on these things, this job is not for you.

Research (assistant professor position only) What I am looking for here is someone who won’t die on the vine. So I’d like to see evidence of:

1. Independence: can you work independently? Can you find your own problems to work on? Do you have collaborators already set (if appropriate)? What I mean: you cannot be too advisor dependent at our job, given its limited resources and heavy teaching load. An advisor’s letter that says “student X suggested this problem to work on” stand out in a positive way.

2. Plan: do you have plan moving forward?

3. Realism: you aren’t going to in a Fields Medal or an Abel Prize at our job. You aren’t going to publish in the Annals of Math. If you have your heart set on working on the toughest cutting edge problems, you will likely fail at our place and end up frustrated. And yes..staying current at the cutting edges of mathematics is all but impossible; they best you’ll be able to do is to tackle some of the stuff that isn’t dependent on heavy, difficult to learn machinery. You simply will not have large blocks of uninterrupted time to think.

March 19, 2019

My brush with mathematical greatness

Filed under: editorial — Tags: — collegemathteaching @ 6:18 pm

Yes, I TA’ed for Karen Uhlenbeck. She was patient with me and nice to me, even though I was a nothing ..actually, a below average graduate student and she was a department superstar..holder of an endowed chair. Karen Uhlenbeck just won the Abel Prize.

December 21, 2018

Over-scheduling of senior faculty and lower division courses: how important is course prep?

It seems as if the time faculty is expected to spend on administrative tasks is growing exponentially. In our case: we’ve had some administrative upheaval with the new people coming in to “clean things up”, thereby launching new task forces, creating more committees, etc. And this is a time suck; often more senior faculty more or less go through the motions when it comes to course preparation for the elementary courses (say: the calculus sequence, or elementary differential equations).

And so:

1. Does this harm the course quality and if so..
2. Is there any effect on the students?

I should first explain why I am thinking about this; I’ll give some specific examples from my department.

1. Some time ago, a faculty member gave a seminar in which he gave an “elementary” proof of why \int e^{x^2} dx is non-elementary. Ok, this proof took 40-50 minutes to get through. But at the end, the professor giving the seminar exclaimed: “isn’t this lovely?” at which, another senior member (one who didn’t have a Ph. D. had had been around since the 1960’s) asked “why are you happy that yet again, we haven’t had success?” The fact that a proof that \int e^{x^2} dx could not be expressed in terms of the usual functions by the standard field operations had been given; the whole point had eluded him. And remember, this person was in our calculus teaching line up.

2. Another time, in a less formal setting, I had mentioned that I had given a brief mention to my class that one could compute and improper integral (over the real line) of an unbounded function that that a function could have a Laplace transform. A junior faculty member who had just taught differential equations tried to inform me that only functions of exponential order could have a Laplace transform; I replied that, while many texts restricted Laplace transforms to such functions, that was not mathematically necessary (though it is a reasonable restriction for an applied first course). (briefly: imagine a function whose graph consisted of a spike of height e^{n^2} at integer points over an interval of width \frac{1}{2^{2n} e^{2n^2}} and was zero elsewhere.

3. In still another case, I was talking about errors in answer keys and how, when I taught courses that I wasn’t qualified to teach (e. g. actuarial science course), it was tough for me to confidently determine when the answer key was wrong. A senior, still active research faculty member said that he found errors in an answer key..that in some cases..the interval of absolute convergence for some power series was given as a closed interval.

I was a bit taken aback; I gently reminded him that \sum \frac{x^k}{k^2} was such a series.

I know what he was confused by; there is a theorem that says that if \sum a_k x^k converges (either conditionally or absolutely) for some x=x_1 then the series converges absolutely for all x_0 where |x_0| < |x_1| The proof isn’t hard; note that convergence of \sum a_k x^k means eventually, |a_k x^k| < M for some positive M then compare the “tail end” of the series: use |\frac{x_0}{x_1}| < r < 1 and then |a_k (x_0)^k| = |a_k x_1^k (\frac{x_0}{x_1})^k| < |r^k|M and compare to a convergent geometric series. Mind you, he was teaching series at the time..and yes, is a senior, research active faculty member with years and years of experience; he mentored me so many years ago.

4. Also…one time, a sharp young faculty member asked around “are there any real functions that are differentiable exactly at one point? (yes: try f(x) = x^2 if x is rational, x^3 if x is irrational.

5. And yes, one time I had forgotten that a function could be differentiable but not be C^1 (try: x^2 sin (\frac{1}{x}) at x = 0

What is the point of all of this? Even smart, active mathematicians forget stuff if they haven’t reviewed it in a while…even elementary stuff. We need time to review our courses! But…does this actually affect the students? I am almost sure that at non-elite universities such as ours, the answer is “probably not in any way that can be measured.”

Think about it. Imagine the following statements in a differential equations course:

1. “Laplace transforms exist only for functions of exponential order (false)”.
2. “We will restrict our study of Laplace transforms to functions of exponential order.”
3. “We will restrict our study of Laplace transforms to functions of exponential order but this is not mathematically necessary.”

Would students really recognize the difference between these three statements?

Yes, making these statements, with confidence, requires quite a bit of difference in preparation time. And our deans and administrators might not see any value to allowing for such preparation time as it doesn’t show up in measures of performance.

August 28, 2018

Commentary: what does it mean to “graduate from college”?

Filed under: editorial — collegemathteaching @ 1:21 am

Recently, an Oregon university touted graduating someone with Down’s syndrome:

Walking across the stage at graduation was more than just a personal accomplishment for Cody Sullivan as he became Oregon’s first student with Down syndrome to complete four years of college.

Sullivan, 22, received his certificate of achievement at the Concordia University graduation ceremony last month, declaring that while assignments and curriculum were modified for his learning abilities, Sullivan completed all the relevant coursework to make him an official college graduate.

It is every interestingly worded: “certificate of achievement” and “assignments and curriculum were modified for his learning abilities”.

This represents a different point of view than I have.

When a teach a course, getting a certain grade in a course requires that the person getting grade to master certain concepts and skills at a certain level. Those requirements are NOT modified for someone’s learning ability. And getting a degree in a certain subject means (or should mean) that one has established a certain competency in that said subject.

But, well, I wonder if we are moving toward a “meeting a certain competency level isn’t relevant” anymore and just giving “you were here and did stuff” certificates.

There was a time when I thought “aptitude matters” but, well?

August 3, 2018

A rant about academic publishing

Filed under: academia, editorial, research — collegemathteaching @ 12:52 am

Ok, it is nearing the end of the summer and I feel as if I am nearing the end of a paper that I have been working on for some time. Yes, I am confident that it will get accepted somewhere, though I will submit it to my “first choice” journal when it is ready to go. I have 6 diagrams to draw up, put in, and then to do yet another grammar/spelling/consistent usage check.

Part of this “comes with the territory” of trying to stay active when teaching at a non-research intensive school; one tends to tackle such projects in “modules” and then try to put them together in a seamless fashion.

But that isn’t my rant.

My rant (which might seem strange to younger faculty):

A long time ago, one would work on a paper and write it longhand and ..if you were a professor, have the technical secretary type it up. Or one would just use a word processor of some sort and make up your Greek characters by hand. You’d submit it, and if it were accepted, the publisher would have it typeset.

Now: YOU are expected to do the typesetting and that can be very time consuming. YOU are expected to make camera ready diagrams.

And guess what: you aren’t paid for your article. The editor isn’t paid. The referee(s) isn’t (aren’t) paid. But the journal still charges subscription fees, sometimes outrageously high fees. And these are the standard journals, not the “fly by night” predatory journals.

This is another case where the professor’s workload went up, someone else’s expense went down, and the professor received no extra benefit.

Yes, I know, “cry me a river”, blah, blah, blah. But in this respect, academia HAS changed and not for the better.

June 20, 2018

Editorial: one major disconnect between us and much of the public ..

Filed under: editorial — Tags: , , , , — collegemathteaching @ 1:52 am

The University of Chicago decided to stop requiring the ACT/SAT of its applicants. Now never in a million years would I give a suggestion to the University of Chicago (or any other elite school) as to what their admissions/applicant policies should be.

But there is a broader “scrap the college entrance exams” movement out there and much of the justification you hear is just complete nonsense. Example: “we have data that says the high school gpa is a better predictor of X”. (X meaning “first year success”, or “graduation”) Now that may be true, but why stop with just one bit of information if the second bit, taken together, increases predictive power?

And there is a second claim from those who admit that not all high schools are created equal, and an A in, say, high school calculus in one school might mean less than an A from another school: admitting that the quality of high schools vary means that you are just punishing the students from the academically weaker high schools a second time when you use a college entrance exam.

That claim misses the point entirely. Many schools (like ours) uses the score, at least in part, for placement purposes (we aren’t that tough to get into). And we have have decades of data that shows that, yes, the math ACT score matters, in terms of success in first year calculus. This isn’t our school (it is the University of Michigan), but we have very similar results.

And this brings us to the disconnect in attitudes.

1. We use scores to determine if the student has a reasonable probability of success in, say, a freshman calculus course. Now of course, sometimes someone under the cut-off has success. But if you give too much benefit of the doubt to prospective students, your DFW rate (D’s, F’s, Withdraws) will climb and administrators such be made aware of the trade-off.

2. We also understand that aptitude matters. There are many (more than you think) that aptitude has no role, or a very minor role (“you can do anything you want to do if you put your mind to it”, etc.) and some who embrace “blank slate” thinking (to them, aptitude is a fiction).

I suppose that people who REALLY believe “2” believe that, say, recruiting plays no role in the success of college sports team..a good coach can just draw from the student body and win games.

3. Part of the role of, say, the calculus sequence is to identify those who have a good probability of success in certain majors. Let’s face it; if you really can’t calculate \frac{d}{dx}sin(2x) you have no business being an engineer. Yes, on rare occasion, I’ve had students flunk my class in science/engineering calculus class because they really could not do that.

June 18, 2018

And my “clever proof” is dashed

Filed under: complex variables, editorial, knot theory, numerical methods, topology — Tags: , — collegemathteaching @ 6:03 pm

It has been a while since I posted here, though I have been regularly posting in my complex variables class blog last semester.

And for those who like complex variables and numerical analysis, this is an exciting, interesting development.

But as to the title of my post: I was working to finish up a proof that one kind of wild knot is not “equivalent” to a different kind of wild knot and I had developed a proof (so I think) that the complement of one knot contains an infinite collection of inequivalent tori (whose solid tori contain the knot non-trivially) whereas the other kind of knot can only have a finite number of such tori. I still like the proof.

But it turns out that there is already an invariant that does the trick nicely..hence I can shorten and simplify the paper.

But dang it..I liked my (now irrelevant to my intended result) result!

February 11, 2018

Posting went way down in 2017

Filed under: advanced mathematics, complex variables, editorial — collegemathteaching @ 12:05 am

I only posted 3 times in 2017. There are many reasons for this; one reason is the teaching load, the type of classes I was teaching, etc.

I spent some of the year creating a new course for the Business College; this is one that replaced the traditional “business calculus” class.

The downside: there is a lot of variation in that course; for example, one of my sections has 1/3 of the class having a math ACT score of under 20! And we have many who are one standard deviation higher than that.

But I am writing. Most of what I write this semester can be found at the class blog for our complex variables class.

Our class does not have analysis as a prerequisite so it is a challenge to make it a truly mathematical class while getting to the computationally useful stuff. I want the students to understand that this class is NOT merely “calculus with z instead of x” but I don’t want to blow them away with proofs that are too detailed for them.

The book I am using does a first pass at integration prior to getting to derivatives.

August 1, 2017

Big lesson that many overlook: math is hard

Filed under: advanced mathematics, conference, editorial, mathematician, mathematics education — Tags: — collegemathteaching @ 11:43 am

First of all, it has been a very long time since I’ve posted something here. There are many reasons that I allowed myself to get distracted. I can say that I’ll try to post more but do not know if I will get it done; I am finishing up a paper and teaching a course that I created (at the request of the Business College), and we have a record enrollment..many of the new students are very unprepared.

Back to the main topic of the post.

I just got back from MAA Mathfest and I admit that is one of my favorite mathematics conferences. Sure, the contributed paper sessions give you a tiny amount of time to present, but the main talks (and many of the simple talks) are geared toward those of us who teach mathematics for a living and do some research on the side; there are some mainstream “basic” subjects that I have not seen in 30 years!

That doesn’t mean that they don’t get excellent people for the main speaker; they do. This time, the main speaker was Dusa McDuff: someone who was a member of the National Academy of Sciences. (a very elite level!)

Her talk was on the basics of symplectec geometry (introductory paper can be found here) and the subject is, well, HARD. But she did an excellent job of giving the flavor of it.

I also enjoyed Erica Flapan’s talk on graph theory and chemistry. One of my papers (done with a friend) referenced her work.

I’ll talk about Douglas Arnold’s talk on “when computational math meets geometry”; let’s just say that I wish I had seen this lecture prior to teaching the “numerical solutions for differential equations” section of numerical analysis.

Well, it looks as if I have digressed yet again.

There were many talks, and some were related to the movie Hidden Figures. And the cheery “I did it and so can you” talks were extremely well attended…applause, celebration, etc.

The talks on sympletec geometry: not so well attended toward the end. Again, that stuff is hard.

And that is one thing I think that we miss when we encourage prospective math students: we neglect to tell them that research level mathematics is difficult stuff and, while some have much more talent for it than others, everyone has to think hard, has to work hard, and almost all of us will fail, quite a bit.

I remember trying to spend over a decade trying to prove something, only to fail and to see a better mathematician get the result. One other time I spent 2 years trying to “prove” something…and I couldn’t “seal the deal”. Good thing too, as what I was trying to prove was false..and happily I was able to publish the counterexample.

December 28, 2016

Commentary: our changing landscape and challenges

Filed under: calculus, editorial — collegemathteaching @ 10:34 pm

Yes, I haven’t written anything of substance in a while; I hope to remedy that in upcoming weeks. I am teaching differential equations this next semester and that is usually good for a multitude of examples.

Our university is undergoing changes; this includes admitting students who are nominally STEM majors but who are not ready for even college algebra.

Our provost wants us to reduce college algebra class sizes…even though we are down faculty lines and we cannot find enough bodies to cover courses. Our wonderful administrators didn’t believe us when we explained that it is difficult to find “masters and above” part time faculty for mathematics courses.

And so: with the same size freshmen class, we have a wider variation of student abilities: those who are ready for calculus III, and those who cannot even add simple fractions (yes, one of these was admitted as a computer science major!). Upshot: we need more people to teach freshmen courses, and we are down faculty lines!

Then there is the pressure from the bean-counters in our business office. They note that many students are avoiding our calculus courses and taking them at community colleges. So, obviously, we are horrible teachers!

Here is what the administrators will NOT face up to: students frequently say that passing those courses at a junior college is much easier; they don’t have to study nearly as much. Yes, engineering tells us that students with JC calculus don’t do any worse than those who take it from the mathematics department.

What I think is going on: at universities like ours (I am NOT talking about MIT or Stanford!), the mathematics required in undergraduate engineering courses has gone down; we are teaching more mathematics “than is necessary” for the engineering curriculum, at least the one here.

So some students (not all) see the extra studying required to learn “more than they need” as wasted effort and they resent it.

The way we get these students back: lower the mathematical demands in our calculus courses, or at least lower the demands on studying the more abstract stuff (“abstract”, by calculus standards).

Anyhow, that is where we are. We don’t have the resources to offer both a “mathematical calculus” course and one that teaches “just what you need to know”.

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