# College Math Teaching

## 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 3, 2018

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.

## August 1, 2017

### Numerical solutions to differential equations: I wish that I had heard this talk first

The MAA Mathfest in Chicago was a success for me. I talked about some other talks I went to; my favorite was probably the one given by Douglas Arnold. I wish I had had this talk prior to teaching numerical analysis for the fist time.

Confession: my research specialty is knot theory (a subset of 3-manifold topology); all of my graduate program classes have been in pure mathematics. I last took numerical analysis as an undergraduate in 1980 and as a “part time, not taking things seriously” masters student in 1981 (at UTSA of all places).

In each course…I. Made. A. “C”.

Needless to say, I didn’t learn a damned thing, even though both professors gave decent courses. The fault was mine.

But…I was what my department had, and away I went to teach the course. The first couple of times, I studied hard and stayed maybe 2 weeks ahead of the class.
Nevertheless, I found the material fascinating.

When it came to understanding how to find a numerical approximation to an ordinary differential equation (say, first order), you have: $y' = f(t,y)$ with some initial value for both $y'(0), y(0)$. All of the techniques use some sort of “linearization of the function” technique to: given a step size, approximate the value of the function at the end of the next step. One chooses a step size, and some sort of schemes to approximate an “average slope” (e. g. Runga-Kutta is one of the best known).

This is a lot like numerical integration, but in integration, one knows $y'(t)$ for all values; here you have to infer $y'(t)$ from previous approximations of %latex y(t) \$. And there are things like error (often calculated by using some sort of approximation to $y(t)$ such as, say, the Taylor polynomial, and error terms which are based on things like the second derivative.

And yes, I faithfully taught all that. But what was unknown to me is WHY one might choose one method over another..and much of this is based on the type of problem that one is attempting to solve.

And this is the idea: take something like the Euler method, where one estimates $y(t+h) \approx y(t) + y'(t)h$. You repeat this process a bunch of times thereby obtaining a sequence of approximations for $y(t)$. Hopefully, you get something close to the “true solution” (unknown to you) (and yes, the Euler method is fine for existence theorems and for teaching, but it is too crude for most applications).

But the Euler method DOES yield a piecewise linear approximation to SOME $f(t)$ which might be close to $y(t)$ (a good approximation) or possibly far away from it (a bad approximation). And this $f(t)$ that you actually get from the Euler (or other method) is important.

It turns out that some implicit methods (using an approximation to obtain $y(t+h)$ and then using THAT to refine your approximation can lead to a more stable system of $f(t)$ (the solution that you actually obtain…not the one that you are seeking to obtain) in that this system of “actual functions” might not have a source or a sink…and therefore never spiral out of control. But this comes from the mathematics of the type of equations that you are seeking to obtain an approximation for. This type of example was presented in the talk that I went to.

In other words, we need a large toolbox of approximations to use because some methods work better with certain types of problems.

I wish that I had known that before…but I know it now. 🙂

## April 12, 2016

### At long last…

Filed under: academia, editorial — Tags: — collegemathteaching @ 9:17 pm

I’ve been silent on this blog for too long. Part of what is happening: our department is slowly morphing into a “mostly service courses” department due to new regulations on “minimum class size” (set to 10 students for upper division courses). THAT, plus a dearth of “mathematics teaching majors” is hurting our “majors” enrollment.

So it has been “all calculus/all the time” for me lately. Yes, calculus can be fun to teach but after close to 30 years…..zzzzzz….

And it would be unethical for me to try something new just because I am bored.

But I finally have something I want to talk about: next post!

## February 9, 2016

### An economist talks about graphs

Filed under: academia, economics, editorial, pedagogy, student learning — Tags: , — collegemathteaching @ 7:49 pm

Paul Krugman is a Nobel Laureate caliber economist (he won whatever they call the economics prize).
Here he discusses the utility of using a graph to understand an economic situation:

Brad DeLong asks a question about which of the various funny diagrams economists love should be taught in Econ 101. I say production possibilities yes, Edgeworth box no — which, strange to say, is how we deal with this issue in Krugman/Wells. But students who go on to major in economics should be exposed to the box — and those who go on to grad school really, really need to have seen it, and in general need more simple general-equilibrium analysis than, as far as I can tell, many of them get these days.

There was, clearly, a time when economics had too many pictures. But now, I suspect, it doesn’t have enough.

OK, this is partly a personal bias. My own mathematical intuition, and a lot of my economic intuition in general, is visual: I tend to start with a picture, then work out both the math and the verbal argument to make sense of that picture. (Sometimes I have to learn the math, as I did on target zones; the picture points me to the math I need.) I know that’s not true for everyone, but it’s true for a fair number of students, who should be given the chance to learn things that way.

Beyond that, pictures are often the best way to convey global insights about the economy — global in the sense of thinking about all possibilities as opposed to small changes, not as in theworldisflat. […]

And it probably doesn’t hurt to remind ourselves that our students are, in general, NOT like us. What comes to us naturally probably does not come to them naturally.

## February 8, 2016

### Where these posts are (often) coming from

Filed under: academia, linear albegra, student learning — Tags: , — collegemathteaching @ 9:57 pm

Yes, my office is messy. Deal with it. 🙂 And yes, some of my professional friends (an accountant and a lawyer) just HAD to send me their office shots…pristine condition, of course.
(all in good fun!)

Note: this semester I teach 3 classes in a row: second semester “business/life science” calculus, second semester “engineering/physical science” calculus and linear algebra. Yes, I love the topics, but there is just enough overlap that I have to really clear my head between lessons. Example: we covered numerical integration in both of my calculus classes, as well as improper integrals. I have to be careful not to throw in $\int^{\infty}_{-\infty} \frac{dx}{1+x^2}$ as an example during my “life science calculus” class. I do the “head clearing” by going up the stairs to my office between classes.

Linear algebra is a bit tricky; we are so accustomed to taking things like “linear independence” for granted that it is easy to forget that this is the first time the students are seeing it. Also, the line between rigor and “computational usefulness” is tricky; for example, how rigorously do we explain “the determinant” of a matrix?

Oh well…back to some admin nonsense.

## January 26, 2016

### The walk of shame…but

Filed under: academia, research — Tags: , — collegemathteaching @ 9:07 pm

Well, I walked to our university library with a whole stack of books that I had checked out to do a project…one which didn’t work out.

But I did check out a new book to get some new ideas…and in the book I found a little bit of my work in it (properly attributed). That was uplifting.

Now to get to work…

## September 23, 2015

### Intelligence doesn’t show outwardly….

Filed under: academia, editorial — Tags: , — oldgote @ 12:02 pm

This semester has been the “semester from hell” in that I am teaching a class in actuarial mathematics and I have never seen the material before. So I am doing a “self-study” course on my own just ahead of the students.

I’ve done things like this before, but almost always it has been in classes where at least I understood both the notation and the point of the material fairly well.

The upside: I am learning something new.
But one consequence is that I have had little to share on this blog this semester.

I will make one comment though:

I am giving the first exam back in my “calculus II” (of 3) courses. This is the “off semester” which means that I’ll have students who placed out of calculus I and I’ll have those who have either flunked this course once (or several times) or I’ll have some who have been through our remedial calculus preparation program.

Hence, my grading curve looks like a “bathtub” curve.

But, time and time again, I am fascinated by the fact that all of the students, both the smart ones and the not-so-smart ones, “look alike” in that you can not distinguish them by appearance.

This is just the opposite from sports.

In a 5K race, if I see some tiny, slender but muscular person I know that I won’t see them after the start of the race. In the gym, if i see some guy who looks like he was carved out of marble, I know that I’ll be lifting about half of what he will.

But intelligence just doesn’t show in the same way.

## September 18, 2015

### Teaching a class that one is unqualified to teach…

Filed under: academia, editorial — Tags: , — oldgote @ 11:39 am

I haven’t posted much lately. I might post some this weekend, IF I ever get caught up. I have a couple of homework sets and one set of exams to grade.

What is going on: originally, I had a 3 preparation schedule: second semester calculus (the usual), first semester “business” calculus and numerical analysis. The latter is a time suck, but I’ve taught this course multiple times and have the details reasonably well worked out.

Note: my research specialty is topology though I’ve published an elementary analysis paper as well.

Anyway, it turns out that our part time instructor who teaches our “theory of interest” and “life contingencies” class got called away and we had no one to cover a class that had 18 students enrolled. A call for volunteers was put out and I said “if no one else….” BIG MISTAKE.

I am ok with it being an evening class.

But:

1. The amount of preparation time is incredible; basically I am teaching myself this material about 1 week (if that) ahead of walking into the class. I do ALL the homework to make sure I can do it correctly.
2. While the nuts and bolts are elementary on mathematical grounds, I have very little extra insight to offer. In the other classes, I can give a bit of perspective on “what is out there”. Not so in this class. I can teach “how to use the table”.
3. I am one who needs to know the stuff really, really well (at almost an unconscious level) to be comfortable in the class room. I don’t “fake it” well.

On the other hand, this is one of the things about having earned a Ph. D. and continuing to do research (though not this semester): I know how to learn and how to test my own knowledge on something. That is an ability that allows for me to “sub” in an area that I am not qualified to be in.

Of course, I still think that our university is obligated to hire a qualified person in this area if it wants to offer an actuarial program, though our increasingly corporate administration disagrees.

## August 21, 2015

### Just shoot me. Now. (personal…and my screw up…)

The upcoming semester line up:

1. My original schedule called for science/engineering calculus II, “business calculus” I, and numerical analysis. No, my specialty is pure math (topology) but I got roped into teaching this course a few years ago, and because no one complained…guess who is stuck with it now? 🙂 (and yes, as an undergraduate, and once as a part time graduate student, I made C’s in this class)

2. But a part time faculty who taught our actuarial mathematics classes got called away so..I said “if you can’t find anyone else…” and so I got stuck with that class (in lieu of the “business calculus” class)

3. I get an e-mail about the class; I read the Fall 2014 syllabus and so prepare based on that (since it is Fall 2015)..but the topics for that class go “spring, fall, off, fall, spring” instead of “fall/spring”…so I had prepared FOR THE WRONG CLASS and ordered THE WRONG BOOK.

I caught that a week prior to classes starting..hence frantic e-mail to the department chair and secretary….and I’ll have to really work some this weekend.

Fortunately much of the stuff in this topic (“life contingencies”) is like reliability engineering and I’ve had that class. Things like the “survival function” and “bathtub curve” are familiar to me. The mathematics won’t be hard; I’ll have to focus my self study on definitions and notation.

Still, this is more interesting than I’d hoped that it would be.

The positive: I’ll have learned some new mathematics (I always learn something new every time I teach numerical analysis) and new applications of mathematics (the life contingencies …and I’ve already learned a little bit of interest theory by preparing for the class that I thought that I was teaching..)

The university
But our university (6000 students, 5000 undergraduates) is suffering from an enrollment slump for the second year in a row. We are at about 85-88 percent of what would be a “healthy” enrollment. The place is in turmoil; we lost our athletic director, provost (left) and president, the flagship basketball team has hit rock bottom and things are in disarray.

So we have a second “small” class but this time: enrollments are UP in our remedial sections. UP. And many who couldn’t place into our regular calculus sequence have been admitted…by …engineering. Seriously. They are hurting that badly, and when they hurt, WE hurt.

I’ll be shielded from much of that in the classroom because of the classes I am teaching BUT with these changes come “changes in major”; we are going to try to make our major easier to navigate by trying to maximize flexibility by making our “required courses” less prerequisite dependent. It will water down the major somewhat but hopefully make it more likely that we keep a major.

Oh well…this is what I get for taking my Ph. D. in pure mathematics instead of applied. 🙂

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