# College Math Teaching

## 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!

## April 24, 2018

### And I trolled my complex variables class

Filed under: advanced mathematics, analysis, class room experiment, complex variables — collegemathteaching @ 6:34 pm

One question on my last exam: find the Laurent series for $\frac{1}{z + 2i}$ centered at $z = -2i$ which converges on the punctured disk $|z+2i| > 0$. And yes, about half the class missed it.

I am truly evil.

## April 5, 2018

### A talk at University of South Alabama

Filed under: advanced mathematics, knot theory, topology — Tags: — collegemathteaching @ 3:27 pm

My slides (in order, more or less), can be found here.

## March 12, 2018

### And I embarrass myself….integrate right over a couple of poles…

Filed under: advanced mathematics, analysis, calculus, complex variables, integrals — Tags: — collegemathteaching @ 9:43 pm

I didn’t have the best day Thursday; I was very sick (felt as if I had been in a boxing match..chills, aches, etc.) but was good to go on Friday (no cough, etc.)

So I walk into my complex variables class seriously under prepared for the lesson but decide to tackle the integral

$\int^{\pi}_0 \frac{1}{1+sin^2(t)} dt$

Of course, you know the easy way to do this, right?

$\int^{\pi}_0 \frac{1}{1+sin^2(t)} dt =\frac{1}{2} \int^{2\pi}_0 \frac{1}{1+sin^2(t)} dt$ and evaluate the latter integral as follows:

$sin(t) = \frac{1}{2i}(z-\frac{1}{z}), dt = \frac{dz}{iz}$ (this follows from restricting $z$ to the unit circle $|z| =1$ and setting $z = e^{it} \rightarrow dz = ie^{it}dt$ and then obtaining a rational function of $z$ which has isolated poles inside (and off of) the unit circle and then using the residue theorem to evaluate.

So $1+sin^2(t) \rightarrow 1+\frac{-1}{4}(z^2 -2 + \frac{1}{z^2}) = \frac{1}{4}(-z^2 + 6 -\frac{1}{z^2})$ And then the integral is transformed to:

$\frac{1}{2}\frac{1}{i}(-4)\int_{|z|=1}\frac{dz}{z^3 -6z +\frac{1}{z}} =2i \int_{|z|=1}\frac{zdz}{z^4 -6z^2 +1}$

Now the denominator factors: $(z^2 -3)^2 -8$ which means $z^2 = 3 - \sqrt{8}, z^2 = 3+ \sqrt{8}$ but only the roots $z = \pm \sqrt{3 - \sqrt{8}}$ lie inside the unit circle.
Let $w = \sqrt{3 - \sqrt{8}}$

Write: $\frac{z}{z^4 -6z^2 +1} = \frac{\frac{z}{((z^2 -(3 + \sqrt{8})}}{(z-w)(z+w)}$

Now calculate: $\frac{\frac{w}{((w^2 -(3 + \sqrt{8})}}{(2w)} = \frac{1}{2} \frac{-1}{2 \sqrt{8}}$ and $\frac{\frac{-w}{((w^2 -(3 + \sqrt{8})}}{(-2w)} = \frac{1}{2} \frac{-1}{2 \sqrt{8}}$

Adding we get $\frac{-1}{2 \sqrt{8}}$ so by Cauchy’s theorem $2i \int_{|z|=1}\frac{zdz}{z^4 -6z^2 +1} = 2i 2 \pi i \frac{-1}{2 \sqrt{8}} = \frac{2 \pi}{\sqrt{8}}=\frac{\pi}{\sqrt{2}}$

Ok…that is fine as far as it goes and correct. But what stumped me: suppose I did not evaluate $\int^{2\pi}_0 \frac{1}{1+sin^2(t)} dt$ and divide by two but instead just went with:

$latex $\int^{\pi}_0 \frac{1}{1+sin^2(t)} dt \rightarrow i \int_{\gamma}\frac{zdz}{z^4 -6z^2 +1}$ where $\gamma$ is the upper half of $|z| = 1$? Well, $\frac{z}{z^4 -6z^2 +1}$ has a primitive away from those poles so isn’t this just $i \int^{-1}_{1}\frac{zdz}{z^4 -6z^2 +1}$, right? So why not just integrate along the x-axis to obtain $i \int^{-1}_{1}\frac{xdx}{x^4 -6x^2 +1} = 0$ because the integrand is an odd function? This drove me crazy. Until I realized…the poles….were…on…the…real…axis. ….my goodness, how stupid could I possibly be??? To the student who might not have followed my point: let $\gamma$ be the upper half of the circle $|z|=1$ taken in the standard direction and $\int_{\gamma} \frac{1}{z} dz = i \pi$ if you do this property (hint: set $z(t) = e^{it}, dz = ie^{it}, t \in [0, \pi]$. Now attempt to integrate from 1 to -1 along the real axis. What goes wrong? What goes wrong is exactly what I missed in the above example. ## February 22, 2018 ### What is going on here: sum of cos(nx)… Filed under: analysis, derivatives, Fourier Series, pedagogy, sequences of functions, series, uniform convergence — collegemathteaching @ 9:58 pm This started innocently enough; I was attempting to explain why we have to be so careful when we attempt to differentiate a power series term by term; that when one talks about infinite sums, the “sum of the derivatives” might fail to exist if the sum is infinite. Anyone who is familiar with Fourier Series and the square wave understands this well: $\frac{4}{\pi} \sum^{\infty}_{k=1}$ $\frac{1}{2k-1}sin((2k-1)x) = (\frac{4}{\pi})( sin(x) + \frac{1}{3}sin(3x) + \frac{1}{5}sin(5x) +.....)$ yields the “square wave” function (plus zero at the jump discontinuities) Here I graphed to $2k-1 = 21$ Now the resulting function fails to even be continuous. But the resulting function is differentiable except for the points at the jump discontinuities and the derivative is zero for all but a discrete set of points. (recall: here we have pointwise convergence; to get a differentiable limit, we need other conditions such as uniform convergence together with uniform convergence of the derivatives). But, just for the heck of it, let’s differentiate term by term and see what we get: $(\frac{4}{\pi})\sum^{\infty}_{k=1} cos((2k-1)x) = (\frac{4}{\pi})(cos(x) + cos(3x) + cos(5x) + cos(7x) +.....)...$ It is easy to see that this result doesn’t even converge to a function of any sort. Example: let’s see what happens at $x = \frac{\pi}{4}: cos(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}$ $cos(\frac{\pi}{4}) + cos(3\frac{\pi}{4}) =0$ $cos(\frac{\pi}{4}) + cos(3\frac{\pi}{4}) + cos(5\frac{\pi}{4}) = -\frac{1}{\sqrt{2}}$ $cos(\frac{\pi}{4}) + cos(3\frac{\pi}{4}) + cos(5\frac{\pi}{4}) + cos(7\frac{\pi}{4}) = 0$ And this repeats over and over again; no limit is possible. Something similar happens for $x = \frac{p}{q}\pi$ where $p, q$ are relatively prime positive integers. But something weird is going on with this sum. I plotted the terms with $2k-1 \in \{1, 3, ...35 \}$ (and yes, I am using $\frac{\pi}{4} csc(x)$ as a type of “envelope function”) BUT…if one, say, looks at $cos(29x) + cos(31x) + cos(33x) + cos(35x)$ we really aren’t getting a convergence (even at irrational multiples of $\pi$). But SOMETHING is going on! I decided to plot to $cos(61x)$ Something is going on, though it isn’t convergence. Note: by accident, I found that the pattern falls apart when I skipped one of the terms. This is something to think about. I wonder: for all $x \in (0, \pi), sup_{n \in \{1, 3, 5, 7....\}}|\sum^{n}_{k \in \{1,3,..\}}cos(kx)| \leq |csc(x)|$ and we can somehow get close to $csc(x)$ for given values of $x$ by allowing enough terms…but the value of $x$ is determined by how many terms we are using (not always the same value of $x$). ## 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 28, 2017 ### Integration by parts: why the choice of “v” from “dv” might matter… We all know the integration by parts formula: $\int u dv = uv - \int v du$ though, of course, there is some choice in what $v$ is; any anti-derivative will do. Well, sort of. I thought about this as I’ve been roped into teaching an actuarial mathematics class (and no, I have zero training in this area…grrr…) So here is the set up: let $F_x(t) = P(0 \leq T_x \leq t)$ where $T_x$ is the random variable that denotes the number of years longer a person aged $x$ will live. Of course, $F_x$ is a probability distribution function with density function $f$ and if we assume that $F$ is smooth and $T_x$ has a finite expected value we can do the following: $E(T_x) = \int^{\infty}_0 t f_x(t) dt$ and, in principle this integral can be done by parts….but…if we use $u = t, dv = f_x(t), du = dt, v = F_x$ we have: \ $t(F_x(t))|^{\infty}_0 -\int^{\infty}_0 F_x(t) dt$ which is a big problem on many levels. For one, $lim_{t \rightarrow \infty}F_x(t) = 1$ and so the new integral does not converge..and the first term doesn’t either. But if, for $v = -(1-F_x(t))$ we note that $(1-F_x(t)) = S_x(t)$ is the survival function whose limit does go to zero, and there is usually the assumption that $tS_x(t) \rightarrow 0$ as $t \rightarrow \infty$ So we now have: $-(S_x(t) t)|^{\infty}_0 + \int^{\infty}_0 S_x(t) dt = \int^{\infty}_0 S_x(t) dt = E(T_x)$ which is one of the more important formulas. ## 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. 🙂

### 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.