# 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!

## 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”.

## August 11, 2016

### Post Promotion Summer

Filed under: editorial, topology — Tags: — collegemathteaching @ 12:02 am

This is my first “terminal promotion” summer. And while I have something that I have “sort of” written up…I just don’t like the result; it basically fills in some gaps in a survey article. But I think that my thinking about this article has lead me to something that I can add to the paper so that I’ll actually LIKE what I submit.

Then again, my quandary can be summed up in this tweet:

If I wait until I am absolutely in love with my work before I send it out, it will never get sent out.

Hopefully, I’ll have more material to add to this blog this semester.

What I am working on: equivalence classes of simple closed curves; these are one to one, continuous images of the unit circle in 3-space. The objects that I am studying are so pathological that these curves fail to have a tangent at ANY point. One of these beasts can be constructed by taking the intersection of these nested, solid tori.

## 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

### Bedside manner

Filed under: editorial — Tags: — collegemathteaching @ 7:53 pm

One of the things I’ve had to change is how I related to students.

I grew up playing football and wrestling. I went to a service academy and served in the Navy. Then, of course, I got pushed in graduate school.

I cannot treat my students the way that I got treated; they would break down rather than get motivated; in general they have trouble handling even a hint of anger.

### 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.

## January 20, 2016

### Congratulations to the Central Missouri State Mathematics Department

Filed under: advanced mathematics, editorial, number theory — Tags: — collegemathteaching @ 10:43 pm

The largest known prime has been discovered by mathematicians at Central Missouri State University.

For what it is worth, it is: $2^{74,207,281} -1$.

Now if you want to be depressed, go to the Smithsonian Facebook page and read the comment. The Dunning-Kruger effect is real. Let’s just say that in our era, our phones are smarter than our people. 🙂

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