# College Math Teaching

## September 29, 2013

### Post Sabbatical Post

Filed under: academia — Tags: , — collegemathteaching @ 1:09 am

Even though it’s only been a month since I moved back to Maine, pre-tenure leave seems like a distant memory. I expected that the change would be abrupt, especially since I was traveling and trying to do research pretty much until the day I got in my car in Austin, Texas to drive back to Portland. But I did not expect to feel this busy and overwhelmed. In this post, I share a few thoughts on the return to my regular life after my magical sabbatical.

When I was thinking about writing a blog post about my rude post-sabbatical awakening, there were a couple of things causing me writer’s block. The first, ironically enough, is that I felt overwhelmed and it’s very hard to write when your mind is thinking of all the things you need to do. The second was that I didn’t really see a point in writing a post where I complain without offering anything to the reader (I like my rants to be useful, to some extent). So I needed to think about why I wanted to write about being stressed out after sabbatical. Why do I feel this way, and what lesson can be learned from all this?

Ok, I am not in the same situation. I am NOT young, my job is a 11-12 hour teaching load job (101 students total this semester) and my sabbatical was “full pay for one semester”, which, of course, is better than zero.

On sabbatical, I spent some time merely getting up to enough speed (reviewing technical stuff) to be able to write a paper, which was submitted. I also worked on an already accepted paper that needed revisions.

Challenges? Well, I left having had taught older, more mature students.

Back: I have an “off semester” calculus II course which consists of some very talented students who placed out of calculus I, some who just got to calculus II because they needed a year of remedial work and some who have flunked one (or more) of their previous calculus courses. So I have a mix of students including some sharp ones and some who literally cannot compute $\int e^{3x} dx$

Then in differential equations: some highly talented students, some motivated students, and one case that I talked about earlier. One often doesn’t think about having to “manage” a classroom at the college level, but that can be reality in the days of “mainstreaming special needs students” and not a challenge that I anticipated.

Then there is the usual committee work; some of it important.

BUT: I had a sabbatical and that is a luxury. And I have a job.

## September 22, 2013

### Mathematics journal articles: terse but is it the author?

Filed under: advanced mathematics, point set topology, research — Tags: , — collegemathteaching @ 11:54 pm

Via Recursivity:

It’s a sad truth, but the mathematics research literature is very tough going for beginners. By “beginners” I mean bright high-school students, or university students, or beginning graduate students, or even professional mathematicians who are trained in an area different from the article he/she is trying to read. […]

Things like this permeate the mathematical literature. Take compactness, for example. Compactness is a marvelous tool that lets you deduce — usually in a non-constructive fashion — the existence of objects (particularly infinite ones) from the existence of finite “approximations”. Formally, compactness is the property that a collection of closed sets has a nonempty intersection if every finite subcollection has a nonempty intersection; alternatively, if every open cover has a finite subcover.

Now compactness is a topological property, so to use it, you really should say explicitly what the topological space is, and what the open and closed sets are. But mathematicians rarely, if ever, do that. In fact, they usually don’t specify anything at all about the setting; they just say “by the usual compactness argument” and move on. That’s great for experts, but not so great for beginners.

I really wonder who was the very first to take this particular lazy approach to mathematical exposition.

Hmmm, often it is the reviewer, referee or editor. They accept your paper, but make you take out some details (and, to be fair, add others)

A colleague and I are thinking of starting a journal called “The Journal of Omitted Details”.

But yes, this practice makes some mathematics very difficult for the non-expert to read.

Note: the usual definition of a compact set (given some topology) is: $X$ is compact if, given any collection of open sets $U_{\alpha}$ where $X \subset \cup_{i \in \alpha} U_{\alpha}$,there exists a finite number of the $U_{\alpha}$ where $X \subset \cup^{k}_{i=1} U_{\alpha i}$. That is, any open cover has a finite subcover. This is equivalent to saying that any infinite set of points in $X$ has a limit point, and in a metric space this means that $X$ is both closed and bounded.

## September 20, 2013

### Stupid Integral Tricks….

Filed under: calculus, integrals, integration by substitution — Tags: — collegemathteaching @ 9:35 pm

Ok what is the worst way to perform $\int sin(2x) dx$ correctly?

How about $\int sin(2x)dx = \int 2sin(x)cos(x)dx = sin^2(x) + C$ ?

Yes; if you compare $sin^2(x)$ to the expected $\frac{-1}{2} cos(2x)$ you’ll see that $sin^2(x) = \frac{1}{2} - \frac{1}{2}cos(2x)$. So the answers differ by a constant and therefore correspond to the same indefinite integral.

I wanted so much to take off a point for “style” but didn’t. 🙂

### Ok, have fun and justify this…

Filed under: calculus, popular mathematics, Power Series, series, Taylor Series — Tags: — collegemathteaching @ 7:59 pm

Ok, you say, “this works”; this is a series representation for $\pi$. Ok, it is but why?

Now if you tell me: $\int^1_0 \frac{dx}{1+x^2} = arctan(1) = \frac{\pi}{4}$ and that $\frac{1}{1+x^2} = \sum^{\infty}_{k=0} (-1)^k x^{2k}$ and term by term integration yields:
$\sum^{\infty}_{k=0} (-1)^k \frac{1}{2k+1}x^{2k+1}$ I’d remind you of: “interval of absolute convergence” and remind you that the series for $\frac{1}{1+x^2}$ does NOT converge at $x = 1$ and that one has to be in the open interval of convergence to justify term by term integration.

True, the series DOES converge to $\frac{\pi}{4}$ but it is NOT that elementary to see. 🙂

Boooo!

(Yes, the series IS correct…but the justification is trickier than merely doing the “obvious”).

### Only a narrow view of the students on a campus

Filed under: basic algebra, editorial — Tags: — collegemathteaching @ 4:52 pm

My recent experiences on teaching college mathematics has shielded me from a significant segment of the student population. While I have taught across the curriculum, mostly I’ve taught courses designed for science and engineering majors.

Today, I sat in a so-called “college algebra” course (remedial) to evaluate a new faculty member. This faculty member was getting excellent class participation; I was favorably impressed.

He was teaching them how to graph a polynomial that has been factored; example: $p(x) = (x+1)(x-1)(x-2)$. He wanted them too see if the graph of the polynomial was above or below the $x$ axis; in particular he was interested in the graph of $p$ between $x = 1$ and $x = 2$. So he chose the test point $x = \frac{3}{2}$ and asked the question “is $\frac{3}{2} - 1$ positive or negative?

Many sang out “positive”; a few said “negative” (seriously) but……one student said “$\frac{3}{2} - 1$ can be positive or negative.”

I started to laugh out loud but had to work to stifle it.

Later, when talking to this faculty member, I asked if the person who said “it can be either positive or negative” was making a joke. The faculty member looked down and said “uh….no.”.

So, what is going through the mind of a student who says such a thing? I don’t know for sure, but I think that it might be something like this:

At Cal, he was among the hardest workers in the dorm, but he could barely keep afloat.

Seeking help, he went at least once a week to the office of his writing instructor, Verda Delp.

The more she saw him, the more she worried. His writing often didn’t make sense. He struggled to comprehend the readings for her class and think critically about the text.

“It took awhile for him to understand there was a problem,” Delp said. “He could not believe that he needed more skills. He would revise his papers and each time he would turn his work back in having complicated it. The paper would be full of words he thought were academic, writing the way he thought a college student should write, using big words he didn’t have command of.

Sometimes students are so lost, they don’t realize that they are lost; they don’t understand that the material WOULD be clear to them if they understood it. They don’t see that there IS something to understand; it is almost as if the responses that they hear the other students given are random phrases with certain key words and key phrases in them. That these key words and key phrases actually have meaning is lost on them.

As far as whether these students should even be admitted to begin with is beyond the scope of this blog; personally, I am a fan of the “prep school” approach that the service academies use (a year to address a student’s academic deficiencies prior to being admitted to the main campus) but I haven’t studied the data there.

The issue to me: what does one do with these students? Some MIGHT reach the point where they realize that there is a point to it all, but many won’t.

## September 19, 2013

### Modern college teaching: social media

Filed under: academia — Tags: , — oldgote @ 5:44 pm

Here is another difference between current college teaching and college teaching at the time I started (1991): social media.

As a rule, I do NOT:

1. accept friend requests on Facebook
3. accept any social media connections whatsoever

from current undergraduate students at MY institution.

Personally, the social media is my playground and my time, and I’d rather not have to walk on eggshells there.

### What we mean about poor algebra skills…

Filed under: basic algebra, calculus, student learning — collegemathteaching @ 4:47 pm

Yes, mathematics professors have been complaining about their students lack of algebra skills as long as there have been calculus courses.

No, we aren’t talking about a student who, in a moment of panic, decided to write $\int \sqrt{x^2+1} dx = \int \sqrt{x^2} + \sqrt{1} dx$ because they were stuck on an exam. And yes, I once saw a professor walk into an analysis class, write $\sqrt{x^2+1} dx \ne \sqrt{x^2} + \sqrt{1}$ on the board (while grinding the chalk into the board) while saying “the next person who makes this mistake will get an F for this class, ON THE SPOT! 🙂

But the weakness is more of the following: in class today, I wrote
$\int (sec^2(x) - 1)tan(x) dx = \int (sec^2(x)tan(x) -tan(x)) dx$ $= \frac{1}{2}tan^2(x) - ln(|sec(x)|+C$

The student actually understood the integration, but didn’t understand where the first equality came from! I said “it is just algebra” and he STILL didn’t get it.

I have a hard time believing that this student doesn’t understand the distributive axiom of algebra; what I think is going on is that they don’t have the concept as a regular working part of their math/science/engineering mind.

## September 14, 2013

### Reality of modern college teaching: students with Asperger’s syndrome

Filed under: academia, mathematics education, student learning — Tags: — collegemathteaching @ 4:43 pm

One of the major changes I’ve encountered since I started college teaching (first as a teaching assistant in 1986; then as a new professor in 1991) is that students with Asperger’s syndrome have been showing up.

Most of the time, it isn’t a big deal; the worst I’ve had is one of these students became completely disoriented when he got to class and someone was sitting in “his” seat (no, I don’t make seat assignments; this is college).

This semester, I have a transfer student (not sure why he transferred); in spots he is “disruptive to a minor degree”: you have to remind him that there are 34 OTHER students in the class; this isn’t a one-on-one dialogue just for him.

Also, I sometimes make side remarks (to explain a point to another student) and use analogies; that just confuses the heck out of him. But I am not going to stop being effective with the other 34 students just for him; I just tell him “see me in office hours” or “don’t worry about this”.

On the other hand, he is relatively easy to work with in office hours; the one-on-one exchanges are usually reasonable and pleasant.

Hence, when I see he is getting confused, I tell him “for this point, see me for office hours.”

I’ve searched the internet to see what is out there; most of it is what I already know and much of it is a series of tired cliches, finger wagging, etc. I haven’t found much of the following: “I had these issues in my calculus class; here is how they were resolved” or “these issues COULDN’T be resolved.” Sometimes they aren’t up to the task of being in college.

But, overall, it seems to be this way: we are told to be “more productive” which means more students per semester (105 students in 2 sections of calculus and 1 of differential equations). So no, one cannot tailor lessons and work to the learning style of a specific student, especially if that student is an outlier. One has to teach to a type of average or to the class as a whole; one can adjust for a class full of, say biology students, or one full of engineers or one full of computer science majors.

These students require time, more attention and resources and these COST MONEY. This is where some of the increased educational expense is coming from (some from technology as well). At times, it appears as if colleges and universities are being tugged in different directions.

## September 13, 2013

### Partial Fractions Expansion: sometimes complex numbers can save time.

Filed under: calculus, differential equations, integrals — Tags: , — collegemathteaching @ 8:56 pm

One sometimes needs to do a partial fraction expansion when one is integrating or when one is doing Laplace transforms. Most people know the standard methods: either gather terms and compare coefficients, or use selected (real) values for $x$.

But sometimes, (NOT all of the time), one can speed things up by using complex numbers.

Here is an example: expand $\frac{1}{(x+1)(x^2 + 1)}$.

Solution: set this up as $\frac{1}{(x+1)(x^2 + 1)}=\frac{A}{(x+1)}+\frac{Bx+C}{(x^2 + 1)}$.
Now clear denominators to obtain $1 = A(x^2+1)+ (Bx+C)(x+1)$.

Setting $x = -1$ yields $1 = 2A$ which means $A = \frac{1}{2}$

(Yes, I know that we used a number not in the domain of the original fraction…but why can we get away with that? :-))

Now set $x = i$ we obtain $1= (Bi+C)(i+1)=(B+C)i +C-B$. By comparing real and imaginary parts, we obtain $-B = C$ and then $C = \frac{1}{2}, B = \frac{-1}{2}$.

Here is a second, more complicated case. Expand $\frac{1}{x^3 -1} = \frac{A}{x-1}+ \frac{Bx + C}{x^2+x + 1}$.

Clear denominators again to obtain $1 = A(x^2+x+1)+ (Bx+C)(x-1)$. Trying $x = 1$ yields $A = \frac{1}{3}$.

Now use a primitive complex 3’rd root of unity: $x = e^{\frac{2\pi i}{3}}$; this causes the first term to vanish. The second term becomes immediately:
$B(e^{\frac{4\pi i}{3}}-e^{\frac{2\pi i}{3}}) + C(e^{\frac{2\pi i}{3}}-1)$ which simplifies to: $\sqrt{3}i(B + \frac{1}{2}C) - \frac{3}{2}C = 1$.
Comparing real and imaginary parts again: $C = \frac{2}{3}$ and $B = -\frac{1}{3}$.

Caveat: one has to be very comfortable with complex arithmetic to use this method, but some engineers and physicists are.