College Math Teaching

November 22, 2014

One upside to a topologist teaching numerical analysis…

Yes, I was glad when we hired people with applied mathematics expertise; though I am enjoying teaching numerical analysis, it is killing me. My training is in pure mathematics (in particular, topology) and so class preparation is very intense for me.

But I so love being able to show the students the very real benefits that come from the theory.

Here is but one example: right now, I am talking about numerical solutions to “stiff” differential equations; basically, a differential equation is “stiff” if the magnitude of the differential equation is several orders of magnitude larger than the magnitude of the solution.

A typical example is the differential equation y' = -\lambda y , y(0) = 1 for \lambda > 0 . Example: y' = -20y, y(0) = 1 . Note that the solution y(t) = e^{-20t} decays very quickly to zero though the differential equation is 20 times larger.

One uses such an equation to test a method to see if it works well for stiff differential equations. One such method is the Euler method: w_{i+1} = w_{i} + h f(t_i, w_i) which becomes w_{i+1} = w_i -20h \lambda w_i. There is a way of assigning a method to a polynomial; in this case the polynomial is p(\mu) = \mu - (1+h\lambda) and if the roots of this polynomial have modulus less than 1, then the method will converge. Well here, the root is (1+h\lambda) and calculating: -1 > 1+ h \lambda > 1 which implies that -2 >   h \lambda > 0 . This is a good reference.

So for \lambda = 20 we find that h has to be less than \frac{1}{10} . And so I ran Euler’s method for the initial problem on [0,1] and showed that the solution diverged wildly for using 9 intervals, oscillated back and forth (with equal magnitudes) for using 10 intervals, and slowly converged for using 11 intervals. It is just plain fun to see the theory in action.

November 19, 2014

Tension between practitioners and theoretical mathematicians…

Filed under: academia, applied mathematics, mathematician, research — Tags: — collegemathteaching @ 2:01 am

I follow Schneier’s Security Blog. Today, he alerted his readers to this post about an NSA member’s take on the cryptography session of a mathematics conference. The whole post is worth reading, but these comments really drive home some of the tension between those of us in academia :

Alfredo DeSantis … spoke on β€œGraph decompositions and secret-sharing schemes,” a silly topic which brings joy to combinatorists and yawns to everyone else. […]

Perhaps it is beneficial to be attacked, for you can easily augment your publication list by offering a modification.

[…]

This result has no cryptanalytic application, but it serves to answer a question which someone with nothing else to think about might have asked.

[…]

I think I have hammered home my point often enough that I shall regard it as proved (by emphatic enunciation): the tendency at IACR meetings is for academic scientists (mathematicians, computer scientists, engineers, and philosophers masquerading as theoretical computer scientists) to present commendable research papers (in their own areas) which might affect cryptology at some future time or (more likely) in some other world. Naturally this is not anathema to us.

I freely admit this: when I do research, I attack problems that…interests me. I don’t worry if someone else finds them interesting or not; when I solve such a problem I submit it and see if someone else finds it interesting. If I solved the problem correctly and someone else finds it interesting: it gets published. If my solution is wrong, I attempt to fix the error. If no one else finds it interesting, I work on something else. πŸ™‚

November 11, 2014

About that Texas Tech “Politically Challenged Video”

Filed under: editorial — Tags: , — collegemathteaching @ 11:45 pm

I became aware of this video from the blog College Misery and Why Evolution is True. An anguished quip from the College Misery blog:

While grading essays, exams and reports, you may wonder, “Do these morons know anything?”

That’s a darn good question.

The video (yes, it is UGLY)

I thought to myself: “surely my students aren’t THIS clueless.”

So, I tacked on three questions to last week’s weekly quiz (which covered relative maximums and minimums and critical numbers for functions of one variable); I did NOT give them warning that I would ask them these questions, though I gave them a tiny amount of “extra credit” if they answered the questions correctly.

The data (discounting 2 international students): 2 sections of “engineering/science” calculus, a total of 53 non-international students. Median mathematics ACT: 28, Mean: 28.15.

The questions:
1. Who won the US Civil War? (94.3 percent got it right)
2. Who is the current US Vice President (84.9 percent got it right)
3. Who did the US attain its independence from? (100 percent correct)

For question 1, I counted “Union” or “North” as correct. For question 2: it had to look like “Biden” or “Joe Biden” to be correct. For question 3: though the correct answer is England, I counted Great Britain or UK as correct as well.

1. 2 left blank, one said “US” which I didn’t count as correct. 3 misses out of 53. 94.3 percent correct.
2. 8 misses; 2 admitted that they didn’t know, 2 said “Pat Quinn” (our governor), 2 said “Dick Durbin” (our senior US Senator), 1 said “John Boehner” (Speaker of the House), 1 said “Reagan”. (He is popular in this area as he graduated from nearby Eureka College). 84.9 percent.
3. zero misses.

So, the above video is not representative at all. Now one might find it troubling that ANYONE missed 1 or 2….that’s for sure.
Caveat: yes, these were written questions, but they came without warning and they were attached to a mathematics quiz. And one can note the math ACT scores and wonder if, say, a very low level freshman mathematics class would have done as “well”. (say, “college algebra”, which is really a remedial course)

Math professor FAIL: CHECK THE OBVIOUS first.

Numerical analysis: I had the students run a “5 step predictor, 4 step corrector, variable step size” routine to solve a differential equation. This was a homework problem.
A student reported that the routine failed.

So, I looked at it…the interval for the solution was [1, 1.2] and the initial step size was…. 0.05 and what is 5 times .05? Yep, the correction part overshot the right endpoint and the “next h” was indeed negative.

This came to me when I was eating dinner…and sure enough..

OMG, I am an idiot. πŸ™‚

November 1, 2014

Ok Graduate Student, do you want a pure math Ph. D.???

Filed under: academia, calculus, editorial, research — collegemathteaching @ 2:19 am

appliedvspuremath

This slide made me chuckle (click to see a larger version). But here is the point of it: it is very, very difficult to earn your living by researching in pure mathematics.

Is it a reasonable expectation for you?

Ask yourself this: look at your advisor. Is your advisor considerably smarter than you are, or even moderately smarter than you are? If so, then forget about earning your living as a research professor in pure math. It. Is. NOT. Going. To. Happen.

Yeah, you might get a post-doc. You might even manage to get one of those “tenure track with little hope for tenure” jobs at a D-I research university…maybe (perhaps unlikely?).

I’ve been on search committees. I’ve seen the letters for those who didn’t get tenure; often these folks had decent publication records but didn’t get large enough external grants.

It is brutal out there.

If you get a pure math Ph. D. and you aren’t your advisor’s intellectual equal, about your only hope for a tenured academic job is at the “teaching intensive” universities; basically you’ll spend the vast majority of your time attempting to teach calculus to students of very average ability; after all, most of the teaching load in mathematics is teaching service courses rather than majors courses.

It does have its charm at times, but after 20+ years, it gets very, very old. I’ll discuss how to alleviate the boredom in a responsible way in another post. (e. g., it is probably a bad idea to, say, spice it up by teaching integration via hyperbolic trig functions or to try to teach residue integrals).

So, ask yourself: is your passion research and discovery? Or, is it teaching average students? If it is the latter: well, go ahead and get that theoretical math Ph. D.; after all, there ARE jobs out there, and we’ve hired a couple of people last year and might hire some more in the next couple of years.

IF your passion is research and mathematical discovery and you aren’t your advisor’s intellectual equal, either switch to applied mathematics (more demand for such research) OR enhance your education with sellable skills such as computer programming/modeling, software engineering or perhaps picking up a masters in statistics. Make yourself more marketable to industry.

Blog at WordPress.com.