College Math Teaching

September 24, 2012

Advice to students: LEARN from your homework assignments!

Filed under: editorial, how to learn calculus, mathematics education, pedagogy — collegemathteaching @ 9:14 pm

This is going to sound a bit banal, but I think that this is sometimes overlooked.

College students are sometimes under time pressures; therefore it is common for them to view a homework assignment as a task to be “checked off” when completed and a hoop to jump through for a grade.

But that is exactly the wrong approach to take with regards to homework.

Homework is designed to help the student learn the material; that is, the student who does a homework problem should be just a bit smarter and more knowledgeable after completing the assignment (and each problem, for that matter!) than they were prior to starting the assignment.

So, my advice to doing the problems: be sure to do a few problems with your book shut and notes closed; that is how you learn if you really know the stuff or not. Then ask yourself: “why did I do the first step?” “Why did I do the next step?” “Why this approach and not another one?”

After doing the problem (or a set of problems), ask yourself: “what did I learn from this?”

Then, when it comes to review, I suggest writing down some problems on sheets of paper, scrambling the sheets, and then shutting the book and closing the notes. Why?
Many times, 75 percent of the problem is knowing WHAT technique to use.

Remember: homework is designed to help you learn.

September 21, 2012

A an example to demonstrate the concept of Sufficient Statistics

A statistic U(Y_1, Y_2, ...Y_n) is said to be sufficient for \hat{\theta} if the conditional distribution f(Y_1, Y_2,...Y_n,|U, \theta) =  f(Y_1, Y_2,...Y_n,|U) , that is, doesn’t depend on \theta . Intuitively, we mean that a given statistic provides as much information as possible about \theta ; there isn’t a way to “crunch” the observations in a way to yield more data.

Of course, this is equivalent to the likelihood function factoring into a function of \theta and U alone and a function of the Y_i alone.

Though the problems can be assigned to get the students to practice using the likelihood function factorization method, I think it is important to provide an example which easily shows what sort of statistic would NOT be sufficient for a parameter.

Here is one example that I found useful:

let Y_1, Y_2, ...Y_n come from a uniform distribution on [-\theta, \theta] .
Now ask the class: is there any way that \bar{Y} could be sufficient for \theta ? It is easy to see that \bar{Y} will converge to 0 as n goes to infinity.

It is also easy to see that the likelihood function is (\frac{1}{2\theta})^n H_{-\theta, \theta}(|Y|_{(n)} where H_{[a,b]} is the standard Heavyside function on the interval [a,b] (equal to one on the support set [a,b] and zero elsewhere) and |Y|_{(n)} is the Y_i of maximum magnitude (or the n'th order statistic for the absolute values of the observations).

So one can easily see an example of a sufficient statistic as well.

September 16, 2012

Commentary: New Ph.D. in math? Is a “teaching institution” for you?

Filed under: academia, advanced mathematics, calculus — collegemathteaching @ 10:25 pm

I remember MathFest (the MAA summer meeting) and how I enjoyed many of the talks. But one of the talks was rather enlightening. The content of the talk was good enough: it was about the applications of topology to science and engineering problems. The speaker wondered how we might get the research engineers to be more conversant in topics like algebraic topology and wondered if we might…introduce such concepts in our multi-variable calculus and linear algebra courses! Yes…

He made a remark on how calculus was, well, one of those “less-than-desirable” courses for faculty to teach. Oh yes, the speaker was from an Ivy league institution. I had to snicker; at my place the “regular calculus” sequence is considered one of the more desirable courses; the unpopular courses tend to be the non-calculus based statistics, “math for poets” and our “business calculus” (aka “faux calculus”) courses.

So this made me reflect on my career and describe my experience at a “teaching institution that has a research requirement”. I do this to give the new Ph. Ds a chance to determine if, barring a failure to obtain a post-doc position or a tenure track position at a research school, they want to go to academia or retrain for industry or some other area. Life at an undergraduate oriented institution is very different from life at a research institution.

Caveat: I realize that my experience at an “average” undergraduate institution (median ACT of entering freshmen: about 25) is but one data point. People going to an elite undergraduate oriented institution or to open admission caliber places are likely to have a different experience.

My background: Ph. D. in topology from a D-I research place; part of my thesis got published in the Proceedings of the American Mathematical Society. My adviser solved a big problem and has graduated elite students; I wasn’t one of them. This was back in 1991; at the time the old Soviet Union and the market was flooded with Russians with very strong credentials. I feel very fortunate to have landed a tenure track job; I had a couple of nibbles at post docs but nothing worked out.

My university: about 5500-6000 total students; we have a business and engineering department. So we have 15 full time faculty members and most of our load is service courses. Usually we teach two “calculus of some sort” courses (both regular and fake calculus) and one differential equations/linear algebra or upper division course.

Teaching life: students are very needy; they are sold on the idea of “individual attention”. They use office hours. (note: this isn’t a bad thing, but it does eat up time; that is reality). Often most of the students in the class have to take it; most have minimal motivation to learn the material for its own sake and many don’t have a lot of ability. The classes vary; I did have one “fake calculus” class with a median ACT of 22; another one has 10 of the 21 on academic probation AT THE START of the semester.

On the other hand, I’ve had a few classes in which the students were naturally fired up and ready to learn…and had tons of ability to boot. I remember teaching the lesson on Green’s Theorem (integrals) and having a student ask “but…isn’t it true that not all two dimensional surfaces can be embedded in the plane?” Yes, we were ready to go on to Stokes Theorem (ok experts, I know that ALL of the integral theorems are really Stokes Theorem but…)

The stages I went through:
1. Denial: I figured that I’d get started, work my butt off on research (on my own time) and get some publications and get to that post-doc position.

Reality: well, there was a reason I didn’t get the available post-doc positions out of graduate school: ABILITY. I wasn’t as smart as those who landed those coveted positions. Add to that that those in those post-doc positions were teaching lower loads and even teaching advanced classes on occasion (e. g., perhaps an algebraic topology class).

Reality: there were stacks of grading, committees, students seeking help often for problems like \frac{d}{dx} sin(2x) = ?etc.

Reality: I also had to worry about tenure; that meant at least doing a few problems that I had a good shot of solving and getting published. But these publications were the more “routine results”; throw-aways for the big guns but not the kind that would get you noticed at those D-I research places.

Reality: the longer I stayed away, the more techniques I forgot…and the more my discipline left me behind. Staying current is a full time job in itself, and all but impossible to do for someone teaching 12 hours and doing departmental work.

Reality: there were also the “been there forever” colleagues with questionable doctorates (or less) who, well, didn’t see the difference between their courses and those taught at MIT. Seriously.

Reality: some of the old time “researchers” indeed had a long list of publications: in those low-level foreign journals; this was, well, the quality of research that I had to do to ensure that I had something to get tenure. And some of them didn’t understand why research I faculty didn’t consider them to be scholarly equals.

2. Acceptance: many of my colleagues understood and reminded me that we are all in the same boat; I did the same for newer colleagues. They also reminded me that post tenure research is important…for one’s own sanity. Teaching calculus to, well, average students year after year will kill your brain cells if you don’t keep at least some intellectual engagement. You won’t be publishing at Annals of Mathematics and you are highly unlikely to be a lead speaker at a research conference.

And, after tenure, you can afford to do some “fun” filling in the gaps of your own mathematical education and perhaps even publish a paper or two in a new area (not cutting edge research, of course).

Get to some conferences, keep perspective (yes, your discipline HAS marched on without you…yes that hurts but that is reality) and have fun. Don’t pretend to be something you aren’t; that takes too much energy!

A comment made me rethink what I said: I said what I meant but I was incomplete. Here are a few upsides to this situation:

1. We teach across the curriculum. Hence I ended up learning things (or relearning some forgotten undergraduate mathematics and building on it) that I might not have learned as a full time researcher in my field. For example: I never realized how fascinating some of the numerical methods can be! So, when I teach something like Simpson’s rule in calculus, I have a bit more perspective and can point out that the relationship \frac{2}{3} MID(n) + \frac{1}{3}TRAP(n) = SIMP(2n) is really part of a scheme in which previous integration approximations can be averaged to produce a still more accurate approximation (Romberg Integration).

2. I have smart colleagues; they too have had to produce at least a little bit of research and we talk to each other about it. In fact, more than once, discussions with colleagues have lead to minor papers.

3. There are the math major classes too. While these students are not ready for cutting edge stuff, having a bit of research background helps one to provide some perspective on the various major theorems (e. g., what are the ramifications of the various types of convergences of sequences of functions?)

So, there are lots of upsides too and I don’t want to downplay those.

September 11, 2012

Two Media Articles: topology and vector fields, and political polls

Topology, vector fields and indexes

This first article appeared in the New York Times. It talks about vector fields and topology, and uses finger prints as an example of a foliation derived from the flow of a vector field on a smooth surface.

Here is a figure from the article in which Steven Strogatz discusses the index of a vector field singularity:

You might read the comments too.

Note: the author of the quoted article made a welcome correction:

small point that I finessed in the article, and maybe shouldn’t have: it’s about orientation fields (sometimes called line fields or director fields), not vector fields. Think of the elements as undirected vectors (ie., the ridges don’t have arrows on them). The singularities for orientation fields are different from those for vector fields. You can’t have a triradius in a continuous vector field, for example.

Comment by Steven Strogatz

Our local paper had a nice piece by Brian Gaines on political polls. Of interest to statistics students is the following:

1. Pay little attention to “point estimates.”

Suppose a poll finds that Candidate X leads Y, 52 percent to 48 percent. Those estimates come with a margin of error, usually reported as plus or minus three or four percentage points. It is tempting to ignore this complication, and read 52 to 48 as a small lead, but the appropriate conclusion is “too close to call.”

2. Even taking the margins of error into account does not guarantee accurate estimates.

For example, 52 percent +/- 4 percent represents an interval of 48 to 56 percent. Are we positive that the true percentage planning to vote for X is in that range? No. When we measure the attitudes of millions by contacting only hundreds, there is no escaping uncertainty. Usually, we compute intervals that will be wrong five times out of 100, simply by chance.

Note: a consistent lead of 4 points is significant, but doesn’t mean much for an isolated poll.

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