I’ve taught a variety of math classes, and statistics has been the hardest to teach. The thing I find most challenging is coming up with homework problems. Most exercises are either blatantly artificial or extremely tedious. It’s hard to find moderately realistic problems that don’t take too long to work out.

The course I’ve found easiest to teach has been differential equations. The course has a flat structure: there’s a list of techniques to cover, all roughly the same level of difficulty. There are no deep analytic or philosophical issues to skirt around as there are in statistics. And it’s not hard to come up with practical applications that can be worked out fairly easily.

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Is there any way to turn it into teaching statistics via computation? Then the tedious stuff wouldn’t be nearly so tedious, and plus, you can examine some significantly more complicated and advanced topics that are much easier (or possible) to tackle numerically than analytically. You can explain Markov processes with demonstrations of them, and then just about anything numerical is easy to represent via Monte Carlo algorithms measuring Markov processes.

Neil: I had in mind mathematical statistics, i.e. theoretical foundations. A data analysis course would have more computation. Still, I’ve experimented with programming assignments that illustrate theory. I think that has worked well, though I haven’t come up with many of them.

The book Flaw of Averages may be like what you have in mind. The author advocates replacing nearly all theory with simulation.

In college, the hardest math class I ever took was in differential equations:

1) Nothing but a boring list of techniques to learn. I couldn’t believe math could be so ugly, a mere catalog of tricks and techniques to memorize.

2) All had roughly the same level of difficulty. I didn’t feel like I was progressing. You feel that you’re learning when you can solve ever harder problems.

3) There were no deep analytic or philosophical issues to make it exciting.

It didn’t feel like math, it felt like mechanics.

Carlos: It’s not an easy class to take, but it’s an easy class to teach. Because there’s so much emphasis on technique, it’s easy to prepare for class, easy to grade, etc. I’m not advocating classes that strip out the conceptual material, but if you’re asked to teach such a class, it’s easy. (Also, I didn’t say it was

funto teach, only that it waseasyto teach. But some of it I did find fun to teach, particularly applications to mechanical and electrical vibrations.)I had a very theoretical ODE class as an undergraduate. I didn’t learn some of the problem solving techniques until I taught the class. I found it kinda interesting, in part because by that time I could see some connections with other areas of math that wouldn’t be apparent the first time through.

Many people would say that a first course in ODEs is ad-hoc and ugly, but Lie groups are systematic and beautiful. And yet the two are closely tied! Lie groups came out of an abstract view of techniques for solving differential equations, seeing the tricks as exploiting a kind of symmetry.

I’ve found any class with data analysis difficult to teach. The easiest for me was statistical computing (e.g. EM algorithm, MCMC, optimization). That course pretty much teaches itself.

Let me disclaim this by saying I have only taught stats to high school seniors, and only as part of a year-long math course that had to cover a lot of other material, and way back in ’92-’94.

That said, to make it more interesting why not let the students pick their data set, so that the first set of exercises involves recording the data set they will use for the rest of the semester? They need parameters for what kind of recordings to do (e.g., minimum number of measurements, type of measurements, etc.), but it allows for variation of interest. Some possibilities include taking traffic stats at particular intersections, collecting sports statistics from prior years, recording game stats, grabbing census data, collecting internet stats (using APIs from Twitter, Facebook, Flickr, etc.), and so on.

As long as they have a good template to follow in data collection, you don’t really NEED the samples to be truly random or events to be absolutely independent. In fact, you can discuss why they might or might not have those characteristics. If they have just enough data points in just enough samples, you can introduce the stats while letting them debate what their results might mean and the philosophical issues about why their results are or are not necessarily illuminating.

Carlos: I found DiffEq to be my favorite math class as an undergrad and thoroughly enjoyed my graduate-level PDE class as well. But, I majored in Physics and liked the practical applications. I had a prof who emphasized the applications, as opposed to my linear algebra prof who was focused solely on the theory.

John: Just read your post today about Mathematics being like the Hawaiian islands. That’s certainly part of its beauty. I’d never connect differential equations with Lie groups. I’m just sorry that my college teachers couldn’t show the interesting side of differential equations.

Joel: Diff’rent strokes for diff’rent folks. As a physicist, it’s natural that you should be excited about the practical applications of DE. Me, I’m bored to death by mechanical vibrations and such. Likewise, I’m sure that stuff that excites me will make you yawn.

John: By the way, thank you for one of my favorite blogs.

Any suggestions for a introductory (undergrad) book on Mathematical statistics for self-study?

Peter: I’m sorry, but I don’t know of such a book that I’d recommend.

My teaching experience was with mostly social-science undergraduates, and many of them were math-phobic. The hard part was getting them to get past the mechanics of algebra to think about the meaning of basic statistical inference.

My own first statistics class was rather mechanical. It wasn’t until afterwards that I began to think statistically.