15 September, 2005

Refuge of the despairing

I didn't do a good job teaching standard deviation on Tuesday. The problem wasn't that I said anything wrong, or that I didn't understand the mathematics well. In fact, I made a comparison of the whole square-root-of-the-sum-of-squares bit to the distance formula that was very insightful, even if I say so myself.

Rather, in my excitement over understanding the theory, I didn't think to prepare an example. (Pride, pride pride... I looked it over and thought, "It's easy!") I chose a random example from the end of the chapter during class, and before you knew it, I was in trouble. I could solve it just fine, but it was too complicated and the students didn't understand a thing.

I did another example immediately afterwards, a much better example. The students claimed to understand and follow, but the damage was done. You have to keep in mind that these are students who cannot solve a moderately complicated algebraic equation, even if they are sitting in a sophomore-level math class, and even if they have officially "passed" algebra.

It's depressing. They're not morons; some of them are quite clever, really, especially when it comes time to debate how I graded a test problem. The problem is attitude. During lecture, students who don't understand something promptly judge it negatively, and to comment on it publicly. Sometimes they judge it negatively even when they do understand it, simply because it requires careful effort.

The attitude problem doesn't rest merely with statistics students; it cuts across the board. Just yesterday, one of our math majors in the Linear Algebra class pronounced with undisguised disgust, "It's stupid." This guy is our department's créme de la créme.

Was it always like this? I don't remember saying such things as a math major. I used to get frustrated, sure; I might even have spoken ill of the book's author. Heck, I still do that. But I wouldn't have thought (let alone said) that the material was stupid.


Returning to that section of statistics today, it didn't surprise me that the majority of students couldn't compute a single standard deviation. I led them through one of the homework problems, then had them do another homework problem immediately afterwards. I encouraged them to work together. Usually, they don't.

Today, however, those that understood surprised me by helping others. Some of them were even rising excitedly to help two or three students at a time. I couldn't believe it! on the day that the division chair had come to observe me, no less!

I "wasted" the first half-hour of my class this way, wondering nervously what my division chair was thinking, and writing in his notes. The class was loud, since there was a lot of talking; I looked around several times, though, even walked around, and they looked all the world as if they were discussing the math. So it wasn't a waste at all; the students helped each other, and helped themselves, and I helped the handful who called me over. (One or two were later helping others. Yeah!)

One student walked up to me after class and said, "I understand a lot better after today." I'd liked to think that the subsequent lecture on Chebyshev's theorem and the practical meaning of standard deviation was what she meant, but I have to be honest: she meant the time I gave them to work on the "homework".

I can (and have) told them that the whole point of attending a "teaching" college is that the students have a huge opportunity for one-on-one instruction from the professor. I can encourage them all I want to work on the homework in groups. They won't hear it, though. After all, that would impinge on their free time. If they don't have it figured out by the end of class, their obligation to learn it ends there, no?

Sacrifice some class time, though, and give them the opportunity for that one-on-one, either personally or from another students, and they actually learn something for a change.

After the class, the division chair approached me and asked, "Does the class normally run this way?"

"Not really - well, what do you mean by, 'this way'?"

"It looked like there was a lot of collaborative learning going on."

"I don't usually give them time in class to work on the homework, but I didn't do a very good job teaching standard deviation on Tuesday" - here I explained about the bad example, which amused him - "so I was pleasantly surprised. I do encourage it, but I don't normally have this response."

At the same time, I thought, "That's 'collaborative learning' for you: the refuge of the student who despairs of understanding his teacher."

Maybe I should start confusing them on purpose? :-)


qkl said...

I am not an expert in the way humans learn but I have listened to a show that was very interesting about learning math.

They say the teacher should start with an example that everyone can relate to. Than reason by showing the proof and than have students cosolidate or memorize the information.

Here is the link to the show, it is in french though:


jack perry said...

They say the teacher should start with an example that everyone can relate to

You are absolutely correct, and this is a difficulty I have. The trouble is that I have to learn the material first (I'm not a statistician). So I spend a lot of time just trying to learn and understand the material, and I don't usually have the time to find material that everyone can relate to.

However, I am planning to start the next lecture with data on the salaries of math professor. That's something I can relate to, and they should find it interesting, if not particularly relevant to their own futures. :-)