### Class Size: whose perspective?

I was reading the most recent College Mathematics Journal and found an article written by a former professor of mine, Allen Schwenk.

Yes, I had this guy for third semester calculus, freshman year. And yeah, I made a "C"; "weak" academic performance he said.

Why? Well, at that time, I simply wasn't intellectually mature enough to learn from the way I was being taught; I'd have loved him for my upper division courses.

Lesson: don't judge a professor by the opinion of a freshman! Other students did quite well in his classes.

Eventually, he moved from Annapolis to Western Michigan University.

Anyway, back to his article (and I always enjoy his articles):

When one shops for universities, one often encounters the statistic "average class size"; here is an example.

So, what does this mean?

Let's look at a hypothetical example: we have a small school of 500 students. Each student takes 5 classes: math, language, history, political science, and science.

Math is taught in 25 sections of 20 students each (500)

Language is taught in 50 sections of 10 students each (500)

History is taught in 20 sections of 25 students each (500)

Political Science is taught in 5 sections of 100 students each (500)

Science is taught in 2 sections of 250 students each (500)

So the school would say: we have 25 + 50 + 20 + 5 + 2 = 102 sections

And the class enrollment is: 500 * 5 = 2500

So the "average class size" is 2500/102 = 24.5 students per class.

And, this is what the faculty (one per section) would experience.

But what is it like for the student? In this simple example, the student would see:

5 classes, of size 20, 10, 25, 100, and 250, or each student "sees" 405/5 = 81 students per class.

If this seems strange, think of it this way: those 2 sections of 250 students each are experienced by 500 students, but only by 2 faculty members. Hence this receives "heavier weight" when one does the calculation from the student's perspective.

In the article, Dr. Schwenk takes this beyond the hypothetical example, and shows that the student will always see a class size at least as large, or larger than the faculty does.

He gives a couple of proofs; if you wish to try this yourself (and this is an elementary, but tricky problem; it requires some cleverness to set up) and you want a hint, think: "Cauchy Schwartz" inequality, or think of the proof from statistics that says the sum of the squares is always at least as great, or greater than the square of the sums.

Yes, I had this guy for third semester calculus, freshman year. And yeah, I made a "C"; "weak" academic performance he said.

Why? Well, at that time, I simply wasn't intellectually mature enough to learn from the way I was being taught; I'd have loved him for my upper division courses.

Lesson: don't judge a professor by the opinion of a freshman! Other students did quite well in his classes.

Eventually, he moved from Annapolis to Western Michigan University.

Anyway, back to his article (and I always enjoy his articles):

When one shops for universities, one often encounters the statistic "average class size"; here is an example.

So, what does this mean?

Let's look at a hypothetical example: we have a small school of 500 students. Each student takes 5 classes: math, language, history, political science, and science.

Math is taught in 25 sections of 20 students each (500)

Language is taught in 50 sections of 10 students each (500)

History is taught in 20 sections of 25 students each (500)

Political Science is taught in 5 sections of 100 students each (500)

Science is taught in 2 sections of 250 students each (500)

So the school would say: we have 25 + 50 + 20 + 5 + 2 = 102 sections

And the class enrollment is: 500 * 5 = 2500

So the "average class size" is 2500/102 = 24.5 students per class.

And, this is what the faculty (one per section) would experience.

But what is it like for the student? In this simple example, the student would see:

5 classes, of size 20, 10, 25, 100, and 250, or each student "sees" 405/5 = 81 students per class.

If this seems strange, think of it this way: those 2 sections of 250 students each are experienced by 500 students, but only by 2 faculty members. Hence this receives "heavier weight" when one does the calculation from the student's perspective.

In the article, Dr. Schwenk takes this beyond the hypothetical example, and shows that the student will always see a class size at least as large, or larger than the faculty does.

He gives a couple of proofs; if you wish to try this yourself (and this is an elementary, but tricky problem; it requires some cleverness to set up) and you want a hint, think: "Cauchy Schwartz" inequality, or think of the proof from statistics that says the sum of the squares is always at least as great, or greater than the square of the sums.

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