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Comment by nerdponx | Hacker News Reader

I remember my real analysis professor explicitly telling us that he ran the numbers on one of our exams and found bimodality. Real analysis is not what I would call an earlier class. I'm not sure if we were an outlier or a representative data point, however.

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I can easily believe it was representative.

I would expect ability driven bimodality to show up in courses that present new types of challenges that were not in previous courses. That is where you'll get students who either do or don't master key concepts, and where they go from there depends on their mastery of those concepts.

For computer science, one of those key concepts is reasoning about state manipulated through a layer of indirection. The infamous "getting" pointers issue.

In the case of real analysis, it is an issue of switching from "Can you apply known operations to produce the right answer?" to "Do you understand the structure of the reals well enough to prove it?".

> n the case of real analysis, it is an issue of switching from "Can you apply known operations to produce the right answer?" to "Do you understand the structure of the reals well enough to prove it?".

Is this really the case in general? My math degree curriculum started with 100% proof-driven classes almost immediately, long before most people got around to taking real analysis. It was my impression that that kind of approach was fairly ubiquitous.

In the USA and Canada, it is standard for the professor to stand at the board and present proofs of everything, but students are not expected to understand them. If you can learn the formulas and apply the techniques that are in the homework sets, you can pass the class.

I strongly question the value of having the professor recite proofs that the students do not really understand. However it is traditional to do that.

The first course where students are expected to write their own proofs varies. My personal experience was that the transition happened in third year courses aimed at math majors. So I began writing proofs in abstract algebra and real analysis. But not in my third year differential equations course that was required for engineering and physics majors.

When I was in grad school at Dartmouth, they introduced it earlier in a linear algebra course. I taught the last linear algebra course that did that, the next quarter they took proofs out of it.

Hm interesting. My undergrad degree at Berkeley was pretty much solely the professor running through proof after proof, and our homework was more proofs. It actually suited me just fine, it was one of the few curricula in my life that I found presented any challenge.

My impression was that it worked fine for the majority of the other students as well, but I don't have tons of data on that or anything.

I'm not surprised that Berkeley is an outlier in this regard.

Talk to people who went to other universities and I'm confident that you'll find my experience to be more common.

I think the idea is that people on the lower mode of grading would self select themselves out of a CS at the 100 most often. This reduces the bimodality for the 200 level. Where again students self select themselves out, then the 300 level bi-modality is even further reduced.

I think the 100 level is the place where such a bimodality is most likely to exist and investigation ought to be focused there.

Your parent's point is that this hypothesis ("bimodality reduces as students progress through the curriculum" is possibly contradicted by the fact that many late-stage mathematics courses are strongly bimodal.

(FWIW my analysis course was also bimodal.)

Based on some limited experience upper-level math and (I assume) physics probably have a stronger element of "get it" vs. "don't get it" than most majors. Way back when I was an undergrad a number of the math majors I knew considered it to be a relatively easy major and there were definitely people who pretty much breezed right through.

At the same time, I was quite convinced that I could no more have successfully graduated as a math major--no matter how hard I worked--than I could have successfully flown through the air by flapping my arms. By contrast, I'm reasonably confident that I could have gotten through pretty much any other majors, whether engineering or something else.

If there are two major topics in a test (say Topic 1 and Topic 2), then bimodality could arise simply because

* some people studied both Topic 1 and Topic 2

* some people studied Topic 1 more than Topic 2

* some people studied Topic 2 more than Topic 1

This would cause bimodality even when inherent "genetic" skill is the same.

This seems unlikely across multiple students and years (though it is possible so more than one schoool should be studied).

You comment also presumes there are no students who can study well enough to pass all exams. Considering the ease of 100 level courses CS courses acing that and another 100 level is not exactly hard.

If this were be turned into a more rigorous MCMC style simulation and fed basic societal trends of high school and early stage college on courses taken, interest, aptitude, etc you might be able to get an estimate on the size of effect of these factors. Hmmm...

aside: real analysis classes are rarely large enough to do a good analysis of this in, unless you incorporate many years.

While real analysis alone is probably not enough to do a real representative analysis, if the class were the first real analysis class in the undergrad sequence, then I'd call it an earlier class. Eg, 1st upper division real analysis class, assuming the lower divs were the ones the pre-meds and CS majors took.

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