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naraburns  nihil supernum  1yr ago (text post)   4345 thread views

Small-Scale Question Sunday for December 4, 2022

Do you have a dumb question that you're kind of embarrassed to ask in the main thread? Is there something you're just not sure about?

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Why is calculus in university so computationally hard?

I tutor university students in introductory math courses as a side hustle, and a pattern I noticed is that you can more or less approximate how "prestigious" a university is by ranking the difficulty of their calculus 1,2 courses/exams. My observations should apply to universities in North America and Canada because most of the exam papers I see are from there.

For example, if you want to integrate a polynomial fraction, the most common questions involve completing the square, long division, and partial fraction decomposition. In my observation, LR unis will have questions requiring only one technique. MR will have two. And HR will often have all 3.

However, this trend does not hold for any other math course. Let that be Linear Algebra, Stats, or Diff Eq. A good chunk of LR unis has much harder stats and differential equation classes than the MR ones. HR unis are consistently difficult.

For example here is a Lin Algb exam from the University of Waterloo. An HR/MR (Engineering HR no questions about it) university. This Lin Algb exam is about the same difficulty I had in my MR uni. But their calculus exams are way harder.

If this pattern is true, is this some administrative artifact? Cal 1,2 are common "weed out" courses. And I assume given the large number of students from various departments that have to take them, there are more voices than the math department deciding the course content?

Moreover, why are so many calculus questions testing algebra skills? I went to an MR uni, and I never had to use anything more complex than a partial fraction decomposition when solving an integral in a higher-level course (For everything else it was Laplace/Fourier transform all the way down). And lin algb and stats and complex analysis or any nonintroductory math classes did not rely as much on raw algebra skills as cal 1,2.

Moreover, why are so many calculus questions testing algebra skills? I went to an MR uni, and I never had to use anything more complex than a partial fraction decomposition when solving an integral in a higher-level course

partial fraction decomposition is algebra though

For example, if you want to integrate a polynomial fraction, the most common questions involve completing the square, long division, and partial fraction decomposition. In my observation, LR unis will have questions requiring only one technique. MR will have two. And HR will often have all 3.

Which one of these tasks is computationally hard? Linear algebra and statistics have been the most computationally intensive courses I've taken. Multiplying two 4x4 matrices is 16 * (4 + 3) = 112 additions or multiplications. Computing standard deviation for a sizable dataset is, again, a lot of arithmetics. And in uni calculus most questions have an answer that reduces to something neat, so if you get an unwieldy polynomial for an answer you've probably made a mistake. A matrix or a sigma looks just like any other value you can get.

I meant relatively. Subjectivity withstanding.

None of them are particularly hard. But I have seen polynomial fractions that look fairly unwieldy and need 2 of the following to simplify. Subjectively that is more computationally intensive than just multiplying and adding a lot of numbers.

Also, your professor is insane for putting matrix multiplication on exams, it's just wasting time; if you can multiply 2 2x2 matrices, you can multiply any mxn matrices. There is nothing gained other than knowing how to add/multiply numbers fast. My lin algb exams were very theoretical and didn't require crazy computation.

If you're looking at linear algebra as the study of matrices then you're thinking about it in completely the wrong way. Linear algebra is the study of linear maps from vector spaces to other vector spaces, end of. The book "Linear Algebra done right": https://www.amazon.co.uk/Linear-Algebra-Right-Undergraduate-Mathematics/dp/0387982582 is a very good teaching aid, it explains what's actually going on properly while minimising matrix bullshit.

I think so as well. Which is why I said its insane to include large matrix multiplications in a lin alg exam.

However, this trend does not hold for any other math course

WRT "weed-out course": does this hold over time? If what some posters here have mentioned is true- that high school is far too easy (I remember being shown a scatter plot of "high school math grades" vs. "Calc 1 grades" in a pointless university class a while ago; there was no correlation)- we should expect to see the exams from 1960 be easier computationally than they are in 2020. Is that the case?

Why is calculus in university so computationally hard?

Because the university was bad at naming a course that should have just been called "advanced principles of algebra"? The fundamental theory behind calculus is relatively easy to understand to the point that even today's grade school calculus courses cover it in its entirety; there's very little to expand upon after that. And all the other courses are generally just applications of calculus, taught by people that know those applications, and by that point you're out of the academic hazing ritual anyway, so...

If the school is not huge, there is minimal coordination above Calc 2 so it’s mostly run to the instructor’s taste. Instructors vary a lot.

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