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The Most Common Errors in Undergraduate Mathematics (2009)

math.vanderbilt.edu

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Re: The Most Common Errors in Undergraduate Mathematics (2009)

#51

Post by jcranmer »

LanceH wrote: post #5

Bad handwriting. At age 40 I started taking a masters in stats and had situations where I had exponents on exponents. This lead me to buy a higher resolution tablet for reading pdf's with tiny math. I also bought a finer point pen and this helped me improve my handwriting a lot . Closing loops on "o's" or backtracing the upward line of a cursive "t" to not make a loop. With the finer point, I was able to see my impre…

In the Intro-to-C class I TA'd, we dedicated several minutes in lab to instructing students on how to properly draw an ampersand (&) character, and the other ways of drawing it so potentially confusing (especially since a + means something completely different).

Personally, the +/t handwriting issue has never been an issue for me: make sure you get a good tail on the t, and it's pretty easy to distinguish from a +. It's the x/× distinction that was always the most painful to distinguish for me.

Re: The Most Common Errors in Undergraduate Mathematics (2009)

#52

Post by 6gvONxR4sf7o »

konjin wrote: post #10

Mathematical notation is terrible. I don't mean this in the usual "we need to invent new symbols to make it clearer" way. I mean it in the way that it has implicit typing that gets coerced constantly. Using Haskell types D :: (R -> R) -> (R -> R). Yet it gets used on things like D 2 = 0 which implies D :: (R -> R). What you've actually done is an implicit conversion of 2 :: R to 2(x) = 2 :: R -> R and 0 is not 0::R i…

I agree that your example isn’t very convincing, for an additional reason. You can treat 2x as a function, or you can treat it as an expression, where x is its own special kind of thing: a symbol.

With how the notation is typically used, it’s at least closer to my mental model.

Treating the expression as a function under substitution is one way to look at it, but not the only way.

Re: The Most Common Errors in Undergraduate Mathematics (2009)

#53

Post by benrbray »

lmilcin wrote: post #47

I studied theoretical mathematics. The absolutely biggest and most common error and really the only one worth to be on list is not internalizing and understanding entirety of material as the course progresses. Everything else should be basically to achieve goal of understanding the entire course material. It is ok to forget later, as long as you are sure you understood it at least when there was a lot of discussion a…

I would agree, but qualify by saying that not all the blame lies with the student. Precisely because mathematics is so interconnected, beginning students really do require an experienced guide to navigate through the tangled web of concepts.

A particularly bad or lazy professor can be worse than no professor at all.

Re: The Most Common Errors in Undergraduate Mathematics (2009)

#54

Post by aesthesia »

voxl wrote: post #44
tgb wrote: post #42

Earlier quoted context omitted.

It's a completely valid work to do to figure out how to write a proof, but is not a proof itself. The section in the article states all my thoughts on that matter more clearly than I can. https://math.vanderbilt.edu/schectex/commerrs/#Backward

As a PhD student myself, I would die on this hill, and you would lose. Moreover, the first sentence: "This is an unreliable method of proof" implies that it can be a proof if done correctly.

Well, it's a proof as long as you stipulate somewhere that all the logical connectives are "if and only if." This doesn't mean that a student who writes down a long series of equations beginning with some identity to be proven and ending with some known fact has proven the identity.

Re: The Most Common Errors in Undergraduate Mathematics (2009)

#55

Post by at-fates-hands »

LanceH wrote: post #5

Bad handwriting. At age 40 I started taking a masters in stats and had situations where I had exponents on exponents. This lead me to buy a higher resolution tablet for reading pdf's with tiny math. I also bought a finer point pen and this helped me improve my handwriting a lot . Closing loops on "o's" or backtracing the upward line of a cursive "t" to not make a loop. With the finer point, I was able to see my impre…

I had a soccer teammate who was a grad student in mechanical engineering. I was having problems in my Trigonometry class and asked for his help. I was absolutely shocked at how precise his handwriting was when he was writing stuff down and explaining it to me. It looked like he had typed out his notations, it was crazy.

He said something similar to you, "I'm not a doctor writing prescriptions, I need to be precise. If my writing is precise, its easier to read, easier to understand now and later."

Re: The Most Common Errors in Undergraduate Mathematics (2009)

#56

Post by 6gvONxR4sf7o »

voxl wrote: post #31
tgb wrote: post #6

The "working backwards" one was what I always noticed students doing. It's super frustrating as a teacher since they're simultaneously so close to being right (when they manage to avoid an irreversible step) and yet also have completely misunderstood a basic idea in what it means to even attempt to prove something.

"Working backwards" is a completely valid method of proof, used all the time in automated reasoning. If a student successfully reasons backwards then there is no issue, it is a proof.

Nobody’s saying it’s totally invalid. I think you’re reading that into what they’re saying and you’re talking past each other.

The complaint is that students work backwards without understanding why working backwards is different than working forwards. So if they happen to do it correctly (e.g. by using reversible steps), then it’s only success via accident.

Done correctly, and on purpose, with the different care that isn’t required working the other way, yeah it’s legit, but that’s not what’s being complained about.

Re: The Most Common Errors in Undergraduate Mathematics (2009)

#57

Post by hansvm »

smabie wrote: post #11
whimsicalism wrote: post #8

Earlier quoted context omitted.

I stopped handwriting math to turn in my freshman year of college, LaTex was just too good looking.

that seems like a ton of work, especially for problem sets

I had a few professors require LaTeX solutions. After a little practice it was only slower than handwritten answers for some symbol-heavy assignments. So much time was spent comparatively on solving the problems that it was a negligible overhead regardless.

Re: The Most Common Errors in Undergraduate Mathematics (2009)

#58

Post by hyperpallium2 »

jrib wrote: post #4
sparklingmango wrote: post #3

When I was in undergrad, I declared a math minor quite late - just before my junior year. This led to me taking up to 9 hours of math credits in a semester. The reason I declared my math minor was because I was interested in going for a PhD in economics and my econ professors all recommended that I try to take as many math classes as possible to prepare. Reader, I was not a 'numbers person' at all. However, in all of…

> I would go to office hours and the teachers were seriously so helpful. As a grad student that taught undergrads I always encouraged students to attend office hours. Most instructors do want you to succeed and one-on-one time is one of the best ways to learn. Definitely take advantage of your instructors' office hours; ask questions about anything that you aren't 100% sure about!

https://wikipedia.org/wiki/Bloom's_2_sigma_problem

  the average student tutored one-to-one using mastery learning techniques performed two standard deviations better
A tutor can notice gaps. You can't.

Re: The Most Common Errors in Undergraduate Mathematics (2009)

#59

Post by throw9128928 »

commandlinefan wrote: post #34
chapium wrote: post #28

Most of these appear to be errors caused by stress or tiredness.

That's what my (junior in high school) son keeps saying when he makes simple arithmetic errors that cost him points on tests. I disagree, actually - these mistakes are made by lack of practice. If you've practiced solving enough integrals, it doesn't really matter how tired you are, you're going to get the right answer just as you won't read words incorrectly if you're tired.

Which is it? Stress? Or tiredness? Both?

Capacity to deal with either actually grows over time, you know that?

Re: The Most Common Errors in Undergraduate Mathematics (2009)

#60

Post by kkylin »

benrbray wrote: post #53
lmilcin wrote: post #47

I studied theoretical mathematics. The absolutely biggest and most common error and really the only one worth to be on list is not internalizing and understanding entirety of material as the course progresses. Everything else should be basically to achieve goal of understanding the entire course material. It is ok to forget later, as long as you are sure you understood it at least when there was a lot of discussion a…

I would agree, but qualify by saying that not all the blame lies with the student. Precisely because mathematics is so interconnected, beginning students really do require an experienced guide to navigate through the tangled web of concepts. A particularly bad or lazy professor can be worse than no professor at all.

Not arguing with your point, because it has merit, but do want to add that part of the problem with math education (at least here in the US) is structural: for most students, to start with what they know when they enter and to graduate in 4 years with a engineering / science / math / CS degree that means something, there is just too much to "cover." So we tend to cover "methods and tools," e.g., calculus first, with the motivation (e.g., electrical circuits or statistical regression or what have you) later. This lack of context makes it very hard to see how things fit together. For my own part, I usually learned the material from one course when I needed to use it in a subsequent course -- it's in going back to review the material when I really learn it.

Edit: also want to add that providing context in early courses isn't always easy, because students are much more diverse now in their career plans. It used to be (this tells you how old I am) that many of the students taking calculus would also be taking (or have taken) a calculus-based physics course. Those days are long gone.

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