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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)

#41

Post by dr_dshiv »

nemo44x wrote: post #13
jrib wrote: post #4

Earlier quoted context omitted.

> 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!

> I would go to office hours and the teachers were seriously so helpful. This was the only way I survived. I learned that you don't have to be a 'numbers person' at all to be good at math. You just have to understand what's going on under the hood, and this may require lots of questions and outside help. But most importantly, you just have to practice I agree with all of this and it's how I got through extensive math…

> I made sure to get to the office hours and I was surprised that there was almost no one there most of the time. I figured the cost of tuition is best justified by using as many hours of one-on-one time with an expert in the subject to better understand it yourself

Truth!

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

#42

Post by tgb »

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.

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

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

#43

Post by ccortes »

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…

That sounds more like a lazy mathematician tbh. Do you have any other resource about the link you provided?

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

#44

Post by voxl »

tgb wrote: post #42
voxl wrote: post #31

Earlier quoted context omitted.

"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.

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.

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

#45

Post by godelski »

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 find this concept of being a numbers person weird to being good at math. Most math is about symbol manipulation and numbers are really only at the end.

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

#46

Post by bpyne »

bitxbitxbitcoin wrote: post #30
bpyne wrote: post #21

My biggest problems as a math major at the undergraduate level were proofs. I could muddle my way through abstract algebra proofs but real analysis just didn't click. The oddity is that I could read proofs for both subjects: the reasoning made sense. But I couldn't develop a proof.

I had a similar experience doing my math minor. All the way through Caluclus classes and Diff EQ and up until the first half of Linear Algebra, everything is plug and chug. After the first half of Linear Algebra when proofs started making an appearance, I came to the realization that proofs are another type of mental activity entirely. Survey of Algebra, Basic Real Analysis, and even the dedicated proof writing cours…

My university realized the need for a proof writing course a little too late to help me. Students who did well in the more abstract math classes, aside from the outlier "gifted" mathematicians, formed study groups. I wasn't mature enough at the time to realize how valuable those groups were so I went it alone and my grades reflected it.

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

#47

Post by lmilcin »

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 about that specific part of course material.

If you don't understand something today, it is very likely the material tomorrow will refer to it and you will not understand that either. Even if you try to make up for it in couple of days, it will be rushed, will require more effort and your brain connections will not be the same quality.

Every person will have a different way to achieve that goal. Some people need to ask questions, some like to figure out by themselves, some will want to read, solve exercises for as long as necessary.

I have observed many people drop off and the reason was almost invariably the same. A bit of material passes by or maybe the load is too much, and the slippery slope of not understanding starts.

Especially when you start to study mathematics, the first couple semesters are foundational and if you don't understand something it is like a pyramid, the further in time the more connections with the material that you did not understood and the more trouble you are in.

I have seen people drop off because they thought it is the same as in their past or in other study areas. It is not. Mathematics is extremely connected internally and extremely intolerant of ignorance.

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

#48

Post by godelski »

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!

I TA and tell my students that if they don't come to office hours or ask questions that they might as well be getting their degree on YouTube.

I don't mean this in the sense that they have to ask for every class, but the advantage of attending school (besides the piece of paper) is to get direct help. But I'm not attending a top 10 university and honestly most topics are covered by one of those universities and they post their lectures on YouTube or their own OCW.

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

#49

Post by 6gvONxR4sf7o »

Jtsummers wrote: post #32
ARandomerDude wrote: post #23

Earlier quoted context omitted.

They're great...just not for math. Use a decent pencil instead, like a Graphgear 1000 and some HB lead.

Pens are fine for math. I often used a fine point gel pen in college. However, you have to have a high degree of confidence in what you're writing. If you make a lot of mistakes (either actual mistakes or just wrote down something incorrectly or with a misspelling), then pencils are definitely better.

Only if you make sure to have the right kind of lead, eraser, and paper.

A crossed out line followed by a legible one is better than a line written over gray mush. Or mush from papers rubbing against each other over months.

But pens also assume that scratch space isn’t a limitation.

Pens seem crazy until you get used to writing that way, then you can’t turn back.

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

#50

Post by LegitShady »

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 took a course taught by a nice lady with a PhD in pure mathematics, no English skills, and handwriting that made differentiating between mu, u, and w very difficult. It was not a good time.
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