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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)

#81

Post by v64 »

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 also strugged with analysis. I always felt like epsilon-delta proofs involved pulling some absolutely strange value for delta out of your ass that happens to work out in the end, and I never developed an intuition for that. Same with integrating by parts, oh it just works out so nicely if you rewrite u in this totally obtuse way.

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

#82

Post by InitialLastName »

jcranmer wrote: post #51
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 +. I…

I had an English calc teacher in high school who drew his 'x' as approximately ")(" (With the arcs touching-ish) and the cross in the normal way, which made it much easier to distinguish the two.

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

#83

Post by bitxbitxbitcoin »

vharuck wrote: post #69
bitxbitxbitcoin wrote: post #30

Earlier quoted context omitted.

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…

>the dedicated proof writing course What was yours like? The one I took didn't cover much on the actual writing of proofs. The professor accepted reasonable essays with high-school level notation. Instead, he gave us a toolbox for proving things: pairing terms in a series to find the sum, rewriting recursive equations, etc. It was mostly to show that clever tricks are how mathematicians prove new things. But that mig…

It has been over a decade so I don't exactly remember - I looked it up and the class was Introduction to Mathematical Reasoning.

What I personally recall was following along in class while the professor walked through classic proofs emphasizing what each new notation meant as well as the difference between Direct, indirect, and induction proofs. Tests and assignments were essentially recreating proofs cherry picked to be similar to ones we walked through in class.

I realize that's kind of how all my higher level math classes were run. It's just that once the cherry picking becomes looser, intuition doesn't necessarily catch up :(.

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

#84

Post by spijdar »

wolfram74 wrote: post #68
benrbray wrote: post #64

Earlier quoted context omitted.

Oh yeah, I'm with you there. Teaching quality at both my undergrad math program and my graduate compsci program was abysmal, with the exception of a handful of dedicated teachers. Both at well-ranked research institutions, which is the problem. The whole system is crazy. Why on earth do we allow tuition money to be spent on anything else but full-time teaching staff?!

I'm now interested in a list of transactions ordered by participant dissatisfaction, like, paying for gym memberships and not getting the outcomes they want, therapy or medical attention to mostly be ignored by the provider, ISPs in the US, online shopping to get something wildly different. Is education especially bad compared to these things? or just that it's such a large expenditure we just-world-fallacy ourselves…

Is it really a logical fallacy to assume that we should have better outcomes from spending tens to hundreds of thousands of dollars on academia vs e.g. spending thirty dollars on amazon?

At least naively, that money could (should?) be going towards paying for staff and a support system/hierarchy to ensure a higher quality. That's at least nominally why accredited, non-profit education is so expensive, because it's nominally supposed to be capable of paying decent salaries to intelligent people dedicated to teaching, compared to say amazon or the gym where the incentives are to cost cut and maximize profit.

In reality, I suspect the same applies to colleges, but I don't think it should, in some ideal world.

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

#85

Post by bitxbitxbitcoin »

bpyne wrote: post #46
bitxbitxbitcoin wrote: post #30

Earlier quoted context omitted.

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.

I hear you! I wasn't mature enough to join study groups or go to office hours during undergraduate, either, and my grades also reflected it.

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

#86

Post by JJMcJ »

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…

Mathematics is high on the list of "you can't fall behind and cram".

This is true not only within a single class but from class to class.

Aren't fluent in manipulating equations? Calculus is going to be misery. Don't know calculus like the layout of your bedroom? Differential equations aren't going to happen.

All the way back to `2+3 = 5` in first grade. If that doesn't go well, the rest will likely not, either.

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

#87

Post by siltpotato »

godelski wrote: post #45
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.

The concept itself or the name of it?

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

#88

Post by abdullahkhalids »

benrbray wrote: post #64
hawkice wrote: post #62

Earlier quoted context omitted.

> students really do require an experienced guide to navigate through the tangled web of concepts. Consider how much people pay for education and don't get this, and how much some people are paid to be teachers who do not provide it.

Oh yeah, I'm with you there. Teaching quality at both my undergrad math program and my graduate compsci program was abysmal, with the exception of a handful of dedicated teachers. Both at well-ranked research institutions, which is the problem. The whole system is crazy. Why on earth do we allow tuition money to be spent on anything else but full-time teaching staff?!

> Why on earth do we allow tuition money to be spent on anything else but full-time teaching staff?!

Because the culture of innovation that part-time researchers/part-time teacher bring is far more important to society in the long run than the short term gains gotten by full-time teaching staff.

Most fields in the world are rapidly changing, and what's taught in them changes every decade or so. Most full time teachers will atrophy because they don't have any real incentive to keep their knowledge and skills at the cutting edge. After a decade or so, your full time educational institute will be teaching worse than a research+teaching institute.

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

#89

Post by milesvp »

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…

> the first couple semesters are foundational and if you don't understand something it is like a pyramid

I was taking a math/cs hybrid major in college, and the upper level math courses just felt like they were getting harder and harder in ways that didn't make sense in my previous math experience. Tests more and more were weighted towards verbal proofs, and I hadn't yet developed the vocabulary to deal with the precision necessary to use English to prove something mathematically. It wasn't until after graduation that I learned that there are some sophomore math courses at my university that Math majors usually take that were unofficially known as "intro to proofs". I have no doubt this was missing foundation that made 400 level math courses so painful for me.

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

#90

Post by hyperpallium2 »

LanceH wrote: post #76
hyperpallium2 wrote: post #63

Earlier quoted context omitted.

\tangent can you recommend what to look for in a tablet for mathematics? I've got a lenovo M8 - only 8", 800p, but I think that's OK; it's the insufficient contrast of IPS screen that makes my eyes tired (suggesting AMOLED or epaper) - but I don't know the cause for sure... What did you find?

I mostly use an Amazon Fire 10" for math. It shows that it is IPS, but it really looks good, imo. Do you have a big contrast between your screen and your room? That's a big deal. I run a blue light filter at night. I do prefer eink and anything oled looks fantastic, but the fire looks really good, well beyond its $150 price tag. Better than my computer monitors (which are nice). Are you reading tiny, detailed print?…

Thanks! I hadn't considered room lighting - though I also need it bright enough for pen and paper.

Resolution seems ok - I zoom in when needed, which is rare. Some pdf's have lines too long to fit on the screen, for the text to be large enough - but they are also rare. I think a 10" would solve that (held lanscape, 10" is basically A4 portrait width, if you zoom out the margins). So it doesn't feel like resolution is a problem - although some people say fuzzier text tires eyes.

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