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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)

#71

Post by dzolob »

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…

There is no absolute here. While I agree that a great degree of understanding is desirable, lots of concepts and techniques don’t sink in until hours, days or even years after being presented.

I still remember powering through lots problems without much clarity and achieving the right answer. Every attempt, failed or successful, got me closer to really understand what I was doing.

It’s ok not to understand a topic the first or nth-time time you come across it. If you follow through, something will stick, and sooner than later you’ll get a good enough picture.

They are like seeds, you know.

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

#72

Post by arduinomancer »

k__ wrote: post #67

To me the biggest problem was conversion of terms/formulars. I knew all the rules, but when I needed to reach a specific form I somehow ended up running in circles. Calculus felt more like learning chess than learning math. You needed to think three conversions ahead to get a result and that had to be practiced.

Yup, I think the subject is cool and interesting but I hated how the questions were almost like LeetCode where if you don’t know “this one simple trick/identity” to rearrange the equation then you won’t have a chance of solving it.

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

#73

Post by ubercow13 »

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 honestly never found this an issue. Shorthand and type coercion is used a lot but at least when taught, it's usually made very explicit what notation means and when things are being excluded. After that point, the notation is the least challenging part.

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

#74

Post by btilly »

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 write the times as two crossing lines, and an X as two vertical arcs that touch in the middle. Now they are easy to distinguish. It isn't until I went looking for an image of it that I figured out how weird this is.

I also draw a y with one stroke so that it has a visible curve. That keeps a bad y and bad times from looking like each other.

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

#75

Post by voxl »

aesthesia wrote: post #54
voxl wrote: post #44

Earlier quoted context omitted.

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.

Sure, that's fine, but to say that it is an invalid method of proof is wrong.

If I want to prove A, and I prove A B for some proven statement B, then I have proven A! There is no question! Feels like crazy pills to think otherwise. This is a kind of backwards reasoning!

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

#76

Post by LanceH »

hyperpallium2 wrote: post #63
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…

\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? Is it just a resolution thing?

Lastly, are you getting older (40ish) and need reading glasses? That happened to me when crossing 40. I could still focus sharply at reading/screen distance, but I got tired. I went through that a couple years and got reading glasses -- actually just weaker versions of my nearsighted glasses. I had the doctor dial in the focal distance for screen distance, then got another pair for 12 inches are so (for a reading a book up close).

So I have three pairs of glasses, one for distance (everything sharp from 3 feet outward). These strain my eyes from 3 feet inward. You don't need reading glasses if your arms are long enough...

I have computer classes dialed in at roughly 28 inches. I can read up close, and see nearly perfectly at distance. I'm probably 20/20 at distance, but with my regular prescription I'm around 20/10.

Then the reading glasses for up close things. These are fine for computer as well, and just slightly more blurry at distance.

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

#77

Post by tgb »

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.

Say a student is completing a problem set and is asked to prove P. If the student writes "P therefore Q therefore True since Q was proven in class. QED" then the student has committed a significant error in reasoning. If the student instead just writes "P. Q. True. QED" then they are writing the above in a more terse manner and have equally committed an error in their reasoning. If they first write that but then amend in some "iff"s between each step I'll give them full credit (assuming iffs are valid) but be slightly worried that they just learned that you need to put an "iff" in those spots to keep the teacher happy but don't understand why.

If a strategy is not guaranteed to give proof, then you need to verify afterwards that the putative proof is in fact a proof. Just as if you get a potential solution to an equation (via solving a more general equation, perhaps), you have not "solved" the original equation until you actually check that solution, even if your putative solution is the true one. If a student does not do the "check if steps are reversible" part, then they have not written a proof even if every step is reversible! That's what's lacking form their proof.

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

#78

Post by watwut »

6gvONxR4sf7o wrote: post #56
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.

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 tha…

When you are doing equality and such all steps are reversible.

Working backwards is exactly the same as working forward. You develop reasoning where you see it easier until to figure out to connect sides.

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

#79

Post by ubercow13 »

arduinomancer wrote: post #72
k__ wrote: post #67

To me the biggest problem was conversion of terms/formulars. I knew all the rules, but when I needed to reach a specific form I somehow ended up running in circles. Calculus felt more like learning chess than learning math. You needed to think three conversions ahead to get a result and that had to be practiced.

Yup, I think the subject is cool and interesting but I hated how the questions were almost like LeetCode where if you don’t know “this one simple trick/identity” to rearrange the equation then you won’t have a chance of solving it.

I think that unlike with LeetCode, in maths you usually do end up using those 'one simple tricks' a lot later on.

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

#80

Post by LanceH »

at-fates-hands wrote: post #55
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. I…

I was really blown away by how much a finer point on the pen helped me improve my writing -- something which seemed like a lost cause.
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