The Most Common Errors in Undergraduate Mathematics (2009) • Hacker News Board

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

#101

Post by dkarl » Thu, Nov 19, 2020, 9:09 PM UTC

But the part that I really want to discuss here is the other part -- i.e., the phrase "if k is any constant."

To most teachers, that additional phrase doesn't seem important, because in the teacher's mind "x" usually means a variable and "k" usually means a constant.

Most teachers get this right, using these conventions as a redundant booster to their verbal and written communication. Some are grumpy and lazy and want as much as possible to go unsaid. "For the next couple of weeks, when I say 'ring' I'm going to mean a commutative ring." Just say commutative ring. The time you save by glossing over these details doesn't add up to anything.

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

#102

Post by mcguire » Thu, Nov 19, 2020, 9:25 PM UTC

"Loss of invisible parentheses. This is not an erroneous belief; rather, it is a sloppy technique of writing. During one of your computations, if you think a pair of parentheses but neglect to write them (for lack of time, or from sheer laziness)..."

"The great number of sign errors suggests that students are careless and unconcerned..."

Sounds like someone thinks they should have a better class of students.

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

#103

Post by ghaff » Thu, Nov 19, 2020, 9:28 PM UTC

kkylin wrote: post #60

benrbray wrote: post #53

Earlier quoted context omitted.

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…

>It used to be (this tells you how old I am)

Why do you say that? Maybe it's atypical but MIT seems to have the same calculus and physics sequence, taken in parallel, that they did ages ago.

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

#104

Post by viburnum » Thu, Nov 19, 2020, 9:44 PM UTC

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…

When I was 14 my math teacher requested a conference with my parents. I was mortified. All he said was that I needed to write neatly. He was correct. That's all it took to go from a B student to A+. I make a ton of mistakes if I don't write out every step (turns out, the very mistakes on this list).

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

#105

Post by bpyne » Thu, Nov 19, 2020, 9:46 PM UTC

v64 wrote: post #81

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.

Yes! That was my experience exactly.

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

#106

Post by bigger_cheese » Thu, Nov 19, 2020, 10:05 PM UTC

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.

Stress/pressure has an effect on your performance for sure. In high school I didn't have many problems but at University I found I would often choke during exams despite knowing the material and being well prepared.

I would get extremely nervous in the lead up to my university exams, I'd have difficulty falling asleep the night before, I'd be tossing and turning all night going over everything in my head and I'd feel anxious and sick (my stomach would be incredibly queasy) to the point I'd have trouble eating anything in the morning. So I'd go into my exams tired and irritable (because I hadn't eaten much) and I'd make stupid mistakes.

Eventually I got over this but it took me a year or two and it definitely had an effect on my grades.

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

#107

Post by konjin » Thu, Nov 19, 2020, 10:12 PM UTC

euiq wrote: post #16

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's not a very convincing example, given that numeric literals are polymorphic in Haskell.

I never said Haskell was a good language. Q_0 is a minimally acceptable math notation.

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

#108

Post by chapium » Thu, Nov 19, 2020, 10:15 PM UTC

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.

Absolutely. Its a lesson hard learned if the problem reaches the undergrad level.

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

#109

Post by lmilcin » Thu, Nov 19, 2020, 10:17 PM UTC

dzolob wrote: post #71

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 th…

I guess what you mean by "sink in" is a form of deeper more intuitive knowledge. Like when you suddenly figure out how everything is connected.

If that's what you mean, then yes -- this is something that is going to happen over time.

What I mean is that your teacher will want you to understand some concepts and learn some skills before you move to next topic. Very deep understanding is probably not on the list and it is expected to only form over time. But there are topics that are explicitly expected and you have to understand because otherwise what is going to be on the board the next day is not going to make whole lot of sense.

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

#110

Post by konjin » Thu, Nov 19, 2020, 10:20 PM UTC

6gvONxR4sf7o wrote: post #52

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.

The point is that 2 and (λ (x) 2) are vastly different objects, yet in standard math are treated the same.

Also you don't need a symbol type to deal with variables, you just need to allow type variables to be used in standard expressions. (define x R) (^ x 2) is just as easy to manipulate as (λ (x) (^ x 2)), the only issue is that your rewrite rules how aren't just looking at syntax, but semantics too.