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The most common errors in undergraduate mathematics

math.vanderbilt.edu

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Re: The most common errors in undergraduate mathematics

#51

Post by DanBC »

lisper wrote: post #50
reikonomusha wrote: post #44

Earlier quoted context omitted.

It seems tempting to have a single unambiguous notation for mathematics. But In constructing such a language, one will quickly realize that doing mathematics becomes an intensely arduous task. This is not unlike recent discussions about the conlang Ithkuil on HN. For better or for worse, math notation has for the most part been optimized for writing on paper or a blackboard, using context to eliminate "inessential de…

> Have you ever attempted to write Lisp on a blackboard What's a blackboard? Seriously, blackboards are pretty obsolete technology. Designing a notation to optimize for blackboards in this day and age is kind of like designing roads to accommodate horses.

> Seriously, blackboards are pretty obsolete technology.

What? Blackboards are excellent. You're just working in the wrong places.

Re: The most common errors in undergraduate mathematics

#52

Post by scott_s »

lisper wrote: post #50
reikonomusha wrote: post #44

Earlier quoted context omitted.

It seems tempting to have a single unambiguous notation for mathematics. But In constructing such a language, one will quickly realize that doing mathematics becomes an intensely arduous task. This is not unlike recent discussions about the conlang Ithkuil on HN. For better or for worse, math notation has for the most part been optimized for writing on paper or a blackboard, using context to eliminate "inessential de…

> Have you ever attempted to write Lisp on a blackboard What's a blackboard? Seriously, blackboards are pretty obsolete technology. Designing a notation to optimize for blackboards in this day and age is kind of like designing roads to accommodate horses.

Lots of people, particularly people who do lots of math, still communicate with each other over a whiteboard.

Re: The most common errors in undergraduate mathematics

#53

Post by nilkn »

SIGALRM wrote: post #34

I'm surprised that the author made no mention about how big O,theta,and omega is commonly misunderstood.

I studied mathematics as an undergraduate and found that these topics aren't covered in pure math classes typically until students are sufficiently experienced that they can grasp the fully technical definitions.

It's more common for CS courses rather than math courses to rush the definition of Big O out the door to students who will struggle to understand the formal definition.

Re: The most common errors in undergraduate mathematics

#54

Post by krastanov »

silentvoice wrote: post #48

One of the most bizarre errors I've seen undergraduates make is in induction proofs. I saw it so many times that I decided a TA must have been telling them to do this. They would always show the base case correctly. Then they assume it's true for all positive integers up to a fixed positive integer "n." So far so good, now we need to prove it's true for "n+1." This is where weirdness happens, I have seen a hundred as…

Wait, I am missing something here. If the reduction to a true statement is done through operations that can be performed in both direction (which are most of the trivial operations undergrads use) then this is a perfectly fine approach, right?

Re: The most common errors in undergraduate mathematics

#55

Post by Grue3 »

Xophmeister wrote: post #29
Grue3 wrote: post #17

I disagree that 0^0 is undefined. I would argue should be 1 and the function 0^x is not continuous at zero. The basic definition of exponentiation for integers is n^m is a product of m instances of n. Because the multiplicative identity is 1, a product of zero numbers is always 1. Therefore 0^0=1.

The limit of x^x does equal one when approached from the right (i.e., when x is positive, reducing to zero). From the left, it also approaches one, but is only continuous over the complex numbers. Anyway, while the limit approaches one from both sides, the function cannot be evaluated at x=0 and so is undefined.

Just because the function has different limits at some point does not mean it is undefined at that point. Consider the signum function. It has limit -1 when approached from negative side, limit 1 when approached from positive side, but is actually defined as 0 at zero.

Re: The most common errors in undergraduate mathematics

#56

Post by lutusp »

akater wrote: post #37
lutusp wrote: post #21

Earlier quoted context omitted.

It's more of an open question than your reply would seem to indicate: http://mathforum.org/dr.math/faq/faq.0.to.0.power.html It's less controversial when the terms are integers, but even there an argument can be made for indeterminacy. As the linked article says, "There is no one definition that always works well for 0^0" I can use 0^0 = 1 to prove that 1 = 2, but I'm sure you can anticipate the argument's form.

> "There is no one definition that always works well for 0^0" Yes there is. :-) 0^0 = 1. Actually, the only case for claiming it to be an “indeterminate”, comes from so-called “continuous exponents”. Which, arguably, something that never occurs in reality — only in exam sheets by lazy calculus teachers. Whenever you meet an algebraic equation with sum over 0 always only true when 0^0 = 1. I don't claim I've seen them…

> What exactly was that argument for indeterminacy you were talking about?

http://www.math.vanderbilt.edu/~schectex/commerrs/#Infinity

From the bookmark given above, search for the phrase "That reminds me of a related question that seems to bother many students", followed by a brief and instructive exposition on the indeterminacy of 0^0.

Also http://mathforum.org/dr.math/faq/faq.0.to.0.power.html, already given.

It's nice to be so sure of oneself, but in this case it's misleading.

Re: The most common errors in undergraduate mathematics

#57

Post by kaitai »

lisper wrote: post #50
reikonomusha wrote: post #44

Earlier quoted context omitted.

It seems tempting to have a single unambiguous notation for mathematics. But In constructing such a language, one will quickly realize that doing mathematics becomes an intensely arduous task. This is not unlike recent discussions about the conlang Ithkuil on HN. For better or for worse, math notation has for the most part been optimized for writing on paper or a blackboard, using context to eliminate "inessential de…

> Have you ever attempted to write Lisp on a blackboard What's a blackboard? Seriously, blackboards are pretty obsolete technology. Designing a notation to optimize for blackboards in this day and age is kind of like designing roads to accommodate horses.

Mathematicians still fight college building committees and IT people to get blackboards in newly-built classrooms. (Why IT people? Dust + classroom computer system/projector/etc.)

Re: The most common errors in undergraduate mathematics

#58

Post by Grue3 »

silentvoice wrote: post #48

One of the most bizarre errors I've seen undergraduates make is in induction proofs. I saw it so many times that I decided a TA must have been telling them to do this. They would always show the base case correctly. Then they assume it's true for all positive integers up to a fixed positive integer "n." So far so good, now we need to prove it's true for "n+1." This is where weirdness happens, I have seen a hundred as…

All horses are of the same color.

By induction, suppose there are only n=1 horses. Obviously they are all the same color.

Suppose we had proven the statement up to n. For every n+1 horses, the first n horses must be the same color. The last n horses too, must have the same color. Clearly all of them have the same color. QED.

Re: The most common errors in undergraduate mathematics

#60

Post by nmrm »

krastanov wrote: post #54
silentvoice wrote: post #48

One of the most bizarre errors I've seen undergraduates make is in induction proofs. I saw it so many times that I decided a TA must have been telling them to do this. They would always show the base case correctly. Then they assume it's true for all positive integers up to a fixed positive integer "n." So far so good, now we need to prove it's true for "n+1." This is where weirdness happens, I have seen a hundred as…

Wait, I am missing something here. If the reduction to a true statement is done through operations that can be performed in both direction (which are most of the trivial operations undergrads use) then this is a perfectly fine approach, right?

I have a feeling that the problem is students show "If the equation is true, then 0=0" instead of "Note that 1=1; now, we derive P".

Even if all facts used in the proof are true "in both directions", it's still a serious breach of modern mathematical style to start with what you're trying to prove.

It's also possible that they don't explicitly note that the facts they're using are "true in both directions". edit: to be explicit, in that case I'd still consider the proof wrong

Anyways, if a student presented a proof like that, I would at least take a couple points off for the awkward style unless it was explicitly justified somehow.

It's also kind of a weird proof technique to start with 0=0 or 1=1 and bother to explicitly state this fact...

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