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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

#101

Post by iak8god »

BeoShaffer wrote: post #41
euroclydon wrote: post #36

I studied undergraduate mathematics, and I think it's a field which requires the student to spend as much of their own time as needed to understand a concept. Perhaps this could be said of all fields, but there is just limited time in a class to absorb, e.g. Euler's Formula, because there are several concepts in play, and if the student doesn't have a firm grasp or recollection of one of them, then they get lost at o…

This is why I really like learning math through MOOCS (with pause/rewindable videos) despite them having a number of other drawbacks.

Could you recommend some good maths MOOCS at the level of undergraduate math major courses?

Re: The most common errors in undergraduate mathematics

#102

Post by lutusp »

lutusp wrote: post #56
akater wrote: post #37

Earlier quoted context omitted.

> "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'…

Ingenious solution -- faced with contradicting evidence, and rather than debate the topic on its merits, just downvote the post and walk away.

Re: The most common errors in undergraduate mathematics

#103

Post by lisper »

carstimon wrote: post #100
lisper wrote: post #83

Earlier quoted context omitted.

Look, I have nothing against blackboards, just as I have nothing against horses. Horses are really handy in some situations. If you're in the wilderness and you need to cross a stream, a horse can be just the thing. There's no technology that can compete with a horse in that case. But to constrain your infrastructure (notation in the case of mathematics, roads in the case of horses) according to the needs of a blackb…

The great thing is we're not constraining our notation. As reikonomusha said, the standard notation is easier to read. Other notation is better for programming or certain things, and that's what we use there. Re: your last paragraph. I only went to the end because I knew that there must have been a good example in the questions. If you want I can give you examples from the middle of a talk.

> the standard notation is easier to read

Only because you're used to it. In fact, standard notation is much harder to read because it's ambiguous, often to the point of actively introducing errors. See:

http://mitpress.mit.edu/sites/default/files/titles/content/s...

> If you want I can give you examples

No, I don't dispute that blackboards are useful. What I dispute is that their utility is so high that we ought to design mathematical notation around their limitations.

Re: The most common errors in undergraduate mathematics

#104

Post by Chris_Newton »

duaneb wrote: post #64
lisper wrote: post #63

Earlier quoted context omitted.

I honestly can't tell if you're being ironic or not. But if you're not: what makes blackboards so excellent?

Blackboards don't generally crash, or have parse errors, or have encoding errors, or have usability problems. If you have chalk and a blackboard and have some semblance of an ability to write, you can use it to its fullest extent.

Blackboards don't generally crash

But they suffer from “memory exhaustion” rather easily, to the point where even several blackboards on a funky roller system might be insufficient for a one hour lecture developing a complicated proof.

And the garbage collection causes a serious interruption and sometimes misses things.

have parse errors

Your lecturers obviously had much more legible handwriting than some of mine!

have encoding errors

Well, an encoding where P, p, and ρ all occupy the same code point might be considered ill-advised, and one where m, n, r, u, v and w may variously appear distinct or not depending on the display device in use is downright mischievous.

have usability problems

Does blocking half the lecture theatre’s view every time you write up a new formula count as a usability problem?

Re: The most common errors in undergraduate mathematics

#105

Post by akater »

lutusp wrote: post #56
akater wrote: post #37

Earlier quoted context omitted.

> "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'…

I'm sorry but your links lead to the same continuity example that has nothing to do with real world (and real mathematics, as well).

Can you prove (0^0 = 1) -> (1 = 2), as you claimed above?

And no, we don't need “temporary definitions”. For any combinatorial investigation imaginable leads unambiguously to 0^0 = 1:

https://en.wikipedia.org/wiki/0%5E0#Zero_to_the_power_of_zer...

The only concept a general expression of the form x^y could possibly represent is the space of mappings y -> x. There's one and only one such mapping when x and y represent empty set. Arguing this to be wrong as in “but it can't be 1 because there are no mappings to empty space from non-empty sets!” ( = “0^x = 0”) is plain ridiculous. This definition simply does not fall apart. 0^0 is not a special case for it in any way.

Enumerating numbers one can represent with 0 digits put in a string of length 0 leads to the same conclusion. It's clear that there is one and only one string of length 0, unless you demand it to contain more than 0 digits, in which case there are none. [However, this combinatorial problem is not an independent one: it's equivalent to enumerating mappings from space of strings to space of chars.]

In other words, not only there are definitions that work well for all cases, including 0^0, there's actually only one such definition.

Re: The most common errors in undergraduate mathematics

#106

Post by lutusp »

akater wrote: post #105
lutusp wrote: post #56

Earlier quoted context omitted.

> 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'…

I'm sorry but your links lead to the same continuity example that has nothing to do with real world ( and real mathematics, as well). Can you prove (0^0 = 1) -> (1 = 2), as you claimed above? And no, we don't need “temporary definitions”. For any combinatorial investigation imaginable leads unambiguously to 0^0 = 1: https://en.wikipedia.org/wiki/0%5E0#Zero_to_the_power_of_zer... The only concept a general expression…

> I'm sorry but your links lead to the same continuity example that has nothing to do with real world (and real mathematics, as well).

I guess that would explain why it's located in a litany of common student errors compiled by math educators, as well as the other reference I provided.

But you know what? I'm not interested in posting to a thread that downvotes posts with a probability proportional to their accuracy and relevance.

From: http://www.wolframalpha.com/input/?i=0%5E0

Result: 0^0 = indeterminate

Re: The most common errors in undergraduate mathematics

#107

Post by enthdegree »

noelwelsh wrote: post #18
enthdegree wrote: post #4

"An equation such as ∫ 3x^2 dx = x^3+C says that we add together uncountably many infinitesimals, and we get a medium-sized number." Yelp! My buddies Riemann & Lebesgue had quite a different story on the matter...

Non-standard analysis (e.g. http://www.sjsu.edu/faculty/watkins/infincalc.htm -- another example of awesome web page design) does formalise the concepts of infinitesimals. It's quite neat to have a complete alternate formalism for calculus.

That's really cool. However it's important to note that it nonstandard analysis does not 'live' in the real numbers. It turns out the hyperreals aren't even a metric space!

I'm not intimately familiar with them though, and have heard that they are a lot more lucid abstraction for some things when compared to calculus on the reals.

Re: The most common errors in undergraduate mathematics

#108

Post by j2kun »

Chris_Newton wrote: post #104
duaneb wrote: post #64

Earlier quoted context omitted.

Blackboards don't generally crash, or have parse errors, or have encoding errors, or have usability problems. If you have chalk and a blackboard and have some semblance of an ability to write, you can use it to its fullest extent.

Blackboards don't generally crash But they suffer from “memory exhaustion” rather easily, to the point where even several blackboards on a funky roller system might be insufficient for a one hour lecture developing a complicated proof. And the garbage collection causes a serious interruption and sometimes misses things. have parse errors Your lecturers obviously had much more legible handwriting than some of mine! ha…

> But they suffer from “memory exhaustion” rather easily, to the point where even several blackboards on a funky roller system might be insufficient for a one hour lecture developing a complicated proof

This is more of a problem with people or time constraints, not the medium.

Re: The most common errors in undergraduate mathematics

#109

Post by smorrow »

userbinator wrote: post #6

By the way, write your plus sign (+) and lower-case letter Tee (t) so that they don't look identical! From my experience, I'm one of the very few people who write manually in a serifed font; in particular, I write 1, l, I, and | very distinctly and use a slashed zero so as to distinguish it from the letter O. I wish more people would do this since it is still sometimes necessary to communicate on paper, and trying to…

Don't forget the slashed z so it's different from 2.

Re: The most common errors in undergraduate mathematics

#110

Post by j2kun »

wtallis wrote: post #99
j2kun wrote: post #95

Earlier quoted context omitted.

Essentially, that is the argument. If the point is to allow students to manipulate the lecture as data, then it must be error-free. That is literally putting a parser between the lecturer and the students. And your condescending comment does not address my point, which is that a lecture would not benefit from (and is actively harmed by) the "features" being suggested. Real time error flagging would be extremely distr…

Enabling multiple users to interact with the equations in realtime as they are being written is hardly the only possible benefit of using computers to communicate math, and even so it only requires that the equations be tokenized to be manipulable, not that the whole expression be completely error-free and the parser be running in an enforcing mode. And your claim that writing mathematics as source code is too slow i…

I'd like to hear your ideas for other benefits.
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