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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

#31

Post by PeterisP »

nmrm wrote: post #15
acbart wrote: post #2

My god, if someone created a similar list for undergraduate Computer Science education, I would be astoundingly happy.

Shouldn't be too difficult. I haven't done much teaching and could write a short list. Unfortunately, I don't think it would be very surprising to most people (just as this list probably isn't surprising to most math majors, since we've all made at least one of these "stupid mistakes" before, even when we knew better, especially in the early years. The difference -- and this is reiterated several times -- is that goo…

What are the functional equivalents of off-by-one?

Re: The most common errors in undergraduate mathematics

#32

Post by nmrm2 »

PeterisP wrote: post #31
nmrm wrote: post #15

Earlier quoted context omitted.

Shouldn't be too difficult. I haven't done much teaching and could write a short list. Unfortunately, I don't think it would be very surprising to most people (just as this list probably isn't surprising to most math majors, since we've all made at least one of these "stupid mistakes" before, even when we knew better, especially in the early years. The difference -- and this is reiterated several times -- is that goo…

What are the functional equivalents of off-by-one?

Folding with the wrong initial value. Recursive functions where the base case is improperly defined. Stuff like that I consider 'functional off-by-one' because the student understand the general shape of the correct solution but gets a detail about iteration/recursion incorrect at the boundary.

edit: oh, also, I guess literally off-by-one errors are also possible in any language that allows any sort of side effect, but that's kind of a stupid degenerate case :-) (edit: possible, definitely not popular... wrong word there, sorry).

Re: The most common errors in undergraduate mathematics

#33

Post by astrobe_ »

One realises how confusing and complicated are the standard notations when one implements a programming language. You would prefer Lisp/Scheme or RPN notations, but they both scare away a lot of people, who prefer "natural" notations. People don't realise that they probably hurt them as much as they help them.

Re: The most common errors in undergraduate mathematics

#35

Post by jackmaney »

JoshTriplett wrote: post #12

I've seen many instances of the "undistributed cancellation" error, most commonly in even simpler cases than the one in the article: Given a fraction similar to: x + y ----- x I've seen many people cancel the "x" on top and bottom, leaving y.

Yes, but if we cancel the 6's in 64 / 16, we get 4/1 or 4. :P

Oh, and incidentally, sin(x) / n == 6.

Re: The most common errors in undergraduate mathematics

#36

Post by euroclydon »

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 one step on the chalk board, and the rest of the derivation is useless to them. This happened to me many times, and I'd just have to make a note to go figure something out, and the rest of the class was basically useless to me. Once it became more acceptable to have a laptop in class, I could just just look it up right then, and not have to wait till later, but I still just tuned out the professor.

Re: The most common errors in undergraduate mathematics

#37

Post by akater »

lutusp wrote: post #21
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.

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 all but really, try to find a counterexample. What exactly was that argument for indeterminacy you were talking about?

Those equations come from reasoning about meaningful entities, not chimeras of “x^x”, or “x^y”, or worse. Again, try to find, say, a physics paper with x^y in it. I haven't read much physics papers but I'm pretty sure you'll find precisely 0.

As one mathematician I knew put it, “hard analysis is the primary source of all obscurantism in mathematics out there”. ;-)

Re: The most common errors in undergraduate mathematics

#38

Post by prof_hobart »

aaronbrethorst wrote: post #9
trentmb wrote: post #7

Earlier quoted context omitted.

> are laid out in columns. Because they can't be resized on the fly?

Because the optimal line length for readability on a monitor seems to be between 50-95 characters per line[1]. On my 1920x1080px screen, I'm seeing about 300 characters per line in this article. [1] http://baymard.com/blog/line-length-readability and http://psychology.wichita.edu/surl/usabilitynews/72/LineLeng... - and no, the irony is not lost on me on the second link

On most platforms, it's relatively easy to resize your browser to whatever width suits you best.

Personally, I tend to find columns wider than the 95 character thing easier to quickly scan - I had this site set to about 130 chars. When someone fixes the width of the text on the website, there's no sensible option I have to widen it to my preference.

Re: The most common errors in undergraduate mathematics

#39

Post by trentmb »

itazula wrote: post #10
trentmb wrote: post #7

Earlier quoted context omitted.

> are laid out in columns. Because they can't be resized on the fly?

No, for readability. Paul Graham's essays are a good example. For example, http://paulgraham.com/copy.html

Right - I can resize my browser so that the web page is most readable to me, because it defaults to full width. I can't resize my newspaper or magazine, so they don't default to full width.

Re: The most common errors in undergraduate mathematics

#40

Post by trentmb »

aaronbrethorst wrote: post #9
trentmb wrote: post #7

Earlier quoted context omitted.

> are laid out in columns. Because they can't be resized on the fly?

Because the optimal line length for readability on a monitor seems to be between 50-95 characters per line[1]. On my 1920x1080px screen, I'm seeing about 300 characters per line in this article. [1] http://baymard.com/blog/line-length-readability and http://psychology.wichita.edu/surl/usabilitynews/72/LineLeng... - and no, the irony is not lost on me on the second link

Right - I can resize my browser so that the web page is most readable to me, because it defaults to full width. I can't resize my newspaper or magazine, so they don't default to full width.
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