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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

#111

Post by smorrow »

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.

The horse thing seems like a bad analogy. Whether you're using a blackboard or some fruity iPad or whatever it is, it still must go through your eyes before you read it. So you don't "optimize for blackboards", optimize for people is what you do.

What's the equivalent in the horse/car thing?

Re: The most common errors in undergraduate mathematics

#112

Post by carstimon »

lisper wrote: post #103
carstimon wrote: post #100

Earlier quoted context omitted.

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…

Ok, I can agree that we shouldn't design notation around their limitations. And I do like what they do in SICM. But, I'm just trying to root for the point up-thread:

>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.

When talking about math, our notation doesn't have to be precise, and that's ok.

> No, I don't dispute that blackboards are useful. Just obsolete :P

Re: The most common errors in undergraduate mathematics

#113

Post by jdkaplan »

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…

On the topic of handwriting, I remember this link[1] circling around during my senior year of high school. I pretty much took all of the advice.

1. http://www.johnkerl.org/doc/ortho/ortho.html

Re: The most common errors in undergraduate mathematics

#114

Post by lisper »

smorrow wrote: post #111
lisper wrote: post #50

Earlier quoted context omitted.

> 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.

The horse thing seems like a bad analogy. Whether you're using a blackboard or some fruity iPad or whatever it is, it still must go through your eyes before you read it. So you don't "optimize for blackboards", optimize for people is what you do. What's the equivalent in the horse/car thing?

> optimize for people is what you do

I guess that depends on what you think math is for. If you think the purpose of math is merely to provide humans with intellectual stimulation then yes, it makes sense to optimize the nation for human consumption. But if you think that math is actually good for something besides being a distraction from existential despair then rendering math for human consumption might not be the thing you want to optimize for. Instead you might want to use a notation that, while it can be rendered for human consumption, isn't optimized for that, but is instead optimized for, say, automated error detection, or automated compilation into some other form, like an executable program or a design for an FPGA.

The closest I can come to an analogy in the horse/car world is that in the horse world it makes sense to dispense water in troughs to make it easy for horses to drink. But despite the fact that water and gasoline are both liquids, it might make not make sense to dispense gas in the same way you dispense water.

Re: The most common errors in undergraduate mathematics

#115

Post by nmrm2 »

gizmo686 wrote: post #96
nmrm wrote: post #60

Earlier quoted context omitted.

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…

From reading math research papers, it is not that uncommon to start with what you are trying to prove and proceed with a series of reverisible operations. This is gennerally done to as a first step to convert the proposistion into something that fits more naturally into the proof. This also tends to be done in prose. It saddens me that so little of what goes in in math papers makes it into textbooks, or math students…

I don't see any pressing need to freshman to write their very first non-euclidean proofs in the same style as professional mathematicians.

Obviously, it's a matter of style. I still maintain few mathematicians would write canonical discrete math style induction proofs in reverse order.

But then, a mathematician would totally not write out induction proofs the same way we teach in freshman discrete math courses.

So in some sense the style question is completely irrelevant, and what matters is that the student's answer demonstrates unambiguously an understaning of the concept.

Re: The most common errors in undergraduate mathematics

#116

Post by silentvoice »

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?

If the fact to be proved is an equation, then this could be conceivably a reasonable approach - assuming it is correctly written. Our class was a math class, so we weren't just giving them equations that hold for natural numbers (which usually have a more illuminating non-inductive proof anyways).

I can't now remember the specifics of a particular paper because I have seen so many, but the main thing is they believed that if they arrive at some fact that happens to be true then their original hypothesis must be true. It is a classic affirming the consequent fallacy, just shrouded in lots of sophisticated symbol manipulations.

Re: The most common errors in undergraduate mathematics

#117

Post by silentvoice »

nmrm wrote: post #60
krastanov wrote: post #54

Earlier quoted context omitted.

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…

Like I said above, induction can be used to prove facts which are not equations. In fact most equations are best proved without induction, at least in my opinion. I was getting this argument even when the fact to be proved was not an equation, it is an affirming the consequent fallacy.
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