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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

#91

Post by wtallis »

j2kun wrote: post #87
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…

To constrain your infrastructure according to the needs of a computer is also silly. Imagine if, in order to hum a tune, one needed to write sheet music using a programming language. Or if every spoken conversation were halted the instant a word is used incorrectly. Mathematics (and a lecture on mathematics) is closer in nature to a conversation than a road. [edit sp]

Those are stupidly irrelevant examples. Nobody's suggesting that you would need to have a Coq parser between your keyboard and your display. You can type an incomplete or invalid expression just as easily as you can write one, but only with a computer can you have your statements automatically and reliably checked and errors flagged in realtime.

Re: The most common errors in undergraduate mathematics

#92

Post by tokenadult »

j2kun wrote: post #78

When I teach calculus I give an hour-long-talk version of this document. I particularly emphasize that infinity is not a (real) number, so any arithmetic I see them doing with infinity will be automatically wrong; and that "equals" will be overloaded, and that most professors/TAs/tutors will not point out that there are different kinds of equals signs. The stream-of-consciousness notation section also rang true. It's…

I particularly emphasize that infinity is not a (real) number, so any arithmetic I see them doing with infinity will be automatically wrong

I agree with you for the purposes of teaching undergraduate beginning level courses that involve mostly the real number system (and only incidentally the complex number system). That said, our learned fellow participant impendia here on HN, a professor of mathematics, has strenuously disagreed with me by pointing out the specialized number systems that do treat infinity somewhat like a number. On my part, for the students I encounter, I stick to discussions like "All about Infinity"[1] (formerly titled "Infinity Is Not a Number - It's a Free Man") Katherine Körner, another astute mathematician. I have frequently seen discussions of infinity as a (real) number here on HN that essentially boil down to the error of treating the quotient upon dividing by zero as a real number.

[1] http://nrich.maths.org/2756

Re: The most common errors in undergraduate mathematics

#93

Post by lmkg »

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…

- Missing the base case (or not realizing there are multiple)

- Not understanding/using variable scoping

- Insufficiently-exhaustive case checking (especially if there are multiple variables)

- Using built-in data-types that do not have the functionality or invariants that you need instead of making custom types

--- (subcase) Using the wrong number type: Using floats for any non-integer, when sometimes you want Decimal or need to hand-roll a fixed-point number; Not thinking about the size of your integer type.

--- (subcase) Encoding data as strings, parsing out and concating in as needed. For some reason, this is surprisingly common in 101-level beginners.

Re: The most common errors in undergraduate mathematics

#94

Post by gizmo686 »

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.

Blackboards might be obsolute (especially given whiteboards), but free form written math is not. You run into many of the same constraints regardless of if you are writing on a blackboard/whiteboard/paper/tablet.

Re: The most common errors in undergraduate mathematics

#95

Post by j2kun »

wtallis wrote: post #91
j2kun wrote: post #87

Earlier quoted context omitted.

To constrain your infrastructure according to the needs of a computer is also silly. Imagine if, in order to hum a tune, one needed to write sheet music using a programming language. Or if every spoken conversation were halted the instant a word is used incorrectly. Mathematics (and a lecture on mathematics) is closer in nature to a conversation than a road. [edit sp]

Those are stupidly irrelevant examples. Nobody's suggesting that you would need to have a Coq parser between your keyboard and your display. You can type an incomplete or invalid expression just as easily as you can write one, but only with a computer can you have your statements automatically and reliably checked and errors flagged in realtime.

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 distracting, and writing mathematics as source code would be tediously slow and again distract from the point of understanding the mathematics.

Re: The most common errors in undergraduate mathematics

#96

Post by gizmo686 »

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…

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 that do not study original papers themselves.

Re: The most common errors in undergraduate mathematics

#97

Post by pmiller2 »

GFK_of_xmaspast wrote: post #81
LaikaF wrote: post #66

Earlier quoted context omitted.

What about white boards?

You don't have to pick up a dozen pieces of chalk in order to find one that sort of works if you hold it just right.

Chalk boards also feel nicer to write on, are easier to erase, and the dust comes off of clothes and hands easily.

Re: The most common errors in undergraduate mathematics

#98

Post by j2kun »

tokenadult wrote: post #92
j2kun wrote: post #78

When I teach calculus I give an hour-long-talk version of this document. I particularly emphasize that infinity is not a (real) number, so any arithmetic I see them doing with infinity will be automatically wrong; and that "equals" will be overloaded, and that most professors/TAs/tutors will not point out that there are different kinds of equals signs. The stream-of-consciousness notation section also rang true. It's…

I particularly emphasize that infinity is not a (real) number, so any arithmetic I see them doing with infinity will be automatically wrong I agree with you for the purposes of teaching undergraduate beginning level courses that involve mostly the real number system (and only incidentally the complex number system). That said, our learned fellow participant impendia here on HN, a professor of mathematics, has strenuo…

I usually level with my students about this, but with the caveat that any serious use of infinity involves a precise and technical definition, in particular one that is not guessed on the fly as students tend to do. To illustrate, I then talk about how in elementary calculus infinity is a precise and technical shorthand for the idea of unbounded growth.

[Edit:] I also recently had the wonderful opportunity to talk to really advanced high school students about projective geometry, and I contrasted the difference between the way people talk about infinity (the point on the horizon! I'm just making it up!) versus the precise and technical definitions (a specific point in a quotient of a vector space).

Re: The most common errors in undergraduate mathematics

#99

Post by wtallis »

j2kun wrote: post #95
wtallis wrote: post #91

Earlier quoted context omitted.

Those are stupidly irrelevant examples. Nobody's suggesting that you would need to have a Coq parser between your keyboard and your display. You can type an incomplete or invalid expression just as easily as you can write one, but only with a computer can you have your statements automatically and reliably checked and errors flagged in realtime.

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 is very much lacking in proof. All we can say with confidence is that syntax like LaTeX markup on a standard keyboard layout is too inefficient for realtime use. This does not mean that realtime use is impossible if you allow for a more complicated IME and for a different final notation on-screen than the current standard math notation.

Re: The most common errors in undergraduate mathematics

#100

Post by carstimon »

lisper wrote: post #83
carstimon wrote: post #73

Earlier quoted context omitted.

There are two situations which black/white boards are awesome for. -Giving lectures in subjects which desperately need drawing and writing. Look at what happens at 1:16:27 here https://www.youtube.com/watch?v=BPSEpDq6QYc The speaker can just go draw a picture, in response to a question. I know of no alternative which can do something like that nearly as well. -Collaborating in subjects which need drawing and writing.…

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.

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