I just realized I have forgotten how to add fractions! I don't think i've ever had to do this in the day job. I've been out of school for 10 years.
The most common errors in undergraduate mathematics
71–80 of 117 posts
Re: The most common errors in undergraduate mathematics
#72Earlier 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.
You're forgetting the many populations in the world that have no access or limited access to modern technology, yet need or want to learn mathematics.
Re: The most common errors in undergraduate mathematics
#73Earlier 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.
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. I can stand next to someone and talk over a problem. They write some formula on the board. I insert some additional bits and pieces that they missed. We draw pictures.
I'd like to hear what you think replaces black/white boards.
The black vs. whiteboard thing is a whole different issue :)
Re: The most common errors in undergraduate mathematics
#74Earlier quoted context omitted.
Lots of people, particularly people who do lots of math, still communicate with each other over a whiteboard.
Because their notation is optimal on whiteboards and onerous on computer screens.
Going forward, the ideal notation would be suitable for both digital and analog formats while being concise but comprehensible.
It's hard to hit all of those points. Lisp falls short (neither concise nor suitable for analog) as does the existing notation (not as comprehensible and not suitable for digital formats).
Re: The most common errors in undergraduate mathematics
#75Earlier quoted context omitted.
> Seriously, blackboards are pretty obsolete technology. What? Blackboards are excellent. You're just working in the wrong places.
I honestly can't tell if you're being ironic or not. But if you're not: what makes blackboards so excellent?
No weird security bugs that you need to be careful of.
Re: The most common errors in undergraduate mathematics
#76My 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…
There are lots of incredibly common mistakes people make in theoretical CS as well:
- Confusing asymptotic upper bounds with worst-cases,
- Assuming asymptotic bounds can only refer to running times,
- Not being clear about what "n" represents in O(f(n)),
- Not specifying a model of computation before talking about complexity,
- Confusing (complexity-class) hardness with completeness,
- Confusing Turing recognisability with decidability,
- Confusing decision problems with function problems,
- Assuming proofs of decidability must be constructive,
- Assuming O(2^n) == O(3^n),
- Not recognising O(lg(n)) == O(ln(n)).
For programming, I think simple type mistakes are a biggie...
Re: The most common errors in undergraduate mathematics
#77One 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…
I'd guess it's probably because they don't actually understand proofs in the first place. There are a lot of computer science undergrads who go in with math only up through calculus somewhere and get totally lost when it gets to this weird wibbly-wobbly place where you have to make logical statements using previously unknown Latin notation instead of just solving for x.
Re: The most common errors in undergraduate mathematics
#78When 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 a huge frustration and time waster for everyone involved.
Re: The most common errors in undergraduate mathematics
#79By 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…
> slashed zero
I'd also consider a dotted zero, or a reverse-slashed zero. That way it can't be confused with a null-set symbol (or phi, or a Danish vowel.)
Re: The most common errors in undergraduate mathematics
#80Earlier quoted context omitted.
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.)
What about white boards?
I have no hard data on this, but dry-erase markers seem to be used up more quickly than chalk.
Chalk is also less expensive. As one data point: 48 sticks of chalk for $4.30[1] versus 12 markers for $7.74[2]. I'm not sure about the bulk-price comparison, or if colleges can get dry-erase markers for cheap. Although, many of my professors carried their own supplies as markers left in rooms were usually stolen.
There could be a health comparison to be made between dry-erase marker fumes and chalk dust, but I know nothing about it off-hand.
[1]: http://www.amazon.com/Crayola-Non-toxic-Anti-Dust-Chalkboard... [2]: http://www.amazon.com/Expo-Chisel-Erase-Markers-80001/dp/B00...