Unlearning Math

June 7, 2012 11:20 AM

Learning and Unlearning Math explores topic in mathematics from the perspective of a teacher. The author has developed series of articles on such varied topics as Multiplication, Quantity, Central Tendency, Math in Comics, and things not taught in schools. More recently, the blog has focused on the conventions for Mathematical Notation.
posted by borges (20 comments total)

Normal notation is so weird, no wonder students have problems. Is 1/2 a number or a division problem? Why do we use the same - for subtraction as for negating a number? Yes, I know you can come up with "answers" but it's because you already speak this language.
posted by Obscure Reference at 12:36 PM on June 7, 2012 [1 favorite]

There are problems with the minus sign even once you "speak the language". The minus sign is the only annoying unary operator in Haskell, for example. I suppose they should probably use unary operators for strictness annotation, but that's off topic.

We should not challenge the fact that 1/2 represents both a division and its result however. We gain so much power from ignoring distractions like that. I would not teach assembler for an introduction to programming course either.
posted by jeffburdges at 2:13 PM on June 7, 2012 [1 favorite]

It is both a division and a number; both a negation and a subtraction. They are all numbers. They are all expressible with operations.

Don't listen to me. I've become Lisp bigot and am nearing the point where I can't even do math unless it's in symbolic expressions.
posted by wobh at 2:15 PM on June 7, 2012 [1 favorite]

There is a good book and blog about mathematics education by a coauthor of mine.
posted by jeffburdges at 2:18 PM on June 7, 2012

I both teach mathematics at the college level, and have an elementary school age student who has been introduced to negative numbers, and I've never seen the superscripted - that the author talks about.

I like to entertain myself once a semester or so explaining to my students about conventions for variable assignment. Clearly, i, j, k are integers, and iterators/indices, at that, but x, y, z, w should be continuous variables, while m and n are again, typically integers. Using \alpha, \beta, \gamma for integers would be weird (unless it would make sense, instead): they're probably continuous (or angles!). And u, v, w are also frequently vectors. In linear algebra, V is a vector space and W (or sometimes H, K, don't ask me why) is a subspace.

I tend to use u, v, w for points and L, M, N for lines in my research, which frustrates at least one of my coauthors, who thinks the capitalization should be reversed.

Using \theta for something other than an angle in a context where you're working with angles would be super-weird.

Mind you, these conventions may all be in my head.
posted by leahwrenn at 2:29 PM on June 7, 2012 [1 favorite]

While it is worth asking whether one is just thinking the conventions work because of inertia, I'm not sure I buy his equals sign argument, for example. For a start, some people already use := for their definitions, admittedly, I think, not in elementary school, something which he seems to be totally unaware of. But he's confusing equality and assignment while claiming to distinguish them. (I actually don't strenuously object to his suggestion of using \equiv instead of = when there are variables are involved. Well, maybe. I'm not sure creating distinctions that aren't actually there is a good thing.)

Is 1/2 a number or a division problem?

Both? When it's a number, it's a division problem that we don't need to 'solve'. (I do, in saying that, run the risk of committing same sin as my students do, one which drives me up the wall, namely assuming that decimals are 'real' numbers and that there's something wrong with fractions. They also have a weird affinity for mixed numerals, something I swear I never used in high school.)
posted by hoyland at 2:40 PM on June 7, 2012

Is 1/2 a number or a division problem?

1/2 is an equivalence class, all of whose elements are written as a/b such that a/b = .5. Damn these statistics and finance courses I'm currently taking; I miss writing proofs.
posted by The Great Big Mulp at 2:51 PM on June 7, 2012

1/2 is an equivalence class, all of whose elements are written as a/b such that a/b = .5.

Disagree. We don't need decimals for the field of fractions of the integers.
posted by hoyland at 3:08 PM on June 7, 2012 [1 favorite]

You're right, I wasn't thinking of it correctly. Well, then, it's an equivalence class, all of whose elements are written as a/b such that b=2a.
posted by The Great Big Mulp at 3:13 PM on June 7, 2012

A fraction is a division problem waiting to happen...

To change a fraction to a decimal, do the division problem.
posted by wittgenstein at 3:49 PM on June 7, 2012

Damn these statistics and finance courses I'm currently taking; I miss writing proofs.

Don't know about finance, but you're doing statistics wrong.
posted by Mental Wimp at 4:31 PM on June 7, 2012

You need stochastic calculous to derive the Black-Scholes equation from Brownian motion, Mental Wimp, similarly the Heat equation going the other direction in time. You might therefore prefer the students know Lebesgue measure when teaching mathematical finance, ditto thermo dynamics I suppose, but everybody finds shortcuts for obvious reasons. There is a cute little book by Baxter & Rennie that does this non-rigorously but correctly enough that the intuition behind the rigorous approach is fairly clear to mathematicians.
posted by jeffburdges at 5:18 PM on June 7, 2012

Yeah, I actually already know that financial and econometric models require rigorous mathematics to derive and justify, and that there is significant overlap with statistical models. I guess I was being a bit too coy.

Is "calculous" the British spelling? I'll see myself out...
posted by Mental Wimp at 5:39 PM on June 7, 2012

V is a vector space and W (or sometimes H, K, don't ask me why)

But we so so want to know.

A site that gave a comprehensive review of all the many math notations would be a fine addition to the web. I can still feel the stigma of a math teacher, while she didn't use my name, telling about a student that pronounced the natural log symbol "ell enn". Or the difference between Σ and ε (just being silly there). Hmm wikipida is pretty good, but not hyperlinked enough, I know I've see Π (capitol π) used for several different operations in differing fields.
posted by sammyo at 5:43 PM on June 7, 2012 [1 favorite]

Somebody explained to me once that things like imaginary numbers, irrational numbers, fractions and negative numbers were invented so that some kinds of problems would have answers. At the time I didn't think much of it, but since then it's become the prevailing way I understand these things.

     -1 = 0 - 1
    1/2 = 1 / 2
sqrt(2) = 2 ^ (1 / 2)
      i = (0 - 1) ^ (1 / 2)

With just positive whole integers, subtraction, division and exponentiation you can express so much of what I painfully had to learn of algebra. Sure it's more verbose, but the rules for it are dead simple: try to evaluate the expression to a positive whole integer and if you get to the point where you can't, stop.

I suspect there are an infinite amount of perfectly good algebraic numbers I can't express this way but don't have the chops to prove it or find the minimal set of operations that would do the job but there you go.
posted by wobh at 5:52 PM on June 7, 2012 [1 favorite]

Is 1/2 a number or a division problem?

This isn't a mathematics problem. I was trying to explain to my class how to evaluate an expression, And I had to confront whether this was an expression or was already evaluated. Similarly for -1. In the context of the order of operations in evaluating expressions, how do you know when you're through evaluating?
posted by Obscure Reference at 6:24 PM on June 7, 2012

Wikipedia has some interesting ones, like this List of mathematical symbols and this Table of mathematical symbols by introduction date with the first author to use the symbol.
posted by borges at 6:25 PM on June 7, 2012

I was trying to explain to my class how to evaluate an expression, And I had to confront whether this was an expression or was already evaluated.

I don't see this as a problem with the notation, but as a problem with the idea of a bright-line distinction between expressions and "actual numbers". Nothing we write is a number; it's all representations of numbers using one system or another. Some representations are more direct than others, but there's no bedrock "really truly direct representation of the number in question" which is the goal of evaluation. (Except, perhaps, in those rare cases where the number in question has a name — but even then there is an underlying definition, etc.)
posted by stebulus at 9:12 AM on June 8, 2012

V is a vector space and W (or sometimes H, K, don't ask me why) is a subspace

H and K are subgroups (which all vector subspaces are). We skip I and J because they might be indices or index sets. I don't often see them as subspaces but as subalgebras or Lie subalgebras (though those are often lower case Fraktur), where the connection to groups is clearer.
posted by eruonna at 9:33 AM on June 8, 2012

Here, then, is my proposal: introduce the asterisk “*” fairly early on in school as a variant of “×”, right around the same time that decimal numbers come into play. Don’t make a big deal of it, don’t require its use, just get students used to seeing it as meaning the same thing as “×”.

Why not start with * right off the bat? Switching symbols over the years is just confusing and intimidating.

And I've never seen that superscript negative sign, either.
posted by Gordafarin at 1:59 PM on June 8, 2012