Vlorbik on Math Ed brings us the pencil rant, in which you expect him to say -- but he actually doesn't -- that math should be done in pencil, at least by students. Personally, I don't care what my students use, so long as their work is readable and not full of cross-outs. (It is surprisingly difficult to get them to do this; they seem to believe that there is no value in making one's work be presentable.) I use pen, myself, mostly because pens are lower-maintenance than pencils -- they don't need to be sharpened. Also, things I wrote in pen are still legible months or years later, while pencils smudge. Most importantly, if I erased everything I did that I thought was wrong but later turned out to be right, I'd never get anything done because I would constantly be retracing my steps! A teacher of mine in middle school once took off points because I took a quiz in pen. (If I remember correctly -- and I might not -- he had never explicitly said to use pencil.) My father was outraged when he learned about this; he said that the teacher ought to have given me extra points for being confident enough to write in something non-erasable.
Calculating the Word Spurt from MathTrek. Certain words are easier for children to learn than others (what makes a word easy to learn isn't entirely clear, but it seems to depend on a large number of factors, so the "difficulty" of learning a word is normally distributed). A child needs to hear an "easy" word less times than they need to hear a "hard" word in order to learn it. Thus a chhild will start out by learning a trickle of words, but when they get to the point when they've heard the medium-difficulty words enough times to learn them, that's when the flood comes. From what I can gather from the coverage I've read of this (I can't see the actual article), this particular theory applies only to little kids, and considers words independently of each other. But I would imagine that if you know certain words, it's easier to learn others. The obvious example is a lot of technical terminology which has explicit definitions which use other technical terminology, but non-techhnical natural language could be the same way. People talk about figuring out vocabulary by context. If there's one word in a sentence you don't know, you might be able to figure out what it means; if there are two words you don't know, probably not. (However, if you hear those two words in two sentences you might be able to figure it out; I'm seeing flickers of an analogy which identifies sentences with equations and unknown words with variables.) This last analogy has some interesting (and probably nearly trivial) ramifications for mathematics education, indeed education of all kinds -- try not to introduce too many new concepts at once. I have had professors who might have, say, ten things they want to say in a given lecture, and they cram them all into the first ten minutes, or the last ten minutes. By simply reordering what they say they could probably do a better job of facilitating the learning process. Similarly, giving a definition of the form "An X that has properties P1, P2, ... P10 is a Y", although logically sound, isn't cognitively sound -- it's better to break the definition up into chunks. "An X which has properties P1, P2, and P3 is said to be R. An X which has properties P4, P5, and P6 is said to be S. An X which has properties P7 through P10 is said to be T. Something which is R, S, and T is said to be Y." Those of us learning are not computers; we are humans, with human brains.
This sort of "chunking" probably comes about naturally if one does not talk from meticulously prepared notes, which is Vlorbik's suggestion at Jazz Math Ed. The human brain won't remember that list of ten things, but it will remember the chunked version, so the chunked version is what will come out. I believe that one of the worst sins of some mathematicians is to write everything as if it is reference material for those who already know it, therefore making it incredibly difficult to digest. (I almost called this blog "fuck Bourbaki", in fact, since that's a hallmark of the Bourbaki style, but having an obscenity in the title seemed unwise.)