This is math.
It all started over with teaching Math to Monkey. Only he didn't get the memo, so he had no idea that math is a "chore." That boy just inhales everything I show him about numbers. Has since he was pretty little. And because he was (is) so little, I've been careful to keep plenty of physical objects around to play with as we discuss the numbers. We've been as much playing as anything, often the numbers have been relatively incidental. For a while we were moseying through the book - we're working with Math Expressions - but lately I've been feeling like a skipped stone: no sooner do I land us in a section of the book than he comprehends it and we're off again, trying to keep up with where he's at. I asked him the other day if I'd even given him "hard math." He said no. So off I went to my forums. What do you do with a mathy kid anyway??
Turns out, you change your thinking.
I'd seen, and felt drawn to, cuisenaire rods whenever I saw them mentioned in the forums. Some people that are clearly very mathy seem to like them. A lot. But I never could figure out what you are supposed to do with them. They're sort of mysterious. So you build a staircase. That's nice. Now what?
I dunno. So I never bought them. But when Monkey started moving so quickly through our math book that I started feeling like a skipped stone, I went and searched the forums for information about them. There's plenty. I ended up reading a thread from last January.
One of the women asked her daughter, "Define 8." Now, me, I would have said something absolutely brilliant, had someone asked me that: "8 is... 8." That's all it is, what's to define? Her daughter said a whole lot more:
5+3, 6+2, 7+1, 8+0, 4+4 ("and all the opposites"), 1+2+2+3, 2x4, 4x2, 16/2, 24/3, 32/4, 40/5, 18-10, and eight ones.
It blew my mind, as did the discussion of "sixness" and how to go about teaching it - or if it was even necessary to fully explore "sixness." These people were discussing, as givens, ideas I'd never encountered, and the whole shape of my perception of math and numbers started to change as I was reading. I had some questions, so I started my own thread and learned some more. The people who responded were as concerned about the relationships between the numbers as they were about the answers to the questions. What a different attitude from the "show your work - and it better be the same as the book or it's wrong" type instruction I received! Then they offered me a link to a mathematician's comments on the matter. Being a musician, the opening had my attention right away:
A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory. “We are helping our students become more competitive in an increasingly sound-filled world.” Educators, school systems, and the state are put in charge of this vital project. Studies are commissioned, committees are formed, and decisions are made— all without the advice or participation of a single working musician or composer.
Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music.” It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music, let alone composing an original piece, are considered very advanced topics and are generally put off until college, and more often graduate school. ...
In their wisdom, educators soon realize that even very young children can be given this kind of musical instruction. In fact it is considered quite shameful if one’s third-grader hasn’t completely memorized his circle of fifths. “I’ll have to get my son a music tutor. He simply won’t apply himself to his music homework. He says it’s boring. He just sits there staring out the window, humming tunes to himself and making up silly songs.” ...
“To tell you the truth, most students just aren’t very good at music. They are bored in class, their skills are terrible, and their homework is barely legible. Most of them couldn’t care less about how important music is in today’s world; they just want to take the minimum number of music courses and be done with it. I guess there are just music people and non-music people. I had this one kid, though, man was she sensational! Her sheets were impeccable— every note in the right place, perfect calligraphy, sharps, flats, just beautiful. She’s going to make one hell of a musician someday. ...”
This I can understand. Music as an art, as something that must be felt, experienced, and enjoyed is a concept I readily understand and agree with. Lockhart goes on to say that Math is the same way.
The first thing to understand is that mathematics is an art. The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such. ... Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depend heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood. (pg 3)
Ah-ha! This is what motivates my brother when he talked about solving a millennium problem. This is what makes people major in math! They're playing! (Incidentally, I saw one of those math people commenting on how much she enjoys her homework now. I believe that she just changed her major to mathematics, rather than math education.)This is what brings my brother-in-law so much enjoyment that he learned how to move a number to base 3 and then base 2... in his head. (He did that the day that I was teasing him about having a huge number of kids. If you change bases, then 8 isn't so many!)
The trick, then, is to discover how this is done so that Monkey and I can do it together.
I think this lady is onto something: