I occasionally (or not so occasionally) come across people who believe that the way they were taught math before all this problem-solving mumbo-jumbo was superior, not because it worked better at the time but because, after learning the steps without understanding them, they developed the ability to problem-solve and reason mathematically later on. Their reasoning: kids don't need to learn to reason mathematically from the outset; they'll learn to think procedurally and then as their brains develop, they'll fit their procedural knowledge into a growing problem-solving framework. The end result will be adults who reason well AND are good at arithmetic (these are usually the people who believe that New Math creates kids who can't add or multiply, to which I say: then the person who was teaching it didn't really know what they were doing.)
I used to think this way, because in math, this is what I did. School taught me the traditional algorithms, my dad taught me to add and multiply quickly in my head, and I made connections between concepts without being encouraged to. (When I first learned fractions, they were already old friends - I'd been seeing them on sheet music for two or three years at that point.)
It occurred to me last night, in conversation with wggthegnoll
, that there is a better metaphor in my life for old math versus new math: music.
Now, I have some natural talent for music. There was never a time when I couldn't carry a tune, and some of my earliest memories are of singing interesting and difficult music at the Salvation Army, sometimes before I could read. I started taking piano lessons at the age of eight, and continued taking them until I was sixteen, at which point I started teaching piano instead. My training on the piano was classic. I was given a piece of music in a book geared to the level I was supposed to be at. I read through that music with my teacher there to point out anything I didn't get or some new technique for fingering. Then I went home and practised that piece of music until I could play it well. When I was done most of the songs in that book, I moved on to the next one. My education in music theory was mostly taken care of at music camp, and I ended up skipping Grade One Rudiments entirely (though I did do the Grade Two exam - it was necessary to get a high school credit for my piano lessons, along with my grade eight piano exam.) I learned my scales, I did fingering exercises, I knew how to play tonics and dominant sevenths and minors harmonic and melodic.
However, I was not the kind of kid who thought outside the box that was presented for me. I didn't start listening to the radio until years after my friends did, and I was fifteen and already had my grade eight exam under my belt before it occurred to me that I could play music that didn't come out of a Royal Conservatory Repertoire book. I rarely tried to figure out for myself how something was played, and other than a few hymn tunes, I had little experience with playing anything that wasn't classical. In short, I never learned to play by ear, and more importantly, nobody ever drew for me the connections I would have needed to develop that talent (which, btw, I have when I'm singing, though it's undeveloped.) I never learned about modes; no one pointed out to me what standard chord progressions were and how I could play with them to make them sound different using those arpeggios and scales I'd memorized so assiduously. Though my dad pointed out that I didn't have to play a hymn tune exactly as it was written, I could break up the notes for better rhythm, no one showed me what the chord notations were or how to read them or how they connected to each other. The advanced harmonics, which I was perfectly capable of playing, were not taught to me as something I could reason my way through.
I was taught the arithmetic of music. I was taught, "If you follow these steps, you can play any piece of music you pick up. You'll learn how they work later." I was not taught to think critically about the form of the music I was playing. All of that was saved for after I had the rudiments; it would have come about during Grade Three Harmony, which I started but never finished. It was assumed I'd pick it up on my own - but I never did.
Because of all that, I'm a very limited pianist. I know the most basic chord structures, the ones that show up in most pop music and a wide variety of hymn tunes, and I can generally figure out how to make them sound cohesive if I work at it. I can read music, though I can't sight-read very well - that is, if you put a piece of difficult music in front of me and ask me to play it, it's going to take me a week of painstaking work to get to the end of it and be able to play it back to you. (That one is a confidence issue: I don't like to keep going if I make a mistake at something, but when you're sight-reading, that's exactly what you need to do so that the person you're accompanying doesn't have to stop in confusion when you break rhythm.) I still know very little about modalities and harmonies. I still can't play by ear or even chord by ear most of the time, though I can come up with vocal harmonies with no trouble at all. I can hear in my head what it is supposed to sound like, and I'm frustrated trying to bridge the gap between how it should sound and how I'm able to make it sound.
I'm going to bring this up with my uncle when he starts teaching Elizabeth in the fall. I've taught the way I was taught, and passed on these same errors out of ignorance, but I'd prefer that my daugthers' talent for music ends up better developed than my own, so I'm going to talk to him. I want to figure out how to apply New Math strategies to elementary music, and I won't take the platitudes - they'll figure it out on their own later, don't worry about it, one step at a time, basics before critical thinking - at face value. While it may work for some kids, it didn't work for me. I don't settle for, "They'll pick it up later" when I'm teaching math. Why should I settle for that when teaching music?