Okay, so I've been thinking a lot about math. Teaching math, learning math, using math. I've just completed a free online course from Stanford University called "How to Learn Math" and my mind = BLOWN. All I can say is sometimes a little inspiration goes a long ways. It's taken me about two months to work through the class material and in that time I've done a lot of thinking and digesting and pondering.
Our general view of math is so warped. Actually, not just warped, but limited and stunted. How far would we get in music if we first had to spend years and hundreds of hours studying music notation before we could start to make music? Dang that would be boring. I would quit. I'm guessing we'd have a lot less beautiful music in the world. When you decide to build a pole barn or a house or a garden shed do you spend years learning the names of your tools and pounding hundreds of nails into boards? No, you start your project, figure out what you need, and if you weren't very good at hitting a nail with your hammer in the beginning, well, you're gonna be improving that skill, aren't you. If you fix a car, but it still won't start, do you write it off and go on to the next car? No, you're going to back up and find your problem and keep going until it works. Do we require extensive courses in parenting and all the things you'll need to know about children before having a baby and raising it into an adult? Some might think that would be a good idea, but no, it's definitely a learn-as-you-go endeavor.
In no part of life do we think we need to have all the tools and understanding of them before we start. It's not possible. Our brains simply don't work that way. We get bored, we can't keep track of information we don't use, and we simply don't have that kind of time.
And yet with math, we think the proper way to learn it is to spend years drilling, memorizing, and practicing, learning the tools in order, going through the steps before we use any math, if we actually ever do. It's no wonder it's a classic, stereotypical question, whined by school children across the world: "When are we ever gonna use this stuff?"
And then of course, we're all well-drilled in the idea that mistakes are wrong! and bad! and must be purged out of our mathematical work. If you make a mistake, you get a big red check mark and your grade goes down, and if that happens enough, you can't get into a good college and you'll be doomed to menial, minimum-wage jobs for the rest of your life. Avoid those check marks! No mistakes! What about real life? In real life when we make a mistake, we learn what not to do. We learn what doesn't work, and we move on to find what does. Are you getting this? When we make mistakes, we learn. Isn't that what school is supposed to be? Learning? But mistakes are not allowed in school. That's a non sequitur if I've ever seen one.
I've come to the conclusion that we just have a very poor understanding of what math actually is. I really like this quote by British mathematician Keith Devlin for describing what I (and most of the rest of the population, I gather) are not seeing:
"Mathematical notation is no more mathematics than musical notation is music. A page of sheet music represents a piece of music, but the notation and the music are not the same; the music itself happens when the notes on the page are sung or performed on a musical instrument. It is in it's performance that the music comes alive; it exists not on the page, but in our minds. The same is true of mathematics."How beautiful and inspiring is a sheet of music? Eh... meh. But when we hear music? Amazing! I have to say that I honestly do not know what math "sounds like" or "looks like" or however else it can be beautiful. But I've been told that it is. It's a way of expressing our experience of creation, but most of us don't "get it" even though we've spent many years learning math.
I recently read a portion of Plato's Republic and was astonished when Socrates said that math "leads naturally to reflection, but never [has] been rightly used; for the true use of it is simply to draw the soul towards being." That may be a little esoteric, but this is Socrates we're talking about. We know that math has lots of practical applications, but if kids are getting neither practical application nor beauty and meaning and excitement, then it is a dry subject indeed, and I can understand why so many kids love to hate math.
So. This is all well and good, but where do we go with this idea? Kids still have to learn math; can't get through life without it; it's useful in our society; need those shiny college credits... etc. Well, that's what I'm trying to find out. I want to learn to teach math to my kids without textbooks and worksheets. And I'm no mathematician, so I think I'm gonna have to do some digging.
I saw an interview with Sarah Flannery, a young mathematician who has won awards for her work. She said that she learned math because her father gave her math puzzles to work on. As she worked on the puzzles, she discovered the tools she needed by asking questions and collaborating with her family and then she learned how to communicate the solutions. I've also watched videos of classrooms situations where the students are all solving tangible problems together and learning together as they go. And then I've seen how my own 8 year old son lights up with real-life problems or entertaining math games and how he wilts at the prospect of a whole page of repetitious addition and subtraction combinations.
I really think that the most important part of learning math like this lies in key #3 of the Seven Keys of Great Teaching: inspire, not require; and #7, you, not them. Sarah Flannery's father did puzzles with her. The kids in the videos I saw had a teacher engaged in the problems with them. Jonah loves to work out math problems or games that I am working on. If I shudder at the thought of a math worksheet, why should I expect it of my child? But if I'm learning to use a Japanese soroban abacus instead of a calculator, or Nathan and I talk about a math puzzle at the supper table, or I'm figuring out how many gallons of paint I need to buy for the room I want to paint, or I'm calculating the right amount of pectin and sugar to use in my quintuple batch of strawberry jam, you better believe my child will be wanting to use that math, too! So my goal, for the time being, is rather than make them do math everyday, to show my kids math everyday and help them be excited about the solutions.