Wednesday, September 18, 2013

Math Buzzing

Blog?  I have a blog?  Oh yeah, I do have a blog.  And it's been a while, hasn't it?  And you're probably hoping I'm going to post some more photos of this sweet little lump of baby flesh that is wriggling in my lap right now, but alas, I have no intention of doing that today.  I have a bee buzzing in my bonnet and I'm going to work it out right here, right now, while my supper cooks and my boys play and my baby soaks me in drool.  You see, I have a marginal intellectual life that peaks out from under the laundry pile every now and then.

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.


  1. But...but....but!

    Math is my weakest subject. Which made me suck at chemistry and physics also. So I probably shouldn't say much on this. My understanding is that math is the ultimate in orderliness and a black and white world. In math, something is either completely right or completely wrong. No middle or almost. Thus the red check marks when one little mistake leads to a completely wrong answer. Which is frustratingly discouraging.

    I've heard all the "practical life application" arguments with the un-schooling movement and it seems to me that that approach leaves too many holes. Don't we need to be methodical and progressive in understanding so nothing is left out or missed? Dry and boring, yes.
    The other argument I've been given is that higher math -that you don't use every day unless you are an electrician or a scientist- teaches you to THINK. Otherwise, really, we don't need math at all now that we have digital devices to do the calculations for us. Though we would need to understand WHICH calculations to use.

    Just my random thoughts...

    1. Do you remember how to do quadratic functions? Do you know how to USE them? No? You're missing something. You have a hole in your math. Do you remember half of what you learned about math methods? You have LOTS of holes in your math. But you do okay, right?

      Math absolutely does teach us how to think, but only if we actually use it to think. Most of the time in math class, we just apply the formulas and methods we're told to use for a certain situation. If we encounter something in real life requiring that method, we have no idea what to do. I have experienced that. I got good grades in math, but that doesn't actually mean I'm good at it. I don't know how to use a good portion of it. I just have a good enough memory to get the answers "right" and pass tests.

      There was an interview with a guy who works in robotics and actually invents math to solve his problems. He said that it doesn't really matter WHAT math kids are learning as long as they are learning to use it. He said it's far more important to develop mathematical curiosity and and intuitive relationship with math because that is what gets him through his work.

      Of course math instruction needs to be somewhat progressive in an age-appropriate way, but we don't need to take that too far. If something is "missed" it can be picked up later when necessary. We tend to take the right vs. wrong issue so far that we end up thinking there is only one way to solve a problem, when in fact there can be many ways to arrive at the same answer, but we don't see or encourage that (and THAT is where the thinking comes in). We are so uptight about "right answer" that we call collaboration "cheating", when in fact learning math as a group helps with the thinking process and allows us to learn to understand math from other people's thinking. Maybe math is about more than "right answers".

    2. I've seen this with Samuel lately. He has a hard time memorizing steps and he is always trying to come up with his own way to do things. It seems his way is always a lot harder and more circuitous and doesn't always lead to the right place. But he definitely thinks that way. So the question is HOW does someone develop the intuitive relationship with math? How does someone avoid all the time consuming rabbit trails in the process that could also be discouraging?

  2. Now let's have a post and some more photos of that sweet little lump of baby flesh!