Originally posted 3/08"A musician wakes from a terrible nightmare....
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 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."
From the introductory paragraphs of this article:
"A Mathematician's Lament," by Paul Lockhart.
If you love mathematics, read this article. If you have hated mathematics since you stepped foot in your kindergarten classroom, read it. If you were dumbfounded by division, befuddled by algebra and confused by geometry, read it. If you're teaching a young person (or people) mathematics, or if you have a child who is or will be learning mathematics, read it.
Actually, if you're teaching a young person anything, read this article because it's got a lot to say about teaching. And about math. And about
learning.
Read it right now. Really.
Go ahead. I'll wait for you.
Who knew that I could be blown away by an article about teaching, about
learning mathematics? For real. I'm not exaggerating for effect. I nodded my head. I laughed. I scribbled all over the paper. I said "Yes!" and "Exactly" right out loud a couple times. I read the article during a fine young gent's gymnastics class, and I'm sure the people nearby were a little leery of the crazy lady muttering to herself as she read.
I was inspired to be a better teacher. By a math article. Go figure.
I was always a bit of a math-o-phobe myself. Until I got to college and stopped attending my math classes. Instead I read the textbook, muddled through on my own....and loved it. Statistics, trigonometry, the other math classes I took that I can't remember the names of. Suddenly I
got it, I understood math, all by myself. I turned in all of my homework, but I only attended class when I didn't understand the assigned lesson. It usually took ten minutes to figure out what I'd missed, then I'd work on my English papers until the end of class because I thought it would be rude to get up and leave while all of the other students sat and listened politely and quietly. (A side note: Handing back a midterm test my trigonometry instructor pointedly remarked that three people scored above 90%, and he was happy to report that the two people who scored above 95% actually came to class regularly. He looked right at me while he said it, to where I was hiding away behind the tall guy in the back of the classroom. I got a 93.) My point is that I found math
classes excruciating. But I found out that I kind of liked
doing math.
Back to the article, the analogy with which Lockhart begins. Can you imagine teaching music this way? Not allowing children to sing or play music, an activity that comes to most of us as naturally as breathing or speaking, without learning the theory and notations first?
Or painting? (Another analogy used by Lockhart.) Having to learn color theory and perspective before you pick up a paintbrush?
Science? Memorizing the scientific method and the periodic table before you can look through a microscope or mix baking soda and vinegar?
Yet we rarely explore mathematics playfully with our children as an art, as an experience, as a way to play with the ways the world works. Mathematics is a part of our daily lives: Cooking, drawing, jumping, building. Children naturally count things, measure things, create meaningful patterns in their play. And we don't take advantage of it because we don't know to recognize and point out the mathematics in our daily world, because the way we were taught didn't give us a grounding in playful mathematics and exploring math ideas. (The way I was taught, at least...and though I can't speak for the general "we," I suspect that many of you had similar educational experiences.)
Lockhart points out that mathematics is a discipline, as much of an art as music or painting, and in our culture we've reduced the teaching of math to math facts and how-to without addressing the theory, the why, the art behind the drills and the formulas. He writes:
"There is no question that if the world had to be divided into the 'poetic dreamers' and the 'rational thinkers' most people would place mathematicians into the latter category.
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 found any), and allows more freedom of expression then poetry, art, or music (which depend heavily on the properties of the physical universe)." (p. 3)
And later ("Exactly!" I said as I scribbled notes on the paper, drew a big red star in the margin, repeat as I re-read it today):
"By concentrating on
what, and leaving out
why, mathematics is reduced to an empty shell. The art is not in the "truth" but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is
the art of explanation. If you deny students the opportunity to engage in this activity--to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, to cobble together their own explanations and proofs-- you deny them mathematics itself." (p. 5)
Lockhart is passionate about mathematics, and he's passionate about teaching mathematics. He's not just sounding an alarm, a call for change. He's not blaming teachers or students or parents. He's asking us all to look at mathematics differently, to see its beauty and complexity, and to find ways to help our children experience math without killing their natural passion for learning. He writes about how to teach mathematics as an art...and how not to. How to engage students. How to help them make sense of it all for themselves and how to inspire a meaningful relationship with mathematics. How to create a sense of discovery.
And that's what teaching is all about. No matter what the subject.
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Books to Inspire Playful MathematicsBuy them for your children. Check them out from the library. Leave them laying around without saying a word. Read them yourselves. Pick a day or two to do math puzzles instead of workbooks. Play the games. Tell the jokes. Work out the riddles.
It's fun. I promise.
Math for Smarty Pants (Brown Paper School Book) by Marilyn Burns.
I Hate Mathematics! (Brown Paper School Books) by Linda Allison, Marilyn Burns, and David Weitzman.
The Number Devil: A Mathematical Adventure by Hans Magnus Enzensberger, Rotraut Susanne Berner, and Michael Henry Heim.
Family Math for Young Children: Comparing (Equals Series) by Grace Coates, Jean Kerr Stenmark, and Brian Gothberg
Family Math (Equals Series) by Jean Kerr Stanmark, Virginia Thompson, and Ruth Cossey
Family Math : The Middle School Years, Algebraic Reasoning and Number Sense by Karen Mayfield-Ingram and Virginia Thompson.
Math By All Means series (Various authors).
Math Wizardry for Kids by Margaret Kenda, Phyllis S. Williams, and Tim Robinson
Janice VanCleave's Geometry for Every Kid: Easy Activities that Make Learning Geometry Fun
by Janice VanCleave
Janice VanCleave's Math for Every Kid: Easy Activities that Make Learning Math Fun by Janice VanCleave
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More Fun Math (and links and book recommendations) at Poohsticks here:
MathGo figure! The Fascinating World of Mathematics: "Links to math games, activity ideas, puzzles, articles, learning and teaching aids, freebies, math in daily life, 'unschooling math,' overcoming math anxiety, and much more..."
Bridges: Amazing Structures to Design, Build & Test (Kaleidoscope Kids), by Carol A. Johman.
Third-grade gent started with a single sheet of paper across two books. Will it hold a car? He took notes on his failures and successes. He ended up with an accordion-fold sandwiched between two flat pieces to give his Hot Wheels a sturdy smooth bridge.
We also got the idea for this bridge of straws and pins from a Bll Nye video:
The gents wanted to explore and discover construction using household materials, too. They made a maze for the Minotaur, a ziggurat, pyramids, and refrigerator box forts. They built bridges using stacking cups, dominoes, marbles mazes, pick-up sticks and (of course) wooden blocks.
I stepped around and over block bridges, mazes and houses for weeks.
Or gumdrops and toothpicks?
Or gingerbread and candy?
And we've been doing challenges from one of our favorite blogs, Think! No pictures, so no posts, but we've kept ourselves busy. Eventually we'll finish our Bridges! book, but not (I think) our bridges study.
I didn't see him for nearly an hour.
