Scene: Harriet’s car. Harriet and AJ are en route to the local ice rink, as they are at this time every Friday morning.
AJ: You know what I think?
Harriet: What?
AJ: I think that if there’s no end to how big numbers can get, then there’s no end to how small they get either.
Harriet: Interesting theory. Why do you think that?
AJ: Well, because if a number can always have a bigger number, then there should always be a smaller number too. But I can’t figure out zero.
Harriet: (parking in front of the ice rink). There’s another way to think of it. What if you were trying to go somewhere but you could only go halfway at a time?
AJ: Then it would take you two times. You’d go halfway and then you’d go the other half.
Harriet: But what if you could only go half of the distance to the door. Here, I’ll show you what I mean. (They grab their skates and gear and lock the car). We’re going to walk halfway to the door of the skating rink and then stop.
AJ: O.K. (He eyeballs the distance and walks to the halfway point). I’m halfway.
Harriet: Good. Now we’re going to walk halfway between here and the door.
AJ: Oh, I see. And then we’ll go halfway again.
Harriet: That’s right. What’s going to happen?
AJ: We’re never getting to the door. The door is zero!
Harriet: What happens to the distance we walk?
AJ: It gets smaller and smaller to infinity!
[AJ runs into the ice rink and spots his friend D., who is waiting for him inside the glass doors.]
AJ: Hey, D.! I just ran infinity!
* * * * *
Blazingstar recently left me a link to a contentious debate about the teaching of math in elementary school that is being waged on YouTube. One of the subjects of the debate is Everyday Mathematics, the curriculum that has been adopted by AJ’s school. The first of the videos can be seen here. There is also a series of responses, most of which I’ve had a chance to watch yet. The basic problems the first video has with Everyday Mathematics is that 1) it doesn’t teach the traditional algorithms for mathmatics, 2) it teaches math in a way the parents don’t always understand 3) it forces the students to come up with more than one way to do a problem (which apparently prevents “mastery” that is better acquired through rote memorization) and 4) it relies on calculator use.
Everyday Mathematics’ take on the teaching of curriculum as well as the rebuttals to the video, believe that the problem with teaching rote memorization is that we’re not teaching comprehension but the following of instructions. Everyday Math seeks to teach children how to problem solve. They spend a lot of time on the practical uses of math, having children figure out how to find math problems in everyday life.
I find the debate interesting, because I was one of those kids who balked at rote memorization. I’m still like that. If I didn’t know why I was supposed to memorize something, then I just wouldn’t remember it. AJ is the same way. AJ is, however, quite good at traditional algorithmic-based math. He likes practicing counting by 3s or 5s or 10s. He likes doing pages of problems. And he also craves the sense of mastery that the completion of a task, like the memorization of times tables or finding quick and accurate solutions to a page of problems provides. The big question I have is why does asking “why” and showing “how” necessarily mean that no memorization is happening? Can't they be used in tandem?
I feel like AJ’s current way of learning math is ideal for him. It is “Everyday mathemetics” at its fundamental sense. He thinks about something he notices in the world and we try to figure it out together through numbers. Numbers, to AJ, are yet another way of exploring meaning, the same a language or art or music. He plays in the real world with numbers, but he also likes to play with the numbers themselves to see what they will do. It's like the way he plays with Legos, sometimes building buildings -- a house, the Sears Tower-- sometimes just experimenting to see how he can put them together
After the conversation about infinite smallness, I found myself noticing AJ’s mathematical mind at work on the ice rink. Or rather, what I noticed was that other kids run into each other all the time but AJ almost never does, unless someone gets him from behind, out of his line of vision. Why? Because AJ triangulates with amazing accuracy. I’m sure this is more an issue of his attention than his ability – he doesn’t like to get run into. He’s never been the kind of kid who hurls his body around without fear of the consequences. But it struck me what a complicated skill it is to figure out how you, a moving object, might intersect another moving object, where that intersection is likely to take place, and then adjust your own speed and direction to avoid an intersection. That could easily be another math problem from the everyday, although I’m not sure I’m qualified to do the calculations.
But it is still a mathematical skill, and a much more useful application of math than pretty much anything I remember ever doing in an elementary math class. You do the math, you stay on your feet. You fail the problem, you hit the ice. It’s that simple.
[Crossposted at Spynotes]
Friday, March 16, 2007
Thursday, March 15, 2007
March Book Review
My daughter Dusty is a budding artist. I’ll admit the deck has been stacked in her favor since before birth. My parents are both artists as are my sister and brother-in-law. So it’s in the genes. But, I’ve always felt art education as important as the ABCs so there have always been art supplies on hand since she was old enough to hold a crayon and scribble (at 16 months old, but who’s counting?). Once, I bought huge rolls of brown paper which were cut, in four- or five-foot increments, and taped to the floor so she could color and draw and there was room for us to draw as well.
Turns out, Dusty’s got talent and an eye for color. Her drawings are, developmentally, beyond her peers. She’s discovering perspective and point of view.
Recently, we read “The Artsy Smartsy Club” by Daniel Pinkwater. In it, the children create an art club and visit a museum to educate themselves about art and the artists who created the pieces. One museumgoer discusses Vincent Van Gogh with the children and they become acquainted with two of his most famous paintings, “Starry Night” and “Sunflowers.”
Because I am a hopeless nerd (or a librarian wanna be), I went looking for copies of the paintings for us to look at as we read (and it will not surprise you to learn that we had books in the house with those paintings – in Dusty’s room, no less!). We talked a bit about Van Gogh but I couldn’t let it rest.
I went to the library soon after and found a couple of books to further our impromptu study of Van Gogh, the Impressionists, and art in general. Here’s what I found:
The Starry Night by Neil Waldman is about a young boy (Bernard) in New York City who meets a painter named Vincent. Vincent has just arrived in New York and is looking for things to paint. Bernard gives him a tour of his favorite places – Statue of Liberty, Brooklyn Bridge, etc. “Vincent’s” paintings grace the pages. Only the painting themselves are in color. The rest of the book is done in brown ink on brown paper. They go to MOMA and find a painting that looks just like the ones Vincent’s already painted. As soon as Bernard asks whether this is one of his, Vincent disappears. Bernard decides to try his hand at painting. He’s been officially inspired. The story is simple but effective. It helps lay the groundwork for an appreciation of art. The endpapers of the book showcase the work of school children: their renditions of “Starry Night.”
No One Saw – Ordinary Things Through the Eyes of an Artist by Bob Raczka is a bit more elementary in nature. Truly an basic introduction to art and artists and while good for the beginner, may be a little light for the more advanced art appreciater in your family. Each page presents a painting by a famous artist, one that represents the kind of work they are most known for. To the side, in one sentence, is a statement about what this painter does best. “No one saw flowers like Georgia O’Keefe” by a painting of lilies. “No one saw mothers like Mary Cassatt.” 16 artists are represented. 16 different styles ranging from Van Gogh to Grant Wood to Andy Warhol. Biographical notes for each artist are appended.
This book is a nice one to have in a classroom. It might work well as an introduction for an art lesson but it doesn’t offer much beyond that. If you have a young art lover at home, especially a fairly precocious one with a long attention span, I’d recommend a larger coffee-table type book that covers a range of artists within a particular style that offers more paintings by each artist with more biographical information. I’d also recommend a book that includes more than just paintings.
Picture This! Activities and Adventures in Impressionism by Joyce Raimondo hits closer to what I was looking for for Dusty. This is a particularly good book for homeschoolers and parents who want to present “art lessons” in a more structured format. There is some discussion about Impressionism – what it is and how it was received during its heyday. There is discussion about color, color wheels, experimentation, and observation of nature.
Paintings by Monet, Renoir, Degas, Pissarro and Cassatt are used as a jumping-off point for further study. Study this painting. What do you see? Can you tell what time of day it is? What season is it? How can you tell? Is the wind is blowing? From what direction? What techniques does the artist use to show this? Now, paint your own favorite outdoor place.
There are step-by-step instructions for making “art inspired by Monet,” as well as using different methods and materials to do things like watercolor, tissue paper, pastels, plastic foam painting, plaster gauze, printing with sponges. You can even try your hand at pointillism using q-tips or experimenting with facial expressions and movement. There’s a lot to this slender book and it’s my favorite “how to” art book so far. It’s not too advanced for a kid like Dusty nor is it too basic. I may end up buying it so we can have it as a reference.
All that being said, Dusty still tends to do her own thing. These books are more used for general inspiration and reference. They’re good to just have around the house. Like a box of band aids, you never know when you’re going to need one so it’s important to stock up.
Turns out, Dusty’s got talent and an eye for color. Her drawings are, developmentally, beyond her peers. She’s discovering perspective and point of view.
Recently, we read “The Artsy Smartsy Club” by Daniel Pinkwater. In it, the children create an art club and visit a museum to educate themselves about art and the artists who created the pieces. One museumgoer discusses Vincent Van Gogh with the children and they become acquainted with two of his most famous paintings, “Starry Night” and “Sunflowers.”
Because I am a hopeless nerd (or a librarian wanna be), I went looking for copies of the paintings for us to look at as we read (and it will not surprise you to learn that we had books in the house with those paintings – in Dusty’s room, no less!). We talked a bit about Van Gogh but I couldn’t let it rest.
I went to the library soon after and found a couple of books to further our impromptu study of Van Gogh, the Impressionists, and art in general. Here’s what I found:
The Starry Night by Neil Waldman is about a young boy (Bernard) in New York City who meets a painter named Vincent. Vincent has just arrived in New York and is looking for things to paint. Bernard gives him a tour of his favorite places – Statue of Liberty, Brooklyn Bridge, etc. “Vincent’s” paintings grace the pages. Only the painting themselves are in color. The rest of the book is done in brown ink on brown paper. They go to MOMA and find a painting that looks just like the ones Vincent’s already painted. As soon as Bernard asks whether this is one of his, Vincent disappears. Bernard decides to try his hand at painting. He’s been officially inspired. The story is simple but effective. It helps lay the groundwork for an appreciation of art. The endpapers of the book showcase the work of school children: their renditions of “Starry Night.”
No One Saw – Ordinary Things Through the Eyes of an Artist by Bob Raczka is a bit more elementary in nature. Truly an basic introduction to art and artists and while good for the beginner, may be a little light for the more advanced art appreciater in your family. Each page presents a painting by a famous artist, one that represents the kind of work they are most known for. To the side, in one sentence, is a statement about what this painter does best. “No one saw flowers like Georgia O’Keefe” by a painting of lilies. “No one saw mothers like Mary Cassatt.” 16 artists are represented. 16 different styles ranging from Van Gogh to Grant Wood to Andy Warhol. Biographical notes for each artist are appended.
This book is a nice one to have in a classroom. It might work well as an introduction for an art lesson but it doesn’t offer much beyond that. If you have a young art lover at home, especially a fairly precocious one with a long attention span, I’d recommend a larger coffee-table type book that covers a range of artists within a particular style that offers more paintings by each artist with more biographical information. I’d also recommend a book that includes more than just paintings.
Picture This! Activities and Adventures in Impressionism by Joyce Raimondo hits closer to what I was looking for for Dusty. This is a particularly good book for homeschoolers and parents who want to present “art lessons” in a more structured format. There is some discussion about Impressionism – what it is and how it was received during its heyday. There is discussion about color, color wheels, experimentation, and observation of nature.
Paintings by Monet, Renoir, Degas, Pissarro and Cassatt are used as a jumping-off point for further study. Study this painting. What do you see? Can you tell what time of day it is? What season is it? How can you tell? Is the wind is blowing? From what direction? What techniques does the artist use to show this? Now, paint your own favorite outdoor place.
There are step-by-step instructions for making “art inspired by Monet,” as well as using different methods and materials to do things like watercolor, tissue paper, pastels, plastic foam painting, plaster gauze, printing with sponges. You can even try your hand at pointillism using q-tips or experimenting with facial expressions and movement. There’s a lot to this slender book and it’s my favorite “how to” art book so far. It’s not too advanced for a kid like Dusty nor is it too basic. I may end up buying it so we can have it as a reference.
All that being said, Dusty still tends to do her own thing. These books are more used for general inspiration and reference. They’re good to just have around the house. Like a box of band aids, you never know when you’re going to need one so it’s important to stock up.
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