Sunday, January 24, 2010

"What's Math Got to Do with It?" by Jo Boaler


Jo Boaer has written a book about "Helping Children Learn to Love Their Most Hated Subject...". In the first chapter s tells us about a math class she once attended. "The students were given problems that interested and challenged them... As the students filed out of the room at the end of class one of the boys sighed, 'I love this class.' His friend agreed."

"Unfortunately, very few math classes are like (this)... (and) Far too many students in America hate math and for many it is a source of anxiety and fear." It's not just students who hate math, "...In 2005, an Associated Press--America Online (AOL) news poll showed that ...twice as many people hated math (when in school) as any other subject."

However, recent "...trends suggest... (that while)...school math is widely hated, ...the mathematics of life, work, and leisure is intriguing and much more enjoyable." And so, "Our task is to ...get (students) excited about math..." by introducing them to the beauty, intrigue, and fun that math can be.

I had a great experience at work this past week. I've been helping my sixth graders to understand and work with fractions. They're good kids and work hard, but some of them are still having difficulty with some of the basics such as finding common denominators and simplifying fractions.

Why is this so hard? Haven't they compared 2/8 to 1/4 and seen that it is the same? Haven't they had experiences working with fraction circles? Fraction bars? Visualizing pizzas and pans of brownies?
Magnetic Fraction Circles
images copied from: www.lakeshorelearning.com
Hands-On Math Tiles

I think it's because we don't always emphasize the importance of the pieces being exactly the same size. Face it, it's difficult dividing circles be hand and arriving at equal sized pieces. How many times do we say, "Now, imagine that these are the same size..." ? When we use rectangles or circles to illustrate fractions, do we always draw them each the exact same size? How can we say that we are comparing fractions, if the whole is a different size?

There is a sequence to all learning: hands-on, illustrations, verbal, then symbolic. After we've had experience working with the actual manipulatives, we draw them, then we are able to visualize them and mentally manipulate them. Can you do this easily with circles or rectangles? I can't.

For this reason, I like to use pattern blocks when working with fractions. For those of you who don't know about pattern blocks, they are, for the most part, regular polygons: a yellow hexagon, a red trapezoid, a green equilateral triangle, and a blue rhombus.

image copied from: www.lakeshorelearning.com

  • 6 green equilateral triangles = 1 yellow hexagon
  • 3 blue rhombuses = 1 yellow hexagon
  • 2 red trapezoids = 1 yellow hexagon
They've created two new polygons that make these blocks even more versatile: a brown right trapezoid that is exactly half the size of the regular red trapezoid, and a purple right triangle that is exactly half the size of the green equilateral triangle. So now you also have:
Overhead Fraction Pattern Blocks
image copied from: Didax
  • purple right triangles = 1 yellow hexagon
  • 4 brown right trapezoids = 1 yellow hexagon
In this way, you can represent 1/2's (red trapezoid), 1/3's (blue rhombus), 1/4's (brown right trapezoid), 1/6's (green equilateral triangles), 1/12's (purple right triangles), and, of course, 1 whole (yellow hexagon).

When you move from working with these blocks to drawing them, you know the relationship between the triangle and the hexagon; the rhombus and the hexagon; the triangles and the rhombus; etc. You've felt this relationship.

Then there's the color. I can visualize these blocks much easier than I can rectangles or circles, because they are color coded. Mention the yellow pattern block and not only do I see the yellow hexagon in my mind, I can see all of the other blocks as well. Thus I can visualize and mentally manipulate these blocks.

Not only that, but I can change the size of the whole simply by combining or decomposing the blocks. One time the whole might be a hexagon and a red trapezoid. I call this my fish cake. I call it a cake so that the students will begin to see it as a whole and no longer a group of blocks.

Or the whole might be a hexagon minus a green equilateral triangle. This I call my rabbit cake, because it looks like there are 2 ears.

The whole might be a smaller block, such as the blue rhombus. How does this change all of the relationships?

A lot depends on what the whole is. A LOT depends on what the whole IS. I am continually asking what is the whole? Or stating, this blue rhombus; or hexagon and green triangle; or 2 green triangles, this is the whole cake, so...?

Students begin to really reason, to think, to make sense of fractions and wholes, as well as mixed numbers. They use the blocks, solve difficult problems, then I show them how what they just did with the blocks might be represented by numbers and operational signs. There is an "AH-HA!" moment. THEY GET IT! THEY REALLY GET IT!

I just had to share all of this with you because it made my day, my week, maybe even my month, when a student said recently, "I LIKE FRACTIONS!" This student said this with surprise, delight, and joy. Wow, Math Was Fun Today! Isn't that a hoot?

What do you do to make fractions; decimals; or math in general fun? Please write to me and let me know. --Lauren

For more fun with pattern blocks go to:

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