Monday, January 29, 2007

Cookie Connections

One of the common themes you hear around homeschoolers is that so many lessons can be taught through real life. One that irritates me is "teach writing by writing thank-you notes." Unless we're getting married or having a baby, most of us just don't have to write that many thank-yous, and I assume that most of our children are in neither position. The other one is fractions-by-cookies. "Make cookies. Look at what they're learning."

Well, yes. But I have an issue with the "just make cookies" idea, besides dental objections. It's true I had my own first exposure to fractions by making cookies with my mother. ("It says put in 1, funny line, 2 cups of peanut butter. What's that mean?") But there's more to math than just recognizing what 3/8 of a cup looks like, or even that if you put two of those together you get 3/4 of a cup. There seem to be an awful lot of excellent cooks out there who still get nervous around fractions. Some of them are homeschooling moms, and that worries me. 

 One of the most interesting arithmetic concepts--that my teachers somehow forgot to point out in school until we did algebra--is that multiplication, division, and fractions are all interchangeable. Connected. Once you understand this, arithmetic gets so much easier. Consider 2/3 of 5. In Miquon Math you learn that the word "of" can be written "x." As in "times." If something doesn't make sense to you with the word "times," try substituting "of." Or the other way around. So 2/3 x 5 is the same as 2/3 of 5, or 5 of 2/3. If you don't know what 2/3 of 5 is, you can figure it out with multiplication. Everybody knows how to multiply fractions, right? (much easier than learning to add them) So 10/3, or 3 1/3. Simple. 

Little kids can get "of." You write "1/2 x 10," and they say 5. They've just multiplied fractions. 

And then there's that cancelling-out maneuver. When you add this to your arsenal, you have some powerful arithmetic tools going for you. You know what I mean, right? Like 3/10 x 5/9. Of course you can multiply the tops and the bottoms, and you end up with 15/90. And then you can fool around reducing, and you get 1/6. But sometimes that's a lot of work. So you can cancel out the numbers that criss-cross; and you know why, don't you? Because 3/10 x 5/9 is the same as 3 x 5 over 10 x 9 (I'm not sure how to get those to line up properly). 

And you could write that 5 x 3 over 10 x 9; and you could split those back up and write 5/10 x 3/9 . And if you reduced the fractions before you multiplied, you'd have 1/2 x 1/3 = 1/6. Well, just in case you need a reminder on this--you don't need to go through all that moving around. You can do the same cancelling out by checking the numbers that are criss-cross with each other in the original equation. The 3 and the 9 cancel out, and the 5 and the 10. 

The third point I wish my teacher had remembered to pass on is that fractions are also division. The "funny line" is not just a fraction marker, it's a division sign. 2/3 means 2 divided by 3, or how many 3's in 2, or how much pizza do 3 people get if they split 2 pizzas? Obviously they each get 2/3 (you could have figured that out even without doing fractions), but isn't that still kind of mind-boggling? You say 2 divided by 3, you write 2/3, and you already have your answer. 

 And what's 5/3 of 2? Obviously, still 10/3. 3 1/3. What's 5 divided by 3? How many 3's in 5? 5/3, or 1 2/3. What's 10 divided by 3? How many 3's in 10? 10/3, or 3 1/3. What's 1/3 of 10? 10/3. Multiply 1/3 x 10/1. 10/3. Fractions? Division? How come we're doing all this multiplying all of a sudden? Zing: connections. 

This is why I like Miquon Math. I like bigger ways of looking at things than the "this year we do multiplication, next year we do division" approach. And that's why I think you do need to go beyond cookies--unless you have an awful lot of them and a very sharp knife.


Anonymous said...

love your posts! I,too, think that teaching my kids math by baking cookies is a little too short-sighted. We learn fractions and then when we bake, it all becomes more connected in their minds ... and bellies! Today it is choc. chip ( more to do with my cycle than math, let me tell you ).
And, fyi, in the blizzard of '77 I was huddled in a sleeping bag next to my brother and my Mom, listening to an episode of the muppet show, hoping that Daddy would actually be able to make it home from work.
the next day, I skated all over the driveway and down to the neighbours houses ... and we all walked to the nearest Community centre ( in St. Thomas, Ont.) and ate beef-o-ghetti together, cooked on Coleman stoves. The only sad part was using powdered milk. Still makes me think of the blizzard to this day!

Lydia Netzer said...

Darnit, you mean I can't get my kid into college with the back of the brownies box!? This changes everything!

Fortunately, baking will still probably give my three year old something to ponder. I think I will modify this recipe to include 1/3 the scorn! ;D

Mama Squirrel said...

Scorn? Here? Never.

Three-year-olds picking things up by baking: dandy. (And your two are very sweet, particularly the book-reporting two-year-old.) But there comes a point when a bit more is needed in the mixture.

And what irritates me is that the cookie-making illustration seems to be the one that's constantly trotted out, just the same as cookie-stealing is about the only thing some people can think of when they talk about juvenile sinfulness.