Wednesday, January 21, 2004
How to measure 5/6 cup of oil and other math follies
I'm not quite sure how to approach this subject. Spring is home schooling The Kids and has spent quite a bit of money on school books (captive audience, much like college textbooks). But on the other hand, The Kids would have these textbooks if they were going to regular school, so I figure it would be fair (if expensive) game to write about the quality. It's not like there are other text books out there Spring could get.
Like I said, captive audience.
[Evaluating schoolbooks] was a pretty big job, and I worked all the time at it down in the basement. My wife says that during this period it was like living over a volcano. It would be quiet for a while, but then all of a sudden, “BLLLLLOOOOOOWWWWW!!!!”—there would be a big explosion from the “volcano” below.
The reason was that the books were so lousy. They were false. They were hurried. They would try to be rigorous, but they would use examples (like automobiles in the street for “sets”) which were almost OK, but in which there were always some subtleties. The definitions weren't accurate. Everything was a little bit ambiguous—they weren't smart enough to understand what was meant by “rigor.” They were faking it. They were teaching something they didn't understand, and which was, in fact, useless, at that time, for the child.
Finally, I come to a book that says, “Mathematics is used in science in many ways. We will give you an example from astronomy, which is the science of stars.” I turn the page, and it says, “Red stars have a temperature of four thousand degrees, yellow stars have a temperature of five thousand degrees …”—so far, so good. I continues: “Green stars have a temperature of seven thousand degrees, blue stars have a temperature of of ten thousand degrees, and violet stars have a temperature of … (some big number).” There are no green or violet stars, but the figures for the others are roughly correct. It's vaguely right—but already, trouble! That's the way everything was: Everything was written by somebody who didn't know what the hell he was talking about, so it was a little bit wrong, always! And how we are going to teach well by using books written by people who don't quite understand what they're talking about, I cannot understand. I don't know why, but the books are lousy; UNIVERSITY LOUSY!
Anyways, I'm happy with this book, because it's the first example of applying arithmetic to science. I'm a bit unhappy when I read about the stars' temperatures, but I'm not very unhappy because it's more or less right—it's just an example of error. Then comes the list of problems. It says, “John and his father go out to look at the stars. John sees two blue stars and a red star. His father sees a green star, a violet star, and two yellow stars. What is the total temperature of the stars seen by John and his father?”—and I would explode in horror.
“Judging Books by Their Covers,” “Surely You're Joking, Mr. Feynman!” by Richard Feynman.
So I see one of The Kid's math book on the desk and I remember Richard Feynman's rant on text books from the 60s. Universally bad then, are they still just as bad now?
From a math text book no less!
- 6 green onions
- 6 ripe tomatoes
- 1 clove of garlic
- 1 green pepper
- 1 slice of bread
- 2 tbsp olive oil
- 2 tbsp lemon juice
- pinch of salt
- ½c cold water
That's it! That's all the recipe they give!
It's certainly more colorful than math books I had in school (although to tell the truth, I don't really remember any prior to sixth or seventh grade—this might be a fifth grade book, which marks it as The Older's book) and it's very Politically Correct with kids of all nationalities and ethnicities present. Scanning through the Table of Contents, I see that Chapter 10 (“Working With Fractions”) has a “food” theme. Fractions, food. Yea, I can kind of see that … maybe. I flip to Chapter 10 and start paging through it.
I get to the section about dressing the salad, which is about adding fractions with different (“unlike”) denominators:
Paul likes to mix his own salad dressing. He adds 5/6 cup of olive oil and ¼ cup of vinegar to the contents of a packet of seasoning to complete the mixture. How much liquid does Paul add?
My first thought was, I usually mix equal parts oil and vinegar. Interesting to see what the actual recipe is. Then, as I looked at the picture, my next thought was, Five sixths? How the hell did Paul measure out five sixths? Okay, slight digression. 2/6 is 1/3, and 1/3 cup measures I have. So 2/3 brings us almost there, but the final 1/6? I had to pour through several cookbooks to find five tablespoons and one teaspoon is equivalent to 1/3 cup, so to halve that you need … 2½ tablespoons and ½ teaspoon. I have ½ teaspoon measures, multiple in fact, but a ½ tablespoon? I have a single 1½ tablespoon measure. So Paul here uses a 1/3 cup measure (or 2/3 cup measure if he has one) a tablespoon measure, a 1½ tablespoon measure (if he has one) and a ½ teaspoon measure to get 5/6 cup of oil. And Paul, who is anal enough to dirty four measuring devices in order to get exactly 5/6 a cup of oil, resorts to using prepackaged seasonings? Some chief!
Or rather, out of sixteen authors, not one can cook! Or even cares enough to use realistic measurements if they're going to bother to use cooking as a theme!
Lourdes wants to learn to cook Portuguese soup. Her grandmother said that the chorizos—spicy Spanish sausages—are the secret to making this soup. Chorizos can be bought at markets that have Spanish or Latin American foods.
About how many cups of beans are needed for Portuguese soup?
- 3 lb beef roast
- 2 tbsp oil
- 3 onions
- 6 c water
- 1 1/8 c dried kidney beans
- 1 7/8 c dried pea beans
- 3 or 4 chorizo sausages
- 4 c cabbage
- 3 or 4 c broccoli florets
- 1 lb spinach
What? You expect cooking directions from a math book?
We never hear about chorizos again, so why put it in? Remember, this is a math book, not a cook book, so that first paragraph appears to be a non-sequitur.
Anyway … getting back to about how many cups of beans are in Portuguese Soup. Notice in the recipe how you have to have 1 1/8 c dried kidney beans, but oh … 3 or 4 sausages. Not 3 1/3 sausages. 3 or 4. But never five. And not two. Unless you then proceed to three, or four, your choice. But remember to measure out an additional 1/8 cup of dried kidney beans (hint: two tablespoons).
Now, the point of this section is to teach rounding of fractions. And yes, while I can immediately see that 1 1/8 + 1 7/8 is about 3 (for exact values of 3) but a fifth grader might not see that, so I can see the value in learning estimation. But a soup recipe is already a bunch of estimates! I bet the original recipe called for a cup of kidney beans and two cups pea beans and one of the sixteen authors fudged the numbers a bit to make this example, obviously oblivious to the fact that 1/8 cup measures don't exist and even if they did, the margin of error in measure beans exceeds the exactness of the very measure!
I can begin to see where Feynman was going. At first thought, using cooking to teach fractions does seem like a good idea, so I'm happy there. But I'm a bit unhappy that the authors pick unrealistic measures for some of the ingredients, and I'm very unhappy that if they are going to waste space listing ingredients at least finish up the recipe so that if a student is intrigued enough to try the food, they can (and learn just how unrealistic this math book—which on second thought, is probably why they didn't include the rest of the recipes!).
Also, if I were writing the text, I'd ask how much cabbage is needed; it's a trick question since broccoli is a type of cabbage (muahahahahahaha—methinks I've been watching too much Real Genius lately).
- Suppose you need to make 6 pounds of fruit salad.
- You already have 2¼ pounds of strawberries, melon, and grapes. How much more fruit do you need?
- You decide to buy blueberries, kiwi, and raspberries. You buy 1½ pounds of blueberries and 3/4 pounds of kiwi. How many pounds of raspberries do you need to buy?
- You want to double the recipe. How much will you have of each kind of fruit?
- Okay …
- Trick question—you already have too much, you need to remove fruit.
- Is this instead of the existing fruit? In addition? You still have either way too much, or you really like raspberries.
- Let's see … you have equal parts strawberries, melon and grapes, but we haven't fully established if we are adding blueberries, kiwi and raspberries or not, so it's hard to establish rations of these additions; who the hell wrote this question anyway?
What finally clinched it, and made me ultimately resign, was that the following year we were going to discuss science books. I thought maybe the science books would be different, so I looked at a few of them.
The same thing happened: something would look good at first and then turn out to be horrifying. For example, there was a book that started out with four pictures: first there was a wind-up toy; then there was an automobile; then there was a boy riding a bicycle; then there was something else. And underneath each picture is said, “What makes it go?”
I thought, “I know what it is: They're going to talk about mechanics, how the springs work inside the toy; about chemistry, how the engines of the automobile works; and biology, about how the muscles work.”
I turned the page. The answer was, for the wind-up toy, “Energy makes it go.” And for the boy on the bicycle, “Energy makes it go.” For everything, “Energy makes it go.”
Now that doesn't mean anything. Suppose it's “Wakalixes.” That's the general principle: “Wakalixies make it go.” There's no knowledge coming in. The child doesn't learn anything; it's just a word!
“Judging Books by Their Covers,” “Surely You're Joking, Mr. Feynman!” by Richard Feynman.
Update on Thursday, January 22nd, 2004
“Sean,” said Spring, “what did you mean by there being too much fruit?” She obviously just finished reading this entry.
“There were 2¼ pounds each of strawberries, melon and grapes,” I said.
“I think they meant 2¼ pounds total of strawberries, melon and grapes,” she said. “Other than that, you were spot on.”
“Oh,” I said. “I guess I need to work on my reading comprehension skills.”
“And your spelling. There were quite a few mistakes.”
“Well, I didn't take them to task for spelling; it was a math book after all.”
So the question about the fruit salad does make some sense now, but still, the third part is quite involved—there is no indication of how much each of strawberries, melon and grapes you have, so you have to assume an equal portion of each, and is a fifth grader smart enough to divide 2¼ by 3 and get 3/4?