It's another day: I did warn you.
Oct. 21st, 2007 05:33 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
rabidsamfanHow do we learn about abstract concepts? Well, if you've ever watched a parent with a toddler, you know. We give lots and lots and lots of concrete examples of the concept and hope that the child will pry the idea of "red" off of the red crayon and the red shoes and the red car and the red whatchamallit. And eventually the child does just that. What works with colors works with very small numbers like one, two and possibly three, but beyond that we have to resort to counting objects, be they legs or marbles or crayons or fingers until the sequence of numbers is memorized and in its wake the abstract values of the numbers too.
Most everybody gets that far. And if you've ever worked with anyone who has trouble with math, chances are pretty good you've seen them resort to counting on fingers. There are elaborate methods of calculating on fingers, or abacus beads, and they work, but they are no more the process of doing addition than pushing the buttons on a calculator is. You see, counting is concrete and adding is abstract.. Once you can do the addition in your head without drawing pictures or using fingers you've left the real world and you're playing the Math Game.
And of course, to teach an abstract, what do we do? Give lots and lots of examples. Two crayons plus three crayons is five crayons. Two dolls plus three dolls is five dolls. We show kids examples of two and three, and (if they're learning in the Boston Public Schools) make them draw pictures of boxes or triangles or whatever and count them up to reach five, in the hope that they'll pry the idea of 2+3=5 off of the examples the same way that they pried off the idea of two from all the examples they had then.
Except for one teensy little problem. There are two concepts that need to be pried off of adding, and the second one is hardly ever articulated to children who are under the age of five or six. Or rather, hardly ever articulated in a way that isn't a confusing metaphor.
You Can't Add Apples and Oranges.
Sure you can. Except when you do you have to give the answer as "pieces of fruit", or "round things", or even just "things". And if you're counting the objects in front of you and there are two apples and three oranges, it's absolutely the easiest thing in the world to count "one two three four five" and look up for approval from your teacher when you haven't understood adding unlike things requires a change of vocabulary. After all, when you count on your fingers and toes, you don't stop counting just because you reached eleven. And if all you need to know is whether all five children at your table are going to get something to eat, what does it matter if all of the fruits are the same?
In the real world, you can usually find a word which will describe a heterogeneous group, but in the Math Game, adding numbers which don't have the same "tag" is absolutely not allowed. That's why you have to find the lowest common denominator when you do fractions, and why you have to include the units in every calculation you do in science class. "You can't add apples and oranges," we chirp to ourselves in imitation of the adults who taught us, but it takes most kids a while to understand what that means and some of us are still fighting to pry the abstract concept loose from the metaphor years later.
And now for the bad news.
You can't add ones and tens either.
(Saywhat?)
Come on, you know this! If you have 10 and 1 they don't add up to 2, they add up to eleven! And eleven is written as 11, which means 1 ten and 1 one. Well, you know that if you understand "place value".
But place value is another abstract idea, and we don't start even trying to teach it to kids until we get to subtracting. This is a little like first mentioning the alphabetic principle -- that letters stand for sounds -- when we start teaching kids to read two syllable words. We could teach it sooner, but we'd have to start by going backwards.
Symbols and meaning
You see, when we write down words we call it "writing," and when we write down mathematical ideas we call it "notation" and the two things aren't the same. In spite of the quirks English has acquired by borrowing words and letters from hither and yon, writing is essentially putting symbols which stand for sounds down in order, and reading is saying the sounds back in order (either aloud or in our heads) and listening to what we're saying. Some writing systems have symbols for syllables instead of phonemes, some writing systems put the order the other way, but even Chinese writing, at its root, is about the sounds of a language.
Mathematical notation, on the other hand, is about meaning. The symbol [2] stands for the meaning /2/ which we read as "two" and understand. The symbol [+] stands for the meaning /added to/ even if we read it as "plus". Most of the time, anyway. [+] can also stand for the meaning /positive/....
Sorry, getting ahead of myself there. But for right now, hang onto this: When math is written down, half the trouble you have in understanding the concepts is understanding the notation. And to make things worse, once mathematicians developed notation, they began to do things with numbers and operations and ideas that simply can't be represented in the concrete world. Mathematical notation is full of shorthand and shortcuts: it uses symbols to stand for more than one thing in different contexts, and it records the same concepts in multiple ways. And modern mathematical notation is a fairly recent development. If Pythagoras showed up in a modern classroom where his theory was being taught, he'd be just as confused as most of the kids in the class.
More tomorrow.
Most everybody gets that far. And if you've ever worked with anyone who has trouble with math, chances are pretty good you've seen them resort to counting on fingers. There are elaborate methods of calculating on fingers, or abacus beads, and they work, but they are no more the process of doing addition than pushing the buttons on a calculator is. You see, counting is concrete and adding is abstract.. Once you can do the addition in your head without drawing pictures or using fingers you've left the real world and you're playing the Math Game.
And of course, to teach an abstract, what do we do? Give lots and lots of examples. Two crayons plus three crayons is five crayons. Two dolls plus three dolls is five dolls. We show kids examples of two and three, and (if they're learning in the Boston Public Schools) make them draw pictures of boxes or triangles or whatever and count them up to reach five, in the hope that they'll pry the idea of 2+3=5 off of the examples the same way that they pried off the idea of two from all the examples they had then.
Except for one teensy little problem. There are two concepts that need to be pried off of adding, and the second one is hardly ever articulated to children who are under the age of five or six. Or rather, hardly ever articulated in a way that isn't a confusing metaphor.
You Can't Add Apples and Oranges.
Sure you can. Except when you do you have to give the answer as "pieces of fruit", or "round things", or even just "things". And if you're counting the objects in front of you and there are two apples and three oranges, it's absolutely the easiest thing in the world to count "one two three four five" and look up for approval from your teacher when you haven't understood adding unlike things requires a change of vocabulary. After all, when you count on your fingers and toes, you don't stop counting just because you reached eleven. And if all you need to know is whether all five children at your table are going to get something to eat, what does it matter if all of the fruits are the same?
In the real world, you can usually find a word which will describe a heterogeneous group, but in the Math Game, adding numbers which don't have the same "tag" is absolutely not allowed. That's why you have to find the lowest common denominator when you do fractions, and why you have to include the units in every calculation you do in science class. "You can't add apples and oranges," we chirp to ourselves in imitation of the adults who taught us, but it takes most kids a while to understand what that means and some of us are still fighting to pry the abstract concept loose from the metaphor years later.
And now for the bad news.
You can't add ones and tens either.
(Saywhat?)
Come on, you know this! If you have 10 and 1 they don't add up to 2, they add up to eleven! And eleven is written as 11, which means 1 ten and 1 one. Well, you know that if you understand "place value".
But place value is another abstract idea, and we don't start even trying to teach it to kids until we get to subtracting. This is a little like first mentioning the alphabetic principle -- that letters stand for sounds -- when we start teaching kids to read two syllable words. We could teach it sooner, but we'd have to start by going backwards.
Symbols and meaning
You see, when we write down words we call it "writing," and when we write down mathematical ideas we call it "notation" and the two things aren't the same. In spite of the quirks English has acquired by borrowing words and letters from hither and yon, writing is essentially putting symbols which stand for sounds down in order, and reading is saying the sounds back in order (either aloud or in our heads) and listening to what we're saying. Some writing systems have symbols for syllables instead of phonemes, some writing systems put the order the other way, but even Chinese writing, at its root, is about the sounds of a language.
Mathematical notation, on the other hand, is about meaning. The symbol [2] stands for the meaning /2/ which we read as "two" and understand. The symbol [+] stands for the meaning /added to/ even if we read it as "plus". Most of the time, anyway. [+] can also stand for the meaning /positive/....
Sorry, getting ahead of myself there. But for right now, hang onto this: When math is written down, half the trouble you have in understanding the concepts is understanding the notation. And to make things worse, once mathematicians developed notation, they began to do things with numbers and operations and ideas that simply can't be represented in the concrete world. Mathematical notation is full of shorthand and shortcuts: it uses symbols to stand for more than one thing in different contexts, and it records the same concepts in multiple ways. And modern mathematical notation is a fairly recent development. If Pythagoras showed up in a modern classroom where his theory was being taught, he'd be just as confused as most of the kids in the class.
More tomorrow.
