More math...
Oct. 22nd, 2007 08:45 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
rabidsamfanOne, two, three, many, LOTS!
I'm reading a book about numbers right now that's so old the recommendation on the back was written by Albert Einstein. (Well, I'm reading the 1954 edition, he probably read the original edition in 1930.) It's cool because it's not only talking about the idea that primitive peoples don't understand numbers past three, it's also citing the anthropological research that makes the assertion. Chances are that no matter where or when you grew up, your brain didn't come wired to comprehend numbers over three either. Neither do most other creatures' brains. Some birds, some insects, and people -- the "number sense" doesn't even show up in dogs, and when it does show up, it's limited. (Although I wonder what the research since the book might tell us about dolphins.) What even "primitive" peoples DO understand though, is the concept of "just enough", or what the math geeks call "one to one matching," and that -- plus our fingers, is what led us to the numbers we have today. (Very simply, if you have a bunch of chairs and a bunch of people, you have "just enough" chairs if every person is sitting and there are no extra chairs or extra people.)
When people started having possessions that they needed to keep track of (like sheep) they did it by making tally marks -- you put one cut into a tree for each sheep, and when you wanted to see if you had all your sheep you matched marks to sheep. If you had more marks than sheep it was time to go looking for the sheep that had gone astray.
This worked, but it was awkward, and keeping track of the tally marks was confusing. And if you moved your sheep to a new area, you couldn't carry the whole tree along and even carrying part of the tree meant that it might get lost. Fortunately, the answer to both problems was hanging off the ends of your arms. You could use fingers instead of tally marks for low numbers. And because you used your fingers for this job a lot, gradually the names of the fingers became the names of the values instead. (It's as if you counted "thumb pointer middle ring" so often that [ring] came to mean /four/.) Gradually, the number values of /one/ /two/ /three/ /four/ etc. began to have names. Once number values had names it became possible to talk about adding "two" to "three" to get "five" in the abstract sense without having to put pebbles in piles or even point to fingers, although we probably still did it. With number names around, you didn't have to carry a stick with marks on it to remember how many pieces of fruit to bring back to the family, you could use your fingers and count up to seven. You could even add two more pieces of fruit for visitors.
Using fingers instead of tally marks also leads to the idea of using "handfuls" when you need to represent more sheep than you have fingers for -- in other words, if you're using a [/] to represent the value of /one/, then you can come up with a new symbol to represent the value of /handful/ or /five/ and add your fives and your ones together to find out the total number of sheep.
This is not an easy idea. One to one matching makes sense. One to five matching? I'm guessing that it started when someone decided to use a slash across four vertical lines to represent four fingers and a thumb. (Try it with your left hand -- with the four fingers together, touch the tip of you thumb to the tip of your pinky. See the similarity to five hashmarks on a page?) Start doing hashmarks instead of tallies and pretty soon you would come to the idea of counting off five sheep and drawing a diagonal line to save time -- especially if you're carving the lines into a rock. And counting up the diagonals leads to simple multiplication, doesn't it? Once we've got a symbol for any number besides "one", we can think about numbers in entirely new ways because we can put down those symbols and look at them in new ways. We just took a concrete idea and gave ourselves a way to think about it abstractly.
But once you've started creating symbols to stand for number values, why stop with one and five? Why not symbols for all the other numbers values too? Well, the human brain isn't bad at understanding symbols. Put a red octagonal sign on a stick at the end of a piece of road and drivers are going to hit the brakes, even if you didn't bother to put the letters [S][T][O][P] on the sign. But there's a limit to how many symbols we can arbitrarily attach to meanings. If we tried to create a symbol for each number value we'd overload our brains before we even reached a hundred. We have to make bunches and add them (or multiply them) to keep the number of number symbols (called "numerals") reasonable.
There's more than one way to skin a cat, or what you should learn about Roman numerals in first grade and why you don't want to
The Greeks used their letter-shapes to stand for numbers too. The shape called "alpha" stood for a sound /a/ when it was used as a letter and a value /one/ when it was used as a numeral. Beta stood for a sound /b/ and a value /two/ and so forth. This worked, but it led to some confusion, as well as sidepaths like numerology and Gematria, and it isn't very useful when you want to teach first graders who are still working out what the letter-shapes stand for when they're writing, much less in math. (There's a reason that Greek mathematicians made the most progress when they were talking geometry and drawing pictures.) But you can adapt the Roman system to illustrate one of the early versions of arithmetic and number-writing even with kids.
The Romans also used letter shapes to represent their numbers, but they didn't use the letters in the order they came in the alphabet. Their system looked a lot more like hash marks. They used a vertical line [I]-- like a tally mark -- to stand for /one/, For the handful, they turned the hand sideways and left the thumb out in a neutral position (try it!) and wrote /five/ as [V]. Fewer lines, takes less space on the page. Then two hands together are two "V"s attached to each other for an [X] and that's a ten. They could have done what a lot of other peoples did and made a special symbol for /twenty/ too, but they didn't think it was all that important. (Maybe their shoes hid their toes!) Still with three symbols, the Romans could easily write and add numbers. Small numbers anyway. And because X doesn't look like V and neither X nor V look like I, you don't have to worry about mixing them up. The Romans could have put VXXIII to mean /twenty-eight/ just as easily as XXVIII. (They didn't but I'll talk about why further down.)
If a Roman wanted to add two numbers: XXII and VIII, for example, he could take all the [X]s and write them down. XX..., and then all the [V]s and write them down next to the [X]s, XXV... and then put all the [I]s at the end: XXVIIIII, like so. But of course if you've got five [I]s in a row you can turn them into a [V] which makes it XXVV and two [V]s in a row is the same as an [X] which makes it XXX.
When the Romans got up to five [X]s, they saw it as a handful again and created a new symbol. They were already using letters, so they just chose another letter for /fifty/ [L] and when they got two [L]'s together (or two handfuls of [X]s) they made [C] stand for a /hundred/ (which at least made sense since the word for /hundred/ started with "C".) Presumably they could have kept on doing this kind of bunching indefinitely, but in practice they only did it once. [D] stands for /five hundred/ and [M] stands for /one thousand/. (They did have a way of indicating even bigger numbers, but we'll skip over that.)
Now if the Romans had stuck to piling up symbols into bunches and making another symbol each time, we'd want to teach first graders to add and subtract with Roman numerals. What better way to get across two concepts? First, that you have to add like things together (ones to ones, fives to fives, tens to tens) and second, that you can take a bunch of individual things and describe them as a bunch, or take a bunch and split it into individual things again. We saw what happens when you add above -- subtraction might look like this:
If you start with XXX and you want to take away VIII, you need some [V]s and [I]s in the first number of the problem (the minuend). So you take one [X] and break it into two [V]s. Now your problem looks like XXVV - VIII. You still need some [I]s, so you take one of the [V]s and change it into five [I]s. Now your problem looks like this: XXVIIIII - VIII. Get out your eraser and erase one [V] and three [I]s from each number and you have your answer: XXII.
But the Roman scribes got bored with writing down four marks of anything and decided to take a shortcut. Instead of writing /four/ as IIII they decided to write it as IV, instead. Saves time, as long as everyone understands that now the positionof a symbol is as important to the meaning of the symbol as the shape is. An [I] before a [V] or an [X] means /take one away from the next bunch/. Now you had to write Roman numerals in a strict order in order for them to make sense.
This is too much for first graders. Heck, it's too much for some fourth graders, judging from the scratching of heads I regularly see when the topic of Roman numerals wanders into the library. But "place value" (remember place value from yesterday?) works on a similar principle. We don't know the value of a numeral in a written number unless we consider both the basic value of the symbol [4] for /four/ for example, AND its position in the number. A [4] in 345 represents a different quantity than the [4] in 45,653. "Place value" is a shorthand for a kind of addition that we do all the time. We add the ones to the tens to the hundreds to the thousands to the ten thousands and so forth and so on.
Symbols and shortcuts
Writing about math is repetitive. You have to use the same words over and over. The scribes in medieval Europe got tired of writing "minus" in the and started abbreviating it as an [m] with a sort of tilde [~] over it, and they weren't the first to take a shortcut by any means. Whenever mathematicians have figured out a way to use a few symbols to express a lot of meanings, they've grabbed it, and the position of symbols matters a lot! "Place value" is the most familiar example, but it's not the only one. And like any kind of shortcut, there's a potential for getting lost. When we say numbers or read them back from the numerals we don't articulate the whole addition problem, at least not in English, and especially not with the numbers under twenty, and as a result "reading" numbers can obscure the connection between the spoken number and the written numeral for the beginner, hiding the concept of place value.
(Next time: Tom Lehrer!)
I'm reading a book about numbers right now that's so old the recommendation on the back was written by Albert Einstein. (Well, I'm reading the 1954 edition, he probably read the original edition in 1930.) It's cool because it's not only talking about the idea that primitive peoples don't understand numbers past three, it's also citing the anthropological research that makes the assertion. Chances are that no matter where or when you grew up, your brain didn't come wired to comprehend numbers over three either. Neither do most other creatures' brains. Some birds, some insects, and people -- the "number sense" doesn't even show up in dogs, and when it does show up, it's limited. (Although I wonder what the research since the book might tell us about dolphins.) What even "primitive" peoples DO understand though, is the concept of "just enough", or what the math geeks call "one to one matching," and that -- plus our fingers, is what led us to the numbers we have today. (Very simply, if you have a bunch of chairs and a bunch of people, you have "just enough" chairs if every person is sitting and there are no extra chairs or extra people.)
When people started having possessions that they needed to keep track of (like sheep) they did it by making tally marks -- you put one cut into a tree for each sheep, and when you wanted to see if you had all your sheep you matched marks to sheep. If you had more marks than sheep it was time to go looking for the sheep that had gone astray.
This worked, but it was awkward, and keeping track of the tally marks was confusing. And if you moved your sheep to a new area, you couldn't carry the whole tree along and even carrying part of the tree meant that it might get lost. Fortunately, the answer to both problems was hanging off the ends of your arms. You could use fingers instead of tally marks for low numbers. And because you used your fingers for this job a lot, gradually the names of the fingers became the names of the values instead. (It's as if you counted "thumb pointer middle ring" so often that [ring] came to mean /four/.) Gradually, the number values of /one/ /two/ /three/ /four/ etc. began to have names. Once number values had names it became possible to talk about adding "two" to "three" to get "five" in the abstract sense without having to put pebbles in piles or even point to fingers, although we probably still did it. With number names around, you didn't have to carry a stick with marks on it to remember how many pieces of fruit to bring back to the family, you could use your fingers and count up to seven. You could even add two more pieces of fruit for visitors.
Using fingers instead of tally marks also leads to the idea of using "handfuls" when you need to represent more sheep than you have fingers for -- in other words, if you're using a [/] to represent the value of /one/, then you can come up with a new symbol to represent the value of /handful/ or /five/ and add your fives and your ones together to find out the total number of sheep.
This is not an easy idea. One to one matching makes sense. One to five matching? I'm guessing that it started when someone decided to use a slash across four vertical lines to represent four fingers and a thumb. (Try it with your left hand -- with the four fingers together, touch the tip of you thumb to the tip of your pinky. See the similarity to five hashmarks on a page?) Start doing hashmarks instead of tallies and pretty soon you would come to the idea of counting off five sheep and drawing a diagonal line to save time -- especially if you're carving the lines into a rock. And counting up the diagonals leads to simple multiplication, doesn't it? Once we've got a symbol for any number besides "one", we can think about numbers in entirely new ways because we can put down those symbols and look at them in new ways. We just took a concrete idea and gave ourselves a way to think about it abstractly.
But once you've started creating symbols to stand for number values, why stop with one and five? Why not symbols for all the other numbers values too? Well, the human brain isn't bad at understanding symbols. Put a red octagonal sign on a stick at the end of a piece of road and drivers are going to hit the brakes, even if you didn't bother to put the letters [S][T][O][P] on the sign. But there's a limit to how many symbols we can arbitrarily attach to meanings. If we tried to create a symbol for each number value we'd overload our brains before we even reached a hundred. We have to make bunches and add them (or multiply them) to keep the number of number symbols (called "numerals") reasonable.
There's more than one way to skin a cat, or what you should learn about Roman numerals in first grade and why you don't want to
The Greeks used their letter-shapes to stand for numbers too. The shape called "alpha" stood for a sound /a/ when it was used as a letter and a value /one/ when it was used as a numeral. Beta stood for a sound /b/ and a value /two/ and so forth. This worked, but it led to some confusion, as well as sidepaths like numerology and Gematria, and it isn't very useful when you want to teach first graders who are still working out what the letter-shapes stand for when they're writing, much less in math. (There's a reason that Greek mathematicians made the most progress when they were talking geometry and drawing pictures.) But you can adapt the Roman system to illustrate one of the early versions of arithmetic and number-writing even with kids.
The Romans also used letter shapes to represent their numbers, but they didn't use the letters in the order they came in the alphabet. Their system looked a lot more like hash marks. They used a vertical line [I]-- like a tally mark -- to stand for /one/, For the handful, they turned the hand sideways and left the thumb out in a neutral position (try it!) and wrote /five/ as [V]. Fewer lines, takes less space on the page. Then two hands together are two "V"s attached to each other for an [X] and that's a ten. They could have done what a lot of other peoples did and made a special symbol for /twenty/ too, but they didn't think it was all that important. (Maybe their shoes hid their toes!) Still with three symbols, the Romans could easily write and add numbers. Small numbers anyway. And because X doesn't look like V and neither X nor V look like I, you don't have to worry about mixing them up. The Romans could have put VXXIII to mean /twenty-eight/ just as easily as XXVIII. (They didn't but I'll talk about why further down.)
If a Roman wanted to add two numbers: XXII and VIII, for example, he could take all the [X]s and write them down. XX..., and then all the [V]s and write them down next to the [X]s, XXV... and then put all the [I]s at the end: XXVIIIII, like so. But of course if you've got five [I]s in a row you can turn them into a [V] which makes it XXVV and two [V]s in a row is the same as an [X] which makes it XXX.
When the Romans got up to five [X]s, they saw it as a handful again and created a new symbol. They were already using letters, so they just chose another letter for /fifty/ [L] and when they got two [L]'s together (or two handfuls of [X]s) they made [C] stand for a /hundred/ (which at least made sense since the word for /hundred/ started with "C".) Presumably they could have kept on doing this kind of bunching indefinitely, but in practice they only did it once. [D] stands for /five hundred/ and [M] stands for /one thousand/. (They did have a way of indicating even bigger numbers, but we'll skip over that.)
Now if the Romans had stuck to piling up symbols into bunches and making another symbol each time, we'd want to teach first graders to add and subtract with Roman numerals. What better way to get across two concepts? First, that you have to add like things together (ones to ones, fives to fives, tens to tens) and second, that you can take a bunch of individual things and describe them as a bunch, or take a bunch and split it into individual things again. We saw what happens when you add above -- subtraction might look like this:
If you start with XXX and you want to take away VIII, you need some [V]s and [I]s in the first number of the problem (the minuend). So you take one [X] and break it into two [V]s. Now your problem looks like XXVV - VIII. You still need some [I]s, so you take one of the [V]s and change it into five [I]s. Now your problem looks like this: XXVIIIII - VIII. Get out your eraser and erase one [V] and three [I]s from each number and you have your answer: XXII.
But the Roman scribes got bored with writing down four marks of anything and decided to take a shortcut. Instead of writing /four/ as IIII they decided to write it as IV, instead. Saves time, as long as everyone understands that now the positionof a symbol is as important to the meaning of the symbol as the shape is. An [I] before a [V] or an [X] means /take one away from the next bunch/. Now you had to write Roman numerals in a strict order in order for them to make sense.
This is too much for first graders. Heck, it's too much for some fourth graders, judging from the scratching of heads I regularly see when the topic of Roman numerals wanders into the library. But "place value" (remember place value from yesterday?) works on a similar principle. We don't know the value of a numeral in a written number unless we consider both the basic value of the symbol [4] for /four/ for example, AND its position in the number. A [4] in 345 represents a different quantity than the [4] in 45,653. "Place value" is a shorthand for a kind of addition that we do all the time. We add the ones to the tens to the hundreds to the thousands to the ten thousands and so forth and so on.
Symbols and shortcuts
Writing about math is repetitive. You have to use the same words over and over. The scribes in medieval Europe got tired of writing "minus" in the and started abbreviating it as an [m] with a sort of tilde [~] over it, and they weren't the first to take a shortcut by any means. Whenever mathematicians have figured out a way to use a few symbols to express a lot of meanings, they've grabbed it, and the position of symbols matters a lot! "Place value" is the most familiar example, but it's not the only one. And like any kind of shortcut, there's a potential for getting lost. When we say numbers or read them back from the numerals we don't articulate the whole addition problem, at least not in English, and especially not with the numbers under twenty, and as a result "reading" numbers can obscure the connection between the spoken number and the written numeral for the beginner, hiding the concept of place value.
(Next time: Tom Lehrer!)
