Still on the Quest

Why did I spend a whole essay on place value? Because it's the most inescapable example of a simple truth -- the written language of math is not the same as the spoken language of math. The history of math is a history of symbols, and in order to keep the number of symbols to a manageable level, position and context have become as significant as shape. Compared to mathematical notation, even the jumble of written English is positively straightforward. We might have more than one way to write the sound /f/, but at least "ph", "f" and "gh" always do stand for a sound, and you always read from right to left. If writing English was like writing numbers, the "k" in "ark" would stand for a phoneme, the "k" in "eke" would stand for a syllable and the "k" in "key" would stand for a whole word. And that's without a whiff of grammar!

Take a "2" and put it under the line in a fraction, it means "halves". Put it in superscript on the left hand side of the "x" and now it's an exponent -- which is an abbreviation for multiplication. Leave it standing alone and you've still got the option of interpreting it as "two over one". And of course, it changes value by order of magnitude in 182, 821, and 218, and that's before we even kick in a decimal point.

Along with the number symbols we've got + and - for addition and subtraction, which are both (fairly) consistent. But multiplication and division can be indicated in (ahem!) multiple ways.

3 x 4 is the same as 3*4 is the same as 3(4).

3 ÷ 4 is the same as 3/4 is the same as 3:4 and I don't know how to put a long division symbol in, but you'll have to believe me...


Adding to the fun, there are a few exceptions to the general rules. Mixed numbers don't just confuse us when we're learning about fractions and we get told that putting the whole number next to the fraction means adding, they come back to haunt us in Algebra, when we're told that juxtaposition means multiplication. So is "2" set down right next to a fraction that reads "x/y" equal to "2 + x/y" or "2x/2y"?


Don't like Algebra? I know whose name you should be cursing!

François Viète. (He published under the Latinate version of his name, Franciscus Vieta, which I'll use because it's faster than adding the special characters.)

Vieta was an amateur mathematician, and rich enough to pay for publishing his ideas, and luckily for us, they were mostly good ideas. In 1591 he published a treatise called In artem analyticam isagoge. In it he suggested that a regular system of notation be used to distinguish the true unknowns in an equation from the "coefficients" -- the numbers which would be known in a specific equation, but are undetermined in a general equation.

Like this...

4(unknown) + 7 = 35, 9(unknown) + 5 = 50, and 220(unknown) + 100 = 300 are all specific equations, but they all have a similar pattern and are solved for the unknown in the same way, (by subtracting the second term from both sides and then dividing both sides by the coefficient.) Vieta suggested using vowels for the unknowns and consonants for the knowns. So all three of those equations would look like BA + C = D. Fifty years later, Descartes used the same principle, but different letters -- and we still use Descartes' convention today. Letters near the beginning of the alphabet stand for knowns, letters near the end stand for unknowns. The modern general form of the equation would be ax + b = c.

A modern mathematician, told to "solve" ax + b = c wouldn't even begin to think of asking if that meant solve for "a" or "b" -- the only unknown there is the "x" -- the other letters stand for things which are unspecified. It's obvious! Isn't it? Just like it's obvious that black is the color of mourning! Er... well, except in places where the color of mourning is red, or yellow, or purple... That's the trouble with conventions -- they're artificial, and they need to be explained to the newbies. (Who are usually busy trying to grasp fifteen other concepts at the same time...)

But the really dreadful thing about Vieta's contribution to the world of mathematics is the effect it had on what was considered possible and what was considered impossible. Paraphrasing from Dantzig, prior to Vieta, most medieval algebrists could divide equations into "possible" and "impossible" depending on what they knew about the numbers involved.

x + 4 = 6 is possible, x + 6 = 4 is impossible.
2x = 8 is possible, 2x = 5 is impossible.
x² = 9 is possible, x² = 15 is not.

That's because before Vieta, negative numbers weren't considered fair play by anyone who wasn't a banker, numbers could only be divided by numbers which went into them evenly, and only perfect squares had square roots. And sure enough, at first the general equations made in Vieta's literal notation had little qualifiers. x + a = b if b is greater than a, ax = b can only be solved if b is a multiple of a, that sort of thing.

But if you're looking at an equation which is all made up of symbols, there's no obvious reason why those limitations should remain. It's faster, easier and better to change your ideas about what numbers are allowed to "exist". Bonus points if changing your ideas about numbers mean that some intractable problems from antiquity are suddenly solvable.

x = a - b works fine if you allow for negative numbers.
x = b/a is no problem at all if you don't mind your answer being a fraction, improper or proper, rather than an integer.
And the square root of a number that isn't a perfect square is acceptable if you don't mind being a little irrational about the whole thing.

Vieta's idea -- using symbols to stand for undetermined numbers as well as unknown quantities -- freed mathematicians from having to use words to describe what they were thinking about, and at the same time, elevated the symbols into things which could be manipulated in the same way that numbers were. You might want to define what the symbols stood for at the start of the problem, but once you had, you could gleefully ignore those definitions until you'd finished manipulating the symbols around.

Of course, the same economy of symbols which led to place value also led to mathematicians preferring to use old symbols rather than create new ones. Some new ones got created, of course -- the radical for square roots, which I can't construct in html, was one -- but for anything which indicated a sequence the number symbols were already around. (Like exponents!) And since they looked like numbers, some bright bunny started to think about whether they could be manipulated and... well, pretty soon you've got a page of symbols that ordinary mortals look at and think "I don't even know how to begin to read that out loud!"

(Not that reading equations out loud sounds much like what they mean anyway. Who on earth looks at a² + b² = c² and says, "The square of the hypotenuse is equal to the sum of the squares of the other two sides" without a lot of prompting?)

Manipulating symbols around lets mathematicians to leave the concrete entirely and play in the abstract world. Ironically, it's the ability to play in the abstract world which has created the most useful applications of math back in the real world. Vieta freed math from the tyranny of words, but in the process things got more than a bit confusing.

If it's consolation, even people who like math get that feeling when the waters are deep. Dantzig took a look at what Bertrand Russell and Alfred North Whitehead were doing with symbols and said " ... my repeated efforts to master their involved symbolism have invariably resulted in helpless confusion and despair."


Next time, Math Words

If you've never heard Tom Lehrer's song, "New Math", you can pop over to YouTube and give it a listen -- this is the most complete version -- even if that isn't Tom Lehrer mouthing the words. There's also a fun bit with Ma and Pa Kettle that is enlightening too.

There's an ongoing argument, at least in America, about how to teach math. It swings back and forth from time to time, but essentially comes down to folks on one side saying that children should understand what they're doing, and folks on the other side saying that children should be able to get the right answer. Near the middle of the last century, this argument led to a huge change in how most teachers taught arithmetic. In the Old Math, you memorized your addition facts up to ten plus ten (or twelve plus twelve) by chanting them over and over in first grade. You also did a lot of problems where you added things together, and between the chanting and the examples, if you were lucky you began to get a handle on adding, and if your teacher had you solve problems like: "5 + ? = 7" you actually were likely to begin to get a handle on subtraction too.* Gradually the numbers in the problems got larger, and you learned about "carrying the one" and adding more than two numbers together, and by second grade you were ready to do subtraction and even bigger sums, and begin to work on learning your times tables and manipulating simple fractions. But in New Math, the emphasis was on discovering how the arithmetic worked. Which you would think made things easier, but noooo....

Confusion in the Classroom, or Why Place Value is a Critical Concept.

I got onto this math kick because of a question over at ozarque's journal. And her experience of being lost in the second grade makes me think that she was on the cusp of the "New Math" which Lehrer skews so neatly. To save myself some typing, I'll quote a note I sent her.

Thinking about your numb spot, and I'm guessing of course, but I think that what your teacher did in second grade was equivalent to yelling at a kid who had learned how to read by the whole word method for not being able to sound out a familiar word – and without anyone ever mentioning to the child that letters stand for sounds, either.

If it was 1942, you were on the leading edge of the movement that eventually got called "New Math", where a lot of emphasis was put on the idea of "place value". If you'd ever heard Tom Lehrer's song about New Math, you'd recognize it, and probably think it was even funnier than I do. I was in second grade in 1965, and by then it was completely entrenched in the schools. But by 1965, the teachers really understood it, and I'm betting that your teacher didn't.

You see, if I'm right, when she was trying to teach you how to subtract by borrowing and carrying, she was also meant to be teaching you about place value – and I'll bet you a nickel she never made sure that you really understood how numbers are put together. By second grade, you had already internalized what are called "number families" now. 4+5=9, 5+4=9, 6+7=13. 7+6=13, etc, all the way up to 10+10=20. Your first grade teacher probably had you chant the "addition facts", and if you were a student who learned best by hearing, you had them solid. But you didn't think about why a written ten had two digits and a seven had one. They just did.

That you both spoke English only complicated the problem. In some languages the numbers between ten and twenty translate to ten-and-one, ten-and-two, ten-and-three, and so forth. But in English those 9 numbers have specialized names. Eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen. And all the number families you need for subtraction fall under twenty. Who needs place value?

American money didn't help. If we only had pennies and dimes and dollars, you would have figured out place value just from counting money. But quarters and nickels obscure the decimal system. When it came to money, you just continued the strategy you used to twenty, learning through repetition the "change" from 100 for any amount below that. 63+37=100, so a subtraction problem that looks like money is as easy as falling off a log. Percents make sense to you because they look like a dollar in pennies. Money is often written as decimals, which gave you a handle on decimals that works well enough for most things, and you get by, at least for adding and subtracting and multiplying.

But long division requires you to understand place value. And the easiest way to convert a fraction or ratio into a decimal number is long division. Decimal numbers were definitely not invented by anyone with a numb spot for place value when it comes to math. And statistics are full of decimal numbers because they're full of ratios. Ratios are comparisons of two numbers, but they're also division problems.

Now I was trying to figure out how someone who is in the habit of using number families would divide. Paralleling the "subtracting by adding" strategy I get to dividing by multiplying... Is this similar to your your warble for division?

3585 ÷ 75 =

75 x 10 is 750 and 750 x 2 is 1500 so that's 20.

1500 x 2 is 3000 so that's 40

75 x 6 is 450 but we can get one more 75 to get to 525 so the answer is 47 with a remainder of 60 (because 25 plus 60 equals 85.)

And then I stop, and stare at the remainder and say "Now how do I turn that into a decimal number?" and bump up against MY numb spot. I just can't imagine how to do it. Time to haul out a calculator.

Ah, well...

Prior to the twentieth century students also learned how to do "compound addition." There's a lovely snarky description of it over here that I don't think I can top, but essentially (in case that link dies) compound addition is the process of adding compound numbers -- i.e., numbers where the relationship between the "columns" isn't related to the numeration system you're using.

Here's the most important bit from the link:


So, you need to solve 18 + 12 + 15. You begin by putting together 8 + 2 + 5 = 15. You then divide 15 by 10 and get a quotient of 1 plus a remainder of 5, so you put down the remainder and carry the quotient into the tens column, making you then add 1 + 1 + 1 + 1. The answer is thus 45 apples.

This is what you do every time you add things up manually instead of using a calculator or computer to do it for you. What's with the dividing and remainder business? Well, we just so happen to have a system of numbers that when any two digit number is divided by 10, the first digit is the quotient and the remainder is ten. Makes things quite simple, wouldn't you say? That's why this is simple addition.

The thing is, this works because each column of digits is ten times bigger than the column to the right of it, forever in both directions. You can apply this same principle to bases other than ten, which I'll leave up to you as it's such a simple manner. Just replace "divide by ten" with "divide by ". The more interesting thing is when you want to add numbers that aren't regular like this... that have weird relationships between each of the columns.

(Good grief, but I wish I'd had the "quotient and remainder" explanation when I was doing problems in base 12 or base 8. It would have made life soooo much easier.)

Here's a quick example of compound addition:

2 weeks, 3 days, 17 hours, 45 minutes, 22 seconds plus
3 weeks, 5 days, 12 hours, 26 minutes, 45 seconds

You could convert everything down to seconds, add and then convert back out to weeks, days, hours and minutes, but you'd want a lot of paper or a calculator. With compound addition you can do it in your head.

22 seconds plus 45 seconds is 67 seconds, divide by sixty for minutes get 1 and 7 left over. Carry the one and put down "7" in the "seconds" column.

For minutes: 1 + 45 + 26 = 72 divide by 60 to get hours, get 1 and 12 left over. Carry the one and put down a "12" in the "minutes" column.

For hours: 1 + 17 + 12 = 30 divide by 24 to get days, get 1 and 6 left over. Carry the one and put down a "6" in the "hours" column.

For days: 1 + 8 + 5 = 14, divide by 7 to get weeks, get 2 and none left over. Carry the two and put a "0" in the "days" column.

For weeks: 2 + 2 + 3 = 7.

The answer is 7 weeks, 6 hours, 12 minutes, 7 seconds (with the "0 days" understood by its omission.)

Now Isaac Asimov found a Revolutionary War period book called "Pike's Arithmetic" which had lots of examples of compound addition, and other kinds of compound math. Here's a sample problem I found on the web:

Bought 9 Chests of Tea, each weighing 3 Cwts. 2 qrs. 21 lb. at 4£ 9s per Cwt what came they to?
Here are some conversion factors: 4 qrs (quarters of a hundred weight) = 1 Cwt (hundred weight) ; 28 lb = 1 qr ; 20s (shillings) = 1£ ; 12d (pence) = 1s.]**

Not exactly easy, is it?

Abraham Lincoln learned his numbers from Pike, which tells you how long the book was popular. And Asimov used the book as part of an argument in favor of going over to the metric system. His essay was called "Forget It!" and was largely about how not having to convert pounds to shillings to pence made life easier for us, so why did we still convert pounds to ounces, etc.? This, Asimov said, would make teaching math much easier.

Asimov was wrong. Well, fudging. First of all, he described Pike's Arithmetic as an "elementary" textbook, when in fact it was the only math book a student would need from childhood until just before college. And at 500 pages it included lots of example problems, many of them very difficult, because it was the only book on math a student might have all those years. Second of all, Asimov failed to see the utility of compound addition. Do a lot of compound addition, and place value just becomes a simpler form of it. And you need place value when you do long division.

Let's take our answer from the addition problem and say it represents the amount of time that can be spent at a certain website. 8 children want to play there, so to apportion the time out evenly we need to divide the time by 8 as closely as we can.

7 weeks, 6 hours, 12 minutes, 7 seconds divided by 8.

You can't take 8 into 7, so you convert the 7 weeks into 49 days. 8 into 49 goes 6 times with 1 left over. Put down 6 in the "days" column, and convert the 1 into 24 hours and add it to the 6 hours already there. Next you divide 8 into 30 hours, which is 3 with a remainder of 6. Put 3 in the into the "hours" column and convert the 6 into 360 minutes. Add that to the 12 minutes you began with. Now it's 8 into 372 minutes goes 46 and 4 left over. Put 46 into the "minutes" column, convert the 4 into 240 seconds. Add the seven seconds we began with. 8 into 247 goes 30 and 7 left over. Put "30" into the seconds column and just don't tell the kids that there are 7 seconds going begging.

The answer, to recap, is 6 days, 3 hours, 46 minutes and 30 seconds, with a remainder of 7 seconds.

What has that to do with long division?

Well, it has to do with how we write numbers down. If someone writes 945, it's actually shorthand for nine hundred, plus four tens, plus five. Where if the same person wrote 495 it would be shorthand for four hundreds plus nine tens plus five. If we wrote out regular the same way we wrote out the time values it would look like this when we added:

9 hundreds, 4 tens, 5 ones plus

4 hundreds, 9 tens, 5 ones

Five ones and five ones is ten. Ten divided by ten (the conversion factor to the tens column) is one and no remainder. Put a 0 into the ones place and carry the one into the tens column. One ten and four tens and nine tens is fourteen tens. Fourteen tens divided by ten (converting to the hundreds) is one and four left over. Put the 4 into the tens column and carry the one into the hundreds column. One hundred and nine hundreds and four hundreds is fourteen hundreds. Fourteen hundreds divided by ten (converting to thousands) is one and four left over. Put a 4 into the hundred column, and carry the 1 into the thousands column. The answer is

1 thousand, 4 hundreds, 4 tens, and 0 ones. Or 1440.

Lets divide our answer by 5.

I look at 1 thousand and I can't divide it by five so I convert the thousands into hundreds and get ten hundreds to add to the four I've already got in that column. 5 into 14 hundreds is two and four left over, so I put a 2 into the "hundreds" column and convert the remainder of four into tens. Forty tens added to the four I already had is 44 tens and five can go into that eight times with four left over. I put an 8 into the "tens" column and convert the leftover four into ones. Forty ones plus zero is still forty and five goes into that eight times with no remainder so I put an 8 into the "ones" column and I'm done. The answer is 288.

Because we use the shorthand of place value to indicate a conversion factor of ten in both directions from column to column, we can line up the columns and do long division by a rote procedure, getting the answer correct and baffling anyone who doesn't understand why the 2 had to go into the hundred column and not the thousands column. But we're really doing the same thing we did before with compound numbers. We add by starting with the smallest bunches and making bigger bunches, and we divide by starting with the biggest bunches and breaking them down into smaller bunches so that we can start parcelling things out into shares.

(Next time, it's all about the shorthand, man...)

*Modern methods often eschew memorization of addition facts, preferring manipulatives, and teaching "easy" parts of the table first. But I like the older method. Just as it's easier to put together a puzzle when a copy of the picture you're trying to build is lying under the pieces, it's easier to understand how to put together the facts you learn from the manipulatives and problems when you've got a framework to place them in.

**So, doing the multiplication out first:

27 Cwts, 18 qrs and 189 lbs. converts to 27 Cwts, 18+6 qrs and 21 lbs. which is 27+6 Cwts, 21 lbs. or 22 Cwts, 21 lbs. So for the 22 Cwts you get 88 pounds 198 s or 31 pounds and 18 shillings. For the 21 lbs, hmmm. 1 Cwt = 112 lbs, so 21 lbs is 3/16 of a Cwt. 4 pounds 9 s = 189 shillings = 2268 pence times 3/16 = 425 pence and a farthing. which is 21 shillings, 5 pence and 1 farthing, which is 1 pound, 1 shilling, 5 pence and 1 farthing, added to the 31 pounds 18 shillings we had before we get 32 pounds, 19 shillings, 5 pence and 1 farthing.

One, 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.)

( Handfuls, bunches, and the beginning of numbers )

How 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.

( Counting is not Adding )

Okay, so you now know that dividing fractions works the way your teacher said it did, but you still don't understand WHY it works.

( Well, I can tell you that too. )

(or what happens when I'm having a week I don't want to think about:)

I can pretty much guarantee that whenever I get a young teenager in the library who is desperately trying to make sense of pre-algebra or algebra, a little discussion will show me that the child doesn't understand dividing fractions. (Given, of course, that the kid isn't still trying to count on fingers, but that's a rant for another day.)


Now, I believe that there are people who are "number blind", in the same way that people can be color blind. And for those folks, a thousand different explanations will only make a slight dent. They might understand for a while, but the comprehension won't stick for long. But for a lot of folks – kids and adults – the trouble is that the explanation which would make sense of dividing fractions hasn't been given to them yet. And they need an explanation, because dividing fractions is, as the present jargon puts it "non-intuitive."


( You can see what's coming, can't you? )


( Cut to the chase )