Math Symbols Are NOT All Created Equal

I have a friend named Dave Refro who writes and edits questions for one of those high stakes tests that is used for admission into certain graduate programs and uses his job as an excuse for his fascination with archiving old math journal articles. Some folks garden, Dave archives. He spends hours pouring through journals and abstracts and fits together articles with a common theme. If you read almost any math discussion on line, you will probably have come across one of Dave's responses to a question with numerous links to how the question was addressed, discussed, and argued over through history.

Fortunatly for me, Dave sometimes finds an article that he thinks might be of interest to me, and when he gets a stack of them, I get a big present in the mail and my wife knows I will be taking my meals in the den for a few days. In a stack of journal articles he sent recently, (THANKS Dave!) there was one particularly interesting article by Florian Cajori from 1923. In the article Cajori points out two interesting things about the equal sign that every one uses; and that is one of the interesting things he points out, is that EVERYONE uses it. Even in 1923, it was one of the most ubiquitous math symbols in the world and today there are still only about four math symbols that you could write and they would not only be understood, but written exactly the same way whether you found yourself in darkest Africa, the Far East, or downtown Los Angeles. It seems like the perfect symbol, and as Robert Recorde said when he created the symbol in 1557 in his "Whetstone of Witte", the use of "a pair of paralleles, or Gemowe(twin, from the same root as Gemini) lines of one length ... bicause noe 2 thynges can be moare equalle." In fact, Recorde's equal sign had much longer lines than is common today, sort of like == but longer .

Indeed, one wonders why it hadn't been thought of years before, and assume that it immediatly became the most common of mathematical notations....ahhh, but not so. The other thing Cajori commented on that I think would surprise young students is that it took over a hundred years for the symbol to become accepted. So what symbol did mathematicians use before the good old == signs? Well, many of them used nothing. The early development of algebra occurred with a very rhetorical approach. When people wanted to write 7x+5 = 26, they would say," the product of seven and some quantity when added to five will equal twenty-six." Ok, they probably said it in Latin, and sometimes they did write numbers in place of the words for numbers, but for equals, they often wrote out the Latin aequales or some variation of it. Frequently they used abbreviations instead of full words and so "p" would stand for plus and "m" for minus...and they would shorten aequales as "aeq" or just "ae". By the time that Recorde had his inspirational stroke, lots of other people had decided THEY had a really good symbol. A pair of vertical lines, ||, was used by Xylander (Wilhelm Holzman) in his translation of Diophantus, Arithmetica only a few years later, and Regiomontous had used a single horizontal line for equality almost a century earlier. Descartes used a which was probably drawn from the "ae" abbreviation for aequalis. Descartes symbol became a popular competitor on the continent, finding favor with Huygens and the Bernoulli's, while many of the he English mathematicians, Wallis, Barrow, and Newton, followed Recorde's lead. Others used the "gemowe" lines of Recorde for other meanings, Descartes used it to mean +/- in his Geometrie, and Johann Caramuel used them where we would use a decimal point, so Pi would be 3==1415 etc.

So what brought the divided world into a common accord? It took a brand new idea, a reveolutionary idea, the calculus. As if by divine providence, the two great minds that created the calculus, almost in unison, tended to publish their versions with a common symbol for equality, Recorde's "gemowe" lines. They disagreed on almost every other symbol they used, but in the last half of the 1600's and the early 1700's the = sign rose to world dominance. In Cajori's words, "The fact that both Newton and Leibniz used Recorde's symbol led to its general adoption."

If I can get my students to understand how long and difficult it is to get mathematicians to accept a symbol, perhaps they will not be too surprised if their College Prof goes into a rant when they use the symbol "ln" for the natural log... and if they accept that the symbol exists (honest, they don't all accept it's use), I can't begin to imagine how they will react if you pronounce it differently than they would. Wait for them to say it first!

Proof Without Words

I have stated previously how much I like "napkin" techniques that give quick calculations or estimates of a problem. I also like things like visual displays which essentially prove some mathematical idea. The one at the top of the page is from the cover of Roger Nelsen's "Proofs without words II.." which is way too expensive for a paperback, but I will probably break down and buy it.

The problem, is that, except for mathemtaiticans who already know the proof, it seldom convinces "without words". Most of my high school students will not look at this image and be able to explain easily and clearly why it shows that the limit as n goes to infinity of 1/4 + (1/4)² + (1/4)³+ ... +(1/4)ⁿ is equal to 1/3. If I'm wrong, not generally, but in your particular case, then stay with me and read what's written, and tell me if you see it the way I do.

I think they will be able to see that the triangle is divided into thirds... by the colors, 1/3 purple, 1/3 orange, 1/3 white. But I don't know if they can see the series of powers of 1/4 going off to infinity. That's why they have high school teachers... and so here are some words to help make it more "visual"

Look at the largest white triangle... can you see it is 1/4 of the largest (outside) triangle?... Ok, now look at the line across the top of the biggest white triangle, it connects the midpoints of two legs of an equilateral triangle, so the triangle above this medial segment, the one with multiple smaller triangles in it, is exactly congruent to the Biggest white triangle and is 1/4 of the total area of the outside triangle also. This upper triangle is a scale model of the original outside triangle,with all the same colors in the same positions and the white triangle in it is 1/4 of the area of this upper replica. So the second largest white triangle is 1/4 of the area of the Largest white triangle.... its area is 1/16 or (1/4) ². Now the line above the second smallest white triangle is a medial segment of the upper triangle, and so the triangle above it, which is also a scale model of the original biggest triangle, is also 1/16 of the total area... and the third smallest white triangle, is 1/4 of that, so its area is 1/4 of 1/16 or (1/4)³... OK, now you see it, and as you move out each white triangle is 1/4 of the previous one... and sure enough, all the white triangles add up to 1/3 of the total area.

If you are one of my students, next year remind me to say "similar triangles" a lot...if you are going to be in calculus, don't worry, we will

More Stuff That is “Easier Than You Think”

A couple of days ago, I mentioned that I had been shown how to test divisibility by seven, and by using the same method, I could develop a similar approach for any of the small (but difficult primes).. So today I want to show you how that works, and if you read carefully, you will be able to mentally determine if any number is prime up to 9000 ....go ahead, impress yourself.

first, how the seven rule works…. I’m going to talk about what big people call modular congruence, but don’t let that scare you. It just means that two numbers have the same remainder when divided by some specific quantity (in the first case, seven). So 9, 16, 23, etc are all congruent mod seven because they all have a remainder of 2 when we divide them by seven… see…easy..and if they are congruent to zero mod seven, that means they have no remainder…they are divisible by seven

Now a rule that should make sense if you think about it; is that any two numbers that have a congruence of zero, when added together will have a sum that is congruent to zero… In mod seven we could think of 14 and 21. Both are divisible by seven, so if we add then together we get another number divisible by seven. I know it is hard to believe you can come up with much from such a simple rule, but watch.

Lets think of some number n, and for a moment let us break it into two parts, the number made up of all the digits except the units digit, call that one A, and the units digit, call it B. Then N = 10A + B… As an example with 324 we would say A is 32 and B is 4 so 324 = 10( A) + B…. got it?

Now we don’t know if N is divisible by seven, but we notice that if we take 2 times N it will equal 20 A + 2B…. yeah, I hear you saying “So, what?”… patience. Now 20 A is equal to 21 A – A…. you know that… so we can write 2N =20 A + 2B = 21 A – A + 2B. NOW, we have something, because 21 A is divisible by 7, and so if –A+2B is divisible by seven, then N must be also by the rule above in italics. Now it doesn’t matter if –A+2B is positive or negative, so since A may be a multi-digit number it would be easier to do A-2B. So we want to know if 2947 is divisible by seven (it is). We take 294 (A) and subtract 2 x 7 and get 280. Still a big number so we can do it again. 28 – 2x0= 28…and HEY, I know that is divisible by 7, so the original number, 2947 must be.

Ok, can we make that trick work for other primes. We can skip 11 because it already has an easy divisibility rule… (see casting out nines and elevens). What about 13? Well 3x13 is 39, which is almost 40. That is the secret of these prime tests; we want a multiple that is one more or one less than an even multiple of ten (one less is easier as we will see, no subtraction). So we go back to N=10A + B, and we want to know if it is divisible by 13… Well, set 4N = 40 A + 4B and that is the same as 39A + A + 4B. Now the 39A is a multiple of 13, so if the A+4B is a multiple of 13, then 4N, and hence N is a multiple of 13. So we just use the rule A+4B to test a number…. Like 403. 40 + 4x3 = 52… and if you didn’t recognize that as a multiple of 13, you could do 52 as 5 + 4(2) = 13..hey, we ALL recognize that is divisible by 13.

Here are some tests for other primes I worked out by the same approach, and they are pretty easy to remember by just two rules of thumb. Find a multiple of the prime that is one more or one less than a multiple of ten. If it is one more then you have to subtract some multiple of B from A, and if it is one less, then you add some multiple of B to A. And the multiple of B, it is just the number of tens that we are one more or one less than (don’t get confused, it is NOT how many times we multiply the prime to get there). Try making sense of how these were done, then you can make your own for even higher primes (although they start to have some big multiples, but 39 should be an easy one)…

Test 7 by using A- 2B
Test 13 by using A + 4B
Test 17 by using A- 5B
Test 19 by using A+2B (see, it is already just one less than twenty)
Test 23 by using A+ 7B (OK, that is the hardest one in the group, sorry)
and test 29 (one less than 30) by using A+3B.

Any number less than 9000 must have a prime factor less than 30 if it is not prime, since 30² = 9000, so if you come across some big number like 8531 and wonder if it is prime, just try the rules one at a time.

3 sevens is 21 one more than two tens so the rule is A-2B… 853-2=851… 85-2 = 83 and 7 won’t go into that evenly so it is NOT divisible by 7.

Not divisible by 11 so we try 13, 3x13 = 39 which is one less than 40, so we add A + 4B… 853+4 = 857… 85+4(7)= 109… 10 + 4(9) = 46…nope, 13 won’t go into that… on to 17

3 x 17 is 51 so we need to subtract A – 5B…… 853-5(1) = 848… 84-5(8)= 44, nope 17 won’t go into 44 evenly… let’s try 19.

19 is already one less than 20 so we test with A + 2B… 8531.... 853+2(1)= 855 .... 85+2(5)= 95 ... 9+2(5) = 19…hey, it is divisible 19, and in fact, 8531 = 19 x 449. ….”easy peasy” as the British folk say.

Collateral Damage in the Math Wars

I'm a victim of friendly fire in the math wars...wait, scratch that... I'm a victim of used-to-be friends fire.... you apparently have to be on one side or the other, and recently I am catching hell from both sides.

I guess it is my own fault. For the last ten-plus years I simply refused to enter into any dialog involving the "New Math/intuitive math/discovery math" camp on the one side and the "gimme that old time religion/long division or die/calculators are evil" camp on the other. The problem, at least as I see it, is that each group is assuming that it has to be all one or all the other. As one blogger put it, it is a choice between "why" vs. "how".(No one apparently envisions that a teacher might actually try to teach both how AND why.) California has fallen aside as the chief battleground after a victory, more or less, by a coalition of anti-reform elements (many of whom would not be caught dead championing the excess of the most conservative among them), and now Washington state seems to be the new battleground..(see A University View)

I can understand parents getting upset... as one observer of the process wrote, "Parents .... don’t understand what the specific tasks are, or how to perform them, or even why their kids are being taught this way instead of the older one. And again the proponents of this style (rehashing the why vs. how discussion) come off as arrogant. Their concerns aren’t for the parents’ ability to follow along with what their children are learning — and something tells me these are the same educators who insist that parental involvement is key — but just that the parents aren’t screwing up all their hard work." I admit I was shocked to read the following from the teachers resource packet for a program called "Everyday Learning"...
"The authors of Everyday Mathematics do not believe it is worth students’ time and effort to fully develop highly efficient paper-and-pencil algorithms for all possible whole-number, fraction, and decimal division problems. Mastery of the intricacies of such algorithms is a huge endeavor, one that experience tells us is doomed to failure for many students. It is simply counter-productive to invest many hours of precious class time on such algorithms. The mathematical payoff is not worth the cost, particularly because quotients can be found quickly and accurately with a calculator." I am reminded of a quote from my wall at school,” For every problem there is a solution which is quick, easy, and wrong! "

For me the intuitive learning and process learning support each other; the kids learns the multiplication algorithm, then he sees in algebra that two digit multiplication is the same as the "foil" method he is learning, and the rigor and intuition feed each other. One guy calls it the Mr. Miyagi method (from the Karate Kid movies). OK, that means teach the algorithm, but don't expect every kid to achieve 100% accuracy on the most difficult problems in the fourth grade, and then show them the intuitive approach that helps them understand WHY the algorithm works

Strangely, most of the good HS math teachers I know support a mixture approach; learn the long division algorithm, learn to multiply fractions, then use the mastery to understand concepts and big ideas..(see"Easier Than You Thought")Education Professors keep pushing the idea of concept based learning without foundation training (I think that is wrong) and Ultra conservatives keep pushing the abstract calculation as if mentioning a use of math was a sin (I think that's wrong too, most of us learn math to use it, only a very few use pure math untainted by application, and then when the least expect it, we find an application for what they did.)

So now I have two ex-correspondents who have cut me off, each because I endorse their view too weakly, or more likely, because I am too tolerant of the opposition viewpoint. But I will go on teaching long division and multiplication, factoring and all the other taboo subjects and trying my best to balance them with an intuitive understanding of why these ideas work, why the are actually esthetically beautiful ..at least until some administrative zealot of one or the other side of the math wars tells me to pack my bags and leave. But even them, look for me in your public park standing on a soap box selling my "evil" to anyone who will listen. But if you get caught in the Math wars, my best advice is from those old Uncle Remus stories by Joel Chandler Harris..."Ol’ Brer' Fox, he don't say nothing. He just lay low."