If you came here to find the answers to the puzzles I gave in last month's column, you'll find them further down the page.
Talking of the new year, why do mathematicians think that this is an ideal year?
I'll answer that teaser next month.
First up for 2010, however, I want to return, briefly, to a topic I covered extensively in this column in 2008. Namely, the clear distinction - long recognized by mathematicians, and much emphasized by leading mathematics education scholars going back to Piaget, and in at least one leading industrial country (Japan) mandated for teachers to respect - between multiplication and repeated addition. (For newcomers, I'll begin by saying it one more time: multiplication is not, repeat not, repeated addition. Not even for positive whole numbers. This is the one case where you do get the same answer, but getting the same answer to two procedures does not make them the same procedure.)
You'll find the story so far in Devlin's Angle for June 08, July 08, September 08, with related articles in December 08 and January 09.
The series generated a huge response in my email inbox and on the blogosphere. Some of it applauded me for creating greater awareness of several hundred years of advances in mathematics and over sixty years of research in mathematical education that had somehow failed to migrate into the everyday educational world. In contrast, a lot of it amounted to little more than people calling me an idiot or a crank and repeating, Canute-like, the mantra "Multiplication is repeated addition" over and over again. All very edifying and scientifically convincing, I must say.
After three columns, with the final one, September 08, being a long piece citing some of the scholarly research on which I based my articles, and then the two somewhat related follow-up pieces, I felt it was time to put the topic to bed as far as my involvement was concerned. But thanks to search engine technology, from time to time someone comes across my articles and writes to me about the issue.
One question that comes up frequently generally goes like this. "Sure, the repeated addition definition doesn't work for multiplying fractions, let alone irrational numbers, but can it cause later problems if multiplication of positive whole numbers is introduced (to young children learning math) as repeated addition?" Now, some of the research I cited in my September 08 column indicates that it certainly can. But the issue comes up sufficiently often that I wish I had given just one more example in my originals articles. Here it is.
For those who claim that multiplication of positive whole numbers really is just repeated addition, let me ask you how you would explain to a small child the mathematics that describes this simple activity.
Take a piece of elastic, and tie two knots in it, one near each end. Ask the child to measure the distance between the knots. Suppose it turns out to be 5 inches. Now, as the child watches, slowly stretch the elastic until the distance between the two knots is 10 inches. Get the child to measure it again. Now ask the child to write down a mathematical description of how the new length depends on (is related to) the original length. I would hope the child writes down
10 = 2 x 5In more general terms, what you did was double the length, or, as an equation:
new length = 2 x old lengthI would be very surprised if the child wrote down
10 = 5 + 5corresponding to
new length = old length + old lengthand if he or she did, you would have your work cut out trying to put them right before they have serious trouble in the math class. Sure, these addition equations are numerically correct. But so what? What you have just shown the child when you stretched the elastic has absolutely nothing to do with addition and everything to do with multiplication.
Wikipedia actually gets it right (at least it did when I checked on January 1 at 2:00 PM PST) when it says:
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic (the others being addition, subtraction and division).Unfortunately, it then immediately gets it wrong by going on to say, "Multiplication is defined for whole numbers in terms of repeated addition", which is totally at odds with the first statement, and an illustration of the dangers of the wisdom of the crowds approach behind Wikipedia. (Saying "can be defined" is a bit better - and historically correct it you go back several thousand years - but, as the research shows, it's unfair to the child to present it that way, since it will likely cause problems later. Besides being flat wrong. Multiplication is not defined in that way, and has not been for millennia. As Wikipedia's first statement says, multiplication is a basic arithmetic operation; as such, it is not defined in terms of anything else.)
ASIDE: Actually, as a factual observation about the world we live in, at least in our cocoon-like United States, that second part of the Wikipedia entry (the "multiplication is defined" statement) is all too frequently distressingly correct: it often is defined that way. If you want to bring the repeated addition connection in to the multiplication scene (and it surely makes good educational sense to bring it in pretty soon, or even better guide the kids to discover it for themselves), then the correct statement is that, "Multiplication gives you a fast, efficient way to calculate the answer to a repeated addition sum." (Saying it the other way round is inexcusable. Repeated addition is most definitely not an efficient way to do multiplication. If you don't believe me, let's have a timed competition to multiply 1,000 by 999.)
Still, for all its failures, Wikipedia is somewhat better than many of the other websites you come up with when you google "multiplication", not least one by an entrepreneur who is savvy enough to name his company in a way that will likely get his product at or near the top of your search (full marks for that, Sir), but whose product gets multiplication flat wrong from the getgo, and who may be setting himself up for a class-action lawsuit when users of his product subsequently find themselves with real problems in math due to such a misleading start.
Okay, enough of that. Now for the answers to last month's puzzles.
"Of the birds that remained, a third were finches, a quarter were budgies, a fifth were canaries, a seventh were mynah birds, and a ninth were parrots."
However, the reporter got one of the fractions wrong. How any parrots were left?
(This problem comes from Fibonacci's Liber abbaci, published in 1228 (the surviving second edition), the book that brought breeding rabbits and Hindu-Arabic numbers to western Europe. The "denaro" was a unit of currency in Medieval Italy.)
Let R be the number of birds that remained in total, and let F, B, C, M, P be the number of finches, budgies, canaries, mynah birds, and parrots that were left. Suppose the reporter had got all the facts correct. Then we would have:
F = R/3, B = R/4, C = R/5, M = R/7, P = R/9So,
R = 3F, R = 4B, R = 5C, R = 7M, R = 9PIf correct, these five equations tell us that R is a common multiple of 3, 4, 5, 7, 9. So R is at least 4 x 5 x 7 x 9 = 1260.
But we know the reporter got one fraction wrong. Thus the most we can know is that R is a common multiple of four of the numbers 3, 4, 5, 7, 9.
What are the possibilities? The common multiple of 3, 4, 5, 7 is 420; that of 3, 4, 5, 9 is 180; for 3, 4, 7, 9 it is 252; for 3, 5, 7, 9 it's 315; and for 4, 5, 7, 9 you get 1260. Thus, depending on which number the reporter got wrong, we can conclude that R is at least 420, 180, 252, 315, 1260, respectively.
There were exactly 300 birds at the start, and over 100 birds escaped. So the number of the birds that remained is less than 200.
Of the different possible minima for R, only 180 is less than 200. So R has to be 180. (In order words, the reporter got the fraction of minah birds wrong.) So the number of parrots that stayed is 180/9 = 20.
So much for the first puzzle. What about Fibonacci's teaser?
Let x be the number of partridges, y the number of pigeons, and z the number of sparrows. Then the information you are given leads to two equations:
x + y + z = 30 (the number of birds bought equals 30)Hmmm. As everyone learns in the high-school algebra class, you need three equations to find three unknowns. So what to do next?
3x + 2y + z/2 = 30 (the total price paid equals 30)
Well, in this case you have one crucial additional piece of information that enables you to solve the problem. First, double every term in the second equation to get rid of that fraction:
x + y + z = 30Subtract the first equation from the second to eliminate z:
6x + 4y + z = 60
5x + 3y = 30
Notice that 5 divides the first term and the third, so it must also divide y. So y is one of 5, 10, 15, etc. But y cannot be 10 or anything bigger, since then it could not satisfy that last equation! Thus y = 5. It follows that x = 3 and z = 22. Neat, eh?