Dear Verity

Abstract

Ms. Verity Keane Local University U.S.A. Hi ,Verity- Sorry I haven't been in touch for a while, but school is very hectic. In addition to class in elementary mathematics methods, I am also taking a course on teaching elementary music. Have you ever played a record-er? Trust me when I tell you that you'd rather not hear me playing one. Even my dog goes into other room when I do. That's why I am so thankful for math; it is so much easier. Well, at least that's what I used to think. We in class today, and you will never guess what happened! First, let me say that I do know how to multiply. Well, at least I have mechanics of it down straight, that sort of thing. My educational experience in this area was the facts - also affectionately called tables - all done by rote memorization over lots and lots of time, until they were sufficiently drilled into our heads. We could practically recite them in our sleep. But in class today, boy, did I get a shock! And we have an assignment to do on it, so there may be another delay before I write again. Our professor was smart. He didn't say very much. He just quoted some research by Thomas O'Brien and Shirley Casey, who had looked at children's understanding of multiplication.1 Evidently, when these children were asked if they 6 x 3, they all responded 18. But when asked to set 6 x 3 in a real-life context, they couldn't do it. Well, about 74% couldn't. So our professor tested it out on us. Here we are in a college-level mathematics methods class for future educators, with our average age being about 29+, and he asks if we know 6 x 3. Can you believe it? Of course we know - it's 18! I have to tell you, we were a bit put off. Then, instead of telling us answer, he had us get into our cooperative groups and come up with a situation for 6 x 3 =18. That was fun. My partner and I had a bit of a heated debate over what to do, but we settled on students with three Twinkies each. You know, lots of It was obvious, really, and didn't take us long, so we had a little chat. But here's kicker. After all that, our professor never asked us what we had decided. Honest, he never asked! Instead, he had us look up meaning of multiplication in our dictionaries that we had brought with us. For heaven's sake, Verity, we all knew it was six lots of three. No dissent. It is not something you vote on. Mathematics is mathematics is mathematics - all over world! Why waste time looking it up? So we all looked up in dictionary, and wow! You should try it. What a shock! Well, actually at first it didn't help much, as Living Webster Encyclopedic Dictionary defined as the act or process of multiplying; state of being multiplied; operation by which any given number or quantity may be added to itself any number of times. Not very clear, don't you agree? Then we found Webster's II New Riverside Dictionary, which said was addition in which one number indicates how many times another is to be added to Not a great help either. We wondered which number or quantity is to be added to itself. But Verity, read this one from Webster's 3rd Edition College Dictionary, and read it carefully. It totally blew us away: Multiplication: process of finding number or quantity (product) obtained by repeated additions of a specified number or quantity (multiplicand) a specified number of times (multiplier); symbolized in various ways (ex. 3 x 4 = 12 or 3 [middle dot] 4 = 12, which means 3 + 3 + 3 + 3 = 12, to add number three together four times) (my emphasis). We also had a handout about Euclid. You remember him, mathematician from around 350 BCE - one who wrote Book of Elements? Seems an article in Arithmetic Teacher in 1966 had quoted Euclid as having defined like this: One number is said to multiply another when number multiplied is so often added to itself, as there are units in number, and another number is produced. …