I know, but think about your answer before you read on, OK? What do you think multiplication is? I ask because I want you to have your own sense of it in your mind before you read the folks I'm going to quote.
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It seems like a simple question. But when you get involved in discussions about how to teach, it isn't. Many elementary teachers present multiplication as repeated addition, and really, it's much more than that - areas, combinations, stretching, and more. (Here's a cool poster, created by Maria Droujkova.) Many math education experts think that calling it repeated addition is a big problem. [Keith Devlin's articles started this discussion. Jason Dyer just posted on this issue from a computer science perspective.]
I personally think that this is one tiny facet of the real problem, and in a comment at Rational Mathematics Education, I said, "I see plenty of problems in the way math gets taught, but this would not be in my top ten list." (After saying that, I decided to figure out what my top ten list would be - I'll post on that soon.) The top two problems, in my opinion, are that so many elementary teachers don't like math, and that they don't have a deep understanding of the math they teach.* We have a vicious circle going, where those who dislike math teach the young to dislike it - and that's a hard thing to change.
Back to the question at hand. Devlin says 'multiplication is not repeated addition'. I agree that it's not just that, but he and others say it's not that at all, and that saying it is messes kids up. I think it's more a case of the translation between English and math-language being rough sometimes. I trust that if we haven't gotten a student to give up thinking, they'll eventually construct their own definition of multiplication, as it becomes clear to them through what they do with it.
Here's a scenario: My son (7 years old) wants to know how much 5 dimes are worth. He says 10, 20, 30, 40, 50 while holding up fingers. The process he's using to figure it is skip counting, which feels like repeated addition to me. But what he's thinking about is 5 dimes x 10 cents for each dime, which is multiplicative reasoning. So the repeated addition is the process he uses to solve his multiplication problem.
It's important to note here that I didn't suggest this 'problem' to him. It was something he wanted to know. I didn't tell him he was doing multiplication, and I don't plan to 'extend his learning' with other problems that involve multiplicative reasoning. I expect his natural curiosity will lead him to explore lots of situations where he'll reason in whatever way helps him figure out what he wants to know. It's important to me not to push. I've noticed plenty of kids of mathematicians who don't like math, and I really want to give him the space to develop his own relationship with the beauty in math.
Most people who write about this are imagining a conventional classroom, where all the students are supposed to be 'learning' the same thing at the same time. (An impossibility, no?) When I imagine that classroom, I see a little child coming up to the teacher after class, worried about this multiplication thing, and the well-meaning teacher trying to be reassuring, and saying, "Don't worry, it's just repeated addition." What the teacher is doing is connecting the new material with something old and familiar. This is how our brains work; through connections. The teacher is also recognizing the child's concern about what they will have to do, and since math lessons are so procedurally-based in this country (on my top ten problems list), she's telling the child s/he can do the multiplication problems by repeated addition. So there are positive aspects to this sort of response, but there are also ways in which it's somewhat problematic.
As we learn new concepts, we go through a phase where we feel confused. Recognizing that, and even celebrating it, is important. (Thanks, Maria, for that insight. It's hard to celebrate our confusion, though, when we're worried about grades.) I'm trying to think of an example that most kids would feel at home with... It's not confusion exactly, but when you learn to ride a bike, it feels all wrong, until suddenly, it feels right. Learning something new can be like that.
Maybe that's a part of what I'd tell that worried child. I might also refer to the repeated addition metaphor to help them feel calm, since I know plenty of people shut down when confronted by the mysteries of math. But I'd also give them an easy area model to think with, so they'd see a basic 'real' multiplication problem. Repeated addition can get at it, and yet it's really something new - a shape made with 3 rows of 5 squares is also 5 rows of 3 squares. But (and here's my problem with imagining that 'conventional' sort of classroom) I think it's better to be playing with areas enough that the kids will tell me, "Oh wow, look at this! 5 threes is the same as 3 fives!"
Devlin also says exponentiation is not repeated multiplication, and functions are not processes. He says you're starting with a lie if you explain these concepts using these metaphors. I disagree. We start out thinking of exponents as meaning repeated multiplication, and then we expand and extend that, to see exponential growth in a more continuous sense. (A 4% annual growth rate can be helpfully seen as multiplying by 1.04 each year, but the growth doesn't happen at one point in the year - it's smooth.)
Here's Devlin on functions: (Dec 08)
...a significant proportion of university mathematics students do not have the correct concept of a function. Do you? Here is a simple test. ... Consider the "doubling function" y = 2x (or, if you prefer more sophisticated notation, f(x) = 2x.) Question: When you start with a number, what does this function do to it?I think seeing functions as processes is a fine perspective to start from - and very few students will go far enough in math to need another point of view. I also think Devlin's insistence is likely to make people think math is stranger and harder and less knowable than it really is. If our elementary teachers were well-educated mathematically, they could weigh in with their own opinions on this subject. I'm concerned that Devlin's tone sets up the notion that there is one right answer to this. (And his question was quite a setup, wasn't it? "What do functions do?" "Gotcha! They don't do anything.") Real mathematicians ask why, which is what I'm doing, along with some of the people I respectfully disagree with. But others are focusing more on the 'right answer' to this pedagogical question than on the reasons, which encourages the wrong approach to math and its pedagogy.
If you answered, "It doubles it," you are wrong. No, no going back now and saying "Well what I really meant was ..." That original answer was wrong, and shows that, even if you "know" the correct definition, your underlying concept of a function is wrong. Functions, as defined and used all the time in mathematics, don't do anything to anything. They are not processes. They relate things.
Here's one more part I'd like to think about (Keith Devlin, July 08):
Part of the problem, I suspect, is that many people feel a need to make things concrete. But mathematics is abstract. That is where it gets its strength.I don't believe that explicit metaphors like these get in the way, unless just one metaphor is used all the time. I agree with Devlin's claim that the strength of mathematics lies in its ability to use abstraction, but I disagree that starting from the concrete is dangerous or even problematic. I'll address that issue in a future post.
Where does the "abstracted from everyday experience and developed by iterated metaphors" mathematics end and the "rule-based mathematics that has to be bootstrapped" begin?
What if the mathematics that has to be bootstrapped in order to be properly mastered includes the real numbers? What if it includes the negative integers? What if it includes the concept of multiplication (a topic of three of my more recent columns)? What if teaching multiplication as repeated addition (see those previous columns) or introducing negative numbers using an everyday (explicit) metaphor (such as owing money) results in an incorrect concept that leads to increased difficulty later when the child needs to move on in math?
The real questions for me are broader: Are students getting a chance to explore lots of different multiplicative relationships? Are they maintaining their curiosity and rage to learn? Is math presented as a tool they can develop to help them think? I want schools in which: teachers are respected for the hard work they do, they're given time daily in which to have professional discussions with their peers about what they are trying to help students learn, and they come in ready to approach math with comfort and joy.
[Edited on 3-1 to add: Surprisingly to me, the discussions on this topic have often become hostile. It's important to me that people treat each other decently here at my bloghome, and I turned comment moderation on when I first posted this, to enforce that. I am rejecting any comments that don't meet this standard. Here's what you see when you post a comment:
I would like this blog to be a safe place for people to disagree. Please do not attack the integrity of the person you disagree with. (Any comments which do so will not be accepted. If I can email you with my concern, I will. My email is suevanhattum on the hotmail system.)Perhaps I should have said it more thoroughly. I will ask you to rewrite if you treat another person badly, or if you malign the intelligence of people on 'the other side'. etc. One comment has been rejected so far.]
Comments with links unrelated to the topic at hand will not be accepted.
*Note to any elementary teachers reading this: I think a good K12 teacher is a saint. You work harder than I do, you have less autonomy, and you get paid less. If you can really reach kids, you also make a bigger impact than I do. I'm guessing the fact that you're here means you either like math or want to do better with it. I'm grateful for all you do. Say hi in the comments, email me, point me to things I should know.