Two quotes stood out for me. The first is about the nature of "doing" mathematics:

I think many students give up on mathematics because they don’t see how they could possibly come up with the solutions to problems they see in their textbook, or which the teacher gives on the board, or which some of their classmates produce. What they don’t understand is that the clever argument they have just been presented with was not arrived at by deliberate, rational thought. It was constructed after-the-fact. And so the student misses the crucial lesson that the secret to doing mathematics is not an unusual brain but sheer persistence, trying one thing after another and failing each time until eventually the light comes on.

and the second about the "multiplication as repeated addition" furor:

Using a brittle metaphor (multiplication is repeated addition, for example) inevitably leads to problems later, when the metaphor no longer holds but gets in the way of a better understanding of the concept. It’s hard enough grasping the abstractions of mathematics without compounding the issue with brittle metaphors. One problem is that metaphors inevitably lead to natural inferences. For instance, thinking of multiplication as repeated addition leads to the belief that multiplication makes things bigger. This false belief often persists throughout people’s lives. It’s particularly hard to eradicate since it is often something the child observes him or herself, and as we all know, knowledge we generate ourselves tends to stick like glue. When it comes to mathematics, I think it is probably always unwise to use metaphors as “interim definitions”, which is what often seems to be done, since they all break sooner or later. Rather we should present the student the same instances, but as motivational and illustrative examples of, not metaphors for. Mathematics is abstract. It does students no good in the long term to present it as something concrete. Moreover, there is no need to do so. There is plenty of evidence that children can handle abstraction, particularly when the learning is scaffolded by a range of concrete examples.

Do you think that we make math too tidy for students? Do you have a favourite example of a "brittle metaphor"?

## 8 comments:

Both of those extracts are interesting, Ross. Too much thought for a Sunday afternoon. I can't offer anything original but one quote that Professor Ross Honsberger (author of Mathematical Gems 1 and 2 which I own both) has always stuck with me.

If someone is good at baseball, how do they get better?

Practice, Practice, Practice

If someone is good at football, how do they get better?

Practice, Practice, Practice

If someone is good at mathematics, how do they get better?

Do the odd questions on Page 37 like everyone else and then go outside and play baseball.

Where does the stigma of mathematics originate? Does multiplication as repeated addition symbolize our attempt of dumbing down mathematics so that nobody feels the stigma?

I had Ross as a professor and felt very fortunate.

Devlin sort of implies that anyone can persist and do wonderful mathematics. I am not convinced that that is true. I think I could persist a long time at singing and never perform at Massey Hall. Honsberger's quote starts with the premise that someone is good at mathematics and wants to get better. Like the baseballer and footballer, someone who wants to get better will do so more easily if they find other similarly talented folks to hang around with...

Actually, I say explicitly that it is not the case that everyone can do well at mathematics. :)

@Keith Devlin

I am very happy that you commented. In some math circles, the notion of "Math for all" is promoted. What do you think is Math that is appropriate for all (or nearly all) students and how is that different from the Math for mathematically talented students?

This was the subject of my NCTM Annual Meeting talk in Philadelphia a few years ago. (The video is on the NCTM website.) Then president Johnny Lott invited me to talk - provocatively - since he wanted the issue discussed among teachers.

My argument was based on considering the possible evolutionary origins of mathematical ability. Roughly (and this is a rough summary), achieving a level of competency in any mathematics that is a direct abstraction from the real world should be within everyone's grasp, but once you are at two levels of abstraction, you may lose many people. So numbers, arithmetic, elementary geometry, trigonometry, and the like should be doable. Algebra would be on the border, since it requires abstracting across arithmetic. Uri Leron has carried out studies that seem to support this, but there is scope for a lot more empirical work.

Of course, just because something is going to be difficult does not mean it should not be tried. But we should set realistic expectations. Right now, the main outcome of mathematics education for a lot of people is hatred of math, and that isn't good for anyone.

The first step toward setting realistic expectations would be to carry out a lot more empirical studies - which as it happens is already well underway!

I actually agree with Devlin that multiplication is not repeated addition. However, I do not believe that the multiplication-makes-bigger is the result of this "brittled metaphor." This belief, I think, is an over generalization students make based on their experiences. The vast majority of their experiences with whole number multiplication involve the multipliers other than 0 and 1. If the multiplier is 0 or 1, it is not true that multiplication makes the number bigger. However, those are exceptions and children don't encounter them much at all. So, because they can think, they make this over generalization - and I think they will keep making it even if we use a different metaphor.

I imagine there are many children who thinks addition-makes-number-biger, too.

Looking at the title of this post, I'm reminded that "What is Mathematics?" is the title of a classic exposition by Courant and Robbins, revised some years ago by Ian Stewart. Their 'model' for integer multiplication, for what it's worth, is rectangular arrays of dots. Later they proceed to

definemultiplication of rationals as a/b * c/d = (a*c)/(b*d) : implicitly admitting that the preceding model doesn't work on fractions. If even eminent mathematicians are guilty of swapping definitions like this, perhaps we shouldn't be so hard on teachers!In fact, as I've commented at length over at Good Math, Bad Math, the real problem we face is that we only have limited ways available to 'model' multiplication when the only numbers available to us are positive integers. Of course this doesn't excuse teachers from being as correct as possible - which, on all previous discussion, may just boil down to needing to exhibit as many different models as possible.

Where I really disagree with Keith Devlin, though, is in regard to exponentiation. Not only because I was taught exponentiation through repeated multiplication, but also because I consistently got the same approach from my real analysis textbooks and university lecturers - at least up to the point in the logical development where exp(x) is introduced. In my experience, most mathematicians regard the exponential

functionas more fundamental than the general power operation. Certainly in many contexts (including polynomials and power series) exponentiation really would appear to serve as no more than a shorthand for repeated multiplication - and is perhaps even best understood that way.The address that Keith Devlin refers to was at the 2004 NCTM Annual in Philadephia "Defining Mathematics for All". It is available as a RealPlayer stream. Devlin is introduced at at about the 18:15 mark.

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