I think Devlin was serious – and seriously wrong.
Sometimes people – even very smart people – harbor these little bugs — an “animus” (plural animi?) — that make them irrational when their little bug acts up. I think that’s what’s going on.
And riffing off the previous post, imagine a student taking a test: please define multiplication — answer: just something you can do with numbers; define exponentiation — answer: just something you can do with numbers; define integration — just something you do on functions… and so on.
By the way – how *do* we teach integration then? I always thought of it as a kind of fancy addition … (and yes there are more abstract definitions… )
Back before Euclid, mathematics was empherical, which means table driven. The priests had the tables, so you had to go to a priest to get the answer to things like muliplication problems.
We learned our multiplication tables. Sure, it was easier to add stuff up over and over again, but we learned the tables, and the tricks. At some point the tables became implicit, and we never had to think about them or adding stuff up.
I’ve never gotten to the point where I instantly know what 23 squared happens to be.
I do remember stopping at a convenance store late at night where a guy was learning the GED version of percents. They have to remember three different equations and the situations leading to the use of one or the other. Algebra makes the cognative load so much lower. I can construct the formulas quick enough, but they are not implicit for me. They might actually become implicit to him.
Likewise integration of trig functions. Just memorize the stuff. I didn’t and it has taken me forever to do what is now, post memorization, easy enough.
At some point understanding the system at one level gets in the way of understanding it at another. It works both ways known to the yet to be known, and known to the forgotten–learning, or teaching.
Jonathan,
English may have an advantage, in that we have a simple, two-letter preposition to represent the idea, but every language must have some way to talk about measurement of a continuous quantity by a scale unit:
AxB = “A units of size B.”
Joe,
Allow me to present my understanding of good pedagogy:
One clear definition
+ Multiple representations or applications
= Understanding
It seems to me that you are saying the multiple representations alone are enough, but I think that will leave the students (except for a few who can abstract their own definition) wallowing in confusion. The teacher needs to have in mind and to present a single, clear definition which will stick in the student’s mind — and then all those multiple representations and applications need to be grounded back to the definition.
Part of the teacher’s job is to find the strongest, clearest definition possible. For years, elementary and middle school teachers have used “multiplication is repeated addition” not simply as one representation among many, but as the foundational definition of multiplication for their students. I am arguing that it does not work well in that role.
I am not concerned at this point about teaching complex numbers. I just want to teach my kids to handle the real number system with confidence and think their way through problems within that system. I’d even be satisfied to bring them to true competence with the rationals. Let tomorrow worry about complex numbers.
Denise,
in some declined languages grammatical case plays the role of “of.” In Russian, the case will depend on the number, with different choices for 1, 2 – 4, and 5 and above.
In Turkish, I think the closest you will find to “of” is a nonce word, “tani” which means something like “pieces” or “occurrences” and probably falls closer to “times” than “of.”
Let others chime in, correct me if need be, but I would be surprised if this model holds for more than a few related languages.
Jonathan
Denise,
I could respond at length – I don’t think I really disagree with you in principle, (although I do disgree with Devlin in principle on matters of truth and falsity,) and I think clear definitions are essential at some point (but not very early on.)
But let’s cut to the chase – what is your one clear definition of multiplication that works for reals (or even just rationals) that you can present to a 8 year old as their essential introduction to multiplication?
I’m not trying to put you on the spot – but the stuff that has been said about scaling and “of” doesn’t quite fulfill my expectations for a definition. So I’d really like to see a more definitive presentation of this crucial point. If you want your diagram to be part of the definition I’m fine with that. It kind of presupposes that the child already can conceive of a number as a length (or an area) and I’m OK with that too – though that implies that these related concepts are taught in some sequence that makes sense (as always).
Given that definition, we can perhaps all explore whether we think it will really hold up in the real world of teaching *all* children (not just above-average children).
I mean, part of the uproar (only part) is caused by Devlin’s lack of a truly clear, compelling, and decently worked out alternative, that seems like it will fly in the classroom.
——–
On Text Savvy’s article (Devlin’s Right Angle) – I have to point out that the claims of the opposing side have be distorted. Quoting Text Savvy he says “all multiplication IS repeated addition” – Nobody is saying that! What I am saying is that for whole number (or for integers), yeah, you can define it that way and it works fine. You could also define it other ways and it can work out fine. Argue with Giuseppe Peano if you think it doesn’t work. When it comes to rationals, yes, you need a new definition — but that’s a valuable lesson in itself about how mathematics works — not an ugly about-face and defeat.
By the way, my pointing out that Devlin’s fuzzy suggestion for how to teach “the truth” fails for complex numbers is to point out a matter of principle. In this matter I’m concerned with how Devlin is using (or abusing) language, his authority as a mathematician and public figure, not with what “works best” for elementary or middle school students.
He is abusing language when he talks about the “truth” and “falsity” of various approaches, and then goes on to suggest his way is “true” and the way he doesn’t like “false” – even though his way “fails” in the exact same way (albeit at a later point in time). It’s hard to fathom whether he is really just that confused or just that arrogant.
That abuse of language and authority is both damaging and confusing. And I suspect his students may be very confused, not just because of their prior education, but also by what they may hear from Devlin on these sorts of meta-mathematical issues.
Full disclosure -
I learned multiplcation (for whole numbers) at age 5. My dad drew dots in an array to explain. Saw it immediately. No problems then or later on when approaching fractions.
Anybody else remember exactly when they were introduced to multiplication?
My definition: To me, the number line seems the best representation of the real numbers, so my definitions of the arithmetic operations are based on the number line. I described my definition of multiplication as scaling in the last section of the article above, “Edit: A comment from Devlin.”
I do not teach that definition in that way to early elementary students, because they are not yet used to the number line. But it is always in the back of my mind, and I tie each multiplication calculation into it. For example:
= “3 of the 4s.”
= “ of 10.”
In middle elementary, bar diagrams like those shone above represent a “thickened” chunk of the number line. I have talked about how I use those in several past articles.
Gosh, I must be old. I have trouble remembering last year, let alone when I was five!