## Friday, February 26, 2010

### What is Multiplication?

"Oh, come on, Sue! What kind of a question is that?"

***     ***     ***     ***     ***     ***     ***     ***     ***     ***     ***     ***

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?
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.
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.

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.
...
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?
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.

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.)

Comments with links unrelated to the topic at hand will not be accepted.
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.]

_______
*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.

1. "Multiplication is repeated addition," understood LITERALLY, would fall under your #2 worry, namely, not having deep enough math understanding. However, used as a metonymy, it's passable. Mathematicians can sometimes be almost autistic in refusal of tropes, and this faucet of the math culture is something people need to know and appreciate, if not embrace at all times.

I have a big worry list too, and my related bigger worry is... Why on Earth does everyone in this conversation quietly assume kids always learn addition before multiplication?! When, in fact, it's crucial for kids to learn multiplicative, additive and exponential math at least simultaneously - and each based on its own set of models, not through one another!

2. >Why on Earth does everyone in this conversation quietly assume kids always learn addition before multiplication?!

Maybe because you haven't helped us all see the beautiful thinking kids do, when they get to be part of a really math-rich environment, like what you provide at your math clubs?

3. While I mention computer science, I would call my post more a meta-post trying to explain why people are fighting in the first place.

4. Got it. (I also should have made clear that your position was in opposition to Devlin's, but I saw your post just before I finished, and didn't really get that part right.)

So. Anyone who's reading here - go read Jason's post, so you can see his angle for yourself. :^)

6. Burt, thanks for mentioning Alison's multiplication posts (2 show up with your link). I went back and re-read them, and the comments on the first one really got me thinking. I commented over there.

7. I am one of those elementary teachers. I'm certified K-8 in math and as my name suggests, I more than just like math. I'm constantly appalled at how elementary teachers are viewed as less intelligent (in general) than those who teach at the higher levels and are completely incompetent when it comes to math. However, I know that many of my colleagues (especially at the elementary level) don't truly understand math to teach it well. Some of the things I've seen teachers tell their students during math lessons has made me cringe and scream on the inside.

Of course, I have no idea how these math inadequacies can be fixed. I think the first step is the teachers realizing that often their understanding isn't sufficient. The second step would be them taking actions to rectify the situation.

8. We are generally not able to keep people from making assumptions about the intelligence of a group people as a whole or any individual in that group based on membership in it. This is the nature of bias and prejudice.

Based on my work with (or as a/an) upper elementary, middle school, high school, community college, university, and adult education teachers, no group there has the market cornered on "intelligence" (whatever that might mean) or effective mathematics teaching (which is much more of concern to me).

Same goes for having one's heels dug in on any given educational issue surrounding the teaching and learning of mathematics: there are very open- and very closed-minded members in each group.

It's very interesting, however, to note how many MORE people seem to dig their heels in on the issue that Devlin raised back in 2008. It is one that really seems to have a deep emotional component to it, far more than most I can think of. Even some folks I generally see as open minded have a hard time considering that it might not be so that "multiplication IS repeated addition" (MIRA) or that we would do well not to START teaching multiplicative reasoning by grounding multiplication in repeated addition.

Please note well: no one I am aware of - not Keith Devlin, not Joshua Fisher, not Burt Furuta, not Catherine Sophian, not Barbara Dougherty, not Terezinha Nunes, not I, nor anyone else I've read - objects to the use of the obviously true fact that one can get the results of repeated addition by multiplying (when we make the multiplier a whole number, this is trivially easy to show/model, etc.; it gets more and more convoluted and awkward as one moves to integers, rational numbers, and finally irrational numbers: I'm pretty much stymied trying to model e * pi as either "e iterated additions of pi" or "pi iterated additions of e," but then I never have claimed to be the smartest person in the room or the most clever or imaginative. Your mileage may well vary on modeling or visualizing such products as sums).

But there's a world of difference between how we arrive at calculations and what the problems the calculations solve are about. As well as what the operations involved entail.

Rather than reiterate (or repeatedly add) previously made arguments here or elsewhere, I'll bow to Burt Furuta's writing on this subject as the clearest and most sensible (to me) I've seen thus far. His comments on my blog are really good, and I've now got a revised version of a private post he made to me two weeks ago sitting in my in-box that I need to chew over. I hope he'll let me post it somewhere or will do so himself.

Now, I owe it to Sue to read HER post here carefully before running off at the mouth/keyboard any further.

9. (chuckling)

Hey Michael, just as you (and yours) are not objecting "to the use of the obviously true fact that one can get the results of repeated addition by multiplying...", no one here is arguing that multiplication is repeated addition. I think we're almost on the same page.

Tell me what you think when you're done reading. ;^)

10. Having read your post, I'm now able to comment on the parts that inspired reactions from me that I would like to share (there were a few things, like about Devlin's "trick question" on functions, I'll need to think about further, though maybe that will still produce a comment below from me).

First, I think it's a mistake to start with the question "What is multiplication?" unless one is prepared to come to a conclusion along the lines of "It is what it does" and then investigate what it is multiplication does (particularly what it does that addition does not). I suspect that will get one closer to understanding why multiplication isn't repeated addition (even if it can get one the result of repeated additions, most readily when dealing with situations where one of the two factors being multiplied is a whole number. I thing it gets harder, more convoluted, and more torturous, when one only has negative integers, rational numbers, or irrational numbers as factors, to maintain a good grip on "multiplication is repeated addition" (MIRA).

11. (cont) But moving to your son, I think you added something that most MIRA fanatics don't seem to talk about at all: units of measure. You say that he is "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."

To which I say "Bravo!" to your son for getting that key point, and to you for noting it. The magic words for me are "10 cents for each dime." Once you say that and you have "5 dimes," addition is simply out of the question. The two "arguments" don't have the same units. But they have some sort of sensible connection between their units that allows multiplication to take them and spit out - not "dimes" and not "cents per dime" but a new unit - cents. Addition doesn't do that. Subtraction doesn't do that. But multiplication and division do. And that's one very telling reason that MIRA fails to make sense.

Of course, your son gets his numerical result (without units) by skip counting. He doesn't yet know that 5 x 10 is 50, so he finds a comfortable and perfectly valid and sensible strategy to compute. He will later see, in all likelihood, that it looks like a product and the result of an iterated sum produce the same number. But because he has that units of measure thing going for him, he may also be a little suspicious if someone tries to sell him on MIRA. And if he's lucky, you or someone will push him to think about this difference, particularly as he moves towards rational numbers and beyond and he may have a lot less difficulty as a result.

Moving on to the 3 x 5 array being the same as a 5 x 3 array. On my view, they are not the same unless you are free to rotate them. Yes, they have the same number of elements, but that's because of the commutative property of multiplication. As I've posted in several places, if the only tickets left for a concert are in the last row, I hope it's in an 8 x 26 seat theater, not a 26 x 8 theater. And if I have to buy seats at an extreme end of the first row, I want the 26 x 8 seat theater. I suspect this is not the only real-world example where we care more about the arrangement than the total number of seats (from the perspective of being a patron), and more about the number of (hopefully full) seats, if we're the theater owner.

12. (completed) As for what Devlin called in his columns "brittle metaphors," I tend to agree that as teachers (or learners) we need to treat them with caution. The problems Devlin addresses are, I believe, the result of the very careless and thoughtless use of such metaphors. The closer teachers come to saying or believing or intending or settling for the MIRA metaphor as definitive, the more disservice they do their students.

The question I keep raising is: why would anyone, realizing this possibility or strong likelihood, not exercise some reflective thought about what might be a way to ground multiplication in something other than repeated addition, then allow (nay, encourage) the very natural realization that indeed we can calculate products of whole numbers quite simply (albeit often not in the least bit quickly) via iterated addition. That is, of course, very neat. But it's not "mindlessly neat." That is, we need to think about how and why that is. We need to encourage our students to do the same. We need to problematize the notion of multiplication-as-repeated-addition in ways that make it very difficult indeed for students to believe, "Oh, multiplication? It's repeated addition. No big deal."

You find me teachers willing to do that and I won't worry too much if they slip occasionally and say things that aren't quite so. As long as they are reflective and willing to keep learning, and don't get so thrown by Devlin and others setting traps that force them to challenge their own learning and the assumptions that learning may have led them to believe are unquestionable. (See, I did get to Devlin's trick questions. I know for a fact that he enjoys stirring things up, and I think his ability to do so is invaluable).

I think I'm comfortable with your concluding paragraph. And I'm quite sure that very few teachers are aware that this MIRA business might be worth thinking about. Just as I'm quite sure that not very many are giving their students multiple ways of conceptualizing what multiplication DOES (or 'is,' if people insist): only, at best, different ways to compute a product (as students get to multi-digit multiplication).

I fear that even some of the progressive programs may not challenge TEACHERS to challenge the comfortable idea that MIRA, and to spend meaningful time during the very beginning of multiplication looking at the conceptual, rather than computational, aspects.

The fact that there ARE other ways to approach arithmetic that make it natural to see multiplication initially other than as mere calculation with whole numbers (and hence not as repeated addition as the initial impression students are handed about it) makes me wonder why the work of Davydov and colleagues, and those Americans and British and other math educators who have developed programs based on his work, are not making more impact thus far. Except that I know that the main reason is the undue influence of test-mad traditionalists who have fallen in love with a very narrow band of programs (particularly those they don't see as challenging their own assumptions and biases, most specifically when it comes to pedagogy and political/social content in the word problems - e.g., environmentalism, social justice, diversity, and all that jazz). I find it intolerable that conservative politics is allowed to trump all other concerns when it comes to US education. But such continues to be the case, even under "socialist" Obama. :(

13. "Math itself is the authority - not the curriculum, not the teacher, not the standards committee."

Indeed.

14. @Joshua Fisher: I just now realized you put that comment on this post. Yes, mathematics is the authority, and since mathematicians disagree on this issue, it seems odd you'd say 'indeed', unless I've convinced you that the issue is more layered than Devlin lets on. (Somehow, I doubt that I've convinced you.)

15. @Michael: I probably won't be able to respond to all you said. Here are a few notes...

1. >I say "Bravo!" to your son for getting that key point, and to you for noting it.

Of course we don't know what he 'got'. I'm hypothesizing about what he was thinking, and I'm sure he didn't think the little x I wrote. ;^) What do we think before a concept is solid in our heads?

2. >Moving on to the 3 x 5 array being the same as a 5 x 3 array. On my view, they are not the same unless you are free to rotate them.

Of course I never said they were, and the shape I imagined my students exploring was made of squares, so the rotation would make them the same.

You are pointing to something kids would enjoy thinking about. That's cool.

3. >makes me wonder why the work of Davydov and colleagues, and those Americans and British and other math educators who have developed programs based on his work, are not making more impact thus far.

Actually, I don't think the problem here is the 'test-mad traditionalists'. (My traditionalist friends will point out that not all are test-mad; many are on your side on the testing issue.)

I think it's just a big change, and even most reformers haven't thought about it yet. The U.S. is full of superficial choices and superficial changes, but a big change like this would take more training than our system can imagine. (So do many of the other reforms, but it's easier not to notice that with the others.)

I like the idea of measuring as our starting point into math. But it won't work until we get elementary teachers excited about learning math deeply.

16. @MathAddict: Welcome! And thanks for letting us know there's at least one elementary teacher here.

What do you think is the comfort level with math of most of the elementary teachers you know?

I have not yet found any hard numbers, but I think 4 out of 5 people are not comfortable with math, and I'd guess that holds for elementary teachers, too. What do you think?

17. Perhaps my son reads my mind. I have not been discussing this with him, honest.

Him: "60 plus 60 is 120, right?"
Me: "Yep."
Him: "60 plus 60 plus 60 plus 60, ... 4 times 60,..., is 240... [pause], yep, that's right."

18. Hi Sue

I love this article (and the subsequent discussion) and so have taken the liberty of submitting it to the Carnival of Math. Let Dan or I know if you'd prefer it not to be included.

Best Wishes,
Mike

19. Sure, that would be great.

I need to edit the article to include the discussion boundaries I set in my comment moderation info, and perhaps to explain in more detail. I've seen too much rancor in other discussions, and it's important to me that the tone stay civil here. I have rejected one comment on that basis.

20. skylab gallery downtown jon stimmel 57 e. gay suppose we let
a, b \in N
and then... a far
*wilder* supposition...
suppose we go on
to define

a*b := #{
(1,1), (1,2),(1,3),...,(1,a),
(2,1), (2,2),(2,3),...,(2,b),
.
.
.
(b,1),(b,2),...,(b,a)
}

then multiplication "is"
the function... let's call it
M: NxN...
defined by
M(p,q) = p*q
(for p and q
integers greater than
three, say, so that our
ps'code will be, at least
a little bit, less pseudo.

one then works out examples like
4*5
(=,
for the truly obsessed,
#{
11,12,13,14,
21,22,23,24,
31,32,33,34,
41,42,43,44,
51,52,53,54
}=
[scroll down... i know...
the suspense must
be awful]
20
).

for this to be "the"
meaning of multiplication
would require, for one thing,
that the "counting" function
"#" be defined... or, anyway,
*understood*... for finite sets.
moreover, and i might be
verging on a *point* here,
it's the "finite sets" framework
that makes defining "*"
and "M" for N...
which, not entirely by the way,
i take to be
N:= {0,1,2,3,...}
;
note that 0 \in N
(though i haven't used it here);
this is "natural" in the
"finite sets as foundations"
universe i was brought up in
(and still, almost entirely
on some days, inhabit).

the real point inspiring this...
um... free verse? love letter
to the natural numbers?...
was that, first of all,
"THE" definition of multiplication
necessarily depends on
*context*.

multiplying *what* by *what*?

for me, whatever multiplication "is",
it's a function of two variables.

if we're gonna get all *formal*
about it, we'll need to have
a definition for *real* #'s...
based on line segments, say,
as i hope at least one reader
will have anticipated...
where A*B gives the "right"
this can then go on to be compared
with our definition for finite sets
to show that iso-morphing
natural-numbers
(obtained by counting sets)
to certain line-segments
(whose lengths are
integer-multiples-of-given-unit)
gives isomorphic
"product structures".
(i.e.,
two ways of getting the same
multiplication table
when only natural numbers
... considered as certain
finite sets *or* as lengths
of certain line segments...
are taken ito account.

there is a general feeling abroad
that all this is too "formal" for
five- and six- year olds. one says
"the 'new math' of the sixties"
and everybody's all "it doesn't work".
they say the same about hippie values
too from the same era but they're
rapidly coming around ("going green").

i don't know if math is easier or harder
to understand when you base everything
on definitions and sets; i've never tried
any other way and it's too late now.

of course various concepts will have to be
presented in less than full generality;
various defintions will be suppressed;
etcetera. nobody told me what a *number*
was until grad school (and i was, no
surprise here, lazy to swot it up
on my own). i got by somehow.

moreover. the powers of confusion will try...
and almost always succeed... to blame
"this is too mathy for 'em; let's go
so it gets tiring.

21. corrigenda

omit
"skylab gallery downtown jon stimmel 57 e. gay"
(starting at line 1).

[remark: these are notes for yesterday's post
in an unrelated blog.]

(under the first "display"),
one should have
M: NxN--->N
... our author has omitted the "target" set
(sometimes, confusingly, called the "range").

that same paragraph ends
without its closing paren:
substitute ")." for ".".

all coherence is lost in paragraph 5.
verging on a point evidently
disagrees with me. anyhow, i
did *not* go on... presumably
because i somehow fooled myself
gone on... to say that the "finite
sets" framework makes giving
our definition *possible*...

i get caught in these digressions
all the time but i hope to edit
most of the worst lapses out
before posting.

22. Mathematicians can sometimes be almost autistic in refusal of tropes, and this faucet of the math culture is something people need to know and appreciate, if not embrace at all times.

--maria d
(who should resume blogging)

to a math-head... we who learn to
refuse tropes... this is a matter of
being able even to understand
what is meant by a "definition".

real life offers no absolutes
but definitions in mathematics
(here in plato's cave...):
we mean *exactly*
[such-and-such code]
and *nothing* else...
so whatever "tropes"
anybody comes up with
will be judged by their
tendency to make the
*actual code* easier or
harder to work with.

if anything in life is worth
being a bigot over, here it is.

you can't *do* math
rightly-so-called
*without* definition "autism"
in at least some mild form.

23. ...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?
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.

thus far devlin as quoted in the post at hand.

it's no *wonder* people get all emotional
about stuff like this... it's sheer bollocks
for one thing so taking it seriously is *perilous*.

"Consider the doubling function y=2x (or if you prefer a more sophisticated notation, f(x) = 2x)"
is itself very simply
*not at the level
of formal correctness*
that
"Functions, as defined and used all the time in mathematics, don't do anything to anything"
gestures at (in order that
it may be ignored again
one sentence later).

y=2x, like
f(x)=2x,
is not a function
but an equation.

the function... by the "definition",
if there *were* one... would be
something like { (x,2x) }_x\inR:
a "set of ordered pairs"
(as thousands of my students have
been urged to learn to regurgitate,
by me, mostly without any
benefit to the student that
i could detect... mea culpa).

which, right, doesn't
*do* anything.

*including*, for the love of
fresh air, "relate" things.
it just sits there in set-space
like the set it "is".

our author appears not to know
that "relate" is a verb. this isn't
even *trying* to be math ed
or philosophy or even good
*journalism* from anything
i can see here... though it
would be pretty silly of me
at this point to deny that
it's *interesting*...

it's just a rhetorical trope: show business.
magic! look over here!
[pulls bigfat shoe out of face]
hah! thought it was over there,
didn't-cha! gotcha!

assumes a certain good-faith
attitude on the part of its reader
that there'll be a common-sense

it's like pointing at a guy's tie
and slapping him in the face.
then you chide him for looking.

which is all well and good for
syndicated columns and whatnot
but makes pretty thin gruel
for anybody who actually
wants to go out and, you know,
actually *explain* something.

24. Oh wow, missed a lovely development on functions-as-actions vs. functions-as-things. Doublethink ftw!

Also, there is an ongoing war about that in the parallel universe of computer science, where people love procedural OR functional languages... a lot.

25. Okay I am going to risk 2 cents even though I didn't have time to read every comment in detail. I had a lot of thoughts and feelings about the Devlin piece which I won't go into. But, grist for thought:

Devlin is saying "multiplication is not repeated addition so stop telling kids that it is repeated addition."

It seems to me that I could make more or less the same argument with pi, and with trig functions, and with negative numbers, and a lot of other stuff we teach - a deep mathematical structure is first introduced with a definition that doesn't tell the whole story:

"Pi is not the ratio of a circle's circumference to diameter so stop telling kids that it is."

"The sine of an angle is not the ratio of the side of a right triangle opposite the angle to the hypotenuse, so stop telling kids that it is."

Etc.

First I learned pi was the ratio circumference/diameter.

Later, I learned that the sine function was the ratio opposite/hypotenuse.

Still later, I learned that the sine function was actually the ordinate of the point on a unit circle whose arc to the x axis is the given angle. (this allowed a generalization to sines of angles greater than 90, and less than 0.)

Later still, I learned that sin x was actually the limit of a series:
x - x^3/3! + x^5/5! - ... (This allowed a generalization to complex arguments for the function.)

Finally, I learned that what pi actually is, is not the circle ratio but the first positive zero of the sine function, defined as a series. (We need this kind of a definition in order to prove pi's irrationality / transcendentalness, etc.)

It's helpful to know that the initial definition isn't telling the whole story, sure. But does the fact that the initial definition doesn't tell the whole story constitute a case against telling kids the initial definition?

26. These have been very helpful with my tutoring sessions. Thank you.