Math Teachers

Who knows how multiplication is taught? Not me, not Keith.

This is about an argument about nothing.

A respected math columnist went after teachers for saying that multiplication is repeated addition, but it turns out that he doesn’t know if many teachers do this. I called him on it. And his response came up short.

Background

Yup. One more Devlin post. Synopsis so far for those of you who weren’t watching the whole multiplication vs repeated addition follies.

Keith Devlin, back last Fall, wishes that he could stop teachers from saying multiplication is repeated addition. He elaborates, big time, in “It Ain’t No Repeated Addition” in July. Denise, who teaches math, thinks about it, and asks, then how should we teach multiplication? That’s when the comments get a bit out of hand. Denise posts again. Some other people post. Even I post.

Mostly the posters and commenters were yelling and screaming about whether or not multiplication is repeated addition. In all of this, the question that matters - how should we teach, was pretty much buried.

Question pops up

Fast forward a few days. I am in New Orleans, setting up classrooms. And I stop to skim a variety of elementary and middle school math texts. And I don’t find the error Devlin is chasing. Instead I find books discussing and introducing multiple meanings of mathematics.

Could there be some texts that say Repeated Addition = Multiplication? Sure. But my unscientific sample didn’t find them. Could some teachers ignore the texts and teach Repeated Addition = Multiplication. I know that some do. But I don’t really know if it is very many. So I wondered out loud if Devlin was jousting with a straw man.

Devlin’s rebuttal

His recent column, he’s making one more go of it, attempts to rebut 6 arguments. It is longer because he will “be quoting from some of the leading mathematics education scholars of the twentieth and twenty-first centuries…”

But when he comes to my arguments, um, no. He provides next to nothing. There is one British ed journal article that says teaching multiplication as repeated addition is a problem (from ten years ago, directed to British national policy, looks like the research was a small study in London.)

And his coup de grace? Studies (one British, one Canadian) that show adults, when asked to define multiplication, respond with repeated addition.

(To look for yourself, find the heading “The Problem Is WIdespread” about three quarters of the way down)

Now, think for a moment. Of the various models we may use in teaching multiplication, isn’t repeated addition the strongest? Isn’t that exactly what you would expect an adult, 15 or 30 years removed from grade school to recall first? They remembered what we should expect them to remember - but that doesn’t tell us what they were taught.

Could he have cited something else? Yup. If he found state or national standards telling teachers to teach RA = M, but I don’t think they exist. If he had found studies that said, “teachers do this a lot”… If he could show us texts that do the same… maybe they are there. Josh at TextSavvy might know?

Two things went wrong here.

Like the engineer who comes to a school knowing math but not knowing how to teach it, Keith Devlin arrived to a topic (math ed) that he remembers. He was a student. And he probably remembers better than most. But we are talking memories, not current knowledge here.

And second. Something I recognize. Stubbornness. Look how well he writes. Pick any other column. Pick his recent interview. There’s intellect, there’s quality of expression. He hasn’t poorly defended his position because he argues poorly; it’s just stubbornness without facts supporting it.

I’d be interested in recommendations about multiplication should be taught, but as for this topic, I think this will be my last post.

New math blog: f(t)

A nice addition to the links here: f(t) is a new (from July) high school math teacher blog. Kate teaches in Syracuse, New York. (I’ve been there!)

So far she’s posted problems, lesson ideas, and a little bit about her work. Nicely written, easy to read.

Best of luck!

Multiplication? Addition? Comments?

Kenneth Devlin wrote “stop telling your pupils that multiplication is repeated addition” and all hell broke loose.

There was a storm at Let’s Play Math, and then Denise wrote a second post. There is still a storm raging at Good Math Bad Math. And a bunch of places I don’t normally go. And then Josh at Text Savvy has written 11 posts (they start here) - but he disabled commenting, which turns the conversation into an echo chamber. (Josh, it looks bad if you complain about your comments not being published if you run a site where comments are not allowed)

So, my two cents.

Multiplication is not repeated addition.

Multiplication can represent repeated addition.

Devlin’s point was directed to how we teach little kids math, and he blew it. So we stop telling kids that x = + (rep) and we tell them what exactly instead? Don’t ask Devlin. He devoted a second column to the issue, and never got there.

Doubling back, what was his objection? That we say “math is repeated addition” and that somehow this ruins kids’ ability to handle arithmetic: “Multiplication simply is not repeated addition, and telling young pupils it is inevitably leads to problems when they subsequently learn that it is not.” He’s wrong.

What should we do?

Teach multiplication through repeated addition, skip counting, counting arrays, finding unit areas of rectangles, Cartesian product, scaling… That’s too many to start, isn’t it? Pick one, then one more. And tell the kids that multiplication will manifest itself in many other ways as well… Add more… and remind them… No big deal here.

By the way, congratulations to Denise for handling this in an intelligent way. And for, ever so briefly, becoming the central blog in a math ed tempest.

Integrated Algebra Conversion Chart - Later Today

Later today, if all goes well… Later this morning, if all goes very well, New York State Education Department will be posting a conversion chart for the Integrated Algebra.

The conversion chart will probably be here. Also, please complete the Teacher Evaluation of the Exam.

For previous posts on Integrated Algebra: General - Procedures - Scoring

For previous posts on the Regents in General: from me - from a retired NY State math guy

Finally, I was on the Integrated Algebra Measurement Review Committee. (In Albany in April, and then again Tuesday). Some of what we did remains confidential until after the score chart comes out. Trust me, I will have more to say, soon.

Bad Math B questions (3)

The June 2008 Math B exam is available as a PDF from JMAP.

I already posted three bad problems, and then three more. There is ongoing math teacher discussion of individual questions at a Math A/B listserve run by the Association of Mathematics Teachers of New York State)

Here’s the last two that really bother me are 33 and 34, the 6 point free response questions, most valuable on the exam:

#33 Solve for x: $log_3(x^2 - 4) - log_3(x + 2) = 2$

So here’s the deal: TeX Embedding failed!
$\frac{x^2 - 4}{x + 2} = 9$
$x^2 - 4 = 9(x + 2)$

That’s what the State expected (then solve the quadratic, reject the bad root. Talk about failure to anticipate!

Most kids seem to have done this, instead:

(more common answer, and the worst problem, below the fold)

$log_3\frac{x^2 - 4}{x + 2} = 2$
$log_3\frac{(x + 2)(x - 2)}{x + 2} = 2$
$log_3(x-2) = 2$
$x - 2 = 9$

And the state was not prepared for this. The only guideline that comes close calls for a 50% deduction (3 of 6 points) Clearly the illegal cancellation is a mistake, unless the kid restricts x. But just as clearly, deducting 1 of 6 points would be appropriate. So what’s the state do? In panic, clearly, they say to mark it completely right…

Helps my students, but doesn’t make me happy.

#34. This is the worst.

Gerardo and Bennie are pushing a box. Gerardo pushes with a force of 50 pounds in an easterly direction, and Bennie pushes with a force of 39 pounds in a northeasterly direction. The resultant force forms an angle of 32° with the 39-pound force.

Find the angle between the 50-pound force and the 39-pound force, to the nearest tenth of a degree.

Find the magnitude of the resultant force, to the nearest pound.

So, let’s hear it for meaning what they mean and not what they say. “Easterly” means due east… 0 in polar coordinates. “Northeasterly”? Guess what? it doesn’t mean 45° . Any student taking physics resolves the vectors into components, using the regular meanings of the words, and gets lost.

Bravo, SED. Your rubric fails to specify how many points to deduct from the child who does not misread the way you intended.

Integrated Algebra Regents, a little on procedure

It seems that a few people are interested in yesterday’s Integrated Algebra Regents. For those of you not from New York State, you might like to skip this post.

I will post a link to the exam and the answer key and the conversion chart when they go on line.

The conversion chart will be available the 25th (they said before) or the 26th (they say now) or the 27th (according to people who don’t trust the state)

The exams were done by noon Tuesday. They were graded Tuesday afternoon (but there were no answer materials in the boxes. The State put them on-line a bit after noon) and this (Wednesday) morning, and were sent to the vendor by UPS, today. The schools were supposed to call for UPS, but in many cases UPS showed up unannounced and early and demanded the tests immediately. The vendor was going to send the exams to Iowa, but, you know, FLOODS, they sent them to Texas instead.

The vendor is going to post-equate the exams (I think) and a committee will propose a scale, another will review the scale, and the State will release the scale. All in the next few days. And then the answer sheets will come back to your schools (I think), and the schools will convert the grades, and you know what? Might be too late to get them on this term’s report cards? Check the transcript in September…

Links:

What I wrote yesterday
Official NY State procedures
What I’m writing next (score issues on some questions) (link tb added)
Assoc of Math Teachers of NY State
Math Teacher Listserve (exam discussions)

NY State - Integrated Algebra

I will have a lot to say about this exam, but most will need to wait until after the Conversion Charts are published. I signed a confidentiality agreement…

Until then, you might like to follow the discussion on the Association of Mathematics Teachers of New York State (AMTNYS) listserve, here. Teachers and administrators from across the state are asking questions about wording and scoring, discussing problematic questions, asking advice and making suggestions, appealing rulings to NY State Education Department. and venting a little.

Am I a Fellow?

No, but…

As I have been writing about recruitment and Teaching Fellows over the last few days

I have been thinking about my own start teaching.

I came to teaching before the Fellows existed, but looking not all that different from today’s Fellows. I was young (looked young?), white, smug, know-it-all, from out of town.

On the other hand, I did not possess a fancy degree (not very impressive public college), I wasn’t really that young. I grew up around union organizing, I’d met old-timers and heard stories, first hand, about how hard people fought for their rights, fought for unity. I may not have seen teaching as my ultimate career, but I was giving it a chance (no other immediate options). My uncle, 30+ year teaching veteran, 3-term chapter leader in Brooklyn (not Unity, not New Action), urged me to give it a try.

I may have been too quick to dismiss some other teachers, but on the other hand I gained a support network of informal mentors, senior teachers, who kept me out of trouble, taught me some vital management skills, and taught me to make a good worksheet.

There were 8 people hired in my department in 1997. Me. Four out of teaching within a few terms (change of career, change of career, couldn’t pass LAST, quit before being U’ed). One administrator. One teaching in the suburbs. And one passed away.

How was I different? Pro-union. Informal mentoring from senior teachers. Wasn’t thinking about my next career.

How much of that can we get to the current Fellows?

Some Teaching Fellows I know

I’ve been writing about Fellows, and I will continue tomorrow. For now, this is a reprise of a post from last June: Math Camp. It adds some personal context to

In the intervening year one Math Camper (my sorely missed colleague) moved to a system out of state, but is still teaching math. Two more are considering moving elsewhere in New York (one as a teacher, one as a teacher or an admin). Another is moving to a neighboring state, and is thinking about leaving teaching (but staying in education). Still, wrapping up Year 5, getting ready for Year 6, that’s 18 teaching math, at least 13 (maybe more) in New York City out of a group of 25.

Last year’s post:

You won’t catch me saying bad things about all Teaching Fellows. Yup, in many cases they don’t last in the system very long. Many come in with anti-union animus. Some of them treat their older colleagues with insufficient (no?) respect, or can be know-it-all-ish when they know very little.

25 started in 2003. 19 coming back for their fifth year; most in the Bronx

But there’s also Math Camp.

I met the 25 math campers four years ago, just before they started teaching. And yesterday I went to their fourth annual picnic, a few blocks from West Farms.

I don’t know that they did to make things work. They were young college grads, but also change of career-ers. And a few retirees from one career looking for a brand new one. More women than men, mostly white, but not 100% anything. Certainly not all from the same class, background, region or social group. But in that first summer of math camp, back in 2003, they must have bonded in some strange way.

(more, and stats, below the fold —&gt

Saturday, sixteen of them were there, and more wanted to but couldn’t make it. At least nineteen of them will be teaching math next year, their fifth year.

Here’s the breakdown: 6 in Bronx middle schools (one is a coach), 7 in Bronx high schools (but one is going to be an admin), 3 in Manhattan high schools, 1 out in Queens. Two more will be teaching out of the city: one in the suburbs, one back in her homestate. Of the other six, three didn’t make it out of the first year, two finished three years (Teaching Fellows commitment) and left, and one we lost track of.

And they are overall a good group of teachers. They are mostly a pro-union group. They collaborate with colleagues. And are appreciated by kids.

Once a Teaching Fellow has a job, they are a NYC teacher, with all of the rights, protections, etc. It is easy to treat them as if they are in some sub-teacher category. But when we do that, we divide ourselves, we break our own unity. And why? Because of some stereotype of what TFs are? Just remember Math Camp.

These are our colleagues, and they deserve to be treated as such. They get all the protections they are entitled to, even if sometimes they don’t think they need them.

(How do you illustrate math camp? Probably not with the photo I grabbed from a Florida Disaster Medical Assistance Team training exercise, above. Pretty dramatic stuff. What kind of tent would math campers put up?)

Challenge: Explain why the divisibilty rule for 3 works

Little challenge for some of you. On my return from Scotland, I will write up an explanation of why the divisibility rule for 3 works. I will try to make it accessible to kids.

Do you have ideas? Suggestions? My target date to post the result is no later than Monday evening.

Outline: use base 10 numbers to model something that looks like adding remainders. (maybe 2 or 3 examples with real numbers, then a “challenge,” then one abstract line about base 10. Then generalize to “other bases” (just for an instant) then reapply to modulo 3 with examples with numbers, and then abstract to “the rule”

If you have other ideas, or modifications, or warnings about terminology, or really anything to say at all, please, do.