up the down staircase

Sara (not sure who Sara is!) sends a link to this Everyday Math problem posted on the Well-Trained Mind forum.

Up is negative; down is positive.

That strikes me as a bad mnemonic to teach kids who are going to be encountering coordinate planes just a few years from now.

She got the answer wrong, too.


Up the Down Staircase

Why we can't trust math professors...

At least on k12 math education...

(Cross-posted at Out In Left Field) 

Consider the following math professors: Keith Devlin (Stanford), who wants grade schools to de-emphasize calculation skills; Dennis DeTurck (Penn), who wants grades schools to stop teaching fractions; and Jordan Ellenberg (University of Wisconsin) and Andrew Hodges (Oxford), who criticize grade school math as overly rote and abstract. And consider all the math professors featured in these three recent Youtube videos (an extended infomercial for the so-called "Moore Method," yet another re-branding of guide-on-the-side teaching and student-centered discovery learning--thanks to Barry Garelick for pointing me to these!).

Every last one of these math professors sounds like yet another apologist for Constructivist Reform Math. Each one of them can be--and in some cases is--readily cited by Reform Math acolytes and by the Reform Math-crazed media as such. And, yet, had these professors actually been subjected to Reform Math when they were students, it's hard to imagine that any of them would have enjoyed the subject enough to pursue it beyond grade school.

Indeed, unless the mathematician in question has actually sat down and looked at the Reform Math curricula in detail, and imagined him or herself subjected to it, year after year, in all its slow-moving, mixed-ability-groupwork, explain-your-answer-to-easy problems glory, we should not trust what he or she has to say about it. Indeed mathematicians in general, unless (like Howe and Klein and Ma and Milgram and Wu) they take the time to examine what's going on with actual k12 math students in actual k12 classrooms, are especially unreliable judges of current trends in k12 math. Here's why:

1. Grownup mathematicians remember arithmetic as boring and excessively rote, and tend therefore to downplay the importance of arithmetic in general, and arithmetic calculations in particular, in most students' mathematical development.

2. Unable to put themselves in the shoes of those students to whom math doesn't come as naturally as it does to them, they tend also to downplay the importance of explicit teaching and rote practice. Some mathematicians take this a step further, imagining that if one simply gave grade schoolers more time to "play around" with numbers, they'd make great mathematical leaps on their own.

3. As more and more college students show worsening conceptual skills in math, mathematicians tend to fault k12 schools for failing to teach concepts and "higher-level thinking," not for failing to teach the more basic skills that underpin these things.

4. In upper-level college and graduate math classes, attended disproportionately by those who are approaching expert-level math skills, student-centered learning is much more effective than it is in grade school classrooms, where students are mathematical novices. Not enough mathematicians read Dan Willingham and appreciate the different needs of novices and experts.

Now there are two specific ways in which mathematicians can provide valuable insights for k12 math instruction:

1. They are perhaps the best source on what students need to know in order to handle freshman math classes.

2. To the extent that they remember what it was like to be a math buff in grade school, and to the extent that they take a detailed look at what's going on right now in k12 math classrooms, they can offer insights into how well suited today's curricula and today's classrooms are to the needs and interests of today's budding mathematicians.

But the fact that mathematicians are really, really good at math does not, in itself, make them reliable sources on what works in k12 math classes. Quite the contrary.

Preconceived notions about place value

(Cross-posted at Out in Left Field--with some great comments)

One thing that struck me about the math talks given at this weekend's New England Conference on the Gifted and Talented was the emphasis on manipulatives and the concerns about whether children understand place value. Are these the most appropriate things to be focusing on when it comes to students who are gifted in math? The mathematically gifted kids I know grasp place value and other aspects of arithmetic with only minimal exposure to manipulatives, and quickly advance to higher levels of abstraction by the time they hit first or second grade. But the education establishment seems bent on convincing itself that children--however gifted--don't understand place value.

Why would you want to convince yourself of this? Because it gives you an excuse not to teach the standard algorithms of arithmetic. If children don't understand place value, then they can't understand borrowing and carrying (regrouping), let alone column multiplication and long division. And unless they understand how these procedures work from the get-go, educators claim (though mathematicians disagree), using them will permanently harm their mathematical development.

So, given how nice it would be not to feel any pressure to teach the standard algorithms (because, let's admit it, they are rather a pain to teach), wouldn't it nice to convince ourselves that our elementary school students, however gifted in math, don't understand place value?

But how do you convince yourself of this? As that ground-breaking math education theorist Constance Kamii has shown, it's child's play. All you have to do is ask a child the right sort of ill-formed question. Here's how it works:

1. Show the child a number like this:

27

2. Place your finger on the left-most digit and ask the child what number it is.

3. When the child answers "two" rather than "twenty," immediately conclude that he or she doesn't understand place value.

4. Banish from your mind any suspicion that a child who can read "27" as "twenty-seven" might simultaneously (a) know that the "2" in "27" is what contributes to twenty-seven the value of twenty and (b) be assuming that you were asking about "2" as a number rather than about "2" as a digit.

cranberry on the real world

We also don't want them to ask us how to spell for them. "The teacher won't spell for you" is a steadfast rule made clear to students in the beginning of the school year. It liberates the teacher, who has more important responsibilities, and liberates the students, too!

That once meant that the students were to look up the word in the dictionary. Now, everyone's busy with "more important responsibilities."

Just once in such writings, I'd love to see the glimmerings of the concept that perhaps parents are right when they implore schools to teach proper spelling, grammar, and traditional algorithms. Many teachers progress from college straight into the classroom, without a sojourn in the "outside" world. In the world outside the classroom, spelling, grammar, and penmanship count. Courtesy, good manners, perseverance and punctuality count too. The parents who are able to function in the professions, by and large, would prefer that their children leave school with these old-fashioned skills, because these skills are important.


b-ass ackwards
what do authors do?
Four Blocks by Doug Sundseth
Vlorbik on what authors do
cranberry on the real world
Writing Block
Sifting and Sorting Through the 4-Blocks Literacy Model

Why More Mathematicians Don't Oppose Reform Math: and why we desperately need them to

(Cross-posted at Out In Left Field).

Yesterday's NPR Weekend Edition Saturday featured an interview with Stanford University professor Keith Devlin on the importance of Algebra, and while I listened to it, it suddenly occured to me why more mathematicians don't oppose Reform Math.

Here's what I posted on the NPR website:

Keith Devlin suggests that, given calculators, students should focus less on accurate arithmetic calculations, and more on algebraic reasoning. But, as Devlin's fellow mathematicians (e.g., Howe, Klein, & Milgram) have argued, mastering the basic algorithms of arithmetic is essential preparation for algebra. And while the most mathematically inclined students--including Devlin himself--may be able to master these algorithms without much hands-on, numerical practice, the vast majority do need lots of practice, and striving for correct answers is an essential part of that practice.

Today's arithmetic, unfortunately, has been seriously watered down by the new "Reform Math". More mathematicians need to examine this curriculum and speak out against it; ironically, because they can get by without much arithmetic practice, and because so many of them found arithmetic boring, too few mathematicians have considered the potentially dire consequences that the latest trends in grade school math present to the rest of the population (and to the country as a whole).

When our most prominent, accomplished mathematicians, who themselves may well have gotten by without developing accurate arithmetic skills, discount the importance of teaching such skills to the general population, they do a terrible disservice to elementary school math education (and may themselves be horrified by the results, years later, when today's grade school students enter their classrooms).

Consider what one other NPR poster has taken away from the Devlin interview. As she writes in her post:

I want to thank Dr. Devlin for a great quote that I plan to post at the front of my classroom. "Mathematicians often make mistakes in elementary arithmetic because we have our minds on higher things." That will come in very handy!

Yikes!

How does it all stack up?

A parent and I recently started up a Continental Math League team at our school, which uses Investigations math.

The response?  Enthusiasm from students and parents; skepticism from teachers.

Specifically, about "stacking." (Today's word for how we used to add, subtract, and multiply numbers by placing one number on top of the other.)

Kids love it. And not just the ones on our team. As as friend writes:

When I showed one of my sons how I had learned addition, i.e. the "stacking" method, he was very impressed. "Wow, that's so cool! That works great! I wonder if my math teacher knows about this?" was his innocent comment.

Yes, she does, and she doesn't like it. At least if she resembles the teacher who approached me after math practice yesterday and recounted the dismay she felt when she caught one of her students stacking numbers, thus abandoning the more "meaningful" and "faster" way he used to solve problems.

My co-coach and I tried to explain that the Continental Math League numbers are big enough, and random enough, that Reform Math's methods aren't faster and more meaningful, but inefficient and confusing. It's one thing to add 48 and 39 by reasoning that:

48 is 2 less than 50, and 39 is 1 less than 40, so add 40 and 50 and get 90 and then count backwards by 3 and get 87."

But take one of the problems we did at Continental Math League practice yesterday: 825 - 267. Restricting myself to the kinds of calculation that these second and third graders are able/expected to do in their heads, here's the most efficient non-stacking method I can come up with:

The closest friendly number to 825 is 800, and the closest friendly number to 267 is 250. 825 is 25 more than 800. 250 is 10 more than 260, and another 7 gets you 267. 10 plus 7 is 17. So 267 is 17 more than 250. So subtract 250 from 800. Well, 800 minus 200 is 600, minus 50 more is 550. Then subtract 17 from 25 by counting up from 17. Seventeen plus 3 more is 20 plus 5 more is 25. 3 plus 5 equals 8. Add 8 to 550* to get 558."

*By this point in the problem, how many people remember what they should be doing with this 8?

Anyone with a more efficient non-stacking method for subtracting 267 from 825 (no calculators allowed!) is invited to share it here.

(Cross posted at Out in Left Field).