Yes, Keith, and China is the weakest example of the three.

]]>Classic empty assertion. “The Chinese” is a meaningless phrase in this context and I know you have no reliable data, if any data whatsoever to support a claim that “X has no problem with them,” for any national group, but particularly not for “the Chinese.” And you know perfectly well why not: China is not an open country. No American has reliable access to educational data there. That’s one reason that Liping Ma’s Chinese data, while interesting, can’t be presumed to represent the nation’s entire school system. Rural schools in China are reputed to be awful, but there’s nothing in her book to reflect that.

As for “The Japanese” and “The Koreans,” I trust you understand that they don’t exist in any generic way, either, though we can probably feel more confident in data from Japan. By Korea, of course, you need to specify which one. Do you think the massive malnutrition that’s been reported there might have any impact on educational outcomes over time?

So please, Ze’ev, try not to sneer quite so readily when you don’t really know for sure what goes on in these countries. Do any S. Koreans or Japanese kids struggle with math? Why? Do you know? Do the Japanese know? They have lovely texts and wonderful teachers, but there’s no way in hell that all the kids do well.

And of course, your bluster doesn’t address a much deeper question: why do Japanese and S. Korean educators feel that they have a hard time getting many students to excel in being able to do more than score high on exams? That should be a worrisome consideration to anyone who realizes that doing well in school is the booby prize if you graduate without being able to be good (however that’s measured) in the real world outside of academics. You should know that, of course, not being an academic.

Be honest, Ze’ev: the issue for you is that you have a narrow politicized viewpoint of what it means to be good at math (and, as an engineer, you may not have quite the same one that typical professional mathematicians hold). You do know mathematicians who have the same views of K-12 math education you hold, so you can always cite a James Milgram here or a DIck Askey there to “prove” that you’re right. But please: Keith Devlin, Hyman Bass, and many others who hold contrasting views to those are not exactly chopped liver.

So while I know every argument you’re going to make, the intensity and sarcasm you bring isn’t quite the same as proof. Or logic. Making a “cutting” comment to Keith Devlin about his mathematical proofs should be beneath you, but, sadly, it isn’t. And thus, you lose.

]]>I have traveled to Japan and Korea many times and colleagues there tell a different story. I have no direct information about China, but you are down to one example.

]]>The Chinese seem to have no problem with them. Nor the Japanese or Koreans. So is it a genetic fault of the Americans, or is it bad teaching?

I hope you do better proofs in math.

]]>Those methods are in fact the original algorithms! The name algorithm is derived from al-Khwarizmi, the 9th century mathematician in Baghdad whose book on those methods introduced them to the Muslim Empire, from where they spread to Europe in the 13th Century.

The fact that they do not work well educationally is made abundantly clear by the fact that, though they were used as the standard educational procedure for arithmetic for hundreds of years, the majority of people, even today, are not proficient in arithmetic and exhibit little real understanding of the place-ten number system.

They may be “easy to teach”, as you note, but the overwhelming evidence is that they are difficult to learn, and indeed, most students * do not learn them*! (They certainly did not when I was in elementary school, and I am now in my mid-sixties! I remember being one of the few students in my class who “got them”, and only after a long, hard struggle.)

But still, isn’t the use of those methods a good way to instill the “powers of 10” idea? I saw a YouTube video of a student demonstrating a “new way” of adding 3 several-digit numbers (during which she said “we weren’t allowed to stack them”) by drawing and combining rectangles. After 4 or 5 minutes, she got the wrong result. I don’t see this as a great leap forward.

I will certainly agree that the standard methods are not the most efficient or elegant – just ask Dr Benjamin of Harvey Mudd. But they are simple, they work, and they’re easy to teach.

For those of us who have left them far behind, and rely on the ubiquitous calculator, they’re relics. The problem is that the calculator gives us a false sense of accuracy (when precision is something different).

]]>This thinking is why I support school choice. Do you? I’m part of a big world out there that is not going to hop on any bus, progressive or traditional; democratic or republican. I know what needs to get done in math education to keep all STEM doors open. With the PARCC test for CCSS, many have shown their true colors. It’s not progressive or traditional. It’s low expectations for all.

I’m a big fan of El Sistema, which many claim to be very progressive. But is it? Many won’t “get it” if they are stuck with a traditional versus progressive mindset. Something else is going on. “Music for the poor should not be poor music”. Likewise, academics for the poor should not be poor academics, no matter how much anyone talks about understanding and critical thinking.

]]>Chip Hedler: that’s a fine comment you’ve made, one which hits many key points about what’s been going on here and on the NYT site. You’ve also nailed the obfuscating abuse of Wittgenstein by Crary and Wilson, something I recognized but felt unqualified to argue convincingly (not to mention that Americans fear and loathe philosophy at least as much as they do mathematics).

What resonates most is your last paragraph. I don’t know that I could articulate those points any better than you’ve done. Those with the ears to hear and eyes to see may just get something from contemplating what you’ve written. Unfortunately, that pretty much eliminates the vast majority of those who continue to equate the learning of mathematics with suffering and tedium. And think that both build a child’s character.

]]>One big point to get clear on: the debate about algorithms is NOT about the role “basic facts” play in computational efficiency or deep conceptual understanding. I can’t imagine that either Keith Devln or Michael Goldberg would not want all students (or mathematicians) to have relationships such as 6*7 immediately available without reconstructing them through repeated addition. As for expressions such as 2/3 divided by 4/7 and 1/x^2 divided by (x-1)^3/2x, I hope both of them and both of you would want students to understand how to simplify them efficiently without error. The “debate” about algorithms, when you boil it down, is really about something else: methods for computing the results of arithmetic operations involving multiple digits. As Keith Devlin points out, the “standard algorithms” were designed for rapid handwritten computation. Another extremely efficient alternative to them in other cultures (with a traditional standing that goes back much farther than the mid-20th-century textbooks most “math traditionalists” use as a reference) has been the abacus, the use of which requires at least as much genuine conceptual understanding as the “standard algorithms.” Never mind that many of the adults who claim that 50s-style U.S. math instruction was “good enough for them” and “if they could do it, so should today’s kids” can’t, or won’t, do the math when it comes to environmental change or economics.

Steve, our district dropped Everyday Math for many of the reasons you mention. When our district adopted it, teachers who had enjoyed teaching 1st- and 2nd-graders to love math using truly progressive (and very far from traditional) teacher-developed approaches such as Math Their Way for more than a decade suddenly found themselves struggling to keep kids who were learning to dread math in touch with a bewildering maze of specialized workbook presentations of a curriculum that felt to students and teachers alike miles wide and barely inches deep. When it was clear we needed a different program, we quickly rejected Mathland for a multitude of reasons, choosing Investigations not for its progressive-sounding name but because it offered intellectually rigorous, developmentally-appropriate activities that provided conceptual discovery, reflection, and practice to strengthen skills in applying concepts. Investigations isn’t perfect. And our district is now trying to figure out how Investigations will fit into meeting the challenges of the Common Core standards, not to mention the tools being developed to evaluate student progress (a whole ‘nuther HUGE topic). One of my beefs (beeves?) with many of the commentors to the NYT column is that they tend to lump everything that isn’t their idea of traditional math program, from the New Math of the late 50s to Common Core, together as if it was a monolithic juggernaut bent on promulgating some subversive progressive educational (read: political/cultural) agenda. That’s like lumping everything from free-form post-bop jazz through atonal symphonic pieces through advertising jingles to rap music together, calling it “modern” and criticizing it because it’s not music from the canon of European classical composers.

It’s notable that the references by Crary/Wilson to Dewey, Sellars, Wittgenstein, and Plato didn’t get much reaction pro or con from academicians or practitioners in the field of philosophy as opposed to math/education — but I strongly suspect that Wittgenstein at the very least would drop his jaw in disbelief at how his extremely unconventional approach to the intellectual basis for mathematical thought had been mis- or re-interpreted and reduced to a couple of one-liners to defend the use of the “traditional algorithms” against what Crary/Wilson want us to believe is a simplistic progressive interpretation of them as “merely mechanical.” And I feel even more strongly that neither Crary nor Wilson would want any part of a math curriculum as envisioned by Wittgenstein.

Perhaps the most troubling implications of viewpoints such as the Crary/Wilson column is what may not be articulated explicitly, but hinted at by the stress on terms such as “traditional” and “progressive” which are code words for a much larger debate within American society. This debate is really about submission to personal authority. The stereotype of the mid-20th century teacher is the battleax who drills the students on math facts and algorithms whether or not it’s easy or makes sense or is fun with suitable punitive measures as needed to keep thing moving because “it’s good for them, whether they know it or not.” Actually understanding the methods and goals of “progressive” math approaches isn’t important — all you need to know comes from buying into the symbolic image: progressive math = adults wanting to please kids (who “know” little) instead of kids wanting to please adults (who “know” best). It’s spookily reminiscent of the analysis Swiss psychologist Alice Miller made of destructive family authority dynamics in Books such as “For Your Own Good.” Fear begets, and wants to beget, fear.

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