" I think the methodology of Singapore is easy to learn. What's been difficult for American elementary school teachers, from my research so far, seems to be their lack of understanding of mathematics at any deep level.

It's interesting to me that with Singapore math being in some ways similar to reform math (although in my opinion much better) it still requires lots of drill beyond the school day.

It seems that when reform math was implemented here, that was another blind spot. It didn't occur to anybody that kids would need a whole lot of drill beyond the school day.

And so parents, not having been given warning, are tutoring as a band-aid instead as a way to complete the program. They're doing so unevenly, and some parents don't even realize its necessity.

There are myriad reasons, including what I've explained above, that reform math is failing in our country.

At least in Singapore it's built into the culture that kids need tutoring for drill outside of school. Not to mention that in Singapore they don't expect the kids to discover all this deep understanding on their own."

What is problematic here is that the Eclectic Educator ( in fact Ridgewood, NJ blogger and anti-reform activist, Linda Moran, someone who has been central to the recent storm over the use of the INVESTIGATIONS IN NUMBER, DATA, AND SPACE curriculum and the withdrawal of acceptance by Martin Brooks of the superintendency for the Ridgewood Public School district), seems more than willing to accept reality when so doing explains and excuses difficulties in implementing a favored program, Singapore Math, for which she openly advocates, but will not accept this same limitation as a reasonable explanation for difficulties associated with INVESTIGATIONS, EVERYDAY MATH, or other programs which she dislikes.

Most mathematics educators in the US acknowledge that the vast majority of elementary school teachers here lack what Liping Ma has called "profound understanding of fundamental mathematics" (PUFM). I would agree on this assessment, based on all my experiences as a student, teacher, parent, teacher-educator, researcher, and field-supervisor for student-teachers. Very few of our teachers or would-be teachers can, for example, give an appropriate word-problem to illustrate division by a fraction. Ma, drawing on research by Deborah Ball, now the dean of the University of Michigan School of Education and professor of mathematics education, showed that some of our teachers cannot even do the indicated operation when asked to solve and create a context for 1 3/4 divided by 1/2. Not surprisingly, many wind up with problems that would be appropriate for dividing the number IN half (it's important to note when using examples like this whether the problem is given orally, because it's not clear in such cases if the solver has misheard him/herself being asked the wrong thing. If the problem is written down, that explanation doesn't obtain).

Of course, one issue that arises in trying to understand why K-12 students, teacher education students, or in-service teachers struggle with fraction division is the mechanical way arithmetic has been TRADITIONALLY taught in this country, going back at least as far as the 1950s, when I was in school (but in fact, from most evidence, much further than that). We are told early on, for example, that multiplication "makes things bigger" and that division "makes things smaller." This perspective becomes very deeply ingrained in the constructs we all carry with us about arithmetic. So while it is sensible to say that for positive dividends and divisors greater than one, the quotient will be smaller, but things get more complex when one allows for rational and negative numbers, and the various combinations and general rules for what happens to the quotients actually require some thought. If one divides -3 by 2, the resulting quotient, -3/2 is actually greater than the initial dividend. If one divides 3 by 1/2, the resulting quotient, 6, is also greater than the dividend. But 3 divided by -1/2? The quotient is -6 which is of course smaller than 3. And so on. The various possibilities can be very confusing for adults, let alone for young children. However, it is imperative that teachers understand what's going on and why, and that they have sharpened number sense so that none of these outcomes strikes them as at all surprising if they are going to be able to explain, model, and teach these concepts to kids.

Unfortunately, as Ball's and Ma's research makes clear, such is far from the case. But this isn't new (though it may be news to some), and it didn't emerge suddenly the day the 1989 NCTM Standards was published, or in the 1990s when the first NSF-funded curriculum projects were being developed and piloted. Far from it.

Why won't the anti-reform extremists admit this? Because to do so would be to admit that the entire approach that has been used in the United States to teach the vast majority of our students is woefully mechanistic, grounded in blindly-taught and blindly-followed (often poorly in each case) procedures, rather than deeply thought-about and investigated concepts in order to master strategies and algorithms. A huge part of the Math Wars is over this issue, but sadly one gets mostly heat and virtually no light if one reads only what the anti-progressive haters say on the subject.

To read most of them is to be taken to an imaginary land and time in which all American children were taught by highly-knowledgeable, patient, insightful teachers who knew the math and understood children (all children, of course) and how they learned (and it goes without saying that all children learned the same way, at the same pace, and at the same age). If you're reading this and think I'm exaggerating, I can readily point you to thousands of posts to various math discussion lists that are based on assumptions like this. Further, there seems to be an enormous emotional investment on the part of some people to ignore the fact that millions of Americans were ill-served by the mathematics education they were offered in the post-war era (and, likely, before that). It's now a cliche, but nonetheless true, to point out that mathematics teachers are told all the time when they state what they do for a living, "Oh, I was always awful at math. I always hated math. I never could do math," etc., something English teachers are almost never told openly by adults regarding reading and writing. Illiteracy is a mark of shame in this country, but innumeracy is nearly a badge of honor: "Yes, I've been brutalized by math, just like you, and I'm STILL bad at it. Ain't it awful? (Nudge, nudge, wink, wink)."

Thus, it is an odd aspect of the Math Wars that not only some mathematicians and advanced end-users of mathematics (engineers, physicists, etc.) think that the way they were taught was great because clearly it worked for them (and hence must be the best, only valid way for math to be taught to everyone), but also those who themselves suffered and withered in math classes will decry so-called fuzzy methods because "I know math, and that isn't math." Of course, it IS math, but it may not be the same approach to it that they didn't connect with when they were in school.

Not everyone is blinded by either nostalgia or a kind of "frat boy" or "no pain, no gain" mentality ("I suffered in math class and my kid will too, if s/he knows what's good for her/him!"). Some very successful mathematicians and math users recognize the limitations of the methods they saw as students and have generally used themselves as teachers. Some who failed to thrive in math realize now that it was the limited methods and limited teachers who failed them, not their innate ability to do math. Which brings me back to Ms. Moran and her comments on Singapore Math and her blind, unrelenting hatred of INVESTIGATIONS and other progressive math programs.

I will not attempt to interrogate her motives in my currrent entry, but I will point out that everything she says about the weaknesses of American elementary school teachers in mathematics is directly relevant to understanding the difficulties we face in moving forward towards more effective mathematics teaching and learning in the United States. Hypocritical selectivity in to whom one gives a pass (e.g., Singapore or Saxon Math) and whom one takes over the coals (e.g., INVESTIGATIONS, EVERYDAY MATH, TRAILBLAZERS, and a host of middle and high school programs) serves our children badly, while serving to make any reasonable examination of what works, for whom, and possibly why it does just about impossible.

I knew before I began this blog that: a) I would come under personal attack from people both anonymous and on occasion honorable enough to give their real names or identities; b) that I would be accused of being a flack for one or more programs or for NCTM, or something of the kind (I've never worked for any of the projects or publishers, though I have worked for and with school districts that use progressive curricula, and I have taught a couple of high school programs that Ms. Moran would no doubt criticize as fuzzy and the like); c) I would be criticized for not being "fair" in my entries or my responses to those whose comments I find myself in disagreement with: while I never promised anyone a rose garden OR fairness here, I have yet to edit or block a single response from anyone. I've made the existence of this blog as public as possible, and expect that sooner or later it will draw much harsher criticisms and comments than it has thus far (at which point I suppose I'll face some interesting decisions). But the blog IS, after all, mine, and I will respond to critics as I see fit, just as I expect they would do were I to post in spaces they control. As stated, that's not been an issue for me, so far, but if I'm supposed to "roll over" "play dead," and generally be nice because someone thinks I'm being a bit too slanted, I will only point out the obvious fact that this is a blog, and my slant is precisely the one I'm planning to offer.

That said, I think that EM, INVESTIGATIONS, and many other reform programs are valuable first- and second-generation attempts to carve out new directions for mathematics education. (And I see good things in Singapore Math we can use, and even some useful things, though far fewer of them, in Saxon Math). Reform programs have been around long enough to be criticized responsibly and to have undergone revisions (for which, ironically, they are ALSO attacked by the extremist critics, as if other textbooks emerge from the heads of their authors and publishers perfect and inviolable and never need corrections, rewriting, or wholesale changes).

They are not, however, the final word on where to go with mathematics teaching and learning in K-12. Nor will anything ever be. Math grows and changes. So do our uses for it. So does our culture, our technology, and the tools we have available. The very nature of school is changing, though often painfully, almost glacially slowly.

There are interesting things going on that don't involve textbooks at all, something that should be utterly unsurprising if you're reading this on line. There are projects that see computer science and programing and applications of programming as central to learning mathematics effectively. Buckminster Fuller long ago predicted approaches to learning that anticipated the availability of video lectures "on demand" to people who would not need travel to a central location like a university to gain access to a college education, and it is not unreasonable to think that Mr. Fuller would have found math education via books alone, viewed during lectures in traditional K-12 classrooms unbelievably archaic and anachronistic were he alive today.

The bottom line isn't INVESTIGATIONS vs. Saxon or Singapore Math vs. EM, but rather an overall understanding of how kids learn and how we can best help them learn and decide what they want and need to learn. The Math Wars, regardless of what the anti-progressives tell you, have never been about the math content, but rather about education and what it means in a democracy. We have so many good choices, but also so many bad ones, and the ones that have for the most part failed us in the past are not the way we should be pointing now. If schools and math teachers aren't to go the way of the evening newspaper and even network nightly news and the reporters and anchors whose livelihoods depend upon them, they need to look towards the future, not the past alone. The lack of PUFM among our teachers, the low value we place upon K-12 teaching and especially K-5 teaching, and our cultural attitudes and beliefs about mathematics teaching and learning are far more central to whether we will raise the level of numeracy in this country than are the choice of math texts. This isn't to say that such choices don't suggest a lot about the vision of those making the choices, but I am sure that with or without a text, or with a text one doesn't completely favor, a competent, motivated, imaginative teacher can bring mathematics to life, and a student willing to engage that teacher fairly will have every fair chance to reach his or her maximum potential in mathematics and beyond.

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