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One glaring weakness in many American mathematics classrooms is the nature of questions teachers ask. This manifests itself in two pedagogical realms: first, the kinds of mathematical questions or tasks teachers tend to pose are often of a closed nature which call of nothing more than a calculation (or short series of calculations) and for which the answer is a single number that requires no further interpretation or thought on the part of the student. The second area of difficulty arises in the types of questions teachers ask when they want to help students who are having difficulty performing a given task (which generally consists of something like that mentioned previously, although this problem arises in more complex problem situations). My focus here will be on the the first situation: what kinds of tasks might comprise "good questions" for math students in elementary school? Mathematical Tasks In their book MAKING SENSE: Teaching and Learning Mathematics With Understan...

The following appeared in a video at a Ridgewood School Board meeting about the INVESTIGATIONS IN NUMBER, DATA AND SPACE elementary mathematics program in use there: "The literature of reform math has changed history, painting a dichotomy wherein its proponents state that all former math teaching required memorization of algorithms without understanding and that students had to “check their brain at the door” before math class. I didn’t invent that expression. The official TERC Investigations web site reads, “otherwise intelligent and curious children who check their brain at the door as math time begins.” That’s how they characterize 'math before TERC.'" Is this an accurate quotation? Yes, as far as it goes, and up to the word 'begins.' But the last sentence is an outright lie, as is her claim in her opening sentence. Look at the context of the quotation as it appears on the TERC site: "Teachers see it all the time: otherwise intelligent and curious c...

The following recently appeared on a blog of one of those fiercely opposed to reform mathematics instruction in Ridgewood, NJ. "GK" stands for "Gifted Kid" GK prefers fractions and division to simple multiplication A mom of a Gifted Kid gave him some multiplication drill. She keeps it to a minimum, but once in a while it's a good idea. However, it turns out she didn't give him a hard enough problem. Here's the problem she gave her 9 year old and the expected partial products and final product: 4.2 X .24 ---------- 168 840 ---------- 1.008 But that's not how GK worked the problem. Instead, he did all this in his head: 21/5 X 6/25 = 126/125 = 1 1/125 = 1.008 Just like with reform math, Mom asked him to explain his reasoning, but not because of any ideology about language and math, but rather, because she had no idea what he had done until he spelled it out for her. S-L-O-W-L-Y. Then she asked him why he didn't use the standard way, and...

The following appeared on-line today: "Math. Throughout my childhood, adolescence, and most of my adult life, the very thought of the subject struck genuine fear in me. I thought about fractions and decimals or even addition or subtraction and would actually feel pinpricks of sweat break out on the back of my neck and the palms of my hands." I found myself wondering, as I read one person's glib comment that it "is unfortunate that Denise Noe did not have the same type of teachers that I had," if the terms "math anxiety" and "mathphobe" aren't misnomers to some extent. That is to say, the implication seems to be that there's something about mathematics that is inherently anxiety-producing. While anything is possible, and unlike my esteemed colleagues on the other side of the Math Wars, I don't profess to know any great universal truths, I suspect that "math anxiety" is not something that would occur naturally in many pe...

There is a group of numbers for which the process previously described won’t work. For example, try to use it to find the square root of 100. Grouping as before: 1 | 00 Subtracting 1 from 1 = 0. Write 1 above the 1, bring down the next pair of digits, 00, and append to the 0. Multiply 1 x 10 and add 11 = 21. Can't subtract 21 from 0. Hmm. Although we know the answer is 10, to make things work, we can note the following, which is Rule #3: If you want the square root of a whole number that ends in two or more zeros, write the number as a product of a number and an even power of ten. So 100 = 1 x 10^2. We get that the square root of 1 = 1, append one zero for every pair of zeroes in the original number, and Bob's your uncle. (Or something like that). For example, to find the square root of 3,610,000, remove two pairs of zeroes from the original number, then apply the original procedure: Group: 3 | 61. Subtract 1 from 3 = 2 Can't subtract 3 from 2, so write 1 above the 3, brin...

One algorithm for doing square roots by hand was recently mentioned (though not actually yet described in any detail) on math-teach@mathforum.org. It was averred that finding square roots required estimation. I posted part of an algorithm I taught to teacher-education students last year that does not rely on estimating, but instead uses subtraction of successive odd integers. What follows is part of the story using two examples that illustrate most of the situations that might arise. The third situation, as well as how to deal with the square roots of non-perfect squares will be posted later. If there are typos, let me know: As usual, there are more ways to the woods than one. Here's one in which estimation is NOT necessary. As mentioned in a previous post, I used this with elementary education students in a math content class and we most definitely DID explore why it works. If anyone is interested, I can send the labs that were used to explore this approach, first in base five (Fe...

Recently, on math-teach@mathforum.org, the issue of whether changes in the standard K-12 curriculum are a natural, reasonable reflection of changes in both society and education, or whether they are yet another harbinger of the coming of the Apocalypse. The debate centered on an opinion piece by Stuart Wachowicz in which he begins, “Pride in craftsmanship obligates the mathemati- cians of one generation to dispose of the unfinished business of their predecessors.” -E.T. Bell, The Last Problem The above statement most accurately describes the legacy of one generation of mathematicians to the next. However, on might be tempted to ponder whether this will continue to be possible in North America. The dis- cipline of mathematics, as we have known it, is clearly under threat. The threat is a consequence of allowing cur- riculum writers to change the centuries-old definition of mathematics and what needs to be learned based on utili- tarianism, combined with the current practice of allowing ...

The following appeared recently on the Eclectic Educator blog (6/14/07) and is an excellent example of some of the countless contradictions that seem to inform so many of the opinions one reads about mathematics education from the anti-reformers: " 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 par...

It's amazing the depths some folks will plumb to smear, defame, and intimidate decent people, while justifying their anonymity because of the repercussions they maintain will fall on their heads from teachers and administrators. In this regard, it seemed timely that Yahoo! reported yesterday that police departments in Boston have set up a way for citizens to use text messaging anonymously to offer tips about crimes. Not that I oppose responsible citizens having a safe way to report real criminal activity, but this seems like the perfect way for our Ridgewood friends and many like them to further attempt to assassinate their perceived enemies with impunity. http://news.yahoo.com/s/nm/20070615/us_nm/usa_crime_telephones_dc

I see that I've erred in stating that it was Greg Goodknight rather than the more carefully "shielded" 'Haim Pipik' who accused me yesterday of paranoia. My apologies both to Greg for mistakenly accusing him of both calling me that and for playing at amateur psychologist in this regard. In fact, Greg accused me of "projection" yesterday, in his actual failed attempt to be a, as he put it, "shrink"; and of course I apologize deeply to "Haim Pipik" as well, for not properly giving him credit for this particular bit of idiocy. It's tragic that when being attacked by two such mental giants as was the case for me yesterday that I relied on my short-term memory rather than simply checking back through the flood of hate e-mails I got from these two charming individuals.

Yesterday, I was falsely accused of paranoia on the math-teach@mathforum.org list by amateur psychologist and professional know-nothing "Haim Pipik," yet another anonymous foe of progressive politics and education. While I and others believe we know precisely who uses that nom de net, efforts to get him to drop his charade and simply admit who he is have been in vain. Of course, in the wake of the travesty of democracy we have witnessed in Ridgewood, NJ this week, thanks to the clandestine efforts of a tiny group of activists, some of whom don't even live in New Jersey, let alone in Ridgewood, "paranoia" becomes a very relative term. By making anonymous calls and sending anonymous e-mails, this little group of brown shirts undid the legitimate hiring of a top-notch superintendent because he'd written favorably and intelligently about constructivism and was perceived to support the legitimately selected K-5 math program in Ridgewood, INVESTIGATIONS IN NUMBE...

As promised elsewhere, I'm going to look carefully at what Everyday Math offers about multi-digit multiplication, how it advises teachers to teach this topic, and what I've seen in practice (and how I attempted to rectify problems with how it was being taught in some classrooms). First, from p. 140 of the first volume of the EM 4th grade teacher's manual, the very first thing it says about multiplication. If you've never seen this before, it's no doubt because no one who attacks EM would dare quote it to you. "The authors believe that most students should learn the basic facts for addition and multiplication to the point of instant recall. Mastery of the basic facts will give students surprising power in making quick estimates and operating with larger numbers." I hope you were sitting down when you read that. Surprising, isn't it, if you've been getting your propaganda feed from various hate blogs or the Mathematically Correct/HOLD websites and th...

Dear Dr. Brooks: I'm very saddened to learn that you've chosen not to take the Ridgewood, NJ position. I grew up next door, in Fair Lawn, and graduated high school in 1968. I'm now a mathematics educator based in Ann Arbor, MI. I am an active supporter of good, high-quality mathematics teaching and follow the "math wars" closely and with great interest. I became aware of Linda Moran and her band of activists earlier this year, and then learned through reading her lists and blogs that you were taking the job there, much to my pleasure. I've read your book on constructivism with pleasure and had looked forward to seeing you keep things going in the right direction in Ridgewood. Unfortunately, your withdrawal, which I can understand must have been an unpleasant personal decision, has given a great deal of satisfaction to the anti-progressives in Ridgewood. Surely you understand that they represent a small minority. Had you come, you would have found great support...

I saw Liping Ma speak at Rutgers University on July 19th, 2000. I took extensive notes during her talk, and in thinking about the many misrepresentations I have read of her actual thinking about mathematics education, both here and in Mainland China, I decided it was time to re-read KNOWING AND TEACHING ELEMENTARY MATHEMATICS, the book that shot her into the limelight in the midst of the on-going fight over what mathematics teaching and learning is about. It has disturbed me to no end to see here balanced and reasonable ideas distorted and "reinvented" and misused to justify what is usually referred to as "traditional" math teaching in the United States, both by activists in national groups like Mathematically Correct and NYC- HOLD, and on local anti-reform blogs and discussion lists such as Linda Moran's beyond-terc and many others. One would think that Professor Ma had at some point signed a secret accord with such folks, but my best guess is that she remains ...

I urge everyone to read the petition and seriously consider signing it. Forward it, link to it, get the word out. Michael From: Susan Ohanian Date: June 10, 2007 Subject: Re: [eddra] Announcing the Rational Mathematics Education blog Educator Roundtable invites all of you to join the grassroots movement. Sign the petition to end NCLB. http://www.educatorroundtable.org/petition.html Buy 100 booklets explaining, in dispassionate, factual fashion, just what is IN NCLB. The booklet, written by a teacher/researcher, Elizabeth Jaeger, is written in the hopes that activists will distribute information to the community. We can start the revolution if each of you hold a conversation with 100 people. We are selling the booklet at cost: 100 for $50. Our object is not to sell booklets but to start those conversations. You can send me your snail address, and I will send you the booklets. Susan Ohanian

I found this quotation on an anti-Everyday Math site this morning: "To have a great idea, have a lot of them." Thomas A. Edison Could anything be more ironic? This site has links to the NYC-HOLD "Math Myths" piece I'm currently interrogating, and the blogger makes clear that those "multiple methods" for doing arithmetic that are developed in the Everyday Math books (and in other curricula such as INVESTIGATIONS IN NUMBER, DATA AND SPACE (aka, "TERC" by its detractors, who apparently find it hard to say or type "INVESTIGATIONS") are anathema to its predictably anonymous author. I imagine the wrath of the entire state of Connecticut and the mythical "Education Mafia" will come down on the author's head and the heads of the author's child(ren) for criticizing Everyday Math. :) In any event, Edison calls for lots of ideas in order to have a great one. The reform haters want ONE idea about EVERYTHING in and out of math...

As readers familiar with the Math Wars know, one of the most active groups who attack reform and progressive ideas in mathematics education (and other educational areas) calls itself HOLD, which ironically stands for "Honest, Open, Logical Debate." That is a grand slam of linguistic abuse, given that there is nothing honest or open about this group of angry professionals and parents, they do NOT welcome debate, and while there is sometimes logic in their arguments, just as there is a kind of logic in MOST opinions, that doesn't make what they assert true. If you start with false assumptions, you can draw some lovely logical conclusions that are utterly false, even if they are logically valid. While HOLD began in California as part of the assault on whole language and reform math, it leapt across the country to form a second (and as far as I can tell, the only active branch), NYC-HOLD. What has made this offshoot effective is that many of its members are professional mathe...