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Some day, my princess will come. And she'll be passionately interested in discussing questions of actual teaching practice in mathematics classrooms. Actually, of course, many such exist, but getting them together on this blog or in open, active public forums, having the time to participate actively, etc., isn't always a trivial task. For full-time teachers, time is always at a premium, and while there are many good resources on the Internet, tracking down what would really be productive for one's own work can be difficult. From my perspective, one of the shortcomings of even the best conversations about practice, however, is the lack of access to specific classroom data, particularly video with supporting documentation (teacher journals, samples of student work, assessment instruments, observer notes, etc.) that would allow other practitioners to engage in deep analysis of what goes on in mathematics teaching based on common access to the same materials. As a former litera...

Above is a picture of John N. Warfield , "the author of two U.S. patents on electronic equipment, and the inventor of Interpretive Structural Modeling, Interactive Management, and General Design Science." Professor Warfield's work has had a significant influence on the contributions of G.S. Chandy to the math-teach@mathforum.org list. In connection with my recent blog entries about "important open questions in mathematics pedagogy," (" Is Mathematics Teaching A Closed Book? "), Mr. Chandy has tried to offer an approach to discussing these and related points of contention on that most contentious of lists that is grounded in the work of Professor Warfield. I do not profess to know Warfield's work, but I found what Mr. Chandy posted to be sufficiently heuristic to warrant posting what he wrote in its entirety in hopes of generating interest among readers of this blog, as well as those on various lists to which I contribute who many not have ventured ...

In response to one of my previous posts on the issue of meaningful open questions about mathematics pedagogy, an anonymous contributor wrote in part: "How can I teach a student at level N who hasn't mastered all the work of level N-1 (or maybe even N-2 for some things)? What can be done to help with the learning of the current content while remediating gaps in learning?" This is a particularly nagging problem for most mathematics teachers in K-12 and beyond, though at the college level it is easier perhaps for instructors to be blithely condescending and dismissive of the problems of students who come to class ill-prepared for the level of coursework expected of them. College is, after all, a choice, and professors are not obligated to provide remediation, although there are generally non-credit classes offered that do just that, or at least purport to do so. I wrote on math-teach@mathforum.org in response to the above question: The first question is nearly ubiquitous in ...

The multiplication issue continues to live on in various quarters, and now Keith Devlin, who started the current round of debates, discussions, and arguments, has posted a third column on the subject, " Multiplication and Those Pesky English Spellings ." I'm still chewing over what he says there, but I can report with pleasure that this blog, along with three others, is mentioned favorably in his new column: Of the blogs I looked at, which had threads devoted to my "repeated addition" columns, the following all had some good, thoughtful comments by their owners and by some of their contributors - by no means all agreeing with me - though the discussion in "Let's Play Math" soon descended to uninformed and repetitive name calling, and the owner eventually closed the thread, which unfortunately soon reappeared elsewhere. * http://letsplaymath.wordpress.com * http://www.textsavvyblog.net * http://rationalmathed.blogspot.com * http://homeschool...

While some ideologues are busy denying that there are any open questions in mathematics pedagogy, lots of bright folks are actively exploring many such meaningful questions. Only those who do not teach (and some who do) could believe that there are neither meaningful questions about how to more effectively teach mathematics nor people actively engaged in pursuing answers to them. But such is the nature of the Math Wars and the current American adaptation of The Big Lie strategy: small cadres of dedicated and destructive individuals would sooner invest in trying to undo the good work of others than to do anything original and creative themselves. As Kandinisky said so well in " On The Problem of Form ," This evolution, this movement forward and upward, is only possible if the path in the material world is clear, that is, if no barriers stand in the way. This is the external condition. Then the Abstract Spirit moves the Human Spirit forward and upward on this clear path, which ...

Hans Freudenthal A. Dean Hendrickson It's never a good idea to get involved in a fight with an Internet "ghost." But if you play in the sty formerly known first as nctm-l@mathteach.org and now called, ironically, math-teach@mathforum.org, you can't avoid it. One of the more prolific voices there is a fellow who posts under the name of "Haim Pipik " (if you don't "get" the Yiddishism, you're not missing much). However, a few of us who've been around the Math Wars for more than a decade know him better as Edmond David (6th from left standing), a resident of Brooklyn, erstwhile member of NYC-HOLD, self-identified "NYC parent," and a fellow who doesn't mind slinging mud, pushing his quasi-libertarian, transparently right wing and vehemently anti-liberal agenda behind his pseudonym. All in all, not the most courageous guy in the Math Wars or any other battle: by their lack of courage shall ye know them, I suppose. Ordinarily,...