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In 1998, as California was in its first year of a counter- revolutionary mathematics curriculum framework thanks to efforts led by anti- progressive mathematics education groups like Mathematically Correct and HOLD, Richard Rothstein's THE WAY WE WERE? The Myths and Realities of American's Student Achievement was published as a Century Foundation Report. The first chapter should be required reading for anyone who thinks s/he knows about the history of American public education. It explores and debunks many widely-believed myths about "falling" achievement in literacy, general academic knowledge, and the whole notion that our public schools are failing us, kids today are far less competent in basic skills than were those of previous generations, and that our lousy schools are paving the road to hell for our kids and our nation. In particular, Rothstein examines two currently hot-button topics - social promotion and bi-lingual education - and reveals many surprising r...

Last time, I explored some ideas about what comprises good mathematical questions in mathematics classrooms taken from MAKING SENSE: Teaching and Learning Mathematics With Understanding, by Hiebert, et al. In this entry, I want to introduce ideas from a book by Peter Sullivan and Pat Lilburn: Good Questions for Math Teaching: Why Ask Them and What to Ask [K-6] Sullivan and Lilburn's Criteria In the introduction to their book, Sullivan and Lilburn list three main criteria for good questions: a) They require more than remembering a fact or reproducing a skill; b) Students can learn by answering the questions, and the teacher learns about each student from the attempt; and c) There may be several acceptable answers. The first of these features, while not something everyone sees as important in all mathematics classrooms, is essential to non-routine questions. It takes no teacher skill to come up with questions that only ask for students to show that they have memorized a fact (e.g...

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...