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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 ...