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November 14, 2006 [feather]
Crunching numbers

How a parent discovered that her child was not being taught math properly in school:

"When my oldest child, an A-plus stellar student, was in sixth grade, I realized he had no idea, no idea at all, how to do long division. ... so I went to school and talked to the teacher, who said, 'We don't teach long division; it stifles their creativity.'

The New York Times is all over the reform math curriculum that has, for more than a decade now, "crippled students with its de-emphasizing of basic drills and memorization in favor of allowing children to find their own ways to solve problems." The article is worth reading to see how the debate is shaping up, and also to get a sense of how much more involved parents need to be in their kids' education.

The mother who found that her sixth grade son could not do long division is an involved mother--and when she learned that her son was being groomed by his teachers for a lifetime of innumeracy, she struck back by founding Where's the Math, a parents' group dedicated to ensuring that kids get decent math instruction in schools. But she was a bit slow off the mark -- I learned long division in an Indiana public school in the fourth grade, when I was nine. Granted, that was during the seventies, but my guess is that in states that still believe in long division, that's still when kids learn it. Parents need to be checking in much more regularly and inquisitively with their kids about what they are learning when--and what they aren't learning at all.

And parents shouldn't only be concerned about math instruction. They should be looking hard at the reading and writing parts of their kids' educations, too. Are they learning grammar? Can they spell? Punctuate? Understand what they are reading? Most of the Ivy League English majors whose writing I grade have trouble in these areas, which suggests to me that most everyone their age does. I tend to assume that the students I see are among the most linguistically competent students of their generation--but there are still a lot of issues with things such as run-on sentences, comma splices, murky phrasing, limited vocabulary, dangling modifiers, spelling, and so on. That's the legacy of a pedagogical attitude toward literacy that mirrors the one the mother above encountered when she inquired why her son wasn't being taught basic math skills. When I taught high school English in a boarding school a couple of years ago, I found that a great many students there had abysmal language skills. Some bordered on functional illiteracy. When I asked whether the school taught grammar at any point, the head of school told me that teaching grammar thwarted students' creativity and stifled their interest in reading. The utter inadequacy of that outlook really hits home when you realize that it amounts to lying to parents and kids about their kids' abilities, and that it involves sending kids off to college without the skills they will need to succeed there.

posted on November 14, 2006 7:45 AM


This is an old story. The clearest signpost of official sanction of content-free math was the publication of the National Council of Teachers of Mathematics (NCTM) Standards in 1989. These standards dealt as much with method as with content. Skills, the importance of correct answers, and traditional methods were 'de-emphasized.' Discovery Learning was favored over direct teaching. Students were to discover the methods and great ideas of mathematics for themselves. Teachers were supposed to go from being the 'sage on the stage' to being the ' guide on the side.' Practice to mastery was denigrated as 'drill and kill.'

These ideas, similar in underlying philosophy to the Whole Language method of teaching reading, found, and still find, numerous supporters in education schools and school administrations.

Unfortunately, each new generation of parents has to discover for itself just how foolish this system is. In 1994 my older children were in seventh grade and exposed to their first overtly fuzzy math class. I and others founded Mathematically Correct a year later, while others in Palo Alto founded HOLD in response to similar circumstances. Among the most active groups now is NYC-HOLD, which has been very active in New York.

Although these groups have had success, changing the course of education is like turning the Titanic. By the time California changed substantially, my kids were in college. In most other places, parents couldn't get the support of go-along school boards that bought into administrators' assurances that fuzzy was best. Ultimately, parents, who aren't paid to fight bad instruction, move on, while the administrators remain.

There has also been a major failure of leadership by elected and appointed officials. Notable in this are those administrators who have been brought into school districts because of their success in non-education endeavors. These people come in with can-do attitudes and immediately start to shake things up, usually by irritating teachers. Unfortunately, they almost always are unwilling to consider the critical nature of curriculum and what actually happens in each classroom in education. Instead, being more generic administrators than clear thinkers, they poll 'experts' about whom to bring in as assistant superintendent for instruction. Since these experts are usually leaders of various professional education organizations or schools, the recommendations result in hiring of a well connected advocate of discovery learning methods. Such people usually complement the new superintendent's my-way-or-the-highway attitude and brook no deviation from a strict application of the very methods that many parents object to.

I don't expect to see the cycle end soon.

Posted by: Mike McKeown at November 14, 2006 9:26 AM

For an interesting, iconoclastic look at math instruction, see Rick Garlikov's Teaching Math to Young Children. I think the mistake of the NCTM in the late 1980s was not simplifying the issues involved, which are essentially three: how do you communicate and teach the central concepts of math, how do you communicate and teach efficient algorithms such as long division, and how do you get children to understand what the nature of an algorithm is (and why there's usually more than one way of solving a problem)?

There's a fairly efficient alternative to the standard way of doing long division -- Jakow Trachtenberg's algorithm which relies entirely on addition (plus the rest of Trachtenberg's program). And there's the so-called "Vedic" way of multiplying single-digit numbers so you don't have to memorize anything over 5x5 (and all other claims about so-called Vedic math). But absolute speed is less important than automaticity of *some* things and children understanding that an algorithm is designed for efficiency, not as the One True Way of doing math.

Then again, few ways of teaching math try to make those three things distinct. And I'm not a math methods person. And I learned a lot from set theory (New Math). So maybe I'm all wet on this.

Posted by: Sherman Dorn at November 14, 2006 9:55 AM

It is very sad to see the idea of "discovery learning" being reduced to silliness by both its supporters in today's schools and its detractors.

The concept was first formulated by Jerome Bruner in his book, *On Knowing: Essays for the Left Hand*, although its basic principles were established in *The Process of Education*, which grew out of the historical Woods Hole Conference and the post-Sputnik desire for better American training in math and science.

Bruner argues that we need to think about curriculum in terms of our future goals. That is, if we want our children to eventually master calculus, we need to give them the proper early training. This never meant getting rid of arithmetic, nor did it mean the highly abstract teaching style of New Math. Instead, it meant challenging the idea that young kids aren't bright enough to learn certain ideas. Bruner gives a simple example. In his research, he took a group of very young kids and taught them the quadratic equation. He did it not by giving them the equation and a set of problems to practice on, but by having the kids play with a balance beam and build squares out of wood slats. The kids kept a careful record of how they balanced objects on the beam, and how they built squares out of slats. By organizing these notes in a certain way, the students basically arrived at the equation without knowing it abstractly.

The point was this: no idea is too advanced for any student. The teacher simply needs to meet the student where s/he is in his/her learning. Young students rarely grasp abstract principles, and they do better with stories, object manipulation, and visual representations. So Bruner said, let's design a curriculum that prepares students for complex ideas by breaking them down into simpler ideas which young kids can grasp at their stage of learning. Multiplication is repeated addition, which means that if a kid can add, s/he can multiply. They might not *know* they can multiply, but they can be prepared for multiplication by training in addition in a certain way. This is what Bruner called scaffolding, or the spiral curriculum (the latter idea was borrowed without citation in E. D. Hirsch's Core Curriculum, although Hirsch constantly quotes Bruner's famous comment that all idea can be taught in some intellectually responsible to any school age child).

For Bruner, discovery learning was about the experimental method and inductive learning of abstract concepts. For example, rather than simply lecturing an English class on Northrop Frye's theory of genres, I've had students build up the theory of genres from their knowledge of TV shows, movies, books, comics, and so on. Or rather than giving students a few bulletpoints about what a Romantic poem looks like, we might give students a few Romantic poems and have them cosntruct their own idea of the shared traits.

Bruner's discovery learning was very much tied to content. He was challenging the idea that some ideas are too advanced for young kids. Furthermore, Bruner's own examples of discovery learning show that he still saw the need for intense training in basic skills like addition or multiplication. He talks about the need for automaticity in order to master higher order concepts.

At the same time, Bruner wanted children to understand what they were doing when they were performing a mathematical or scientific operation. I know I was trained in pre-calc simply to follow a procedure for finding a derivative, but I was never taught exactly what the hell I was doing when I did this. (Luckily, my physics teacher showed me that the concrete ideas we studied in physics were basically the same as the pre-calc ideas.) One reason I didn't pursue math as a profession despite doing very well in it was that all I was ever doing was memorizing and following a procedure with no real understanding. It wasn't until I studied statistics that I both learned how to *do* math and learned what I was *doing* when I did it. And that's because statistics is usually taught through induction and discovery: flipping coins, dealing card hands, playing roulette, picking cards out of a sack, etc.

Basic skills need rote training ('tho this is not drill and kill: you can play games to achieve rote training; you don't have to have kids sit at their desks and write "2+2=4" two hundred times). But conceptual understanding requires experimentation and inductive thinking rather than simply giving students equations and having them plug numbers into them.

New Math failed to give young kids the *type* of training they needed. It explained higher order concepts at high levels of abstraction, rather than allowing students to scaffold up from, say, addition to multiplication, or from balancing weights to the quadratic equation. Partly, this was Bruner's fault. While he stressed the need to meet students where they were cognitively, he also stressed the need for experts in a field to set the curriculum. And when science or math (or English) eggheads set a curriculum, they tend toward abstractions and away from the type of enactive training Bruner used with first graders. The New New Math is even worse: it's empty both of basic skills and real understanding. I tutor middle school kids who have never been taught the multiplication table. And my fifth grade class had a really fun time memorizing the times tables because we played fun games out on the playground that trained us in mutiplication.

I was pleased to see in my recent research on Hirsch's Core Curriculum schools that teachers were using progressive methods to teach Hirsch's core content -- and they were having excellent results ('tho not statistically significant improvements over the average public school). It seemed common-sensical to me: give students exciting, challenging content and deliver it in a challenging way that sparks students' natural curiosity and desire for mastery. Which is exactly what Bruner proposed back in 1960.

(Mike McKeown rightly mocks the "guide on the side" versus the "sage on the stage" language. At the same time, think about the non-school training kids receive that is clearly effective: sports coaching and piano lessons. In neither case do you have kids sitting silently taking notes as a football coach tells them about how to kick a fieldgoal or as a piano teacher tells them about how she plays a Bach fugue. Instead, each is, often literally, a guide on the side: the coach sets up a problem situation (here's a certain offense and certain defensive) and let's the team work out the situation through real play. Likewise, the piano teacher gives basic skills instruction (scale practice) but then presents the student with a real composition for problem-solving (here's a segment of Bach, how will you approach it, how will you solve the issue of hand placement and dynamics and so on). And yet in school education, we think that having a teacher tell students about science works better than having them practice science, or having a teacher talk about analyzing a poem works better than having the students practice such analysis. Meanwhile, piano teachers and football coaches produce winning teams year after year . . .)

Posted by: Alvin Lucier at November 14, 2006 2:03 PM

"The great 20th-century composer Igor Stravinsky wrote, in The Poetics of Music, "You cannot create against a yielding medium." Stravinsky's innovations were nothing if not revolutionary, but he knew that he could not have produced them if he had not be constrained by the traditions of music and the mathematical strictures of tone. "Let me have something finite, definite," he wrote. "My freedom will be so much the greater and more meaningful the more narrowly I limit my field of action and the more I surround myself with obstacles. Whatever diminishes constraint, diminishes strength.""

Nicholas Carr in Strategy & Business

Posted by: david foster at November 15, 2006 9:57 AM


Posted by: jonny at November 17, 2006 7:23 PM

The majority of my freshman cannot put together a coherent sentence; they have no idea what that means or how to do it. Their handwriting is atrocious as well. I guess teaching grammar and neat handwriting both "stifle creativity." I'm really unclear as to how public schools expect these students to survive, never mind thrive, in the real world.

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