Constructivism is rearing its ugly head again. With this kind of help, students are not going to "own" the algebraic method of solving this problems. They might not even "own" the inefficient guess-and-check method. If you are being taught properly and practice sufficiently to the point of overlearning, then you are more likely to "own" math knowledge than when left groping in the dark.
A farmer sends his daughter and son out into the barnyard to count the number of chickens and pigs. When they return the son says that he counted 200 legs and the daughter says she counted 70 heads. How many pigs and chickens does the farmer have?
A student well versed in algebra might do the following to set up the problem: p = pigs, c = chickens. p + c = 70 (heads) 4p + 2c = 200 (pigs have 4 legs and chickens have 2 legs). These two equations may be used to solve the problem. Students might solve this problem by "guessing and checking," or drawing pictures. Some methods of solving problems might be considered more "efficient." That may be true, but the correct answer can be found using multiple methods. Children think about mathematics in different ways depending on their prior experiences at home and school. By allowing students to think flexibly about numbers, we encourage them to "own" the math forever, instead of "borrowing" until class is over. (Answer: 40 chickens and 30 pigs)
Wednesday, November 30, 2005
Saturday, November 26, 2005
In what sounds like a three stooges routine, the three "learning specialists" are alarmed by what they see as a trend toward academic achievement:
Perhaps now is a good time to ask this question: What are schools supposed to do for our children? As learning specialists, we see an alarming trend: Our education system increasingly is focusing not on developing children’s aptitude for learning—their ability to absorb new information quickly and solve problems creatively—but on their academic achievements—their mastery of particular subjects and skills as proven by performance on standardized tests.Silly me. I would have thought that academic achievement demonstrates at least an "ability to absorb new information quickly," the purported goal of these specialists.
The authors of the anti-achievement piece do manage to offer "sobering" examples of imperial decline due to memorization and academic achievement. The cause-and-effect scenario painted here does seem a bit fishy to me. I kind of doubt that China went into a tailspin because a few mandarins had the ability memorize Confucian philosophy. Would we as a society suddenly have to live in caves in the unthinkable event that some of our bureaucrats (say, at the board of ed) suddenly had the urge to memorize a few poems by Whitehead and Tennyson?
Snippet from the trio's commentary:
This is a serious concern for our kids and our society. History offers sobering examples of what can happen when standardized achievements are elevated over open-ended abilities.The essence of the anti-academic achievement position as far as I can distill it is a false dichotomy between learning academic subject matter on the one hand and "creativity, innovation, and independent thinking" on the other. Educationists worship at at the altar of ignorance in the name of "creativity" but ignorance is not a prerequisite for "creativity".
During the 18th and 19th centuries, imperial China—once the most technologically advanced civilization in the world—fell into decline as power passed into the hands of a mandarin class of bureaucrats selected for their ability to memorize Confucian philosophy. More recently, Japanese authorities have begun dismantling an education system that long relied on a uniform national curriculum and after-hours classes at juku “cram schools.” The Japanese believe this approach has stifled creativity, innovation, and independent thinking, contributing to the stagnation of the Japanese economy.
We worry that America is heading down a similar path. If promoting our children’s achievements becomes our sole focus, both our children and our society will suffer.
Saturday, November 19, 2005
I would label myself a political liberal and an educational conservative, or perhaps more accurately, an educational pragmatist. Political liberals really ought to oppose progressive educational ideas because they have led to practical failure and greater social inequity. The only practical way to achieve liberalism’s aim of greater social justice is to pursue conservative educational policies.In combating the dominant ed creed, propopents of quality education must have a developed sense of how the language employed by the dominant ed creed is manipulative, charged, disparaging and stacked against them. Some of the phrases that come to mind are "sage on the stage, guide on the side," "chalk and talk," "drill and kill" "teach the whole child," "teach the child, not the subject," "less is more," "up is down," "freedom is slavery" (well, I made up the last two or, rather, stole the last one from Orwell). There is nothing comparable for those who value expository instruction and domain knowledge.
[Fortunately, not all educationist phrases are catchy. There is also some pretty dreary stuff. I culled this from a constructivist site: "This article discusses several active-learning techniques that instructors can use to help students construct knowledge, such as think-pair-share, guided reciprocal peer questioning, jigsaw, and co-op co-op."]
Other words deployed by the dominant ed creed to disparage the notion of imparting knowledge are "lecture", "active" and "passive". "Lecture" has the negative connotation of droning on without regard for the audience's (in this case the pupils') level of understanding and its capacity to follow while the captive audience sits by "passively". No teacher worth anything would teach that way. (Adherents of the dominant ed creed forget that listening attentively to explicit instruction is also being active, but the followers of the creed claim to have a monopoly on "active").
But the disparagement of any explicit instruction by labeling it "lecture" is so strong that explicit instruction is proscribed in many places. That is detrimental to achieving quality education. For example, good math instruction should consist of modeling (interactive modeling if appropriate) followed by guided practice, independent practice and review. In other words, there must be talk as required by the circumstances. And how would one teach history and other subjects without talking?
I coined the impressive, albeit cumbersome, phrase "empathetic, interactive expository instruction" as a counterpoint to the dreaded "lecture". Alas, it can't compete with the catchiness of "talk and chalk" or "drill and kill". (Now I have to put my phrase in rhyme form).
I include "empathetic" in my phrase to stress the importance of discovering and being being sensitive to Vygosky's fabled zone of proximal development (a jazzed-up way of saying that our instruction must be geared to the pupils' ability and level of comprehension).
In his article, E.D. Hirsch takes apart some of the conceits of educationists:
Unfortunately, many of today’s American educators paint traditional education as the arch-enemy of "humane" modern education. Even everyday classroom language unfairly pits the two alternatives against one another. Here are some typical descriptions used by progressives to compare old and new methods:You can read the entire article here.
Traditional vs. Modern
Merely verbal vs. Hands-on
Premature vs. Developmentally appropriate
Fragmented vs. Integrated
Boring vs. Interesting
Lockstep vs. Individualized
Parents presented with such choices for their children’s education would be unlikely to prefer traditional, merely verbal, premature, fragmented, boring, and lockstep instruction to instruction that is modern, hands-on, developmentally appropriate, integrated, interesting, and individualized. But of course this is a loaded and misleading contrast. Let’s look at those simple polarities one at a time.
Sunday, November 13, 2005
From the NYT's Penfield article:
By last spring, these parents had discovered one another and their common exasperation with constructivist math. Jim Munch's father, Bill, a software developer at Kodak, drew up a petition asking the Penfield schools to offer pupils the option of taking traditional math. Nearly 700 residents signed it. Last June, the Board of Education turned down the request.Giving parents a choice between fuzzy and real math is the democratic thing to do and is also a good political strategy. It should satisfy everyone. It's an inoffensive offensive. But I doubt that zealous educationists in a position of power will go along (and don't as the example above shows). Being responsive to reasonable popular wishes is not their thing. I also suspect that many parents are not conversant with the real vs. fuzzy math issues and won't know what to do with choice.
Just yesterday I talked to a parent of a sixth grader I am tutoring in math who had no clue of fuzzy math. (I tutor disadvantaged kids after hours in addition to my regular classes.)
I was helping the kid do homework. Part of the homework required the young girl to cut a sheet of paper into strips to make various fractions. The parent was aghast and thought it was a time-waster. I had to explain the purpose of the exercise. It was all news to her.
The school the kid is in uses the fuzzy math series Connected Math. The homework assignment was quite demanding and way beyond this pupil's abilities. She had neither a conceptual understanding of the task nor the requisite tools (computational skills, procedural knowledge, math facts) to accomplish the task had she had a conceptual understanding of the problem.
This is a key problem with fuzzy math. It is quite pretentious on the one hand, and refuses to teach the necessary skills on the other. The result: the kid was hopelessly drowning and getting straight F's.
Now what was the task? It was a real-world problem.
A class was holding a fundraiser to raise $300.00 in ten days. The progress was shown in the form of thermometers showing progress in two-day increments. The thermometers were all 8 1/2 inches long and showed the money raised so far on the various days in red. The fraction strips were to be used to determine the amount of money raised so far on the various days and then to plot the progress in a coordinate plane. The pupil was to make the strips and mark fractions from 1/2 to 1/12 on the various strips, and then use the strips for measurement.
Making fractions strips of 1/2, 1/4 and 1/8 is of course easy. It's not so easy to come up with 1/3, 1/5, 1/7, 1/9, 1/12. You could do time-consuming trial-and-error folding. Or you could divide 8 1/2 by the denominator of the various fractions if you know how (the child didn't know).
Even if you can get the numbers they don't work well with an inch ruler. You could approximate. Suppose you (meaning the kid) could accomplish all that. Then what?
The goal of the assignment is to come up with dollar amounts derived from the thermometers and then to plot these amounts over time (fundraising progress).
How are you going to derive dollar amounts if all you have is an 8 1/2 in long thermometer (representing $300) and a red bar on the thermometer without any numbers, marks or gradations (the length of the red bar represents the money raised so far)?
You could measure the red bar in inches, form a ratio (the red bar to total thermometer length ratio), calculate the decimal, multiply the decimal by 300.
However, the assignment calls for measuring the red bar with the "fraction ruler". Then you would know what fraction of 8 1/2 the red bar represents. You can then multiply the fraction by 300 to get the dollar amount. All this without instructions in CMP and without computational skills and procedural knowledge.
This is too complicated and frustrating for a math-challenged child who needs to learn at her level and make steady progress.
No wonder the kid is drowning. What a tragedy.
After citing example after example of math cripples, the article has this gem that shows the arrogance of educationists:
LAST spring, when he was only a sophomore, Jim Munch received a plaque honoring him as top scorer on the high school math team here. He went on to earn the highest mark possible, a 5, on an Advanced Placement exam in calculus. His ambition is to become a theoretical mathematician.
So Jim might have seemed the veritable symbol for the new math curriculum installed over the last seven years in this ambitious, educated suburb of Rochester. Since seventh grade, he had been taking the "constructivist" or "inquiry" program, so named because it emphasizes pupils' constructing their own knowledge through a process of reasoning.
Jim, however, placed the credit elsewhere. His parents, an engineer and an educator, covertly tutored him in traditional math. Several teachers, in the privacy of their own classrooms, contravened the official curriculum to teach the problem-solving formulas that constructivist math denigrates as mindless memorization.
"My whole experience in math the last few years has been a struggle against the program," Jim said recently. "Whatever I've achieved, I've achieved in spite of it. Kids do not do better learning math themselves. There's a reason we go to school, which is that there's someone smarter than us with something to teach us."
By last spring, these parents had discovered one another and their common exasperation with constructivist math. Jim Munch's father, Bill, a software developer at Kodak, drew up a petition asking the Penfield schools to offer pupils the option of taking traditional math. Nearly 700 residents signed it. Last June, the Board of Education turned down the request.The superintendent haughtily dismisses parent concerns. Many of the parents have extensive math backgrounds:
Susan Gray, the superintendent, attributed the criticism of the math program to "helicopter parents" who are accustomed to being deeply involved in all aspects of their children's lives. "Because the pedagogy has changed, the parents who knew the old ways didn't know how to help their children," she said. "They didn't have the knowledge and skills to support their children at home. There's a security in memorization of math facts, and that security is gone now."
YET many of the dissident parents have extensive math backgrounds and thus the ability to criticize the curriculum. It is also true that most of them tolerated the constructivist program for its first several years, until bitter experience drove them into rebellion.
"Mathematics achievement in America is far below what we would like it to be. Recent "reform" efforts only aggravate the problem. As a result, our children have less and less exposure to rigorous, content-rich mathematics. The advocates of the new, fuzzy math have practiced their rhetoric well. They speak of higher-order thinking, conceptual understanding and solving problems, but they neglect the systematic mastery of the fundamental building blocks necessary for success in any of these areas. Their focus is on things like calculators, blocks, guesswork, and group activities and they shun things like algorithms and repeated practice. The new programs are shy on fundamentals and they also lack the mathematical depth and rigor that promotes greater achievement."
One way out of this disastrous state of affairs is to create national standards. This is, of course, fraught with danger as past experiences show, e.g. the experience with the Gary B. Nash history "standards". Such a project would attract the usual suspects. But I think the risk has to be taken. The advocates of sound, rigorous standards must organize and mobilize and be in a position to counter the inevitable attempt to dilute the standards and to institute educational lunacy.
It is therefore gratifying to read that Diane Ravitch is calling precisely for such standards. [Diane Ravitch's article was first published in the NYT but the article will soon disappear into the paid archives.]
The release last month of test results by the National Assessment of Educational Progress, which is part of the Department of Education, vividly demonstrated why varying state standards and tests are inadequate. Almost all states report that, based on their own tests, incredibly large proportions of their students meet high standards. Yet the scores on the federal test (which was given to a representative sample of fourth and eighth graders) were far lower. Basically, the states have embraced low standards and grade inflation.
Friday, November 11, 2005
So why don’t I like it? Why do I belittle it as kitsch? Kitsch, after all, usually refers to something of tawdry design or appearance created to appeal to nondiscriminating tastes—for example, those portraits of Elvis or bullfighters painted in neon colours on black velvet. But the Woolfolk text can hardly be accused of tawdry design or appearance. Indeed, measured against its competitors, it is presumably the crème de la crème.
Miseducative kitsch. In name-calling the Woolfolk text “kitsch,” I have in mind a less obvious, but perhaps more definitive, meaning of the term suggested by writer and critic Robert Fulford. In a recent CBC radio interview, Fulford dismissed so-called “victim-based art” as kitsch, but not necessarily because of any cheap or garish aesthetic qualities. Rather, Fulford argued, this “art” is kitsch because it seeks, by design, to compel the viewer to experience certain predetermined responses to it—in this case, sorrow, sympathy, compassion, and, perhaps, guilt. Fulford went on to liken victim-based art to the kitschy Saturday Evening Post cover art of Norman Rockwell, the artist whose slice-of-Americanlife paintings are typically unambiguously and irresistibly “cute” and, hence, admit of no other viewer response. For Fulford, what makes both victim-based and Norman Rockwell’s “art” quintessentially kitsch is that both contrive to over-determine and, consequently, to limit the viewer’s range of intellectual and emotional response. Neither allows for any interpretive or responsive ambiguity; both attempt to coerce thought and feeling.
Sunday, November 06, 2005
The teacher appreciates multiple perspectives and conveys to learners how knowledge is developed from the vantage point of the knower.What is it with these "multiple perspectives?" It seems to me that at the bottom of this excitement is this postmodern notion that evidence, scholarship, objectivity and truth are illusory. There are only perpectives -- influenced, no doubt, by some "power" considerations.
The teacher appreciates and values human diversity, shows respect for students’ varied talents and perspectives, and is committed to the pursuit of “individually configured excellence.”
This is dangerous drivel. What would stand in the way to conferring to creationism and to Holocaust denial the status of just another, equally valid, "multiple perspective?"
It would be much more appropriate to talk about "multiple ignorances."
UPDATE: See Chris Correa for a thoughtful examination of dispositions.