from my own past:

Right-triangle trig was difficult for me because I resisted memorizing the ratios. I resisted memorizing the functions of common angles. I didn't see why radians were necessary. I was trying to understand and going nowhere. I finally just memorized SOHCAHTOA and the reciprocals. Things became abundantly clear and so incredibly simple. Now, every similar situation in the wood shop became obvious, every real-world problem became a piece of cake. (and the angle on that piece of cake needed to be 45

^{o}- more would be greedy.) Then I practiced. Okay, the teacher gave us a bunch of questions, but they were easy because he had taught, I memorized some basics, I practiced and then I knew. I'd still be there if I'd had to come up with it myself.

"If I have seen farther than others, it is because I have stood on the shoulders of Giants." At what point do our students get to stand on the shoulders of giants? We say that people learn better if they come up with it themselves. I say that's a flippin' waste of their time. Math geeks spend their lives figuring this stuff out and you want a bunch of kids to do it in 9 months?

Anyway, here's the article from over at ednews ...

“Student-centered” Learning (or “Constructivism”)

By Laurie H. Rogers, author of "Betrayed"

Columnist EducationNews.org

Constructivism and lack of practice

Here are two of the clues to America's current mathematics problem:

1."Student-centered" learning (or "constructivism")

2.Insufficient practice of basic skills

"Student-centered" Learning (or "Constructivism")

In an October email, Spokane's secondary mathematics coordinator reaffirmed this district's commitment to a "student-centered" approach to teaching (also sometimes called "discovery learning" or "constructivism"). In this approach, students often work as partners or in groups, and teachers act as "facilitators" rather than as "instructors." Students are encouraged to come up with their own multiple solutions to problems and to ask fellow students for help before asking the teacher.

Reform math curricula are typically built around a constructivist approach, probably because the 1989 Standards document from the National Council of Teachers of Mathematics calls for it (Stiff, 2001c; "Curriculum," 2004). Proponents say the approach leads to "deeper understanding," helpful collaboration and better student enjoyment of the process. Others say a dependence on it can hinder the learning process and frustrate students.

A local parent told me this story about when his daughter took a math class that used reform math curriculum Connected Mathematics:

Students were told that "Juan" was mowing a lawn in a right-angle triangle. He wanted to figure out the length of the diagonal. The term "Pythagorean Theorem" (a2 + b2 = c2) wasn't presented. The students were to work in groups and figure out a way to get the answer. Finally, one student who knew the theorem provided it to her group. (Her group was the only one to get the right answer.) Incredibly, the teacher "chastised" the student for using the formula.

"A lot of parents don't believe it at first," the parent said to me. "Like, their kids are younger, they don't know, and they feel that parents are exaggerating, but it is the honest-to-God truth, and these stories get worse."

In small doses, constructivism can provide flavor to classrooms, but some math professors have told me the approach seems to work better in subjects other than math. That sounds reasonable. The learning of mathematics depends on a logical progression of basic skills. Sixth-graders are not Pythagorus, nor are they math teachers.

Meanwhile, anti-reform advocacy group Mathematically Correct provides an amusing take on constructivism ("What Is," 1996):

"This notion holds that students will learn math better if they are left to discover the rules and methods of mathematics for themselves, rather than being taught by teachers or textbooks. This is not unlike the Socratic method, minus Socrates."

Insufficient Practice of Basic Skills

Another problem in math classrooms is the lack of practice. Instead of insisting that students practice math skills until they're second nature, educators have labeled this practice "drill and kill" and thrown it under a bus.

I wish I had a dollar for every time I heard that phrase. It's a strange, flippant way to dismiss a logical process for learning. Drilling is how anyone learns a skill. Removing drilling from the learning process is like saying, "We'll just remove this gravity. Now stay put." Everyone drills – athletes, pianists, soldiers, plumbers and doctors. Drilling is necessary.

It isn't good or bad – it's simply what must be done.

Imagine if I told chess players they had to figure out the rules of chess on their own, in fits and starts, by trial and error and by asking their fellow players. Imagine if I expected them to win games when they hadn't had a chance to practice.

In American education, the "worm" is not yet turning, but it might be looking over its shoulder. In its March 2008 report, the National Mathematics Advisory Panel reintroduced the notion of practicing the basics:

"Practice allows students to achieve automaticity of basic skills – the fast, accurate, and effortless processing of content information – which frees up working memory for more complex aspects of problem solving" ("Foundations," 2008, p. 30).

But children in the system now are stuck with a process that asks them to work in diverse groups to reinvent thousands of years of math procedures that they then don't get to practice.

Some people enjoy puzzles on logic and process, where things might not be what they seem and where they've got to figure out subtle differences and new ways of thinking. But this esoteric, conceptual approach to math, with a constant struggle to understand the process, doesn't seem like a logical approach for children. Children are concrete thinkers who tend to appreciate concrete ideas. Children want instructions, direction and things that make sense. Many don't appreciate the daily grind of writing about math, of having to figure out what they're doing, of having to count on classmates for guidance, of trying to remember things they've done just once or twice and several weeks ago.

It's ironic that proponents of reform math criticize traditionalists for supposedly not knowing "how to teach math to children." The reform method seems completely oppositional to how children learn best.

I asked a Spokane student if she prefers the Connected Mathematics she gets in school over the Singapore Math she gets at home. She said, "In a way, Connected Mathematics is easier because you don't have to know as much math, but in a way, it's harder because you have to know more. You have to know exactly what they want."

She gave me an example of the classroom approach: Students are to gather in groups to discuss a problem. The problem might be a complicated twist on simplistic math, or it might be a concept they've never seen before. As the groups muddle around, they don't always agree on what's required. Sometimes, they don't have the necessary underlying skills. Some students become frustrated or bored. Trying to help each other, some confuse the others. They might come up with the right answer, or they might not, but – without practicing the new concepts – the class moves on to something new.

Singapore Math, on the other hand, "might be harder as far as the math goes," she said, "but at least you know what they want."

I told her I thought her answer was articulate and enlightening. "I've spoken to a lot of people now," I said, "and you explained things very well."

"That's because they teach it," she replied, "but I'm the one who has to learn it."

Published November 9, 2008

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