Fast Learners

Walstein is known for highly structured classes, in which he carefully plots out each day's lesson, culminating with a weekly exam. He methodically sets aside his Saturdays to grade tests and carefully assess where each student stands. But, like most admired teachers, he's also a bit of an entertainer, say his students and colleagues. Walstein pushes his students excruciatingly hard, until he notices their eyes start to glaze over. And then he quickly lifts the mood with some lively intellectual jousting on nonmath topics, such as sports, that he knows interest them. His ability to joke around and speak their language, even as he pushes them to reach their mathematical potential, tends to endear him over time to his older students. On the other hand, new students often fear his demands, and colleagues note his sometimes confrontational demeanor in meetings with administrators.

"Walstein ranges from disgruntled silence to really vociferous," says Ralph Bunday, a science teacher who retired this summer from the magnet program and counts himself among Walstein's fans. "He can get really outraged. And he's big."

Bunday remembers a meeting in the spring about procedural changes in the magnet program, during which Walstein stood up before a teachers' union representative backing the county position and "shook his finger in her face . . . a whole charade," Bunday says. "He has some very valid issues, and sometimes he shoots himself in the foot because of this emotional reaction. But once he's subdued and confronted, he can actually be a very intelligent and thoughtful man."

Walstein's outspokenness stands out, but he is not alone in believing that the county is moving too many students through the math curriculum too fast. "You would have a hard time finding one math teacher in this county who supports the scope and sequence of the way math is taught," says Billie Bradshaw, the math and science magnet program coordinator at Poolesville High School. She, too, is seeing a difference in the skills of the students entering upper-level math classes. "The kids are not as prepared as they have been in the past," she says, and are being "accelerated too fast."

Admittedly a bit of a curmudgeon, Walstein says he has resigned himself to school administrators' unwillingness to reevaluate the way math is taught. So, he says, he focuses his energies on working with his students to show them what they've missed and fill in the gaps. "I'm saying to them: Here's a lesson, here's where it comes from, and do you realize this is what you were missing by just pressing all these buttons?"

Those would be calculator buttons. Walstein has a particular gesture that he often uses to express disdain for the county's calculator-based approach to math: With his right index finger, he punches imaginary buttons on his left palm, then flips the left hand around so the palm faces out, as if to display the instant result.

The problem with calculators, he explains, is that the kids can get the answers without learning the underlying principle, and how and why it works. "Equations are not being derived for the kids," he says. "They're just being told: Push these buttons, get this graph. Everybody looks at me, and they say, 'Walstein, how come you're not using technology?' And the answer is, technology is another level up." When you're an engineer and you need to calculate something, the calculator is an almost indispensable tool. But first, he says, you need to know why you're doing the operations you're doing on the calculator.

Asked how the use of the calculator has supplanted important lessons, Walstein mentions factoring. Ideally, he says, students would be required to break an equation into its component parts before solving it. "One of the things kids have a hard time doing is reducing [algebraic] fractions," Walstein says. "The way I approach it is, I do numerical examples, show them how, and they all accept it; then I jump over to the algebraic form. If we're going to do the same thing, then we have to factor it . . . Factoring is important because it's the key to upper mathematics. Because of the way we evolve, we learn linear equations first. The next thing we do is quadratic equations." (A linear equation is an algebraic equation in which each term is either an unspecified fixed number or the product of an unspecified fixed number and the first power of a single variable. A quadratic equation is an equation in which the variable is squared. Here is a linear equation: y=x+2. Here is a quadratic equation: y=x2+4x+4.)

"Well, how do I solve a quadratic equation?" asks Walstein. "Well, what you're going to do is you're gonna take the harder situation, and you're gonna reduce it to what you know, and what you know is linear. And so, consequently, to take the quadratic equation and factor it reduces it to two linear pieces. So what you're doing essentially is taking a difficult exponent and reducing the power, so you're turning it back to something you know how to do. What I'm telling you is not said to these students. And that's why the kid last week said to me: 'Why do I want to factor? What's the purpose? Who cares if I factor?' This was a kid coming into the magnet." (When factored, the quadratic equation above looks like this: y=(x+2)(x+2))

Brown takes a different view on the calculator, whose use in the classroom has been debated nationally. "The whole point of the calculator is to make sure the kids learn the concepts behind the equation . . . And the calculator actually becomes a tool for teaching the concepts," she says. She gives an example: By using a graphing calculator when studying linear equations, she says, students "can learn the relationship of an equation to a graph. They can see that by changing the variables in an equation, what that does on a graph {lcub}hellip{rcub} They can see the linear relationship. And they can understand the concept of slope before anyone even gives them that long formula."

Using the calculator, Brown maintains, is "more than just pushing buttons. It's really getting the kids actively involved in the learning."

Allowing the use of calculators may drive up participation in accelerated classes by making them easier, Walstein notes, but, he asserts, it's done at the expense of crucial basic understanding. The curriculum shortchanges geometry, as well, Walstein contends. What about all those proofs, he asks. Gone, he says. "Are they learning geometry?'' he asks. "No. Are they learning a handful of facts related to geometry? Yes."