Fast Learners
Montgomery County officials say accelerating students in math will better prepare them for college, but a revered teacher says it's time to step on the brakes.

By Emily Messner
Sunday, August 3, 2008

It's the day before final exams start at Montgomery Blair High School in Silver Spring, and Eric Walstein is teaching a class he calls "a travesty."

It's not that he minds teaching Algebra II, but these students are in Blair's acclaimed math and science magnet program, and traditionally the magnet hasn't bothered with the course -- the kids were smart enough, and their grounding in the fundamentals of algebra strong enough, that they could proceed directly from geometry in middle school to precalculus in high school and pick up the additional algebra they needed along the way. But the precalculus teachers found so many freshmen struggling that the magnet created an Algebra II section on the fly in the middle of this past school year. And so here sits Walstein, 63, in a faded green polo shirt, his glasses perched on the tip of his nose, as he reviews algebra with students who were in precalculus just a few months earlier. It's not that these kids aren't as smart as those who came before them. So, what's going on?

When asked why she was taking the Algebra II course, one student, Manisha Sarkar, said it was "because we were not given the proper background" in middle school math. Lisa Ma said that while she did very well on tests in Algebra I, she didn't feel she'd really learned the material.

This complaint is nothing new to Walstein and his students. At end-of-school-year parties hosted at Walstein's Brookeville home, the trajectory of math education in the county is a familiar topic of conversation. At his 2007 barbecue, a bunch of students from Walstein's advanced calculus, origins of math and complex analysis classes sat around an oblong table on his back deck. But they weren't talking about complicated mathematical theorems; they were talking about the fundamentals. They had younger siblings and friends at other schools, windows into middle school math classrooms throughout the county. Walstein shuffled over to a big wooden armchair at the head of the table. He was dismayed -- but not at all surprised -- by what he heard.

The students swapped stories of little sisters, brothers and cousins who were taking above-grade-level math and getting good grades, yet did not seem to have a firm grasp of the material. The curriculum is being "narrowed and shallowed," Walstein said. "The philosophy is that they squeeze you out the top like a tube of toothpaste. That's what Montgomery County math is."

Several students nodded their heads. This thesis has become Walstein's obsession: In its drive to be the best, please affluent parents and close the achievement gap on standardized tests, the county is accelerating too many students in math, at the expense of the curriculum -- and the students. The average accelerated math student "thinks he's fine. His parents think he's fine. The school system says he's fine. But he's not fine!" Walstein declares on one occasion. On another, Walstein is even less diplomatic. " 'We have the best courses and there's no achievement gap and everything is wonderful,' " he says, parroting the message he believes county administrators are trying to project.

"The problem is, they're lying!"

It's not surprising that those county administrators whom Walstein loves to mimic don't agree with him.

Developed in the 1990s, the county math curriculum is based on national research and better prepares students for calculus and for college, says Betsy Brown, the county's director of curriculum and instruction. The debate in Montgomery reflects a discussion "we're experiencing across this country," Brown says. "What's the best way to teach and what's the best way to learn? And it's not just in math, it's in all disciplines."

Walstein isn't just any math teacher -- he's arguably the most highly regarded high school math teacher in the county. He was hired by Blair's math and science magnet in 1986, when the program was just a year old, after teaching for 12 years at two top county middle schools and at Winston Churchill High School in Potomac. He has been coaching the county math team since its inception more than 30 years ago, and, under Walstein's leadership, students have turned in some stunning performances in events such as the Math Olympiad (the test to choose who would be on the equivalent of the national Olympic team for mathematics) and the American Mathematics Competitions. He's also a three-time winner of a national award for distinguished high school mathematics teaching, and his former students include some of the more highly regarded mathematicians of their generation, such as MIT's Jacob Lurie and Jordan Ellenberg of the University of Wisconsin.

"He knows more about mathematics from kindergarten through college than anyone I know," says Blair magnet program coordinator Dennis Heidler. "He has an understanding of how the concepts thread themselves through the years."

Christina Zou, a 2008 Presidential Scholar who named Walstein her most influential teacher, made him the subject of a college application essay. Zou joined the county's math team in seventh grade and plans to major in math at Harvard. "Teachers have standards for their students, and his for me are higher than any other teacher has ever set for me," Zou said. "I feel like with those standards in place I have more to [strive] for."

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."

Walstein says he is not opposed to acceleration; when he finds exceptional math minds, even in kids as young as 7, he will tutor them, he says. In fact, both supporters and critics of the county math curriculum agree that building in flexibility to accommodate students who can handle learning math at a fast rate is a good thing.

Edward Nolan, the math department head at Albert Einstein High School, says he generally supports the county math curriculum, particularly because it allows many more students the opportunity to take algebra in seventh grade than was the case 15 years ago. "Compacting means you do things faster, and you don't get the depth that we did in the past, so, in that sense, I see it as students not being as well prepared as they may have been in the past. But that's in part the role of the teacher, to reach back and remind students of what they did before," Nolan says.

Walstein says it's more than reminding these days. While he says some teachers apparently manage to provide the necessary background, increasingly students coming into the magnet have not been taught certain math concepts in enough depth for the concepts to become ingrained. In addition to acceleration, the compacting of the algebra curriculum can be traced to Maryland's High School Assessment exam for algebra, a substantial portion of which focuses on non-algebraic data analysis, including basic probability and statistics. To cover these lower-level objectives, teachers must take time away from teaching algebraic concepts. The result, say Walstein and others, is that explanations that connect concepts and reinforce the learning are lost. Yet the reasoning skills developed by making these connections and finding their logical extensions are crucial to success in higher levels of math.

In addition, Walstein says, some children are not cognitively ready for algebra as early as the county wants to give it to them. "Have they passed from the concrete into the abstract? {lcub}hellip{rcub} Different kids' brains are ready at different times," he says.

While many Montgomery parents support the county curriculum and push to have their children accelerated, not all of them do.

"There are several parents at our school, my husband and I included, who have decided that our child will repeat the previous year's math class rather than promote to the next class," wrote Jean Hickey of Silver Spring, in a response to a query on a parent listserv. Hickey said her daughter this year will repeat the fifth-grade math she took in fourth grade. "We want her to truly understand the concepts . . . I wonder how many other parents in the county are now taking a stand like this rather than letting the county dictate the pace of the child's progression?"

The evolution of the county's math program parallels nationwide changes in the way math is taught, Brown says. In the 1980s, research by Harvard University and organizations focused on math education led to making calculus more concept-based, as opposed to focused on process. Next, algebra instruction was reevaluated, and part of what came out of that was the increased use of calculators, she says.

The county further refined its math curriculum after a 2000 audit by Phi Delta Kappa International, a nonprofit organization for education professionals, urged creating a consistent curriculum throughout the county's schools to address an excessive achievement gap on standardized tests between white and Asian, and black and Hispanic students, Brown says. The previous year, Jerry Weast had been hired as county superintendent, and he made closing the achievement gap a top priority. Weast championed opening up advanced courses to more students, and the county's goal is now that 80 percent of students complete Algebra I in middle school by 2010, well above the current 56 percent. The Fairfax County public school system also is accelerating more students. Nearly half of Fairfax students have taken Algebra I by the end of eighth grade, and the school board has set a goal to increase that number of students, though a target has not been established, according to Craig Herring, the system's high school math specialist.

The current math curriculum "involved a real deliberate and real thoughtful approach to seeing what it is kids need to be able to do in calculus {lcub}hellip{rcub}and with a real conceptual emphasis," Brown says. "And then we have gone step by step back through each of the courses to make sure that we have put into place what kids need to learn to be successful at the next step." The county took the requirements of the Maryland standardized tests into account, she says, but went well beyond them, or students "wouldn't have been ready for college."

One result of the county's new curricular strategy is that while classroom math teachers are still involved in shaping the curriculum, experts in math education theory now play a much larger role.

Walstein and Bunday believe this change has led to a less effective curriculum. "It's top down . . . the county has it all figured out," Bunday says. "They know how to raise test scores . . . and we have to respond to directives that don't always make sense."

Responds Brown: "We can say that our curriculum is requiring a new way of thinking and approaching mathematics, and it's highly likely that we're asking teachers to teach in a way that's different from the way they learned math or {lcub}hellip{rcub} were trained to teach math."

So, is this just Walstein's version of that standard grandfatherly tale of woe? ("When I was your age, I walked 10 miles every day, in the snow, uphill -- both ways! -- just to get to school. You young whippersnappers have no idea how easy you have it!")

There might be a little of that. Walstein admits that few get as agitated about the issue as he does. But if it's not such a big deal, he asks, what explains the need to create that emergency Algebra II class for magnet students at Blair? And what explains the gaps in understanding he sees among his students?

Montgomery County Council member Mark Elrich, who taught math at Rolling Terrace Elementary School in Takoma Park for 16 years, says Walstein's observations are spot on. "It looks good because you can say . . . look at all the kids who have gotten algebra," Elrich says. Montgomery students do well on the state's standardized tests, but when some of them take the SATs, "the truth comes out."

As usual, Walstein puts it more bluntly.

Sitting in his classroom, he bangs his fist on a desk in frustration and booms, "What they're doing to these kids is wrong!"

Emily Messner is a writer and researcher in Arlington. She can be reached at emessner@stanford.edu.