Hunched over tables, peering over one another’s shoulders, a group of 5th graders is attempting to conquer some of the most difficult—and essential—material in elementary school math.
They are not alone.
On one side of their classroom, about 30 adults are sitting on risers, watching the students closely. They jot down notes. They listen to the students’ comments and questions, broadcast to them over a sound system. And when the students leave the room for a break, the adult observers move in to peruse the answers the children have scrawled in their notebooks.
This unconventional classroom arrangement is part of the Elementary Mathematics Laboratory, a forum held here over a two-week period this summer at the University of Michigan’s school of education. The lab, now in its second year on this campus, brings together teachers, college students preparing for the teaching field, and academic researchers from across the country to observe and discuss the challenges elementary educators face in trying to help students struggling in math.
The teacher working in the lab’s spotlight is Deborah Loewenberg Ball, the dean of the school of education and a well-known math scholar, who helped design the program along with other faculty members at the university.
Every day for two weeks, Ms. Ball led a group of about 26 children, who will be 5th graders this fall, through a variety of math topics. Most of the students are from the Ypsilanti school system, a nearby working-class district of 4,000 students. The main focus is on lessons that give struggling students the most trouble, such as fractions.
Two video cameras are set up in her class. Images are transmitted into other classrooms for researchers and other observers outside the lab to watch. The university has also begun streaming images onto a password-protected Web site. The students’ written work is digitally copied, with their parents’ permission, as are lessons plans and handouts for future study.
Classroom “laboratories” aimed at testing new ideas or allowing observers to study exchanges between teachers and students have long had a place in schools. Some observers liken the Michigan lab to “lesson study,” a research method that originated in Japan and was popularized in the United States in the 1990s, in which teachers collaboratively plan, test, and review lessons over months or even years.
The Michigan faculty members are trying to offer something different. Their goal is to provide educators at all levels with a forum to pick apart and examine every aspect of elementary math instruction, including the “invisible,” or unexpected and overlooked, challenges that teachers in those grades face every day. They hope to expand the program eventually to include secondary school teaching and the study of other subjects.
“It’s a way of incubating and developing some very concrete ways of improving teaching and learning,” Ms. Ball said.
The program, she added later, seeks to answer a question, “What’s the math you need to know to teach?” and in doing so “expose the myth people have that teaching elementary math is easy.”
Teaching the Unsolvable
The Elementary Mathematics Laboratory grew out of an effort hosted for three summers at the Park City Mathematics Institute, a research center in Utah, where Ms. Ball led a math project. After she became the education school dean at Michigan in 2005, she sought to move the program here both for logistical reasons and so that she and her university’s faculty could revamp it.
Much of Ms. Ball’s scholarly work has focused on “math knowledge for teaching,” or the particular content knowledge and skills educators need to make the subject understandable to students. While Ms. Ball and her colleagues, such as mathematician Hyman Bass, have crafted the lab’s lessons, many of them could be found in almost any elementary classroom.
The lab work begins shortly after 8 a.m., when a few dozen observers, in sandals, shorts, and polo shirts, crowd into a second-story classroom at the education school for the mandatory preclass briefing.
Ms. Ball, who taught elementary school for 14 years before moving into academia, outlines the problems she will present to the Ypsilanti students, which today focus heavily on fractions and the number line. The lab observers include Ypsilanti teachers, who ask questions and sometimes challenge her plans for the day.
The give-and-take is evident when Ms. Ball describes an extended problem she will present the students over several days. A fictional customer is asking the class to construct a five-car train, with specifications to use different-size cars, which can be broken apart to meet his needs.
The problem comes with an unexpected twist: It is not possible, mathematically, to build the train the customer has requested, given the conditions. The goal is not only to show the students, who will work with miniature blocks representing train cars, that some math tasks cannot be solved— but also to show them that proof of that impossibility can, in fact, be a solution to certain math problems.
Ms. Ball explains that this problem can work to build a number of math skills at once, such as math vocabulary, through the use of words such as “exactly,” which give the problem precise meaning.
Some of the teachers, though, wonder if the assignment will confuse students.
“It’s satisfying to work on a problem and get an answer,” says one, adding that not finding a numerical solution “is going to be new for them.”
Another educator also questions the point. When a college professor responds by arguing that mathematicians deal with unsolvable problems all the time, a teacher counters: “But they’re mathematicians—not 5th graders.”
The teachers and researchers offer suggestions on how Ms. Ball can set up and present the problem. Shortly after 9 a.m., she and the educators move into the lab itself to see how their planning will play out. The observers take seats on blue chairs mounted on a three-deep row of risers in the back of the room, only steps from where the students will work. Whispering to each other, some unpack laptop computers, while others take out notebooks. A sign reminds them to silence cellphones and computers.
A few minutes later, the Ypsilanti students filter into the classroom and take seats around tables arranged in a U-shape.
The adults go quiet.
Ms. Ball reviews the previous night’s homework, then begins with a warm-up problem, in which fractions are represented as shapes and then as numbers. She moves to a number line, labeling positive and negative numbers, and then fractions on it.
The video cameras roll. An audio technician adjusts knobs, zooming in to pick up students’ voices and their interactions with Ms. Ball, which are captured by microphones set up on the tables.
At one point, Ms. Ball asks students to describe a shape presented on a poster as a fraction. A girl answers “two-eighths,” and Ms. Ball asks her to explain how she knows that. The girl’s answer comes through over the microphone.
“It takes eight of those green rectangles to cover the whole thing, and they’re all equal,” she says. Two of them are shaded in, she adds.
“That was a superb answer,” Ms. Ball says. “I want you to repeat it.”
Better Understanding
University officials recruited the Ypsilanti students, targeting those who have struggled in math. Some parents were attracted by the opportunity to send their children to a University of Michigan program; others by the chance to get their children help in math that they themselves felt unable to provide, said Imani Goffney, a Michigan graduate student in teacher education who helped recruit parents.
Students sign “contracts” agreeing to attend the lab and take it seriously. Transportation and meals are provided. After spending the morning in class, the children take part in an arts program in the afternoon and then engage in more one-on-one or small-group math activity.
Ms. Goffney’s daughter, Bria, takes part in the program. The 9-year-old, who says math is one of her stronger subjects, confesses that when students’ minds wander, they’ll point out adults in the crowd and gossip about them. But usually they’re too busy to think about the audience, she said.
“Most of the time, you feel like it’s kind of just you and your best friend” in class, Bria observed.
Parents have responded enthusiastically to the program, and district officials have seen strong math gains among 5th graders who took part last summer, Ypsilanti Superintendent James Hawkins said. The lab can sow good teaching and learning methods in the early grades, which he says is crucial as the state raises academic requirements for high school math.
“You really don’t start to shore up math literacy by just starting at the secondary level,” Mr. Hawkins said.
One way his district’s teachers are building their skills is through workshops at the education school, where they discuss the lab and their own teaching challenges.
Joel Grambau, a 5th grade teacher at Estabrook Elementary School, said he hopes the lab will help him take problems on fractions and other topics “in five or six different directions” so that he can cover important content while also meeting state standards. The 28-year-old said he’s also seeking tips on how to have students explain their answers in writing.
“It gives you an understanding of what they understood,” Mr. Grambau said, “and what they didn’t.”
Joining the Ypsilanti teacher in lab each day was Geoffrey Saxe, a professor of education at the University of California, Berkeley, who attended with a team of his graduate students. Mr. Saxe is using the lab and interviews with students to develop a 5th grade curriculum on fractions and integers, organized around the number line. He found the opportunity to work side by side with K-12 teachers appealing.
“This is really an interdisciplinary group,” he said. “It serves to embellish our project in an extraordinary way.”
While lesson studies sometimes bring together teachers and university researchers, Michigan’s program seems to go much further in that regard, said James W. Stigler, a professor of psychology at the University of California, Los Angeles, who has studied teaching methods used in other countries, including Japan. The program appears to resemble an element of lesson study known as an open house, in which participating schools invite outside observers to watch and discuss their lessons, he said.
The Michigan approach seems to encourage discussions of teaching that “are grounded in a common observation,” Mr. Stigler observed in an e-mail. “The more this happens, the better.”
At the end of the day at the Michigan lab, the eclectic group of observers meets with Ms. Ball for a debriefing. The participants bounce from topic to topic. One visitor praises Ms. Ball’s technique, but another says she noticed a girl losing interest, putting her head down on her desk.
“Is she lost or did she not get it?” the participant asks.
Another observer has a broader question: How can we choose the most effective math lessons?
Ms. Ball says she tries to weigh the time it takes to set up a task against how much students will gain from it mathematically. The advantage of an extended problem, such as the train-car exercise, is that a teacher can weave several different concepts within the same narrative.
“There’s a big ‘liftoff’ cost with every problem,” Ms. Ball tells the group. “I try to find one where there’s a big payoff.”
Coverage of mathematics, science, and technology education is supported by a grant from the Ewing Marion Kauffman Foundation, at www.kauffman.org.
