Math Project Takes Wings
Paul Scott strides to the blackboard and writes, If the answer is 24, what is the question? Forty 3rd graders, sitting in teams of four, pull out their calculators and begin scribbling on pads: What is 14 + 10? What is 24 - 0? How many is two dozen eggs? Working in their teams, the students try to come up with as many problems as they can. Scott and his colleague, Ann Rouse, circulate around the large, open classroom providing assistance.
The brain-teasing exercise is part of a mathematics curriculum that is being piloted in St. Mary's County, Md., as part of an ambitious effort to revamp elementary schools for the 21st century. In 1992, the New American Schools Development Corporation funded researchers from Johns Hopkins University, local teachers and administrators, and officials from the Maryland Department of Education to create break the mold schools that could be replicated nationwide. Known as Roots and Wings, the project's goal is to keep all children performing at grade level in the regular classroom. The program also pushes students to move beyond the basic skills by applying ideas and concepts to solve real world problems. (See Education Week, June 9, 1993.)
Four elementary schools in this rural community green Holly, Carver, Lexington Park, and Ridge Have volunteered to serve as laboratories for the project. Last year, teachers began introducing new methods for teaching reading and writing. They also began using an innovative social-studies curriculum that asks students to solve real-life problems by drawing on their knowledge of science, history, math, and the arts.
This year, 22 teachers are pilot-testing the math curriculum, known as Math Wings, in grades 3 through 5. Next year, it will be expanded to grades 1 and 2.
Building a Bridge
Like the other components of Roots and Wings, Math Wings is designed to move students beyond the rote application of basic skills. The curriculum is based on the standards developed by the National Council of Teachers of Mathematics.
Students enter school with a great deal of mathematical knowledge, argues Robert E. Slavin, the project director and a professor of education at Johns Hopkins. They know about combining and separating, halves and wholes, and so on. What they need is a bridge between their pre-existing knowledge and the formal representation of this knowledge in mathematical symbols.
Math Wings is designed to provide that bridge through the use of hands-on materials, cooperative learning among students, and an emphasis on problemsolving.
Its classes come in two varieties. Whole-class units last four to six weeks and focus on a general concept, such as measurement. During the units, students work largely in teams of four that are mixed by skill level. A variety of hands-on materials and real-world problems exposes all youngsters to core mathematical ideas.
The daily routine includes facts checks rapid-fire exercises to improve students' agility in addition, subtraction, multiplication, and division and problem-solving to develop higher-order skills.
Math Wings, Slavin says, is meant to be a balanced approach. We're trying to have a program that emphasizes creative problem-solving and discovery, but that also insures that kids can get the right answers.
People used to think you had to master the basics before you could talk about concepts, he adds. We're saying that's exactly backward. Kids need to have the big picture of what these numbers are doing for them in the real world ... before they have any interest in how the numbers operate.
All This Good Stuff'
In between the whole-class units, students spend two-week intervals working to hone their math skills at their own pace. During the individual units, children work with a same-ability partner based on a skills inventory that is given before the unit begins. The teacher also provides mini-lessons to small groups of youngsters every day that are geared to their particular needs.
The innovative structure enables all students to become proficient at math, without holding back the entire class or limiting the exposure of less able students to the depth, breadth, and beauty of mathematics.
What's happened in the past is our slower kids never got all this good math stuff the concepts, the exposure to problem-solving, the number sense that they're getting now, says Gail Holt, a Roots and Wings facilitator at Green Holly Elementary School, where Scott and Rouse team-teach.
On the day before the Thanksgiving break, Green Holly's walls are festooned with paper Pilgrims and cardboard turkeys. And most students already have one eye on the door. But Scott and Rouse are plowing valiantly ahead with a whole unit lesson in subtraction for their 3rd graders. This particular lesson is designed for use by a substitute teacher and so uses few manipulatives. But it follows the general structure of a Math Wings class.
During the first 15 minutes, the student teams take out their materials and then quickly do a series of problem-solving activities. Scott also has children write in their logbooks, daily journals that help students think and write mathematically.
Today, the children are given a sheet with a few word problems and their supposed answers and are asked to decide as a team whether the answers are reasonable. The first problem states, Last year, Bert weighed 24 kilograms. Now, he weighs 35 kilograms. How many kilograms has Bert gained? The answer on the paper says, Bert has gained 59 kilograms.
The children spend a few minutes thinking. Then small heads converge around desktops and the discussion begins. I think it doesn't make sense, says one boy to his team. It's adding, when it's supposed to be subtracting. Another girl notes that Bert couldn't have gained 59 kilograms if he weighs only 35 kilograms in total. After a few minutes, Scott asks teams who think the answer is a reasonable one to hold their thumbs up, and those who think it's not to turn their thumbs down. A few students are asked to explain their team's reasoning.
The next 40 minutes of the class are devoted to action math: active, whole class instruction by the teacher on the day's concept and the use of hands-on materials by the children. During one 4th-grade lesson on number sense, for example, students use their spatial sense to estimate the number of raisins in an individual-serving-sized box. They measure the sides. Open the lid and eyeball the raisins lying on top of the heap. Shake the contents up and down.
Then, they collect data by counting the raisins in their boxes and recording and comparing the information for all of the boxes in the classroom. At the end of the lesson, the students are asked to look at the graphs on the walls and make predictions about the contents of other raisin boxes, based on the data they have gathered.
The last five minutes of the lesson are devoted to explaining the homework assignment, putting away the materials, and writing in logbooks, if the students have not done so earlier. Throughout the unit, nightly homework and periodic assessments keep track of how children are faring.
Team score sheets provide the groups with an incentive to work together and to support their less able peers.
More of a Participant'
Sharon Cook, a 4th-grade teacher at Green Holly, says, The kids are definitely more actively involved than in a traditional math class. There's more discussion going on between the students about what it is they're working on. If they have a problem, they can ask someone else on the team for help.
Signs on the wall urge students to rely on each other and not to ask teachers for help until they have exhausted other options.
It's not that the content is different, says Diane Berry, another 4th-grade teacher. It's just that it's presented in a different way that allows them to be more of a participant in what's going on.
Teachers particularly praise the almost daily use of hands-on materials that help students make the connection between the concrete and abstract worlds of mathematics. In addition to boxes of raisins, on any given day students may be wielding geometric solids, cubes, measuring spoons, protractors, dice, measuring cups, play currency, compasses, rulers, and thermometers.
Stephanie Haines, a 5th-grade teacher at Ridge Elementary School who first pilot-tested one of the fraction units last spring, says: Eighty percent of it was hands-on stuff that the kids made and used and manipulated. Rather than telling the kids what the concepts were in the beginning, they had to discover the ideas for themselves.
Those kids that usually had a difficult time in math seemed to understand and grasp this a lot faster, she says.
Today, Patricia Baltzley, the former president of the Maryland Council of Teachers of Mathematics who is spearheading the development of Math Wings, is visiting Green Holly to demonstrate a lesson from the individual units.
The units are based on an earlier model developed by Slavin and his associates at Johns Hopkins, called Team Accelerated Instruction, that stresses the self paced development of math skills.
As usual, Baltzley begins the class with an open-ended problem-solving exercise that the children do in teams. Then, she hands out a series of brown cardboard folders, one per student. Each folder contains an information sheet that reviews the skills the child will be working on, a set of practice pages and practice tests, and the answer sheets. Baltzley explains to the children that the information in their folders will differ.
Using the information sheets as a reference, the students work through the practice pages and the practice tests at their own pace, before taking a mastery test away from their notes. As part of the routine, children stop periodically to check their answers. Based on how they are doing, they either move on to the next practice page, continue with the one they are working on, or ask the teacher for help.
As the children continue to work independently, the teacher calls up small groups of youngsters for mini-lessons that last about 20 minutes each. Additional enrichment activities are available for students who progress at a rapid clip.
Theoretically, the structure enables teachers to zoom in on a child's strengths and weaknesses in an efficient manner. But teachers are nervous about the sheer mechanics of monitoring 25 children who progress at their own speed. They also worry about how much assistance nonreaders will need simply to make it through the written instructions.
Baltzley tries to be reassuring. And as she circulates around the room, helping individual students and chatting quietly, it all looks rather effortless. The six teachers, who sit off to one side, take notes.
The first year of Math Wings has not been without its glitches. The team scoring sheets, which were supposed to encourage and reward teamwork, were too cumbersome, discouraging some teachers from using them. Baltzley is going back to the drawing board to come up with a solution.
Some of the whole-class lessons could not be completed in 60 minutes or used too many manipulatives, leaving teachers feeling frustrated and overwhelmed.
But perhaps most troubling was the initial reaction from parents, who had not been prepared for how different their children's homework would look. There were phrases used that parents didn't understand, says 4th-grade teacher Wendy Flaherty. Children were using calculators to tackle problems that they couldn't do on their own. The parents didn't know how to do the homework, and it frustrated them, Rouse adds.
After parents expressed their dissatisfaction at a meeting of the parent teacher association, Green Holly decided to send home information packets before the start of each new unit that would include a glossary of terms. Researchers at Johns Hopkins have decided to develop similar materials for the program district wide. The packets will explain how Math Wings approaches a particular topic, such as division; include sample problems and their solutions; and explain terminology.
It's a shift in the way mathematics has been done, Baltzley says, and, in retrospect, we didn't do enough to pave the way for it.
Teachers are also struggling to master the cooperative-learning techniques that undergird Roots and Wings. And there are lots of management problems still to be resolved. Many of the student teams, for example, have not coalesced. As a result, some children are working steadily while others are sitting back and letting their partners do the work.
We're still not getting that team cooperation going, says Scott, who pulled out a manual on the rudiments of cooperative learning recently and then worried whether he was the only one having problems.
A workshop this month was designed to address such concerns by focusing on some of the more sophisticated aspects of teamwork, such as how to reach consensus.
It is a problem and one that we're just continuing to work on, Slavin says. Some teachers are doing a beautiful job with it. Some are not. These teachers are trying to incorporate major changes in the curriculum at the same time as major changes in pedagogy, and it's a lot.
Taking a Chance'
Despite the problems, teachers say the initial results have been encouraging. When I went through my grade book, Scott says, 95 percent of my students made a C or above on the report card, and that is an increase over the last 10 years that I've taught math. So I think that the children are learning more.
Flaherty agrees. Out of the 25 or 26 students in her class, she says, only three earned less than a C last marking period, and that's just incredible compared to what I've usually gotten.
I find that my kids are more willing to take a chance, she says, and say, Well, this is what I did to solve a problem,' even if they aren't correct. But she acknowledges that the process adds to her workload. I won't lie and say that it's not overwhelming, she says, because it is.
Students also seem to like the approach. One student, Kathleen, writes in her logbook: What I like most about math class is it's fun. What I like least is it's hard. Amanda, another student, writes: I like problem-solving because you get to add and minus. And you can think and have time to think.
As the students in Scott's class put away their materials for the day and prepare for the long holiday weekend, a few chat enthusiastically about the games they have played and the paper money they have used. Then, they put away their Pilgrim-shaped erasers and head toward the door.