Doin' What Comes Naturally
"How did you know not to count the mother?'' Yttri asks.
"Because you said, 'How many babies,' '' a classmate interjects.
Ben, on the other hand, sees no need to use his blocks.
"I counted by fives--five, 10, 15, 20--because there were four mothers, and they had five babies,'' he says.
Another little boy solves the problem a different way. "I knew five and five was 10, so 10 and 10 must be the right answer,'' he says. "Then I counted it out on my fingers.''
Without having been taught formal computation, all these students have used perfectly reasonable strategies to arrive at the right answer. And that, in large part, is the governing principle behind Cognitively Guided Instruction--the approach that drives the mathematics curriculum in Yttri's class.
Developed over a dozen years by Thomas Carpenter, Elizabeth Fennema, and their colleagues at the Wisconsin Center for Education Research, located at the University of Wisconsin at Madison, Cognitively Guided Instruction is a research-based philosophy for teaching mathematics. A central idea of the approach is that children do not come to school as blank slates, that they already have certain intuitive understandings about math and can use a variety of fairly predictable strategies to solve problems--even multiplication problems like those Yttri's students were tackling.
If teachers understand those strategies and understand how children think mathematically, Carpenter, Fennema, and their colleagues maintain, they can teach in ways that help their students build on what they already know and learn from each other.
Yttri knows, for example, that Emily will learn in time that she can substitute her fingers for the blocks she uses to model her problem. And the boy who used his fingers to count from 10 to 20 will eventually learn how to count by fives, as Ben did. Her job is to give her students the scaffolding they need to make those leaps in understanding on their own.
"I really know what my students are able to do and how they think about numbers,'' Yttri says. "I see them in action, and I can see them progress. So, for me, it's really satisfying. It's also good education.''
Cognitively Guided Instruction grew out of research Carpenter conducted with James Moser in the 1970s and early '80s on how young children learn basic arithmetic ideas. They studied more than 150 1st, 2nd, and 3rd graders, interviewing them three times a year and giving them word problems to solve each time.
They discovered marked similarities in the way children come to understand mathematics. For one, most children, like Emily, first solve problems by modeling them with concrete objects. Eventually, they realize they can find the same answer more efficiently by counting. As they solve problems, they also come to know certain number facts more quickly than others. They easily learn, for example, doubles--such as 6 plus 6--or sums up to 10 without having to be drilled in them. "There seems to be a natural move to use more efficient strategies,'' Carpenter says.
The researchers also identified types of word problems designed to reflect those kinds of intuitive understandings. For example, "joining'' problems--those in which groups are joined or added together--are easier for children than problems in which they are given a total and a part of the total and then are asked to find the remaining part. An example of this type would be: There are 20 trees in the park. Twelve are oak, and the rest are pine. How many are pine? Although adults may think of this as a subtraction problem, many young children would solve it by counting from 12 to 20, keeping track of the number of counts they make as they go.
"We decided to see if we could combine the research base on children's thinking with what we know about how teachers make instructional decisions,'' says Fennema, who, like Carpenter, is a professor of curriculum and instruction at UW-Madison.
Toward that end, Fennema and Carpenter held a four-week summer workshop to share the research on children's thinking with 20 1st grade teachers and to give them an opportunity to figure out how they might use that information in their own classes. They then selected 20 teachers who had not participated in the workshops and sent observers to all 40 classrooms.
They found that teachers who had been through the program did change their instructional practices. They used problem solving more than did teachers in the control group, and they spent half as much time drilling students on number facts. These teachers also encouraged their students to use a variety of strategies to solve problems, and they asked students to share those strategies with their classmates--all this even though the researchers had never prescribed any specific instructional practices for them. "The kinds of instructional decisions teachers make are much too complex to script,'' Carpenter says.
Unlike many teachers in the control group, those using the cognitive approach believed they could base their math instruction on students' existing knowledge.
As for the students, the researchers found that by year's end the program 1st graders were better than the others at problem solving and understanding arithmetic ideas. They also expressed more confidence in their problem-solving abilities. And even though they had not been drilled as heavily in number facts, they recalled them more accurately than the other 1st graders in the study.
"The debate in mathematics has been: Should we teach skills first, then teach the understandings?'' Fennema says. "We think it makes a lot more sense to get the understanding first and build the skills on them.''
Spurred on by the findings, the researchers decided to take a closer look at the cognitive approach's potential in the classroom. This time working with Megan Franke, the third principal co-investigator in the project, they recruited 21 teachers from 1st through 3rd grades from three demographically diverse schools in the Madison area. Over the course of three years, they gave the teachers a series of workshops on children's thinking. Each school also had a teacher who had already used the program and that teacher was given one release day a week to work with the teachers in the study.
Again, teachers altered their classroom practices, and the emphasis on problem solving and discussing solutions in their classrooms increased over time. But the researchers went one step further: They looked at how the achievement of five teachers' students compared with that of students the same teachers had taught the previous year. On average, the program children's scores on word problems and tests of mathematical concepts were higher by a full standard deviation than the other students' scores--a sizable amount by research standards.
When the researchers expanded the program to include kindergartners, they discovered something new: Five-year-olds are capable of solving more sophisticated mathematical problems than is commonly assumed.
In that study, 32 of 70 kindergarten children were able to use valid strategies to solve nine problems that involved a whole range of mathematical operations--including addition, subtraction, multiplication, and division. Most of these operations are not traditionally part of the kindergarten curriculum and were not even included in the first draft of national standards for teaching mathematics developed in 1989. "As you talk to teachers about this and show them the videotapes, they'll say, 'I never knew my kids understood this,' '' Fennema says.
Now, center researchers are working with teachers of 3rd through 5th grades to see whether the approach holds up as children begin to deal with fractions, decimals, and other more complex ideas.
All of the fieldwork has resulted in dozens of articles and papers. It has also put Cognitively Guided Instruction among "the top crop of leading research projects that have gone on in this area,'' says Raymond Hannepel, a former project officer for the National Science Foundation, who oversaw $6 million in grants that supported the research.
What's more, the work has helped build up a network of teacher converts. "Once you see the beauty of what this does for kids, you can't teach another way,'' says Mazie Jenkins, a Madison teacher who, after 12 years with the project, has become a resource on the method for other teachers in her district and across the country.
Still, most teachers who use the approach acknowledge that their conversion has not been easy. Some say it was years before they became comfortable and adept at listening to their students' thinking; others say they're still not there. "I feel like after two years I should have perfected it by now,'' says Pam Thomas, who teaches 3rd grade at Washington, D.C.'s Watkins Elementary School, where Jenkins is lending a hand. "When Mazie asks questions I can see my kids' light bulbs light up. Sometimes they're dim when I'm talking to them.''
Fennema and Carpenter have taken great pains not to tell their teachers how to teach. "It seems to me that what you want children to do, you also want teachers to do,'' Fennema says. "And that is to solve the problem of teaching their class.''
For that reason, Cognitively Guided Instruction in Yttri's kindergarten looks different than it does in Thomas' 3rd grade. And both those classrooms look different still from Madison teacher Annie Keith's combined 1st and 2nd grade classroom. Keith often has her students work on various problems at math centers. She also asks her students to choose their own numbers for problems, encouraging them to select what is "just right'' for them rather than something that is "too challenging'' or "too easy.''
"I'll say, 'Did you challenge yourself?' '' she says. "As we start to get to know the kids, we know which ones to push for a little more challenge.''
Students in Keith's class also write their own word problems and keep math journals. Some teachers prefer to have students work in groups; others stick to classroomwide instruction. Likewise, some teachers introduce mathematical symbols to the class, while others never do. They prefer to watch their usage spread slowly, usually because one student has seen a division or fraction symbol somewhere else and has started using it.
Regardless of their individual approach, teachers say cognitively guided teaching is in many ways harder than more traditional teaching methods. For one thing, they must keep track of where students are in their understanding. And, while they once might have relied on a textbook to dictate their lessons, teachers using the method now spend a lot of time preparing their own problems.
Teachers aren't the only ones who find the approach challenging; it's hard for many students, as well. Rather than plug numbers into formulas on work sheets, they must come up with their own formats and solutions.
"I didn't do math like this last year; this is hard,'' says Rachel McCrea, one of Thomas' Washington, D.C., 3rd graders. "But I'd rather have it hard because once you learn it hard, you understand it.''