# It All Adds Up

Madison, Wis.

It's time to do "mouse problems" in Mary Jo Yttri's kindergarten class at Lapham Elementary School. The problem she poses to her students is this: If there are four mother mice and each has five babies, how many baby mice are there altogether?

Emily, a brown-haired girl with paper mouse ears perched crookedly on her head, carefully arranges four piles, each made up of five plastic blocks. Atop each pile, she places a sixth block. This, she says, is the mother mouse. She counts the blocks underneath and decides the answer is 20.

"How did you know not to count the mother?" Yttri asks. "Because you said, 'How many babies,"' an eager classmate interjects.

Ben, on the other hand, sees no need to use his blocks.

"I counted by fives--5, 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 5 and 5 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 classroom.

Developed over a dozen years by Thomas P. Carpenter, Elizabeth Fennema, and their colleagues at the Wisconsin Center for Education Research here at the University of Wisconsin, 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, as many educators suppose. They already have certain intuitive understandings about math, and they 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 help them 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, "and I see them in action, and I can see them progress so, for me, it's really satisfying."

"It's also good education," she adds.

## Finding Universals

Cognitively Guided Instruction grew out of research Carpenter conducted with James M. Moser in the 1970's and early 1980's 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 explains.

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 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.

Later studies showed that all of these problem-solving patterns were found regardless of whether children lived on Oneida Indian reservations, in African villages, in France, in Korea, or in large urban school systems here in the United States.

"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 the university.

## Testing a Theory

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 paired those teachers with another group of teachers who had not participated in the workshops and then sent observers to all 40 classrooms to watch what would happen over the course of the next school year.

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. They 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.

And, more than the control teachers, the teachers using the cognitive approach believed that they could base their instruction on students' existing knowledge.

As for the students, the researchers found that, by year's end, they were better than the other 1st graders 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."

Inspired by their results, the researchers decided to take a closer look at the cognitive approach's potential in the classroom. This time, working with Megan L. 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 teachers in the study.

Again, teachers changed 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 they had taught in that grade the previous year. On average, they found, children's scores for the same grade level on word problems and tests of their grasp of mathematical concepts were higher by a full standard deviation by the end of the first year in the study--a sizable amount by researchers' standards.

Center researchers also expanded the program to include kindergartners and discovered something new: Children that age 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 today 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, who, as aformer project officer for the National Science Foundation, oversaw more than $6 million in grants that supportedthe research.

It has also helped build up a network of working teachers who now consider themselves converts to the approach.

"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.

But teachers also 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 are still not there.

"I feel like after two years I should've perfected it by now," says Pam Thomas, who teachers 3rd grade at Watkins Elementary School in the District of Columbia, where Jenkins is working with a handful of teachers who are trying out the approach.

"When Mazie asks questions, I can see my kids' light bulbs light up," she adds. "Sometimes they're dim when I'm talking to them."

## No Blueprints

Part of the reason for the long change process is that Fennema and Carpenter have taken great pains not to tell their teachers how to teach.

"It seems to me what you want children to do, you want teachers to do," Fennema says. "And that is to solve the problem of teaching class."

For that reason, Cognitively Guided Instruction in Yttri's kindergarten looks different than it does in Thomas's 3rd grade. And those classrooms look different still from Annie Keith's combined 1st- and 2nd-grade classroom back in Madison. There, during a recent visit, children were working at a variety of problems at math centers.

Keith also asks her students to choose their own numbers for problems, encouraging the children 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 also write their own word problems in her class and keep math journals. Some teachers prefer to have students work in groups; others stick to classroom-wide 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.

In many ways, teachers say, Cognitively Guided Instruction is also harder than more traditional teaching methods. Teachers 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 say they spend a lot more time preparing their own problems now.

"I think about where I want them to go, and I play around with what certain numbers will do," says Susan Gehn, a 3rd-grade teacher at Nichols Elementary School just outside Madison.

She asked her pupils, for example, to find out how much a family of seven would spend on sodas at the Super Bowl if each drink cost $2.25. She chose those numbers, she says, because she knows many of her students can easily multiply seven times two and others would know that four quarters add up to a dollar.

"I'm trying to get my kids to use the knowledge they already possess to solve problems," she says. "A lot of them don't have that confidence yet."

As Cognitively Guided Instruction matures to the point where it will become more widely used, Fennema worries that some schools and educators might not have the patience to stick with the philosophy for the time it takes to bring about real change.

"For so long, children have been so worried and so frustrated about mathematics, and teachers have been so worried and so frustrated about mathematics," Fennema says. "I'm worried they'll think it's a quick fix."

But, if it's harder for teachers to teach mathematics this way, it also is in some ways harder for students. Rather than plug numbers into formulas, they must come up with their own solutions.

"I didn't do math like this last year," says Rachel McCrea, one of Thomas's 3rd graders. "This is hard."

"But I'd rather have it hard because once you learn it hard, you understand it," she adds.

Further information on this topic is available from:

Wisconsin Center for Education Research 1025 West Johnson St. University of Wisconsin Madison, Wis. 53706

Carpenter, T.P., Ansell, E., Franke, M.L., Fennema, E., & Weisbeck, L. (1991). Models of problem solving: A study of kindergarten children's problem-solving processes. Journal for Research in Mathematics Education, 24(5), 427-440.

Carpenter, T.P., & Fennema, E. (1992). Cognitively guided instruction: Building on the knowledge of students and teachers. In W. Secada (Ed.), Curriculum reform: The case of mathematics in the United States. Special issue of International Journal of Educational Research. Elmsford, N.Y.: Pergamon Press Inc.

Fennema, E., Carpenter, T.P., & Franke, M.L. (In press). Mathematics instruction and teachers' beliefs: A longitudinal study of using children's thinking. Journal for Research in Mathematics Education.

Fennema, E., Franke, M.L., Carpenter, T.P., & Carey, D.A. (1993). Using children's mathematical knowledge in instruction. American Educational Research Journal, 30(3), 555-583.

Fennema, E., Carpenter, T.P., Franke, M.L., & Carey, D.A. (1992). Learning to use children's mathematics thinking: A case study. In R. Davis & C. Maher (Eds.), Schools, mathematics and the world of reality. Needham, Mass.: Allyn and Bacon.

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