Elementary-age students bring lots of things to school with them — besides huge backpacks stuffed with supplies. They bring ingenuity, intuitive knowledge, and mathematical insight. They sometimes amaze their teachers with innovative ways to solve problems. When mathematics teachers link classroom instruction to students’ intuitive knowledge, students can go a long way.

What is CGI?

In a typical CGI math problem, young students are presented with a problem like this: Robin has $5. How many more dollars does she have to save to have enough money to buy a puppy that costs $12? The teacher asks students to think about ways to solve the problem. Students use a variety of strategies to solve this problem, such as using plastic cubes to model the problem, counting on fingers and using knowledge of number facts to find the answer. The teacher then asks the students to explain their reasoning process. They share their explanations with the class. The teacher may ask the children to compare different strategies. Children are expected to explain and justify their strategies, and they, along with the teacher, take responsibility for deciding whether a presented strategy is correct.

How unlike rote instruction this is. For one thing, it puts more responsibility on the students. Rather than simply being asked to apply a formula to several virtually identical math problems, they are challenged to find their own solutions. Second, they are expected to publicly explain and justify their reasoning to their friends and the teacher. Third, teachers are required to open up their instruction to students’ original ideas, and to guide each student according to his or her own developmental level and turn of reasoning.

Expecting students to solve problems using strategies that haven’t been taught to them and to explain and justify their thinking has a major impact on their learning. Not only are students learning specific ways to solve problems, but they are also increasing their knowledge of the fundamental principles of mathematics. For example, students who learn the standard addition algorithm often learn little more than a procedure to find the correct answer. Students who develop their own strategies for solving addition problems are likely to intuitively use the commutative and associative properties of addition. Students who use their own strategies to solve problems and justify them also contribute to a positive disposition toward learning mathematics.

CGI is based on research that shows that children come to school with rich informal systems of mathematical knowledge and problem-solving strategies that can serve as a basis for learning mathematics with understanding. A major goal of CGI is to help teachers build on this informal mathematical knowledge so that they understand the new ideas that they are learning. Because this teaching method is innovative, CGI offers classroom teachers help in understanding how children’s mathematical ideas develop. The focus is on children’s thinking, not on specifying specific teaching procedures or curriculum materials. Teachers learning to use the CGI method are given intensive professional development. They bring in examples of their students’ work. They share them with other teachers, examining and discussing student work, sometimes realizing they have underestimated their students.

CGI Continues to Grow

Graduates of the CGI research and development team are now professors, instructors and consultants, helping teachers implement CGI in more classrooms.

One CGI alum is Linda Levi, now an education consultant based in Madison, Wis. While a researcher at WCER, Levi observed strong demand for CGI professional development. She determined to do something to increase the availability and quality of CGI professional development.

“WCER had no central clearinghouse for workshops and no control over the quality of what others were doing,” Levi said. She now provides systematic professional development, working with the nonprofit Teachers Development Group in Oregon, which specializes in professional development for kindergarten through grade 12 mathematics teachers. She supervises 15 people who provide workshops across the country.

Why this demand? In part, because the No Child Left Behind Act (NCLB) pressures states to meet Adequate Yearly Progress (AYP) goals. Levi points to Arkansas and Iowa as two proactive states that are seeking research-based instruction and professional development in mathematics. Educators in these states are working at the state level to develop local capacity so they can provide their own CGI professional development.

Another CGI alum, Megan Franke, is an associate professor of Urban Schooling at UCLA. Franke’s current research builds on earlier CGI studies that showed that students in CGI classrooms solve a wider variety of word problems, use a wider range of strategies, and recall their number facts better than their peers in control groups. Franke said that listening to students’ mathematical thinking has transformed the teachers in her studies into learners. Her work aims to create new communities of practice with teachers, administrators and her research team.

Working with one school over a 4-year period enabled Franke and her colleagues to observe communities of practice and shifts in participation within them. The teachers in the work groups became much better at detailing students’ mathematical thinking: They not only detailed the strategies the students used but also could analyze the pedagogical practices that supported that student thinking.

The teachers also developed ways of talking with each other about the relationships across the students’ strategies that highlighted the mathematical ideas being developed. Teachers could talk about the relationship between a strategy used in multiplication and a strategy a child used in solving a multi-digit addition problem.

These teachers develop what’s called generative learning; that is, a sense of themselves as learners. In their work groups, they talk about their experimentation with problems, and they describe and enrich their learning from their expectations.

The Beginnings of CGI Math

Beginning in the 1980s, WCER researchers Tom Romberg, Tom Carpenter and Jim Moser studied how early number concepts develop in young children. Their work showed that children enter school with rich informal knowledge about mathematics. This informal knowledge could provide a basis for understanding basic mathematical concepts and skills. The researchers developed a list of problem types, which provided a framework for understanding the processes children use.

Eliz Fennema, Tom Carpenter and Penelope Peterson studied how this work on children’s thinking could be applied to improve mathematics instruction. They learned how teachers could use this knowledge to help children connect their own informal systems of mathematics to the formal symbols and procedures they learn in school. They based this approach on the premise that the teaching-and-learning process is too complex to specify in advance. Instruction needs to be mediated by teachers’ decisions. They reasoned that significant changes in instruction occur when teachers make informed decisions, rather than when they are programmed to perform in a particular way.

To assess students’ knowledge, teachers must understand the general stages that children pass through as they acquire the concepts and procedures, the processes children use to solve different problems at each stage and the nature of the knowledge that underlies these processes.

Most teachers can distinguish among many of the problems and the strategies that children use to solve them. But this knowledge is not always organized into a coherent framework that relates distinctions among problems, children’s solutions and problem difficulty to one another. CGI professional development helps teachers develop frameworks for understanding children’s thinking and learn to base instructional decisions on these frameworks.

Tapping and nurturing that ingenuity and intuition is the idea behind Cognitively Guided Instruction. CGI is an elementary-level mathematics professional development program developed at WCER in the 1980s and 1990s by education professors Eliz Fennema, Tom Carpenter, Megan Franke, Linda Levi, Susan Empson, Victoria Jacobs, and colleagues.