## CGI Math Encourages Ingenuity and Reasoning

June 15, 2007

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 their classroom instruction to students’ intuitive knowledge, students can take classroom instruction a lot farther.

### 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. A variety of student-generated strategies are used to solve this problem, such as using plastic cubes to model the problem, counting on fingersand using knowledge of number facts to figure out 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 the children, along with the teacher, take responsibility for deciding whether a strategy that is presented 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 with strategies that haven’t been taught to them. and asking students to explain and justify their thinking, has a major impact on students’ learning. Not only are students’ learning specific ways to solve problem, 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 to solve addition problems are likely to intuitively use the commutative and associative properties of addition in their strategies. Students using their own strategies to solve problems and justifying these strategies also contributes 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 method of teaching 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 saw a large 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 quality over what others were doing,” she says. She now provides systematic professional development, working with the nonprofit Teachers Development Group in Oregon, which specializes in kindergarten through grade 12 mathematics teacher professional development . 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 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 says 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 four-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.

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