Programs that Work
Cognitively Guided Instruction (CGI)
Children Succeeding in School
Students performing at grade level or meeting state curriculum standards
Age of Child
Early Childhood (0-8)
Middle Childhood (9-12)
Type of Setting
Type of Service
Type of Outcome Addressed
Cognitive Development / School Performance
Evidence Level (What does this mean?)
Cognitively Guided Instruction (CGI) is a professional development program that increases teachers’ understanding of the knowledge that students bring to the math learning process and how they connect that knowledge with formal concepts and operations. Developed by education researchers Thomas Carpenter, Elizabeth Fennema, Penelope Peterson, Megan Loef Franke, and Linda Levi, CGI is guided by two major theses. The first is that children bring an intuitive knowledge of mathematics to school with them and that this knowledge should serve as the basis for developing formal mathematics instruction in primary school. This thesis leads to an emphasis on assessing the processes that students use to solve problems. The second thesis is that math instruction should be based on the relationship between computational skills and problem solving, which leads to an emphasis on problem solving in the classroom instead of the repetition of number facts (e.g., practicing the rules of addition and subtraction).
Students in kindergarten through sixth grade
In 1989, Carpenter et al. studied the effects of CGI in a sample of 40 first-grade teachers from 24 schools located in Madison, Wisconsin, and in four smaller communities near Madison. Twenty teachers were randomly assigned to the CGI program (by school), and 20 were assigned to the comparison group. Twelve first grade students (six girls and six boys) were selected randomly from each class to serve as target students for analysis (excluding those children with special learning needs). In the two schools in which there were fewer than 12 first grade students in the classroom (due to first/second grade combinations), all first grade students were included in the sample. Student outcomes were evaluated using the computations and problem-solving subtests of the Iowa Test of Basic Skills (ITBS), along with a problem-solving instrument developed by the CGI research team. Pretest results were collected in September and posttests in April/May.
A second study (Villasenor and Kepner, 1993) examined CGI in a large Midwestern urban school district. Two first-grade teachers were voluntarily recruited from six schools in which there was at least a 50 percent minority population, resulting in a total of 12 CGI classrooms. Comparison schools were then identified that matched the treatment schools’ population characteristics, and 12 classrooms were selected to match the treatment classrooms. The CGI teachers participated in a 19-hour summer workshop, a 2-hour review in September, and two additional support sessions. Comparison teachers participated in a staff development program that focused on problem solving in elementary school mathematics but did not include CGI principles or research. To compare group outcomes, 12 students were randomly chosen from each class (six boys and six girls), for a total of 144 students in the CGI group and 144 in the control group. Student outcomes were assessed on a 14-item arithmetic word-problem test (Carpenter et al., 1989), which was administered to students in early October and again five months later. In addition, students were interviewed individually to assess the processes and strategies they used to solve both written (word) problems and number-facts problems. An analysis of pretest scores indicated that the CGI group significantly outscored the control group on the written-problem solving pretest; therefore analysis of posttest outcomes were conducted using pretest scores as covariates.
Key Evaluation Findings
The Carpenter et al. (1989) study of 20 CGI teachers and 20 control teachers found the following:
- CGI students scored significantly higher than control students on the complex mathematics addition and subtraction portion of the ITBS; with a mean score of 8.6, compared with 7.8, out of a possible score of 12.
- Similarly, CGI students scored significantly higher than control students on the ITBS problem-solving interview, with an average score of 5.61 versus 5.38, out of a possible score of 6.
- CGI students also significantly outscored controls on the ITBS number-facts problem scale, with a mean score of 2.26 versus 0.78, out of a possible score of 5.
- No significant differences were found between groups for the ITBS problems, advanced problems, or computation scales.
Villasenor and Kepner's (1993) study of 144 CGI students and 144 control students reported:
- CGI students scored significantly higher than control students on the written problem-solving test (9.41 versus 3.18, out of a possible score of 14).
- CGI students significantly outscored control students on the word problems portion of the interview (5.44 versus 2.93, out of a possible score of 6).
- The CGI students also scored significantly higher than control students on the number-facts section of the interview (4.68 versus 3.00, out of a possible score of 5).
Public and private elementary schools
CGI for kindergarteners was originally developed with funding from the National Science Foundation. CGI for older elementary school students (Integrating Arithmetic and Algebra) was developed with funding from the National Science Foundation and the U.S. Department of Education's Office of Educational Research and Improvement through the National Center for Improving Student Learning and Achievement in Mathematics and Science.
CGI provides a basis for identifying what is difficult and what is easy for students to comprehend in their study of math. It also provides a way for dealing with the common errors students make while learning. The emphasis is on what children can do, rather than on what they cannot do, which leads to a very different approach regarding incorrect answers. With the CGI approach, teachers focus on what students know and help them build future understanding based on present knowledge. The program aims to improve children's mathematical skills by increasing teachers' knowledge of students' thinking, by changing teachers' beliefs regarding how children learn, and by ultimately changing teaching practices.
In 1996, CGI was extended into the upper elementary school levels to assist first through sixth grade teachers in integrating the major principles of algebra into arithmetic instruction. The program is based on the premise that children throughout the elementary grades are capable of learning powerful unifying ideas of mathematics that are the foundation of both arithmetic and algebra. Learning and articulating these ideas enhance children's understanding of arithmetic and provide a foundation for extending their knowledge of arithmetic to the learning of algebra.
There is no set curriculum. Teachers use the CGI framework with existing curriculum materials, or they use CGI principles to help develop their own math curriculum.
Training for CGI includes an initial workshop for teachers led by program developers or experienced CGI teachers, along with additional support that often lasts for several years in the form of booster workshops, technical assistance, and mentoring.
Issues to Consider
This program received a "promising" rating. Evaluations demonstrated significantly superior mathematics achievement for CGI students when compared with control students on several outcomes.
While the Carpenter et al. (1989) study reported significant differences between CGI and control students when looking at performance on smaller subgroups of test problems, there were no significant differences between the treatment and control groups on the total score of the ITBS. CGI students scored significantly higher than control group students on problems identified as number-facts problems or complex addition/subtraction problems, but not on simple addition/subtraction or advanced problems. This finding suggests that CGI may be limited to improving student outcomes only on certain types of mathematics skills. This outcome is of particular interest because the ITBS is the only measure used in any of the evaluations that was not designed by the program developers.
The evaluations also imply that successful implementation of CGI requires a substantial amount of training and supervision. The success of the first two pilot studies was linked to intensive involvement with staff trainers, who were often the original program developers. While there have been several attempts at scaling up CGI, none of those efforts has included systematic evaluations of the effectiveness of the program in terms of the children’s math performance.
CGI training programs have been established in Wisconsin, Minnesota, Texas, North Carolina, Arizona, and Ohio. There are also several start-up programs in Phoenix, Arizona; Fargo, North Dakota; Dearborn and East Lansing, Michigan; and other sites in California, Alaska, and New Zealand.
University of Wisconsin-Madison
Wisconsin Center for Education Research
1025 West Johnson Street
Madison, WI 53706
phone: (608) 263-4267
Three CGI publications are available to assist teachers with implementation of the program. Available through the National Council of Teachers of Mathematics (NCTM) Web site (http://my.nctm.org/), the first (Carpenter et al., 1999) is designed to help teachers understand children's intuitive mathematical thinking and how students can build up their concepts from within. The guide and accompanying CDs provide a framework for assessing children's thinking in whole number arithmetic and allow readers to look inside real classrooms implementing the CGI technique.
- Carpenter, Thomas P., Elizabeth Fenneman, Megan Loef Franke, Linda Levi, and Susan B. Empson, Children's Mathematics: Cognitively Guided Instruction, The National Council of Teachers of Mathematics, Inc., 1999, Heinemann: Portsmouth, NH.
Also available through NCTM, Carpenter et al. (2000) presents a CGI professional development program—a series of workshops along with work in classrooms with children—in which participants learn about the CGI structured framework of mathematics, children's thinking, and the way children's thinking evolves.
- Carpenter, Thomas P., Elizabeth E. Fennema, Linda Levi, Megan Loef Franke, and Susan B. Empson, Children's Mathematics: Cognitively Guided Instruction—A Guide for Workshop Leaders, The National Council of Teachers of Mathematics, Inc., 2000, Heinemann: Portsmouth, NH.
The third publication (Carpenter, Franke, and Levi, 2003) is available through Heinemann (http://books.heinemann.com/products/E00565.aspx), and it provides a foundation for the teaching and learning of algebra.
- Carpenter, Thomas P., Megan Loef Franke, and Linda Levi, Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School, The National Council of Teachers of Mathematics, Inc., 2003, Heinemann: Portsmouth, NH.
Carpenter, Thomas P., Elizabeth Fennema, Penelope L. Peterson, Chi-Pang Chiang, and Megan Loef, "Using Knowledge of Children's Mathematics Thinking in Classroom Teaching: An Experimental Study,"
American Educational Research Journal,
Vol. 26, No. 4, 1989, pp. 499-531.
Villasenor, Albert, Jr., and Henry S. Kepner, Jr., "Arithmetic from a Problem-Solving Perspective: An Urban Implementation," Journal for Research in Mathematics Education, Vol. 24, No. 1, 1993, pp. 62-69.