In a nutshell, Cognitively Guided Instruction (CGI) is learning through problem solving. The developers of CGI believe that students come to school with a substantial amount of informal mathematical experiences that can act as the foundation for learning formal mathematics curriculum (Carpenter, 4). The goal of this approach is not to deliver direct instruction but to provide students with opportunities that will further develop each child's problem solving strategies. This approach aims for students to be able to think flexibly and mentally. The students are expected to use what they've learned in order to successfully complete a mathematics task and are given the option to use which ever manipulatives (blocks, rods, linking cubes, etc) and strategies (making a list, drawing a picture, number sentence, etc) that are most comfortable for them. Once they have completed a problem, they are encouraged to explain their solution strategies to the class while other students listen and ask questions about their strategies.
The structure of CGI is an inquiry-based, student-centered approach in its truest form. The teacher's role is not to provide knowledge but to move students along in the learning process. Students are actively participating in their learning which occurs within each student's zone of proximal development. In this way, the teachers are truly the facilitators in each child's learning as they design and develop mathematical tasks that will guide students towards the highest level of thinking in problem solving. This type of approach fosters a student-centered environment in which students' needs, abilities, and learning styles are at the center of mathematics instruction. Students are expected to interact with a variety of problem types as well as interacting with other students through the problem solving process. Multiple solution strategies are welcomed so that each child is able to process and apply more effective techniques as they are guided to the next level of development.
The levels of development within this approach is divided into three stages: (1) direct modeling, (2) counting on, and (3) number facts. Direct modelers use manipulatives to actively model each step of the problem in an effort to find a solution. These students often have trouble creating a plan for their solution strategy and can only complete one step at a time. In addition, this assists students to successfully complete only a limited number of problem, specifically problems that require a starting amount and representing the problem with concrete objects (join result unknown, part-part-whole whole unknown, and separate result unknown). Eventually, students substitute their modeling strategies with counting strategies, which allows them to further develop their understanding of numbers. These students have the ability to represent numbers more abstractly and can be more efficient. As this strategy becomes easier to apply to a great variety of problem types (join and separate start unknown and compare difference unknown), the students will naturally develop more flexible ways of thinking out of necessity. As learners encounter larger numbers in their problem solving, they know to learn number facts so that they can solve problem efficiently and accurately. As students encounter various problems types they will reflect on their current strategies and search for alternative strategies that better meet their needs.
Research and Support for CGI
"CGI is based on an integrated program of research focused on the development of students' mathematical thinking; on instruction that influences development; on teachers' knowledge and beliefs that influence their instructional practices; and on the way that teachers' knowledge, beliefs, and practices are influenced by their students' mathematical thinking" (105). Through research conducted by Carpenter and his colleagues from 1985 - 1996, they discovered that:
- teachers had valuable insight into students' learning about mathematics but this insight did not always influence teachers' decision making (105).
- children can solve an assortment of problems at an early age by using inventive strategies when given the chance to do so;
- teachers' understanding of their students' thinking positively correlated with student achievement.
As a result, CGI was designed to help teachers better understand and guide students mathematical thinking by educating teachers about the important role that modeling and counting strategies play in children's learning about mathematics.
Additionally, CGI supports the National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics. Teachers are expected to understand what students' know and need to know for students to be successful and support their mathematical thinking. Students are expected to play an active role in their learning by building new knowledge and sharing their thinking. CGI primarily focuses on the number and operations strand which includes understanding and representing numbers, relationships among numbers, meanings of operations, and computational fluency (NCTM, 2004). CGI aims for students to build mental and computational fluency in order develop efficient and accurate mathematical strategies. As physical modeling strategies give way to more abstract counting strategies, students begin to rely on number facts as they gain better comprehension of mathematical operations. Through these methods, students become more skilled at identifying different ways to solve various problems by recognizing how operations and strategies overlap.
Furthermore, CGI not only addresses the number and operations strand but also the following principles established by the NCTM:
- Problem Solving. Problem solving is an embedded part in the CGI approach. "Problem solving is not only the goal of learning but also a major means of doing so" (NCTM, 2004). Students are expected to critically think about a mathematical task, ask questions, draw conclusions, and stimulate new learning. Students expand their strategy base for solving problems and can make adjustments to their solution strategies.
- Reasoning and Proof. Logical thinking and planning is expected as students share their thinking with one another. This helps students to learn from one another and rationalize their solution strategies.
- Communication. Communication is at the core of problem solving as students are expected to organize, consolidate, and convey their mathematical thinking and ideas. They are obligated to listen to, build on, and evaluate the ideas that they hear from their classmates to extend their learning.
- Connections. By solving different types of problems through the CGI approach, students are able to discover similarities and differences between these problems and identify overlapping strategies.
- Representations. Students are participants in creating representations in ways that make sense to them. This may include physical manipulative, expressions, symbols, diagrams, etc. Students also learn common representations so that they are able to communicate their thinking to others. CGI supports by leaving the learners responsible for choosing how to represent their understanding.
Key Components of CGI
The CGI approach clearly has several components that parallel with the NCTM Principles and Standards. The framework for CGI approach includes: (1) learning base-ten concepts, (2) students learning within their zone of proximal development or ZPD, (3) the ability of multi-digit computation, and (4) verbal discourse.
Learning base-ten concepts is a fundamental idea to understanding place value. As students "encounter base-ten concepts, they have the opportunity to develop basic principles that are essential to understanding base-ten numbers. As the learn base-ten principles, they only have to deal with the special characteristics of grouping by ten and how such groupings are related..." (Carpenter, 45). This knowledge fosters students ability to solve grouping and partitioning problems "...provide a context for them to develop a rich understanding of mathematics in a way that is meaningful to them (46). Teachers are encouraged to develop word problems that support the development of groups of ten.
"CGI teachers provide problem solving experiences that enable each child's knowledge to grow" (104). As multiple solutions strategies are shared, students are encouraged to question, adopt, and employ strategies that they hear from their peers. The students are able to adopt strategies that are developmentally appropriate for them as they complete mathematical tasks and work within each student's zone of proximal development.
The development of multi-digit computation increases the establishment of base-ten concepts (63). Students do not need to be taught rote facts or skills in order to successful complete multi-digit problems but are able to build upon and create strategies that are meaningful to them. Students may choose to apply a variety of strategies to obtain an accurate solution to a problem.
It is evident through the CGI approach that students are encouraged by what they hear and say to others. This is based on the belief that 'the more problems I solve using basic facts that I need to know, the more I will remember them better and can be more comfortable with those facts.' As previously mentioned, students are actively engaged in their learning as they interact with problems and classmates in their mathematical environment. This constructivist approach using verbal discourse not only as a mental tool but also a method of communicating important ideas, employing Vygotsky's social learning theory.
These components of the CGI approach are also evident in Math Expressions (ME), the recently adopted mathematics program in Wake County. Similarities between CGI and ME include the following:
- The social interaction and discussion component is evident in each program. Originally constructed based on Vygotksy's theory of social development, participants of CGI are expected to participate in tasks that require them to play an active role in their learning. Students contribute to their learning by building upon prior knowledge through contact and communication with peers and teachers. Likewise, ME offers a ‘math talk’ component which students ‘solve, explain, question, and justify’ their thinking to peers and teachers. This verbalization of concepts helps students to internalize skills and concepts and deepens students’ understanding of mathematical ideas.
- · Flexible mental strategies are a focus. Both programs encourage the dvelopment of efficient thinking strategies that elicit alternate ways of thinking. CGI offers students the opportunity to generate solution strategies that they understand since they are invented, adapted, and extended by them. This foundation allows for taught structures in single-digit concepts to be applied to multi-digit structures, enabling students to build mathematical competency in place value concepts. This mental flexibility provides students with a natural progression of alternative solutions. Similarly, ME is structured so that students are expected to demonstrate their knowledge through proof drawings, reasoning, and fluency tasks.
- 'Student-centered' approach is central to both programs. CGI teachers are expected to create problems and provide opportunities that meet the needs of the students on a regular basis. The teacher acts as the facilitator in the classroom as students engage in their own learning, develop, and teach their strategies. In the same way, Math Expressions aims to create a shift from teachers as the source of ideas to children having a stronger influence on the teaching and learning that happens in the classroom. Though the teacher is still facilitator in the children’s learning, students are expected to contribute to the learning environment and their knowledge and insight is at the forefront of instruction.
One major difference between CGI and other programs such as ME is the way that students establish their mathematical foundation. In other words, the "learning through problem solving" approach inverts the traditional process that mathematics is taught. Traditionally, students are taught the skills and knowledge needed in order to learn and build upon new concepts. Teachers then give students several chances to practice and apply these strategies in story problems. In the CGI approach, the heart of learning new concepts and skills is through solving word problems. There is no direct teaching of specific strategies by the teacher but through inventive strategies created and shared by the students.
Pros and Cons
As with all programs, the CGI approach has its strengths and weaknesses. Below are a few of the benefits (+) of CGI as well as limitations (-).
(+) Opportunities for learning multiple solution strategies are enormous. Students learn from one another through discussion and interactions with one another as they solve different types of problems.
(+) This approach provides a great deal of flexibility in teacher choice about what types of problems the students focus on. These allows teachers to have more input in what students learn, a component that is often lacking in school curriculums.
(+) The student-centered approach allows for natural differentiation of learning math strategies and ideas. This sets the stage for the teacher to provide scaffolding at each child's individual level.
(+) This approach breaks down different types of problems to ask children which helps teachers develop problems that are more beneficial to student learning.
(-) CGI does not cover all strands of math that students learn. Though the number and operations strand is integrated in all other strands, it can not stand-alone program. It must be used in conjunction with a holistic mathematics program in order to ensure student success. With that said, key components such as verbal discourse can be applied to all strands and is applied in ME through 'math talk.'
(-) Vocabulary skills are not specifically taught using this approach. This may make it difficult for students to effectively communicate with one another, and may present further difficulties as the students progress through the grades because a 'common language' may not have been established.
(-) The lack of structure and knowledge of resources can be daunting for students that need more guidance from the teacher. Students that struggle to grasp mathematics concepts may require direct modeling of specific strategies to be successful. The variety of problem types and strategies can be overwhelming and students make struggle to make sense of what strategies to use.
Integrating CGI into the Regular Classroom
"It is not simple to describe a typical CGI classroom because each one is unique and can appear to be quite different from other CGI classrooms" (95). This flexibility allows the format of CGI to be done one-on-one, in small groups, or whole class. However, there is a general sequence that CGI classrooms follow:
- Teachers pose a problem which they or the children have written, children are given time to solve the problem while the teacher moves around the classroom and ask questions, and children report how they solved the problem. This sequence is then repeated (96).
Afterwards, the teacher may elicit several problem solving strategies for completing the problem. All learning is done through problem solving for story-like problems and each child is actively involved in choosing how to complete a task. Because of this, student discussion is prominent and serves to promote evaluation and reflection of strategies, build comprehension, and allows the teacher to assess each child's metacognitive strategies. They may communicate verbally, use pictures and/or number sentences, or with paper and pencil. There is no preferred strategy that all students must understand and apply. Accepting and encouraging a variety of multiple solution strategies lets students know that their contributions are valued and they realize that "there is no best or 'right' way to solve any problem" (99).
It is also important for teachers to emphasis the relationships of numbers and operations. They must create questions that focus on number fact relationships so that they can better understand the overlap between different types of problems and solutions. By understanding relationships, students can move toward more effective strategies and consolidate their problem solving process. "A CGI teacher's role is active. CGI teachers continually upgrade their understanding of how each child thinks, selects activities that will engage all children in problem solving and enable their mathematical knowledge to grow, and create a learning environment where all children are able to communicate about their thinking..." (101). Effective teachers are active in providing diverse learning opportunities for all students, are able to provide scaffolding, differentiate, facilitate, and offer opportunities for student interaction and communication. This active role is not unique to CGI and can/should be implemented into the regular classroom.
REFERENCES:
Carpenter, T. P., Fennema, E, Loef Franke, M., Levi, L., & Empson, S. B. (1999). Children's Mathematics: Cognitively Guided Instruction. National Council of Teachers of Mathematics: Portsmouth, NH
NCTM (2004). "Standards for School Mathematics." Retrieved on April 8. 2010, from http://standards.nctm.org/document/chapter3/numb.htm.
