Problem-Solving Strategies of First Graders
Hoosain, Emam, Chance, Renee H., Teaching Children Mathematics
The significance of problem solving in the K-12 mathematics curriculum has been well documented over the years by professional organizations and individuals. The National Council of Supervisors of Mathematics (NCSM 1977) and the National Council of Teachers of Mathematics (NCTM 1980, 1989, 2000) endorse the inclusion of problem solving at all grade levels. For example, NCTM (1989) states, "The development of each student's ability to solve problems is essential if he or she is to be a productive citizen" (p. 6). In the same publication, NCTM asserts that "problem solving should be the central focus of the mathematics curriculum. As such, it is a primary goal of all mathematics instruction and an integral part of all mathematical activity. Problem solving is not a distinct topic but a process that should permeate the entire program and provide the context in which concepts and skills can be learned" (p. 23). NCTM reiterates its position on problem solving in Principles and Standards for School Mathematics (NCTM 2000).
Cognitively Guided Instruction (CGI)
Cognitively Guided Instruction (CGI) is an example of a research-based, problem-solving approach to teaching mathematics. As the name implies, an understanding of the learner's thinking guides instruction. The idea originated at the University of Wisconsin with Thomas Carpenter and his colleagues. Although it is intended primarily for grades K-3, CGI is such a broad philosophy of teaching that it can be adapted for use at almost any grade level.
In this approach, the teacher presents students with mathematical word problems set in the context of their environment and allows students the freedom to create their own strategies for solving the problems using available resources. They can use physical, pictorial, or symbolic representations and are encouraged to explore different ways to solve a particular problem. In the process of solving problems, children invent their own algorithms and share their problem-solving strategies with the rest of the class. The process of obtaining an answer is more important than the answer itself.
Based on a student's response to a problem, the teacher prepares appropriate follow-up problems. For example, if a student has not solved a problem satisfactorily, the teacher may give him or her a less difficult problem, perhaps one with lesser numbers. The CGI approach easily addresses NCTM's Process Standards (Problem Solving, Reasoning and Proof, Communication, Connections, and Representation); this is one of its greatest benefits. More important, learners become adept at these processes.
The CGI teacher is a facilitator of learning rather than a disseminator of knowledge. He or she avoids imposing adult thinking on students. The teacher spends a lot of time questioning and listening to student's explanations in order to determine how they are thinking. Therefore, the teacher must possess good questioning skills. Once the teacher has determined how the learner is thinking, he or she tries to build on the learner's ideas. CGI is based on the premise that children come to school with a great deal of mathematical knowledge; the task of the teacher is to uncover this knowledge and build on it.
CGI places great emphasis on individualization, conceptual understanding, and higher-order thinking. More emphasis is placed on formative evaluation than on summative evaluation. The approach does not use any teaching materials such as text-books, so the teacher must prepare his or her own problems. This and other responsibilities such as classroom management make teaching challenging for the CGI teacher. For a discussion of the characteristics of a CGI classroom, see Chambers and Hankes (1994).
Classroom management issues are important concerns for the CGI teacher, especially if the class is large, because CGI is an individualized approach. …