TEACHER CHANGE IN THE CONTEXT OF COGNITIVELY GUIDED INSTRUCTION: CASES OF TWO NOVICE TEACHERS

Nancy Nesbitt Vacc UNC Greensboro nnvacc@uncg.edu

Anita H. Bowman UNC Greensboro abowman@acme.highpoint.edu

George W. Bright UNC Greensboro gwbright@uncg.edu

Abstract: Two teachers who joined a five-year Cognitively Guided Instruction [CGI] project during their first year of teaching seemed similar at the beginning of the project but exhibited different patterns of change in instruction and beliefs across four years of implementing the principles of CGI in the classroom. Change was documented in the areas of discourse, children’s thinking, and instructional planning through analysis of transcribed annual interviews, teachers’ written responses to a variety of instruments, and classroom observations with post-observation interviews. By the end of the project, Ms. A provided students with opportunities to solve a variety of problems but she did not use what children shared to make instructional decisions. In contrast, Mrs. D’s instructional planning appeared to be driven by her knowledge about individual children’s mathematical thinking. Their Belief Scale scores also differed significantly.

Instructional decision making for teachers who implement the principles of Cognitively Guided Instruction [CGI] (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989) is largely dependent on research-based knowledge of children’s mathematical thinking. CGI teachers focus on students’ understanding and then adjust instruction to match the thinking of individual students. This study focused on how two first-year teachers changed as they gained experience in implementing the principles of CGI across a four-year professional development program (National Science Foundation Grant ESI-9450518). Specifically examined were changes relative to discourse, children’s thinking, and instructional planning using the theoretical framework provided by Fosnot’s (1996) principles of learning derived from constructivism: learning is not the result of development, learning is development, disequilibrium facilitates learning, reflective abstraction is the driving force of learning, dialogue within a community engenders further thinking, and learning proceeds toward the development of structures.

Method

Project

Through the CGI project (Jan 95-Dec 99), 5 teams of mathematics educators (originally 2 teacher educators and 6 teachers on each team) learned to use CGI as a basis of mathematics instruction. Workshops were held in May 1995 (3 days), July 1995 (10 days), June 1996 (8 days), June 1997 (7 days), June 1998 (4 days), and June 1999 (2 days). Between workshops, teachers implemented CGI in their mathematics instruction, each team met approximately once a month, each teacher was visited approximately once a month by one of the team's teacher educators, and project staff visited each teacher once each semester to provide general support.

Instrumentation

Data sources were transcribed annual interviews, written responses on several annually administered instruments, field notes taken during classroom observations on two consecutive days in Spring 1998, Fall 1998, and Spring 1999, and transcribed debriefing interviews that followed each observation.

Subjects

At the beginning of the project, both Ms. A and Mrs. D were first year teachers with K-6 licensure. Ms. A was teaching third grade and Mrs. D was teaching second grade. Ms. A changed to first grade in the second year and continued to teach at that grade level through the remainder of the project; Mrs. D continued to teach second grade. Both of their principals participated in part of the summer workshops. During the year, Ms. A had limited within-school support from her principal and peers at her school. Mrs. D enjoyed ongoing enthusiastic support from her principal, assistant principal, and peers, with collaborative support each year from another teacher at the school who was also in the project. By the project’s end, Ms. A appeared to be at level 3 on the Fennema, Carpenter, Franke, Levi, Jacobs, and Empson (1996) ‘levels of instruction’ scale. She provided students with opportunities to solve a variety of problems, but she did not use what children shared to make instructional decisions. Mrs. D seemed to be at level 4b: instructional planning appeared to be driven by her knowledge about individual children’s mathematical thinking.

Analysis

Data from the teachers’ reflections on teaching, classroom observations, and debriefing interviews were grouped into three categories (i.e., discourse, children’s thinking, and instructional planning) using Fosnot’s principles and the principles of CGI. The extent to which the teachers attended to the three components simultaneously in creating coherent mathematics instruction was determined through classroom observation data.

Results and Discussion

Discourse

At the beginning of the project, Ms. A’s mathematics lessons involved giving students a problem and then telling them the steps that they needed to follow to solve it. By the end of the project, she encouraged students to construct their own solution strategies and share them with the class. However, the general pattern of sharing seemed to involve multiple monologues rather than a dialogue between the teacher and a student or among students. At the beginning of the project, Mrs. D also taught by demonstrating procedures and explaining steps, but by the end of the project, her references to discourse became characterized by increased emphasis on student-to-student discussions of important mathematical ideas facilitated by teacher input in the form of questions. Specific purposes for asking varying types of questions became more clearly defined and detailed as she applied her increased knowledge of problem types and students’ solution strategies. She described her class as a learning community where she was a partner with her children in making sense of mathematical ideas.

Children's Thinking

Across the project, Ms. A referred to children collectively and appeared to focus little if at all on the importance of mathematical thinking of individual students. Her reflections relative to assessing children’s understanding centered on strategies that are global in nature (i.e., “seeing” how children solve problems in general and “reading” the looks on their faces). She seldom referred to explicit solution strategies used by individual children nor did she seem to consider the effect of different problem types on children’s understanding. By the end of the project, Mrs. D assessed children’s thinking based on the solution strategies they constructed and the content of their explanations. At the beginning of the project, she believed that students struggle with mathematical ideas only when instruction has been insufficient; by the end, she realized that, by letting students struggle to develop their own solution strategies and to make sense of strategies shared by others, her students were developing confidence in their ability to understand mathematical concepts. She appeared to recognize how knowledgeable and capable her students were and, subsequently, set higher expectations of their mathematical growth.

Instructional Planning

Ms. A’s instructional planning at the end of the project was similar to her pre-CGI planning. In general, she planned lessons that focused on having children memorize basic facts and learn to do standard arithmetic algorithms. She viewed CGI in a limited way; she “taught CGI” once a week and “taught math” the rest of the week. In addition, she selected students to share strategies dependent more on whether they had been on task than the strategies used and how they might be helpful to other students. Mathematics lessons followed the textbook and the state-curriculum’s organization rather than important levels of thinking that students must go through in order to develop concept understanding. At the beginning of the project, Mrs. D focused on “students’ prior knowledge” and “learning styles” as her basis for instructional decision making. Individual instruction was provided for students who struggled during the whole-group instruction. By the end of the project, she based instructional decisions on specific knowledge of the children’s levels of thinking. She differentiated instruction within lessons for individual students by varying numbers in problems, types of problems assigned, and the choice of manipulatives available for student use. She also grouped students who were at similar levels of thinking and encouraged students to pose and solve their own problems. She intentionally challenged students to struggle with making sense of mathematical concepts because she saw the need to create an instructional environment that continually encouraged children to move toward more efficient strategies.

Mathematics Instruction

Ms. A and Mrs. D changed their mathematics instruction across the project, but in different ways.During observational visits, Ms. A posed CGI-type problems during at least one lesson, but there often seemed little “connection” between the 2 consecutive lessons. For example, Ms. A followed a lesson on open and closed plane figures with a lesson based on two-digit addition and subtraction without regrouping. During the latter lesson she focused mainly on the students’ factual knowledge with little apparent use of the information gained as children shared their solutions. She seldom sought further clarification concerning the level of their mathematical thinking. During another visit, Ms. A indicated that she taught the day’s lesson because it was the next one in the textbook. Mrs. D typically planned instruction based on her knowledge of student thinking. For example, during one lesson, a student shared the following solution to a word problem involving the sum of 5 and 7: “I know that 5+2 is 7, 5 and 5 more is 10, and 10+2=12.” Ms. D had not yet introduced the concept of place value, but because of this student’s solution strategy, she posed a problem the next day that involved the sum of 6+4+7 to see if the children might begin to think in terms of 10 as a unit. This use of children’s thinking when planning the next mathematics lesson was evident across Mrs. D’s observed lessons.

Summary

Although the two teachers were similar in their teaching experience at the beginning of the project, attended the same workshops each summer, and had supportive classroom visits each year, they changed across the project in different ways. Ms. A’s instructional activities often were inconsistent with the principles of CGI, while Mrs. D clearly and effectively based mathematics instruction on CGI principles. Why did these two teachers conclude the project with such different CGI implementation levels? This question may be answered, in part, by differences in the amount of support each received. Ms. A’s principal supported her use of CGI, but she was mainly concerned about how CGI might help improve student scores on standardized testing. Most teachers at Ms. A’s school were not interested in observing in her classroom or learning about CGI, and they were concerned about whether her students would have the procedures and facts that “first graders are supposed to learn.” This view by her colleagues may explain (a) why Ms. A distinguished between teaching CGI and teaching mathematics and (b) what appeared to be a lack of confidence in her own knowledge and competencies. In contrast, Mrs. D received strong support for her CGI implementation and, in addition, a system-level supervisor arranged for some of Mrs. D’s lessons to be videotaped for system-wide professional development sessions.

Consideration also needs to be given to what appear to be critical differences in beliefs about teaching and learning. Mrs. D’s baseline total scale score on the Beliefs Scale (Peterson, et al., 1989) was 27 points higher (more constructivist) than Ms. A’s score. Belief Scale scores at other times during the project support the notion that Mrs. D’s “buy in” to the project stabilized early. Ms. A’s scores increased some but then declined across the project. In fact, Ms. A’s total scale score in Spring 1999 was 7 points below Mrs. D’s baseline score (May 1995).

In applying Fosnot’s principles to the data, differences in teacher change may be related to the amount of support each experienced. Mrs. D received ongoing direct support and encouragement in her school setting and as a result continued to develop as she learned; i.e., learning is development. Ms. A, however, may have seen little need to learn about children’s thinking because of the lack of support and encouragement that she experienced. As a result, she may not have developed to the level that she might have if she had experienced support similar to that which Mrs. D enjoyed. Once Ms. A added problem-solving activities to her instructional planning and having children share their solution strategies, she seemed content with her instructional planning and may have seen little need to struggle with finding out about individual children’s mathematical thinking. Thus, she did not experience disequilibrium, which in turn would have facilitated further learning. Without ongoing collegial support and interaction, Ms. A may not have been challenged to a reflective abstraction level that could have served as an impetus for further learning (i.e., reflective abstraction is the driving force of learning). Further, she did not have an opportunity for dialogue that might have engendered further thinking. In total, these limitations may have affected the development of structures that were needed for Ms. A to reach a higher level of CGI implementation. For example, implementing CGI principles only once a week would limit the extent to which most teachers could learn about children’s thinking, let alone plan instruction based on the students’ mathematical understanding.

References

Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C., & Loef, M. (1989). Using knowledge of children's mathematics thinking in classroom teaching: An experimental study. American Educational Research Journal, 26, 499‑531.

Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., & Empson, S. B. (1996). A longitudinal study of learning to use children's thinking in mathematics instruction. Journal for Research in Mathematics Education, 27, 404-434.

Fosnot, C. T. (1996). Constructivism: A psychological theory of learning. In C. T. Fosnot (Ed.), Constructivism: Theory, perspectives, and practice (pp. 8‑33). New York, NY: Teachers College Press.

Peterson, P. L., Fennema, E., Carpenter, T. P., & Loef, M. (1989). Teachers' pedagogical content beliefs in mathematics. Cognition and Instruction, 6, 1-40.