Teacher Change in the Context of Cognitively Guided Instruction:

The Case of Mrs. R

George W. Bright UNC - Greensboro gwbright@uncg.edu

Nancy Nesbitt Vacc UNC - Greensboro nnvacc@uncg.edu

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

Abstract: We documented Mrs. R’s change across four years of implementation of the principles of cognitively guided instruction. Data included annual interviews, her written reflections on instructional issues, and our observations of her mathematics instruction. Her beliefs shifted immediately toward a constructivist view and remained stable throughout the project. By the end of the project, Mrs. R (a) saw student-student interaction as critical to development of mathematical thinking, (b) viewed students’ struggles with mathematics ideas as desirable, (c) helped students reflect, (d) made explicit decisions about when children would share solutions, and (e) focused questions to help children see mathematical structures.

Cognitively guided instruction, or CGI, (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989) is an approach to teaching mathematics in which knowledge of children's thinking is central to instructional decision making. Teachers use research-based knowledge about children's mathematical thinking to help them learn specifics about students and to adjust instruction to match students' performance. The tenets of CGI fit well within Fosnot’s (1996) principles of learning derived from contructivism: (a) learning is not the result of development, learning is development; (b) disequilibrium facilitates learning; (c) reflective abstraction is the driving force of learning; (d) dialogue within a community engenders further thinking; and (e) learning proceeds toward the development of structures (pp. 29-30). These principles were our theoretical framework for understanding teacher change.

Method

Project. The project (NSF Grant ESI-9450518), conducted from January 1995 to December 2000, included 5 teams (originally, 2 teacher educators and 6 classroom teachers on each team); participants 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 the principles of CGI in mathematics instruction, each team met about once a month to continue learning about CGI and to discuss progress, and each teacher was visited about once a month by one of the team’s teacher educators. Project staff visited each teacher once each semester to provide general support. In 1997, teams began to deliver CGI professional development for their colleagues.

Instrumentation. Data sources were (a) transcribed annual interviews, (b) written responses on several instruments (described in Bowman, Bright, & Vacc, 1997) administered each year, and (c) three sets of classroom observations (two consecutive days each) in Spring 1998, Fall 1998, and Spring 1999. Field notes were taken during each observation, and after each observation there was a debriefing interview; all interviews were transcribed.

Subject. At the start of the project, Mrs. R was a 3rd-grade teacher with 15 years of experience, pre-K to 3rd grade. She was licensed for K-3 instruction and held a master’s degree in early childhood education. She had served as a university-level child development trainer. During the first 3 years of CGI implementation, Mrs. R taught 3rd grade at one school. Mrs. R indicated a general lack of interest in CGI by other teachers in her building. Her principal was verbally supportive of her participation in the project, but the principal did not attend the principal days during the summer workshops and appeared to be most interested in whether the state test scores of Mrs. R’s students increased more or less than the scores of other 3rd-grade students. During the 4th year, Mrs. R moved to a different school and taught 2nd grade. The principal seemed supportive, but there were no opportunities for that principal to attend CGI inservice. By the project’s end, Mrs. R seemed to be at a high level of CGI implementation -- possibly level 4b (Fennema, Carpenter, Franke, Levi, Jacobs, & Empson, 1996).

Analysis. Reflections on teaching were grouped into three categories, based on both Fosnot’s principles and the principles of CGI: discourse, children’s thinking, and instructional planning. The classroom observations served as evidence of the extent to which Mrs. R attended to these notions simultaneously in creating coherent mathematics instruction. Reflections were also grouped into four time periods (i.e., pre-CGI, early CGI, mid-CGI, late CGI) but the limited space here prevents explicit presentation of the evidence from each period.

Results and Discussion

Beliefs. Mrs. R’s total Belief Scale score (240 maximum) increased substantially between the first administration (196 at the beginning of the initial workshop) and the second administration (227 at the end of the first summer workshop) and remained essentially stable (227, 224, 220, 228 on four additional administrations) during the remainder of the project.

Discourse. Mrs. R said little directly about discourse, in terms of either the teacher’s role or students’ roles. She was consistent over the four years in identifying “teacher as facilitator” as a critical component of instruction, and this seemed to be an important part of a teacher’s role in discourse. She was not very specific about what she meant by “facilitation” until toward the end of the project, when she began to identify some specific aspects of facilitation (e.g., carefully sequencing the solution strategies that students were asked to share).

Three changes seem to stand out in Mrs. R’s views about discourse. First, the roles of students became more prominent in instruction, through both more student sharing of solution strategies and more student-to-student interaction. Second, questioning became more useful for helping reveal students’ thinking, partly through use of more high-level questions. Third, classroom instruction became more responsive to children’s needs, for example, through explicit decisions about whether unusual solutions were shared with the entire class.

Children’s Thinking. Helping students develop confidence was a pervasive theme across all of Mrs. R’s reflections. Throughout the project, Mrs. R seemed alert for children’s notions of number sense. These orientations are certainly consistent with the principles of CGI, and it seems that her developing knowledge of problem types and children’s solution strategies reinforced, rather than changed, her views. The specific evidence that she cited about either children’s confidence or understanding of number sense was typically general and generic.

Three changes stand out in Mrs. R’s perceptions about children’s thinking. First, the process of revealing thinking became a joint effort between Mrs. R and the students, rather than being the responsibility only of Mrs. R. Second, thinking was inferred from a wider variety of situations, and it was tied to more specific mathematical concepts. Third, Mrs. R began to talk more about the thinking of individual children and less about the thinking of groups of children collectively.

Instructional planning. For Mrs. R, it was important that CGI held “value” for children, for example, by helping increase students’ knowledge. Specifically, by learning to ask each other questions, children learned mathematics. The notion of creating a risk free environment for children pervaded Mrs. R’s reflections about instructional planning, and she increasingly talked about having patience with children to let their thinking develop.

Three changes stand out in Mrs. R’s perceptions of instructional planning. First, the knowledge-base for her planning moved from general knowledge to specific knowledge, and the specific knowledge-base expanded across the life of the project. Second, her planning became more explicitly designed to move children’s thinking forward. Third, her planning became more responsive to individual students’ needs.

Mathematics instruction. Mrs. R was knowledgeable about the needs and mathematical understandings of individual students in her class. She used this knowledge to adapt problems for individuals. She asked questions and altered problems to probe students’ understanding.

• Carter shared his solution to this problem: In the ocean there was a school of 155 fish swimming around the reef. 50 fish swam away. How many fish were still in the school of fish? He wrote

155

- 50

= 105

and then said, “I went 1 take away 0 and I put 1; 5 take away 5 is 0; and 5 take away 0 is 5.”... Mrs. R gave Carter a new problem: 150 - 49 and asked him to think about that. Carter wrote

150

- 49

= 119

Mrs. R asked Carter if he could take 9 away from 0. He said no, but still wanted to write 9. Mrs. R then said that the problem would be his own challenge to think about. (Classroom Observation Field Notes, May 17, 1999)

Mrs. R was concerned about the “comprehension” of her students. She wanted them to understand problem situations as a whole and not focus on isolated words or phrases.

• I’d say 12 to 13 of them have pretty solid understanding of tens.... [and] comprehension of the problem. There’s a real problem with this group; it’s comprehending what the problem is asking you to do.... When you stop throwing numbers together is when you are willing to look at the problem and try to figure out what you know and what you don’t know. (Classroom Observation Debriefing Interview, November 4, 1998)

• I purposefully try not to put in the problems, like that fish problem ... [the] key words. I use different words so they’re really not comprehending an isolated word but the whole concept of what I was asking. (Classroom Observation Debriefing Interview, May 17, 1999)

One of the problems was a multiplication problem: A rare find was discovered in a deep section of the ocean. It was a group of 12 octopus. Knowing that octopus have 8 arms each, how many arms were there in that group of octopus? In her conversations with individual children, Mrs. R changed the number of octopus to 5 for one child and 10 for another child. As children shared solutions to this problem, they used a variety of skip-counting and counting-on strategies, without any apparent connection to place value (e.g., 10 eights equals 8 tens or 80).

• I know they know eight tens, but they don’t see yet that ten eights are the same as eight tens. I really thought that Stan and Evelyn would pick up on that right away, and maybe James. (Classroom Observation Debriefing Interview, May 17, 1999)

Evidence for Constructivist Principles

Learning is not the result of development, learning is development. During the project, Mrs. R began to encourage more student to student interaction. She saw interaction as critical to development of children's mathematical thinking. Illustrative of this is her focus on not avoiding conflicts among students' answers; good discussion, with resulting increase in understanding, arose from addressing those conflicts. Mrs. R's emphasis on comprehension by the students further illustrates her implicit agreement with this construct. Mrs. R also reorganized her own thinking about instruction. Her interpretations of children's work moved from general development concerns to specific mathematics development concerns. She expanded both the range of situations where she looked for mathematical thinking and the frameworks within which she interpreted children's thinking.

Disequilibrium facilitates learning. Mrs. R was conscious of the fact that students need to struggle in their learning. Struggles were not viewed as stumbling blocks or barriers but rather as opportunities to make sense of mathematical ideas. Mrs. R's realization that making sense of students' understanding was not solely her responsibility, but was a shared responsibility between her and each student, is an indication that she could accept her own limitations. She acknowledged that she, too, had to struggle to understand how to plan and implement instruction.

Reflective abstraction is the driving force of learning. Mrs. R came to understand that having students reflect on their own learning is important. She acknowledged that every child has to reflect in order to learn. Mrs. R herself clearly became much better at reflecting on her own teaching, and she came to realize the importance of doing so. She could recognize the mathematics that occurred in lessons even when she had not planned for that mathematics to be addressed. One result of this was her recognition first, that there were changes in her own teaching that she needed to work on and second, that she needed to learn more mathematics.

Dialogue within a community engenders further thinking. Dialogue among students became a focal point for her planning and implementation of instruction. She made conscious decisions about when to involve the entire class in dialogue and when to involve only selected students. This dialogue was intended to further students' understanding and reflection.

Learning proceeds toward the development of structures. The discussion about 8 tens versus 10 eights indicated how important students' understanding of structure was for Mrs. R. She recognized when structures were being developed, and she explicitly focused her questions to help students see those structures. Mrs. R's structural knowledge of teaching evolved from her growing base of frameworks for understanding children's thinking. By the end of the project, she identified quite a list of things that teachers need to know in order to be effective.

Summary. First, the perspectives that a teacher brings to a professional development project are important and need to be identified and acknowledged. Mrs. R's background in early childhood education seemed critical in framing her development. Second, there are identifiable stages of development as teachers work through significant professional development programs. For Mrs. R, the stages included teacher as teller, students as tellers, and students and teachers working together in a community of learners. Third, a teacher's working environment is an important factor influencing a teacher's development. Mrs. R might have developed faster or differently if she had had collegial support in her building or a stronger professional community with which she could have interacted. Fourth, Fosnot's (1996) constructs provide a framework for understanding not only students' mathematical growth but also teachers' instructional growth. Making this framework explicit to teachers might help them in self-reflection and self-analysis.

References

Bowman, A. H., Bright, G. W., & Vacc, N. N. (1997). Changes in teachers' beliefs and assessments of students' thinking across the first year of implementation of cognitively guided instruction. In J. A. Dossey, J. O. Swafford, M. Parmantie, & A. E. Dossey (Eds.), Proceedings of the nineteenth annual meeting, North American Chapter of the International Group for the Psychology of Mathematics Education (vol. 2, pp. 409-416). Bloomington/Normal, IL: Illinois State University.

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.