Cognitively Guided Instruction

Gwenanne SalkindMathematics Education Leadership

Research Expertise PresentationDecember 11, 2004

Addition & Subtraction

• Problem Types– Joining or separating actions– Comparing situations– Part-whole relations

• Combined with what is unknown

• Carpenter, T. P., & Moser, J. M. (1983)• Riley, M. S., Greeno, J. G., & Heller, J. K.

(1983)• Carpenter, T. P. (1985)

Classification of Word Problems

Result Unknown Change Unknown Start unknown

Join Connie had 5 marbles. Jim gave her 8 more marbles. How many does Connie have altogether?

Connie has 5 marbles. How many more marbles does she need to have 13 marbles altogether?

Connie had some marbles. Jim gave her 5 more marbles. Now she has 13 marbles. How many marbles did Connie have to start with?

Separate

Connie had 13 marbles. She gave 5 marbles to Jim. How many marbles does she have left?

Connie had 13 marbles. She gave some to Jim. Now she has 5 marbles left. How many marbles did Connie give to Jim?

Connie had some marbles. She gave 5 to Jim. Now she has 8 marbles left. How many marbles did Connie have to start with?

Part-part-whole

Compare

Connie has 13 marbles. Jim has 5 marbles. How many more marbles does Connie have than Jim?

Jim has 5 marbles. Connie has 8 more than Jim. How many marbles does Connie have?

Connie has 13 marbles. She has 5 more marbles than Jim. How many marbles does Jim have?

Connie had 5 red marbles and 8 blue marbles. How many marbles does she have?

Connie has 13 marbles. Five are red and the rest are blue. How many blue marbles does Connie have?

Development of Problem-Solving Strategies

• Direct Modeling• Counting• Derived Facts• Recall Facts

CGI Framework

• Analysis of problem types and solution strategies provides a framework for analyzing teachers’ pedagogical content knowledge.

• Will providing research-based knowledge to teachers influence their instruction?

Researchers

• Thomas P. Carpenter, University of Wisconsin

• Elizabeth Fennema, University of Wisconsin

• Penelope L. Peterson, Michigan State University

• Deborah A. Carey, University of Wisconsin

• Megan L. Franke, UCLA• Nancy Knapp, University of Georgia

CGI Workshops

• 40 first-grade teachers• 4-week summer workshops• 1986 and 1987• Familiarize teachers with the findings

of the research on the learning and development of addition and subtraction concepts in young children

• Provide teachers with an opportunity to think about and plan instruction on the basis of this knowledge

Measures of Teachers’ Knowledge• Knowledge of problem types• General knowledge of strategies• Teachers’ knowledge of their own

students

(Carpenter, Fennema, Peterson, & Carey, 1988)

Measures of Student Performance• Number Facts• Problem Solving

Results• Teachers could distinguish between major

problem types and were capable of identifying student strategies

• Teachers were able to predict the success of their own students.

• Most teachers did not have a coherent framework for classifying problems.

• Many teachers did not recognize that problems that can be directly modeled are easier than problems that cannot.

(Carpenter et al., 1988)

Another Study

• 20 of the original 40 teachers• Case studies of two of the teachers• Compared a knowledgeable teacher

with a less knowledgeable teacher

(Peterson, Carpenter, & Fennema, 1989)

Data Collection

• Classroom observations• Teachers’ Belief Questionnaire• Teachers’ knowledge of their own

students• Student Measures of Achievement

– ITBS pretest/posttest– Interviews

(Peterson et al., 1989)

Teachers’ Knowledge & Beliefs

• Teachers’ knowledge of their students’ problem-solving abilities was the best predictor of students’ problem-solving achievement.

• Teachers’ beliefs were significantly positively correlated with students’ mathematics achievement.

(Peterson et al., 1989)

Difference between More Expert Teachers and Less Expert TeachersMore Knowledgeable• Questioned and

listened to students• Believed that

students construct their own knowledge

• Believed that children came to school with a lot of knowledge

• Believed that the role of teacher is as facilitator

Less Knowledgeable• Explained how to

solve the problem• Believed that

children receive knowledge

• Skeptical of students’ entering knowledge

• Focused on knowledge that her children did not have

• Believed that role of teacher is to present knowledge. (Peterson et al., 1989)

A Case Study

• Ms. J• First grade teacher• Participated in the CGI studies• Four years

(Fennema, Franke, Carpenter, & Carey, 1993)

Data Collection• Year 1

– Interviews– CGI Belief Instrument

• Year 2– Classroom observations– CGI Belief Instrument

• Year 3 (Case Study)– Group discussion– Interviews– Classroom observations– Student Interviews

• Year 4– Interviews– Knowledge assessment

(Fennema et al., 1993)

Ms. J

• Ranked near the top of her experimental group on knowledge of CGI framework

• High score on CGI Belief Instrument• Students learned mathematics at a

higher level than most first grade children

(Fennema et al., 1993)

Ms. J’s Instruction

• Frequently questioned her students about their thinking

• Listened to her students more than the other teachers

• Expected multiple solution strategies at a higher level than most of the other teachers

• Expected students to persist in their work, share strategies, and reflect on their own thinking

(Fennema et al., 1993)

Results

• Ms. J had research-based knowledge of children’s thinking and was able to use it to make instructional decisions.

• Study shows evidence that teachers can use CGI research to inform their instruction and increase learning of children.

(Fennema et al., 1993)

CGI After Four Years

• 20 of original 40 participants• All 40 were contacted and asked to

participate• Phone interviews• Interviews collected detailed descriptions

of the participants’ teaching practices.• Teachers varied widely in degree of CGI

use and beliefs

(Knapp & Peterson, 1995)

CGI After Four Years

• Three groups of teachers emerged– Ten teachers used CGI as the main

basis for their teaching.– Four teachers had never used CGI

more than supplementally.– Six teachers had used CGI more

extensively in earlier year, but now were using it only occasionally.

(Knapp & Peterson, 1995)

CGI After Four Years

• Researchers did, as they had hoped, develop an intervention that could result in significant changes in elementary teachers’ practices and beliefs about mathematics.

• The positive effects of CGI intervention seem to have been most pervasive and long lasting in teachers who constructed for themselves more conceptual and flexible meanings for CGI rather than adopting means that were tied to specific procedures from the CGI training.

(Knapp & Peterson, 1995)

Other Studies

• Four year longitudinal study• 21 teachers• Workshops & Support• Classroom observations• Interviews• CGI Belief Scale• Student Interviews• Student computation tests

Fennema, Carpenter, Franke, Levi, Jacobs, & Empson, 1996)

Levels of CGI Instruction1 Provides few, if any, opportunities for children to engage in

problem solving or to share their thinking.

2 Provides limited opportunities for children to engage in problem solving or to share their thinking. Elicits or attends to children’s thinking or uses what they share in a limited way.

3 Provides opportunities for children to solve problems and share their thinking. Beginning to elicit and attend to what children share but doesn’t use what is shared to make instructional decisions.

4-A Provides opportunities for children to solve a variety of problems, elicits their thinking, and provides time for sharing their thinking. Instructional decisions are usually driven by general knowledge about his or her students’ thinking, but not by individual children’s thinking.

4-B Provides opportunities for children to be involved in a variety of problem-solving activities. Elicits children’s thinking, attends to children sharing their thinking, and adapts instruction according to what is shared. Instruction is driven by teacher’s knowledge about individual children in the classroom.

Teachers’ Beliefs

• The beliefs of 18 teachers in the final year were more cognitively guided than were their beliefs in the initial year.

• Beliefs were characterized by the acceptance of the idea that children can solve problems without direct instruction and that the mathematics curriculum should be based on children’s abilities.

( Fennema et al., 1996)

Student Achievement

• Student achievement in problem solving was higher at the end of the study than at the beginning

• There was no change in computation skills.

( Fennema et al., 1996)

Three Case Studies

• Three teachers chosen from the original 21

• Interviews• Classroom observations• Looked at teacher change• Only one teacher showed self-

sustaining, generative change.

Frank, Carpenter, Fennema, Ansell, & Behrend, 1998)

Kindergarten Children’s Problem-Solving Processes

• Carpenter, Ansell, Franke, Fennema, and Weisbeck, 1993– 70 kindergarten children– Teachers participated in CGI course– Student interviews– Children can solve a wide range of

problems, including multiplication and division situations, much earlier than generally presumed.

Other Researchers

• Villasenor & Kepner, 1993– Used a control group– Urban district with significant

minority population– CGI students scored significantly

better on number facts and problem solving tests

– CGI students used advanced strategies significantly more often than non-CGI students

Other Researchers

• Vacc & Bright, 1999– Thirty-four preservice teachers– Two case studies (Helen and Andrea)– University of North Carolina– Two years of professional coursework– Student teaching– Observations– CGI Belief Instrument– Interviews

Other Researchers

• Warfield, 2001– Case study of one kindergarten

teacher– Sixth year as CGI teacher– Classroom observations– Interviews– Looked at beliefs, knowledge, and

instruction– Teacher used CGI framework to learn

about individual children’s mathematical thinking and used that knowledge to make instructional decisions

Summary

• Evidence that knowledge of CGI framework changed teachers’ instructional practices and beliefs

• Evidence that CGI instruction increased student performance in problem-solving and computation.

CGI in FCPS

• Have trained about 70 teachers• Support from instructor/coach• Share research with CGI Instructor• Consider using CGI Instructional

Levels and CGI Belief Instrument