Penelope L. Peterson

Using Knowledge of Children’s Mathematics Thinking in Classroom Teaching: An Experimental Study

This study investigated teachers’ use of knowledge from research on children’s mathematical thinking and how their students’ achievement is influenced as a result. Twenty first grade teachers, assigned randomly to an experimental treatment, participated in a month-long workshop in which they studied a research-based analysis of children’s development of problem-solving skills in addition and subtraction. Other first grade teachers (n = 20) were assigned randomly to a control group. Although instructional practices were not prescribed, experimental teachers taught problem solving significantly more and number facts significantly less than did control teachers. Experimental teachers encouraged students to use a variety of problem-solving strategies, and they listened to processes their students used significantly more than did control teachers. Experimental teachers knew more about individual students’ problem-solving processes, and they believed that instruction should build on students’ existing knowledge more than did control teachers. Students in experimental classes exceeded students in control classes in number fact knowledge, problem solving, reported understanding, and reported confidence in their problem-solving abilities.

Effective Teaching, Student Engagement in Classroom Activities, and Sex-Related Differences in Learning Mathematics

This research identified the classroom activities that were related to the low level and high level mathematics achievement of boys and girls. In December and in May, students in 36 fourth grade mathematics classes completed a mathematics test containing low level and high level items from the National Assessment of Educational Progress. During January through April, the engagement/nonengagement in mathematics activities was observed for six randomly selected students of each sex in each class. Results showed that girls and boys did not differ significantly in either mathematics achievement or in observed engagement/nonengagement in mathematics activities. However, engagement in the following four types of activities was consistently and differentially related to girls’ versus boys’ low level and high level mathematics achievement: competitive mathematics activities, cooperative mathematics activities, social activities, and off-task behavior.

Teachers’ Reports of Their Cognitive Processes During Teaching

Each of 12 experienced teachers taught a social studies lesson to three groups of junior high students under experimental conditions. After each 3-hour lesson, teachers viewed four brief videotaped segments of their teaching and responded to a structured interview concerning their cognitive processes while teaching. Teachers’ responses were audiotaped and coded. Reports were characterized most frequently by a sequence of observing student reactions, judging them to be satisfactory, and continuing teaching. Differences in reports were related to teachers’ cognitive styles and abilities, to teachers’ planning statements, and to student achievement and attitude.

Teacher Planning, Teacher Behavior, and Student Achievement

This study investigated individual differences in teacher planning and the relationship of teacher planning to teacher behavior and student achievement. Each of 12 experienced teachers taught a social studies lesson to three groups of junior high students (N=8 per group) randomly formed from 288 paid volunteers. Before teaching each group, teachers had 90 minutes to “think aloud” and plan the lesson. Teachers’ planning and teaching were recorded and coded. Students completed achievement and attitude measures after the lesson. Results indicated that generally, the greatest proportion of planning statements concerned the subject matter, but substantial differences occurred. Planning differences were related to teachers’ cognitive styles and abilities. Relationships between teacher planning, teacher behavior, and student outcomes also appeared.

Teachers' schemata for classroom events: The mental scaffolding of teachers' thinking during classroom instruction

Abstract Researchers have shown that experts and novices differ in their recall, representation, and analyses of problem situations. This study investigated whether such differences existed between novice and experienced high school teachers in their recall and analyses of problem events during interactive teaching. Experienced and inexperienced teachers (n = 10 in each group) completed cognitive ability tests and were interviewed after viewing three videotaped vignettes of interactive teaching. In contrast to novices, experienced teachers both recalled more classroom events and relied more on procedural knowledge and principles in analyzing classroom events. These differences may indicate that experienced teachers have better developed knowledge structures of “schemata” for classroom teaching than do novice teachers.

The Relationship of Student Ability and Small-Group Interaction to Student Achievement:

This study examined student ability and student behaviors during small-group interaction as hypothesized mediators of the effectiveness of small-group learning. Fifth-grade students (N = 43) completed ability and attitude pretests. Students were stratified on ability and randomly assigned to small groups within class. The small groups were assigned randomly to either a treatment or control condition. The treatment group was trained in small-group interaction. Students received regular classroom instruction in mathematics for 4 weeks. Each day after the teacher presented new material using direct instruction, students worked on assignments in mixed ability groups of four students. Achievement, retention, and attitude toward mathematics were assessed. A Mann-Whitney comparison showed that trained students participated in more task-related interaction than did control students. Results suggested that task-related interaction in the small group enhanced the achievement and retention of high and low ability students but did not facilitate the achievement of medium ability students.

Students’ Cognitions and Time on Task During Mathematics Instruction

This study investigated students’ reports of attention, understanding, cognitive processes and affect during mathematics instruction. Two classes of fifth grade students (N = 38) were taught a 9-day mathematics unit on measurement by one of their teachers. Students were videotaped during instruction and interviewed subsequently using a stimulated-recall procedure. Students completed an achievement test and questionnaires about their attention, cognitive processes, motivational self-thoughts, and attitudes toward mathematics. Results suggest that students’ reports of attention, understanding, and cognitive processes were more valid indicators of classroom learning than observers’ judgments of students’ time on task. Findings also indicate that students’ reported affect as well as cognitions mediated the relationship between instructional stimuli and student achievement and attitudes.

Learning to Teach Mathematics in the Context of Systemic Reform

This article looks at how teachers understand recent mathematics reforms. Case studies of three California teachers in a disadvantaged, urban elementary school are presented. Framing the study are issues of teacher learning and systemic reform. These teachers have access to multiple opportunities to learn about reforms. But access guarantees no common understandings. Teachers’ understandings are influenced by the kinds of students they have, their prior knowledge and experience, their views of mathematics, textbooks, and tests. If teachers understand reforms in different ways, this raises questions about systemic reform and the notion of alignment. But there are other problems as well. Questions about what it means to be aligned, how all students’ needs might be considered, and what the multiple goals of a decentralized system mean for teachers suggest that efforts at systemic reform will be challenged on several fronts.

Doing More in the Same Amount of Time: Cathy Swift

This article examines the perspectives and practice of elementary mathematics teaching of Cathy Swift, a second-grade teacher in a low-SES school in a large California city. Cathy’s mathematics lessons are smoothly and swiftly-paced lessons in the tradition of effective teaching for basic skills. Yet from her perspective, Cathy is implementing a “new” mathematics program connected with the state-level Mathematics Framework. Cathy’s view of the state policy is through her textbook, one of several approved by the state, and through district-level ABS materials developed specifically for low-SES schools. The ABS model includes components of content coverage, pacing, mastery testing and reteaching, maximizing students’ time on task, and use of direct instruction. To these, Cathy has added new elements-using manipulatives, using partner and group work, and emphasizing problem solving. Exploration of Cathy’s perspectives and practice reveals powerful effects of knowledge and beliefs, tangled influences of layers of policy, and multiple uncertainties and conflicts.

Problems in Classroom Implementation of Cognitive Strategy Instruction

The purpose of this chapter is to describe problems that need to be solved before cognitive strategy instruction can be implemented effectively in classrooms. At this point, these problems are not clearly defined because few attempts have been made to train students in cognitive strategies for classroom use. Moreover, in the few studies that have been done, researchers have not attempted to identify the specific variables that are related to unsuccessful classroom implementation of cognitive strategy instruction (see Peterson & Swing, Note 1). The questions that need to be addressed and the problems that must be solved include the following: 1. Can individually administered cognitive strategy training be adapted effectively