The purpose of this study was to document the mathematical knowledge and pedagogical content knowledge teachers used while tutoring elementary school students in equivalent fractions. Specifically, the study defined the scaffolding procedures used by more knowledgeable classroom teachers (those with excellent knowledge of equivalent fractions) in working with individual students. An additional purpose of this study was to define the scaffolding procedures used by more knowledgeable and less knowledgeable classroom teachers in working with individual students. Using a three-phase assessment process developed in this study, the cognitive and teaching patterns of teachers with more pedagogical content knowledge were compared with the cognitive and teaching patterns of teachers with less pedagogical content knowledge. The study combined techniques (cognitive analyses) used in expert/novice research in cognitive psychology, work on assessments (standardized assessments and interview protocols) of teacher content knowledge, and transcripts of individual tutoring scores.
Sixty elementary teachers from a large suburban school district participated in a summer mathematics staff development program. In Stage I, as part of the staff development program, fifty-three teachers completed the twenty-item Equivalence Assessment A and a Demographic Questionnaire. Thirty-two teachers volunteered for Stage II of the study in which the teachers completed Equivalence Assessment B, a Written Concept Map and a Verbal Think Aloud Concept Map I. The verbal concept map was modeled after a “think aloud” interview, where Pedagogical Content Knowledge and Scaffolds teachers were asked to state their ideas about the concepts they believed are important. Teachers' verbal concept maps were audio taped, transcribed and analyzed.
Ten teachers were selected for Stage III, five more knowledgeable and five less knowledgeable. They were audio taped during tutoring sessions with individual students on equivalent fractions. After the tutoring sessions were complete, teachers constructed a verbal concept map using two of their transcribed tutoring scripts. These tutoring sessions were examined for pedagogical content scaffolds and these results were compared with the teachers' knowledge of equivalent fractions. Pedagogical content scaffolds were defined by (1) types of models used (area, set or linear); (2) model translations (partitioning, fair shares, fraction strips); (3) use of relationships (part/whole, part/part and relationships among different fractions) and (4) instructional strategies (conceptual or numeric).
The results show that teachers who possess more knowledge about equivalent fractions are able to demonstrate that knowledge across both traditional (Equivalence Assessment A and B) and conceptual (Written Concept Map and Verbal Think Aloud Map I and II) measures as well as instructional (Tutoring Transcripts) measures. Additionally, case studies of four teachers were developed and their data was examined for differences across the measures.
The case studies allowed the differences between the teachers to be examined closely. Teachers with more knowledge were able to demonstrate their knowledge both in assessments of their own knowledge and in their instructional practice. The teachers with more knowledge demonstrated a deeper understanding of equivalent fractions evidenced by their highly connected concept maps, lack of errors in their knowledge base and their ability to use a variety of representations of models when tutoring a student. On the other hand, teachers with less knowledge performed poorly across all measures. The teachers with less knowledge achieved lower scores on Equivalence Assessment A and B (traditional assessments) and the Written Concept Map. All their concept maps contained errors and misconceptions, as well as having fewer categories and fewer pedagogical content scaffolds during tutoring instruction.
The three phase assessment process developed in this study supported the comparison of the cognitive and teaching patterns of these with more pedagogical content knowledge and with less pedagogical content knowledge. In particular, the development of pedagogical content scaffolds used in tutoring and the scoring procedures of the tutoring transcripts themselves are a contribution to the research in these areas.