Ph.D Thesis
Ph.D Student Raveh Irina-Ira Teachers' Knowledge of the Interconnections between the Elementary Algorithms of the Four Basic Arithmetic Operations and their Underlying Mathematical Principles Department of Education in Science and Technology Professor Orit Zaslavsky Professor Boris Koichu Dr. Irit Peled

Abstract

It is well known that schoolchildren have difficulty in performing and understanding four standard arithmetic algorithms, and that the difficulty can partially be attributed to their teachers’ mathematical knowledge related to teaching the algorithms. This study aims at identification of the components of elementary school teachers’ mathematical knowledge for teaching the subject and examination of a particular way of promoting their understanding of mathematical principles underlying the algorithms.

The research sample included 46 elementary teachers. The main data were collected during an intervention lasting 8 meetings, in two parallel groups (about 32 hours), and in a series of semi-structured interviews (about 20 hours). All the meetings and interviews were recorded and fully transcribed. The research data included also the teachers’ work sheets, field notes and the researcher’s diary. The data were analyzed according to methodology of qualitative research.

The findings suggest that while working on various tasks related to the four algorithms, most of the teachers exhibited a rote approach. Teachers saw some connections between the algorithms and their underlying principles of place-value and number regrouping. Specifically, most of the teachers could recognize connections between the addition and subtraction algorithms in terms of their common underlying mathematical principles. In contrast, few teachers recognized similar connections between the multiplication and division algorithms.

After performing the intervention, an improvement was apparent in the teachers’ explanations of the stages of the multiplication and division algorithms. The teachers saw better the similarity between these algorithms. They also successfully recognized the difference between the division algorithm and the other algorithms, related to that the regrouping principle appears in the division algorithm in a different way. It was found that the task dealing with the translation of the standard algorithms to horizontal representation was the task with high potential of promoting the teachers’ knowledge.

The theoretical contribution of this study is in identification of the components of mathematical knowledge for teaching four standard algorithms. The methodological contribution of the study consists of a method of developing operational criteria for identifying various components of the teachers' knowledge; the developed criteria have served for constructing mathematical tasks in the study. The developed method can hopefully be applied for identifying components of teachers' knowledge related to additional mathematical topics. On the practical level, the findings help in formulating recommendations for improving the ways of teaching the algorithms, either in professional development programs for teachers or in school setting.