This guidance report focuses on the teaching of mathematics to pupils in Key Stages 2 and 3. It is not intended to provide a comprehensive guide to mathematics teaching. We have made recommendations where there are research findings that schools can use to make a significant difference to pupils' learning, and have focused on the questions that…

Examines the development of students' understandings about fractions during six weeks of instruction. Reports that all students possessed informal knowledge disconnected from their knowledge of fraction symbols and procedures and that knowledge of rote procedures often interfered with students' attempts to build on their informal knowledge.…

Investigated was the use of cognitive conflict to probe the misconceptions held by preservice elementary teachers that in a division problem the quotient must be less than the dividend. Explains how preservice teachers' reliance on information about the domain of whole numbers and their instrumental understanding support their misconceptions.…

A study on the impact of misconceptions on preservice elementary teachers' processes in solving problems involving multiplication and division of decimals is described. Included are the purpose of the study, a description of the subjects, instruments, and interviews, the findings, and discussion. (KR)

Examined is the situation in which pupils invent mathematics on their own and teachers' reactions to this situation. The assimilation of students' original ideas into correct mathematical concepts is discussed. (CW)

Discussed are preservice elementary teachers' misconceptions and inconsistent beliefs about multiplication and division with decimals. Sources of inconsistencies and recommendations for overcoming inconsistencies are included. (KR)

Described are the conceptions held by United States and Israeli students about multiplication and division that may impede their work with decimals. Included are the introduction, method, division and multiplication task results, and the implications for education and textbook development. (KR)

This paper describes the BUD project which surveyed childrens' conceptions of division, and of fractions and decimals. The lack of connection between counting skills and conceptual understanding is discussed. The expectations for new algorithms and the basic idea in planning the BUD project are summarized. Some previous studies on counting, the…