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Summary of Reference

Yackel, Erna

Perspectives on arithmetic from classroom-based research in the United States of America

Bibliographic data:
In J. Anghileri (Ed.)(2001) Principles and practices in arithmetic teaching (pp. 15-31).  Buckingham: Open University Press.

Yackel examines the dilemma faced by teachers between proficiency in completing algorithms and developing children’s understanding of number ideas. As with mathematics teaching in the United States, New Zealand mathematics education has traditionally emphasised children’s mastery of the standard algorithm for addition, subtraction, multiplication and division. This has been a major focus of primary middle school mathematics programmes in particular. Yackel challenges this traditional view of arithmetic by first discussing why an emphasis on conceptual understanding is important. In this discussion she gives examples of what she means by sense-making and reasoning and gives illustrations of underlying conceptions of number using a missing-addend interview. Her discussion is based on a large body of classroom-based research conducted in American classrooms with her colleague Paul Cobb and others. More recently this group of researchers has collaborated with researchers from the Netherlands involved in Realistic Mathematics Education (RME) at the Freudenthal Institute, including Koeno Gravemeijer amongst others.

In describing the classroom teaching experiments she examines ways in which the design of instructional tasks and the use of a “learning trajectory” might promote the development of thinking strategies. The idea of a learning trajectory is useful for describing the possible route students’ learning might take. A key point in this chapter is the discussion of two possible conceptions of number that Yackel and colleagues’ research found underpinned children’s solutions. She identifies these as counting-based (or sequence-based) and collections-based solutions with both being important. The chapter concludes with a useful commentary on the role of visual imagery in instruction to develop conceptions of number by associating quantities with visual images. The tools to which she refers, such as the empty number-line and the various versions of the tens frames, have been promoted in New Zealand schools as part of the numeracy projects.