ing. Over the past two decades, computational tools have increasingly influenced the kinds of transformations that are important to learn, the kinds of representations, especially graphical ones, that are readily accessible, and the kinds of applications of mathematics that are appropriate to address. One of the biggest shifts has been to emphasize the ideas of pattern, function, and variation.172 This new focus is particularly amenable to approaches that begin in the elementary grades and continue through middle school, and a sizable body of instructional materials has been developed that reflects this emphasis.173 But the long-term impact of these materials is as yet unknown.

Recent research on measurement and geometry suggests that children’s development of geometric reasoning can be greatly enhanced in instructional environments that are specifically designed to promote such understanding and that children’s thinking may fluctuate across stages identified by earlier researchers. Furthermore, computer technologies offer the promise of being able to support developing understanding in ways not available before.

Unlike the domains of measurement and geometry, research on the development of concepts of statistics and probability indicates that, especially for probability, very young children are capable of less than developmental theories might predict. Fundamental concepts in both domains, such as the conventions of scaling in graphs and the makeup of the sample space, need more careful attention in initial instruction. As in the areas of measurement and geometry, technology offers promise for helping to support and link students’ developing conceptions of data and chance. It is still an open question when and how many of the central conceptual structures of probability and statistics should be introduced in the elementary and middle grades.

Notes

1.  

Kieran, 1992.

2.  

Mason, Graham, Pimm, and Gowar, 1985, p. 38.

3.  

Bochner, 1966.

4.  

This characterization of the main activities of school algebra is based on a categorization by Kieran, 1996. A number of different characterizations of algebra can be found in the literature. For example, Usiskin, 1988, listed four conceptions of algebra: generalized arithmetic, study of procedures for solving certain kinds of problems, study of relationships among quantities, and study of structures. The National Council of Teachers of Mathematics, 1997, offers four organizing themes for school algebra: functions and relations, modeling, structure, and language and representation. Kaput, 1995, identified five aspects of algebra: generalization and formalization; syntactically guided manipulations; study of structure; study of



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