RATIONAL AND FRACTIONAL NUMBERS: FROM QUOTIENT FIELDS TO RECURSIVE UNDERSTANDING
Thomas E. Kieren University of Alberta
This chapter contains some suggested elements of models of students' conceptual structures with respect to fractional and rational numbers, of conceptualizations of rational number knowing toward which instruction might be directed, and conceptualizations of curricula based on such elements. These elements are drawn from three major sources. The first is the mathematics of rational numbers whose central formal properties are seen as pointing to central aspects of rational number knowledge building. The second is the construct theory of rational numbers, which is in part seen as a source of applicational knowledge of rational numbers. (A formal quantity analysis of rational number constructs is provided in chapter 2 of this book.) Finally, models for knowing and understanding mathematics are developed to provide insight into student actions as they come to know fractional and rational numbers and into a new recursive consideration of curriculum building to include the various rational number constructs. The thrust of this chapter is to make researchers and teachers aware of the dynamic implicit orders that underlie the particular patterns of behavior by children as they come to know and understand a wide variety of rational number concepts.
But this "firm standing" [Verstehen] must find its appropriate place in the broader context of the flowing movement of intuitive reason.
-- Bohm & Peat, 1987, p. 146
Mathematics is an especially significant example of the interweaving of intuitive reason and formal logic . . .
-- Bohm & Peat, 1987, p. 147
What is number that it is humanly knowable; what are humans that they may know number?
-- Paraphrase of W. McCullough, 1963, p. 1