One, two, three, four. Twoths, I mean halves.
Sheena walks up: Four halves, isn't it?
Betsy: Yeah, four halves! Halves are two parts. So . . .
Jeannie: So we need two cookies and cut them each in half, then we have four halves.
Overhearing this conversation, I realized the distance these girls had come. Beginning with an intuitive, inexplicit, and visual notion of one half that they could draw, use, and write, I had helped them travel into a new domain of numbers. Suddenly, looking back, the familiar looked, for a moment, strange.9 One-twoth? But their comprehension of fractions had evolved into principled understanding of part-whole relationships and the symbolic notation for fractional quantities. And, consequently, a "2" in the denominator was no longer taken for granted: It had taken on explicit meaning. Ahead of these students still lie many excursions in the domain of rational numbers -- into different interpretations and applications of rational numbers, as well as arithmetic with the rationals. They are launched now, with tools and ways of thinking that have built on and challenged the informal understandings they held.
The author gratefully acknowledges the intellectual contributions of Magdalene Lampert, Margery Osborne, James Reineke, Janine Remillard, Kara Suzuka, Sylvia Rundquist, and Suzanne Wilson.
Ball, D. L. ( 1988). Knowledge and reasoning in mathematical pedagogy: Examining what prospective teachers bring to teacher education. Unpublished doctoral dissertation, Michigan State University, East Lansing.
Ball, D. L. ( 1990a). "The mathematical understandings that prospective teachers bring to teacher education". Elementary School Journal, 90, 449-466.____________________