Journal of Mathematical Behavior 22 (2003) 107–111 Teaching and classroom experiments dealing with fractions and proportional reasoning Gary E. Davis * College of Education, Washington State University, P.O. Box 642132, Pullman, WA 99164-2132, USA Keywords: Fractions; Constructivist teaching experiment; Cognitive-scheme; Memory This and the following issue of the Journal of Mathematical Behavior are dedicated to research articles dealing with fractions, rational numbers, part–whole relationships, and ratio and proportional reasoning. The study of ratio and proportional reasoning was already well formalized in Euclid. Yet fractions as ob- jects were not well understood for yet another 1500 years. Abu Ja’far Muhammad ibn Musa al-Khwarizmi, known for short as “al-Khwarizmi,” introduced to the Arabic world in Baghdad around 800 bc the Hindu notion of fraction, and showed how operations on these fractions was more or less mechanical, as distinct from the sophisticated relational thought required to interpret and operate with ratios in Euclid’s sense (Crossley & Henry, 1990). In this work, and in his work on algebra, al-Khwarizmi unknowingly gave rise to the term “algorism,” which has become the modern “algorithm,” or mechanical procedure. What al-Khawrizmi saw as a major intellectual achievement, many students — perhaps most — see as the first significant cabalistic mystery of mathematics: operations on fractions. Freudenthal (1983) once wrote of fractions as the “rape of ratio.” The mechanical operations on fractions — so simple after al-Khawrizmi as to be trivial — seemed to have the effect of turning teachers’ and students’ focus away from the beautiful ideas of proportion and ratio to mechanical manipulation of odd pairs of whole numbers, written as “a/b.” Yet we cannot discard al-Khwarizmi’s achievement and revert to complicated and sophisticated ideas of ratio and proportion for school age children, or as a body of research on proportional reasoning indi- cates, for anyone. How then to introduce fractions as numbers so that they and operations on them make sense? How to reconcile ratio and proportional reasoning with the form of fractions and students’ grow- ing understanding of rational numbers? These problems have been studied extensively, yet the problems remain. The first paper in this volume, by Steencken and Maher is based on a constructivist research paradigm, as are the following articles by Nabors and Sáenz-Ludlow. The primary intellectual influence for Steencken * Tel.: +1-509-335-8853. E-mail address: gedavis@wsu.edu (G.E. Davis). 0732-3123/$ – see front matter © 2003 Published by Elsevier Inc. doi:10.1016/S0732-3123(03)00016-6