FRACTIONAL KNOWLEDGE AND EQUATION WRITING: THE CASES OF PETER AND WILLA Mi Yeon Lee Indiana University at Bloomington miyelee@indiana.edu To investigate relationships between students’ fractional knowledge and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. The students were interviewed twice, once to explore their quantitative reasoning with fractions and once to explore their solutions of problems that required explicit use of unknowns to write equations. As a part of the larger study, a case study of two seventh grade students, Peter and Willa, was conducted. Peter and Willa’s fractional knowledge influenced the linear equations they wrote to represent quantitative situations. In particular, the findings showed that a reversible iterative fraction scheme is important to understand reciprocal relationships between two quantities in writing a basic linear equation of the form ax=b. Also, considering fractions as operators on unknowns is important to write an equation. Implications for issues involved in the Korean mathematics curriculum revision are suggested. Quantitative reasoning with fractions, Algebraic reasoning, A reversible iterative fraction scheme, Use of fractions as operators on unknown, reciprocal reasoning. INTRODUCTION Since the early 1990s, mathematics educators have paid attention to elementary students’ early algebraic reasoning (Bastable & Schifer, 2008; Carpenter, Franke, & Levi, 2003; Kaput, Carraher, & Blanton, 2008). Early algebra researchers acknowledge that in algebraic activities with elementary graders, the use of conventional algebraic notation occurs very gradually, but young students can learn to use letters to stand in for unknown quantities (Bodanskii, 1991; Carraher, Schliemann, & Schwartz, 2008; Dougherty, 2008). Also, early algebra researchers mention that this early experience about unknown quantities is helpful for elementary students to success in algebra such as generating equations. Along with this tendency, the Korean national mathematics curriculum has changed so that equation solving and proportional expressions, which used to be taught in the 7 th grade, are now officially to be taught to 6 th graders (MEST, 2009). Thus elementary school teachers are to teach these topics. This revision is supposed to be implemented from 2011 but it is still controversial in Korea because some math educators are worried about that including equation solving in 6 th grade will cause overburdening of elementary students and teachers (Han, 2010; Seo, Yu, & Jeong, 2003), while the other math educators think that early introduction of equation solving would improve the possibility of students’ success in algebra courses in later years (Carpenter et. al., 2003; Han, 2010). Thus, it is important to think how to effectively teach equation solving without overburdening young students.