227 Students' Retention of Mathematical Knowledge and Skills in Differential Equations Oh Nam Kwon Seoul National University Chris Rasmussen San Diego State University Karen Allen Michigan State University This study investigates students' retention of mathematical knowledge and skills in two differential equations classes. Posttests and delayed posttests after I year were administered to students in inquiry-oriented and traditional classes. The results show that students in the inquiry-oriented class retained conceptual knowledge, as seen by their performance on modeling problems, and retained equal proficiency in procedural problems, when compared with students in the traditionally taught classes. The results of this study add additional support to the claim that teaching for conceptual understanding can lead to longer retention of mathematical knowledge. Why can someone remember a baseball player's batting average for years, but not be able to solve a linear equation several months after instruction? Reten- tion of knowledge is an important area of research that has the potential to inform instructional practices and school learning goals (Bahrick, 2000). In mathematics, however, few studies have specifically addressed re- tention-related questions. Central questions regarding retention of mathematical knowledge include, * Does retention of conceptual understandings differ from retention of procedural understandings? * In what ways can educators facilitate students' retention of mathematical understandings and skills for longer periods of time? * What is the relationship, if any, between instructional approaches and retention of mathematical knowledge? The purpose ofthis study is to contribute to thebody of research in retention of mathematical knowledge. Specifically, this study examines the retention of con- ceptual and procedural knowledge in the context of differential equations 1 year after instruction. Students' retention ofknowledge is compared across a traditional and a reform-oriented instructional approach. Theretention studyreported here is a follow-up to an evaluation studyofstudentlearning of importantconcepts and skills in differential equations (Rasmussen, Kwon, Allen, Marrongelle, & Burtch, 2004). The evaluation study, which took place at the end of the course, compared students' understandings of central ideas and analytic methods for solving differential equations between students in reform-oriented and traditionally taught classes at four undergraduate institutions, three in the US and one in Korea. The main results were that students in the reform-oriented classrooms scored significantly higher on the conceptually oriented items. There was no significant difference on student performance on the procedurally oriented items, despite the fact that analytic solutions were the main focus of the traditional classes. This follow-up study examines retention of knowledge I year after instruction for a subset of the students from the comparison study. Background In this study students' retention of knowledge is framed in terms of conceptual and procedural knowl- edge, because this distinction has been useful for discrimninating the nature of students' mathematical understandings from elementary through undergradu- ate education (see, for example, Darken, Wynegar, & Kuhn, 2000; Siegler, 2003). Hiebert and Lefevre (1986) characterized conceptual knowledge as "knowledge that is rich in relationships. It can be thought of as a connected web of knowledge, a network in which the linking relationships are as prominent as the discrete pieces of information" (p. 3 -4). Procedural knowledge, Volume 105(5), May 2005