Grade 4-6 Student Conceptions And Utilizations Of Informal And Formal Representations Of Variables Across Core Mathematical Tasks
Successful completion of an algebra course, or equivalent, serves as a gatekeeper for students’ future educational, professional, and economic opportunities (e.g., Ball, 2004; National Academy of Science, 2007). Those who successfully complete an algebra course are at a considerable advantage over their peers who have not. This critical role of algebra has also been a focus in several recent high profile reports. Trends in Mathematics and Science Study (TIMMS), Rising Above the Gathering Storm (National Academy of Science, 2007), Mathematical Proficiency for All Students (Ball, 2004), and Foundations for Success (National Mathematics Advisory Panel, 2008) have each placed an increased emphasis on two specific, but related, aspects of learning and teaching algebra. First, students’ early mathematics education must prepare them for success in algebra (Kilpatrick & Izsak, 2008). Second, in order to increase students’ proficiency in algebra their experiences in algebra and beyond must also support all students in attaining this goal.
In addition to the increased emphasis on the importance of students’ success in algebra, is the corresponding emphasis on algebra for all. From a social justice perspective Robert Moses has argued for and worked toward this goal through the Algebra Project (Moses, 2011). The stance that all students can be successful in mathematics in general, and algebra in particular, is also consistent with the Equity Principle in NCTM’s Principles and Standards for School Mathematics (NCTM, 2000). Likewise, Achieve has taken the position that everyone can do algebra (American Diploma Project (ADP), 2004). Further, numerous state legislatures have taken the position that students must complete the equivalent of at least an algebra course in order to receive a high school diploma.
Considerable knowledge exists regarding the teaching and learning of algebra (e.g., Blanton, et al., 2007; Booth, 1984; Brizuela & Schliemann, 2004; Carpenter, Levi, Berman, & Pligge, 2005; Drijvers, 2003; Kaput, 2008a; Kieran, 2007; Kuchemann, 1981; Lee, 2006; NCTM, 2000; Radford, Bardino, & Sabena, 2007). This includes the relatively new and evolving domain of research on early algebra, which has begun to develop a knowledge base addressing the relationships between arithmetic and algebra (e.g., Blanton & Kaput, 2001; Carpenter, Franke, & Levi, 2003; Carraher & Schliemann, 2007; Kieran, 2007; Schliemann, et al., 2003; Van Amerom, 2003; Warren & Cooper, 2008b). One finding consistent across these areas of research is that many students at all levels demonstrate difficulties with the meaning and use of conventional mathematical symbols (e.g., Cooper & Warren, 2008; Fujii, 2003; Kuchemann, 1981).
A subset of the research on students’ meaning and use of conventional mathematical symbols addresses students’ meanings for and subsequent use of variables. In fact, the research (cf., Booth, 1984; Carraher, Brizuela, & Schliemann, 2000; Carraher, Schielmann, & Brizuela, 2001; Ellis, 2007; Knuth, Alibali, McNeil, Weinberg, & Stephens, 2005; Kuchemann, 1981; Lannin, Barker, & Townsend, 2006; MacGregor & Stacey, 1997; Swafford & Langrall, 2000; Warren & Cooper, 2008b) has established a great deal about students’ meaning and use of conventional letter-symbolic representations of variables. This research has demonstrated that many students appear to have different meanings and strategies for dealing with variables and these strategies may vary across task types (e.g., word problems, word equations and equations (Koedinger & Nathan, 2004)). However, the setting for the majority of this research has been at the middle school level and beyond and does not include students’ meanings and use of informal representations of variables, or why or how difficulties arise.
Therefore, this study is examining how 36 grade 4-6 students from one elementary and middle school in a Midwestern US city and one elementary and middle school in a Southern US city conceptualized and utilized formal and informal representations of variables across core mathematical tasks. While algebraic conventions for variables represented with literal symbols (e.g., x and y) have been established for student in algebra classes and beyond, little research into elementary school students’ initial conception(s) for variables exists. This study examined student interpretations of formal (e.g., x + y = 12) and informal representations of variables (e.g., c + r = 12).