Children’s Use of Variables and Variable Notation to Represent Their Algebraic Ideas

Abstract

AbstractIn this article, we analyze a first grade classroom episode and individual interviews with students who participated in that classroom event to provide evidence of the variety of understandings about variable and variable notation held by first grade children approximately six years of age. Our findings illustrate that given the opportunity, children as young as six years of age can use variable notation in meaningful ways to express relationships between co-varying quantities. In this article, we argue that the early introduction of variable notation in children's mathematical experiences can offer them opportunities to develop familiarity and fluency with this convention as groundwork for ultimately powerful means of representing general mathematical relationships. Notes1 In this paper, when we say variable or variable notation we mean both the concept of variable and the notation to represent variables. For ease of reading, throughout the paper, we will use either one of the terms, but mean both.2 In this paper, when we say variable or variable quantity we refer to both varying and fixed unknown quantities (see Blanton, Citation2008; Blanton, Levi, Crites, & Dougherty, Citation2011). An example of the former would be the use of x to represent a varying, unknown quantity in the equation x + 7 = y. An example of the latter would be the use of x to represent a fixed, unknown quantity in the equation x + 7 = 10.3 Because of the age of the participants, we expected them to express these relationships additively rather than multiplicatively. For example, the relationship y = 2x might be represented as y = x + x.4 We chose functions of the type y = x + b instead of y = b + x for two reasons. First, they more closely model the mathematics that children are accustomed to engage in. For instance, for the example provided in this paper, if we have a variable height, and a constant hat height, it is more likely that children would look to symbolically express the following:"what is someone's total height (y) if they (x) put on a hat that is one foot tall (b)," and not this:"what is someone's total height (y) if we take a one-foot-tall hat (b) and put it on them (x)."Second, in our own prior research (e.g., Blanton, Citation2008; Blanton et al., Citation2011; Schliemann et al., Citation2007) with upper elementary school children, we had found that students had no difficulties making sense of functions of the type y = x + b.5 All children's names used are pseudonyms.6 Certainly, the thinking of students who are not visible (verbal) participants in a classroom discussion is an interesting point of study. While it is not our intent to minimize this aspect of learning, it is beyond the scope of our paper.7 Another possible interpretation for Zyla's rejection of Z + 1 is that for her, additions need to be represented in their executed form (such as 3 + 1 = 4). If the result of the operation needs to be made explicit, then just stating Z + 1 makes no sense: "it's just lame." However, in classroom lessons we had routinely used unexecuted expressions as a way to examine problems, so it is unlikely that at this point in our teaching experiment Zyla was questioning the unsolved nature of this expression.8 This phrase is how Leo and the interviewer had come to refer to hatless heights.9 The word "variable" had been used in the classroom before as shorthand for "variable notation."10 Only 6% of the 14-year-olds in Küchemann's (Citation1981) study provide evidence of "letter as variable" understandings in his assessment, and in Knuth et al.'s (Citation2011) study, less than 50% of grade 6 students see a literal symbol as representing multiple values. MacGregor and Stacey (Citation1997) report that none of the 11-15 year-old students in their study provided responses on the written test that could be considered this kind of understanding.Additional informationFundingThe research reported in this paper was supported by the National Science Foundation's DRK-12 Award #DRL 1154355. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.