ROOFS, STAIRS & LINES: MIDDLE SCHOOL STUDENTS’ STRATEGIES IN SOLVING STEEPNESS PROBLEMS

Research shows that middle school students use a variety of strategies to solve proportion related problems (Hart, 1981). Mathematically, there is a connection between proportional reasoning and steepness, since steepness can be measured by the proportion which is the slope of a line. In this study, sixteen middle school students’ solution strategies in solving problems regarding steepness are explored, based upon their abilities to solve proportion related problems. Two tests were administered to students: an adapted version of the Ratio and Proportion Test (Brown, et al., 1981) and a Steepness Test (Author, 2013). This article contributes to literature on early algebraic reasoning exploring applications of proportional reasoning. INTRODUCTION According to assessments administered by the International Association for the Evaluation of Educational Achievement, students internationally struggle with responding to real-life situations involving proportional reasoning and slope (Gonzalez, et al., 2008). Slope is an important concept because it represents a bridge between arithmetic and algebra (Lobato, 1996). The concept of slope draws upon knowledge of proportional reasoning and leads into the more general concept of rate of change, which is applied to increasingly complex types of functions in algebra and calculus. Steepness has been proposed as an intermediary concept to be introduced to bridge the conceptual gap between proportional reasoning and slope (Stump, 2001; Lobato & Thanheiser, 2002; Lobato & Ellis, 2010). Through careful analysis of students’ strategies as they work with proportions, we know a lot about students’ development of knowledge about proportionality (Hart, 1984). And through careful analysis of students’ strategies as they work with linear equations, we know a lot about the development of slope (Schoenfeld, Smith & Arcavi, 1993; Lobato & Thanheiser, 2002). However, there is little research that helps us understand how students make the conceptual leap from proportionality to slope. Mathematically, there is a clear connection between proportionality and slope, but the research base has not yet carefully examined this connection in terms of students’ understandings. The present paper explores this connection through an examination of students’ strategies on a class of problems that are designed to bridge the concepts of proportionality and slope. As described in more depth below, we refer to these problems as “steepness” problems, in that the problems involve determining which of two situations (e.g., two roofs, two staircases, two lines) is steeper. As we argue below, understanding how students approach steepness problems helps the mathematics education community to understand whether and how students are using their knowledge of proportionality as they begin to learn about slope. We begin by examining past studies that have focused on students’ solutions to proportion-related, slope-related, and steepness-related problems; we then discuss the idea that using a ratio to measure steepness is a connecting concept between proportionality and slope. STEEPNESS AS A CONCEPTUAL BRIDGE BETWEEN PROPORTIONALITY AND SLOPE Many studies have found that the development of students’ proportional reasoning follows a common learning trajectory (e.g., Kaput & West, 1994; Lamon, 1993). When first encountering contextual problems involving multiple quantities, some of which are related proportionally, there is evidence that many students have difficulty determining which of the quantities are relevant and which are irrelevant to the proportionality of the context (Karplus & Peterson, 1970). In addition, even when students are able to discern the relevant data, they may coordinate it in non-proportional ways, such as using only two of the three relevant pieces of information or using the three relevant pieces of information but in an additive way (Hart. 1981). Over time, students eventually learn to correctly coordinate the relevant quantities in problems whose solution processes require use of proportional reasoning (Lamon, 1993). Research has also investigated the strategies used by students who are able to reason proportionally. For example, in order to solve missing value proportional reasoning problem (i.e., a problem that could mathematically be expressed using the equation a/b=? /d, where letters represent known quantities and the question mark represents an unknown quantity) and comparison proportional reasoning problems (i.e., a problem that could be mathematically be expressed as determining which of the two fractions, a/b or c/d is greater), Siegler (1976) found that, en route to thinking multiplicatively, students progress from using simple strategies that involve mostly additive reasoning to increasingly sophisticated strategies that take into account multiplicative relationships between quantities. Studies have also determined that the numerical quantities involved in solving problems and the contexts in which the problems are situated may determine students’ success on problems requiring proportional reasoning. For instance, The Online Journal of New Horizons in Education – January 2015 Volume 5, Issue 1