Middle School Students' Steepness and Proportional Reasoning

Middle School Students' Reasoning about SteepnessIn an era of increasing international scrutiny on the educational preparation of students, the International Association for the Evaluation of Educational Achievement has developed assessments that reflect internationally important concepts that students should learn by grades 4, 8, and 12 (Gonzalez et al., 2008). The content of the eighth grade Trends in International Mathematics and Science Study, implemented every four years at the eighth grade level from 1995 through 2007, especially reveals necessary components of middle grades mathematics in the domains of Number, Algebra, Geometry, and Data and Chance. Of the four domains, Number and Algebra produced the lowest number of students in countries who scored at the high benchmark, indicating facility with working with proportional relationships and linear equations (Gonzalez et al., 2008). Thus, there is international interest in improving instruction in proportional reasoning and in algebraic concepts related to linearity.Slope is a central concept in algebra that is not particularly well understood by students internationally in the middle and secondary grades (Stump, 2001; Yerushalmy, 1997). Slope is fundamentally related to the idea of proportionality which is generally introduced to students in the middle grades. However, this relationship is not explored empirically nor do curricula make the connection explicitly (Lobato & Thanheiser, 2002). Slope is also related to the idea of steepness, a physical characteristic of a line which can be determined visually using an angle or analytically using a proportion (Stump, 1999). The goal of this article is to explore the relationship between students' understandings of proportional reasoning and steepness, in an effort to shed light upon the learning of slope in the middle grades.We begin by examining past studies that have focused on students' understandings of proportionality and slope; we then explore the mathematical connections between proportionality and steepness.Students' Understandings of ProportionalityProportionality is a multiplicative relationship which can be represented on the coordinate plane as linear functions that pass through the origin (Lobato & Ellis, 2010). Concepts involved in the multiplicative conceptual field include multiplication and division, linear functions and their graphs, rates, ratios, fractions, and rational numbers. These concepts all involve multiplicative relationships between and within quantities. An understanding of the multiplicative conceptual field entails identifying when situations require multiplicative reasoning, particularly distinguishing between the uses of additive and multiplicative reasoning. It also entails being able to perform operations with fractions, such as writing equivalent fractions and solving for an unknown variable in a proportion. With knowledge of concepts in the multiplicative conceptual field, students should be equipped to see connections between modes of representations of multiplicative concepts such as symbols, tables of data, area models, and written descriptions of real-world situations.A number of parts of the multiplicative conceptual fields are relevant to this study. A proportion is a mathematical relation between quantities that can be represented symbolically as ^ = the ability to mentally process this relation involves proportional reasoning. Reasoning proportionally involves coordinating ratios in multiplicative ways and using rational expressions such as quotients, fractions, and rates.The most common type of proportional relationship is a ratio. In ratios, there is a multiplicative comparison of two or more quantities. Rates, which are a type of ratio, involve a comparison of two numbers that represent different types of quantities, and there are two measure spaces involved, one for each type of quantity. These types of proportional relationships are called 'associated sets' because each of the quantities is a set and they are associated in a multiplicative way (Lamon, 1993; Marshall, 1993). …