By Sybilla Beckmann, Josiah Meigs Distinguished Teaching Professor in the Department of Mathematics at the University of Georgia, and Andrew Izsák, Professor of Mathematics Education in the Department of Mathematics and Science Education at the University of Georgia.
One of the challenges of teaching mathematics is understanding and appreciating students’ struggles with material that to the instructor, after years of thinking about it, may seem straight forward. Once we understand an idea, it may seem almost impossible not to understand if it is presented clearly enough. Yet experienced math teachers know that presenting mathematical ideas clearly, as important as that is, is generally not enough for students to learn the ideas well, even for dedicated and determined students. At the same time, students who struggle can have insightful and productive ways of solving problems and reasoning about mathematical ideas. Research into how people think about and learn mathematics reveals why this surprising mix of struggle and competence can coexist: learners can use what they do understand to make sense of new things, yet ideas that are tightly interconnected and readily available for an expert may be fragmented or inchoate for a learner.
Consider the ideas surrounding slope and rate of change, which are well known to be difficult for students. To the expert, a slope is a number that expresses a measure of steepness. It connects changes in an independent variable to changes in a dependent variable. This connection is multiplicative and explains why non-vertical lines have equations of the form y = mx + b. But even students who appear to be proficient—because they can calculate a slope and use it to find an equation for a line—may be missing some crucial connections. They might not see slope as a number, but instead think of it as a pair of numbers separated by a slash, basically “rise slash run.” If the “rise” is 3 and the “run” is 2, then even if they know that 3/2 is a number, they may not connect it to the geometry and algebra of the situation. They might not see this number as a measure of steepness, and if asked to describe steepness, might prefer to subtract the “run” from the “rise.” Students might not see the “rise” as 3/2 of the “run” and they might not connect this multiplicative relationship between the “rise” and “run” to the point-slope form of an equation for a line. Mathematics education research is examining the fine-grained details of how students think about ideas surrounding slope. It is investigating how certain ways of representing and drawing attention to ideas can help students extend and connect their ideas. Research-based instruction can then take into account known challenges and opportunities for learning.
We thought readers of this blog might be interested to learn a little about approaches to slope and linear equations that we are currently investigating. Proportional relationships—pairs of values in a fixed ratio—provide an entry point into the study of linear functions and are a focus in the Common Core State Standards for Mathematics at grades 6 and 7 (see  and ). So consider the proportional relationship consisting of all pairs of quantities of peach and grape juice that are mixed in a fixed 3 to 2 ratio to make a punch. When graphed, these points lie on a line. One way to think about the slope, 3/2, of this line is that for every new cup of grape juice, the amount of peach juice increases by 3/2 cups. This way of thinking is part of what we call a multiple batches view, a view that has received significant attention in mathematics education research. From this perspective, we may think of 1 cup grape juice and 3/2 cups peach juice (or 2 cups grape juice and 3 cups peach juice) as forming a fixed batch of punch, and we vary the number of batches to produce different amounts in the same ratio. This fits with the image in Figure 1a, which evokes repeatedly moving to the right 1 unit and up 3/2 units. But as indicated in Figure 1b, the general multiplicative relationship, y = (3/2) x, is less evident, especially for x values that are not whole numbers.
Figure 1: Slope from a multiple-batches perspective.
Another way to think about the punch mixtures in a fixed 3 to 2 ratio uses what we call a variable parts perspective. This perspective has been overlooked by mathematics education research, but we are currently studying how future teachers reason with it. In a variable-parts approach, for any point on the “punch line” (see Figure 2), there are 3 parts for the y-coordinate and 2 parts for the x-coordinate, and all the parts are the same size. From this perspective, we vary the size of the parts to produce different amounts in the same ratio. The parts expand or contract depending on the direction the point moves along the line. In a variable-parts approach, the slope 3/2 is a direct multiplicative comparison between the numbers of parts of grape and peach juice: The number of parts peach juice is 3/2 the number of parts grape juice. Put another way, the value 3/2 is the factor that multiplies the number of parts of grape juice to produce the number of parts of peach juice, regardless of amounts of juice in each part. Therefore the y-coordinate is 3/2 of the x-coordinate, so y = (3/2) x.
Figure 2: A proportional relationship viewed from a variable-parts perspective.
Figure 3: Slope and equations from a variable-parts perspective.
We don’t think there is any way to make the concept of slope easy for students. But we suspect that working with both the multiple-batches and the variable-parts perspectives should help students develop a more robust understanding of slope. In particular, the variable-parts perspective might help students connect the slope of a line and an equation for the line. References  and  discuss the multiple-batches and variable-parts perspectives in greater detail.
We are currently conducting detailed studies of how students in our courses for future teachers reason from both the multiple-batches and the variable-parts perspectives on proportional relationships*. Discoveries about how future teachers reason about the interconnected ideas of multiplication, division, fractions, ratio, and proportional relationships, and what is easier and what is harder to learn, will help us identify productive targets for instruction in courses for future teachers. But we also hope that others will try the variable-parts perspective with other groups of students. For example, we could imagine a group of college algebra instructors collaboratively designing lessons that use a variable-parts perspective to help students better understand slope and its connection to equations for lines.
We also think that the variable-parts perspective is potentially productive for trigonometric ratios. From a variable-parts perspective, we can think of the radius of a circle as 1 part of variable size, r. For a fixed angle, its radian measure, sine, cosine, and tangent can all be viewed as a fixed number of parts (although this number is often irrational). With this perspective, equations such as x = cos(θ)r and y = sin(θ)r arise from the very same reasoning that connects slope to the equation of a line.
We think that there are many useful findings of mathematics education research that could help improve mathematics teaching and learning, but that environments and cultures are often not conducive to using the knowledge that we have. We need professional environments and cultures that foster serious discussions about what to teach and how to teach it, where knowledge about teaching and learning mathematics is intertwined with the practice of mathematics teaching, and where knowledge and practice grow together. We applaud the editors and the AMS for starting this blog as a way to nurture and develop such a culture.
[*] We are grateful to the University of Georgia, the Spencer Foundation, and the National Science Foundation, award number 1420307, for supporting our research.
 National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Author. Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
 Common Core Standards Writing Team. (2011). Progressions for the common core state standards for mathematics (draft), 6–7, ratios and proportional relationships. Retrieved from http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf
 Beckmann, S., & Izsák, A. (in press). Two perspectives on proportional relationships: Extending complementary origins of multiplication in terms of quantities. Journal for Research in Mathematics Education.
 Beckmann, S., & Izsák, A. (2014). Variable parts: A new perspective on proportional relationships and linear functions. In Liljedahl, P., Nicol, C., Oesterle, S., & Allan, D. (Eds.). (2014). Proceedings of the Joint Meeting of PME 38 and PME-NA 36 (Vol. 2). Vancouver, Canada: PME. http://www.igpme.org