ZOOMING IN ON THE LEARNING PROGRESSIONS ON PROPORTIONAL REASONING
Among the many topics in the middle-school curriculum, one of the most prominent areas that are conceptually rich and mathematically sophisticated, yet difficult to teach as well as learn is fractions and their counterparts, namely ratios and proportions. The difficulty lies in interpreting the meaning of rational numbers, which requires explanations through connections with other mathematical knowledge, various representations, and/or real-world contexts. Often students do not realize that they have been exposed to fractions in their everyday activities long before they were formally instructed in school. An essential understanding of ratios and proportions is the notion that as one quantity in a proportional relationship changes, so does the other quantity—and by the same factor.
“We use ratio to mean the comparison of two quantities, and we define proportion as an equivalence relationship between two ratios.” PEU p. 115
In grades 3-5, students build on their understanding of creating equal groups and partitioning groups into smaller groups of equal size as a foundation for the multiplicative understanding necessary for working with rational numbers and ratios. Students in these grades also investigate situations in which it is appropriate to use partitioning and create equal-sized groups and situations in which it is not. In grades 6-8, students extend this reasoning to contexts that involve a constant rate, such as situations that involve equal rates of speed, the cost of an item per particular unit, and scale models or drawings. Further, students learn to use ratios to convert measurement units and transform units.
Currently, there is a big gap between constructed knowledge that students bring to the classroom and the instructed knowledge that the teachers try to deliver. Students who are mathematically proficient make the connections between the two bodies of knowledge and students that do not understand may never make this connection. Mathematical modeling is one way to achieve these connections.
Teaching proportional reasoning through mathematical modeling and problem solving requires depth of mathematical knowledge for teaching that includes understanding of general content and the domain-specific knowledge of students. Clearly, a solid understanding of proportional reasoning is requisite for moving from elementary grades on to more complex problem solving in the intermediate and higher grades; else, as students’ progress from one mathematics course to another, some can get deeply and profoundly lost.
Although it is possible to pick up virtually any chapter in a history book and learn something new, without a strong mathematical foundation of proportional reasoning, an advanced chapter of mathematics could be totally incomprehensible to a student who is unfamiliar with, uncomfortable with, or fearful of the concepts. Because an understanding of proportional reasoning is foundational and the risk of frustration is high, there is a need for mathematics teachers who can ably explain these topics and who themselves have a deep conceptual understanding of them. Noting the difficulties of both learning and teaching mathematics topics that involve multiplicative structures, Lamon (2007) suggests:
Of all the topics in the school curriculum, fractions, ratios and proportions arguably hold the distinction of being the most protracted in terms of development, the most difficult to teach, the most mathematically complex, the most cognitively challenging, the most essential to success in high mathematics and science, and one of the most compelling research sites.
Our traditional teaching for computational ability, Lamon (2007) contends, has left us pedagogically bankrupt for an age that values connections and meaning, and requires innovative pedagogy. In order for students to make connections, they need to abandon the rote plug-and-play scenario of implementing a formula and rise to levels of comprehension that would enable them to understand why a certain formula or procedure should be used and the inherent relations between multiple representations (see Figure 9.2).
Figure 9.2 Multiple representations in understanding proportional reasoning. Source: Authors.
Proportional Reasoning is fundamental to many important mathematical concepts and is often regarded as the pathway to performing well in algebra (Confrey & Smith, 1995; Lobato & Ellis, 2010). This topic is difficult for most students, especially for those who do not understand what is actually meant by a specific proportional situation or why a given solution methodology works (Cramer & Post, 1993). Teachers have also been urged to focus students’ attention on the meaning of problems and to help students’ value different mathematically correct solutions to a single problem (NCTM, 1989, 1991, 2000). There is still a great need for research in evaluating the effect of solving one proportional situation via multiple solution strategies for example using “unit rate strategy; repeated subtraction strategy; equivalent fractions strategy; size-change strategy; cross-multiplication using equal rates or ratios strategy, relative and absolute thinking strategy and; reasoning up and down strategy” (Lamon, 2007).
Students use proportional reasoning in early math learning, for example, when they think of the number 10 as two fives or five twos rather than thinking of it as one more than nine. This is an essential part of developmental step for students to transition from additive to multiplicative reasoning (Kent, Arnosky & McMonagle, 2002; Sowder et al., 1998). Although additive reasoning can develop intuitively, multiplicative (proportional) reasoning is difficult for students to develop and often requires formal instruction. It requires reasoning about several ideas or quantities simultaneously. It requires thinking about situations in relative rather than absolute terms. For example, if the number of students in a middle school grows from 500 to 800 and
Text Box 9.2 Common Core State Standards for Mathematics for Proportional Reasoning (NGA Center and CCSSO, 2010)
Understand ratio concepts and use ratio reasoning to solve problems.
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
Understand the concept of a unit rate a/b associated with a ratio a:b with b Ф 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”
Use ratio and rate reasoning to solve real-world and mathematical problems, for example, by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
Analyze proportional relationships and use them to solve real-world and mathematical problems.
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.
Recognize and represent proportional relationships between quantities. CCSS.Math.Content.7.RP.A.3
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
another middle school grows from 300 to 600, a student thinking in absolute terms (or additively) might answer that both schools had the same amount of increase.
On the other hand, a student that is trained to think in relative terms might argue that the second middle school saw more increase since it doubled the number of students unlike the first school who would have needed to be 1000 students to grow by the same relative amount. While both answers seem reasonable, it is the relative multiplicative thinking that is essential for proportional reasoning. This ability to think and reason proportionally is very important in the development of a student’s ability to understand and apply mathematics. It is estimated that over 90% of students who enter high school cannot reason well enough to learn mathematics and science with understanding and are unprepared for real applications in statistics, biology, geography, or physics (Lamon, 2001).
While students may be able to solve a proportion problem with a rote algorithm, this does not mean that they can think proportionally. Therefore, it is essential for mathematics teachers to (a) understand how students develop multiplicative (and proportional) reasoning, (b) build on students’ prior knowledge of concepts such as multiplication and division of whole numbers to strengthen students’ proportional reasoning, and (c) develop learning environments, contexts and experiences for students that encourage multiplicative comparisons to prepare them for higher-level mathematics topics involving proportional reasoning.
The CCSSM (NGA Center % CCSSO, 2010) expects an instructional emphasis on ratio and rate concepts in grades six and seven. CCSSM indicates that grade six students should “understand ratio concepts and use ratio reasoning to solve problems” (p. 42), with a focus on using ratio and rate language in the context of ratio relationships to understand these relationships and conceptualize a unit rate associated with a ratio (see Text Box 9.2).