Content Background
What Proportional Reasoning Is Not
Consider the following problem: A recipe calls for 3 lemon wedges to make 12 ounces of lemonade. How many lemon wedges would you need to make 20 ounces of the same lemonade? To analyze deficits in my students’ proportional reasoning, I will offer three example responses to the lemon problem that represent some of the most common trends in the way past students ultimately approached ratio and proportion problems by the end of the unit.
Example A shows the use of the most widely utilized algorithm to find a missing value in a proportion often called cross multiply and divide, also known as “rule of three”. To utilize this algorithm, students need to find one value when given three. The student here arrived at the correct answer, but we can’t definitively tell if the student utilized proportional reasoning. The only thing we can be certain of is their ability to substitute numbers into an algorithm.
Example B shows the same algorithm being utilized. Here they set up the proportion incorrectly, leading them to an incorrect comparison. This response gives cause for worry, because there is no evidence of attending to and coordinating the different units to make an accurate comparison. A multiple choice test question would likely include this answer as a trap choice due to the common nature of this mistake. The unreasonableness of this answer is also evidence of the student’s narrow and procedural understanding.
Example C shows no evidence of proportional reasoning. Here addition is used to find the missing value; the rationale being that the ounces went up by eight, so the lemons must go up by the same amount. Instead of proportional reasoning, the student utilized additive reasoning by only considering the difference in magnitude of the two quantities. Until 6th grade this type of thinking is of greater focus. (The CCSS Writing Team 2011) To employ proportional reasoning, students need to compare multiplicatively.
What Proportional Reasoning Is and Why it Matters
In my research I came across a wide range of definitions, attempting to explain the complexity of proportional reasoning. The term encompasses a wide range of concepts including ratio, multiplicative comparison, co-variation, proportion, rate, unit rate, and constant rate of proportionality. Having a solid grasp of these concepts is indeed required in proportional reasoning. However to include all of these elements in its definition, clouds the essence of what proportional reasoning is. To simplify and to cut to the core of this term, I will use the following definition:
Proportional reasoning is reasoning about proportional relationships.
Being able to understand the presence or absence of proportionality in the relationship between two quantities is of significant importance. It is a critical form of thinking that requires the firm grasp of elementary fraction, operation and measurement concepts and lays the groundwork for algebra and other higher levels of mathematics. (Lesh et al. 1988) For most of my students, proportional reasoning will be a new way of thinking within mathematics. Because of this, my unit aims to develop proportional reasoning by centering my instruction on understanding proportional relationships. The National Council of Teachers of Mathematics lays out ten different essential understandings of ratios and proportions for grades sixth through eight. I have developed the essential understandings for my unit from four of these. They are as follows:
Essential Understanding 1. Reasoning with ratios involves attending to and coordinating two or more quantities.
Essential Understanding 2. A ratio is a multiplicative comparison of two quantities, or it is a joining of two quantities in a composed unit.
Essential Understanding 6. A proportional relationship is an equality between two ratios. In a proportional relationship, the ratio of two quantities remains constant as the corresponding values of the quantities change. This is the constant rate of proportionality.
Essential Understanding 7. Proportional reasoning is complex and involves understanding that-
- If one quantity in a ratio is multiplied or divided by a particular factor, then the other quantity must be multiplied or divided by the same factor to maintain the proportional relationship
(Lobato et al. 2010)
When students can apply the above understandings flexibly to solve ratio and proportion problems, they will be exhibiting thorough proportional reasoning. To achieve this ambitious goal I will reflect on my past teaching of the subject and use research and my work with this seminar to develop better strategies and approaches.
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