The Importance of Proportional Reasoning
Proportional reasoning is fundamental to the understanding of the mathematics of the secondary school curriculum.
—Litwiller, Preface, 2002 Yearbook of the National Council of Teachers of Mathematics: Making Sense of Fractions, Ratios, and Proportion
For those with a serious interest in the teaching and learning of proportion, there are two readily available and practical resources. The first is the 2002 Yearbook published by the National Council of Teachers of Mathematics (cited above), and the second is the website of the Rational Number Project, http://cehd.umn.edu/rationalnumberproject.
The Rational Number Projectâ€™s website has a variety of types of research and support materials. One of the articles it offers is â€œConnecting Research to Teaching and Proportional Reasoning.â€ Written by Kathleen Cramer and Thomas Post, the article originally appeared in the Mathematics Teacher in 1993 and is of particular usefulness to those needing a primer on proportional reasoning.
Cramer and Post delineate three types of tasks to assess proportionality, and then they discuss, with just the right amount of detail, solution strategies for proportional reasoning tasks. The solution strategies are probably known to and used by teachers, but it is worth enumerating the strategies and comparing them. Ideally, students will be able to use all four strategies, selecting the most appropriate given the task at hand. The following problem and four strategies are based on material from the Cramer and Post article:
Steve and Mark were driving equally fast along a country road. It took Steve 20 minutes to drive 4 miles. How long did it take Mark to drive 12 miles?
- The first strategy is the unit-rate strategy and relies on the â€œhow many for one?â€ question. It is most commonly found in unit pricing in grocery stores. In this case, the student would think: It took 20 minutes to drive 4 miles—so it would take 5 minutes to drive 1 mile; since Steve is driving 12 miles, it would take him 12 x 5, or 60 minutes.
- The second strategy is the factor-of-change strategy. It is a multiplicative strategy. A studentâ€™s thinking would be something like this: Steve will be driving 3 times as far as Mark (12 miles instead of 4); therefore it will take him 3 times as long—3 x 20, or 60 minutes.
- The fraction strategy is the third strategy that the authors explain. It is based on the idea of fractional equivalence: 4/20 would be the same as 12/? The student sees that it is necessary to multiply the numerator and denominator by the same number (in this case, 3).
- The fourth strategy, perhaps the one most widely taught, is the cross-product algorithm.
Cramer and Post note: â€œAs with many standard algorithms, this is an extremely efficient but mechanical process devoid of meaning in the real world.â€
Though not in a strict â€œdevelopmentalâ€ order, there is a kind of hierarchy to the strategies as they appear in the list above. The unit method—how many for one? —is understandable by relatively young students; and even though it sometimes requires multiple steps in solution, the logic is clear for most people. The change-of-factor strategy requires a different kind of thinking (multiplicative) and a clear understanding of the relationships in the problem. The fraction strategy also requires some multiplicative thinking and an understanding of equivalent fractions. The cross-product algorithm should build on the thinking found in the development of the other strategies—but often it is taught procedurally instead (â€œBe sure you have the same units on the topâ€). And often teachers donâ€™t help students see the relationship between the four strategies.
High school teachers need to be aware of all four strategies, and they need to help students use the strategies effectively. Teachers of high school students should intentionally make the point that proportional reasoning is used extensively in algebra, geometry, measurement, and probability and statistics, as well as in social studies and science—and they should illustrate the point by providing contextual problem examples that use proportional reasoning.
It is important to help students see the breadth of situations where proportional reasoning is used. Below are just five examples; there are some notable omissions from the list (e.g., similar triangles), but it is a list you could share with your students—and then ask them for other situations where they would use proportions.
- So far this season, Albert Pujols from the St. Louis Cardinals has 38 hits from his 106 times at bat. Based on the data, what is the likelihood of his getting a hit the next time he is at bat?
- In 1998, active-duty military personnel were apportioned between the services in the following way: Marines: 12%, Air Force: 26%, Navy: 27%, Army: 35%. Make a circle graph of the data.
- The actual distance, as the crow flies, from Columbus to Cincinnati is 125 miles. If the scale of a map is 1 inch for 25 miles, how long is the line drawn on the map from Columbus to Cincinnati?
- Are you more likely to draw a black marble from a bag with 2 black marbles and 8 white marbles or from a bag with 19 black marbles and 81 white marbles?
- Mary mixed 3 pints of green paint with 4 pints of white paint. John mixed 4 pints of green paint with 5 pints of white paint. Who had the darkest paint?
One of the most difficult things about teaching is that very few teachers have the same students for 12 years, which means that each teacher provides only â€œpart of the program.â€ If a student has been taught the cross-product method in a rote way, before using any of the other strategies, it is likely that the student doesnâ€™t understand the concept and wonâ€™t always know when and how to apply proportional reasoning. Intentionally teaching all four strategies in the sequence shown—making students aware of how many different kinds of problems use proportional reasoning—will help them see why having different strategies is important and will also perhaps help them know when it is better to use a unit strategy and when the cross-product algorithm is a better choice.
A final example of how we would like our students to be able to apply their proportional reasoning to solve a multistep problem: In 2005, the National Assessment of Education Progress included the following item for 12th grade students:
The pulse rate per minute of a group of 100 adults is displayed in the histogram above. For example, 5 adults have a pulse rate from 40-49 inclusive. Based on these data, how many individuals from a comparable group of 40 adults would be expected to have a pulse rate of 80 or above?
Think for just a minute about the item. What does it ask students to do? First, students have to understand what is meant by saying that the histogram represents 100 adults; as teachers, we might like them to see if that makes sense. We want them to think something like: Well, it looks like 35 people have a pulse rate between 70 and 80 and another 30 are between 60 and 70. And then there is a group of 5, a group of 10, a group of 15, and another group of 5—35 + 30 + 5 + 10 + 15 +5. Yep, that is a 100. Then we want them to read the problem carefully enough to see that it is talking about a pulse rate of â€œ80 or above,â€ which means people in the range of 80 to 90 (15 people) and people in the range of 90 to 100 (5 people); so we want them to see that in the group of 100 adults there are 20 that meet the criterion. Then—and only then—does student thinking need to involve proportional reasoning. If 20 out of 100 are in that range, then (using one of several methods) we hope that the student can see that it is reasonable to expect that 8 out of 40 would be also have a pulse rate greater than 80.
Unfortunately, 12th grade students did not do well on this item. Six percent omitted it, 81% got an incorrect answer, and 2% were â€œoff task.â€ Only 11% of 12th grade students correctly answered this question—which required proportional reasoning.
Proportional reasoning should develop over time, so that when students are ready to enter high school, it is part of their repertoire of tools. We know, however, that facility with proportionality is not as common as we would like. This is especially true when students must apply proportional reasoning in concert with other mathematical understandings, as in the pulse rate problem above.
Being able to use proportional reasoning is important for students who want to do well on the Ohio Graduation Test—but it is even more important for students who want to be mathematically literate citizens.
Cramer, Kathleen, & Post, Thomas. (1993, May). Connecting research to teaching proportional reasoning. Mathematics Teacher, 86(5), 404-407. Accessed at http://cehd.umn.edu/rationalnumberproject.
Litwiller, Bonnie (Ed.). (2002). 2002 Yearbook: Making sense of fractions, ratios, and proportions. Reston, VA: National Council of Teachers of Mathematics.