

Connecting
Research to Teaching Kathleen Cramer and Thomas Post
PROPORTIONAL REASONING Consider the following proportional situation: Three meters equal 300 centimeters, How many centimeters equal 4.5 meters? The relationship between meters and centimeters is multiplicative and can be expressed in either of two ways, # of cm = 100 (# of m) or # of m = (1/100)(# of cm). The critical component in proportional situations is the multiplicative relationship that exists among the quantities that represent the situation (Cramer, Post, and Carrier, in press). Because of this relationship, all proportional situations can be expressed through an algebraic rule of the form y = mx. PROPORTIONAL REASONING TASKS: TYPES AND DIFFICULTY Assessing understanding of this multiplicative relationship has been done in various ways. Learning tasks devised in research studies can be a rich source of creative problem sets for classroom instruction and assessment. Research reports not only suggest varieties of tasks but give information about the relative difficulty of the tasks and factors (e.g., context, numerical complexity) that influence difficulty. The Rational Number Project developed three different types of tasks to assess proportionality: (1) missing value, (2) numerical comparison, and (3) qualitative prediction and comparison (Post, Behr, and Lesh 1988; Heller et al. 1990). Each problem type was posed to students in four different real world contexts: speed, scaling, mixture, and density (see examples in fig. 1). Missingvalue problems. In missingvalue problems three pieces of numerical information are given and one piece is unknown. Karplus's tallmanshortman problem is representative of this type of problem (Karplus, Karplus, and Wollman 1974). For the tallmanshortman task, students are given a chain of six paper clips and told that this chain represents Mr. Shores height in paper clips. The students are also told that Mr. Short measures four large buttons tall. They are then told (not shown) that Mr. Tall is similar to Mr. Short but is six large buttons tall. Students are asked to find the height of Mr. Tall in paper clips and to explain their answers. The information in a missing value problem can be represented as rates. In the tallmanshortman task, 6 paper clips/4 buttons is a complete rate and __ paper clips/6 buttons is an incomplete rate. Numerical comparison problems. In these problems, two complete rates are given. A numerical answer is not required, however the rates are to be compared. Noelting's (1990) orangejuice task is an example of this type of problem (see fig. 2). Students are told that the shaded glasses represent orangejuice mix and that the unshaded glasses represent water. They are asked to imagine that the orangejuice mix and water are pitcher, Students determine which pitcher has the' strongesttasting orange juice or if the mixtures would taste the same. In studying the results of these tasks, researchers found that students were more successful when one quantity in a complete rate was an integral multiple of the corresponding given quantity of the other rate, as, for example, two parts of orangejuice mix and five parts of water to six parts of orangejuice mix and eleven parts of water. When multiples were nonintegral, students often reverted to additive strategies. For example, in solving problem I of figure 1 (speed example), students would often conclude that the answer was eight minutes, reasoning that six miles is two more than four miles so the answer must be two more than six minutes. This typical error in proportional reasoning is discussed at length by a number of researchers (Hart 1981; Noelting 1980). Qualitative prediction and comparison problems. These types of problems (see problems 3 and 4 in fig. 1) require comparisons not dependent on specific numerical values. Such thinking is a part of proportional thinking and, at least for seventh and eighth graders, is not the same as solving numerical comparison and missingvalue problems (Heller et a. 1990). On the one hand, students may use a memorized skill to solve numericalcomparison and missingvalue problems. Qualitativeprediction and comparison problems, on the other hand, require students to understand the meaning of proportions. Thinking qualitatively allows students to check the feasibility of answers and to establish appropriate parameters for problem situations. Since this type of thinking is often necessary before actual calculations, the inclusion of such problems in teaching encourages students to use such approaches and improves their calculations and problem solving. 

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Importance of context. Researchers found that problem context, as well as the nature of the numerical relationships, influenced problem difficulty. Of the four contexts studied in the Rational Number Project, scaling was significantly more difficult for middlegrades students than the other contexts. This finding was true even though the numbers in the problems were held constant. However, a proportional reasoner should not be radically affected by the awkward numerical relationships or the context in which the problem is posed. Therefore, in both instruction and testing, teachers should vary the numerical relationships and the context of proportionalreasoning problems, Besides furnishing teachers with creative assessment items, research tasks can also be a rich source of instructional activities. The task presented in figure 3 can help students develop proportional reasoning and understand the multiplicative relationship inherent in proportional situations. The activity comes from the Rational Number Project and was one of twelve such experiments used by students to highlight the multiplicative relationship in proportional situations. In this activity, students collect data, form a table, and determine an algebraic rule that represents the data. They then transfer the data to a coordinate axis. The rule highlights the multiplicative nature of proportional situations; the graph models the characteristic that in any proportional situation all the rate pairs fall on a straight line crossing the origin. Additional examples for instruction can be found in Cramer, Post, and Behr (1989) and Cramer, Post, and Currier (in press). 



SOLUTION STRATEGIES FOR PROPORTIONAL REASONING TASKS When seventh and eighth graders attempted missingvalue and numericalcomparison problems, success rate was low. Analysis of students' correct responses showed four distinct solutions strategies: unit rate, factor of change, fraction, and cross product. Table 1 shows the percent of problems correct by grade level and by solution strategy. Each strategy will be illustrated by using it to solve the following problem: 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 unitrate strategy.
As the name implies this is a "how many for one?" strategy. It always involves
two rates corresponding to a given pair of quantities: 20 minutes/4 miles
and its reciprocal, 4 miles/20 minutes. The first pair can be represented
by the unit rate: 5 minutes per 1 mile; the second by the unit rate: 1/5
mile per 1 minute. Since the question about Mark asks for the amount of
time to drive twelve miles, the unit rate describing the length of time
for one mile is used ([5 minutes/ 1 mile] x 12 miles = 60 minutes). If the
question was, How many miles did Mark drive in 60 minutes? The other unit
rate would be used ([l/5 mile/ 1 minute] x 60 minutes = 12 miles).
The factorofchange strategy. This is a "times as many" strategy. A student using this method would reason as follows: "It takes twenty minutes to drive four miles. Since Mark is driving three times as far, it should take him three times as long. So the answer is twenty minutes times three, or sixty minutes." The ease in using this method is related to the numerical aspects of the problem. Students would be less apt to use this method if the factor to be used was not an integer for example, if the factor was 2/7, as it would be if the problem had been, It takes Steve 20 minutes to drive 7 miles, how long will it take Mark to drive 2 miles? (2/7 of 20 is 40/7, or 5 and 5/7 minutes). The fraction strategy. When using rates, the labels for each quantity are usually kept in the expression. This strategy is used when using the unitrate strategy. If students dropped the labels and used ideas of equivalence, it is a fraction strategy. Students treated the rates as fractions, applying the fraction rule for equivalent fractions (multiply the numerator and denominator by 3) to calculate the answer of 60. The crossproduct algorithm. As with many standard algorithms, this is an extremely efficient but mechanical process devoid of meaning in the real world. To solve the problem about Steve and Mark, a student sets up a proportion, forms a cross product, and solves the resulting equation by division: 



The unitrate strategy was used most by seventh graders in the Rational Number Project study. The unit rate seemed to be an intuitive approach building on students' reallife experiences, whereas the crossproduct algorithm used most often by eighth graders seemed more contrived. (What meaning does 20 minutes x 12 miles have?) The more intuitive unitrate and factorof change strategies related more meaningfully to the situation. On the one nonproportional problem included on the assessment, eighth graders were less successful (close to 30 percent inappropriately applied the crossproduct algorithm to this task). Seventh graders, who had not learned this algorithm, were more successful and solved the nonproportional problem using other problemsolving strategies. Knowing how to do a procedure does not mean that a student knows when it can be applied, and value is found in intuitive methods. Since students not instructed in the crossmultiplication algorithm used the unitrate and factorofchange strategies to solve proportion problems, teachers may capitalize on these natural thought patterns and begin instruction focusing on these strategies. A sample lesson can be found in a Mathematics Teacher article by Cramer, Bezuk, and Behr (1989). CONCLUSION Research can have a positive effect on classroom practice. This article suggests that research tasks can and should be used in classrooms. They can function as instructional activities as well as assessment tools. The types of problems generated by research inform teachers of the different ways in which understanding can be assessed. Being a proportional reasoner means more than applying the crossproduct algorithm. Students should be able to solve multipleproblem types, including missing value, numerical comparison, qualitative comparison, and qualitative prediction. Just as researchers analyze the effect of different variables on performance, teachers can consider these same variables as they teach and assess. Two important variables include context and the presence of integral multiples. Instruction should start with familiar contexts and extend to less familiar ones. With the availability of calculators, proportional problems should include "nasty" numbers so that students can encounter nonintegral relationships early on. Research suggests teaching multiple strategies, including unit rate, factor of change, fractions, and the crossproduct algorithm. Teachers should begin instruction with more intuitive strategies, such as the unit rate and factor of change. These teaching suggestions emphasize learning concepts over learning procedures, As textbooks often focus on procedural knowledge, teachers will have to go beyond the content of textbooks to offer meaningful instruction for this important domain. Edited by Anna 0. Graeber University of Maryland College Park, MD 20742 Kathleen Cramer and Tom Post teach at the University of Wisconsin, River Falls, WI 54022. Their research interests involve studying children's learning of, and teachers' conceptions of, rational number and proportionality concepts. The preparation of this article was supported in part by the National Science Foundation, NSF DPE 8470077, Any opinions, findings, conclusions, and recommendations. are those of the authors and do not necessarily reflect the views of the National Science Foundation. REFERENCES Cramer, Kathleen, Nadine Bezuk, and Merlyn Behr. "Proportional Relationships and Unit Rates." Mathematics Teacher 82 (October 1989):53744. Cramer, Kathleen, Thomas Post, and Merlyn Behr, "Interpreting Proportional Relationships." Mathematics, Teacher 82 (September 1989):44553. Cramer, Kathleen, Thomas Post, and Sara Currier. "Learning and Teaching Ratio and Proportion: Research Implications." In Research Ideas for the Classroom: Middle Grades Mathematics, edited by Douglas Owens, Reston, Va.: National Council of Teachers of Mathematics and Macmillan. In press. Hart, Kathleen. "Ratio and Proportion." In Children's Understanding of Mathematics 11 16, edited by Kathleen Hart, 88101. London: John Murray, 1981. Heller, Patricia, Thomas Post, Merlyn Behr, and Richard Lesh. "Qualitative and Numerical Reasoning about Fractions and Rates by Seventh and EighthGrade Students." Journal for Research in Mathematics Education 21 (November 1990): 388402. Karplus, Elizabeth, Robert Karplus, and Warren Wollman. "Intellectual Development Beyond Elementary School IV: Ratio, the Influence of Cognitive Style." School Science and Mathematics 74 (October 1974): 47682. National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics, Reston, Va.: The Council, 1989. Noelting, Gerald. "The Development of Proportional Reasoning and the Ratio Concept: Part 1the Differentiation of Stages." Educational Studies in Mathematics 11 (May 1980):21753. Post, Thomas, Merlyn Behr, and Richard Lesh. "Proportionality and the Development of Prealgebra Understandings." In The Ideas of Algebra, K12, 1988 Yearbook of the National Council of Teachers of Mathematics, edited by Arthur Coxford and Albert Shulte, 7890. Reston, Va.: The Council, 1988. 
