

CHAPTER 9 Learning
and Teaching Ratio and Proportion: Kathleen Cramer, Thomas Post, and Sarah Currier




The problem above was one of two problems given to these students as part of the introduction to teaching ratio and proportion ideas. The other problem involved a money exchange context: 3 U.S. dollars can be exchanged for 2 British pounds. How many pounds for 21 U.S. dollars? This is a proportional situation, and all the students correctly solved this problem using the traditional proportion algorithm. But no one could explain why this problem reflected a proportional situation while the running laps problem did not. Superficially the two problems are similar; each contains three pieces of information with one number unknown. That structure does lend itself to a proportion: a/b = c/x where a, b, c are the known quantities and x is the unknown quantity. The deeper differences between the two tasks highlight what is special about proportional situations. The next section will discuss these differences. At this point it is sufficient to conclude that we cannot define a proportional reasoner as simply one who knows how to set up and solve a proportion. The research on proportional reasoning helps expand our view of what it means to be a proportional reasoner. Having a better understanding of what proportional reasoning entails should influence our classroom instruction.




Mathematical Relationships in Proportional Situations The critical component of proportional situations is the multiplicative relationship that exists among the quantities that represent the situation. In the running laps problem, the relationship between the number of laps Sue ran and the number of laps Julie ran can be expressed through addition or subtraction: Sue's laps = Julie's laps + 6; Julie's laps = Sue's laps  6. In the money exchange problem the relationship between U.S. dollars and British pounds can be expressed through multiplication: pounds = 2/3 U.S. dollars or U.S. dollars = 3/2 British pounds. The running laps problem is not a proportional situation while the money exchange situation is proportional. Gerard Vergnaud is a French psychologist and mathematics educator who for many years has directed a center for research in Paris. Vergnaud (19) has used a model based on the concept of measure space to assist in clarifying the nature of the multiplicative relationships that exist in proportional situations. A measure space can be thought of in terms of physical magnitudes such as length, weight, money, or children. When tile type of physical magnitude is quantified as with 2 cm, 3 kg, or 5 dollars, these values are known as quantities. A proportion then can be viewed as a multiplicative relationship between the quantities in two measure spaces. Vergnaud's notation is as follows:




M 1 and M 2 represent any two measure spaces and a, b, c, and d are the quantities that form the rates in a proportion. For example, let us record the money exchange situation using the measure space notation: If 3 U.S. dollars can be exchanged for 2 British pounds, then at this rate 21 U.S. dollars can be exchanged for 14 British pounds. Translating to measure space notation we have the following:




In proportional situations the quantities between or across measure spaces (3:2, 21:14) always are related by multiplication. If we multiply the number of U.S. dollars by 2/3 we obtain the corresponding number of British pounds. If we extend the table using this multiplication rule, some of the other entries would include the following:




In each case the number of British pounds can be found by multiplying the number of U. S. dollars by 2/3 (6 X 2/3 = 4; 9 x 2/3 = 6; 12 x 2/3 = 8). This constant factor can be used to write an algebraic rule for this proportional situation: y = 2/3 x where y = British pounds and x = U. S. dollars. This constant factor also describes the number of pounds per one dollar: 2/3 pound per 1 dollar. This is called the unit rate. The graph for this situation is shown in Figure 9.1. Note that the line is straight, leans toward the right and passes through the origin. This is true of graphs of all proportional situations. Another interesting characteristic of proportional situations can be shown by recording the different rates as fractions. In the example above, the relationship of British Pounds to U.S. dollars can be expressed as 2/3, 4/6, 6/9, 8/12. These fractions are equivalent and reduce to the same number: 2/3. All rate pairs in proportional situations always reduce to the same fraction, the unit rate. A different multiplicative relationship exists among the elements within each measure space. Starting with 3 U.S. dollars for 2 British pounds, if we multiply the number of dollars by 2 and correspondingly multiply the number of pounds by 2 then we obtain another rate pair in the table: 6:4. If we triple the number of dollars we must triple the number of pounds. This rate pair, 9:6, is also in the table. Intuitively, we can easily see this number pattern going down the table (within a measure space). Note that the factor relating any two quantities within the same measure space is not a constant while the factor relating quantities between or across the measure spaces is a constant. A proportional relationship is just one type of relationship that can exist between two sets of quantities. It is a special class in which multiplication defines the relationship. Another example of a nonproportional situation highlights this point (see Fig. 9.2). A taxicab charges $1.00 plus 50 cents per kilometer. The cost for one kilometer is $1.50; the cost for 2 kilometers is $2.00. Though a relationship between cost and kilometers exists and can be written using a function rule (y = .50 x + 1.00, y = cost, x = kilometers), it is not proportional because both addition and multiplication define the relationship. The graph of this function does not go through the origin.


Methods of Assessing Proportional Reasoning Research has used students' achievements in solving missing value problems as an important index of their ability to think proportionally (8,13,14,15,17).




In a missing value problem three pieces of information are given. The task is to find the fourth or missing piece of information. The measure space notation can be used to describe missing value problems and the solution strategies presented in tile research. The wellknown "Tall Man Short Man" problem (7) has been a popular research task. In this problem subjects arc given a picture of a figure labeled Mr. Short and a chain of paper clips. They are told that Mr. Short's height is four buttons. After measuring the height of Mr. Short as six paper clips, the subjects were told that there was another figure not shown, a Mr. Tall, who was six buttons high. They were asked to find the height of Mr. Tall in paper clips. At this point the subjects had three pieces of information: tile height of Mr. Short in paper clips and in buttons, and the height of Mr. Tall in buttons. The missing information was the height of Mr. Tall in paper clips. The problem is depicted as follows:




To solve this problem the student can find the constant factor that relates the quantities between measure spaces. This is done by finding the number of paper clips that equal one button (6 paper clips ÷ 4 buttons = 3/2 paper clips per 1 button). Multiplying this factor, 3/2 paper clips per 1 button by the quantity 6 buttons solves the problem. As mentioned previously, finding the constant factor that describes the relationship across two measure spaces is the same as finding the unit rate or amount per one. When considering the function rule y = mx, m is not only the constant factor but also the unit rate. Finding the multiplicative relationship among elements within each measure space is another way of reaching a solution. This has been called a factor of change method. In the next problem the constant factor across measure spaces is a noninteger while the factor relating the numbers within the button measure space is an integer.




Students can reason that if the number of buttons is tripled, then the number of paper clips must also be tripled. Knowing that both within and between relationships exist offers students, alternative strategies. Research has shown that students often look for the whole number relationship and solve the task using this integer factor. When the multiplicative relationship between and within measure spaces are both noninteger, students have more difficulty. In this type of missing value problem incorrect additive strategies often occur.




Students often reason that x = 9 because 3 + 4 = 7 so 5 + 4 = 9 or because 3 + 2 = 5 so 7 + 2 = 9. This is all incorrect fallback strategy and is probably utilized because of all inability to deal with the noninteger relationships. This is an example of what Karplus called the "fraction avoidance syndrome" (8).




Understanding of proportional situations must eventually involve generalizing beyond the numerical aspects of the problem to the point that all solutions ore based Oil either the multiplicative relationship within or between measure spaces. Problems without integer relationships seem more difficult but in fact have the same mathematical structure as the integer problems and are solved by identical types of solution strategies. Missing value tasks are just one type of problem used in the research oil ratio and proportion. Numerical comparison problems requiring the comparison of two rates also have been used to assess proportional reasoning ability. Noelting's (13) orange juice problems are wellknown examples of this problem type. Figure 9.3 shows Noelting's problem types. Subjects were to determine the relative strength of orange juice given the number of glasses of juice and the corresponding number of glasses of water that were mixed. In each picture the orange juice is presented first. Students were to imagine that the liquid in each set of glasses was poured into a jug and mixed well; they were then asked to identify the jug whose liquid would have the stronger "orangey" taste. In the specific example shown, 3 parts Orange juice to 4 parts water is stronger than 2 parts orange juice to 3 parts water so the box under the stronger mixture is shaded. 

FIGURE 9.3 Noelting's Orange Juice Task (13, p. 219). Reprinted by permission of Kluwer Academic Publishers 

The numbers used in orange juice tasks were varied to include integer (whole number) and noninteger relationships. For example:




In this example, there is 4 times as much water as orange juice in each rate so the mixtures taste the same. Here there is an integer relationship between each measure space comparison. If the tabular entries are as follows,




the integer relationship would be within each measure space. As with missing value tasks, the numerical comparison tasks with rates having noninteger relationships were more difficult for students than rates with integer relationships. This body of research also showed that the comparison of two unit rates such as 1 to 3 and 1 to 5 made the tasks easier. Comparing unequal rates was more difficult than comparing equal ones. Qualitative prediction tasks and qualitative comparison tasks have also been used to assess proportional reasoning ability (4,15). These problems contain no numerical values but require the counterbalancing of variables in measure spaces. An example of a qualitative prediction task follows: If Devan ran fewer laps in more time than she did yesterday, would her running speed be (a) faster, (b) slower, (c) exactly the same, (d) not enough information to tell. The following is a qualitative comparison task: Mary ran more laps than Greg. Mary ran for less time than Greg. Who was the faster runner? (a) Mary, (b) Greg, (c) same, (d) not enough information to tell. The Rational Number Project (RNP) administered a survey of proportional reasoning tasks to over 900 seventh and eighthgrade students. Questions included missing value problems, numerical comparison problems, and two types of qualitative problems. Integer and noninteger relationships were tested. The study varied the contexts in which the problems were embedded while keeping the numerical aspects the same across contexts. The contexts included speed, buying, density, and scaling. The researchers predicted that the more familiar buying and speed contexts would be easier than the less familiar density and scaling contexts.
Students' Solution Strategies Information from the RNP study is shown in Table 9.1. The types of Correct solution strategies for the missing value and numerical comparison problems are shown as well as the percentage correct for each strategy. Although seventhgrade students had no prior instruction in the standard algorithm (crossmultiply and divide), eighth graders had received such instruction a few weeks prior to the survey. Note the much larger incidence of the algorithmic approach with missing value problems for eighth graders. Also note that eighthgrade students performed better on both the missing value and numerical comparison problems. Other results not depicted here show that there was almost no difference (5%) between the groups on the qualitative questions, an area for which neither group received specific instruction. The unit rate approach was the most popular strategy and one that was responsible for the largest number of correct answers (15). This approach is characterized by finding the multiplicative relationship between measure spaces. The unit rate is found through division. For example, if 3 apples cost 60 cents, find the cost of 6 apples. The cost for 1 apple is found by dividing: 60 cents ÷ 3 apples = 20 cents per apple. This unit rate is the constant factor that relates apples and cost. To find the cost of 6 apples, you simply multiply 6 apples by 20 cents per apple. This method was especially popular with seventhgrade students who were uninstructed in the usual crossmultiply and divide algorithm. This result should not be surprising. Children have made purchases and have had the opportunity to calculate unit prices and other unit rates. It seems a natural way to approach these problems. A student using the factor of change method might reason as follows: "If I want twice as many apples, then the cost will be twice as much." The factor of change method is a "How many times greater" approach and is equivalent to finding the multiplicative relationship within a measure space (15). A small number of seventhgrade students employed what we have called a fraction strategy. This strategy was used by a much larger percentage of eighthgrade students. The fraction strategy is applied devoid of problem context. That is, rate pairs would be treated as fractions by disregarding the labels. A student using this strategy would calculate the answer, applying the multiplication rule for generating equivalent fractions as follows:




The percentage of correct responses for the single item that did not use integral multiples was significantly lower than problems with integer relationships. In this problem, one number of a rate pair was 1. 5 times the other. Although this is still not a particularly difficult situation, differences in the results obtained were rather dramatic. Apparently the presence of a noninteger relationship does two things: first, it significantly decreases the level of student achievement and second, it actually changes the way in which students think about a problem. This is evidenced by the significantly lower percentage of students who used the unit rate and factor methods. As denoted by the last row in Table 9. 1, large percentages of seventh and eighth graders were unable to solve these problems correctly. Data not reported here suggest that scaling problems were significantly more difficult than the buying, speed, or density problems (4). 



Summary




Proportional reasoning involves an understanding of the mathematical relationships embedded in proportional situations. These relationships are always multiplicative in nature. Algebraically, proportional relationships can be expressed through a rule with the form of y = mx. This again emphasizes the multiplicative relationship inherent in all proportional situations. Graphically proportional situations are depicted by a straight line passing through the origin. Recalling an idea from algebra, the W in the equation y = mx designates the slope of the line. This slope m is also the unit rate and the constant factor that relates quantities between two measure spaces. All rate pairs for that situation appear on the line y = mx. Proportional reasoning also involves the ability to solve a variety of problem types. Missing value problems, numerical comparison problems, and two types of qualitative situations are among the types of problems that are important for children to understand. Proportional reasoning involves the ability to discriminate proportional from nonproportional situations. A proportional reasoner ultimately should not be influenced by context nor numerical complexity. That is, students should be able to overcome the effects of unfamiliar settings and cumbersome numbers. The understandings underlying proportional reasoning are complex. We should expect this type of reasoning to develop slowly over several years (18).




A Look at the Curriculum Proportional reasoning abilities are more involved than textbooks would suggest. Textbooks emphasize the development Of Procedural skills rather than conceptual understandings. This approach often encourages role learning and inhibits meaningful understanding of proportional tasks. The following example is typical of a grade 5 text.
To solve this problem type, ratios are identified and a proportion is set up. The equation 2/3 = n/27 is written and solved by noting that 3 X 9 = 27 and 2 X 9 = 18. Many questions arise: What does 18 mean? How is it related to 27? Why not multiply 2/3 by 27?




Why should students assume a multiplicative relationship between the numbers? In other words, why is this a proportional situation? Students need to see examples of proportional and nonproportional situations so they call determine what it is appropriate to use a multiplicative solution strategy. If side issues are not raised, how can students be expected to be aware of and understand them?




In the textbooks for grades 6  8, the traditional crossmultiply and divide algorithm is used to determine whether two rates are equal. This algorithm is also used to set up and solve missing value problems. To solve the apartment problem, for example, the following proportion is set up:




While the crossproduct algorithm is efficient, it has little meaning. In fact, it is impossible to explain why one would want to find the product of contrasting elements from two different rate pairs. The labels are normally excluded. What meaning does "2 students in apartments x 27 students" have? The crossproduct rule has no physical referent and therefore lacks meaning for students and for the rest of us as well (15). Teachers need to step outside the textbook and provide handson experiences with ratio and proportional situations. Initial activities should focus on the development of meaning, postponing efficient procedures until such understandings are internalized by students. The RNP developed such a curriculum and used it with seventh grade students.




Rational Number Project Teaching Experiment Since 1979 the Rational Number Project (RNP) has been conducting research on children's learning of rational number concepts (part whole, ratio, decimal, operator and quotient or indicated division) and proportional relationships. One of the teaching experiments conducted by the Rational Number Project dealt with seventhgrade students' learning of ratio and proportional concepts. A teaching experiment is a research model that focuses on the process of concept development rather than on achievement as measured on a written test of some sort. It is normally conducted with a small number of students, involves observation of the instructional process by persons other than the instructor, and controls the instruction by providing detailed lesson plans, activities, written tests, and student interviews. Data come primarily from student interviews. Instructional materials and interviews developed for these experiments reflected what has been gained from the research discussed in the previous sections of this chapter. We hope the description of the curriculum that follows will influence mathematics teachers in grades 5 through 8 to reconsider how they might approach this topic. The RNP lessons reflected the belief that proportional reasoning involved more than simply setting up and solving a proportion. The lessons emphasized the mathematical characteristics of proportional situations. To do this, students' initial experiences involved physical experiments with proportional and nonproportional situations in which students collected data, built tables, and determined the rule for relating the number pairs in the tables. Proportional situations were defined as those whose rule could be expressed in the form y = mx. Coordinate graphs were used to depict the data from these experiments showing that proportional situations had straightline graphs through the origin. The lessons addressed the issue of context and numerosity. Problem contexts evolved from familiar to less familiar ones. Early emphasis focused on problem situations with whole number multiples within and between the elements of the proportion. Students' understanding evolved to see that the strategy they used did not depend on the presence whole number multiples, Students had opportunities to solve a variety of problem types including missing value problems, quantitative comparison problems, and qualitative reasoning problems. Multiple strategies for solving problems involving proportional relationships were taught. These included building tables to see number patterns, a unitrate approach, a factor of change approach, and a fraction approach. The unitrate strategy was stressed initially because earlier survey results suggested that this interpretation was not only the most natural with students but also the solution strategy that resulted in the greatest percentage of correct responses. Relationships among the different approaches were emphasized. 



Since the standard algorithm (crossmultiply and divide) is not meaningful, its use was postponed until more meaningful, although less efficient, strategies were developed. Observations from this teaching experiment include the following:




The next section presents sample lessons that demonstrate the type of initial concrete experience possible for modeling the mathematical characteristics of proportional situations.
Teacher Notes for Sample Lessons These lessons represent the types of initial experiences with proportional reasoning that students should have. The activity pages found at the end of the chapter present students with proportional and nonproportional situations. For each problem they construct a table based on the particular situation, generalize a rule to describe those data, and then plot the data on coordinate axes. From the tables, students have the opportunity to see the different number patterns within and between measure spaces that are inherent in proportional situations. The graphing aspect of the lesson highlights the idea that the graphs of all proportional situations will form a straight line through the origin. By comparing the number patterns and graphs of proportional and nonproportional situations students can see what makes Proportional situations unique. These activities can be extended to highlight the relationship between the quantities across measure spaces and the unit Tate. For suggestions and for other lessons similar to the two presented here, readers are referred to the Mathematics Teacher (1, 3).




Materials. Red and green Cuisenaire rods, graph paper and activity sheets 1 and 2 for each student (see Figures 9.4 and 9.5). Teachers will need to make a sheet showing four stick figures 6 cm, 12 cm, 18 cm, and 24 cm in height. Label the figures Ms. Adams, Mr. Barton, Ms. Crane, and Mr. Dahl. Objectives.
Directions. For each of the activity sheets, tile teacher should introduce the problem) in a large group setting and discuss tile solution plan. Students complete the activity in small groups and discuss tile questions. The teacher should then discuss tile questions in a large group. Each activity may take more than one class period. For activity 1 (Fig. 9.4) students measure the figures using a 2cm red rod and a 3cm green rod. The rule developed finds the number of greens given the number of red rods: G = 2/3 R. A teacher can draw the figures to develop a rule with an integer relationship. The graph drawn is a straight line through the origin. For activity 2 (Fig. 9.5) the students should use play money to act out the relationship between kilometers and cost of the limousine. Tile rule developed finds tile cost given the number of kilometers: C = 50 + 2K. The graph is a straight line through (0, 50). Students compare the results of the two problems. The teacher should lead students to discover that proportional relationships, which form a special class of problems, always have a straight line graph going through tile origin. Only One of tile two problems represents a proportional situation. The teacher should also point out that tile rule describing the proportional relationship involves only multiplication or division, To reinforce these two ideas, students should be given an opportunity to do similar activities and asked to identify tile proportional situations. 



Looking Ahead . . . Curriculum to develop the deeper understanding of the mathematics taught in the Middle grades needs to be developed. Research on various middle school content areas supports file value of a conceptually oriented curriculum over a procedurally based curriculum. Since textbooks are generally procedurally oriented, textbookdominated programs should become less frequent. There is currently a mismatch between how and what teachers are encouraged to teach and what skills are evaluated. More valid ways to evaluate student outcomes must be found. Evaluating what students are thinking and how they solve problems is more important than a numerical score reflecting the number of items correct.




As teachers are encouraged to teach toward deeper understandings of tile content in the middle grades, we need to determine whether they have the necessary understandings of the content. Research with preservice and inservice elementary school teachers on rational numbers suggests that many teachers do not have this understanding (2, 10). We may be seeing mathematics specialists in the middle grades in the future. This is all idea supported by the National Council of Teachers of Mathematics in the Curriculum and Evaluation Standards (11) and its Professional Standards for Teaching (12). Kathleen Cramer and Thomas Post
Teachers who support the need for change in how ratio and proportion ideas should be taught may wonder how they call institute such changes in their classrooms. Questions arise that inhibit change: Am I not required to cover what is in the textbook? If I do something different in fifth grade, will my students have trouble in sixth grade? Will not students need to know the standard algorithm in fifth grade because it is tested on the standardized tests? Teachers hoping to change need the support of other teachers at their own grade level and of those at other grade levels. They also need the support of principals and parents.




We hope that teachers reading this chapter will be encouraged to use techniques involved in teaching experiments. Finding time to interview small numbers of students after instruction will be 2 worthwhile activity. Interviews are the major data source for research in mathematics education, and teachers will find interesting interview questions in the professional literature. Sarah Currier About the Authors Kathleen Cramer is all associate professor in the College of Education at the University of WisconsinRiver Falls. Dr. Cramer teaches mathematics methods classes for undergraduate and graduate students. She is currently working on a National Science Foundation (NSF) grant revising the curriculum from RNP teaching experiments. Thomas Post is a professor in mathematics education at the University of Minnesota. Besides teaching graduate and undergraduate courses in mathematics education, Dr. Post has been a codirector of the NSFsponsored Rational Number Project since 1979. Sarah Currier is on leave from her sixthgrade teaching responsibilities in the Minneapolis Public Schools. She is currently working oil a master's degree in mathematics education at the University of Minnesota. She particularly enjoys teaching geometry and incorporating cooperative learning techniques in her mathematics classes.
1. CRAMER, K., BEZUK, N., & BEHR, M. (1989). Proportional relationships and unit rates. Mathematics Teacher, 82(7), 537544. 2. CRAMER, K., & LESH, R. (1988). Rational number knowledge of preservice elementary education teachers. In M. Behr & C. Lacampagne (Eds.), Proceedings of the Tenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 425431). De Kalb, IL: Northern Illinois University. 3. CRAMER, K., POST, T, & BEHR, M. (1989). Interpreting proportional relationships. Mathematics Teacher, 82(6), 445453. *4. HELLER, P, POST, T, BEHR, M., & LESH, R. (1990). The effect of two context variables oil quantitative and numerical reasoning about rates. Journal for Research in Mathematics Education, 21 (5), 388402. 5. HIEBERT, J., & LEFEVRE, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case for mathematics (pp. 127). Hillsdale, NJ: Erlbaum. *6. HOFFER, A. (1988). Ratios and proportional thinking. In T. Post (Ed.), Teaching mathematics in grades K8: Research based methods (pp. 285313). Boston: Allyn & Bacon. 7. KARPLUS, R., KARPLUS, E., FORMISANO, M., & PAULSON, A. (1979). Proportional reasoning and control of variables in seven countries. In J. Lochhead & J. Clement (Eds.), Cognitive process instruction (pp. 47103). Philadelphia: The Franklin Institute Press. 8. KARPLUS, R., PULOS, S., & STAGE, E. (1983). Proportional reasoning of early adolescents. In R. Lesh & M. Landau (Eds.), Acquisition of mathematical concepts and processes (pp. 4589). New York: Academic Press. 9. KURTZ, B., & KARPLUS, R. (1979). Intellectual development beyond elementary school VII: Teaching for proportional reasoning. School Science and Mathematics, 79(5), 387  398. 10. LACAMPAGNE, C., POST, T, HAREL, C., & BEHR, M. (1988). A model for the development of leadership and the assessment of mathematical and pedagogical knowledge of middle school teachers. In M. Behr a, C. Lacampagne (Eds.), Proceedings of the Tenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 418424). De Kalb), IL: Northern Illinois University. 11. NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. 12. NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS. (1991). Professional standards for teaching Mathematics. Reston, VA: Author. 13. NOELTING, G. (1980), The development of proportional reasoning and the ratio concept. Part 1: The differentiation of stages. Educational Studies in Mathematics, 11 (3), (pp. 217253). 14. NOELTING, G. (1980) The development of proportional reasoning and the ratio concept. Part II  Problem structure at successive stages: Problemsolving strategies and the mechanism of adaptive restructuring. Educational Studies in Mathematics, 11 (3), 331363. 15. POST, T, BEHR, M., & LESH, R. (1988). Proportionality and the development of prealgebra Understanding. In A. Coxford (Ed.), Algebraic concepts in the curriculum K12 (1988 Yearbook, pp. 7890). Reston, VA: National Council of Teachers of Mathematics. 16. POST, T * , CRAMER, K., BEHR, M., LESH, R., & HAREL, G. (in press). Curriculum implications from research on the learning, teaching and assessing of rational number concepts: Multiple research perspectives. In T Carpenter & E. Fennema (Eds.), Learning, teaching and assessing rational number concepts: Multiple research perspectives. Madison: University of Wisconsin. 17. POST, T, & CRAMER, K. (1989). Knowledge, representation and quantitative thinking. In M. Reynolds (Ed.), Knowledge base for beginning teachers (pp. 221232). Elmsford, NY. Pergamon Press. 18. TOURNAIRE, F., & PULOS, S. (1985). Proportional reasoning: A review of the literature. Educational Studies in Mathematics, 16(2), 181204. 19. VERGNAUD, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 127174). New York: Academic Press. 

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