QUALITATIVE AND NUMERICAL REASONING ABOUT FRACTIONS AND RATES BY SEVENTH AND EIGHTH-GRADE STUDENTS
PATRICIA M. HELLER, University of Minnesota
THOMAS R. POST, University of Minnesota
MERLYN BEHR, Northern Illinois University
RICHARD LESH, Educational Testing Service
There are conceptual as well as procedural linkages between rational number concepts and basic direct proportion situations. Kieren (1977) suggested that the main of rational number concepts consists of interrelated constructs that must understood singly and in concert if an individual is to "understand" rational number. These constructs are part-whole, ratio, decimal, measure, and multiplicative operator. Recently Kieren (1989) identified three processes (recursion, abstraction, and language acquisition) involved in understanding the multiplicative relationships that underlie the entire domain of rational numbers.
Vergnaud (1983, 1988) suggested that the domain be expanded to include multiplication, division, and proportionality. He referred to the new entity as the domain multiplicative structures. Several research teams, including the Rational Number Project (1990), are currently attempting to more precisely define this domain the interrelationships within it. As with other major mathematical structures, students' understanding of the constructs within this domain evolves slowly over extended period of time (Tourniaire & Pulos, 1985). Indications are that many concepts within the domain of multiplicative structures are not well taught nor are they well learned. For example, numerous studies have shown that early adolescents and many adults have a great deal of difficulty with the basic concepts of fraction, rates, and proportion and with problems involving these concepts (Behr, 1987; Hart, 1978; Karplus, Karplus, Formisano, & Paulson, 1979; Kieren & Southwell, 1979; Noelting, 1980a; Vergnaud, 1983). Studies of students' thinking strategies indicate that they often use incorrect and inappropriate qualitative reasoning in solving problems involving these concepts or use additive approaches where multiplicative approaches are required (Hart, 198 1; Karplus, Pulos, & Stage, 1983b; Noelting, 1980b). Students have particular difficulty adjusting to this new domain, which is multiplicatively rather than additively based. It is important to find ways to improve students' understanding within this domain, since it is the foundation of all that is to come.
Rate, Fraction and Proportion-Related Problems
In this paper the term rate is used to denote a comparison between elements in two different measure spaces (e.g., 3 laps/12 minutes). Fraction denotes a context-free situation whose elements are integers (e.g., 3/12). Two common problem situations have been studied in which students must apply their basic concepts of fraction and rates. In numerical-comparison problems, students judge the equality or inequality of the two given fractions or rates (which rate is larger, a/b or c/d, or are they equal). In missing-value problems, three components of two equal fractions or rates are given and the student solves for the fourth component (a/b = x/d, where the position of the unknown x may vary). These two problem situations can be presented as numerical exercises free of any context or as word problems that include a context or story line. There are well-developed algorithms for comparing the size of two fractions or generating sets of equivalent fractions. When a context is present, however, there are the extra steps of interpreting the problem and generating a physical and mathematical representation of the problem. We will refer to these two types of word problems as proportion-related problems.
In this study we investigated two factors that could influence students' performance on proportion-related problems: qualitative reasoning about rates and the extent to which students use their rational number skills to solve the problems. These factors and the associated research questions are described below.
Qualitative Reasoning About Rates
One factor that could affect students' performance on proportion-related problems is their skill in qualitatively analyzing the problem before proceeding to actual calculations and the generation of a numerical answer. Qualitative analysis figures significantly in physics problem-solving performance (Larkin & Reif, 1979; Larkin, McDermott, Simon, & Simon, 1980). The qualitative analysis of physics problems is complex and involves several types of reasoning (Chi, Feltovich, & Glaser, 1981; Chi, Glaser, & Rees, 1981). Nevertheless, the analysis usually includes inferring the direction of change that a particular interaction will produce (e.g., if I put the second box on top of the first box, then the normal force will increase, so the frictional force will also increase). Similarly, in simple proportion-related problems, qualitative analysis may involve inferring in what direction the value or intensive quantity of a rate will change (decrease, stay the same, or increase) when the numerator and/or the denominator of the rate increases, stays the same, or decreases. The rate could be speed, package size, unit price, the concentration of a mixture, or any of the other types of rates commonly encountered in proportion-related problems.
The nine ways that changes in the numerator and/or the denominator of a rate can affect the value of the rate is shown in Table 1. Two of these cases are ambiguous, when the numerator and the denominator both decrease or both increase (e.g., if I run more laps in more time, does my speed decrease, stay the same, increase, or can't I tell?). In such cases, the numerator and denominator can decrease (or increase) proportionally or nonproportionally, so the correct answer to the question "What happens to the value of the rate?" is that there is not enough information to decide.
|In this study we investigated
a new type of qualitative question about rates that may be important in
understanding the development of proportional reasoning skills in adolescents
(Heller, Post, Behr, & Lesh, 1989). These questions, which we call directional
questions, ask students to determine the qualitative direction of change
in the value of a fraction or rate, given specified qualitative changes
in the numerator and/or the denominator. Although this type of question
is not -generally included in the mathematics curriculum, directional reasoning
about rates may be an important prerequisite skill for successful performance
in numerical, proportion-related problems.
Like the missing-value and numerical-comparison problem situations, directional questions can be presented as fraction exercises free of any context or as word problems that include a context or story line, as illustrated in Table 2. When a context is included, two types of directional questions can be asked. When the qualitative change in the numerator and/or denominator refers to different events in time, we will call these qualitative-rate-change questions. When the qualitative change in the numerator and/or denominator refers to different objects or people, we will call these qualitative-comparison questions. If students take advantage of the structural similarities in the three types of directional questions (context-free, qualitative-rate-change, and qualitative-comparison), then they should measure substantially the same skills.
Rational Number Skills
A second factor that could influence students' performance on proportion-related problems is the extent to which students use their rational number skills in solving these problems. Once a mathematical representation has been generated, the procedures used to find a fraction B that is equivalent to a fraction A given one of B's terms (i.e., 8/24 = [ ] /6) are identical to the procedures used to solve a missing-value rate problem where one complete rate pair and one element of the second rate pair are given. Likewise, determining the order relation between two fractions is structurally similar to determining the order relation between two rates as required in numerical-comparison problems. If students are capitalizing on the structurally similar nature of context-free fraction exercises and proportion-related problems, then they should measure substantially the same skill.
In this study, the following questions were posed:
Two tests were designed for this study, a word-problem test and a fraction test. Four versions of the word-problem test were constructed, each version using a single rate type: speed (running laps and driving tractors), mixing (lemonade and paint), linear density (standing in movie lines and hammering nails into a board), or scaling (making a classroom map and reading city maps). Each version of the 17-itern test consisted of four qualitative-rate-change questions, four qualitative comparison questions, four missing-value problems, and four numerical-comparison problems. A distractor addition problem was included to check the validity of comparisons and conclusions. All items were scored as right or wrong. To investigate the relationship between performance on directional questions and performance on the numerical problems, two scales were formed, a directional scale and a numerical scale.
The directional scale of the word-problem test consisted of the number correct on the qualitative-rate-change and qualitative-comparison questions on the test. For each question type we selected only four of the nine possibilities shown in Table I because we were concerned about the limited time and student fatigue in taking the tests. The questions selected had resulted in the greatest difference in difficulty in a previous pilot study. The Cronbach alpha reliability of the directional scale was 0.72 for seventh grade and 0.74 for eighth grade.
The numerical scale of the word-problem test consisted of the number correct on the missing-value and numerical-comparison problems on the test. Because the number structure of proportion problems influences performance (Hart, 1981; Karplus, Pulos, & Stage, 1983a; Karplus, Pulos, & Stage, 1983b; Noelting, 1980b; Rupley, 1981), we limited our investigation to easy numerical rates. Three of the missing-value problems and three of the numerical-comparison problems involved integer multiples both within and between rate pairs (e.g., 8/24 = 2/6); the fourth problem of each type used the simple noninteger relationship of 2:3 (4/6 = 6/9). In two of the comparison problems the rates were equal, and in two problems the rates were unequal. The missing-value and numerical-comparison problems were alternated to alleviate the tedium of solving the same kind of problem one after another. The Cronbach alpha reliability of this scale was 0.72 for seventh grade and 0.77 for eighth grade.
The 24-item fraction test included 11 exercises that paralleled items on the word problem test, as illustrated in Table 2. Of the parallel items, 3 directional exercises used the same direction of change in the numerator and denominator of a fraction as four of the directional questions on the word-problem test. Four exercises requiring students to calculate the missing component in two equivalent fractions (e.g., 4/20 = 12/[ ]) were numerically identical to the missing-value problems on the word-problem test. Similarly, 4 exercises requiring the determination of the order or equivalence of two fractions (e.g., 8/32 ? 2/10) used the same numbers as the numerical-comparison problems on the word-problem test.
Subjects were 467 seventh graders (16 classes and 522 eighth graders (19 classes) in a middle-class suburban junior high school in Minnesota. They included all students in attendance on the day the two tests were administered. About half of each group were girls and about half were boys. The seventh-grade students had not received instruction in proportion problems in their mathematics classes; the eighth-grade students had received textbook instruction in the standard cross-multiplication algorithm 6 to 8 weeks prior to the study. Seventy-seven students who did not respond to half or more of the items on one of the tests were dropped from the sample, leaving 421 seventh graders and 492 eighth graders.
The students completed both the fraction test and the word-problem test in their scheduled 45-minute mathematics class period. One half of the students in each class completed the fi-action test first; the other half completed the word-problem test first. The four different versions of the word-problem test were randomly assigned to the students in each class.
The four research questions were analyzed as follows:
For each factor analysis the scree test (Cattell, 1966) was used to determine the number of factors. Factors with eigenvalues less than 1 were excluded from the analysis. The factors were then rotated using the varimax procedure. Only items with factor loadings larger than 0.40 were considered for inclusion in a factor. The correlations of items within and between factors were then examined to verify that items on different factors could be interpreted as measuring different skills. To be considered distinct skills, the item correlations within factors had to be substantially larger than the item correlations between factors (which ideally should be close to zero). Because of die large sample size, item correlations larger than .09 for the seventh grade and .08 for the eighth grade were significant (p ² .05). Finally, separate analyses were carried out for the seventh and eighth grades to check the validity and stability of the factor structures and interpretations.
Analysis of Directional Items
The factor analyses of the directional questions (eight from the word-problem test and three from the fraction test) resulted in three factors, as shown in Table 3. The factor structure was stable across grade levels, with the same items loading on each of the three factors. The first factor consisted of the qualitative-rate-change and qualitative-comparison items from the word-problem test, except for the decreasing/decreasing and increasing/increasing items. The second factor consisted of the three decreasing/decreasing and increasing/increasing items from the fraction and the word-problem test. The third factor consisted of the remaining two fraction exercises. As indicated in Table 4, the item correlations within factors were consistently larger than the correlations between factors.
Performance on Directional Questions
The directional scale was of medium difficulty for the seventh- and eighth-grade students; overall, they answered about five of the eight questions correctly (seventh grade: M = 4.67, SD = 2.04; eighth grade: M = 5.07, SD = 2.05). Eighth-grade students performed significantly better on the directional scale than did the seventh-grade students, t(911) = 2.95, p < .01. The decreasing/decreasing and increasing/increasing questions were the most difficult items on both the word-problem and fraction tests; only about one-fifth of the seventh-grade students and one fourth of the eighth-grade students answered these questions correctly. A significantly larger proportion of eighth graders (28.5%) than seventh graders (21.4%) answered the decreasing/decreasing word problem correctly, c2(1 , N = 913) = 5.66, p < .05. There was no significant difference in the proportion of eighth graders (26.8%) and seventh graders (21.9%) who answered the increasing/increasing word problem correctly, c2(1, N = 913) = 2.77, p = .10. A significantly larger proportion of eighth graders (18.5%) than seventh graders (13.1 %) answered the increasing/increasing fraction question correctly, c2(1, N = 913) = 4.59, p < .05. The most frequent answer given by seventh and eighth graders was that the fraction or rate remains the same when the numerator and denominator of the fraction or rate both decrease (or increase).
Relation Between Directional Questions and Numerical Proportion-Related Problems
Pooled across all contexts, the observed correlation between the directional and numerical scales was .38 for seventh graders and .45 for eighth graders. On the basis of the scales' reliabilities, a correction for attenuation yields an estimated true correlation of .52 for seventh graders (27% of their variation in common) and .59 for eighth graders (35% of their variation in common). The regression analysis of the scatter plots of numerical versus directional scale scores revealed no significant exponential terms - a linear relationship was the best fit to the data. As shown in Figure 1, a stable relationship exists across grade levels. The slopes of the regression equations are about the same (0.41 for seventh grade, 0.50 for eighth grade), and neither graph goes through the origin.
|Figure 1. Graph of numerical scale means versus directional scale means for seventh- and eighth-grade students, showing the regression lines. The error bars above and below the mean represent one standard deviation.|
Fraction Exercises and Proportion-Related Problems
The eighth-grade factor analysis of the eight missing-value items (four from the word-problem test and four from the fraction test) resulted in only one factor; all items loaded on this factor. The seventh-grade factor analysis was ambiguous - one or two factors could be extracted. If two factors were rotated, the fraction items loaded on the first factor and the proportion items loaded on the second factor. However, as indicated in Table 5, the item correlations within factors were not sufficiently larger than the item correlations between factors to justify two distinct response patterns to the different problems. Moreover, all items from both tests loaded on the first unrotated factor. We concluded that the missing-value items are best conceptualized as loading on a single factor. The average of the between-test correlations was .26 for numerically equivalent items and .23 for the other items.
The seventh- and eighth-grade factor analyses of the eight numerical-comparison items (four from the fraction test and four from the word-problem test) resulted in identical factors, as shown in Table 6. The first factor consisted of the equal-fraction and equal-rate items from both tests. The second factor consisted of the two unequal-rate problems from the word-problem test, and the third factor consisted of the two unequal-fraction exercises. As indicated in Table 7, the correlations within items on the same factors were consistently larger than the correlations between items on different factors. The average correlation between equal-rate and equal-fraction items was .29 for numerically equivalent items and .25 for the other items. There were no significant correlations between unequal-fraction and unequal-rate items, even for the numerically equivalent items.
The directional questions on the word-problem and fraction test were novel for students. Not depicted in the school curricula, they assess transfer rather than directly taught reasoning abilities. Our results suggest that there is very little difference in achievement between seventh and eighth graders on the directional scale. Moreover, the identical factor structures for the directional items suggest that seventh- and eighth-grade students have similar response patterns to these questions.
Of the nine possible ways in which changes can occur in the value of a fraction or rate, only two, the increasing/increasing and decreasing/decreasing items, require a functional understanding of proportionality. That is, they require the student to recognize that both the numerator and denominator of a fraction or rate can increase (or decrease) proportionally or nonproportionally. Thus, it is not surprising that these questions appear to measure a different skill than the other directional items, as shown by the factor analyses and the large number of insignificant correlations with the other items. If, as we believe, these items assess students' functional understanding of proportionality, then our results indicate that only about one fifth of the seventh-grade and one fourth of the eighth-grade students have this understanding. Although it is possible that students are reluctant to choose an item distractor "not enough information to tell," it is more likely that these questions assessed abilities that most students do not possess. Apparently the standard instruction eighth graders receive in proportion problems does not prepare them to recognize that both the numerator and denominator of a rate can decrease (or increase) proportionally or nonproportionally. This functional aspect of proportionality is not being adequately addressed by the standard, procedurally driven mathematics curriculum.
On the word-problem test, the factor analyses suggest that the remaining qualitative-rate and qualitative-comparison questions measure a similar skill. That is, deciding what happens to a rate when the numerator and/or denominator changes in time may not be a substantially different skill from deciding which of two rates is smaller given qualitative differences in the numerators and/or denominators, as long as the changes are not in the same direction. The fact that the word and fraction items load on different factors indicates that they may measure different skills. Deciding what happens to the value of a given fraction when the numerator and/or denominator changes does not appear to be substantially related to deciding what happens to a rate (e.g., speed of runner, taste of lemonade, spacing between people in a movie line, or shade of paint mixture) in a word problem. The presence of a context may evoke different reasoning processes.
Another issue in this study was the nature of the relationship between the directional scale and the numerical scale. The substantial correlation between the directional and numerical scales suggests that the skills required for the two scales are not identical, although there is some similarity. The regression graphs do not go through the origin, indicating that numerical problems can be solved fairly well by students who have low directional scores, but a high directional score assures greater numerical success. Logically, it would seem that proportional reasoning in a numerical problem situation would not be possible without knowing at least the direction of change in a rate when the numerator and/or denominator decreases, stays the same, or increases. It is quite possible, however, for students to employ memorized procedures without understanding. This may explain why students with low directional scores are successful on two to three of the proportion-related problems.
The final issue in this study was the relationship between context-free and context-laden embodiments of structurally similar and numerically identical problems. The process of completing a second fraction equivalent to the first, given three of the four terms, is mathematically identical to finding a solution to a missing-value word problem, given one complete rate pair and half of the second. In a like manner, determining the order relationship between two fractions is mathematically similar to determining the order relationship between two rate pairs embedded in numerical-comparison word problems. If students are taking advantage of these structural similarities, then the fraction and word items for each type of problem should measure substantially the same skills.
The factor analyses indicate that calculating the missing numerator or denominator in equivalent-fraction exercises is not a substantially different skill from calculating the missing value in a word problem. Similarly, deciding that two given fractions are equal is not a substantially different skill from deciding that two rates are equal in word problems. However, deciding which of two given fractions is smaller does not appear to be related to the skill of deciding which of two rates is smaller in word problems. Since unequal-fraction exercises and unequal-rate word problems have the same level of difficulty, students must use different reasoning processes in solving the two types of problems.
Given the structural similarities of the problems, it is surprising that the item correlations are not higher, even for the items that appear to measure similar skills. 'Me percentage of variance accounted for by the common trait assessed by parallel fraction and word items was very low (r2 range of .03 to .14 for the missing value items and .04 to .10 for equal-comparison items). Moreover, the correlations between the numerically identical items on the two tests were not substantially higher than the correlations with the other items. It appears that students are not capitalizing on the structural similarities involved, even when the numerical quantities are identical.
We have argued elsewhere (Lesh, Post, & Behr, 1988) that proportional reasoning is both the capstone of the middle school mathematics program and the cornerstone of all that is to follow. The results of this study support the view that proportional reasoning is a multifaceted construct that encompasses more than the ability to solve missing-value and numerical-comparison problems. We believe that numerical ability, although important, is a necessary but not a sufficient condition for proportional reasoning. A lack of student numerical facility is as much a commentary on the lack of appropriate instruction in the area of rational number concepts (part-whole, decimal, rate, measure, operator, etc.) as it is a reflection of the attainment of proportional reasoning ability. Contextual issues and directional reasoning situations of the type introduced here are two additional variables that should be included in any analysis that purports to define this most important intellectual capability.
We have also argued that proportional reasoning in the larger sense involves recognition of the invariance of certain types of transformations on mathematical objects. These transformations may be qualitative or quantitative. In the case of missing-value and numerical-comparison problems, these transformations involve a sense of covariation in ways that are precisely predictable on the basis of the multiplicative relationships within and between corresponding pairs of numbers (rate pairs) in the problem condition.
The indication that students are not capitalizing on the structurally similar nature of fraction-equivalence and missing-value problem situations and of fraction ordering and numerical-comparison situations suggests a lack of instructional parsimony. Rational number skills, once developed, are not being used or applied to areas of obvious application. Because the rational number skills appear first in the school curriculum, they could be used as the computational framework within which proportional relationships could be investigated.
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PATRICIA M. HELLER, Assistant Professor, Department of Curriculum and Instruction, University of Minnesota, Minneapolis, MN 55455
THOMAS R. POST, Professor, Department of Curriculum and Instruction, University of Minnesota, Minneapolis, MN 55455
MERLYN BEHR, Professor, Department of Mathematical Science, Northern Illinois University, DeKalb, IL 60115
RICHARD LESH, Senior Research Scientist, Educational Testing Service, Research Management Division, Princeton, NJ 08541
This research was supported in part by the National Science Foundation under grant No. DPE-947077 (Rational Number Project). Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the National Science Foundation.