Effect of Two Context Variables on Qualitative and
M. Heller and Thomas R. Post
Numerous studies show that early adolescents and many adults have a great deal of difficulty solving problems that involve rates and proportions (for example, Behr, 1987; Hart, 1978, 1981; Karplus, Karplus, Formisano & Paulson, 1979; Kieren and Southwell, 1979; Vergnaud, 1983). Two common problem types have been studied. In numerical-comparison problems, students judge the equality or inequality of the two given rates (which rate is larger, a/b or c/d, or are they equal). In missing-value problems, three components of two equal rates are given and the student solves for the fourth component (a/b = x/d, where the position of the unknown x may vary). We will refer to these two types of problems as proportion problems.
Recently, directional questions about rates have also been investigated (Heller, Ahlgren, Post, Behr, & Lesh, 1989; Heller, Post, Behr & Lesh, 1990; Larson, Behr, Harel, Post, & Lesh, 1989). Directional questions ask how the value of a rate will change (decrease, stay the same, or increase) when the only information given is that the numerator and/or the denominator of the rate decreases, stays the same, or increases (e.g., If I run more laps in less time, will my speed increase, stay the same, decrease, or can't I tell). Of the nine possible combinations of changes which can occur in the value of the numerator and denominator, only two cases, increasing/increasing and decreasing/decreasing, require a quantitative strategy to determine proportionality. That is, these cases require the student to recognize that the value of the rate will stay the same only if the numerator and denominator increase (or decrease) by the same multiplicative factor. If the only information given is that both the numerator and denominator increase (or decrease), then the correct answer to the question "What happens to the value of the rate?" is that there is not enough information to decide.
Most proportional reasoning studies vary the type of problems used, but keep the same context (the same "story"). A few studies, however, indicate that the problem context influences the problem difficulty (Heller et al., 1989; Jesunathadas & Saunders, 1985; Karplus, Pulos & Stage, 1983b; Lybeck, 1978; Vergnaud, 1980). In their review of the proportional-reasoning research, Tourniaire and Pulos (1985) identified four context variables that influence performance on proportion problems: the use of manipulatives, the presence of a mixture, the presence of continuous quantities, and the familiarity of the context.
Familiarity with the Problem Context
One purpose of this study was to define more clearly the relationship between familiarity with the problem context and seventh and eighth grade students' achievement on both directional questions and proportion problems. To this end, we investigated two aspects of the problem context. The first aspect is the type of rate involved in the problem. Table 1 shows nine types of rates that are found in proportion problems which appear in standard textbooks. Students may be more or less familiar with these different rate types. For example, junior high school students typically have little experience with consumption rates, even with familiar measures such as gallons and hours.
For each rate type, a second pair of context aspects involves the specific objects of the problem and the units typically used to measure the relevant attributes of these objects. We called this pair of context aspects the problem setting. Even with familiar rate types, students may be more familiar with some settings than' with others. For example, most junior high school students have measured teaspoons of lemonade concentrate and ounces of water to mix a lemonade drink. They have not, however, measured grams of acid and liters of distilled water to mix a dilute acidic solution.
The results of an earlier pilot study (Heller et al., 1989) indicate that familiarity with what is called the problem context consists of familiarity with both the rate type and the setting. One purpose of this study was to gain knowledge of the hierarchy of difficulty of proportion problems with different rate types and settings for uninstructed students. This knowledge may contribute to a better understanding of how proportional reasoning skills develop in adolescents and to the design of better instruction for students.
Context and Rational Number Ability
Vergnaud (1983, 1988) suggested that it would be misleading to separate studies of multiplication, division, fraction, ratios, rational number, proportions, n-linear functions, dimensional analysis and vector space, since these constructs are not mathematically independent. Consideration of this larger domain, which Vergnaud called multiplicative structures, has captured the interest of researchers during the past twelve years. A second purpose of this study was to explore the relationship between students' performance on directional questions and proportion problems, and their performance on context free rational number exercises.
We first examined
the relationship between students' general rational number ability and
their performance on directional questions and proportion problems. Do
students with higher rational number abilities perform better than students
with lower rational number abilities? A second type of relationship investigated
in this study involved the operational similarities in solving three different
types of context-free and context-laden problems, as illustrated in
Although there are operational similarities in the three problem types with and without a contextual framework, it is not clear that students will be able to exploit such relationships. Indeed, we originally hypothesized that they would not, because our analysis of standard mathematics curricula indicated that the operational similarities in the problem types are not emphasized. Consequently, we hypothesized that context would have a differential impact on students' performance. In particular, we reasoned that the presence of a context should play a detrimental role in missing-value problem types, a neutral role in numerical-comparison situations, and quite frankly we did not know what to expect in directional situations, since both context-free and context-laden directional questions would be new to students. If students' had prior experiences with the various contexts then the presence of a familiar context should be helpful when responding to directional questions. These issues win be discussed more fully in a later section of this paper.
The purpose of this study was to investigate how different rate types and settings affect seventh and eighth grade students' performance on directional questions about rates and numerical proportion problems. In addition, we investigated the relationship between students' performance on these problems and performance on context free rational number exercises. The following questions were posed:
Four types of rates were examined in this study, speed, mixture, scaling, and linear density. These rate types were chosen because (a) we expected students to have different degrees of familiarity with these rate types and (b) they are used extensively in junior and high school science courses in highly unfamiliar settings (e.g., speed in m/sec, molar concentrations in moles/liter, using scaling to determine astronomical distances, and density in g/cm3). Two of the rate types, speed and mixture, have been studied previously (Heller et al., 1989; Karplus et al., 1983b; Noelting, 1980a,b; Vergnaud, 1983). We expected speed problems to be slightly more difficult than mixture and density problems, and scaling problems to be the most difficult of the four rate types.
Table 3 shows the two problem settings selected for each rate type. The differences in the settings were deliberately kept small. We already knew that we could make the problems difficult by going to highly unfamiliar settings and by increasing the numerical complexity of the problems. We wanted to look for more subtle effects that might be important in future investigations and in designing instruction. If small setting differences made little difference in student performance, then subsequent studies could be somewhat simplified in design.
Two tests were designed for this study, a context test and a rational number test. Eight versions of the context test were constructed, each version using a single context (one of two settings for each of four rate types). Each version of the 17-itern test consisted of eight proportion problems and eight directional questions about rates. 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 hypotheses of this study, two scales were formed, a numerical scale and a directional scale.
The numerical scale of the context test consisted of four missing value problems and four numerical-comparison problems, as illustrated in Table 2. To help students understand the problem situations, an illustrative drawing was included on the front page of each version of the context test. The missing-value and comparison problems were alternated to alleviate the tedium of solving the same kind of problem one after another. There were two problems on a page with ample space for answering.
Since we were interested in the effect of context variables on problem solving performance, we did not want to add a numerical difficulty interaction effect. Consequently, we limited our investigation to problems with 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., 3/15 = 9/45); the fourth problem of each type used the simple non-integer relationship of 2:3 (e.g., 6/9 8/12). In two of the comparison problems the rates were equal, and in two problems the rates were unequal. The Cronbach alpha reliability of the numerical scale was .77.
Two types of directional questions about rates, qualitative-rate change and qualitative-comparison questions, were developed for the directional scale of the context test, as illustrated on Table 2. The qualitative change in the numerator and denominator of the rate refers to different events in time for the qualitative -rate -change questions, and to different objects or people for the qualitative-comparison questions. Both the numerator and denominator of a rate can decrease, remain the same, or increase. Therefore, there are nine qualitative - rate-change and nine qualitative-comparison questions that could be asked. Because we were concerned with student fatigue and the limited time to take two tests, for each question type we selected only four of the nine possible questions. 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 .73.
Finally, a 24-item rational number test was constructed for this study. Thirteen of the items were designed to measure students' ability to operate with fractions (e.g., 1/2 + 1/3 their quantitative notion of a fraction (e.g., write a fraction greater than 2/4 and less than 3/4), and their concept of a whole or unit (e.g., if * * * * * * is 3/2 of a unit, how many are in the whole unit). The remaining eleven items paralleled items on the context test, as illustrated in Table 2. Three qualitative questions asked what effect a numerator and/or denominator change had on the value of a fraction. Four equivalent fraction problems and four fraction ordering problems used the same numbers as the missing-value and numerical comparison problems in the context test. The Cronbach alpha reliability of the rational number test was .82.
Our subjects were 466 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 tests were administered. About half of each group were girls and about half were boys. The teachers reported that the seventh grade students had not received instruction on proportional-related problems in their mathematics classes. The eighth grade students had all received some instruction related to missing-value problems. This instruction generally involved the standard cross-multiplication algorithm and was procedurally rather than conceptually oriented. 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 eight graders.
The students completed both the rational number test and the context test in their scheduled 45-minute mathematics class period. One half of the students in each class completed the rational number test first; the other half completed the context test first. The eight different versions of the context test were randomly distributed to the students in each class.
The three research questions were analyzed as follows:
Results and Discussion
One issue in this study was the effect of the context variables, rate type and setting, on students' performance on the directional and numerical scales. Table 4 shows the mean scores for the students taking the eight context versions of these scales. The numerical scale was of medium difficulty for seventh grade students; averaged across all contexts, they solved about four of the eight problems correctly (M = 3.99, SD = 2.24). The eighth grade students performed better; they solved about five of the eight problems correctly (M = 5.38, SD = 2.28) The directional scale was also of medium difficulty for both seventh and eighth grade students; averaged across all contexts, 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).
Rate type showed a
significant main effect on both the directional scale, F(3,865) = 143.84,
The second context
variable, problem setting, also showed a small but significant main effect
on both directional scale, F(1,865) = 32.03, p < .001, and numerical
scale, F(1,865) = 5.84,
In addition, there was a small but significant interaction of problem setting and rate type for the directional scale, F(3,865) = 5.52, p < .001. The nature of the interaction is illustrated in Figure 1. For the density and mixture rate types, problem setting makes little difference in the directional scores. For speed and scaling, however, the setting has a significant effect on the directional scores. Driving cars is a more difficult speed problem setting than running laps (a mean difference of 1.21 out of 8 problems, or 15%), and drawing scale maps of the classroom is a more difficult problem setting than reading city road maps (a mean difference of 0.95 out of 8 problems, or 12%). Perhaps the stronger context effects on the directional scale allowed this interaction to appear. A similar, although not significant interaction appears for the numerical scale, as shown in Figure 2.
Grade in school showed a significant main effect only on the numerical scale, F(1,865) = 25.70, p < .001. As expected, the eighth graders, who had all received some instruction in solving missing-value problems, performed better than the uninstructed seventh graders (a mean grade difference of 1.39 out of 8 problems, or 17%). There were no significant grade interactions with either context effect. The problem-setting and rate-type effects that were observed were therefore comparable for students in both grades, as can be seen in Figures 2 and 3.
Rational Number Effects
Another issue in this
study was the effect of students' rational number ability on their performance
on the directional and numerical scales. Table 5 shows the mean scores
of students with low, medium, and high rational number ability on the
four rate-type versions of these scales. The trichotomized rational number
factor had a significant main effect on both directional scale, F(2,865)
= 76.32, p < .001, and the numerical scale, F(2,865) = 210.65,
There was also a small but significant interaction of rational number ability level and rate type for the numerical scale, F(6,865) 2.40, p = .026, indicating that the rate-type effects are not comparable for students with different levels of understanding of rational numbers. The nature of this interaction is shown in Figure 3. The order of difficulty of the rate types is different for different levels of rational number ability -- the density and mixture rates are increasingly less difficult compared to speed and scaling rates as rational number ability increases. There was no significant interaction of rational number ability with problem setting, indicating that the observed problem setting effect is comparable for students with different levels of understanding of rational numbers.
Performance on Equivalent Rational Number and Context Items
The final issue considered in this study was whether the presence of a context influences students' performance on equivalent missing value, numerical-comparison and directional problems. The graphs in Figure 4 contrast individual items which are numerically identical and/or structurally similar, one with a contextual framework and one without. Each data point depicts the percentage of students who solved correctly a given rational number problem and the percentage of students who solved correctly the corresponding missing-value problem (Figure 4a), numerical-comparison problem (Figure 4b), and directional question (Figure 4c). An entry below the diagonal (y = x), therefore depicts a situation where the context hinders performance; above y = x context helps performance.
The graphs show a distinct data pattern for each problem type. For the missing-value problem type (Figure 4a), the data points are generally below the diagonal, indicating an overall higher performance on the fraction problems (e.g., 4/20 = 12/[ ]) than on the numerically equivalent rate problems. For the numerical-comparison problem type (Figure 4b), there is a more pronounced clustering of data points around the diagonal that suggests an overall comparable achievement level for this problem type in the fraction form (e.g., which fraction is smaller, 6/24 or 2/6) and in the rate form. Finally, for the directional questions (Figure 4c). the data points are generally above the diagonal for the speed, mix, and density problems, and below the diagonal for the scaling problems. In other words, with the exception of the most difficult rate type, scaling, student achievement is generally higher on the rate questions (e.g., what happens to the spacing between nails if more nails are hammered into a shorter board) than on the fraction exercises (e.g., what happens to the fraction 3/4 if the top number gets bigger and the bottom number gets smaller).
Of the two context effects, rate type had a clearly stronger impact on both directional and numerical problem difficulty than did modest differences in problem setting. Scaling problems were the most difficult for the students in our study. Why should this be the case? There are several hypotheses, some combination of which probably explains the observed differences.
First, consider the numerical-comparison problems, which asked students to determine the relative size of two maps given two inch/mile correspondences. This problem type is not solved in a single step. For other rate types, the decision as to which mixture is darker, who runs faster, which line is more crowded etc. is virtually automatic once the appropriate rates are identified and perhaps simplified. This is a matter of familiarity fostered by past experiences. Such is not the case with scaling. With scaling, the second decision is not automatic and requires a second-order understanding of the problem situation. Consider a typical problem:
The student must first simplify the given rates to determine the order relation between them and then use that information to determine the size of the two maps. The direct size relationship which exists between the intensive value of the rates and the size of the resulting maps (i.e., the larger inch/mile rate results in the larger map, or a larger miles/inch rate results in a smaller map) is neither obvious nor likely to be a type of question which had been asked of students before.
The directional scaling questions involved a similar complexity. Students were given two maps whose longer (same or shorter) lines stand for longer (same or shorter) distances. Students were then asked which map is larger (same or smaller):
Again, students have probably never been asked to decide what happens to the size of a map when the inch/mile or mile/inch rate changes, so this decision is neither obvious nor as automatic as deciding what happens to the speed of a runner, the shade of a mixture, etc. The result was lower levels of student achievement on scaling problems, probably a not unlikely occurrence.
Scaling problems also varied the objects targeted for measurement, while the other rates types did not. This could be another reason why scaling problems were more difficult for students. For example the paint mixture problems always referred to mixing colored tint with white paint. In the classroom scaling problems, however, in some cases the length of objects was measured and in others cases the space between the objects was the primary concern. Subtle differences to be sure, but of potential importance in that each problem required individual analysis rather than permitting the student to establish a 'problem-type mind set' and more readily focus on the numerical aspects of the problem situation. The scaling problems required a greater degree of interpretation of the problem setting prior to any attention to the numerical aspects of the individual problem situation.
Finally, the scaling problems involved less of a sense of previous physical involvement than the other rate types. We hypothesize that this results in the need for a greater degree of image construction than with the other rates. Students probably have experienced running laps, driving in a car, mixing various substances, standing in a crowded movie line or hammering nails into, a board. Such is not the case for constructing and interpreting a classroom or city map. The objects to which one is making inferences are therefore both more remote and more removed from personal experience.
In addition to the rate-type effect, students' rational number ability had a large effect on their performance on both the directional and numerical scale. There was also a small but significant rate-type by rational-number interaction on the numerical scale, indicating that the order of difficulty of the rate types is different for students with different levels of understanding of rational numbers. These results imply that not all word problems in mathematics texts are of equal difficulty. The lumped category of "ratio and proportion problems" obscures important differences about how students with different rational number skills perceive and think about different rate types. A hierarchy of difficulty of rate types should be included as a factor in instructional design. Moreover, several different rate types will need to be mastered before students can be said to have a generalized proportional reasoning skill.
We had hoped that small differences in problem setting could be neglected, which would make the design of instructional examples and equivalent test forms or interview tasks for research easier. But it appears that even rather small differences in problem setting can have a large effect on performance for certain rate types. It may be necessary to research each rate type on Table 1 to determine how different problem settings affect problem difficulty.
Finally, our results suggest that the presence of a context has a different effect on students' performance on the three different problem types, missing-value problems, numerical comparison problems and directional questions. In general, context appears to hinder students' performance on missing-value problems. This result conforms with our original hypotheses, since school mathematics curricula place a heavy emphasis on finding equivalent fractions (or ratios), an operation also used in solving missing-value word problems. Given that students could be expected to be quite adept at the fraction operation, and the fact that all but one of the sets of numbers chosen involved integer multiples within and between the fractions, it seems likely that the additional complexity posed by the contextual situation would only hinder achievement.
For the numerical-comparison problem type, we concluded that there is no predictable difference in achievement when a context is present or not present. An examination of school curricula suggests little attention to fraction ordering activities. It is likewise true that students have not had much experience with this type of word problem. Therefore, for numerical-comparison problems students would not have the comparative advantage suggested above for fraction equivalence exercises as compared to missing-value word problems. One might posit that students at this level should have a firm grasp of relatively simple fraction ordering situations and then conclude that context could only hinder achievement. This did not occur, suggesting less student competency than one would like -- a phenomenon a too common when assessing rational number abilities. We did notice a decided shift in the number of points located closer to the vertical axis in Figure 4b, indicating a lower overall achievement rate when contrasted to the missing-value problems. This is likely due to the lesser degree of curricula attention noted above.
Finally, Figure 4c suggests that except for the most difficult rate type, scaling, the presence of a context assisted student achievement on directional questions. This seems reasonable because students could capitalize on relevant prior experiences. It is also noted that the items were of two very different levels of difficulty. The items types which proved to be the most difficult were those where the answer was indeterminate (e.g., If John mixed less green tint with less white paint than he did yesterday, his green paint would be: a darker shade, a lighter shade, exactly the same shade, or there is not enough information to tell). These were the only items which required a functional understanding of proportionality and hence invoked 'true' proportional reasoning. Results were very disappointing! Neither seventh nor eighth grade students were able to deal successfully with the indeterminate nature of these problems. Two possible explanations are, (1) students do not understand, and/or (2) students are not familiar with problems where answers are not determinable with the information provided, an artifact of textbook dominated programs. They may therefore have been trying to 'force' a response. In either event, results indicate that some attention to this situation is in order.
This study has demonstrated the nature of the influence of two aspects of problem context, rate type and problem setting, on seventh and eighth grade students' achievement on missing-value, numerical comparison, and directional questions involving proportional situations. We have also determined that rational number understandings play an important role. These findings must be consciously integrated with other results (and with those yet to come) if we are to find better ways to help students develop proportional reasoning skills.
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This research was supported in part by the National Science Foundation under grant No. DPE-847077 (Rational Number Project). Any opinions, findings, and conclusions or recommendations expresses in this publication are those of the authors and do not necessarily reflect the views of the National Science Foundation.