The figures for this document are contained in a second browser window.
If that window did not open automatically, please click here.
REASONING: THE EFFECT OF TWO CONTEXT
PATRICIA M. HELLER, ANDREW AHLGREN, AND THOMAS POST
College of Education, University of Minnesota, Minneapolis, Minnesota 55455
Mathematics Education, Northern Illinois University, DeKalb, Illinois 60115
Education Division. WICAT Systems Orem. Utah 84057
This study investigated the effects of two context variables, ratio type and problem setting, on the performance of seventh-grade students on a qualitative and numerical proportional reasoning test. Six forms of the qualitative and numerical tests were designed, each form using a single context (one of two settings for each of three ratio types). Different ratio types appear to have a stronger impact on the difficulty of the qualitative and numerical proportional reasoning problems than small differences in problem setting. However, the familiarity of problem setting did show an increasingly large effect on qualitative reasoning as the difficulty of ratio type increased. We also investigated the nature of the relationships between rational number skills, qualitative reasoning about ratios, and numerical proportional reasoning. Qualitative reasoning appears to be sufficient, but not necessary for numerical proportional reasoning. The evidence for the requisite nature of rational number skills for proportional reasoning was equivocal. The implications of these findings for science education are discussed.
Science is increasingly permeated by mathematics. Indeed, in many contexts they can hardly be distinguished (Steen, 1987). One of the most ancient and fundamental connections between science and mathematics is proportionality. Formally, a proportion is a statement of equality of two ratios, i.e., a/b = c/d. Unit prices, package size (e.g., number of eggs per carton), gas mileage, and speed are ratios1 with which everyone deals. Proportional reasoning is also important to students' success in secondary science courses, especially chemistry and physics, where the list of ratios extends to density, acceleration, concentration, the laws of definite and multiple proportions, speed, power, efficiency, and so on. There is reason to believe that students' difficulty in learning these concepts is in good part the result of their difficulty with proportional reasoning in general and in transferring it to the unfamiliar contexts in science.
Numerous studies have shown that early adolescents and many adults have a great deal of difficulty solving problems that involve proportional reasoning (Behr, 1987; Hart, 1978, 1981; Karplus et al., 1979; Rupley, 1981; Vernaud, 1983). One implication of this difficulty is that teachers should (as many do) question their students' knowledge of ratios before plunging into quantitative topics. A second implication is that science teachers should also begin to concern themselves with whether instruction in mathematics classes is adequate to teach proportional reasoning in a way useful to the learning of science. This article describes the beginning of a cooperative research program involving mathematics and science educators to find out how difficulty is affected by different types of ratios, and eventually, how ratio types can be taught effectively.
Why is proportional reasoning so difficult? What factors affect problem-solving success? Several studies have shown that factors such as problem format, the particular numbers used in the problems, the problem context, and even the immediately preceding problem affect student performance on proportional reasoning problems (Tourniare and Pulos, 1985). In this study we investigate the effect of two aspects of problem context on the level of student performance on proportional reasoning problems, and the arguably prerequisite skills required for it. We also explore some methodological problems in assessing the importance of prerequisite skills and discuss implications for the relationship between science teaching and mathematics teaching.
The Effect of Problem Context
There may be two aspects of "context" that can be usefully distinguished in proportional reasoning problems. One set of aspects comprises (a) the objects in the problem, (b) the variables used to describe the two properties of the objects of interest in the problem (e.g., length, area, weight, time, etc.), and (c) the units of measurement used to specify these variables (e.g., for length - inches, feet, centimeters, kilometers, miles, etc.). We will call this set of context aspects the "problem setting." Students may be more familiar with some problem settings than with others. For example, in speed problems students may be more familiar with running foot races than with driving cars (the objects and events of the problem). Similarly, students may be more familiar with distance measured in laps and running time measured in minutes than they are with distance measured in miles and driving time measured in hours (the variables and units of measure in the problem).
A second aspect of context is the type of ratio involved in the problem. Table I shows nine types of ratios that are used to solve proportional reasoning problems found in standard textbooks. These ratio types might well be organized into a hierarchical scheme, but here we propose only that they may appear distinctly different to students.
Each type of ratio can be used in familiar or unfamiliar problem settings (e.g., buying candy versus buying stocks and bonds). Even with familiar problem settings, however, students may be more or less familiar with the ratio types. For example, junior high school students typically have little experience converting units of measurement, even with familiar measures such as feet, yards, pints, and quarts.
Familiarity with what is called the problem context may consist of familiarity with both the ratio type and the problem setting. Knowledge of the hierarchy of difficulty for uninstructed students of proportional reasoning problems with different ratio types and problem settings may contribute to a better understanding of how proportional reasoning skills develop in adolescents and to the design of better proportional reasoning instruction for students.
Another factor that could affect student performance on proportional reasoning problems is qualitative reasoning skill, which seems to figure significantly in mathematics and physics problem-solving performance (Larkin and Reif, 1979; Larkin et al., 1980). Expert problem solvers are likely to apply a qualitative analysis to the relationships among the variables in the problem before they actually apply quantitative reasoning with algorithms or equations.
Some proportional reasoning studies indicate that many early adolescents use faulty qualitative reasoning or use additive comparisons where multiplicative comparisons are required (Karplus & Peterson, 1970; Karplus et al., 1983; Noelting, 1980). The frequency of these incorrect strategies seems to depend on the problem context (Jesunathadas & Saunders, 1985; Karplus et al., 1983b). However, no systematic research has been conducted to explore students' ability to reason qualitatively about ratios, to determine the effect of different contexts on their qualitative reasoning about ratios, or to determine how qualitative reasoning about ratios contributes to proportional reasoning skills.
In this study we introduce a new type of qualitative question about ratios that may be important in understanding the development of proportional reasoning skills in adolescents. These questions ask in what direction a ratio will change (decrease, stay the same, or increase in value) when the numerator and/or the denominator decreases, stays the same, or increases. Such qualitative directional reasoning about ratios may be an important prerequisite skill for successful performance on numerical proportional reasoning problems.
Ratio Types and Problem Settings
Three types of ratios were examined in this study: speed, exchange (buying), and consumption. These ratio types were chosen because we expected them to have different difficulties and because they have been studied previously (Karplus et al., 1983b; Vergnaud, 1983). We expected speed problems to be slightly more difficult than buying problems, and consumption problems to be the most difficult of the three ratio types.
Table II shows the two problem settings selected for each ratio type. The first problem settings-buying gum, running laps around a track, and the gas mileage of trucks-were chosen because we expected students to be the most familiar with these settings. The second problem settings-buying records, driving cars, and the oil consumption of furnaces-were chosen because we expected these settings to be less familiar to students, but not very unfamiliar (as, for example, science content problems would be).
Numerical Proportional Reasoning Problems
Two formats of numerical problems, missing-value and numerical-comparison problems, were used in this study, as illustrated by the problems below:
Missing-value and numerical-comparison problems have been used extensively in instruction and research. The inclusion of both types of problems in this study complements previous studies with the same ratio types by Karplus et al. (1983b) and Vergnaud (1983).
Two formats of qualitative directional questions about ratios were invented for this study, as illustrated by the questions below:
The qualitative change in the numerator and denominator of the ratio refers to different events in time for the qualitative ratio change questions, and to different objects or people for the qualitative comparison questions.
Both the numerator and the denominator of a ratio can decrease, increase, or remain the same. Therefore there are nine qualitative ratio change and nine qualitative comparison questions that could be asked. Two of these cases are ambiguous, when the numerator and denominator both decrease or both increase. The numerator and denominator can decrease (or increase) proportionally or nonproportionally. So the correct answer to these questions is that there is not enough information to decide. These are the only qualitative questions that require a numerical understanding of proportionality for a correct answer.
Our subjects were 254 seventh graders in a middle-class urban school in Minnesota. They included all seventh-grade 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 students had not received instruction on proportional reasoning problems in their seventh-grade mathematics classes.
Six forms of the directional and proportional reasoning test were designed, each form using a single context (one of two settings for each of three ratio types). Each test was made up of 16 problems. The first section of the test consisted of three missing-value and three numerical-comparison proportional reasoning problems. The numerical values in the six problems, shown in Table III, allow students to solve the problems correctly using integer ratios. The missing-value and numerical-comparison problems were alternated to alleviate the boredom of solving the same kind of problem one after another. There were two problems on a page with ample space for answering.
The second section of the test consisted of five qualitative ratio change questions followed by five qualitative comparison questions. Because we were concerned about student fatigue in taking the test, we selected only five of the nine possible questions of each type. These questions were selected because they had resulted in the greatest difference in difficulty in a previous pilot study. The direction of change for the numerator and denominator of the ratio in each question is shown in Table IV.
The second instrument used in the study was a 20-item rational number test. This test consisted of problems on order and equivalence, finding equivalent fractions, qualitative directional changes in the value of a fraction, operations with fractions, estimating rational number computations, a quantitative notion of a fraction, and the concept of a whole.
The instructions to students included the request to show all of their work on their test papers. The six different forms of the directional and proportional reasoning test were randomly distributed to the students in each class. After each student completed the test, he or she was given the rational number test.
Seventeen students who did not complete the rational number test were dropped from the sample. The items on both tests were scored as right or wrong. a factor analysis of the 20 items on the rational number test revealed only one main factor. Two items did not load appreciably on the first principal component, so they we excluded from the subsequent analysis. The rational number test was difficult for the seventh-grade students in this study; on the average they answered only six of the 18 questions correctly (M = 6.07; SD = 3.74; R = 0 - 17). Nevertheless, the internal consistency reliability (Cronbach alpha) of the rational number test was 0.79.
A factor analysis of the 16 items on the directional and proportional reasoning test indicated that there could be two factors. All but two of the qualitative questions loaded on the first rotated factor. The proportional reasoning problems as well as the less/less and more/more qualitative questions loaded on the second rotated factor. However, since students performed only at the chance level (about 25% on a four choice item) on the less/less and more/more questions, these questions were excluded from the subsequent analysis.
Consequently, two scales were used in the analysis. The Numerical Proportion Reasoning Scale consisted of the number correct on the missing-value and numerical-comparison problems. Students did very well on this scale, on the average solved about four out of the six problems correctly (M = 3.75; SD = 1.77; R = 0 6). The Cronbach alpha reliability of this scale was 0.68.
The Qualitative Directional Reasoning Scale consisted of the number correct (the qualitative ratio change and the qualitative comparison questions (excluding the less/less and more/more questions). Again, students did very well on this scale, on the average answering about six of the eight questions correctly (M = 5.84; SD = 2.18; R = 0 - 8). The Cronbach alpha reliability of this scale was 0.77.
The relationships between measures were investigated by simple correlations and scatterplots. Context effects on performance were investigated by factorial ANOVA. Although forms of the tests were assigned randomly to students, it was possible that there were chance group differences in rational number skills. A one-way ANOVA of rational number scores across the six context groups clearly showed no significant mean differences (F(5,231) = 0.3, p = 0.91). To look for possible interactions of rational number skills with context, the rational number test scores were trichotomized as evenly as possible (27%, 42%, 31%) and included as a third factor in the ANOVA. Thus the analysis for context effects was a three-way ANOVA; ratio type by problem-setting familiarity by rational number skills (trichotomized).
Relationships among Measures
One issue in the study was the nature of the relationship between rational number skills, qualitative directional reasoning about ratios, and numerical proportional reasoning. In particular, the questions were (a) whether the three measures represent distinct skills, and, if so, (b) whether high rational number skills or high directional reasoning were requisite for success in proportional reasoning.
On a simple level, pooled across all contexts, the observed correlation between proportional reasoning and directional reasoning was 0.53. Based on the scales' reliabilities, a correction for attenuation yields an estimated true correlation of 0.73, and thus 53% of their variation in common. On this basis alone, it would appear that the skills required for the two scales were not identical, although they are substantially similar. The same inference holds for the relation of proportional reasoning to rational number skills, for which the observed correlation is 0.50 and the estimated true correlation is 0.68, implying 46% of their variance in common. The relationship between directional reasoning and rational number skills, however, is distinctly weaker. Their observed correlation of 0.35 yields an estimated true correlation of only 0.45 and thus only 20% of their variance in common. Ceiling effects evident on the directional and proportional reasoning scores, and a floor effect on the rational number test, restrict the magnitude of the observed correlations, so the actual scale similarities are likely to be somewhat higher.
Important additional information, however, is available from the scatterplots of one set of scores against another. One possibility for the data of proportional reasoning versus directional reasoning would be to show a bivariate normal distribution with "empty comers" [see Figure 1(a)]. Students with low directional reasoning scores could not have high proportional reasoning scores, and students with high directional reasoning scores would be assured of high proportional reasoning scores. Directional reasoning skills would thus be both necessary and sufficient for proportional reasoning. Another possibility, however, is that directional reasoning would be necessary but not sufficient for proportional reasoning; i.e., some very different, additional understanding would be involved in proportional reasoning. In this case, the data would show a triangular distribution with only one empty corner; the low directional reasoning, high proportional reasoning corner [see Figure 1(b)].
The actual plot of proportional versus directional reasoning is much more like the latter case: 72% of the students are distributed through the lower triangle, only 28% in the upper triangle. The inference is that directional reasoning is helpful (not entirely necessary), but certainly not sufficient for success in proportional reasoning. It also would appear that high performance in proportional reasoning can be achieved by means that do not involve good directional reasoning. The obvious candidates for such means are skill in solving proportionalities by memorized procedures and general rational number skills.
The relation of proportional reasoning to rational number skills is, however, quite surprising. The scatterplot of proportional reasoning versus rational number skills shows 86% of the students filling the upper triangle, with only 14% barely crossing into the lower triangle, similar to Figure 1(c). The implication is that proportional reasoning problems can be solved very well by students who have very low rational number skills, but that high rational number skills assure proportional reasoning success. That is, rational number skills are sufficient, but not necessary for proportional reasoning.
It is important to note, however, that the proportional reasoning test was fairly easy, with a ceiling effect, while the rational number test was difficult, with a floor effect. The combination of the ceiling and floor effects, as suggested in Figure 1(d), could have artificially produced the anomalous "upper triangle" appearance of the data. The results of this study, therefore, are ambiguous with regard to the nature of the dependence of proportional reasoning on rational number skills. The unusual distributions preclude a meaningful multiple correlation analysis as well as more elaborate plots.
Another issue in the study was the effect of the context variables, ratio type problem setting, on directional and proportional reasoning. Table V shows the mean scores for the students taking the six forms of the directional and proportional reasoning test. Context effects on directional and proportional reasoning were investigate three-way ANOVAs: rational number skills(3) by problem setting familiarity(2) by ratio type(3). The results of these analyses are shown in Tables VI and VII.
Ratio type showed a significant main effect in the expected order for both the proportional reasoning (F(2,219) = 10.9, p < 0.001) and the directional reasoning (F(2,219) = 16.0, p < 0.001) scales. Consumption ratios are more difficult than speed, and speed is more difficult than buying ratios. Problem setting, however, showed a significant main effect in the expected direction only for the directional reasoning scale (F(1,219) = 24.7, p < 0.001). The less familiar settings are more difficult.
There was also a small but significant interaction of problem setting and ratio type for the directional reasoning scale [F(2,219) = 3.9, p = 0.02]. The nature of the interaction is that ratio type differences are larger in less familiar problem settings in the more familiar problem settings. Figure 2 displays the group means for the six settings by ratio type groups. For the very familiar buying type of ratio, familiarity of setting makes little difference in directional reasoning scores. For the less familiar speed and consumption ratio types, however, unfamiliar settings have increasingly greater effects on difficulty. Perhaps the stronger context effects on the directional reasoning scale allowed this interaction to appear. A similar, although not significant, interaction appears for the proportional reasoning scale.
Unsurprisingly, the trichotomized rational number scale showed a highly significant main effect. It did not, however, show any interaction with either context effect. The problem-setting and ratio type effects that were observed were therefore presumably comparable for students with different levels of understanding of rational numbers.
Logically, it would seem that proportional reasoning in a problem would not be possible without knowing at least the direction of change in a ratio 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 what they mean. Consequently, the evidence for the necessity of directional reasoning is muddied. It is clear, however, that directional reasoning is not sufficient for proportional reasoning. The ceiling effects realized with the particular item sets used for the directional and proportional reasoning scales do not compromise this finding.
On the other hand, the ceiling effect on the proportional reasoning scale and the floor effect on the rational number test do not allow a confident inference about the relation of these two skills. Future investigations should employ a more difficult test of proportional reasoning and an easier test of rational number skills.
Of the two context effects, ratio type had a clearly stronger impact on both directional and proportional reasoning problem difficulty than did the modest differences in problem settings. Yet the familiarity of problem setting did show an increasingly large effect on directional reasoning as the difficulty of the ratio type increased.
If familiarity is important to directional reasoning, which we suspect to be prerequisite to understanding proportional reasoning (if not always to getting the right answer), then the introduction of new ratio types in science instruction requires considerable care in choosing example problem settings. As for future research, it will be necessary to interpret findings about familiarity in the light of the intrinsic difficulty of the ratio type involved. We had hoped that small differences in problem-setting familiarity could be neglected, which would make the design of equivalent test forms or interview tasks easier. But it appears that even rather small differences in problem setting can have a large effect on performance if the rate type involved is itself unfamiliar.
The long-range relevance of this genre of research may be to increase the responsibility of science teachers in teaching the use of proportional reasoning. It is debatable whether mathematics courses, at least as they are now constituted, would ever be able to prepare students well enough in proportional thinking for them to transfer it to unfamiliar science contexts. The lumped category of "ratio" problems in mathematics instruction obscures important differences about how students perceive and think about different ratio types. The hierarchy of difficulty of ratio types must be included as a factor in instructional design. As long as mathematics teachers see their job as teaching an abstract concept of proportions that only subsequently is used in "applications," there may be no ground for modifying their approach.
There are two possible directions for improving the strategy for bridging the mathematics/science interface. One is to have mathematics instruction begin with real-world situations and develop mathematics to deal with them, rather than use such situations as afterthought applications. This strategy is coming to be termed "mathematical modeling" as distinct from end-of-the-chapter "applications" (Lesh et al., 1987). The training and inclinations of mathematics teachers may militate against this possibility (Pollak, 1986). The other direction is for science teachers to abandon their reliance on proportional reasoning being previously taught in mathematics classes and undertake to teach it as emerging from the very contexts in which they would wish to apply it.
Although further research is clearly needed, we can already progress a little beyond simply advising science teachers to "begin easy," because we have a somewhat better understanding of what constitutes easy. Intuitive understanding of direction of changes in ratios (qualitative reasoning) should precede numerical exercises, and even small increases in familiarity of the objects and units may be very helpful in teaching new types of ratios.
article we use the term "ratio" generally, without the sometimes
special meaning of a unit-free number.
to teachers of English, but we thought it better to use the natural language
of today's students.
*This research was
supported in part by the National Science Foundation under Grant No. DPE-847077.
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.
Behr, M.J. (1987). Ratio and proportion: A synthesis of eight conference papers. In U.C. Bergson, N. Herscovics, & C. Kierat, Eds., Psychology and mathematics education, Vol. II, Proceedings of the Eleventh International Conference, Montreal Canada.
Hart, K. (1978). The understanding of ratios in secondary school. Mathematics in school, 7(1), 4-6.
Hart, K. (1981). Children's understanding of mathematics: 11-16, London: Murray.
Jesunathadas, J. & Saunders, W. (1985, April). The effect of task content upon proportional reasoning. Paper presented at the Annual Meeting of the National Association for Research in Science Teaching, Indiana.
Karplus, R. & Peterson, R. (1970). Intellectual development beyond elementary school. II: Ratio, a survey. School science and mathematics, 70, 813-820.
Karplus, R., Karplus, E., Formisano, M., & Paulson, A. (1979). Proportional reasoning and the control of variables in seven countries. In J. Lochhead & J. Clement, Eds., Cognitive Process Instruction. Philadelphia: Franklin Institute Press.
Karplus, R., Pulos, S., & Stage, E. (1983a). In R. Lesh & M. Landau, Eds., Acquisition of mathematics concepts and processes. New York: Academic Press.
Karplus, R., Pulos, S., & Stage, E. (1983b). Early adolescents' proportional reasoning in "rate" problems. Educational studies in mathematics, 14, 219-233.
Larkin, J. & Reif, F. (1979). Understanding and teaching problem solving in physics. European journal of science education, 1, 191-203.
Larkin, J., McDermott, J., Simon, D.P., & Simon, H.A. (1980). Expert and novice performance in solving physics problems. Science, 208.
Lesh, R., Niss, M., & Lee, D. (1985). Theme group 6: Applications and modality. In M. Carss, Ed., Proceedings of the Fifth International Conference on Mathematic Education. Boston: Birkhauser.
Noelting, G. (1980a). The development of proportional reasoning and the ratio concept: Part I-Differentiation of stages. Educational studies in mathematics, 11, 217-253.
Noelting, G. (1980b). The development of proportional reasoning and the ratio concept: Part II-Problem structure at successive stages: problem solving strategies and the mechanism of adaptive structuring. Educational studies in mathematics, 11, 331-363.
Pollak, H. (1986, November). Summary of Conference. Paper distributed to participants in the conference The School Mathematics Curriculum: Raising National Expectations held by the Mathematical Sciences Education Board of the National Academy of Sciences, University of California at Los Angeles.
Rupley, W. (1981). The effects of numerical characteristics on the difficulty of proportional problems. Unpublished doctoral dissertation, University of California, Berkeley.
Steen, L.A. (1987). Mathematic education: A predictor of scientific competiveness. Science, 237, 251.
Tourniare, F. & Pulos, S. (1985). Proportional reasoning: a review of the literature. Educational studies in mathematics, 16, 181-204.
Vergnaud, G. (1983). Multiplicative structures. In R. Lesh and M. Landau, Eds., Acquisition of mathematics concepts and processes. New York: Academic Press.
Manuscript accepted February 24, 1988.