This chapter presents examples and some general results from a written test on "relations and proportions" that has been used in our Rational Number (RN) and Proportional Reasoning (PR) projects. The RN and PR projects have included: (1) instructional components in which whole classes of fourth through eighth graders have been carefully observed, tested, and interviewed during ''teaching experiments" lasting as long as 20 weeks; and (2) evaluation components in which several thousand second through eighth graders have been tested at a variety of schools in California, Illinois, Minnesota, Pennsylvania, and Utah.
The RN/PR testing program has included three types of tests: pencil-and-paper tests, instruction-mediated tests, and clinical interviews. Each type of test has focused on students' abilities to perform translations of a given idea from one representational system to another, or transformations within a given representational mode. Our chapter of this book on "representations and translations" gives several reasons why we believe these processes are especially important in mathematics learning and problem solving.
In general, we have been most interested in the "mediating" roles that spoken language and manipulative models play in problem solving and in the evolution of proportional reasoning and rational number concepts. However, because of the dominant role that pictures, written language and symbols play in textbook based instruction and testing, we also have investigated these "bookable'' representation systems.
The pencil-and-paper phase of the RN testing program consists of items in which both the questions and the responses use ''bookable" representations. The instruction-mediated testing phase consists of items in which the question presented in spoken language or manipulative models, but the response requires only a ''bookable" mode. The clinical interview phase consists of items in which both the questions and the responses involve "non bookable" modes.
The pencil-and-paper testing battery, from which the examples in this chapter were taken, includes a set of three subtests, each focused on a different aspect of rational number understanding: (1) a ''basic understandings" test focusing on youngsters' abilities to translate from one representational system to another for a single fraction or ratio; (2) a ''relations and proportions" test, in which the items involve equivalence or ordering relationships among pairs of fractions or ratios; and (3) an ''operations" test with items requiring the subject to "operate on'' fraction or ratio ideas embedded in pictures, written mathematics symbols, or written forms of spoken language (e.g., many of the items on this test were either addition or multiplication "word problems" or computation exercises). All the pencil-and-paper items used a multiple choice format.
Details and results from the ''basic understandings" and ''operations" subtests have been reported in Lesh, Landau, and Hamilton (1983). This chapter focuses on results from the "relations and proportions" test.
GENERAL GOALS OF THE RN/PR TESTING PROGRAM
One goal of the RN/PR testing program has been to provide a data base for comparing and integrating results from past research and to provide baseline student-performance information for future R&D efforts. Whenever possible, we have attempted to use items from other large-scale testing programs (e.g., Carpenter, Coburn, Reys, & Wilson, 1978; Hart, 1980) or from past research or development projects (e.g., Karplus, Pulos, & Stage, 1983; Kieren, 1976; Novillis, 1976; Wagner, 1976). For example, the ''relations and proportions" test includes adapted versions of proportional reasoning items from the 1977-78 National Assessment of Educational Progress (1979) and modified versions of "orange juice" items that were used to Noelting and Gagne's (1980) research on proportional reasoning.
Three main goals of this chapter are: (1) to give examples illustrating some of the structural characteristics of students' representational capabilities related to proportional reasoning and rational number understandings; (2) to describe some of the difficulties students frequently experience with representational translations; and (3) to describe several ''rules of thumb" about the relative difficulty of several different kinds of representational translations.
The part of the ''relations and proportions" test that replicated and extended Noelting's (1980) research aimed at identifying Piagetian ''stages" in the development of youngsters' ''proportional reasoning" concepts. Noelting's research is particularly interesting in this chapter because it deals directly with one of the fundamental themes that runs throughout this book. This issue has to do with the educational importance of analyzing students' operational capabilities for particular mathematical ideas and for specific representational systems that facilitate students' abilities to acquire and use these ideas.
Piagetian psychologists have been particularly interested in the major conceptual reorganizations that typically occur in children at about age two, at about age six to seven, and during early adolescence. However, mathematics educators tend to be more interested in the transitional phases between (and beyond) these global transition periods. To create learning experiences that will gradually guide students from easier conceptualizations to those that are progressively more complex and abstract, mathematics educators must be able to anticipate the relative difficulties of specific ideas, or alternative conceptualizations of given ideas. They also must be able to anticipate the relative difficulty of different representational systems (or instructional models) that are available to introduce and illustrate these ideas.
Although Piaget was one of the foremost psychologists to emphasize the inherent structural/constructivist nature of mathematical ideas and representational systems, both he and his followers use the term decalage to refer to situations in which two ideas (or representations for a particular idea) involve similar underlying structures, but one is significantly easier than the other.
During the past decade, some of the most productive research in mathematics education has focused on tracing the gradual evolution of particular ideas, finding the relative difficulty of distinct conceptualizations of given ideas, and identifying characteristics that influence the difficulty of different models (or representations) of these conceptualizations. In other words, some of the most significant research in mathematics education has focused on finding rules to explain or describe predictable regularities that occur within what Piagetians refer to as decalages.
Results from the Relations and Proportions Test
The following results from the ''relations and proportions" test were identical to those we have reported earlier based on the ''operations'' and ''basic understandings" tests (Post, Behr, & Lesh, 1982). Consequently, we simply review them here and give examples to illustrate them using items from the "relations and proportions'' test. Varying a task on any of a number of dimensions (e.g., number size, relationship of numerator to denominator, perceptual characteristics of figural models, etc.) frequently produces dramatic variations in students' performances.
For example, test item 29 (Fig. 5.4) was considerably more difficult than ''similar" items like item 24 (Fig. 5.1), which involved "unit fractions or ratios" (like 1/2 or 1/3), or than items 27, 26, or 60.
Other factors that contribute to item difficulty are related to the "perceptual distractors'' that were inherent in certain representations. For example, only about 53% of our eighth graders got item 36 (Fig. 5.6) correct, and only about 62% got item 44 (Fig. 5.7) correct.
Probably the main factor that made items 44 and 36 so difficult had to do with the fact that such items presuppose an agreement that must be accepted about the basis for calling two situations equivalent. For example, to a child who wants to play an electronic arcade game, four quarters may not be ''equivalent to'' one dollar; similar observations could be made about fractions and ratios in many everyday situations. However, this is not the only factor that seemed to be involved.
In spite of the preceding comments, we should not be too hasty in accepting the notion that students who did not give ''correct" answers to items 44 and 36 did so only because they were thinking of a more restricted interpretation o equivalence. For example, many of the students who responded "incorrectly" to items 44 and 36 used the criteria we were expecting in other situations, like item 43 or 37 (see Figs. 5.8-5.9).
To explain the behavior of these students, it is necessary to explain how the criteria they use to judge "equivalence" is influenced by characteristics of the task. Why are they inconsistent? Our research suggests that students who use one criteria in one situation but abandon it in other ''similar" situations apparently do so because of the strength of "perceptual distractors" (e.g., "extra partitions") that were built into some diagrams more than others (Behr, Lesh, Post' & Silver, 1983; Post, Wachsmuth, Lesh, & Behr, 1985).
An even more dramatic example of the influence of perceptual distractor occurred in the clinical interview phase of the RN/PR testing program. Students were asked to "Give me one-third of this Hershey chocolate bar." The difficulty of this problem depended significantly on whether the student was given a "plain" Hershey bar, or a Hershey bar ''with nuts.'' The plain bar was significantly more difficult, apparently because of the partitions on the plain bar that were not on the bar with nuts.
The fact that students' rational number thinking is often so strongly influenced by "perceptual distractors" suggests that their underlying conceptualizations of many rational number ideas are quite unstable (Behr, Wachsmuth, Post, & Lesh, 1984).
Because each item on our three pencil-and-paper tests can be characterized by one of the following seven translations (see the chapter of this book on "representations and translations" for more information about these seven translation types), it is possible to think of the RN pencil-and-paper testing battery as consisting of seven ''translation subtests''-one for each type of translation: (a) symbols to written language, (b) written language to symbols, (c) pictures to pictures, (d) written language to pictures, (e) pictures to written language, (f) symbols to pictures, (g) pictures to symbols.
If the pencil-and-paper testing battery is thought about in the preceding way, then the order of increasing difficulty of the ''translation subtests" was as just listed, with ''picture to symbol'' translations being most difficult. ln general, if other factors are held constant: (a) Translations to pictures is easier than translations from pictures; (b) translations involving written language (e.g., three fourths) are easier than translations involving written symbols (e.g., 3/4); and (c) the easiest translations are those that only require a student to ''read" a fraction or ratio in two different written forms. The difficulty (measured in % correct) of an item did not necessarily decrease as grade level increased. For example, on item 29 (Fig. 5.4), seventh-grade performance was no better than sixth-grade performance. ln fact, on all three of our pencil-and-paper translation tests, the greatest proportion of performance decreases occurred between sixth and seventh grades, whereas the greatest proportion of "large performance increases" occurred between the fifth and sixth grades. The rational number understandings of seventh graders seemed to be particularly unstable. Why? What explanations might account for such phenomena'?
In Lesh, Landau, and Hamilton (1983), one-to-one researcher-to-child interview results on problems similar to item 29 (Fig. 5.4) suggested that seventh graders are shifting to a new solution mode (i.e., more algebraic/symbolic) than those that are more typical among sixth graders and that this ''shift'' brings with it some new sources of error. Seventh graders may be more likely to use formal/symbolic solution procedures rather than more concrete or intuitive approaches-so they may experience difficulties on items that do not readily fit familiar equations. For example, notice that the statement of item 29 is not of the form ''3 is to 8 as 9 is to x.'' So, students who attempted to use an a/b = c/d style of equation may have been attempting to find a number such that "3 is to 8 as x is to 9 (i.e., 3/8 = x/9). lf this explanation is valid, however, we are again faced with the problem of explaining why other similar similar problems were relatively easy. Again, our follow-up interviews suggest that inconsistent behavior seems to be the result of facilitating or distracting factors that are ''built into" certain diagrams or representations. More examples to illustrate this claim are given next.
Every item on the three pencil-and-paper tests was classified according to the following five dimensions: (1) representational mode, or translation type, (2) rational number subconstruct (i.e., ratio versus fraction), (3) figural attributes (e.g., discrete versus continuous elements, contiguous parts versus noncontiguous parts, etc.), (4) ''Piagetian" level ala Noelting (see next section for a brief explanation of this dimension), and (5) format, e.g., whether the item was: (a) an equation translation problems, like item 34 (Fig. 5. 10), (b) a fraction or ratio translation problem, like items 36 or 44 (Figs. 5.6-5.7), (c) a proportion word problem, like items 24 or 29 (Figs. 5.1-5.4), or (d) a comparison problem, like the ones that are described in the next section of this chapter about Noelting and Gagne's Piagetian Levels.
Distinctions between fractions and ratios, and between discrete and continuous elements are illustrated in item 44. Answer choice g is an example of a ''discrete" model, whereas h is a ''continuous'' model. This discrete-versus continuous distinction has been a task variable influencing item difficulty on all three pencil-and-paper tests and in a number of other past research studies (e.g., Behretal., 1983;Novillis, 1976).
Fraction-ratio confusions also appeared to be sources of difficulty for many items. For example, among the incorrect responses on item 44, 14% selected either answer choice g or h-both of which involve ratios of 3-to-4 between shaded and unshaded parts.
Clear performance differences appeared to be associated with all the aforementioned "task variable dimensions." However, the effect of these variables was not necessarily additive; that is, the impact of changing a single variable in a pair of items may either be much greater than, or possibly even opposite to, a similar change for another pair of items. It seems likely that any explanation that attempts to predict the difficulty levels of tasks like those on the "relations and proportions" test cannot be "one dimensional'' and surely must take into account the facilitating and distracting characteristics of various representational systems.
The next section of this chapter focuses on one scheme, based on the work of Piaget, that attempts to explain the difficulty of one subset of the items on the
Comparison Problems and Noelting's Piagetian Levels
Thirty-six items on the ''relations and proportions" test involved either comparing the "orange flavor" of two orange juice mixes, or comparing the size of fractions represented in pictures or symbols.
Item 21 (next) is an example of a question about making an orange drink by mixing orange concentrate and water. The shaded squares stood for orange concentrate, and the white squares stood for water. The student was to select the true statement following each picture.
Item 21 is an example of a ratio comparison with continuous quantities, and item 65 (Fig. 5.13) is a fraction comparison.
Noelting and Gagne (1980) devised a Piagetian-based scheme to predict the difficulty levels of proportional reasoning and fraction comparison tasks such as those just shown. Their results showed that this scheme was stable with respect to the three types of items, namely orange juice comparisons (e.g., item 21, given earlier), fraction comparison depicted with symbols, and fraction comparisons represented with pictures (e.g., item 65, following).
We used the following scheme, which is mathematically equivalent to Noelting and Gagne's, and which applies to any of the ratio or fraction comparison items on the "relations and proportions" test:
Level 1. two unequal rationals, varying in either the first component or the second component, but not both (e.g., one-third versus two-thirds. or three-fifths versus three-sevenths.)
Level 2A. two equal unit rationals (e.g., one-fourth versus two-eighths).
Level 2B. two equal
nonunit rationals (e.g., three-fifths versus six-tenths).
Level 3B. two unequal rationals, with no between or within component divisibility (two-thirds versus three-fifths, for example).
Table 5.1 [click on this link to open the table in a second browser window] assigns a "Noelting level'' to all the rational number comparison items in the relations and proportions test, with these items partitioned into four categories: "orange juice" comparisons, "picture-picture" comparisons, symbol-symbol comparisons, and "other'' (i.e., comparisons that are not part of the Noelting format, but which can be classified using his scheme.)
Within each category, on Table 5.1, the items are ordered according to difficulty, so that difficulty-comparisons are easy to make, both within categories, and across categories. For example, item 21 (Fig. 5.11), which is an "orange juice'' comparison identified with Noelting level 2B, is positioned at point 63 on the "orange juice" line-because it was the 63rd most difficult item on the relations and proportions test. Similarly, item 65 (Fig. 5.12), which is a picture-picture comparison, is identified with a ''3B" at the point cg on the picture-picture line-because it was the 59th most difficult item on the test.
In a crude wav then. Table 5.1 makes it possible to compare the difficulty of any of the comparison items, or any two items on the test.1; In particular, any "reversals" from Noelting's predictions are apparent. lf our students' performances matched perfectly with the predictions of the Noelting and Gagne model, then, from left to right on each line, one would expect to read the classifications in consecutive groupings (i.e., all the level I items followed by all the level 2A items, then all the 2B items, etc).
Table 5.1 shows that, within the first three categories, the order of difficulty of the items in general correspond to the sequences of difficulty hypothesized by Noelting and Gagne. Indeed, even though a number of reversals can be found on each of the first three number categories, the hypothesized sequences were statistically valid as the first three categories were merged. However, in the ''other" category, which included comparisons not given in the Noelting/Gagne format, the hypothesized ordering did not come close to reaching statistical significance; and if these data were merged with those of the other three categories, then .05 statistical significance was not attained.
Our results, then, are not inconsistent with Noelting and Gagne's finding of statistically significant sequences based on the ''numerical characteristics," as long as attention is restricted to the kinds of tasks used in the earlier studies. However, if a slightly larger class of items is included, where more factors related to particular representations are involved, the hypothesized sequences break down. This suggests that the difficulty-causing dimension identified by Noelting and Gagne probably plays a significant role, but that properties of the representational system also may need to be considered.
Post hoc analyses and follow-up clinical interviews suggested that exceptions to Noelting and Gagne's hierarchy were, in general, explainable on the basis of representational variations. Two examples are given next.
Item 23 (Fig. 5.13), at stage 3b, proved to be easier than all but one of the stage 2B, stage 3A, and other stage 3B orange juice items. As a "2/3 versus 5/8" comparison, the Noelting/Gagne hierarchy predicts that item 23 should be much more difficult than (for example) item 21 (Fig. 5.11), which is a "3/1 versus 6/2" (level 2A) comparison. However, the arrangement of objects in item 23 facilitates one-to-one correspondences between orange juice and water, enabling more intuitive (and correct) judgments than the arrangements for the mathematically less complicated item 21.
For the picture fraction comparisons, item 33 (Fig. 5.14), involves "1/2" and consequently was put into a special "easy'' category according to Noelting's scheme. This is because past research has shown ''halfness" to be a cognitively primitive concept; youngsters typically show earlier proficiency for tasks involving one-half than for other fraction items (Kieren, 1976). Yet, item 33 was much more difficult than item 10 (Fig. 5.15), which Noelting's scheme would classify at stage 2A. Both items 10 and 33 (Fig. 5.14) involved non adjacency of parts however in item 10, the one-third portion in picture A could be (mentally partitioned to form the three one-ninth portions in picture B; that is, the three one-ninth portions can be (mentally) "pushed together" to form a piece congruent to the one-third portion in picture A. In item 33, such a partitioning was not possible. The one-half portion in part A cannot be partitioned to form the three one-sixth portions in part B; "pushing'' the pieces together in picture would produce a one-half portion cut horizontally, not vertically, as in picture A Therefore, successful completion of item 33 requires mapping both picture A and picture B to the idea "one-half," and then making a judgment based on transitivity ("A is one-half and B is one-half, therefore A and B are equivalent.") In item 10, it is possible to map each of pictures A and B to the idea "one third," and then to use transitivity reasoning to complete the comparison, but it is natural (and much easier) to determine equivalence by visual judgment.
In each such case where the Noelting sequence was violated, plausible explanations based on nuances in the item representations seem to be apparent.
Results from the RN/PR ''relations and proportions'' test have been used to illustrate a number of characteristics of students' representational capabilities
We also have identified a number of relationships among students' rational number understandings and their abilities to perform translations among and transformations within various representational systems. Whereas "underlying mathematical structure" and "concreteness of task" are two important variable contributing to item difficulty, the characteristics of particular representational systems in which these mathematical structures are embedded also are important
Mathematics educators are familiar with ''number-numeral" distinctions and similar distinctions related to the idea that mathematics ideas are not ''in" thing but rather are pure structures." Nonetheless, representational systems can b viewed as being transparent or opaque. A transparent representation would have no more nor less meaning than the idea(s) or structure(s) they represent. A opaque representation would emphasize some aspects of the idea(s) or structure(s), and de-emphasize others; they would have some properties beyond those of the idea(s) and structure(s) that are embedded in them, and they would no have some properties that the underlying idea(s) and structure(s) do have. In our research, the opaque nature of representations has been salient; this is one reason why, in real or concrete problem-solving situations, we have found that student often make simultaneous use of several qualitatively different representational systems (e.g., a picture, spoken language. and written symbols), and that the salience of one system over others frequently varies from stage to stage i solution attempts. Capitalizing on the strengths of a given representational sys tem, and minimizing its weaknesses, are important components of ''understanding'' for a given mathematical idea. Representational translations and transformations are important both to the acquisition and use of mathematical ideas
If translation abilities are such obvious components of mathematics under standing and problem solving, why are they so often omitted from instruction and testing? One reason is that many translation types are not easily "bookable" other reasons stem from the fact that so many research questions remain unresolved concerning the exact roles that translations play in the acquisition and us of mathematical ideas, and about the instructional outcomes that can be expected if they are taught effectively (Behr et al., 1983). The following popular misconceptions also are relevant.
Misconception. Translations often are assumed to be easy. Our research shows that concrete problems often produce lower success rates than comparable "word problems'' or written symbolic problems. Lesh, Landau, and Hamilton (1983) includes examples of word problems that become more difficult when additional information is given in the form of concrete materials (which on might naively suppose should have made the problems more meaningful, an perhaps easier).
Misconception. If a student can correctly answer a given type of word problem, he or she surely must be able to solve similar problems in everyday situations. This is because word problems are 'abstract" versions of real situations. Our research shows that purportedly realistic word problems often differ significantly from their real-world analogues with respect to processes most often used in solutions, the types of errors that occur most frequently, and difficulty (Lesh et al., 1983). It is too simplistic to think of word problems as lying on a continuum between abstract ideas and real situations; a word problem may involve understandings that its real analogue does not, and vice versa. More than 40 weeks of student activities have been developed in the RN and PR projects, based on the same theoretical framework as that used to generate items for the testing program. Consequently, profiles of abilities and understandings that can be generated from the testing program can be used to select appropriate learning experiences for individual children; or, the testing materials can be used to measure effects of theory-based instructional treatments. These sorts of possibilities are beginning to be explored using microcomputer-based instructional materials currently being developed at the World Institute for Computer Assisted Teaching (WICAT).
The research of the RN and PR projects was supported in part by the National Science Foundation under grants SED 79-20591 and SED 82-20591. Any opinions, findings, and conclusions expressed in this report are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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1More sophisticated ordering techniques would have to take into account, for example, that the order of difficulty of the items was not consistent across grade level. We elected to use Table 5.1 because of its intuitive appeal in illustrating the general nature of the "degree of difficulty" relationships among items. The comparisons in Table 5.1 are strictly ordinal and did not enter into any statistical analyses that were applied to the data.