

RATIONAL
NUMBER LEARNING AIDS:
TRANSFER FROM CONTINUOUS MODELS TO DISCRETE MODELS Merlyn J. Behr Ipke Wachsmuth Thomas Post 

This paper is concerned with how well children who have head instruction about rational number concepts based on a continuous manipulative aid are able to transfer their knowledge to accomplish tasks based on a manipulative aid which can be interpreted as discrete. The aid used was an egg carton which, if the subject views the holes as salient, is discrete; if, on the other hand, the subject focuses on the proportion of total area enclosed by the edges of each of the holes, then it is viewed as a continuous entity. THE ISSUE The research concerning learning via manipulative materials is somewhat equivocal. The question of what characteristics of manipulative aids best facilitate learning is unanswered. For developing rational number concepts, the manipulative aids most frequently used are called area or measurement models, and set models; equally descriptive of the two categories are the words continuous and discrete, respectively. The work of Payne and his students (Suydam, 1978) suggests that instruction based on the set (discrete) model interferes with children’s fraction understanding previously developed from instruction based on the area model. Novillis (1976) used hierarchical analysis to establish that task based on the set model are more difficult for children than those based on an area model. Behr, Post, Silver, and Mierkiewicz, (1980), on the other hand, suggest that what has been called interference might be more correctly described as appropriate cognitive disequilibrium. Such disequilibirum causes the child to reinterpret the concept by reconciling the differences between models while at the same time observing their similarities or isomorphic properties. This suggestion is consistent with the learning theories of Gagne (1973), Bruner (1966), and Dienes (1971), and with the developmental psychology of Piaget. CONTINUOUS AND DISCRETE MODELS, COGNITIVE DISTINCTIONS From the perspective of the partwhole construct of rational number (Kieren, 1976), there are significant similarities and differences between a continuous model and a discrete model for showing rational number concepts. To represent rational number concepts, each requires: (a) The identification of a unit; for a continuous model, the unit is a set of discrete objects, (b) Partitioning of the unit into parts of equal size (i.e., equal measure); for the continuous model, each part is again a single continuous piece and contiguous to other part(s); for the discrete model, the parts may be a single discrete object, or several discrete objects, and, in general, are not contiguous with other parts. Thus, there seem to be distinctly different cognitive demands for representing fractions with discrete and continuous models. The discrete model involves the need to perceive a set of discrete objects as a unit, one entity. If 15 objects serve as a unit to show fifths, the child must: first, perceive 15 objects as a single entity (i.e., perceive 15 objects as one whole); second, perceive such subset of 3 objects as a subentity (i.e., perceive 3 objects as one subpart). In the majority of instances, objects of the same shape criteria are used, although there are substantial reasons to systematically provide for variation of this. It is not required that the discrete objects which comprise the unit be of the same size or even the same shape, since the appropriate measure of the unit an d subunit relates solely to the cardinality of the sets. For discrete models, the number of items is crucial, not their size and shape; for continuous models, the crucial elements are size and shape., Thus, the perceptual distortion can become even greater with the discrete model than for the continuous model., This can provide for an additional level of cognitive disequilibrium, and provide further opportunity for learning. Finally, a situation which is initially perceived to be discrete, bay, in its analysis and solution, need to be interpreted from both points of view—discrete and continuous. The problem of sharing 14 cookies equally among 3 children can (and will, by some children) be solved by distributing 4 individual cookies to each of the children, the cookies being perceived as discrete objects. To complete the solution, the 2 remaining cookies are treated as continuous objects, each partitioned into thirds, and 2 of the thirds (1/3 +1/3) allocated to each child. THE STUDY The present study was conducted by the Rational Number Project during 198081 (Behr et al., 1980). The Rational Number Project is a multisite effort which began in 1979 with funding from NSF. One focus of the project continues to be to assess the impact of manipulative materials on the development of rational number concepts. SUBJECTS Subjects in the investigation were six children in a 4th grade experimental group in DeKalb, Illinois, five of whom were matched with subjects in a 4th grade comparison group of the same school. In addition, data are available from a group of six 5th graders, which was the pilot experimental group in 197980, and a 5th grade comparison group. Data from a similar group of six experimental children from the Minneapolis site were also analyzed and are included. The 4th and 5th grader experimental subjects from both sites were the students of a teaching experiment conducted over a period of about 18 weeks. Selection of students to participate as experimental and control subjects of an intact class as high, middle, and low mathematics achievers, after having removed form consideration those children at both extremes as well as any with behavior problems of a sort that would inhibit them from providing useful information about children’s mathematical cognition. Four children were randomly selected from each achievement category and then two in each category randomly assigned to the experimental and control groups, respectively. In this way, two groups of six children, matched on mathematics achievement was obtained. ASSESSMENTS Interview assessments for this study were given on a oneonone basis about half way through t e 18 weeks (mid) and another after the completion of the 18 weeks of instruction(final) as part of a more extensive interview assessment. The midassessment took place when children's instructional exposure had been limited to continuous models. INSTRUCTION The teaching experiment provided children with manipulativeoriented, theorybased instruction (Behr et al., 1980). At the time of the (mid) assessment, the children had had experience with the four different continuous manipulative aids: colored fractional parts of circular and rectangular models, paper folding and centimeter rods. Instruction was based on the multiembodiment principle (Dienes, 1967). Students had learned to translate a representation using one of the four aids to a representation using any of the others. They had also learned to translate between different modes of representation, such as manipulative, symbolic, written natural language and oral language. They had associated fraction symbols with embodiments and with word names of fractions. A particular feature of the instruction required children to translate between a familiar and unfamiliar (new) manipulative aid, to explain how the new aid represents the same concepts as the familiar aid, and to observe the similarities and differences between them. Immediately following the midassessment, instruction included activity based on the discrete model, using counting chips. Instruction was then continued on the basic rational number concept, order and equivalence, the concept of unit, and extended to mixed numbers, improper fractions with the like and unlike denominators. CONTROL Children in the control groups utilized the regular textbook program which was highly symbolic and proceeded rapidly through area and set models to focus on algorithmic manipulations. TASKS All tasks, given during the mid and finalassessment, were parts of larger sets of interview tasks. Tow types were utilized. In the first type, children were directed to "put eggs (cubes) into the egg carton so that some fraction of the holes are filled." This direction was intended to suggest a discrete interpretation of the egg carton. The three tasks of the type Egg CartonDiscreterequired children to show 3/4, 2/6 and 1/3 of the holes filled (see figure 1a). The second type of taskEgg CartonContinuous used precut area sections of the egg carton, sized to cover fractional parts of a whole egg carton. The child was asked, "What fraction of the egg carton is covered?" These tasks were intended to suggest a continuous interpretation of the egg carton. The fractional part and cover configuration for the four tasks of this type are shown in figure 1b. Each subject was given each task at the midassessment and again at the final assessment. EGG CARTONDISCRETE Strategies leading to correct solutions are illustrated by student responses classified in the following categories. Category G (Generalized). Responses in the category indicate that the subject abstractly solved the problem by computing n/d of 12 = ____, or used multiplication or division.
Category D (discrete). Responses in this category suggest that the subject converted to and equivalent fraction with twelfths: e.g., covered 3/4 by solving 3/4 = 9/12 and putting in nine eggs.
Category P (Partitioning). The subject's explanation suggested that (s)he first partitioned the carton into sections, then filled holes; partitioning was observed by the subject's overt outlining of sections.
Category LTP (Lowest Terms Partition). Responses in this category suggest that the subject made groups of holes with d in each group, then showed n/d as n of d parts in each group of d.






Category EP (External Partitioning). The subject first solved the task outside the egg carton and then transferred the finished solution to the egg carton.
Strategies leading to incorrect solutions are illustrated by student responses classified into the following categories. Category RDU (Redefines Unit). Subject centered on only a section of the egg carton with holes equal in number to the denominator and filled holes equal in number to the numerator, in that section.
Category I (Isomorphic Representation). Subject made a physical n/d representation in the carton.




Category M (Multiplication). The subject's explanation suggests that (s)he covered holes according to the formula: n x d equals the number of holes to fill.
Category A (Addition). The subject's explanation suggests that (s)he covered holes according to the formula: n + d equals the number of holes to fill
Category DK (Don't Know). Egg CartonContinuous Strategies leading to correct solutions are illustrated by student responses classified into categories as follows: Category C (Continuous). The subject's response suggests that (s)he treated the one or two covering sections as a single entity (area) and mentally generated the lowest terms fraction. No reference was made to the unreduced fraction name. 



Category EF (Equivalent Fraction). Responses in this category suggest that the subject first generated the answer in twelfths and then converted it to an equivalent lowerterm fraction.
Category PLT (Partitioned Lower Terms). Responses in this category suggest the subject treated the covering sections(s) separately or in sub sections (other than the size one) and generated a fraction reflecting that partitioning. IF the subject verbalized n/12 at any time, a lowerterms fraction was given as well.
Category PHT (Partitioned Highest Terms). Responses in this category suggest the subject generated the x/12 form and then explained in terms of 12 holes with x holes filled.
Strategies leading to incorrect solutions are illustrated by student responses classified into categories as follows. Category (Partial Discrete). Response in this category suggest that the subject incorrectly generated the x/12 form in terms of 12 holes with x sections filled.
Category M (Multiplication). Responses in this category indicate that the subject gave a wholenumber interpretation that n/d is n sets of d.
Category R (Ratio). Responses in this category suggest the subject used the ratio of the number of covered parts to the number of uncovered parts.
DISCUSSION Much research in mathematics education has been either experimental or survey (status study) research, inferential or descriptive, respectively. Experimental studies are designed to test previously formulated hypotheses. Status studies describe the state of the art in terms of student, teacher, or content variables; they are not designed to assess the impact of instructional variables on student's conceptual development. Unlike experimental and statusstudy research, teaching experiments such as this are designed to generate hypotheses and to describe student behavior following carefully formulated instruction. This is in contrast to the status study which describes behavior with little or no knowledge of its instructional precursors. Since the data was taken from a small sample, the results should be taken as indicating trends and suggesting hypotheses, rather than giving confirming evidence. The egg cartondiscrete items were production tasks, whereas the egg cartoncontinuous items were interpretive, Inferences made from any differences in performance between the types of tasks must take account of this. Based on the percentages of responses within categories exemplified by strategies leading to correct answers, substantial differences between experimental and control groups were observed. (See Table 3.) This was true at both the fourth and fifthgrade levels, These differences were especially apparent for the egg cartondiscrete tasks. Twothirds (67%) of fifthgrader experimental children used strategies leading to correct solutions for discrete tasks at the midassessment, while none (0%) of the control group did. The corresponding percentages for the final assessment at the fifthgrade level were 67 and 40. Similarly, for the fourthgrade, the midassessment percentages correct were 56 (DeKalb, experimental); 82 (Minneapolis, experimental); and 33 (control). Differences of this magnitude were not observed for the continuous tasks. At the fifthgrade level, both experimental and control students correctly answered all tasks in both the mid and finalassessments., At the fourthgrade level, minor differences were found between the two experimental and control groups. These differences favored the experimental groups (see Table 3). These observations of the data lead to several suggestions. 1. Fourthgrade children who are given instruction on rational number concepts based on a systematic use of continuous embodiments will be more successful with rational number learning and performance tasks based on a discrete embodiment than children whose instruction on rational number concepts is not based on as systematic use of such embodiments; i.e., instruction using continuous embodiments facilitates learning from discrete embodiments. 2. Fourthgrade children who are given instruction on rational number concepts based on systematic use of both continuous and discrete embodiments will be more successful with rational number tasks based on a discrete embodiment, and moderately more successful on rational number tasks based on continuous embodiment, than students whose instruction is not based on a systematic use of embodiments. 3. The difference, stated in 2 above, of tasks based on discrete embodiments will persist through the fifth grade, although the magnitude of the differences will be less. 4. Knowledge of basic rational number concepts gained from instruction employing continuous models transfers as knowledge for performance on basic rational number tasks based on discrete models. The percentage of correct responses in the experimental groups (see Table 3) increased or remained constant from midassessment to finalassessment on the egg cartoncontinuous tasks. IN fact, the performance of all individual children in the experimental group increased or remained consistent (see Tables 4 & 5). Since these children received instruction between assessments based on both discrete and continuous models, this information lessens the strength of Payne’s (1976) claim that instruction based on discrete models interferes with children’s performance on tasks based on continuous models. 5. There is no longterm detrimental effect on children’s knowledge of rational numbers derived from instruction based on continuous models when this is followed with instruction based on a combination of discrete and continuous models. Table 6 summarized information from Tables 4 and 5 and gives the frequencies of pairs of responses on the same item that dropped, stayed the same, or rose in category level from the midassessment to finalassessment. These data only include the frequencies for which responses on the final assessment were correct. Among the discrete tasks, the data suggests a slight trend favoring the experimental groups; and, among the continuous tasks, there appears to be a sharp difference favoring the experimental groups. This together with the observation (see Table 3) that the experimental treatment apparently results in more correct responses suggests: 6. Instruction for rational number concepts arising from instruction incorporating a systematic use of continuous and discrete manipulative aids will result in a higher percentage of correct responses and a higher quality of responses on rational number tasks than a "traditional program" among fourth grade children.












 
Implications The results obtained further reinforce the earlier work on Payne (1976) and Novillis (1976) which suggest that discreteembodiment tasks are more difficult for children than continuousembodiment tasks. This, perhaps, should not be surprising, since the intended unit (and appropriate subunits) from which any fraction derives its meaning is more easily ascertained in a continuous embodiment. In the discrete context, children may have difficulty with the fact that each subunit may be composed of several discrete objects. For example, 1/5 of 15 things is one set of 3 things. Each onefifth is itself a set of 3 separate things. As one student said when describing a related task, egg carton task, "It’s hard to look at it that way, but it is possible…it’s kind of hard to picture three holes as one." With a continuous embodiment, onefifth is one piece of the whole unless a fraction equivalent to onefifth is represented and the task is to interpret the representation as 1/5. The perceptual cues and the associated verbiage is different for the two types of embodiments; the discrete context contains additional variables which must be mentally coordinated by the student. With this additional required mental coordination, it can be expected that students will experience increased difficulty. Our findings suggest several things about teaching children fraction concepts using continuous and discrete manipulative aids. While children do have difficulty interpreting parts of a whole of a fraction with a discrete embodiment, this difficulty has positive learning effect in the long term. The difficulty in interpretation causes the children to rethink the meaning of fraction as parts of a whole. This rethinking, or reconceptualization, deepens the child’s understanding of fractions and rational numbers. When interpreting a fraction representation with a familiar manipulative aid becomes automatic for a child, it is unlikely that new learning takes place. Yet this automatically does not necessarily suggest that the child has full understanding of the fraction concept. By introducing a new manipulative aid into the learning environment which has different perceptual features, such as changing from continuous to discrete embodiments, this automatically is interrupted and new thinking takes place and new insights and understandings can occur. The use of multiple embodiments, and especially ones that differ in perceptual features, is believed to enhance learning and understanding. This claim was originally made by Z. P. Dienes (1971) in his Perceptual Variability Principle which suggests that variation in the perceptual features of manipulative aids is necessary in order for children to observe and abstract the critical attribute represented in the aids; in this case, the attribute of equal parts of a whole. From the research reported in this paper, and from some of our other work (e.g., Behr, Wachsmuth, Post, & Lesh, 1984) we have been able to formulate a procedure for introducing new manipulative aids into the teaching environment in such a way that the learning from multiple aids substantially increases learning over the use of a single aid. The procedure is to work with an aid until children become quite familiar with it and are able to use it to represent the concept of interest without much difficulty. A new aid is introduced in the following way: The teacher demonstrate a procedure for representing a rational number, or other concept, with the new, or the familiar aid; after each step in the demonstration, the teacher asks the children to (a) describe orally what the teacher did, (b) accomplish, on their own, the same thing with the other aid, and finally (c) discuss with the children what is the same and different about the two displays. This activity challenges the learners to do two types of thinking. Bottom up thinking to interpret the teacher’s demonstration, and top down thinking to reconceptualize the concept represented to make it "fit" with the new aid. An example will help to clarify. Suppose the children are familiar with using paper folding to represent fractions. The teacher may wish to introduce centimeter rods as another manipulative aid to show fractions. A first step could be for the teacher (choosing to work with the fraction _) to fold the paper into 4 equal parts. Then the teacher would ask the children to describe what was done took a unit and partitioned it into four equal parts. Next the teacher would ask the children to do the same with the centimeter rods. This would challenge the children to (a) find a rod to use as a unit, (b) recognize that partitioning could be accomplished by putting smaller rods on top of or beside the unit rod (in some cases the teacher will have to suggest a particular rod to use as a unit) and finally, (c) to discuss with the children what is alike and different about the two types of displays, so far. This procedure is repeated until the children complete a representation of _ with the rods. Sometimes it works better for the teacher to demonstrate with the new aid and have children interpret using the familiar aid. The important point is that children are given the opportunity to interpret a representation for a concept and then do a cognitive reorganization of this concept to make it fit another representational devise (i.e., another manipulative aid). This change between top down and bottom up thinking will deepen the child’s understandings of the concept. Additional continuous manipulative aids, and ultimately discrete aids, can be introduced in a similar manner. References Behr, M., Post, T., Silver, E., & Mierkiewicz, D. (1980). Theoretical foundations for instructional research on rational numbers. In R. Karplus (Ed.), Proceedings of the Fourth International Conference for the Psychology of Mathematics Education (pp. 6067). Berkeley, CA: University of California. Behr, M., Wachsmuth, I., Post, T., & Lesh, R. (1984). Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 15, 323341. Bruner, J. S. (1966). Toward a theory of instruction. Cambridge, MA: The Belknap Press of Harvard University Press. Dienes, Z. P. (1971). Building up mathematics (2nd ed.). London: Hutchinson Educational. Gagne, R. M. (1973). The conditions of learning. Worthington, OH: Charles A. Jones. Kieren, T. E. (1976). On the mathematical, cognitive, and instructional foundations of rational numbers. In R. Lesh (Ed.), Number and measurement: Papers from a research workshop (pp. 101144). Columbus, OH: ERIC/SMEAC. Novillis, C. G. (1976). An analysis of the fraction concept into a hierarchy of selected subconcepts and the testing of the hierarchical dependencies. Journal for Research in Mathematics Education, 7, 131144. Payne, J. N. (1976). Review of research on fractions. In R. Lesh (Ed.), Number and measurement: Papers from a research workshop (pp. 145187). Columbus, OH: ERIC/SMEAC. Suydam, M. N. (1978). Review of recent research related to the concepts of fractions and of ratio. In E. CohorsFresenberg & I. Wachsmuth (Eds.), Proceedings of the Second International Conference for the Psychology of Mathematics Education (pp. 296332). Osnabruck, FRG: Universitat Osnabruck.


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