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Behr, M., Post, T., Silver, E., & Mierkiewicz, D. (1980, August). Theoretical Foundations for Instructional Research on Rational Numbers. In R. Karplus (Ed.) Proceedings of Fourth Annual Conference of International Group for Psychology of Mathematics Education (pp. 60-67). Berkeley, CA: Lawrence Hall of Science.




Merlyn J. Behr
Northern Illinois University


Thomas R. Post
University of Minnesota


Edward A. Silver
San Diego State University
Diane B. Mierkiewicz
Carnegie-Mellon University


This paper describes theoretical constructs upon which instructional research concerned with the learning of rational number concepts is being based. The theoretical constructs suggest types of manipulative materials, representational modes, and rational number constructs on which instruction is based. An argument is made that the oral mode of representation is an important intermediary between the manipulative and symbolic modes. Concepts from information psychology are discussed which suggest a basis for observation of student performance.

Reviews of research on the use of manipulative aids in learning mathematics give evidence that their use facilitates learning of mathematical skills, concepts and principles (Fennema, 1972; Kieren, 1969; Gerling & Wood, 1976; (Payne, 1975; and Suydam & Higgins, 1976). Implications for classroom practice is unclear because questions such as how, why, and with whom various kinds of manipulative aids should be used remain uninvestigated.

The current research results reflect little consideration to psychological variables that potentially affect human learning. The work of the project described herein, concerned with the role of manipulative aids in the learning of rational number concepts, derives its theoretical base from three separate but compatible theories. The theories provide perceptual, instructional and information processing models for development of instructional materials and for observation of instructional effects.


The organization and selection of instructional materials used in the project employ two variability principles of Z. P. Dienes. The Perceptual Variability (Multiple Embodiment) Principle suggests that conceptual learning is maximized when a concept is represented in a variety of physical contexts. Exposure to these various representations which differ in appearance but retain the same conceptual structure, is hypothesized to promote mathematical abstraction.

Similar to the Perceptual Variability Principle in that it encourages multiplicity in patterns of exposure, the Mathematical Variability Principle asserts that if a mathematical concept is dependent upon a number of variables, then systematic variation of these is a prerequisite for effective learning of the concept (Dienes, 1967). This principle is hypothesized to promote generalization of mathematical concepts.

Post (1973, 1979; Reys & Post, 1973) have incorporated these principles into a two dimensional matrix each specific to a particular mathematical concept. The project utilized such a matrix designed for the concept of rational number.

The horizontal axis (Mathematical Variability Principle) consists of the five constructs of rational number identified by Kieren (1976); the vertical axis (Perceptual Variability Principle) incorporates the variables of length, area and number and lists manipulative aids upon which project instructional materials are based--counters, paper folding, rods, and the number line.


Reported studies have generally imposed a linear interpretation on Bruner's three modes of representational thought, i.e., first the enactive phase, then the iconic phase, and finally the symbolic phase. More realistically, these phases should be incorporated into an interactive model. Such a model is presented by Lesh (1979) (and the chapter by Lesh, Landau, and Hamilton this volume) who suggests that certain "translation" processes between and within modes of representation may be important in making an idea meaningful. The modes of representation identified in the model are real world situations, spoken symbols, written symbols, pictures, and concrete models. The first two of these represent an extension of Bruner's model. Of major interest in the project is the hypothesis that it is translations among and within certain of these modes which make ideas meaningful for children.


Gagne and White (1978) deal generally with the relationship between memory structures and learner performance. They present a theoretical model upon which research on instruction and learning outcomes can be based. Some elaboration of the memory structures of interest to this research project is given in a later section.


Kieren (1976) has identified five subconstructs of rational number--part-whole, ratio, quotient, operator, measure. Each is important in its own right and in addition contributes to a complete picture of the overall concept. Kieren (1976), Noelting (1979) and Karplus (1970) have all observed the rational number thinking of children at various stages. One concern in this project is development of prototype instructional materials to teach rational number thinking. The current year of our project work has dealt with writing and piloting instructional materials which reflect the rational number subconstructs identified by Kieren. These materials have attempted to emphasize translations within and between modes of presentation (we have elected to use modes of presentation rather than modes of representation because we believe it more clearly conveys the nature of the instructional setting and because it is not possible to previously identify the mode in which an individual encodes a particular experience.)

Initially we conceptualized instruction for rational numbers as suggested in Figure 1.


Figure 1. Conceptual Scheme for Instruction on Rational Numbers


The arrows and dashed arrows in Figure 1 are to suggest hypothesized relationships between rational number constructs, relations and operations on rational numbers. The diagram suggests that (1) partitioning and the part-whole construct of rational numbers are basic to learning other constructs of rational number; (2) the ratio construct is most "natural" to maturate the concept of equivalence and nonequivalence; (3) operator and measure constructs are the most pervasive for maturing the understanding of operations of multiplication and addition of rational numbers. The solid arrows suggest a conviction on our part at this point in our work, while the dashed arrows represent more tentative hypotheses. One important expected outcome of our second year of work is the identification of manipulative aids, which are the most "natural" for developing certain rational number concepts, relationships, skills, and understandings.


The written symbolic mode is the dominate mode in which one generally asks children to demonstrate their knowledge of mathematics. The concrete mode is the dominate mode of presentation. In order for children to meaningfully write and read (decode and comprehend) written mathematical symbolism it is imperative that a strong association between the concrete mode and written symbolic mode be internally established by the learner (Van Engen, 1949). The gap between the concrete and written symbolic modes is greater than earlier perceived (Behr, 1976; Payne, 1975). Persuasive arguments that the oral symbolic mode is an important intermediary can be made.

The project investigators are developing teaching sequences for rational number concepts which emphasize the role of oral verbalization. The teaching sequences require the children to: (1) orally explain and reconstruct the sequential procedure used in the manipulation of the concrete aids; (2) orally verbalize the important mathematical observation to be derived from the objects; (3) record by writing or completing a written English sentence the orally stated mathematical observation. This is done before any attempt is made to introduce mathematical symbolism. In pilot lessons where we have omitted oral activity before going to a written worksheet activity our preliminary observations indicate that children are more likely to have difficulty.


A controversial position among researchers in the psychology of reading (Gleitman & Rozin, 1977) is that reading requires explicit and conscious discovery and building from what one already knows implicitly for the sake of speech. The history of written language indicates a clear trend away from attempts to transcribe meanings directly towards the transcription of meaning mediated by sound. There is increased reliance on speech as a mediator between written symbols and meaning. A linguist Charles Fries (1962) suggests that learning to read in one's native language involves transferring from the auditory signs of language signals which children already know to the new visual signals from the same language signals.

Written mathematical symbolism is different from the written symbolism for our native language. The latter is an alphabetic system while mathematical symbolism is not. Mathematical symbolism is an "abbreviation" of written native language, Because of the interrelations between speaking, reading, and writing, it is reasonable to posit that the oral mode of expression stands as an important intermediary between the ideas of mathematics and the written mathematical symbolization of those ideas.

The Soviet psychologist Vygotsky (1976) addresses the question of the interrelationship between spoken and written language. Vygotsky indicates that

A feature of this system [written language] is that it is a second order symbolism, which gradually becomes direct symbolism. This means that written language consists of a system of signs that designate the sounds and words of spoken language [emphasis ours] which in turn, are signs for real entities and relations. Gradually this intermediate link, spoken language [emphasis ours], disappears and written language is converted into a system of signs that directly symbolize the entities and relations between them (p. 106).

Vygotsky asserts that the primary symbolic representation should be ascribed to speech and on the basis of speech all other sign systems are created.


A review of research on writing by Bereiter (1980) suggests areas where problems might lie for children to express meaningful associations between mathematical ideas expressed via manipulative aids and via mathematical symbolism. The first deals with limitation on children's information processing capacity. Children cannot integrate all of the skills necessary to present a written record of a thought string and retain the thought string simultaneously. Cognitive demands needed to generate the mathematical ideas and simultaneously record the information exceed a child's information processing capacity.

A second involves a developmental sequence of writing skills in children. The simplest system to produce intelligible writing is referred to as associative writing. An apparent precursor to this if transcribed speech. One researcher (Simon, 1973) observed that children in the first years of schooling typically verbalize or subvocalize what they write. This behavior mainly disappears by the 3rd or 4th grade. This seems to be true for narrative writing and also later for expository writing. The problem of writing mathematical symbolism has not been addressed; however, writing fluency with mathematical symbolism becomes intelligible in many learners even later, if at all.


Information processing psychology identifies a number of types of memory structures. Two of particular interest are episodic and semantic memory. Episodic memory receives and stores information about episodes or events which are temporally related as well as the temporal-spatial relations among these events. A perceptual event can be stored in the episodic system solely in terms of its perceptual properties or attributes; moreover, such an ever,' is always stored in terms of its autobiographical reference to contents already existing in the episodic memory store. It is expected that the episodic memory system is susceptible to transformation and loss of information (Tulving, 1972).

A person's episodic memories are located in and refer to his own personal past. It must be translated into the form "I did such And such in such and such a place at such and such a time" (Tulving, 1972). Thus it seems children need to engage in, rather than only observe the manipulation of objects, in order to form episodic memory structures. Implications regarding the use of concrete aids to present mathematical concepts to children are clear.

Semantic memory is necessary for the use of language. It is the part of memory in which the organized knowledge a person possess about words and other verbal symbols is stored. Also stored in semantic memory are the meaning and referents of words and verbal symbols as well as relations among the words and referents and about rules, formulas, and algorithms for the manipulation of the symbols, concepts and relations.

The semantic system permits the retrieval of information not directly stored in it; episodic memory permits retrieval only of information entered on an earlier occasion. The episodic memory system does not include the capabilities of inferential reasoning or generalization: these capabilities are part of semantic memory. By relying on semantic memory it is possible for a person to know something he did not specifically learn.

These observations about episodic and semantic memory have relevance to the question of teaching mathematics based on manipulative aids. Episodic experience which a child gains from concrete aids may not provide retrievable knowledge without semantic information about the episodes and about relationships among different episodic experiences. Verbal interaction by the learner with a teacher or peers to observe the similarities and differences among episodic experiences and the materials on which they are based is probably essential.

This latter contention is supported by Vygotsky's analysis of the process of internalization of knowledge. Vygotsky considers the process of internalization as consisting of a series of transformations:

  1. An operation that initially represents an external activity is reconstructed and begins to occur internally.
  2. An interpersonal process is transformed into an intrapersonal one.
  3. The transformation of a interpersonal process into an intrapersonal one is the result of a long series of developmental events.

It is the second step in this process which is significant in the present context. Every function in children's cultural development, according to Vygotsky, appears twice: first on the social level, and later, on the individual level. That is, first the function appears at the interpsychological level (between people) and then at the intrapsychological level (within the child). This applies equally to various kinds of functions including voluntary attention, logical memory, and to the formation of concepts.

The example which Vygotsky gave to illustrate this process (the formation of a gesture) does not have an exact analog in the context of learning mathematical concepts from concrete materials. However, the following sequence of steps seems compatible with Vygotsky's concept of the process of internalization of knowledge.

Consider the model of paper folding to show fractions: initially the child through imitating the behavior of the teacher, learns to fold and shade the paper to show equal-size parts and represent fractions. Secondly the child orally tells the sequence of steps and explains how the resulting demonstration exemplifies for him/her the mathematical concept represented. As experience of this type continues, the outer speech of the child used to communicate to a teacher or a peer gradually changes to inner speech. In this way the transformations associated with the concrete aids become internalized.

It is in the context of these empirical and theoretical considerations that our research project will attempt to gain some insights into the question of how interactions among and within modes of representation facilitate the learning of mathematical concepts and operations. Emphasis in our research will be on how meaningful symbolism for mathematical concepts can be developed for children.


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* The research was supported in part by the National Science Foundation under grant number SED 79-20591. 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.