

ABSTRACTION, GENERALIZATION,
AND THE DESIGN OF Thomas R. Post Robert E. Reys 





An understanding of variables affecting conceptual development is essential to persons constructing environmental situations within which children will learn mathematics. The vast majority of inschool mathematical experiences emanate from scope and sequence considerations and reflect little consideration of psychological variables which have been hypothesized to affect human learning. "Use manipulative materials" "get the child involved" "provide real problem solving situations," etc., are cliches which all support, but few seriously implement at the classroom level. Cognitively oriented psychologists have reemphasized the role which process considerations might play in the design of learning experiences for children. The current clamor for accountability and in some cases even the generation of a list of minimum mathematical competencies required for high school graduation represent in our minds, a reassertion of the primary concern for product outcomes in the mathematics curriculum. In some ways this is a refutation of the growing acceptance of process oriented objectives which has characterized the last decade. It seems that the parameters of children's mathematical involvements both process and product are determined externally with little or no direct consideration for the psychological aspects of human learning. Researchers are amassing currently a rather impressive data base in support of the use of a variety of manipulative materials in the mathematics program (Suydam 1976). To ask whether children should use physical materials in learning mathematics is analogous to asking whether children learn most effectively from direct experiences or through pseudo experiences. Evidence exists to suggest that the latter type experiences are characterized by symbol manipulation. These simply have been unsatisfactory. What conditions are responsible for the current state of affairs? Surely many variables have been contributory, and it would be impossible to enumerate them. It is our contention that one of the most powerful factors is a fundamental incompatibility between the nature of the learner and the instructional packaging of the mathematics to be learned. In general it is fair to say that there exists a disparity between learning theory and instructional design. Most attention seems directed at the decision to include or not include a particular topic, but considerations as to the appropriate mode of instruction to be utilized are also essential. Assuming a decision to include q particular topic in the mathematics curriculul1J, the next consideration for the curriculum developer is normally concerned with the mathematical development of that idea (sequencing, etings practice exercise, etc. ).Such concerns are obviously legitimate and necessary, but they are not sufficient. If attention to instruction 1 design terminates with content structuring and sequencing, psychological considerations tend to be deleted from plnnin9. In the final analysis, "how" children learn mathematics may be more important than "what" mathematics they do learn. Looking at this issue from a slightly different perspective the broad realm of instructional design can be thought to have two broad types of subcomponents or subareas of concern:
Clearly, these are two separate yet highly related phenomena. The first of these emphasizes the teacher's role in the instructional process, the second emphasizes the instructional sequence from the learner's point of view. The former tends to be extrinsic to the learner. the latter intrinsic. Many attempts to redesign existing mathematics programs or develop new curricula have failed to provide overt recognition to the learning process. Rather, concern has been primarily with the teaching process. Questions such as the following need much more attention: What are the psychological variables hypothesized to promote conceptual development in children? In adolescents? In adults? How can they be mobilized, utilized, or exploited so as to promote more effective learning? Are these variables different across ages? If so, how? Do such variables differ across particular concepts? Are they unique within concepts? etc., etc., etc. And perhaps most importantly how might (should) these variables be accounted for in the construction and sequencing of activities for children? The answers to such questions surely would be useful in future attempts to design instructional activities which do consider both the extrinsic and intrinsic aspects of human learning. The model described in this paper deals with the development of embodiment matrices for various mathematical concepts. It is an attempt to systematically provide for certain psychological aspects of conceptual development in the design of mathematical activities for children. In particular, the model is concerned with abstraction and generalization. It is viewed as a bridge between certain psychological aspects of conceptual development and the design and implementation of instructional activities. The remainder of this paper is divided into five sections
Theoretical Foundations The theoretical foundations of these ideas have been provided by the writings of Z. P. Dienes and E. W. Golding (1971a, 1967, 1971b). Dienes, while generally espousing the views of Jean Piaget, has made contributions to the cognitive psychological view of mathematics learning that are distinctly his own. It is appropriate here to review briefly the major components of the theory of mathematics learning proposed by Dienes. His four "principles" are meant to identify necessary although perhaps not sufficient, since specific teacher methodology is not consideredcriteria for effective mathematics learning. The Dynamic Principle The dynamic principle suggests that true understanding of a new concept is an evolutionary process involving the learner in three temporally ordered stages. 1. The preliminary, or "play" stage, is in evidence when the learner is concerned with activities of a relatively unstructured nature but which provide actual student experience to which later experiences can be related. Although unstructured, the activities here are not random. The child is presented with an "educational ball park," within which the concept{s) are embedded, and given complete freedom to "play" in it. The learner is therefore exposed to basic rudiments of concepts in a very informal manner. Since freedom to experiment is essential, the learners' responses, varied though they may be, must be relatively uninhibited by the teacher and treated with genuine respect. 2. In the "becoming aware" stage, which follows the informal exposure afforded by the play stage, more structured activities are appropriate. It is here that the child is given experiences structurally similar to the concepts to be learned. 3. The third component of the dynamic principle is characterized by the emergence of a mathematical concept with ample provision for reapplication to appropriate environmental stimuli. The concept is not considered to be fully operational until it can be freely recognized and applied to relevant situations. Ideally this practice stage .will serve a dual role: to solidify the newly formed concept in the child's experience and to serve as a play stage for the next concept to be learned. A cyclical pattern emerges which can be depicted as shown in Figure 1. The completion of this cycle is necessary before a mathematical concept becomes operational for the learner. Furthermore, Dienes maintains that this cycle is repeated continuously as new mathematical concepts are formulated. The similarity between Dienes. model of learning and Piaget's description of the processes of assimilation and accommodation is apparent.


Figure 1. Third Component of Dynamic Principle. 

Dienes (1967 1971) has elaborated on this process and referred to it as a learning cycle. The cycle has been subdivided into six consecutive stages each of which is an essential component of effective mathematics learning. Briefly summarized they are as follows: 1. Interaction with the environment. 2. Rule construction of manipulation. This stage is a natural consequence of discovering regularities and of subsequent experimenting with newfound rules of constraints. 3. Comparison of several games or activities having identical structures or rules. (This may be referred to as the search for isomorphisms.) 4. Representation of like (or isomorphic) structures. This involves realization of the "sameness" of the structures in question together with an ability to determine a method of representing all games (activities) having a similar structure. 5. Symbolization. This is characterized by the investigation of the properties identified in 4. At this 1evel such investigation is not dependent on the real world referent although it should be available to corroborate findings generated in symbolic form. 6. Formalization. This means the derivation of other properties of the system from those already identified. This is the process of generating theorems (statements logically deduced) of a system from its axioms (rules of the game or selfevident truths). The process of theorem derivation is known as a proof and is considered to be the quintessence of mathematical activity., These stages describe the general characteristics of a sequence of experiences that result in the appropriate development and subsequent abstraction of a given concept. The remaining components of Dienes' theory are in a sense contained within the dynamic principle and should be considered to exist within the framework already established. The Perceptual Variability (multiple embodiment principle) In an attempt to make provision for individualized learning styles and at the same time promote abstraction of mathematical ideas, Dienes suggests the perceptua1 variability principle as an indispensable element in the process of concept formation. This principle suggests that conceptual learning is maximized when children are exposed to a concept in a variety of physical contexts or "suits of clothes." The experiences provided should differ in outward appearance while retaining the same basic conceptual structure. Children can become sidetracked with irrelevant characteristics of a situation, especially when their grasp of the concept is incomplete. The Association of Teachers of Mathematics (1967) has captured the essence of this phenomena in the following quote:
Dienes believes that the current mode of mathematics instruction promotes learning which is associative in nature. Children are encouraged to associate a particular mathematical process or operation with a particular situation. A "bag of tricks" approach is often used by the teacher, applying established mathematical procedures in a manner which tends to be routine or habitual. The problem with associative learning arises when the student finds himself in a situation for which he does not possess a readymade response pattern. Under these conditions the associative learner becomes bewildered and unable to abstract relevant components of the problem situation. It is, therefore, unlikely that he will be able to reconstruct the problem in a manner which will ultimately lead to a solution. As an alternative to this type of learning, Dienes suggests an instructional setting which would promote abstraction rather than association. (Abstraction is here defined as the ability to perceive a concept irrespective of its concrete embodiment.) Dienes believes that by providing children with the opportunity to see a concept in different ways and under different conditions, the purposes of promoting abstraction will be more adequately served. The Mathematical Variability Principle 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 certain number of variables (quantities which may assume a variety of values), then variation 'of these is an important prerequisite for the effective learning of the concept. For example, a parallelogram is defined as a quadrilateral having its opposite sides parallel. Although its shape can be varied by changing the length of its sides and the size of its angles can be altered, the only crucial factor for a quadrilateral to remain a parallelogram is that its opposite sides remain parallel. If a child were exposed to only those parallelograms with equal angular measure or only to those which have constant proportion relative to the length of its adjacent sides, surely he would not develop a general concept of parallelogram. The mathematical variability principle suggests that in order to maximize the generalizability of a mathematical concept, as many irrelevant mathematical variables as possible (in this example, the size of angle and the length of side) should be varied while at the same time keeping the relevant variables (opposite sides parallel) intact. By the same token, if one is interested in promoting an understanding of place value, it is desirable to vary the base while providing experiences which highlight the consistency of regrouping procedures, the importance of relative positional value, and the appropriate way to record results. When implemented, the mathematical variability principle encourages the student to separate the "wheat from the chaff" by systematically isolating relevant variables in the consideration of a given concept. This could result in conceptual learning with a degree of precision transcending that usually found in the mathematics classroom. The Constructivity Principle Here Dienes identifies two kinds of thinkers: the constructive thinker and the analytical thinker. He roughly equates the constructive thinker with Piaget's concrete operational stage and the analytical thinker with Piaget's formal operational stage of cognitive development. This principle states simply that "construction should always precede analysis." It is analogous to the assertion that children should be allowed to develop their concepts in a global intuitive manner emanating from their own experiences. According to Dienes these carefully selected experiences form the cornerstone upon which all mathematics learning is based. At some future point in time, attention should be directed toward the analysis of what has been constructed however, Dienes points out that it is not possible to analyze what is not there in some concrete form. Summary and Implications The unifying theme of these four principles is undoubtedly that of stressing the importance of learning mathematics by means of direct interaction with the environment. Dienes continually asserts that mathematics learning is not a spectator sport and, as such, requires a very active type of physical and mental involvement on the part of the }earner. In addition to stressing the environmental role in effective conceptual learning (Dynamic Principle), Dienes in his mathematical and perceptual variability principles addresses the problem of providing for individualized learning rates and learning styles. His constructivity principle aligns itself closely with the work of Piaget and suggests a developmental approach to the learning of mathematics which is temporally ordered to coincide with the various stages of intellectual development. This paper will attempt to apply the mathematical and perceptual variability principles to specific mathematical concepts. This application will result in the development of a two dimensional matrix for each mathematical concept considered. The vertical and horizontal axes of each matrix will consist of examples of the perceptual and mathematical variables which are relevant to that particular concept. Before suggesting procedures for the development of such specific models it is appropriate to briefly review some of the more important literature in this area. An Overview of Related Research The model for the design of instructional activities proposed in this paper relies heavily on the "parallel series" organization of instructional activities. This arrangement is best characterized by the term multiembodiment, which will provide the central focus for this brief review of related research. First, it is important that proper embodiments of the mathematical concept be provided. When physical materials are used in instruction, they should provide a concrete representation, or embodiment, of a mathematical principle. When different, yet appropriate, concrete materials are used to develop the same mathematical idea, a "multiple embodiment" is provided. This approach to instruction demands that teachers use a variety of perceptually different materials in developing a mathematica1 concept. Furthermore, it is essential that the physical embodiments be clear and meaningful to the learner if later symbols associated with these embodiments are to be understood (Fennema, 1972). One of the basic assumptions underlying the use of physical materials is that pupils learn best through active involvement with concrete experiences. However, this depends on the concepts involved, the age and previous experience of students, and the particular materials supplied. One cannot say, at least with any degree of certainty, that using physical materials with active learning experiences will be effective in helping all children master all types of mathematical objectives. Too much depends on the complex interaction among and between pupils, teachers, and materials. Consequently, research regarding the value of physical materials has been until recently generally inconclusive (Fennema, 1972i Kieren, 1969i Wilkinson, 1974). That is, a case could be made for either side simply by choosing research reports that support a particular bias. In an unusually comprehensive review and synthesis of current research conducted primarily since 1970, on ActivityBased approaches to mathematics teaching, Suydam and Higgins (1976) have summarized the results of twentythree studies dealing with the use and nonuse of manipulative materials and twentyeight studies contrasting concrete, pictorial, and symbolic types of materials. In addition, this very useful document summarizes results of many other studies dealing with the interaction of multiple embodiments, age, ability, and material use and with the impact of materials on the learning of specific mathematical content. The findings summarized in the Suydam and Higgins paper include the following:
The research reviewed in the Suydam and Higgins document establishes the importance of using manipulative materials' to help children learn mathematical concepts. However, the questions of how specific materials should be organized and sequenced, the subject of interest in the present paper remains essentially unanswered. Several recent reviews of research (Beougher, 1967; Fennema, 1972; Kieren, 1969; Suydam & Weaver, 1970) also have identified investigations related to multiple embodiment. A critical review of published studies reveals a wide range in the quality of research and therefore raises doubts about the credibility of certain findings. Nevertheless, it is clear that the research does not consistently support or refute a mu1tiple embodiment approach to teaching mathematics. In fact, the one common thread among these studies is that learning mathematics depends more on the teacher than on the embodiment used. In addition to research specifically related to mathematics education, there are several more general findings which support a multipleembodiment approach to instruction:
Although none of these statements deal exclusively with mathematics learning, they are consistent with a multipleembodiment approach to teaching mathematics. Many additional issues related to embodiments and levels of concept formation need to be considered. Several questions, related to quality, number, sequence, and time allocation for embodiments, seem particularly important. For a given mathematical concept:
These issues are complex. It seems unlikely that research will provide simple answers. It seems more likely that answers to such questions will need to be qualified in terms of teachers, mathematical content, and embodiments used, as well as by the pupil's ability, back ground, and achievement level. Nevertheless, these issues are significant. Future research studies related to multipleembodiment should be conducted for longer periods of time than the vast majority of the reported studies. Many mathematical concepts simply cannot be learned in three to six weeks. Extended time frames also are crucial if retention as well as skills in generalizing and/or abstracting are to be of major concern. We turn now to a description and discussion of the general procedures which are used to generate a two dimensional matrix for a particular mathematical concept. This discussion is followed by specific examples of such matrices. General Procedures for Specific Model Development All exemplars presented in this paper reflect the two variability principles. The individual examples are topic specific and are not unique. Generally, the specific models are developed in the following manner: The first step is to identify an appropriate mathematical concept. The identification of the mathematical and perceptual variates constitutes the next and perhaps most crucial stage in the development of a particular example. The perceptual and mathematical variates will define the dimensions of a twoway matrix. This can be depicted as: CONCEPT X .


In this example, three "levels" (rows) of the perceptual variate have been identified for the concept in question along with four levels (columns) of the mathematical variate. Each cell in the model defines both a specific physical material (embodiment), and a corresponding level of an appropriate mathematical variate. Once this has been done, it is then reasonable to begin to synthesize existing activities, lessons, practice sheets, etc. The vast majority of such activities are already in existence. This procedure, therefore, does not require a great deal of curriculum development, only the synthesis of existing software, so as to conform with the parameters established by the identification of appropriate perceptual and mathematical variates. This is a particularly important point since the establishment of large scale curriculum development projects appears to be unlikely at this time. The cell entries can be physically represented by "shoebox" activities complete with both hardware and software. These shoe boxes might be used at a mathematics station within the classroom. If conveniently stored, students could be responsible for obtaining these shoe boxes, completing the activity within, either individually or in small groups, then returning it for use by other students. Storage of such materials generally has not been a major concern. Decisions would naturally need to be made as to the extent of student involvement in a particular row or column. Instructional components implied by cells in a matrix will not be experienced always by children in a single (consecutively ordered) series of experiences. It is quite probable that involvement in many mathematical concepts will be spread over several years. All exemplars will contain a row which deals primarily with symbolic representation and/or manipulation. Symbolic representation is considered to be an "embodiment" of a particular concept even though in and of itself it is devoid of any physical context. Symbolic representation, then, is a different kind of embodiment with special useful properties. As such it is deserving of differential treatment. One aspect of such differential treatment is the fact that it will be regularly juxtaposed with other physical embodiments. Symbolic representation is routinely placed in the bottom line to imply that it is the most complex and that students would not become involved exclusively with the manipulation of symbols unless and until they have had a number of different experiences in less abstract form. The reader will note that in the examples to follow not all possible "levels" of either the mathematical or perceptual variables have been included. Perhaps a favorite or highly effective embodiment does not appear. Perhaps an obvious or important mathematical variate is absent. Such omissions are acknowledged. It is not the intent to specify "the best embodiments or levels of the mathematical variate. Rather the point of concern is that the psychological factors (abstraction and generalization) to which the perceptual and mathematical variability principles address themselves are indeed worthy of overt attention and need to be provided for when designing instructional activities for children. The identification and structuring of activities in this manner serves to highlight this overt recognition. One major difficulty in designing the "best" two dimensional matrix for a particular concept which depicts the most effective physical embodiments and the most important mathematical variates is the fact that research in these areas is fragmentary and of uneven quality. That is, the relative effectiveness of various embodiments and the identification of the most important mathematical variates has not been firmly established by empirical research. It is therefore not possible to identify the "best" matrix with any degree of assurance. The exemplars to follow undoubtedly will strike a familiar cord with most readers. The organizational format perhaps is a bit different but the components of individual cells will not be unfamiliar. Each has been fitted to the general paradigm already established. The next section of this paper provides specific examples. These are to be read subject to the qualifications noted above. Exemplars The first example (see Figure 2) deals with the concept of place value. In this instance, the selection of appropriate perceptual and mathematical variates is fairly clear. The concept of "base" or "number system" constitutes the mathematical variable to be systematically manipulated. This is appropriate since operations within all number bases are governed by the same basic regrouping procedures except that the number of pieces involved in a trade will vary with the number base considered. When students realize that the "rules" governing, regrouping procedures are consistent across bases they have in effect generalized the concept, that is, they have developed the ability to apply these rules to any arbitrary system.


Within each base it is also possible to employ a wide variety of physical materials. Children should experience the concept while working with a number of materials that are different in appearance but similar in structure or function. The various materials thus become the perceptual variates or embodiments. When students realize that multibase arithmetic blocks, place value charts (using wooden sticks), and chip trading activities can be utilized in ways which are structurally similar, i.e. can all be used to illustrate the same basic mathematical concept, then they will have abstracted the concept from its physical context. It is not the ability to work with any specific set of materials which is the abstractive component of this mathematical concept, but rather, a realization of their sameness. Obviously if such perceptual variability is not provided within the instructional sequence then opportunities for such abstraction are minimized. One possible arrangement of learning activities concerned with the concept of place value is also depicted by the matrix in Figure 2. Each cell implies a collection of activities dealing with a particular base and a particular type of manipulative material. Unidirectional arrows indicate temporal sequencing. Dual directional arrows imply that interchangeable sequencing is appropriate. Note that in this case work with the "easier" bases is to precede student interaction with base 10. Also note that symbolic representation without some form of concrete referent is considered last within a particular base, although it might be juxtaposed with specific materials earlier in the sequence. Such ordering is consistent with the Dienes' constructivity principle since it does not place children in a position where they are asked (or expected) to analyze in the abstract something that does not mentally exist in some type of concrete form to begin with. In short, students do not work exclusively with symbols unless and until they have had experiences with the concept at the enactive (concretemanipulative) level of operational thought. Certainly all students would not be expected to complete activities within all matrix components. Some students will need more exposure to the concept than others however, all students probably should sample from two or more rows and two or more columns if appropriate conceptual abstraction (promoted by perceptual variability) and generalization (promoted by mathematical variability) are to occur. The ultimate decision as to the number of matrix components considered by a given student is, and should be, left to the teacher. Hopefully research will eventually aid such decisions. Such decisions should be made, with full acknowledgement of the fact that premature student abstraction will only result in future difficulties. Furthermore, these decisions should be made on an individual rather than on a large group basis. Each cell in the matrix implies both a mathematical and a perceptual variate, in this case a number base and a specific type of manipulative material respectively. For example, the "contents" of the checked cell (I) in Figure 2 would consist of a number of activities dealing with the chip trading kit in a "trade for 5" setting. Research is currently underway on this particular model to assess the impact of the variability principles (alone and in concert) on conceptual development with primary grade children. Area Figure 3 illustrates perceptual and mathematical variability dimensions as applied to the concept of area. In this particular model, area is represented by individual tiles, as an enclosed space on a geoboard and as an enclosed space on a piece of graph paper. In addition, of course, abstract formulas are included as the most sophisticated interpretation of the concept. Shape has been identified here as the mathematical variable to be systematically manipulated. As can be seen, the shapes manipulated include four which lend themselves to simple formulas and the fifth (nonregular shapes) for which formulas are not readily apparent. Children can find areas of physical spaces using the technique of super position, using for example floor tiles at a very early age. Certainly one would not expect these same children to become involved in abstract formulas for shapes of a trapezoidal nature. Furthermore, many of the skills developed here within various embodiments will be called upon later to determine surface area of geometric solids.


Figure 3 

The arrows in Figure 3 indicate the order in which activities could (or should) be presented. noteworthy is the unidirectional arrow between the concrete embodiments and the logical formulas. Once again these arrows reassert the constructivity principle as suggested by Dienes, that is, one cannot analyze abstractly what has not been previously constructed in a more concrete form. Note also the bidirectional arrows connecting other matrix cells. Such directionality implies considerable flexibility in the selection of activities. It would be possible to debate the implications of the placement of several of these bidirectional arrows, however, for our purposes here we wish only to suggest that "absolute" sequences probably do not exist. In fact, discussion as to the appropriateness of the relative position of any column or row would no doubt be a fruitful activity for the teacher planning to use such an organizational format. Extending the Natural Numbers to Integers Figure 4 suggests ways of expanding the concept of number to include negative integers. Five different "physical" (or conceptual) embodiments are illustrated. In each case identification of the mathematical variable was limited to three components. Undoubtedly this particular concept has many different manifestations. The intent, however, is not to argue for the uniqueness of appropriate mathematical variates, but rather, to suggest a mental set for teaching particular concepts which systematically manipulates those variables not fundamental to the concept under consideration. In fact, particular rows, columns, and/or cells might themselves be conceptualized as a concept to be taught. In this event, perceptual and mathematical variables might then be identified for that particular sub unit of a given model. For example, in Figure 4 the concept of addition which is identified here in the second column clearly can be identified as a concept in and of itself. Perceptual and mathematical variates appropriate to the concept of addition (or subtraction or counting and ordering) would then be identified. Were one to consider the concept of addition it is likely that Cuisenaire rods, counters, number lines, etc. as well as symbolic algorithms would surface as perceptual variates. Appropriate mathematical variates might be identified also. The fact that the model(s) suggested can be continually atomized is simply a re assertion of the sequential nature of mathematics. Clearly a child should not be involved in the activities implied by Figure 4 if that child does not have at least a passing acquaintance with the concept of addition of natural numbers.


Figure 4 

It is not necessary here to elaborate on the perceptual variables identified in Figure 4. Suffice it to say that all provide a slightly different physical embodiment of the identified concept. Such breadth of exposure should limit the probability of associating the concept with one particular situation. Such overt variation is hypothesized to maximize the abstractive potential of the instructional sequence. with previous models it also should be noted that the row entitled Symbolic Representation in reality will be juxtaposed with other embodiments. Symbolic representation is placed routinely at the bottom line to imply that students would not become involved solely with the manipulation of symbols before having had a number of different experiences in a less abstract form. Figure 5 is concerned with: Equivalent Fractions." In this case the identification of the perceptual and mathematical variates is clear and further discussion does not seem necessary.


Figure 5 

Figure 6, dealing with the Concept of Measurement, is at once different from the previously mentioned models. Originally proposed by two especially innovative graduate students (Koch and Garfield), Figure 6 adds a third dimension this dimension attempts to impose a number of general process or methodological considerations on the structure already established. The implications of the addition of a process dimension is intriguing. At this point, however, the number of additional cells required because of the addition of a third dimension seems unwieldy. Many cell components would require the creation of, rather than the synthesis of existing software. Reality constraints suggest that initially a more conservative approach utilizing only two dimensions be considered. Ultimately the process issue must be addressed. Whether the components of this third dimension are the ones suggested in Figure 6 is less important than the fact that process be considered overtly at all. Our reluctance to develop this idea further is due only to the recognition that large scale curriculum development is a major undertaking and as such transcends the scope of the suggestions made in this paper.


Figure 6 

Additional Thoughts The nonuniqueness of the perceptual and mathematical variables utilized in these illustrations has been noted. Certainly some degree of discretion, professional judgment, and forethought is advised in their selection; but in the final analysis they are clearly not unique. This condition was at first disturbing to us. We naturally wanted the models developed to be "the right ones" for the concepts under consideration. When we found ourselves continually faced with alternative interpretations and/or possibilities, we felt a strong need to make definite decisions as to the inclusion of one or another embodiment or mathematical variate. It was eventually realized that there probably is no single best way to get "from here to there" and that a variety of routes are possible. The ultimate potential in the paradigm suggested here then is surely not in its ability to define once and for all time manipulatives (perceptual variates) to be utilized or all mathematical aspects of a concept which are in need of systematic variation. Its value stems from its use in reminding curriculum developers and curriculum implementers (teachers) that psychological factors should be attended to consciously when attempting to construct and implement meaningful activities for children. Abstraction and generalization are not new terms to persons concerned with the general field of conceptual development. Clearly they are relevant to mathematical concepts as well. It seems reasonable to suggest that a child has truly learned a mathematical concept only when s(he) is able to abstract it from its physical context and generalize it to new situations. The variability principles as originally suggested by Dienes have been adapted from the psychological literature so as to relate more directly to the abstraction and generalization of mathematical concepts. Organizing instructional program components along the lines of the twodimensional matrix suggested herein does not involve a complete re definition of the ideas involved, nor does it imply the need for a vast amount of curriculum development. Individual cell components for any conceivable matrix already exist. What would be required, however, is a rethinking, a resequencing, and perhaps some degree of restructuring of those activities. The synthesis of relevant cell components for a given concept is not a monumental task given 1) the selection of appropriate mathematical and perceptual variables and 2) a collection of activities from existing texts, collections of assignment cards, project materials, or other sources. It also must be noted that instructional design matrices of this type will not always be rectangular; that is to say, for some concepts, appropriate mathematical variates might not exist for each and every perceptual variate. Such a situation would result in an imbalanced matrix. We do not believe this to be a problem. Dienes is not particularly specific in defining the necessary and sufficient conditions for identifying the appropriate mathematical and perceptual variates for a given concept. The concept of perceptual variation has been less elusive than the concept of mathematical variation. Perhaps this is because the latter can be interpreted in terms of its domain, i.e., the mathematical characteristics inherent in the concept itself or in terms of operation, i.e., the mathematical and/or arithmetical procedures which can be carried out over the concept in question. It is not clear which of these interpretations should be used in deciding upon appropriate mathematical variates. It is possible that these should not (must not?) be mutually exclusive and that both need simultaneous consideration. If true, a threedimensional axis^{1} would more appropriately fit this expanded conceptualization. i.e., 

Additional thinking must be done on this important issue. Impressions at this time suggest that some combination of both is probably optional; that is, both interpretations need to be considered when identifying the mathematical variate for a given concept. It has been our experience that once the perceptual and mathematical variates of a concept have been identified, our thinking on the concept has been substantially clarified. The actual construction of such models can be a valuable experience for persons concerned with the teaching of mathematics to children. Preliminary indications resulting from our work with graduate students seem to support this conjecture. Lastly, we must emphasize the need for systematic research on the impact of these variables singly, and together, on mathematical conceptual development. Although the model traces its theoretical foundations to the Psychological literature on conceptual development, this in and of itself does not guarantee its effectiveness in the classroom setting. References Association of Teachers of Mathematics. Notes on mathematics in primary schools. London: Cambridge University Press, 1967. Beougher, E. E. The review of literature and research related to the use of manipulative aids in the teaching of mathematics. Special publication of the Division of Instruction. Pontiac, Michigan: Oakland Schools, 1967. Dienes, Z. P. Building up mathematics (Revised ed.). Education Ltd., 1967. London: Hutchinson Education Ltd., 1967. Dienes, Z. P. & E. W. Golding. Approach to modern mathematics. New York: Herder and Herder Co., 1971. Dienes, Z. P. An example of the passage from the concrete to the manipulation of formal systems. Educational Studies in Mathematics, 1971, 3, 337352. Fennema, E.H. Models and mathematics. Arithmetic Teacher, 1972, 19, 635640. Kieren, T.E. Review of research on activity learning. Review of Educational Research, 1969,39,509522. Lazarus, M. Mathophobia: Some personal speculations. National Elementary Principal, 1974, 53(2), 1622. Post, T.R. A model for the construction and sequencing of laboratory activities. Arithmetic Teacher, 1974, 21, 616622. Reys, R.E. Considerations for teachers using manipulative materials. Arithmetic Teacher, 1971, 18, 509558. Reys, R.E. & Post, T. R. The mathematics laboratory theory to practice. Boston: Prindle, Weber, and Schmidt, Inc., 1973. Suydam, M.N. & Higgins, J.L. Final report: Review and synthesis of studies of activity approaches to mathematics teaching. NIE Contract No. 400750063, Submitted to NIE September, 1976. Suydam, M.N. & Weaver, J. F. Interpretive study of research and development in elementary school mathematics. University Park, Pennsylvania: Center for Cooperative Research with Schools, 1970. Wilkinson, J. D. A review of research regarding mathematics laboratories. Mathematics laboratories: 1m lementation, research and evaluation. Columbus: ERIC/SMEAC Center for Science, Mathematics and Environmental Education, 1974.




^{1}The authors are indebted to Professor Suzanne K. Damarin of the Ohio State University for her valuable suggestions on the issue.  


Several of the models presented in this paper were developed by graduate students in Mathematics Education at the University of Minnesota. The authors are indebted to the following students for their ideational contributions to this paper.
