FLOYD VEST

U S I N G M O D E L S OF O P E R A T I O N S AND E Q U A T I O N S

Teaching the operations and inverse operations on whole numbers in the primary grades is a common and basic concern. It is generally understood today that these basic concepts are taught by utilizing models of oper- ations and families of equations [1, 2]. In teachers' editions, one is often instructed to teach children how a particular family of equations relates to a model. This instruction suggests interesting basic questions: "Just how do models and equations relate? How does one teach this relationship?" The purpose of this article is to develop four general concepts (correlates, analysis of models and families of equations at the general and instance level, oblique correlates, and adequacy of analysis of models and equations) and apply them to answering these and related questions. These concepts will then be used to provide a systematic basis for planning, examining, and improving conventional strategies for teaching number operations. Although the following concepts are applied and illustrated in terms of models of both operations and inverse operations on whole numbers, they are sufficiently general to apply to models of operations in any number system.

1. MODELS OF OPERATIONS

Before the concepts can be developed, there is need to review the concept of a model of an operation or inverse operation on whole numbers. There are at least twenty different models for addition and subtraction of whole numbers and twenty additional models for multiplication and division in use by primary teachers today [1, 2, 5]. As an example, a common model for subtraction is the take-away model. An instance of this model would be a situation in which a subset of two elements is removed from a single set of six elements and a subset of four elements remains. Another model for sub- traction is the comparison model. An instance of this model would be a situation in which a set of six elements is compared with a separate set of two elements and one set has four more elements than the other. Models of operations are interpretations or illustrations which are representative of an operation (loosely isomorphic) [4]. They are classes of situations with each situation termed an instance of the model. Models are used for the foUowing fundamental endeavors in the primary grades: to 'define' the operations, to explain (realize) properties and algorithms, and to

Educational Studies in Mathematics 5 (1973) 147-155. All Rights Reserved Copyright © 1973 by D. Reidel Publishing Company, Dordrecht-Holland

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provide the structure of problems which are solved by use of the operations.

2. CORRELATES

To expand the discnssion of the above instance of the take-away model, one might declare that the numeral '6' of the equation ' 6 - 2 = 4 ' refers to the number of elements in the initial supply of objects. Such a statement is a correlate. Similarly, one could state that the numeral '2' would be associated with the cardinal number of the subset which is being removed. Another correlate in this collection is that the numeral '4' in the equation refers to the cardinal number of the subset which remains. These statements are termed 'correlates' which together indicate the association between the in- stance of the take-away model and the subtraction equation ' 6 - 2 =4' ([3], p. 71),

In general, correlates are correspõndences between elements of models and elements of families of equations and are orten presented in the form of statements or rules. Correlates reflect the correspondence necessary for establishing the isomorphism between the model and the operation [4, 6]. In practice, when presenting correlates to children with the aid of models and equations, the distinction between the level of equations and symbols and their abstract referents is not consistently maintained. Examples of common correlates connecting models or instances and the abstract level are as follows: 'The subtraction of 2 is shown by the removal of the subset of 2 objects. The amount left shows the difference'. For purpose of simplification, the definition of correlates will here be expanded to include statements of connection between elements of instances or models and either the symbolic level or the abstract level of number and operation. This latter level is ad- mittedly not easily characterized in the cognition of the young child.

To establish correlates, one does the following: (a) analyzes the equation, family of equations, or abstract referents into constituent parts, (b) analyzes the instance or model into parts, and (c) establishes rules for making the association between parts of the equation, family of equations or abstract referents and parts of the instance or model.

3. ANALYSIS OF MODELS AND EQUATIONS

To continue with the take-away model, one can analyze the instance under discussion by acknowledging a set of six elements initiaUy, a subset of two elements being removed, and a subset of four elements remaining. Less specifie terms such as 'the number we start with' or 'the number of elements taken away' may be addressed to the instance level by pointing to an instance

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or using qualifying phrases such as 'in this situation', or 'in this picture'. This is an example of what will be termed 'analysis at the instance level'.

An analysis at the 'model level', as addressed to the total class by an appropriate statement, might include acknowledgement that in all take-away situations there is a single initial supply, a subset being removed, and a subset which remains. An analysis at the model level indicates critical attributes possessed by the total class of instances making up the model. An instance under consideration may serve as an exemplar of the total model.

Thus analysis related to models can be accomplished at two levels - the instance level and the model level. Analysis serves two important functions. It describes the attributes of the class of situations constituting the model as well as those attributes used in defining the isomorphism between the model and the abstract operation and thus the correlates to the family of equations.

The analysis to equations can also be accomplished at two levels - the level of a single specific equation and the level of a family of equations. Analysis at the equation level would be in terms of symbols or collections of symbols which are parts of a specific equation. Historically in the tradition of meth- odology for teaching arithmetic, there have been varying analyses of specific equations. A fairly conventional and contemporary equation level analysis of the equation ' 6 - 2 = 4 ' utilizes the parts '6', '2', '4', and '6 -2 ' . Analysis below the level of the total equation will be termed 'proper analysis'. Reference to the equation as a whole will be referred to as 'improper analysis'.

A family of equations is a set of equations sufficient for representing an operation and customarily uniform in symbolic structure. Analysis of equa- tions at the 'family level' would be addressed to the total family by a quanti- fying phrase and would involve such terms as 'addend, difference, or product'. A specific equation under consideration may serve as an exemplar of the total family. These two levels of analysis will be termed 'equation level' and 'family level'.

More precisely, a correlate is a connection between one of the several parts into which an equation or family of equations has been analyzed and one of the several parts into which an instance or model has been analyzed. A single sentence may provide one or more such connections (correlates). Connections are usually indicated by a statement addressed to the model or an instance and the family of equations or equation, and containing any one of a variety of phrases establishing the mood and nature of the connection. In the above examples of correlates, the verbs 'refers' and 'is associated with' have been used. Examination of instructional materials reveals the use of a wide variety of connecting verbs such as 'shown by', 'indicates', 'represents', 'goes with', 'is named by', and 'teils'.

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Recalling that a correlate is based upon analysis of equations, analysis of models, and is a statement of connection, then by count, there would appear to be four types of correlates: equation-instance correlates, equation-model correlates, family-instance correlates, and family-model correlates. For ex- ample, in the presence of an instance of the take-away model and the equa- tion ' 6 - 2 = 4 ' , the statement, "The numeral '6' in the equation refers to the number of elements we start with", is a correlate at the equation-instance level. An example of a correlate at the family-model level is the statement, "In take-away situations the sum indicates the number of elements to begin with". Phrases such as 'in take-away situations', 'in general', 'in situations like this one', referring to the model, and in 'subtraction', 'using equations', 'in general', referring to the family of equations, address the connection to the generality of the family-model level. Without such phrases indicating the level of generality, correlates intended to be at the family-model level may inferentially collapse to the equation-instance level. A correlate stated in terms based upon an equation analysis and a model analysis collapses logically into a correlate at the equation-instance level since reference to a specific numeral in an equation deletes the generality of the terminology related to the model. Similarly, a correlate stated in terms based upon a family analysis and an instance analysis collapses logically to the equation- instance level since reference to a specific instance of a model deletes the generality of the terminology related to the family of equations.

The above passages include the derivation of the second general concept which is that of two levels of analysis of equations and models and two types of correlates at thls stage of analysis: family-model correlates and equation-instance correlates. In the presence of a single instance of a model and an equation, family-model correlates include appropriate phrases indicating that they are addressed to the general level.

4. OBLIQUE CORRELATES

Correlates of the type in the above discussion will be at times termed 'direct'. They include such statements as, "The numeral '6' in the equation refers to the number of elements we start with", and "In take-away situations, the sum indicates the number of elements to begin with". 'Direct correlates' occur when one tells explicitly in general or specific terms which symbols or parts of equations or their abstract referents are associated with parts of models. The equation or part thereof is explicitly indicated verbaUy or by pointing and is in turn connected to a part of a model by connecting verbs. The first of the above correlates is direct at the equation-instance level; the second is direct at the family-model level.

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The concept of 'oblique correlates' arises from observation of teachers' behavior. For example, in a discussion of the take-away model with a drawing of an instance in juxtaposition to the equation ' 6 - 2 = 4', a teacher might use the symbol '2' to stare, "I am now removing a subset of 2 elements, or I am taking away 2 things", without referring directly to the numeral '2' in the equation. The elements of an equation have been simply used in describing the critical attributes of the instance of the model and there is no explicit indication of an equation or abstraction separate from the instance. It is left to the student to infer a direct connection between the equation and the instance. Such an action is termed an 'oblique correlate' since it is usually intended that a student infer a direct correlate from it and concomitant stimuli.

It is common for teachers to use a sequence of oblique correlates and expect the students to infer direct correlates. An interesting research question is then here suggested. What is the relationship between expository instructional strategies and student learning as categorized above? For example, do children efficiently infer direct correlates from a barrage of oblique corre- lates? What relationship exists between verbalization of correlates on the direct, general level and concomitant learning? How do teachers effect expository instructional strategies with children when prompted by various types of commentary from teachers' editions as categorized and analyzed by the above concepts?

The analysis thus far has been based upon equations which do not contain variables although the practice of associating such equations with instances of models is very common. It is also a common practice to utilize instances of models in which not all the numerical information is known. (In the context of models, numerical information is referred to as 'concretization of number' [2].) Concretizations may be cardinal numbers, ordinal numbers, states, processes, operands, patterns or colors. For example, one may have a take-away situation in which the number of elements in the set which re- mains is not known. Thus one may have 'unknowns' in both equations and models. The following derivation will show that variables in equations and unknowns in instances are compatible with the concept thus far developed in this article.

In the case of an equation-instance correlate relating a variable to an unknown concretization in the instance, the variable becomes simply a place holder for the numeral naming the number assigned to the unknown con- cretization by the isomorphism. For example, the variable in the equation ' 6 - 2 = x ' , when associated with a take-away situation in which a subset of 2 is removed from a set of 6 but the number remaining is not known, simply becomes a place holder for the specific numeral naming the number asso- ciated with the unknown concretization. Thus at this concrete level, if the

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place holder concept is understood the variable serves much the same naming function as a specific numeral.

A variable in a specific equation can also be interpreted as an exemplar of a part of the total family of equations and thus it simply indicates family analysis. When a variable serving in this way is associated with an unknown at the model level where the generalized concretization is always unknown, the association simply produces the familiar family-model correlate. When a variable interpreted at the family level is associated with an unknown at the instance level, the association either appears illogical or inferentially collapses back to the equation-instance level discussed above. A correlate involving a variable viewed at the equation level in association with a generalized concretization at the model level either appears illogical or inferentially collapses to the equation-instance level or the family-model level.

Thus to summarize, when connecting a variable in an equation to an unknown concretization, one has the farniliar equation-instance and family- model correlates. It should also be noted that such correlates, as a result of the acknowledgement of the variable in an equation, are themselves direct.

The cases of correlates involving 'unknown' (variable) in the equation and known in the instance and vice versa are yet to be analyzed. It turns out that correlates of the first type are the interesting transitory correlates associated with solving equations in association with models. They are transitory and quickly inferentially change to known-known correlates since the variable serves as a place holder for the known concretization. If the correlate is understood, the variable is then replaced by a numeral naming a specific number and one has familiar known in equation-known in the instance direct correlate.

Correlates involving 'known' in the equation and unknown in the instance are also transitory, and in a similar manner collapse to the familiar known in equation-known in instance correlates. This transition is one of the generalized steps in problem solving involved in translating the solution of an equation back into the concretization in the model.

In summary, it is concluded that correlates involving the four possibilities of known and unknown fall into the earlier cases of equation-instance or family-model correlates. As was indicated at the outset of the derivation devoted to consideration of variables in equations and unknowns in instances, correlates with these qualities are compatible with the three main concepts developed thus rar in this article.

5. QUALITY OF ANALYSIS

The fourth general concept is that of quality of analysis of equations and

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models. Quality of analysis is judged by its adequacy for defining the concept of the model, the concept of the family of equations, as well as the collection of correlates necessary for connecting the model and the family of equations.

6. M E A N I N G OF OPERATION TERMS

In the several levels of exposition generally aimed at teaching operations at the concrete level of models and equations, operation terms such as 'ad- dition, subtraction' are used frequently to refer to a variety of referents. The analysis developed in this article provides a precise classification of many intended referents. Such terms are often intended to mean a model, a family of equations, a coUection of correlates connecting a model and an equation, an abstract operation, an algorithmic process of solving an equation by methods at the model level, or a collection of models representa- tive of an operation.

7. A N EXAMPLE OF A P P L I C A T I O N TO INSTRUCTIONAL EXPOSITION

An endeavor basic to the investigation of questions suggested by the analysis developed in this article is its application to instructional strategies for teaching the operations on whole numbers with the aid of models. One such endeavor would be application to commentary from teachers' editions. The following is an example of such an application to a commentary selected from a Grade 2 teachers' edition which suggests strategies for introducing subtraction as related to the number-line model. Qualities other than repeti- tion and person are left intact. All relevant analysis and correlate information found in the entire teachers' edition is summarized. The commentary involves the first introduction in the program of the number-line model of subtraction ([5], pp. 368-373), and refers to an instance of the number-line model with an initial jump to the right from 0 to 9 and a succeedingjump to the left from 9 back to 6, and with the equation ' 9 - 3 = 6' placed in juxtaposition on the chalkboard. Analyses of the commentary are enclosed in brackets.

The commentary, as paraphrased and adjusted from the teachers' edition, contains the first suggestion that the teacher ask the children to show the subtraction of 3 by tracing the arrow from 9 back to 6. [This instruction analyzes the instance into parts in that it refers to the arrow from 9 back to 6. An indefinite collection of symbols ('3' or " - 3 ' of the equation), or the abstraction 'subtraction of 3' are directly associated with the indicated arrow. Oblique correlates involving the numerals '9' and '6' of the equation are present.]

The teacher is next instructed to emphasize that the first arrow is drawn to

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the 9 and then another arrow is drawn back 3 units, and that we end up at 6. [This instruction contains a careful analysis at the instance level of the types ofjumps and their order as weil as oblique correlates involving '9', '3', and '6', but no direct correlates.]

It is then suggested that the teacher have the class read the equation. [This activity produces no proper analysis of the equation (analysis below the level of the equation as a whole) or direct correlates but obliquely associates the equation with the number-line illustration.]

The next advice is to call attention to the two jumps, a forward jump of 9 and a backward jump of 3. [Again, we have an activity with analysis of the instance and oblique correlates.]

The teacher is to remind the children that the symbol ' 9 - 3' indicates that they should begin by taking a forward jump of 9, then a backward jump of 3. [This statement is a direct correlate connecting the phrase ' 9 - 3 ' to a se- quence of two jumps.]

In addition to the above commentary, the teachers' edition also included the suggestion several times that the teacher points out how the equation relates to the number line, but it failed to characterize the nature of this relationship in general terms and gave no more equation-instance correlates than those given above.

Examination of the above commentary reveals an adequate analysis of the instance into its critical parts, an explicit-proper analysis of the equation involving only ' 9 -3 ' , and a repetition of obtique correlates. There is no general analysis of the family of equations using terms such as 'sum' or 'minuend'. Two direct-equation-instance correlates associate the phrases '9 - 3' and 'subtraction of 3' with parts of the instance. It is left to the teacher to determine if the other related parts of the equation (the symbols '9, 3, 6', and ' 9 - 3 = 6') are to be given meanings directly.

8. SUMMARY

From a general and systematic point of view, this article has attempted to answer the questions, How do models and equations relate?, How does one teach this relationship? In the process, a systematic procedure for analyzing this task of teaching operations with the use of models and developing teaching strategies has been demonstrated. In applying the several concepts of this article to the preparation of a lesson, the teacher can (a) choose models and a family of equations, (b) choose an analysis of the equations into constituent parts involving family or equation levels, (c) choose an analysis of models into their critical attributes, and (d) plan a strategy utilizing direct correlates or oblique correlates for connecting the equations and models at

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a general or instance level. On the basis of such planning, the teacher will develop with a class a knowledge of the parts of the equation, the critical parts of the model, and correlates for connecting the two.

Without utilizing these concepts, teachers have been found to attempt the introduction of the connection between an equation and a new model simply by placing the illustration (instance of a model) and equation to- gether on the chalkboard and reading the equation. (This may seem absurd, but the author is aware of a film designed to present exemplary teaching in which this is done and has observed student teachers overlooking the neces- sity for carefully connecting the model and the equation.) Such practices as these in the context of the common 'telling-teaching' process suggest that teachers' commentaries should provide extensive analyses and direct cor- relates [7].

The four concepts (correlates, analysis of models and families of equations at the general and instance level, oblique correlates, and adequacy of analysis of models and equations) have been referred to as general concepts. They are general in the sense that they can be applied to the use of models of any operation. As was pointed out earlier, this level of generality extends to more than forty families of models of operations and inverse operations on whole numbers. It also extends to the use of models of operations in other number systems.

North Texas State University

Denton, Texas

BIBLIOGRAPHY

[1] Vest, Floyd R., 'A Catalog of Models for Multiplication and Division of Whole Numbers', Educational Studies in Mathematics 3 (1971), 220-228.

[2] Vest, Floyd R., 'A Catalog of Models for the Operations of Addition and Subtraction of Whole Numbers', Educational Studies in Mathematics 1 (1969), 59-68.

[3] Vest, Floyd R., 'A Precaution Applied to the Use of Several Models', Primary Mathe- matics 7 (August, 1969), 68-71.

[4] Vest, Floyd R., 'An Analysis of the Representational Relationship Between the Operations on Whole Nttmbers and Their Models', Journal of Structural Learning 2 (1969), No. 4.

[5] Vest, Floyd R., "Development of the 'Model Construct' and Its Application to Ele- mentary School Mathematics", unpublished Doctoral Dissertation, School of Educa- tion, North Texas State University, Denton, Texas, 1968.

[6] Vest, Floyd R., 'Mapping Models of Operations on Whole Numbers', School Science and Mathematics, May, 1972, pp. 449-457.

[7] Vest, Floyd R., 'Model Switching Found in Lessons on Subtraction in the Elementary Grades', School Science and Mathematics, May, 1970, pp. 407-410.