Mapping Models of Operations on Whole Numbers

Mapping Models of Operations on Whole Numbers

Floyd VestNorth Texas State University, Denton, Texas 76203

The purpose of this article is to present a network of general con-cepts to persons interested in the teaching and learning of math-ematics. The central concept will be called "mapping of models,"and it will be developed in the context of models of operations onwhole numbers which are used in elementary school mathematics.Considerations from current methodology for teaching elementaryschool mathematics, concepts from modern algebra, and recentdevelopments in the theory of structural learning of mathematicswill be drawn together to form the concepts. These concepts are ofuse to psychologists, teachers, and curriculum developers and provideanother element of a theory of’ structural learning of mathematics.It is worth noting that the author has chosen to develop this networkin terms of examples chosen from an area of mathematics generallyconsidered to be basic or essential for mass mathematical education.The development is in terms of operations on whole numbers ratherthan operations in structures such as finite groups or rings, whichare not usually considered essential for mass education. This typeof development of the concepts is then appropriate and useful for awide range of readers.

In preparation for a discussion of mapping models of operations,we will briefly review by example the idea of models. Two very com-mon models related to subtraction of whole numbers will be con-sidered. One of these models is the take-away model in which sub-traction is associated with the action of removing a subset from agiven set. In an application of this model, the equation 6�2=xwould be associated with the removal of a subset of two elementsfrom a set of six elements. Since there would be a subset of four ele-ments remaining, it would be declared that 6�2=4.

Another model is the comparison model. In using this model, theequation 6 �2=x would be associated with a situation involving aset of six elements and another set of two elements. The members ofthe two sets would then be paired until all of the members in one ofthe sets is used. Since there would be a subset of four unpaired ele-ments in the larger set of six elements, it would be declared that6-2=4.These models are just two of more than eighteen different models

of subtraction in use today. The other ^operations" on whole num-bers are also represented by numerous models or interpretations [9].Several of these will be utilized in the following paragraphs in thefurther development of the concepts under consideration.

449

450 School Science and Mathematics

Models can be related to each other by what will be termed a"mapping." In the learning and teaching process, teachers and pupilsoften conduct such mappings. Thus the concept being developed inthis article amounts to a general-level analysis of activities involvedin learning mathematics.Mapping is the process of making correspondences between the ele-

ments of one model and the elements of another model, by whatevermeans required, in such a way that the fact that the models "actalike" or represent the same operation becomes apparent. Themodels are, from an abstract mathematical point of view, identical.They are both relevant to the same operation and to each other [11 ].The realization of this identity (isomorphism) entails the ability todiscard irrelevancies, retain essentials, and at times insert materialextraneous to the immediate models (structures) involved. Thesegeneral concepts are illustrated in the following examples.

EXAMPLES AND TYPES or MAPPINGSThe comparison model and the take-away models for subtraction

presented earlier can be used to illustrate a mapping. The rationalityof an attempt to relate these two models is based on the fact thatthey "act alike" in a certain sense. We can recall that both modelsinvolved the numbers 6 and 2 and resulted in the number 4. In spiteof the fact that they act alike, they are still different in many ways.In the take-away model a subset was removed from a set; no suchaction is implied in the comparison model. There was a total of 6objects involved in the take-away illustration while the comparisonsituation involved a total of 8 objects.On a general level, the take-away model is the class of all situations

where a subset is removed from a set. At the same level, the compari-son model is the class of all situations where two disjoint sets arecompared in the sense of "how many more or how many fewer in oneset than in the other." In mapping these two models one could makethe following correspondences: the number of objects in the "larger"set in the comparison model corresponds to the number of objectsin the single intial set in the take-away model. The number of obiectsin the "smaller" set in the comparison model corresponds to thenumber of objects in the subset which is removed in the take-awaymodel. Finally, the number of objects in the excess subset of the"larger" set in the comparison model corresponds to the number ofobjects in the subset which remains in the take-away model. If oneutilizes this general mapping with different subtraction equations,i.e.: 6�3=3, 5�3==2, etc., it will be discovered that the models actalike, or are isomorphic. One model could be used to predict an out-come under given conditions in the other.

Mapping Models on Whole Numbers 451

To summarize, we have shown by example that a model is a col-lection of instances and that mappings between models can be de-veloped at the general level of the total collection or at the morelimited level of individual instances.We will consider, as another example, a mapping in the area of

models of multiplication of whole numbers. A very common modelfor multiplication is the model involving the union of several equiva-lent disjoint sets which is termed the "set union model for multiplica-tion." In a primitive instance associated with the multiplicationequation 4X3=12, 4 disjoint sets each containing 3 objects wouldbe joined to form a single set of 12 objects. At a general level the setunion model for multiplication would be the class of all situationswhere the union of several disjoint, equivalent sets is formed.The other model in the mapping will be termed the "many-to-one

correspondence model." In an instance associated with the multi-plication equation 4X3= 12, there are two sets involved: one having"3 times as many" objects as the other and a "smaller" set having4 objects as illustrated in Figure 1. The larger set has 12 objects.Thus one factor and the product indicates the number of objects ina set and the other factor indicates the nature of the many-to-onecorrespondence.

FIG. 1. An instance of the many-to-one correspondence model.

The fact that the two models act alike is seen by noting that theyboth involve the numbers 4 and 3 and result in the number 12. Tomap these models onto one another at a general level, the number ofsets in the set union model would correspond to the number of objectsin the "smaller" set in the many-to-one correspondence model. Thenumber in each of the subsets in the set union model would corres-pond to the number indicating the nature of the many-to-one cor-respondence in the other model, and finally, the number of objects inthe union would correspond to the number of objects in the "larger"set in the many-to-one correspondence model. By this general map-ping it can be verified that these two models are isomorphic and arerelated to the same operation.Thus we have presented two examples of what is meant by a

mapping of models. To recapitulate in general terms, a model is

452 Schoo] Science and Mathematics

a class of situations. Mapping is the process of making correspondencebetween elements of one model and elements of another model insuch a way that it is obvious that the models represent the sameoperation or act alike. The models are said to be isomorphic.Another type of mapping, mapping of models onto equations, has

been implicit in the discussion thus far. In the preceeding examples,four such mappings have been implied. To be more explicit, themapping of the many-to-one correspondence model onto the usualtype equation (i.e. 4X3=12) will be outlined. The first factor in anequation is associated with the number of objects in the "smaller"set. The second factor is associated with the number indicating thenature of the many-to-one correspondence, and the product is as-sociated with the number of objects in the "larger" set. By thismapping, multiplication equations are associated with the totalclass of instances of the many-to-one correspondence model. It isalso apparent that alternate mappings between this model andequations are available. Similar mappings exist between the otherthree models used in previous examples and their respective familiesor equations. In general terms, one of the activities of teachers andtextbook writers is the defining of mappings (correlates) from modelsonto families of equations.

Several general concepts have been developed thus far. They are"models of operations," "mapping one model onto another," "map-ping a model and equations," and "isomorphism between models."The concepts are general in that they apply to more than fortydifferent models for operations on whole numbers and hundreds ofmodels of other types of operational structures (i.e. operations onintegers, rationals, reals; and models of groups, rings, etc.). Theseconcepts have been developed at a very primitive level in terms ofthe operations on whole numbers which are so widely used in edu-cational endeavors. Notions from more advanced areas of mathemat-ics have been applied to form a general analysis of learning andteaching activities at a very elementary mathematical level.The examples of mapping models developed thus far will be termed

"non-subsumptive mappings" in that the pairs of models involvedare treated as distinct, separate mathematical entities. Another type ofmapping which is an alternative in teaching strategies and math-ematical thought, will be termed "subsumptive mappings." Thistype relates models in such a way that instead of treating them asdistinct entities, one model is viewed as a subset of the other. Thistype of mapping will be exemplified in terms of the same pairs ofmodels as used above.One can subsumptively map the comparison model into the take-

away model by inserting the extraneous material of pairing and the


removal of a subset of matched objects [4, p. 35]. By way of illustra-tion we consider the following comparison problem: "Mary boughtseven apples and Susan bought five oranges. How many more applesthan oranges were bought?" It is evident that there is no suggestionof a take-away action in this problem and there are initially two setsrather than one set from which a subset is removed.To map this instance of the comparison model into the take-away

model, one might pair the members of the two sets until all of themembers in one of the sets are used. (See Figure 2.) The subset ofpaired elements in the larger set is then removed. Since the results

& o Q & 6w e$

00060FIG. 2. Mapping by inserting extraneous material.

of the removal of the subset of five apples from the total set of sevenapples leaves a subset of two apples, the student has an instance ofthe take-away model and by the usual rules of correspondence (cor-relates) between the take-away model and subtraction equationswrites "7�5=2." Although the pairing and the "take-away" areextraneous to the comparison type problem, the student can betaught to bring this material to comparison problems so that theycan be viewed in terms of the take-away model.By this subsumptive mapping, the class of situations making up

the comparison model is seen as a subset of the class of situationsmaking up the take-away model. The two models are not seen as dis-tinctly different interpretations of the same operation.Using the models of multiplication presented above, we will

illustrate another subsumptive mapping by considering the followingmany-to-one correspondence problem. "Carl has 4 picture cards.Bob has three times as many as Carl. How many does Bob have?"Although the situation is not strictly an instance of the set unionmodel for multiplication, persons not versed in a careful analysis ofthe situational aspects of the many-to-one correspondence modelwould be likely to map this instance into the set union model. How-


ever, those persons whose dominant structure is the many-to-onecorrespondence model (as a result of their past experience and educa-tion) would view the problem solely on the basis of its situationalaspects and not in terms of set union. In mapping this many-to-onesituation into the set union model, each of the 4 cards in CarPs setwould be associated with a set of three cards, and thus it would benoted that there are four sets each containing three cards. The unionof these four sets yields a set of twelve cards.

If this situation were treated solely by the many-to-one interpre-tation with the previously indicated rules for association with anequation (correlates), one would not think ^the union of 4 sets ofthree objects has 12 objects." Instead "4" indicates the number ofobjects in the smaller set, the "3" indicates the three-to-one corre-spondence; and the "12" indicates the number of objects in thelarger set.

It is interesting to note that the above subsumptions were achievedby inserting material extraneous to the model being subsumed (i.e.the matching and taking-away of a subset). Other subsumptivemappings are straight forward in that they are a direct consequenceof the definitions of the models. This occurs in a well-known primarylevel mathematics program in which an ^additive situation"modelfor addition is used. In this model two sets of concrete objects arephysically joined to form a single set, and it is required that one ofthe sets remains immobile [5]. (It appears that the authors of theprogram adopted this model because it provides a non-redundantmodular rationalization of the commutative law of addition.) Thismodel is simply a subset of a more general model in which the setsare allowed to join in any manner. Other mappings by this type ofsubsumption (where one model is simply a subset of another) arebetween certain rectangular array interpretations of multiplicationand the multiplicative situation model [7, p. 333; 8] and between thetake-away model for subtraction and physical decomposition modelsfor subtraction [12].

Another form of mapping by subsumption occurs when one modelsubsumes the other by abstraction. For example, the set union modelfor multiplication involves the formation of the union of severalequivalent, disjoint subsets. This interpretation involves abstractsets which are derivations of "plysically concrete sets." It subsumesby abstraction the multiplicative situation model for multiplication[4] which is limited to concrete sets being physically joined.

It is both interesting and informative to analyze mathematicsprograms to determine the mappings which they utilize. To mentiononly a few examples, the many-to-one correspondence model formultiplication is also frequently mapped into the divisive situation


model with the measurement division interpretation [6], and theCartesian product of sets model for multiplication is frequentlymapped into the set union model.One might wonder whether developers of mathematics programs

are aware (in the general sense of this article) of their mappingactivities and decisions.There is still another type of mapping which unfortunately is

found in classroom practice and even in printed material. We mightterm it a "nonsense mapping." One such nonsense mapping whichoften occurs involves the comparison and the take-away models forsubtraction. Such a mapping can be illustrated with the probleminvolving Mary^s seven apples and Susan’s five oranges. Frequently,the solution to this problem (7�5=2) is explained on the basis ofthe take-away model as "7 apples take away 5 oranges leaves 2apples." One can see that this statement makes little sense.

THE USE OF MAPPINGSAn interesting educational use of several models of a single opera-

tion and non-subsumptive mappings is to force abstraction on thepart of a student [2]. As a result of such a teaching strategy, thestudent would abstract the operation as a property held in commonby the several models. In this way a student sees the operation as amultivalent abstract model and perhaps has begun to think in termsof the powerful and tremendously general multiform concept of"model" used in so many scientific areas today. It is this propertyof multivalency of the operations and numbers that makes them sopowerful and useful even in the mental equipment of the child.The concepts developed in this paper can be used in the study of

children^ thought and learning processes. Mapping of models andstructures is a fundamental pervasive element in their mathematicalthought. Bruner observed this type of mapping behavior in one ofhis experiments. He reported:When they [the children] searched for a way to deal with new problems, the

task was usually carried out not simply by abstract means but also by "matchingup’^ images ... it was interesting that the children would "equate" concretefeatures of one [structure] with concrete features of another [1, p. 329].

With the aid of the concepts developed here, pedagogical pitfalls,such as the nonsense mappings above, can be avoided [10, 13]. Twospecific alternatives to this particular mistake are (a) treating thetake-away and comparison situations under non-subsumptivemapping strategies as distinctly separate areas of application or

(b) subsumptively mapping one into the other.Authors should analyze their materials for different models and


investigate the manner in which they are handled. "New" modelsshould be introduced carefully and not allowed to creep into thematerials without the authors being aware of them. Quite oftenmodels are presented in programs (in the form of the structure of aword problem or rationalization of a property or algorism) withoutcareful introduction, and the children are left with a difficulty. Thisdifficulty consists of their not being able to "do" certain problemsor to "understand" certain explanations (in terms of models) ofproperties and algorisms. In the early grades children have no mathe-matical reason to associate new models with the expected operationsunless they can map them onto the operation or onto a familarmodel. As has already been implied, the solution to this difficulty isto carefully teach the new model to the children in terms of its dis-tinctive attributes and to non-subsumptively map it onto an alreadyfamiliar model, or to execute a subsumptive mapping.There is very little use in introducing certain models unless they

are treated non-subsumptively. This applies primarily to modelswhich have little social pervasiveness�Eg: Cartesian Product modelor certain rectangular array models. (An exception to this is when themodel serves a valid educational function such as motivation,variety, or reflection of important properties of the operation.) Totreat them subsumptively only provides for a form of generalizationthat has little "practical" value. However, to treat them as entitiesdistinct from more familiar models but isomorphic to them providesfor a form of abstraction which has value as cited in the aboveparagraph.

SUMMARYThe purpose of this article has been to present the concepts of

"models of operations" and "subsumptive and non-subsumptivemapping" plus a few of their applications. These concepts aregeneral in that they apply to hundreds of models and dozens ofoperational structures. In spite of the generality and abstractnessof these concepts, this paper has applied them to operations on wholenumbers which are both elementary and important in their universalcharacter in educational endeavors.

BIBLIOGRAPHY[1] BRXJNER, JEROME S., "Some Theorems on Instruction Illustrated with

Reference to Mathematics," Theories of Learning and Instruction, Sixty-Third Yearbook, Part I, Chicago, National Society for the Study of Educa-tion, 1964.

[2] DIENES, Z. P., Building Up Mathematics^ London, Hutchinson and Co.,Ltd., 1964.

[3] DIENES, Z. P., The Power of Mathematics, New York, Humanities Press,1964.


[4] HARTUNG, MAURICE L. AND OTHERS, Charting the Course for Arithmetic,Chicago, Scott, Foresman and Company, 1960.

[5] HARTUNG, MAURICE L. AND OTHERS, Teachers Edition, Seeing ThroughArithmetic 2 and Practice Tablet 2, Chicago, Scott, Foresman and Company,1964.

[6] McSwAiN, E. T. AND OTHERS, Arithmetic 4, Teacher’s Edition, River Forest,Illinois, Laidlaw Brothers Publishers, 1963.

[7] MONTESSORI, MARIA, The Montcssori Method, Translated into English byAnne E. George, New York, Schocken Books, 1912.

[8] School Mathematics Study Group, Mathematics/or the Elementary School,Grade 4, Part L 367 S. Pasadena Ave., Pasadena, California, A. C. Vroman^s.1962.

[9] VEST, FLOYD R., "A Catalog of Models for the Operations of Addition andSubtraction of Whole Numbers," Educational Studies in Mathematics.1-1969,59-68.

[10] ��, "A Precaution Applied to the Use of Several Models," PrimaryMathematics, April/May, 1969, 68-71.

[11] ��, "An Analysis of the Representational Relationship Between theOperations on Whole Numbers and Their Models," Journal of StructuralLearning, Vol. 2, No. 4, 1969.

[12] ��, "Development of the ’Model Construct’ and Its Application toElementary School Mathematics," unpublished doctoral dissertation,School of Education, North Texas State University, Denton, Texas, 1968.

[13] ��, "Model Switching Found in Lessons in Subtraction in the Ele-mentary Grades," School Science and Mathematics, May, 1970, 407-410.

PIPELINE IN ALASKAMADISON, Wis.�Vigorous environmentalist opposition may save Alaska^s

wilderness even if the trans-Alaska oil pipeline is built. The reason�oil com-panies have developed better pipeline designs and construction methods thanthey originally intended to employ.A metallurgist and consultant to environmentalist groups opposing the pipe-

line, points out that the loud and forceful opposition to the pipeline has madeoil companies more than eager not to seriously disrupt the Alaskan landscapeand its fish and wildlife. The companies now realize that to get a constructiongo-ahead and gain from their $4 billion investment they must allay environmen-talist fears that the pipeline will be an environmental hazard. Moreover, theycannot risk the wave of adverse public sentiment that would result if a seriousbreak occurred. Nor can they risk disruption of the landscape during the heavyconstruction work necessary.The pipeline design plans have not yet really satisfied environmentalists but

research under way may provide design and construction methods safe enoughto allow the companies to go ahead. There are still a number of doubtful aspectsto present pipeline plans:�Will the steel pipe withstand the stress of buckling inevitable on the perma-

frost terrain? During earthquakes?�What happens if a break occurs and the oil in the pipe cools during shut-

down to the consistency of molasses? Will it be possible to start it flowing again?�Will the pipeline disrupt fish in�and wildlife along�any of the 350 Alaskan

rivers that must be crossed?These are a few of the problems that oil companies must still solve to the

satisfaction of environmentalists.

Documents

Mapping Models of Operations on Whole Numbers