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DAVID SLAVIT

THE ROLE OF OPERATION SENSE IN TRANSITIONS FROM

ARITHMETIC TO ALGEBRAIC THOUGHT

ABSTRACT. As attention to the development of algebraic understandings at early grade

levels increases, theory and empirical support for these efforts are needed. This paper

outlines a theoretical perspective for studying student understandings of mathematical

operations, with a particular focus on addition. The notion of operation sense is defined

using a perspective that incorporates the construction of mental objects. In the context of

addition, it is argued that operation sense can be used to describe student development of

additive concepts as well as transitions into algebraic ways of thinking. The report of a case

study on the development of a young boy is then provided. The investigation attempts toinstantiate the framework in regard to student development of an understanding of addition.

Evidence was found that his attainment of aspects of operation sense supported transitions

into algebraic ways of thinking, including a finite group setting and use of addition on

unknown and arbitrary quantities. Limitations of the framework are discussed.

1. INTRODUCTION

The purpose of this paper is to present and investigate a theoretical per-spective on the development of understandings of mathematical opera-

tions. The theoretical framework defines operation sense, with a particularfocus on addition. Areas of algebraic understanding that frame the invest-igation and emerge from an operation sense of addition are identified. Todemonstrate the viability of this analysis and to identify kernels of algeb-

raic thought that may be present as numeric and arithmetic understandingsdevelop, I present data involving one childs beginning addition strategiesduring his first and fourth grade year (ages 6 and 9), and interpret thesedata from this perspective.

Evidence is accumulating to suggest that six- and seven-year old stu-dents can value alternate problem solving strategies and opportunities tocommunicate their thinking (Fuson et al., 1997; Franke and Carey, 1997).

In addition, early elementary school students are capable of making sense Portions of this study were presented at the 17th Annual Meeting of the North Amer-

ican Chapter of the International Group for the Psychology of Mathematics Education,

Columbus, Ohio, October, 1995

Educational Studies in Mathematics 37: 251274, 1999.

1999 Kluwer Academic Publishers. Printed in the Netherlans.

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of advanced notions of arithmetic that transcend into algebraic realms

(Kaput, in press). Kaput (1995), in a call for an earlier introduction ofalgebra in current curricula, describes one important aspect of algebraic

thought:Acts of generalization and abstraction give rise to formalisms that support syntactic com-

putations that, in turn, can be examined for structures of their own, usually based in

their concrete origins . . . . These structures seem to have three purposes, (1) to enrich

understandings of the systems they are abstracted from, (2) to provide intrinsically useful

structures for computations freed of the particulars that they were once tied to, and (3) to

provide the base for yet higher levels of abstraction and formalization (p. 77).

Various aspects of algebraic understanding at higher grade levels have been

previously identified, including action-oriented, process-oriented, and ob-ject-oriented understandings (Briedenbach et al., 1992; Confrey and Smith,1995; Thompson, 1994; Slavit, 1997a). Although algebra at early grade

levels is also multi-dimensional (Kaput, 1995), for the purposes of thispaper, early algebraic competence will be primarily restricted to the cognit-ive processes and actions associated with abstracting computation to morestructural realms, commonly referred to as generalized arithmetic. Theseunderstandings are encapsulated in the above discussion by Kaput and can

be manifested by the manipulation of algebraic symbols and equations.Descriptions of the nature of this abstraction and structure are providedbelow.

Mental objects

Numerous theories have been offered on the nature of mental objects in

mathematics (Davis, 1984; Vergnaud, 1988; Cobb et al., 1992; Fuson etal., 1997). The transitioning from action- to object-oriented understandingspresent in the theory of reification can be related to an investigation ofthe manner in which acts of computation are abstracted to more structural

realms. Sfard and Linchevski (1994) describe reification:

The ability to perceive mathematics in this dual way (as an action and as an object) makes

the universe of abstract ideas into the image of the material world: like in real life, the

actions performed here have their raw materials and their products in the form of entities

that are treated as genuine, permanent objects. Unlike in real life, however, a closer look

at these entities will reveal that they cannot be separated from the processes themselves as

self-sustained beings. Such abstract objects like1, 2, or the function 3(x+5)+1 are

the results of a different way of looking on the procedures of extracting the square root from1, of subtracting 2, and of mapping the real numbers onto themselves through a lineartransformation, respectively. Thus, mathematical objects are an outcome of reification of

our minds eyes ability to envision the result of processes as permanent entities in their

own right (pp. 193194).

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OPERATION SENSE 253

Children often perform additive tasks using various counting methods, but

eventually are able to refine this technique towards more efficient methods(Fuson, 1992a). The task of adding 5 + 3 could initially be performed by

counting out a set of 5, and then a set of 3, joining these sets, and countingthe result. Counting up allows the child to start from 5. This action signifies

that the reification stage is beginning since the number 5 no longer needsto be verified through the process of counting. Eventually, children canunderstand that 5 + 3 = 8 without any immediate reference to modellingactions.

This development provides an example of the building of a chain inwhat could be termed the reification cycle. Once a process is reified tothe degree that it can itself be thought of as a mathematical object, thena second operation can be used on this newly-conceived object, which

can later become reified itself. The reification cycle can be a nice toolin analyzing the long-term development of mathematical understanding.

However, caution must be exercised in that reification is not a dichotomousvariable (Schoenfeld et al., 1993; Sfard and Linchevski, 1994) and learningfor understanding is not a totally-ordered process (Kieren and Pirie, 1991;

Hiebert et al., 1996). Mathematical objects in this sense must be describedin light of the motivations for use and meaning-making activities of thestudent involved (Fuson et al., 1997; Slavit, 1997b). In other words, reific-ation can only occur as a consequence of student knowledge construction,

and the results of reification only exist in the context of the studentsexisting conceptual structure.

Operation sense

The notion of mental objects will be used to help explicate the followingdiscussion of operation sense. According to Piaget (1964), an operation is

the essence of knowledge that is central in developing structural under-standings. This perspective will be used in arguing for the importance ofoperation sense in curricula and in discussing what operation sense entails.

Because I am interested in student understanding, and not just problem

solving behaviors, as well as how operation sense might be used to discussalgebraic understanding, I have attempted to develop a theoretical basisthat would be: 1) useful in discussing mathematical operations in general;2) useful in exploring student understandings of addition; and 3) useful in

understanding how childrens early competencies in arithmetic can be seen

as roots of later algebraic forms of thought. I am defining operation sensein an effort to satisfy these requirements.

A base definition of operation sense could involve the ability to use the

operation on at least one set of mathematical objects (such as the ability

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to add positive integers). But this is clearly a minimal conceptualization.

With this in mind, I maintain that operation sense involves various kindsof flexible conceptions which can be interrelated by the learner. These

conceptions involve the operations underlying structure, use, relationshipswith other mathematical operations and structures, and potential gener-

alizations. Such characterizations include the establishment of propertiesthat the operation possesses (Briars and Larkin, 1984), various forms andcontexts in which the operation could exist (Carpenter, 1985), and howthe operation relates to other processes (Fuson, 1992b). An awareness of

these characterizations can delineate features of the operation and lead tovarying degrees of operation sense.

Specifically, the following ten aspects help to clarify the meaning of op-eration sense. While much more could be written about each, the descrip-

tions below provide an adequate picture of the overall notion of operationsense.

1. A conceptualization of the base components of the process. Thisaspect involves an ability to break down the operation into its base com-ponents. Examples include addition as counting, multiplication as repeated

addition, functions as coordinate-by-coordinate mapping over two or moresets, and derivation as a limiting process. Note that this begins as a dynamicunderstanding where the operation is initially thought of as an action.

2. Familiarity with properties which the operation is able to possess. Of

fundamental importance to the development of operation sense is a generalawareness of the group properties of the operation (if they exist). Of these,perhaps of primary importance is an awareness of the ability to reversethe operation (invertibility). The act of undoing provides a map back to

the beginning and, in time, can help make clear the general result of theoriginal process (Wenger, 1987). Other properties, such as commutativity,associativity, and the existence of an identity, may or may not be character-istic of a given operation, but in either case they help to clarify its general

nature.This type of familiarity with properties of operations is supported by

experience with a variety of different mathematical operations acting ondifferent mathematical objects (Riley et al., 1983). For example, an aware-

ness of the commutative property in a mathematical context is usuallyobtained at an early age through additive experiences, but it may not befully appreciated until one also learns subtraction, for which commutativ-ity fails. One consequence of such an understanding of operation properties

could be the promotion of flexibility in thinking generally about computa-tion. For the case of addition, it will be argued that this can lead to algebraicways of thinking.

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OPERATION SENSE 255

3. Relationships with other operations. In addition to the relationships

an operation has with its inverse, the distributive property in any fieldprovides a means of connecting two operations, such as addition and multi-

plication. However, operations can be related in other ways besides the dis-tributive property. Multiplication is often initially understood as repeated

addition, and division as repeated subtraction. Function composition relatestwo or more sets of individual input-output pairs. A triangle is the result ofdrawing three lines that uniquely intersect pair-wise and taking intersec-tions to form the vertices and sides. More advanced examples include the

integration of a function and finding the limit of a sequence of functions.Seeing the interplay between the operations in each of the above examplescan help to enhance ones awareness of each individual operation.

4. Facility with the various symbol systems associated with the opera-tion. Arcavi (1994), in the context of algebra, has defined symbol senseas a list of understandings, feelings, and abilities that allow one to quickly

and instinctively act on a given symbol system. Before this kind of fa-cility with an operations symbols can be achieved, connections must beestablished between the symbols and the underlying meanings associated

with the operation and the objects on which the operation acts (Hiebert,1988). When an operation can be symbolized in more than one fashion,the cognitive load is increased. For example, multiplication is commonlyexpressed using each of the following symbols: x, , ( ). Further, operationproperties such as inverse can be misconstrued due to notations. One couldwonder if there was ever a student who did not misinterpret the symbolsf1(x) upon his or her initial encounter. The failure to develop a user-friendly symbol system for geometric transformations may be one reason

for their marginalization in many curricula.Constructivist perspectives and teaching experiments are beginning tolook at instances in which childrens invented symbol systems can lead tothe establishment of aspects of operation sense. Steffe and Olive (1996)

describe a computer environment in which two ten-year-old children wereable to create their own notations for arithmetic operations out of a parti-tioning activity. The resulting symbolic operational activity both affordedand restricted their ability to develop sense of the act of partitioning. What

is important about this experiment is that the mental operations involvedin the experiential abstraction were symbolic in nature, which is a crucialelement in transforming sensory-motor action into symbolic action. Twolimitations of this environment relate to previously discussed aspects of

operation sense. First, the children had difficulty identifying a unit struc-ture in the operation in the context of their symbolic operational activity,which suggests that the operation was yet to be completely understood in

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terms of its base processes in their symbolic environment. Steffe and Olive

state:

The children experienced difficulty in using the results of symbolized partitioning oper-

ations as material of further operations . . . . We interpret Melissas drawing activity andJoes counting activity as necessary because they needed to actually carry through with

partitioning operations that were only symbolized in order to produce a unit structure that

they could use as input in further operations (p. 129).

Second, the children showed difficulty in reversing the partitioning actionsin their symbolic environment due to inadequacies in their understand-ing of the symbol systems in the context of mental activity. Steffe andOlives examples provide detailed insight into the nature of the relation-ships between symbols and mental objects that are necessary in construct-ing operation sense.

5. Familiarity with operation contexts. Experience with different con-

texts of the operation can provide various perspectives on which a studentcan develop sense about that operation. For example, using join, compare,and part-whole situations has been shown to be useful in the developmentof operation sense of addition (Carpenter, 1985). More recently, the act ofsplitting has been shown to provide avenues for understanding aspects ofmultiplication (Confrey and Smith, 1995). Since it is generally agreed that

transfer is a normally occurring activity, at least to some degree (Hiebertand Carpenter, 1992), increasing the number of lenses a student can useto see an operation will enhance that students ability of identifying itsuse in more general contexts. However, we must remember to distinguish

between knowledge of situational contexts (e.g., joining) and knowledgeof a mathematical operation (e.g., addition).1 Students must make thisknowledge explicit in the context of the operation in order for operationsense to be advanced.

6. Familiarity with operation facts. Knowledge of certain operationfacts has been shown to enable more advanced approaches to a given task.For example, operation facts of addition could lead to the following inven-ted strategy: 7+8 = 15 since 8 = 3+5, so 7+ 8 = 7+ 3+ 5 = 10+ 5 =15.

7. Ability to use the operation without concrete or situational referents.

A student who can perform an operation on abstract numeric values orother mental objects clearly has an advanced sense of the use of that op-

eration. For example, the student who can add numbers without concrete

referents has the ability to perform the operation through internal mechan-isms. When this is accomplished through the use of an operation fact, thennot much can be said about the level of student understanding beyond this

isolated piece of information. But when the student uses invented strategies

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in conjunction with base processes on mental objects, then the operation is

being performed in the context of the operation.Carraher et al. (1988) describe a childs strategy of making change for

a purchase of CR$80 with a CR$500 bill:Child: Eighty, ninety, one hundred, four hundred and twenty.

This add-on strategy was not used in favor of an incorrectly applied al-gorithm when encountering this same problem in a different context: 420+80 =?. One explanation is that this child has not yet made the necessaryabstractions to transfer his knowledge in the experientially-based settingto a task that demands the use of mathematical operations on numbers. Astudent who has this ability may perform the give computation by adding42+ 8, realize that these values represent sets of ten, and obtain an answerof 500. In any case, if the student is able to comprehend the mathematicalmeaning of the operation when only abstract numeric values are used, then

the student is exhibiting an understanding of the operation beyond concreteand situational referents. These understandings are often manifested in thecreation and use of heuristics.

8. Ability to use the operation on unknown or arbitrary inputs. Perhapsa higher level of operation sense is exhibited when the student is enactinghis or her understandings of the operation on quantities that are unknown

or arbitrary. Not only does this involve the use of the operation withoutimmediate concrete referents, but it also illustrates an ability to use theoperation without specific objects being signified by the input referents.Instead, the objects pertaining to the input referents are understood to be

an unknown or arbitrary part of the operation or operational aspect underconsideration. This requires acts of generalization and places the primaryfocus on the operation itself.

The previously discussed aspects of operation sense, particularly fa-

miliarity with properties and symbol systems, are crucial in allowing oneto develop an ability to operate on arbitrary inputs. Not only must theoperation be understood independent of actions on specific inputs, an un-derstanding of the structure of the operation must be used. This structural

understanding would involve facility with a symbol system that illustratesa general operational act as well as solid understandings of the relevantproperties of the operation. Examples of using an operation on arbitraryinputs include combining like terms of a polynomial expression and find-

ing the angle measure of one interior angle of an n-gon. This aspect of

operation sense is often needed in mathematical justifications that requirean examination of an arbitrary case. Because algebra as generalized arith-metic can be viewed as arithmetic on arbitrary inputs, this aspect will be

discussed in greater detail in the following section.

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9. An ability to relate the use of the operation across different math-ematical objects. A student experienced at using an operation on differ-ent mathematical objects (and symbol systems) can create various action-

object schema involving the same operation. For example, addition on con-crete manipulatives such as Base-Ten blocks, integers, fractions, decimals,

variable expressions (symbolic functions), graphs (graphic functions), vec-tors, and sequences all share a fundamental relationship in regard to theprocess, even though the mathematical objects are very different. The abil-ity to see the fundamental similarities with respect to the operation across

action-object systems illustrates a significant amount of operation sense.10. An ability to move back and forth between the above conceptions.

Operation sense involves an understanding of various components andproperties of an operation. Flexibility in these understandings allows for

an ability to move across this conceptual web. It is in seeing the operationthrough the above lenses, either separately or simultaneously, that one is

making the most use of his or her operation sense. A specific example ofthis flexibility will be discussed later in the context of algebraic thinking.

Discussion

The previous discussion of operation sense is a broad attempt at isol-ating specific contextual, symbolic, and mathematical characteristics ofoperations that can support cognitive development and sense of a givenoperation. Although the framework may be flawed in that it does not accur-

ately describe aspects of operation sense for any conceivable mathematicaloperation, it does provide a framework for discussing several commonmathematical operations, including the focus of this particular discussion

(addition). In addition, I make the same caveats as Arcavi (1994) whenhe described his notion of symbol sense; the above list should not beconsidered exhaustive, and there is much more to operation sense than alist, no matter how exhaustive it may be.

2. OPERATION SENSE OF ADDITION

The above discussion of operation sense can be used as a framework toinvestigate student understandings of addition and to observe the origins

of algebraic thinking in childrens developing proficiencies in arithmetic.

Other attempts at describing connections between arithmetic and algeb-raic thinking have been previously made. For example, Filloy and Rojano(1989) investigated how the occurrence of more than one variable in a lin-

ear equation solving situation led to a division among students at different

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OPERATION SENSE 259

levels of algebraic thinking, a phenomenon they referred to as the algeb-

raic divide. Working with middle and high school students, Herscovicsand Linchevski (1994) and Lee and Wheeler (1989) both reported serious

gaps in students coordination between arithmetic and algebraic frames ofmind on similar kinds of tasks.

The following application of operation sense to this discussion takesthe perspective that environments which facilitate cognitive processes as-sociated with generalizing arithmetic make use of particular aspects of theassociated arithmetic operations. A collective sense of these aspects from

the foundation of algebraic thinking of this kind. The previous discussionof operation sense was somewhat general by necessity, and not all aspectswill be relevant to all operational contexts. The following discussion ofhow an operation sense of addition can be used to discuss early algebraic

competence in regard to generalized arithmetic will involve Aspects 1, 2, 3,4, 7, 8 and 10. Because of the relationships that exist between these aspects,

some will be discussed in tandem, and all will be collectively discussed inthe final section.

Aspect 1: A conceptualization of the base components of the process

Connections between addition and counting as its base process, when count-ing has been reified, can correspondingly enrich the meaning of the processof addition. The reification of counting leads to the notion of number as apermanent object, although this development occurs over various stages

and, quite often, a long period of time (Fuson, 1992b; Fuson et al., 1997).Most students begin with sequential understandings of number words thateventually become differentiated and paired with objects. Pieces of thechain are then isolated and embedded within other pieces, establishing a

bidirectional chain. Fuson (1992b) calls complete understandings of thelatter truly numeric counting.

If counting is reified and number is understood as a mathematical ob-ject, then the actions involved in completing an addition task can be per-

formed on object-oriented inputs, rather than inputs which are themselvesviewed as actions. Once this development has occurred to a sufficientdegree the operation of addition can begin to be thought of as a math-ematical object itself. This would seem to signify the beginning stages in

the development of a generalized understanding of addition. Steffe and

Olive (1996) state that a numerical concept minimally includes the oper-ations involved in producing a sequence of units, uniting the units of thesequence together into a composite unit, and decomposing this compos-

ite unit into its constituent parts. Perhaps an algebraic concept, in re-

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gard to generalized arithmetic, minimally includes these same operational

understandings, only on an arbitrary unit.

Aspects 2 and 3: Familiarity with properties and relatinoships with other

operations

Briars and Larkin (1984) present a model of problem solving in the contextof addition that highlights operation properties. For example, commutativ-ity can be used to change a start-unknown additive task (_ + b = c)into a less difficult task (b + _ = c). Reversibility, or inverse, can allowthe same task to be changed into a subtraction task (c b = _). Notonly do these lead to advanced problem solving behaviors, but, from theperspective of operation sense, these actions illustrate roots of algebraic

thought along two dimensions. First, understandings of the properties ofaddition are used to obtain a more general sense of the operation as anobject possessing various properties, thereby leading to a more generalizedview of computation. Second, these problem-solving behaviors eventuallyinvolve acts of computation that are independent of specific input values.By using a generalized sense of the operation of addition to view the taskfrom an alternate standpoint, the student is no longer acting only on theinputs of the operation, but is also acting on the operation itself. The abilityto understand and make use of the properties of addition in these ways

are vital when encountering symbolic algebra in additive contexts. Moreimportantly, from the perspective of operation sense, these two dimensionsillustrate that very young children commonly work in algebraic domainswhen encountering arithmetic tasks.

Aspect 4: Facility with various symbol systems

Childrens facility with an appropriate symbol system is critical in enacting

the generalizations just described relative to the use of additive properties.Although this has been accomplished using standard algebraic notations,with and without meaning (e.g., Matz, 1980; Filloy and Rojano, 1989;Kieran, 1992), children have been shown to make arithmetic generaliz-

ations using a variety of symbol systems (Thompson, 1992; Steffe andOlive, 1996).

As the discussion of the work of Steffe and Olive illustrates, it is criticalthat generalizations of the operation be made in the context of symbolic

activity, and not just sensory-motor activity. In order for the operation of

addition to be generalized to arbitrary inputs, student understanding mustbe constituted in symbolic operational activity. This involves at least thefollowing: 1) the input must be understood as a mental object, and 2) a

knowledge of an appropriate symbol system must be achieved that allows

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for the ability to act on the operation itself. We will discuss how this

is manifested in specific uses of addition on arbitrary inputs in the nextsection.

Aspect 7: Ability to use the operation without concrete or situational

referents

Because generalized arithmetic is, in large part, arithmetic statements aboutthe arbitrary case, Aspects 7 and 8 of operation sense of addition are cru-

cial in developing this kind of algebraic thinking. The ability to utilizethe operation of addition on an abstract symbol system devoid of con-text and concrete representations provides a transition between additionon known values with concrete representations to addition on unknown

or arbitrary quantities with more symbolic representations. Hence, addi-tion without concrete or situational referents does not necessarily involveactions on unknown or arbitrary quantities, but it does place the compu-tational arena entirely within a mathematical domain and symbol system.When the quantitative amounts involved are themselves unknown the levelof abstractness rises. This can force additional computational demands,which are discussed next.

Aspect 8: Ability to use the operation on unknown or arbitrary quantities

Resnick (1992) has identified four types of reasoning that can lead to thedevelopment of operation sense of addition. The first involves childrensuse of addition on informal qualitative amounts, or protoquantities, often

accompanied by words such as big, much, and more. The next level ofreasoning occurs when these activities are performed on known quantities.The third level of reasoning occurs when the student has the ability toreason on specific numbers rather than on physical quantities. The fourth

level of reasoning incorporates the mathematics of operators (Resnick,1992) and deals with structural properties of addition. This level involvesthe ability to not only reason about numbers as mathematical objects, butto also reason about the operations that act on numbers as mathematical

objects in their own right. If algebraic reasoning is defined in terms ofbeing able to generalize arithmetic processes, then this level of operationsense is a benchmark in that development. Not only can the operation beused without an immediate signifier of quantity, but it can also be mentally

manipulated without reference to any fixed amounts.

Realizing that any two numbers can be added in any order illustratesthis level of development. However, a more complex level of developmentassociated with this kind of reasoning involves the ability to reason about

unknown but well-defined numeric values. For example, consider the man-

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ner in which a student could solve the classic handshake problem, which

involves finding the number of handshakes that occur in a room where n

people shake hands with every other person exactly once. A solution could

involve several different operations on several different mathematical ob-jects. Deciding that all n people must shake hands with the other n 1people, and then dividing by 2 to avoid duplicating handshakes, is a verycommon method. The objects are contextually handshakes, but are rep-resented as abstract quantities by the position of the people shaking hands.The operations are multiplication (repeated addition) followed by division.

Oftentimes, students will invent notations, such as AB, AC, AD, . . . , AN,to represent handshakes and then systematically add the total number ofordered pairs, obtaining 1 + 2 + 3 + . . . + (n 1) as a solution. Thisconcrete notation lends familiarity to the situation and eases the burden

of operating on abstract objects, such as handshakes. It also places adegree of certainty on the number of objects being operated on, although

the exact amounts are still not known; the operations are acting on numericvalues that are not fixed. Implicit in this analysis is an awareness of variousproperties of the operation, such as commutativity.

An example provided by Underwood and Yackel (1998) provides amuch richer context for discussing this level of development. Using acandy factory context, they asked elementary students arithmetic tasksin which the number of candies in a given roll was both unknown and

arbitrary. An example of such a task would be If I have 3 rolls of candyand another roll missing 2 pieces, and my friend has 4 rolls and 5 extrapieces, how much more candy does my friend have? The students were inan arithmetic situation, but were being asked to act on quantities that were

arbitrary. Underwood and Yackel created a symbol system that depictedan incomplete roll of candy that facilitated students work on these tasks.Because the amount of candy in a roll is arbitrary, a deep level of operationsense is required to approach this task. In essence, the task is similar to the

base-ten task 4538, only the number of possible values in the tens columnis arbitrary, problematizing the operations on the ones place.

In summary, Aspects 7 and 8 identify three important uses of the oper-ation of addition without the use of situational or concrete referents. The

first involves the ability to perform computation, in a non-rote way, withoutreliance on physical objects. This could involve exclusive use of numbers(8 + 4 = 8 + 2 + 2 = 10 + 2 = 12) or also incorporate base processes(8

+4

=8

+2

+2

=10

+2

=10

+1

+1

=11

+1

=12). The second in-

volves uses of the operation on numeric values that are finite and partially-or well-defined. The third involves the ability to bring to bear flexible un-derstandings of an operation on numeric values that are arbitrary. The latter

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two uses resemble the perspective discussed by Resnick (1992) in that the

operation is raised to noun status. Further, the complexities inherent in theuse of symbols and properties associated with the operation of addition on

arbitrary values go beyond the construction of a mental object associatedwith the operation. These complexities incorporate understandings and ac-

tions that are critical in discriminating arithmetic thinking from algebraicthinking.

Aspect 10: Ability to move back and forth between the above conceptions

Very young children can utilize their counting abilities to investigate and

understand the idea of commutativity. Students can also develop an abilityto use an operation without concrete referents from a series of contextualexperiences and knowledge of operation facts (Baroody and Ginsburg,1983; Vergnaud, 1982). Therefore, aspects of operation sense can elicit

the development of others. However, a richer forum for illustrating thepower of operation sense involves the manner in which it can be utilizedand displayed once such a development has occurred. For example, onecan use the above facets of an operation sense of addition to manipulate

algebraic symbols or, at a higher level, investigate number theory prob-lems involving an operation sense of division. The above discussion of thehandshake problem provides a specific example of a problem solution inwhich several aspects of operation sense are present. Further examples of

relationships between the various aspects of operation sense are providedin a case study discussion of an elementary school student.

Discussion

The previous discussion is not an attempt to clearly define what is algebra

or even what is generalized arithmetic. Rather, the purpose is to providea framework for discussing how young childrens actions with additioncan be used to identify more general understandings that represent roots ofalgebraic competence. The following discussion of a case study, which is

part of a larger project (Slavit, in preparation), illustrates how this perspect-ive can be used to chart this kind of development at the early elementarylevel.

3. CASE STUDY

To illustrate how the framework can be enacted, a case study from an in-

vestigation into early algebraic thought will be presented. The brief discus-

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sion is only intended to provide a snapshot of the development of Mike

between first and fourth grade (ages 6 and 9).Over the course of 5 interviews during his first grade year, Mike worked

a series of tasks designed to measure aspects of operation sense. Amongthese were 20 standard action addition tasks, most of which involved single-

digit values. The placement of the unknown in these tasks was distributedevenly. The tasks were read, and unifix cubes and paper and pencil wereprovided. Mike was successful on 16 of the 20 tasks. More importantly,Mike worked all but four of the tasks without the use of concrete mater-

ials of any kind, and he made frequent use of heuristics in his solutions.For example, Mike displayed an ability to add-on-from-larger during thesecond interview:

DS: Mike has 3 blocks, and does Mike have a brother?

Mike: Yeah.

DS: Ok, Mike has 3 blocks, and his brother gives him 5 more blocks for Christmas, so

how many does he have altogether?

Mike: (reaches for blocks, then pulls back) I can do that without.

DS: Can you do that without?

Mike: Um, he would have 8.

DS: Ok, can you write your answer down for me. How did you do, did you just know

that or did you have to think about that?

Mike: I had to think about that.

DS: Ok, what did you think about?

Mike: (p)

DS: I mean did you count them or what did you do?

Mike: I started with the first number that was larger and I added the lowest number with

it.

DS: Oh, ok, so you went 5 and then 3?Mike: (writes = 8)

Other kinds of tasks were also given to chart the presence of specific as-

pects of operation sense. Mike exhibited facility with and knowledge ofadditive properties, showing awareness and understanding on all 4 com-mutativity tasks and 3 of the 4 identity tasks. During the first interview,Mike was already showing signs of operation sense of addition, as evid-

enced by his use of commutativity in the following discussion:

DS: How many do we have there?

Mike: (counts) 6.

DS: 6, ok. What if I were to give you that many (2) more, how many would we have?Mike: 8.

DS: You did that quick. How do you know 8?

Mike: Because 6 plus 2 is 8.

DS: Ok, how do you know 6 plus 2 is 8?

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OPERATION SENSE 265

Mike: Uh, I memorized it.

DS: Ok, ok, thats good. Let me give you one more like that. Ok, not that many that

time. What is that, just 2?

Mike: Yeah (looking at the blocks).

DS: Ok, what if I give you, whoops, dropped one. What if I give you this many (6) more,how many do you have?

Mike: (immediately) 8.

DS: Howd you get that?

Mike: Uh, because its still 6 plus 2 is 8.

Mikes use of facts, heuristics, and additive properties allowed him to acton the operation, rather than the inputs, to successfully complete several

of the tasks.These abilities allowed Mike to develop additional aspects of operation

sense of addition. For example, Mike was able to successfully apply hisunderstandings to a repeated addition situation involving pairs of shoes.

However, during the third interview when the numbers were larger, Mikeshowed a bit of confusion:

DS: Do you know how many eggs are in a dozen eggs?

Mike: 12.

DS: Ok, how many eggs owuld be in 4 dozen eggs?

Mike: (p) uh 44.

DS: How did you get 44?

Mike: Because you get 40, and you add those extra 40, those ones, and you add the 40, the

ones together equal 44.

DS: Ok, but theres twelve in a dozen, right?

Mike: Yeah.

DS: So, after how many more than 10 is 12 (p) wouldnt there be 2 extras?

Mike: Ok, that owuld be 46.DS: 46. How did you get 46?

Mike: Because if I had extra two, and I had four of them, then it would be 4, 5, 6, the 2.

However, Mike did show an ability to think flexibly about the operation ofaddition with two-digit numbers. The following dialogue, during the third

interview, relates to a task that compares the values of 13+5 and 10+3+5:

DS: Ok, now this is 10+ 3 + 5 and 13 + 5 (pointing to paper).Mike: So, add these together?

DS: Well, you could do it, but which do you think would be the biggest, or would they

be the same?

Mike: The biggest would be these two (pointing to 13+

5).

DS: That would be the biggest?

Mike: If it, yeah, that would be the biggest, because, see, if I didnt have this (the 5), it

wouldnt be the biggest, but if I had the 5 it would be the biggest.

DS: Ok, what difference does the 5 make?

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266 DAVID SLAVIT

Mike: Because, see, this would be 13, and this (13+ 5), that is 18, and see its 13, 14, 15,16, 17, 18, and thats, no, that isnt the biggest, oh, I messed up (pause).

DS: (laughs) Thats ok, just think about it.

Mike: I think theyre the same.

DS: Ok, how do you know theyre the same?Mike: Because 15, 14, because, if that one was a 15 and that wasnt there, and I had 5, and

that already has a ten on it because its a 13, and they would equal the same.

DS: Say that one more time, I think I see what youre saying, but say that one moretime.

Mike: That one (13+ 5) already had 13, and if I added these two (10 and 3 in 10 + 3+ 5)together it would be the same, and then it would equal the same, and thats why its

the same.

DS: Ok, I think I see what youre saying.

Mike: Because these two 5s, the 3 and the 3, and then I added ten and its the same.

DS: Very good.

Mike was able to break down the operational inputs into component pieces

to compare the values of the final outputs. Hence, his sense of addition

was beyond immediate actions on inputs, and his method of analysis didnot depend on a final evaluation of the operational process. Rather, a moresophisticated sense of addition was emerging that allowed Mike to ma-

nipulate both the operation and the inputs on which the operation wasacting.

Specific evidence was also obtained on how Mike used these abilit-ies to generalize his arithmetic understandings in specific situations. Mike

showed an ability to apply the operation of addition on novel mathematicalobjects, as he was successful on a clock arithmetic task (e.g., What timewould it be 7 hours after 9:00?) in both interviews 4 and 5, as well asa task involving arithmetic mod 13. However, an unplanned interaction

provided more revealing evidence regarding his ability to use addition onnon-standard objects and unknown quantities. When Mike accidentallysaw a + c = _ written on the interview sheet, the following dialogueoccurred:

Mike: a + c is 3.DS: It is? (laughs)

Mike: Yeah, by the numbers, the c and b, a. c has 3, and then the a has one. Oh, thats 4.

DS: Oh, ok, so a + c is 4. Whats a + b?Mike: h + b, so that would be a, b, c, d, e, f , g , h, i (while counting on fingers to 9).

a , b , c , d , e , f , g , h (while counting on fingers to 8). That would be h, plus, what?

DS: c.

Mike: c. h+ c. h,h,h,h (pauses) Whats after h, I forgot. a , b , c , d , e , f , g , h (pause) K!Its k!

Mike created his own additive world, including the use of a as the unit in-

crease, by applying his understandings of addition to a symbol system that

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OPERATION SENSE 267

Figure 1. Interview tasks given to Mike at Age 9.

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268 DAVID SLAVIT

normally does not relate to this arithmetic context. His understanding of

counting as the base process of addition was central to his success. Mikesability to apply his understandings in this manner suggests that he had a

flexible operation sense of addition that allowed for generalizations in amanner conductive to the development of algebraic ways of thinking, par-

ticularly in regard to algebra as generalized arithmetic. Clearly, numerousaspects of operation sense were at work during these generalizations.

The narrow scope of a first grade classroom limits the degree to whichan analysis of operation sense of addition can be performed. Therefore,

a follow-up interview was conducted with Mike during his fourth gradeyear that contained several tasks to measure his awareness of aspects 7,8, and 9 and the degree to which he was able to generalize his arithmeticunderstandings. Overall, Mike showed a sense of addition that allowed him

to reason about unknown quantities, but showed a limited ability to operateon arbitrary quantities. In addition, Mike had difficulty coordinating his

algebraic thinking with symbols that depicted arbitrary quantities.Analysis of four tasks2 (Figure 1) will be presented to illustrate Mikes

development in reference to Aspect 8. Task 1 was intended to measure thedegree to which Mike could reason about well-defined but unknown quant-ities. Mikes initial attempts suggested that he had difficulty quantitativelyidentifying the situation, where this is defined as gaining a sense of therelationships between the quantities involved. The unknown value of the

starting amount problematized his ability to utilize the relationship threetimes as many, something he repeated several times. After 6 minutesand two inappropriate solution attempts, Mike was able to quantitativelyidentify the situation, which prompted a guess-and-adjust strategy (e.g.,

7 3 = 21, 21 16 = 7?). After testing 5 and 7, he whispered, I betits 8. However, rather than letting him complete the task, I interrupted inorder to test his ability to work the problem in a less computational mannerand with a different symbol system:

DS: Let me ask you this, um, we dont know how many she picks, so lets just draw

something that looks like maybe a basket.

Mike: A basket (draws).

DS: Yeah, draw something, and that will stand for how many she picked. Now Bobby

picks three times as many apples as Jenny. How could you write down how many

apples he picked?

Mike: Uh, say that again. I didnt understand what you just said.

DS: (restates original question)Mike: (long pause) I still dont understand what youre, um (pause)

At this point, an important difference in the wording of the question is

made:

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OPERATION SENSE 269

DS: Ok, if Jenny picked one basket of apples, how many would Bobby have picked?

Mike: Oh. Three. Yeah, three.

DS: Ok, can you write that somewhere.Mike: Ill just draw three little baskets (draws).

DS: Good. So thats how many Jenny picked, thats how many Bobby picked, and

remember he lost 16. So 16 from here would give you the same amount as there.

Mike: Say that again.

DS: (repeats) So is there a way to write down losing 16?

Mike: Losing 16? He lost two baskets. Of 8. Could he have done that?

DS: Two baskets of 8. What makes you say that?

Mike: Ok, because, ok, these each have 8 in them, and, 24 total, so you could have lost

two baskets to equal one of these, if they both have 8.

With the basket symbol system, Mike was able to apply his operationsense of addition (and multiplication as repeated addition) to act on theunknown quantity. By equating two baskets to the difference amount of16, he was able to quickly obtain a solution. Although he was initially ableto solve the task by a series of computations, the latter solution suggests

that he was able to extend his operation sense into higher algebraic realmswhen provided with an appropriate symbol system and specific guidancein how to express numeric relationships, which involved unknowns, usingthat system.

The final three tasks, adapted from Underwood and Yackel (1998),measure the other facet of Aspect 8; these tasks address the degree to whichMike was able to operate on arbitrary quantities. Mike quickly solved thefirst of these tasks by covering up the two sets of 3 candies, stating, These

are even, so its just this (the remaining pieces), and then by covering 2 of

the 7 remaining pieces of the final set to obtain his solution. Mikes com-ment indicates an ability to think about the arbitrary amount as a compositeunit, as he was able to perform subtraction on the rolls.

On the final two tasks, Mike was unable to overcome the difficulties in-herent in operating on arbitrary quantities. The following occurred duringTask 3:

Mike: (mumbles these will have two extra) Um, two, um, let me think, yeah, I think 7

packages, I mean 7 little pieces.

DS: Ok, how did you get that?

Mike: See, you had this that was crumpled up, so you could take this, um, no, I think it

was one package, um, oh yeah (confidently), so now you have a whole package, ok,

I did that wrong, first Ill take this one away, so its just 3, so 3 and 1

DS: Say that again.

Mike: Ok, I have, I think, 1 package, um, let me do this again (long pause) I think you

have 3 extra, I think.

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270 DAVID SLAVIT

DS: Ok, Im not quite sure how youre getting this.

Mike: Ok, cause he bought another package, right?

DS: Well, you start with 3 wholes and a pack missing two, and then you have four

wholes and another thing of 5.

Mike: I just thought you could add these 2 (2 of the 5 pieces from the second set) to thisone (the incomplete package of the first set), and that would equal this one (fourth

complete package of the second set) and then youd have these three extra.

DS: Ok, so the difference would be 3?

Mike: Yeah.

Mike could have easily solved the equation 4538, the base-ten analogue

of this task. However, because of the arbitrary nature of the tens placein this task, Mike was unable to correctly operate on the numeric valuespresent due to difficulties in quantitatively identifying the situation. It isunknown if Mikes correct initial answer is based on appropriate thinking.

However, Mike used the same reasoning on the final task to obtain an

answer of 2 packs and 2 pieces. This solution reaffirms Mikes ability tooperate with an arbitrary unit, as he successfully canceled out three packsfrom each set. However, the arbitrary amount in a given pack prevented

Mike from correctly using his operation sense of addition on that part ofthe task.

4. IMPLICATIONS

The purpose of this paper was to present the framework of operation

sense as a means of measuring the level of algebraic thought present inchildrens thinking and problem solving schema. The idea of operationsense provides a means of analyzing the development of these structuresat an appropriate level of abstraction (Kaput, 1995). The particular com-

ponents of operation sense allow for specific measures of development.This discussion provides a first step in determining which aspects rep-resent some of the more important benchmarks in this development aswell as the kinds of interactions within the aspects that lead to growth

in understanding.Demby (1997) provides evidence that high school algebra students are

reluctant or not able to make use of operation properties such as com-mutativity when manipulating variable expressions. Case study evidence

presented here shows that many of the understandings necessary to se-

mantically interpret algebraic symbols are present in six-year old children(in the context of a number-addition symbol system). Later instructionwith formal algebraic symbolisms must support the maintenance of this

semantic awareness when encountering novel syntactic aspects (Lee and

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OPERATION SENSE 271

Wheeler, 1989). While some researchers have reported that students en-

counter enormous obstacles when attempting this kind of development(Matz, 1980; Lee and Wheeler, 1989), Davis (1964) showed how stu-

dents sense of addition can be used in the development of algebraic prin-ciples. The evidence presented here suggests that environments promoting

symbolic operational activity are highly important in this development.Students at the age of 6 and 7 are quite capable of developing deep

understandings of mathematical processes and can be well on their way todeveloping algebraic ways of thinking. Studies that address this develop-

ment at more advanced grade levels would provide additional informationon the kinds of understandings that arise from these roots of algebraicthought, and are then necessary to make sense of the abstract cognitivedemands of formal algebra.

NOTES

1 The author wishes to thank Tom Carpenter for making this point clear.2 Task 1 was adapted from work by David Carraher and Analucia Schliemann. Tasks 2

4 were adapted from work by Diana Underwood and Erna Yackel.

ACKNOWLEDGEMENTS

The author wishes to thank Jim Hiebert, Tom Carpenter, Erick Smith, themembers of the Early Algebra Research Group led by Jim Kaput, two

anonymous reviewers, and the editor for their helpful feedback on earlierdrafts of this paper.

REFERENCES

Arcavi, A.: 1994, Symbol sense: Informal sense-making in formal mathematics, For the

Learning of Mathematics 14 (3), 2435.

Baroody, A.J. and Ginsburg, H.P.: 1983, The effects of instruction on childrens under-

standing of he equals sign, The Elementary School Journal 84, 199212.

Briars, D.J. and Larkin, J.H.: 1984, An integrated model of skills in solving elementary

word problems, Cognition and Instruction 1, 245296.

Breidenbach, D.E., Dubinsky, E., Hawks, J. and Nichols, D.: 1992, Development of theprocess conception of function, Educational Studies in Mathematics 23, 247285.

Carpenter, T.P.: 1985, Learning to add and subtract: An exercise in problem solving,

in E.A. Silver (ed.), Teaching and learning mathematical problem solving: Multiple

research perspectives, Erlbaum, Hillsdale, NJ, pp. 1740.

educ807.tex; 16/06/1999; 14:11; p.21


22/24

272 DAVID SLAVIT

Carraher, T.N., Schliemann, A.D. and Carraher, D.W.: 1988, Mathematical concepts in

everyday life, in G.B. Saxe and M. Gearhart (eds.), Childrens Mathematics, Jossey-

Bass, San Francisco, pp. 7187.

Cobb, P., Yackel, E. and Wood, T.: 1992, A constructivist alternative to the representa-

tional view of mind in mathematics education, Journal for Research in MathematicsEducation 23(1), 233.

Confrey, J. and Smith, E.: 1995, Splitting, covaration, and their role in the development of

exponential functions, Journal for Research in Mathematics Education 26(1), 6686.Davis, R.B.: 1984, Learning Mathematics: The Cognitive Science Approach to Mathemat-

ics Education, Routledge, London.

Davis, R.B.: 1964, Discovery in mathematics: A text for teachers, Addison-Wesley,

Reading, MA.

Demby, A.: 1997, Algebraic procedures used by 13-to-15-years olds, Educational Studies

in Mathematics 33(1), 4570.

Filloy, E. and Rojano, T.: 1989, Solving equations: The transition from arithmetic to

algebra, For the learning of Mathematics 9(2), 1925.

Franke, M.L. and Carey, D.A.: 1997, Young childrens perceptions of mathematics in

problem-solving environments, Journal for Research in Mathematics Education 28(1),

825.

Fuson, K.C.: 1992a, Research on whole number addition and subtraction, in D. Grouws

(ed.), Handbook of Research on Mathematics Teaching and Learning, MacMillan, New

York, pp. 243275.

Fuson, K.C.: 1992b, Research on learning and teaching addition and subtraction of whole

numbers, in G. Leinhardt, R.T. Putnam and R.A. Hattrups (eds.), The Analysis of

Arithmetic for Mathematics Teaching, Erlbaum, Hillsdale, NJ, pp. 53187.

Fuson, K.C., Wearne, D., Hiebert, J.C., Murray, H.G., Human, P.G., Olivier, A.I., Car-

penter, T.P. and Fennema, E.: 1997, Childrens conceptual structures for multidigit

numbers and methods of multidigit addition and subtraction, Journal for Research in

Mathematics Education 28(2), 130162.

Herscovics, N. and Linchevski, L.: 1994, A cognitive gap between arithmetic and algebra,Educational Studies in Mathematics 27, 5978.

Hiebert, J.: 1988, A theory of developing competence with written mathematical symbols,Educational Studies in Mathematics 19, 333355.

Hiebert, J. and Carpenter, T.P.: 1992, Learning and teaching with understanding, in

D. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning,

MacMillan, New York, pp. 6597.

Hiebert, J., Carpenter, T.P., Fennema, E., Fuson, K., Human, P., Murray, H., Olivier, A. and

Wearne, D.: 1996, Problem solving as a basis for reform in curriculum and instruction:

The case of mathematics, Educational Researcher 25(4), 1221.

Hiebert, J. and Wearne, D.: 1993, Instructional tasks, classroom discourse, and stu-

dents learning in second-grade arithmetic classrooms, American Educational Research

Journal 30(2), 393425.

Kaput, J.J.: 1989, Linking representations in the symbol systems of algebra, in S. Wagner

and C. Kieran (eds.), Research Issues in the Learning and Teaching of Algebra, NCTM,

Reston, pp. 167194.

Kaput, J.J.: 1995, A research base supporting long term algebra reform?, in D.T. Owens,M.K. Reed and G.M. Millsaps (eds.), Proceedings of PME-NA XVII, Columbus, Ohio,

pp. 7194.

educ807.tex; 16/06/1999; 14:11; p.22


23/24

OPERATION SENSE 273

Kaput, J.J. (ed.): in press, Employing Childrens Natural Powers to Build Algebraic

Reasoning in the Content of Elementary Mathematics.

Kieran, C.: 1992, The learning and teaching of school algebra, in D.A. Grouws (ed.),

Handbook of Research on Mathematics Teaching and Learning, MacMillan, New York,

pp. 390419.Kieren, T.E. and Pirie, S.E.B.: 1991, Recursion and the mathematical experience, in

L. Steffe (ed.), The Epistemology of Mathematical Experience, Springer-Verlag, New

York.Lee, L. and Wheeler, D.: 1989, The arithmetic connection, Educational Studies in

Mathematics 20, 4154.

MacGregor, M. and Stacey, K.: 1997, Students understanding of algebraic notation: 11

15, Educational Studies in Mathematics 33(1), 119.

Matz, M.: 1980, Towards a computational theory of algebraic competenece, Journal of

Mathematical Behavior3(1), 93166.

Piaget, J.: 1964, Learning and development, in R.E. Ripple and V.N. Rockcastle (eds.),

Piaget Rediscovered: A Report of the Conference on Cognitive Studies and Curriculum

Development, Cornell University, Ithaca, NY, pp. 720.

Riley, M.S., Greeno, J.G. and Heller, J.I.: 1983, Development of childrens problem

solving ability in arithmetic, in H. Ginsburg (ed.), The Development of Mathematical

Thinking, Academic Press, New York, pp. 153196.

Resnick, L.B.: 1992, From protoquantities to operators: Building mathematical com-

petence on a foundation of everyday knowledge, in G. Leinhardt, R.T. Putnam and

R.A. Hattrups (eds.), The Analysis of Arithmetic for Mathematics Teaching, Erlbaum,

Hillsdale, NJ, pp. 373429.

Resnick, L.B. and Omanson, S.F.: 1987, Learning to understand arithmetic, in R. Glaser

(ed.), Advances in Instructional Psychology Vol. 3, Erlbaum, Hillsdale, NJ, pp. 4195.

Schoenfeld, A.H., Smith, J.P. and Arcavi, A.: 1993, Learning: The microgenetic ana-

lysis of one students evolving understanding of a complex subject matter domain, in

R. Glaser (ed.), Advances in Instructional Psychology Vol. 4, Erlbaum, Hillsdale, NJ.

Sfard, A. and Linchevski, L.: 1994, The gains and pitfalls of reification The case of

algebra, Educational Studies in Mathematics 26, 191228.

Slavit, D.: 1997a, An alternate route to the reification of function, Educational Studies inMathematics 33(3), 259281.

Slavit, D.: 1997b, Learning as sense-making and property-noticing, in J.A. Dossey,

J.O. Swafford, M. Parmantie and A.E. Dossey (eds.), Proceedings of PME-NA XIX(Vol.

1), ERIC Clearinghouse, Columbus, OH, pp. 649656.

Slavit, D.: in preparation, The development of operation sense.

Steffe, L.P. and Olive, J.: 1996, Symbolizing as a constructive activity in a computer

microworld, Journal of Educational Computing Research 14(2), 113138.

Thompson, P.W.: 1992, Notations, conventions, and constraints: Contributions to effect-

ive uses of concrete materials in elementary mathematics, Journal for Research in

Mathematics Education 23(2), 123147.

Thompson, P.W.: 1993, Quantitative reasoning, complexity, and additive structures,

Educational Studies in Mathematics 25(3), 165208.

Thompson, P.W.: 1994, Students, functions, and the undergraduate curriculum, in E. Du-

binsky, A.H. Schoenfeld and J. Kaput (eds.), Research in Collegiate Mathematics Educa-tion I, American Mathematical Society, Providence, RI, and Mathematical Association

of America, Washington, D.C.

educ807.tex; 16/06/1999; 14:11; p.23


24/24

274 DAVID SLAVIT

Underwood, D. and Yackel, E.: 1998, Developing a concept of an algebraic expression as

a composite unit, Paper presented at the annual conference of the American Educational

Research Association, San Diego.

Vergnaud, G.: 1982, A classification of cognitive tasks and operations of thought involved

in addition and subtraction problems, in T.P. Carpenter, J.M. Moser and T.A. Romberg(eds.), Addition and Subtraction: A Cognitive Perspective, Erlbaum, Hillsdale, NJ,

pp. 3959.

Vergnaud, G.: 1988, Multiplicative structures, in J. Hiebert and M. Behr (eds.), Number

concepts and operations in the middle grades, Reston, NCTM, pp. 141161.

Wenger, R.H.: 1987, Cognitive science and algebra learning, in A.H. Schoenfeld (ed.),

Cognitive Science and Mathematics Education, Erlbaum, Hillsdale, NJ, pp. 217251.

Washington State University,

Vancouver, WA 98686,

U.S.A.

E-mail: [email protected]

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