developing the concept of linear function: one student’s experiences with dynamic physical models

Journal of Mathematical Behavior20 (2002) 337–361

Developing the concept of linear function: one student’sexperiences with dynamic physical models

Ellen Hines∗

Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115, USA

Abstract

Developing a view of functions as systematic processes involving co-variation among variables hasbeen identified as a goal for mathematics learners at the pre-university level. In this investigation I exam-ined the processes used by an eighth-grade student to interpret linear functions originating in dynamicphysical models and the processed he used to link his interpretations to tables, equations, and graphs.The student deepened his understanding of functions by generalizing his view of multiplication beyondthat of products resulting from the multiplication of individual factors or from repeated addition. Hewas enabled to do this by building links between graphic and tabular representations of the functionsgenerated from his exploration with dynamic physical models and by comparing tables of differentlinear functions. This paper suggests that the development of a student’s reasoning about functionsoriginating in dynamic physical models can be interpreted in terms of generalized multiplicative pro-cesses that may occur thorough mapping variations and that a student who interprets such functions asgeneralized multiplicative processes may use notational variations to generate representing equations.© 2002 Elsevier Science Inc. All rights reserved.

Keywords: Linear function; Multiplicative processes; Mathematics learners

1. Introduction

By providing an underlying structure for much of the study of formal mathematics,knowledge of functions is widely recognized as essential for mathematics students at the

∗ Tel.: +1-815-753-6756; fax:+1-815-753-1112.E-mail address: [email protected] (E. Hines).

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338 E. Hines / Journal of Mathematical Behavior 20 (2002) 337–361

pre-university level (NCTM, 1989, 2000). Many studies have focused on student’sinterpretations functions in their various representations (e.g., Monk & Nemirovsky, 1994;Tall & Bakar, 1992; Vinner, 1992, 1983), and generating connections between representa-tions of functions has emerged as an important component in understanding functions (e.g.,Moschovichs, Schoenfeld, & Arcavi, 1993; Eisenberg & Dreyfus, 1986; Even, 1990). Know-ing details of the processes used by students to create and link representations of func-tions reflects the quality of students’ interpretations and provides a basis for instructionaldecisions.

This paper reports on the processes that an eighth-grade student used to interpret linearfunctions originating in dynamic physical models and to link his interpretations to symbolicrepresentations. The physicality of students’ explorations with these devices differs from thatof functions specified in other formats that students will likely encounter, and study of the roleof dynamic physical models in the development of knowledge of functions is warranted in lightof ongoing neurological research aimed at examining how the brain integrates multiple sensoryinputs (Stein & Meredith, 1993). In this study, a student generalized a view of multiplicationthat enabled and supported a deep understanding of functions by building links betweenrepresentations. Findings from this investigation contribute a new measure of detail to ourknowledge of the processes used by students to develop meaning for functions by buildinglinks between their representations.

2. Related research

2.1. Dynamic physical models

Dynamic physical models are mechanical tools that can be used to visualize functions. Theirfeatures include a user-operated domain variable and a separately generated range variable.The specific dynamic physical models used in this study were a spool elevating system, andan Etch-a-SketchTM toy. Dynamic physical models were used to introduce function conceptsto students because of their potential to help students concretely experience variables relatedthrough systematic change.

Many researchers have investigated learners’ interpretations of functions generated throughdynamic physical models. For example, Piaget, Grize, Szeminska, and Bang (1977) docu-mented developmental levels in learners’ knowledge of functions using a winch system con-taining tracks through which metal pieces were pulled by rotating axles of various sizes. Us-ing a similar device containing rotational spools of different circumference to elevate objects,Greeno (1995) analyzed the understanding of linear functions in terms of learners’ attainmentsto constraints based upon quantitative properties of the physical system, affordances of thephysical environment, and students’ abilities to reason about quantitative relationships of thesystem. Greer (1992) suggested that learners’ investigations with a physical system consistingof a bucket attached to a rotational handle by which the bucket was elevated could promote ashift in learners’ views of multiplication away from that of an isolated relationship between

E. Hines / Journal of Mathematical Behavior 20 (2002) 337–361 339

the multiplier, the multiplicand, and product, to a richer view of multiplication as an implicitfunction relationship.

Other researchers have investigated learners’ creation and use of representations for func-tions originating in dynamic physical models. Greeno (1995) considered syntactic and se-mantic constraints and affordances of conventional symbol systems and of the meanings ofsymbols as referents to the system being represented. Using a winch device similar to those de-scribed earlier, Meira (1995) found that meanings created for material representations evolvedover time through learners’ activity and were related to the problem solving strategies thatemerged in the activity. Meira (1998) suggested that learners’ activities with such devicescreate the possibility for increased transparency of the underlying mathematics concepts. ButIzsák (2000), who also used a winch device to document students’ creation of an unconven-tional equation to model a problem originating in the physical system, has shown that thelearners’ previous knowledge of mathematics and of a physical system both supported andconstrained their equation writing efforts. He pointed out that much of the research in devel-opment of knowledge of functions is based upon data analyses that do not consider a learner’sprior knowledge of the physical world, particularly knowledge gained through experienceswith dynamic physical models, and that little research explains how students link such priorknowledge to representing equations.

2.2. Understanding functions

In order to address the question of development of understanding of functions, it is nec-essary to consider how learners might generally understand mathematics. Lesh, Landau, andHamilton (1983) developed a model for the use of representations in which they identifiedfive non-distinct representational modes: real-world situations, manipulative models, pictures,spoken symbols, and written symbols. Translation from one mode of representation to anotheris one way students grow in mathematics “understanding.” Kaput (1989) used the term “rep-resentation system” to describe a learner’s building of correspondences between the notationof one representation to develop “relational” meaning for a second. The dynamical theoryof growth of mathematics understanding developed by Pirie and Kieren (1992) describes aprocess through which learners may develop connections between representations as theyprogress from informal modes of action to formal reasoning levels. The learning process in-volves “folding back” where a learner reasoning at a formal level relies upon images createdat an informal level to support and broaden understanding.

A deep understanding of linear functions centers on the notion of systematic co-variationbetween two related variables in which functions are viewed as generalized processes, ratherthan as collections of isolated ordered-pair matchings. At the secondary level, a student is likelyto encounter a definition of function involving a rule or correspondence between the elementsof two non-empty sets where each element of one set is associated with a unique element ofthe second set. This kind of definition, originally developed by Dirichlet and later framed inset theoretic tradition by Boursbaki, is a powerful tool in the formal mathematics curriculum(Cooney & Wilson, 1993). As an entry point into the study of functions, however, it may


not support the development of a deep understanding of functions as generalized processesinvolving co-variation among variables (NCTM, 1989).

Piaget et al. (1977) described a progression in the development of function knowledge. Withdevelopment, a young child who may initially make no connection between two variables willbecome capable of linking the variables. Later, the child will make multiple pair assignmentsinherent in schemas of action, and as development continues he/she will begin to describepatterns of change in one variable relative to patterns of change in a second variable. Thesekinds of descriptions, which are initially qualitative and later quantified, are a foundationfor the interpretation of functions as generalized processes involving co-variation amongvariables.

Many researchers have found the distinction between process and object conceptions offunctions to be useful in considering students’ understanding of functions (e.g., Breidenbach,Dubinsky, Hawks, & Nichols, 1992; Sfard, 1991). At the process level, an individual considersa function in terms of a sequence of actions that have been interiorized and “condensed”into a relationship of input and output (Sfard, 1991). The student has abstracted from theindividual actions of specific cases to recognize a general pattern of input transformed tooutput (Breidenbach et al., 1992). Through “reification” a learner considers a function as awhole created object, and the process and object conceptions are mutual complements in alearner’s knowledge of functions (Sfard, 1991).

A pre-process conception of function has been described by Breidenbach et al. (1992)in which a learner’s thinking centers on repeatable mental or physical actions. This actioninterpretation may involve, for example, replacement of a symbol by a number followed bycomputation to obtain an “answer,” with little awareness of an overriding process, and eachexample remains isolated from the previous one. Understanding functions involves interplaybetween process and object conceptions that is rooted in the action conception.

Recognition of functions as processes involves awareness of the co-variation among vari-ables in a function, coupled with realization of the consistency of the generalized processrelating variables. Developing a deep understanding of functions as a learner uses vari-ous symbolic representations of functions hinges upon the learner’s interpretation of sym-bolic variables and equations. When students’ interpretations of the symbols in functionequations indicate awareness of consistency in the systematic changes in the values of thesymbols, the students are interpreting the symbols as variables in a function relationship(Küchemann, 1981). Some non-function interpretations of symbols identified by Küchemanninclude ignoring symbols, using symbols as objects, or assigning numeric values to sym-bols. A common error occurring as students attempt to write equations involving symbolicvariables is use of direct syntactic translation of a verbal sequence to a symbolic equation(Clement, 1982). Known as a reversal error, it often appears in attempts to produce lin-ear equations and is indicated when the students multiply the wrong variable by the slopefactor.

Thompson (1994) described the development of learner’s knowledge of ratios, rates, andlinear functions using a computer micro-world to simulate the physical motion of two animalsrunning separately or simultaneously along parallel tracks at assigned speeds. By focusing


on the mental operations used to interpret quantitative relationships, he asserted that a ratioresults through the multiplicative comparison of two measurable quantities. The quantitiesare related by some physical situation and interpretation of the ratio relies upon mainte-nance of the physical aspects of the situation. A rate is a generalized ratio where the sepa-rate measures comprising the rate can be removed from the specific context in which theyoriginally occurred. A ratio becomes a rate when through reflective abstraction a studenthas generalized the relationship and anticipates its applicability to any number of specificcases.

Steffe (1994) defined multiplication in a manner potentially supporting this interpretation offunctions. Steffe (1994) stated that multiplication involves “the coordination of two compositeunits such that one of the composite units is distributed over the elements of the other” (Steffe,1994, p. 19). In Steffe’s description of the implicit concept of multiplication, the major focus ofreasoning is on the number of iterations of a quantity, where each iteration count is exchangedat a consistent rate for a constant number of individual items. It is this focus on the number ofiterations of a quantity, rather than on the counting of individual objects within each iteration ofa quantity, that distinguishes the concept of multiplication from the pre-multiplication notionof repeated addition. Steffe’s interpretation of multiplication can be viewed as an enablingstep in the development of knowledge of rates and ultimately of functions as systematicand generalized processes. How students develop this interpretation for multiplication withinthe context of explorations of linear functions originating in dynamic physical models is animportant question for consideration.

According to Thompson (1994), a learner reasoning about functions from a ratio perspectivefocuses on accumulations made by iterating the ratio. From a rate perspective, the learner isfocused beyond the individual iterations of the ratio, to recognize the underlying consistencyof total accumulations in constant ratio. This interpretation of a rate is a linear function.(Behr, Khoury, Harel, Post, & Lesh, 1997; Thompson, 1994). Thompson’s description of theinteriorization of ratios into generalized rates, which are linear functions, is based within thephysical context of measurable quantities, and provides a framework through which to analyzea student’s interpretation of functions originating in dynamic physical models. Izsák’s (2000)description of how students link constructions of knowledge of winch systems with symbolicequations created as models of the systems has enabled him to suggest initial features ofa model of learning that are based upon the cognitive mechanisms of notational variationand mapping variation. “Notational variation consists of experimenting with and adjustingalgebra symbol patterns until computations on paper match measurements” in the physicalsystem. “Mapping variation consists of experimenting with and adjusting correspondencesbetween algebra symbols and attributes” of the physical system. Through these processes,Izsák (2000) asserts that students develop specialize schemas called forms, that connect aspectsof the physical system with the algebraic notational system in order to account for the role ofprevious knowledge of the physical world in the development of new symbolic knowledge.While Thompson’s model is useful for interpreting how learners think about the functionsoriginating in the dynamic physical models, Izsák’s model is useful to explain how studentssymbolically represent their interpretations of the functions.


2.3. Goals for the investigation

I conducted this investigation to answer the following questions: How do students interpretfunctions created from explorations with dynamic physical models? How do students createand interpret tables, equations and graphs intended to represent functions originating from ex-plorations with dynamic physical models? And how do students interpret variables present intables, equations, and graphs of functions originating from explorations with dynamic physicalmodels? The purpose of this paper is to show how one student developed a view of multiplica-tion to support the interpretation of functions as generalized processes involving co-variationamong variables. The student developed this interpretation by building links between differentrepresentations of functions generated from exploration of dynamic physical models.

3. Method

Under the assumption that a study of students’ learning of mathematics is interactive andinvolves teaching, I used a teaching experiment design to conduct this investigation (Cobb &Steffe, 1983). Two important features of the teaching experiment are first, an interview com-ponent through which the investigator explores a student’s understanding of concepts underconsideration, and second, a teaching component through which the investigator challengesa student to reflect upon and extend his or her understanding of concepts. In keeping withtraditions of the teaching experiment, the nature and types of questioning and instruction thatI used in this investigation were based on the types of responses given by the students. Thispaper is a report of the efforts of a student named Jim to interpret functions originating fromexplorations with dynamic physical models.

3.1. Participant and setting

Jim was an eighth-grade male enrolled in pre-algebra at a public middle school in a Mid-western town of approximately 45,000 residents. His mathematics teacher described him asan average-ability student who was sometimes careless about completing his homework. AtJim’s school, 7.5% of the students were African American. Asian, Hispanic, and Native Amer-ican minorities accounted, respectively, for 2.4, 7.7, and 0% of enrollees, while 82.4% wereCaucasian; 16.7% of the students were designated as low-income; 6.1% were identified aseligible for bilingual education.

I invited Jim to become a participant on the recommendation of his teacher, who had ob-served the consistency in his characteristics with those I had identified using a purposivesampling technique described by Goetz and LeCompte (Merriam, 1988). Specifically, Jimhad little or no previous classroom experience with dynamic physical models, or with relat-ing dynamic physical models to tables, equations, or graphs, or with interpreting functionsrepresented in equations, tables, or graphs. And, Jim showed willingness in the classroom toparticipate and communicate explanations for his mathematics decisions, correct or incorrect.


Because Jim met these criteria he was considered a good candidate for participation. Later, Jimbecame an ideal candidate because of his responses on Chelsea diagnostic tests of the develop-ment of his knowledge of functions (Hart, Brown, Kerslake, Küchemann, & Ruddock, 1985).

Jim and I met in a natural and non-threatening setting to conduct the investigation. Theinterviews and teaching sessions occurred in 20–30 min sessions, two or three times per weekover a period of 2 months in a small conference room located adjacent to Jim’s mathematicsclassroom. Students often used the room for interaction and small group work, and Jim quicklyacclimated to the setting and procedures for the investigation.

3.2. Instrumentation

In order to establish a profile of Jim’s function-related knowledge prior to beginning theinvestigation, he completed the algebra, geometry and proportional reasoning tests of theChelsea diagnostic tests (Hart et al., 1985). The tests were designed “to identify a develop-mental hierarchy of understanding connecting the mathematical concepts commonly taughtin the secondary school. (They) were designed as diagnostic instruments to be used both forascertaining a child’s level of understanding and to identify the incidence of errors” (Hartet al., 1985, p. 1). Jim was instructed to do the best he could and to take guesses in situationswhere he thought he could figure out answers, but not to answer questions for which he felt acomplete lack of knowledge.

To frame the exploration, I developed an initial intended instructional sequence consistingof a series of questions and situations through which Jim explored and reasoned about thephysical models and I observed his reasoning processes. (see Fig. 1 for a sample of the typeof questions asked in the initial intended instructional sequence). In the sessions, I asked Jimto explain orally, in written words, by drawing pictures, creating tables, creating equations,and creating graphs how he reasoned about the various questions and situations. I focused myobservations on how Jim interpreted the function relationships as he experienced them withinthe models and on the processes he used to link his interpretations to tables, equations, andgraphs intended to represent the functions. My questions often deviated considerably fromthe initial intended instructional sequence to get at details of the processes he used to thinkabout the physical models and situations.

3.3. Data collection and analysis

Jim’s spoken, gestured, written, and drawn responses generated during the instructionalsessions with the dynamic physical models provided evidence of his initial interpretations offunctions and evidence of changes in his interpretations as the investigation progressed. Todocument these interpretations and the changes that occurred, I videotaped and transcribedJim’s spoken and gestured responses related to his interpretation of functions and collectedall written and drawn explanations he created during the investigation. These “low infer-ence descriptors” (LeCompte & Goetz, 1982) reduced the possibility for drawing mistakeninferences from the data and were used to substantiate the field notes taken by the investigator.


Fig. 1. Sample of the type of questions asked in the initial intended instructional sequence.

In the teaching experiment the collection of data and an initial phase of analysis occurredat the same time (Cobb & Steffe, 1983; Menchinskaya, 1969; von Glasersfeld, 1987). Duringthe initial phase of analysis I observed and considered Jim’s responses in an effort to under-stand and characterize his conceptualizations of functions and to make decisions about howinstruction should proceed. At the close of data collection, I undertook a two-phased analysis(Merriam, 1988). In the first review of the data, I identified and developed categories for dataitems that appeared pertinent to the development of knowledge of functions. Then, during asecond phase of analysis, I examined these categories to delineate overlapping categories andto combine non-distinct categories. During this process, underlying principles emerged thatsuggested implications for learning about functions.

4. Results and analyses

4.1. Initial interpretations of functions

On the Chelsea algebra test Jim responded correctly only to numerical items or to algebraicsituations where letters could be evaluated, ignored, or treated as objects. He did not use the


letters as specific unknowns, generalized numbers, or as variables in an equation representinga systematic relationship between variables. In some algebraic expressions, Jim evaluatedinappropriately. For example, when asked in one question to add 4 to the expression3x, Jimincorrectly responded, “12.”

Findings from the Chelsea graphs test show that Jim could plot points and identify coordi-nates for points from a graph, but he did not generate or graph ordered pairs from an equation.Nor did he think of a graph as a way to show a relationship. In interpreting graphs, Jim didnot consistently consider two variables changing together on a graph. He did not recognizecurving graphs as indicators of differing slopes but did realize that a graph intended to showa steady rate would be linear. Jim sometimes interpreted graphs involving distances traveledby an individual, as diagrams of the path traveled by an individual.

On the Chelsea proportional reasoning test, Jim showed little or no recognition of the over-riding multiplicative nature of proportional relationships. Although he could take whole-numbermultiples of a quantity, he used a developmentally less mature addition-of-difference strategyto solve some missing value word problems. Jim relied upon doubling for several enlargementproblems where the factor of enlargement was not a whole number.

Jim’s responses on the Chelsea diagnostic tests indicated that he entered the investigationwith little understanding of functions as generalized processes. He was unable to interpretsymbolic variables, equations, or graphs as representations for functions. The limitations inhis use of proportional reasoning suggested that he did not interpret functions as systematicrelationships at the start of the investigation. Because of these characteristics, Jim was anideal candidate for an investigation of how students might use dynamic physical models todevelop understanding of functions and of how they might connect their understandings toother symbolic representations.

4.2. Explorations with the spool elevating system

One of the physical models that Jim explored was the spool elevating system (Hines, Klan-derman & Khoury, 2001) (see Fig. 2). The spool elevating system consisted of an arrangementof spools of varying circumference, a cord with a weighted object attached at one end whichcould be connected to any of the spools, and a ruler to measure the position of the weightedobject. The position of the object was controlled by lengthening or shortening the cord throughturns of the handle attached to the spools’ axle. Each turn of the handle produced a change inthe position of the object. Starting with the object positioned at zero, and treating the numberof handle-turns taken,n, as the initial or input value to a function, the position of the object,p, varied and was dependent on the number of turns taken. For example, when the object wasconnected to the 3-spool, a spool with circumference of 3 in., a complete clockwise turn raisedthe object 3 in. A complete counter-clockwise turn lowered the object 3 in. Similarly, on the4-spool, a complete clockwise turn raised the object 4 in. A complete counter-clockwise turnlowered the object 4 in.

Jim’s introduction to the spool elevating system began with free exploration. I eventuallyasked him to place the object on the 3-spool. As he explored the physical model, I asked


Fig. 2. Illustration of the spool elevating system.

him to identify features in the situation that he saw changing, and features that stayed thesame. Among the changeable things, Jim identified the “height of the thing (object)” and the“movement of the crank” as changeable. Among the things staying the same, Jim identifiedthe length around the spool and the fact that the height of the object goes up by 3 in. with eachrotation of the handle.

Jim eventually prepared a table of values for the actions with the 3-spool in which herecorded various values for the number of rotations of the handle of the system and forthe corresponding heights attained by the object. Through exploration of the system andcompletion of the table, Jim experienced the variability of these variables before he usedsymbols to represent them.

When I asked Jim to choose symbols for the changeable things. He drew a cherry, hereafterreferred to asc, for the number of rotations and a pencil, hereafter referred to ash, for theheight of the object. I directed him to look at his table and to write a mathematical sentencethat told how the number of rotations is related to the height of the object. Jim interpretedmy request for a mathematical sentence as a request for an equation. Jim struggled to write asingle equation to represent the generalized relationship. His interpretation of the relationshipcentered on a ratio ofh, one increment of elevation of the object, toc, one rotation of the handleof the system. He initially wrote the simple equation,h = c, to represent the relationship. Laterhe wrote a sequence of individual numeric equations showing possible individual values forthe variables (see Fig. 3). In these equations his symbols for the variables were used as labelsfor units of measure rather than as symbols intended to represent systematic changes amongvariables.


Fig. 3. Jim attempts to write an equation for the function in the spool elevating system.

The following dialogue occurred between Jim and me as he attempted to generate a math-ematical sentence. Jim’s remarks are prefaced with “J” mine are prefaced with “I:”

J: Height equals the number of cranks, that’s it [writesh = c] . . .

I: If you know the number of cranks, what do you have to do to get the height? Whatis the process?

J: We know one crank equals. . . [writes 1c = 3 in.,h×3, 1c = 3 in. in height]. . . 1c =3 in. = h, 2c = 6 in. = h, 3c = 9 in. = h [writes three equations].

I: What doesc mean?J: The number of cranks.I: Why did you put a 3 inhere [pointing to the 3 in the third equation]?J: So you’d know how many cranks. [The 3 did not represent a rate of elevation factor].


In order to further explore Jim’s understanding of the relationship, I placed the object at anelevation of 15 and asked him to figure out how many rotations of the handle were needed toget the object to that height. Jim indicated that 1 rotation was equal to 3 in. in length and thennodded his head for each group of 3 in. as he counted aloud “3, 6, 9, 12, 15.” I then asked howmany rotations would be required to position the object at a height of 12, and how he couldexplain the process to another person. The following dialogue occurred:

J: Each crank equals 3 in. so see how many it takes.I: Give a general rule to find the position of the object.J: If it’s at 9, give it a crank. It will go up to 12. One crank equals 3. . .

I decided to attempt to reduce the complexity of the multiplication situation. I placed theobject on the 2-spool and asked Jim to figure how that works. After brief exploration, Jimreported that one rotation equals 2 in. on this spool.

I: Could you tell me how many cranks to get to 16?J: [nodding his head for each group of 2 in., counts aloud to 16 and reports the answer

as 8] 2, 4, 6, 8, 10, 12, 14, 16 [pause] 8.

I continued to give Jim different amounts. Each time he counted groups of two to arrive atan answer. In some cases, he used a previous answer and counted on or counted back by oneor two groups of two to arrive at an answer. After several different examples, I asked Jim todescribe the process.

J: 1 crank equals 2 in.I: . . . If you know the number of cranks you wanted to take, how could you predict the

height?J: Without the spools?I: Yes.J: If each crank goes up 2 in. and you want to go to 9, just count to 9 [pause] 2, 4, 6, 8.

It’ll be 4 and 1/2.I: What do you do?J: Just add two each time.I: What if you didn’t know the previous amount? Say a teacher came in and asked,

“How did it get to 14?”J: [nods, and counts groups of two] 7.I: Do you see any way 14 and 7 are related?J: Yeah, times two.

4.3. Analysis of explorations with the spool elevating system

Jim’s interpretation of the spool system is not surprising given his responses on the Chelseaproportions test. He uses a repeated addition strategy to identify particular heights attainedduring operation of the system. In initially writing the equation,h = c, Jim indicates his


consideration of a basic ratio where one increment of elevation is equivalent to one rotation ofthe handle of the system. But in attempting to write an equation involving the rate of elevation,he writes a series of equations, where the symbols for the variables were accompanied byspecific numeric values. Although he has chosen a symbol to stand for the number of rotationsof the handle on the spool system, he wishes to retain his knowledge of specific quantitativevalues for that symbol. He uses the symbol not to stand for the number of rotations of the axle,but as a label for the rotations of the axle. He recognizes that the number of rotations varies,but he does not assign variability to the symbol he had chosen. Furthermore, his attempt towrite a generalizing equation illustrates a reversal error (i.e., 1c = 3 in.,h × 3, 1c = 3 in. inheight) described by Clement (1982). He realizes a basic ratio of one rotation of the handle asequivalent to 3 in. of elevation of the object, but indicates that height, rather than the numberof handle rotations, should be multiplied by 3.

For Jim, each rotation of the handle produces a different height, and Jim thinks of the variousheights as incremental accumulations formed by repeatedly adding the value of each incrementof elevation. Jim establishes a basic ratio, but he did not identify or apply a generalizedmultiplicative strategy. When I ask him to describe the process for finding the number ofrotations of the handle necessary for the object to reach a particular height, Jim sometimesexplains how to arrive at that answer by using a known number of rotations from a previouslyconsidered height and counting on or back. He sometimes arrives at an answer by countingthe total number of rotations from a height of zero. Jim’s thinking focuses on each possibleheight as an individual case. He has not generalized his view of the basic ratios into ratessupporting a co-variation interpretation of functions in a generalized process.

This suggests that he has not interiorized his thinking of the system as a rate in the mannerdescribed by Thompson (1994). His awareness of the number of handle rotations as the numberof increments of elevation suggests a focus in his thinking on the number of iterated groups.This is multiplicative thinking as described by Steffe (1994). But, his thinking of the functionoriginating in the physical system and of the equation representing it, is not separated from hisknowledge of the physical measurements within the system. Neither his view of the operationof the system, nor of his equation is generalized beyond that of the multiplicative relationshipof a ratio as described by Thompson (1994). He is aware of an accumulation of handle rotationsand height increments (i.e., accumulations made by iterating a ratio) but does not focus on theoverriding constant ratio relationship (i.e., total accumulations in constant ratio).

4.4. Explorations with the Etch-a-SketchTM

I decided to have Jim explore the relationship on the Etch-a-SketchTM (see Fig. 4). TheEtch-a-SketchTM is a dynamic physical model where each complete clockwise turn of the leftknob of the toy produces a horizontal segment of length 3 cm from left to right on the screen.Each complete counter-clockwise turn of the left knob produces a horizontal segment of thesame length from right to left. Each complete clockwise turn of the right knob produces avertical segment of length 3 cm from lower to upper position on the screen. Each completecounter-clockwise turn of the right knob produces a vertical segment of the same length from


Fig. 4. Illustration of the Etch-a-SketchTM.

upper to lower position on the screen. If the knobs are turned simultaneously, a diagonalline is produced. For this investigation, the knobs were calibrated into four, quarter-turnincrements, to enable a user to represent different linear functions. This way a specific numberof turn-increments of the left knob could be assigned to a specific number of turn-incrementsof the right knob to create a function. The line produced on the screen when the knobs areturned simultaneously is a graphic representation of the function similar to a Cartesian graph.

I chose the Etch-a-SketchTM because it could be used to show function relationships includ-ing patterns in the changes of two variables that are related to each other without assigningspecific numeric values to the variables. This feature of the Etch-a-SketchTM was appealingbecause it avoided the chance that Jim would focus only on specific sets of variable valueswhich satisfy the relationship and fail to observe the overriding relationship.

Additionally, the Etch-a-SketchTM, having one knob correspond to changes in an inputvariable and the second knob correspond to changes in the output variable, uniquely registersthe relationship between those changes as a single segment appearing on the screen. Althoughthe appearance of the segment is like a Cartesian graph, the action of turning separate input andoutput “knobs” to generate the segment may facilitate realization of an underlying relationshipbetween variables that may be absent in viewing a Cartesian graph.

After some free exploration with the Etch-a-SketchTM I asked Jim to consider how we mightuse the toy to represent the action of the spool system when the object was placed on the 2-spool.I suggested how the knobs of the Etch-a-SketchTM could be turned to show variables increasingor decreasing. Without saying anything, Jim began turning both knobs of the Etch-a-SketchTM

clockwise at the same rate. He indicated to me that one rotation of the spool system producedone increment of elevation of the object and that each increment of elevation was equivalentto 2 in. of elevation. He interpreted the function on the Etch-a-SketchTM as a one-notch turn ofthe left knob matched with a one-notch turn of the right knob. The left knob was used to recordchanges in the number of rotations of the handle on the spool system, and the right knob wasused to represent the changing height of the object. His interpretation of the function using theEtch-a-SketchTM paralleled the thinking he used to create the equation for the spool elevatingsystem (i.e.,h = c). Jim noticed the graph created on the Etch-a-SketchTM, but focused hisattention on the physical increment-by-increment process of creating it. Jim’s interpretation


of the function was based on repeated actions. He showed no particular awareness of the graphas a representation for the function originating in the 2-spool. However, a change in Jim’sinterpretation occurred when I inquired about the appearance of the graph representing theoperation of the 3-spool. The following dialogue occurred:

I: Would it [the line on the Etch-a-SketchTM screen] go up faster on the 3-spool?J: [Moves both knobs on the Etch-a-SketchTM faster to create a graph similar to that

representing the function on the 2-spool].I: The line looks about the same. But, on the spools, this one [3-spool] is going up

faster.J: [Turns one knob clockwise and one counter-clockwise].I: Are you telling me this [object on the spool] goes down?J: Wait! The lines would be the same on the Etch-a-SketchTM.I: Even though this is going up faster?J: Yes.I: You have to tell ’em which is going faster? [if someone unfamiliar with the situation

inquired].J: Yes.I: What if I wanted to show both [relationships] on the screen? How should I turn the

height knob (right knob on the Etch-a-SketchTM) [to show the action of the 3-spool]?J: It’ll just go higher.I: If you saw that, would you be able to tell which one belongs with which spool?J: Oh yeah, this one goes up faster. It has to be on the 3-spool.I: Even though the number of cranks [on the spools] stays the same, this one goes up

faster?J: Yeah, I want to try this one [excitedly pointing to the 10-spool]. It goes up 10 [on

each rotation]. It should go to 20 [in two rotations].

At this point, Jim indicated that the height knob (right knob on the Etch-a-SketchTM) wouldneed to go much faster than the left knob.

4.5. Analysis of explorations with the Etch-a-SketchTM

Jim began his use of the Etch-a-SketchTM as a representation for functions originatingfrom both the 2-spool and the 3-spool by considering each turn increment of the left knobof the Etch-a-SketchTM as a representation for a rotation of the handle of the spool systemto be matched with each turn increment of the right knob of the toy as a representation foreach increment of elevation of the object. The graphs produced with this interpretation arenecessarily the same. Without considering differences in the amount of elevation for the twospools, Jim’s interpretation is consistent with his original equation,h = c. By indirectlyencouraging Jim to consider implications of having the same graph represent the function ofeither spool, Jim is enabled to consider how he might use the graphs to account for differencesin the size of increments of elevation occurring in the operation of the different spools. The


process Jim uses involves mapping variations described by Izsák (2000). As Jim translateshis actions from the spool system to the knobs of the Etch-a-SketchTM, a change occurs inhow he assigns one handle rotation of the spool system to turn increments of the right knob ofthe Etch-a-SketchTM. He first uses a correspondence of one handle rotation for one knob turnincrement. He later uses a correspondence of one handle rotation for several turn increments.

Jim broadens his interpretation of the two functions by expanding his use of graphs to rep-resent the functions. This change does not negate the correctness of his original interpretationsof the functions but enables him to distinguish between different functions by observing theconsistency of the factor of elevation, specifically the slope, within each function and differ-ences in slope between functions. For Jim, this is a new development in his knowledge offunctions. Initially, his initial interpretation (i.e., one rotation of the handle of the spool to oneincrement of elevation) is that of a basic ratio or multiplicative relationship between measur-able quantities as described by Thompson (1994). He now has found a way to acknowledgethe increment of elevation as a factor in the overriding process. The fact that Jim originallyused repeated additions to determine the heights corresponding to various amounts of rotationof the spool had previously suggested his view of individual accumulations made by iteratingthe ratio. This new development positions Jim to be able to consider each of the functionsas systematic processes. By using graphs to acknowledge the difference in the functions pro-vides Jim with a way to observe the overriding consistency within each individual functionby considering total accumulation in constant ratio.

4.6. Further development

Hoping to build on Jim’s newly created knowledge, I presented him with a table (see Fig. 5).The heading symbols,c andp, were the same as symbols Jim had previously used in a tablehe had created to stand for the number of rotations of the spool system and the height of theobject, but I did not mention the spool system at this time.

Jim stated, “c tells you how many 5s you have.” When I asked him what you do to getthe numbers in the right column, he said, “times it by 5,” but explained, “add ‘em up,”suggesting a repeated addition interpretation. I asked Jim to anticipate about how the graph ofthis relationship would look. Using paper and pencil, he carefully sketched a nearly verticalline from lower left to upper right position on a drawing of an Etch-a-SketchTM screen (see

Fig. 5. Table presented for interpretation by Jim.


Fig. 6. First graph created by Jim.

Fig. 6). He indicated that as the left knob changed one-notch, that the right knob would changeone-notch. But upon consideration of the table, he indicated by counting, that as the left knobchanged one-notch, the right knob would change “1, 2, 3, 4, 5” notches.

We then proceeded to generate the graph on the Etch-a-SketchTM. I controlled the “cknob” and Jim controlled the “p knob.” Every time I turned the left knob one-notch, Jimsimultaneously turned the right knob five notches. But Jim remained uncertain about the“correctness” of the resulting graph. I decided to give him a second table that he could compareto the previous (see Fig. 7). Jim stated that in this table the left column was “going by 1s” andthe right column was “going by 2s.”

I asked Jim to predict how the graph would look. Jim replied, “It’ll probably be a little moreoff to the side.” When I asked how he determined this, he referred to the Etch-a-SketchTM

and responded, “This [the right knob] is gonna’ go [just] one more quarter turn [one morenotch] than yours [the left knob].” He sketched the graph as he anticipated its appearance ona drawing of Etch-a-SketchTM screen (see Fig. 8).

When we constructed the graph on the Etch-a-SketchTM, it appeared very near to Jim’sprediction, and he became more confident in the “correctness” of his graphs.

I presented Jim with another table of numeric values (see Fig. 9). The left column washeaded with a triangle symbol, hereafter referred to ast. The right column was headed witha square symbol, hereafter referred to ass. Jim examined the table and reported, “It’s going

Fig. 7. Second table presented for interpretation by Jim.


Fig. 8. Second graph created by Jim.

Fig. 9. Third table presented for interpretation by Jim.

by three each time,” and added, “There’s no zero [for ans-value] at the top.” I asked Jimwhether this table could be used to show the relationship on the spool elevating system. Hethought a minute and responded that the object would be placed on the 3-spool and that itwould be positioned “up two” from the start. I asked, “If we give it a crank, where is it?” Jimanswered, “5.” Following a second rotation, Jim indicated that it would be at 8. When I askedJim to write a mathematical sentence to show the relationship presented in the table, he wrote,“0t = 2s, 1t = 5s, 2t = 8s.” This was similar to his previous attempts to write equations forfunctions.

In an effort to simplify the situation I presented another table of numeric values (see Fig. 10).The left column was headed with a triangle symbol, hereafter referred to ast; the right column

Fig. 10. Fourth table presented for interpretation by Jim.


was headed with a star symbol, hereafter referred to asst. Note thatst was used as a singlesymbol in this situation and was not intended to represent multiplication ofs andt. I insistedthat he search for a single process, consistent for all pairs of values in the table and asked whatshould be “done” to one variable to get the second. With these specific instructions Jim replied,“Oh,” and wrote, “5× 3, 4× 3, 2× 3.” Using input values of 5, 4, and 2, Jim demonstratedhis awareness of the consistent multiplication by 3. When I asked, “What do I have to do tothet to getst?” Jim wrote,

t × 3 = st.

We returned to consider the table in Fig. 9.

I: Before, I make any changes int, this [s] has to be up 2 [from the values illustrated inFig. 10]. What does this mean in terms of the graph?

J: It’s up a little.I: It [Fig. 9] says we’re going from 2 to 5.J: Up 3. It’ll be the same [as for the table illustrated in Fig. 10] because it’s going three

each time.

Jim hesitantly wrote the equationt × 3 + 2 = s.

I: How does the slantiness [a term I introduced] compare?J: It’s the same.I: How do you know the slantiness from the equation?J: It’s the 3.I: What does the 2 do?J: Nothing.I: Are they the same (referring to the two equations)?J: Practically. This one [referring tot × 3 + 2 = s] is up a little.

Jim went on to predict the appearance of the graph of the function and explained how theknobs of the Etch-a-SketchTM would be turned to generate the graph. He even indicated thatthe slanted appearance of the graphs would be the same as that of the function represented inFig. 10.

4.7. Analysis of further development

Jim’s experiences with the Etch-a-SketchTM allow him to distinguish the rate of elevation,slope, in different linear functions without writing symbolic equations, and prepares him toview function relationships in a more general way than he had previously. When Jim states, “ctells you how many 5s you have,” he shows a maturing view of multiplication by focusing on thenumber of iterated groups as described by Steffe (1994). In order to build on Jim’s distinctionbetween different linear functions, I presented two similar relationships in table form. Thetables represent relationships where the output values are consistently three times the inputvalues, but in one, a constant of two is added (see Fig. 9). In the other table no constant is added


(see Fig. 10). Although the tables contain discrete values, Jim interprets them as representationsfor functions that might arise from the spool system. Through comparison of the tables, Jimrecognizes the common overriding multiplicative relationship between variables and that theoutput values differed only by an added constant in one. When Jim states that the relationshipsin the two tables were “the same,” he is focused on the fact that each contains the samegeneralized multiplicative relationship. He uses the table (see Fig. 9) and the equation he hadwritten previously (i.e.,t × 3 = s) to generate an equation for the table illustrated in Fig. 9(i.e., t × 3+ 2 = s). Through these equations, he formalizes his recognition of a generalizedratio, the slope, as a rate of elevation, and realizes that the added constant does not affect theslope.

By asking Jim what should be “done” to the values in the left column to produce the valuesin the right column, he was able to “fold back” (Pirie & Kieren, 1992) to an action imageof the function. Thinking about the consistent action that is taken on an input variable toproduct an output variable, supports Jim’s effort to create symbolic equations. The equationsacknowledge his awareness of the factor of elevation as a rate, and underscore his newly formedinterpretation of multiplication that focuses on a variable number of iteration of a quantity, 3.When Jim had in a previous situation stated that one column “is going by 1s” and the other “isgoing by 2s,” he observed variability but not systematic co-variation. As Thompson (1994)might describe, he was aware of accumulations made by iterating a ratio, but did not focus ontotal accumulations in constant ratio. He now focuses on total accumulations in constant ratioand is aware of systemic co-variation in the variables.

4.8. Addressing the reversal error

Jim’s thinking of the functions had changed. I decided to revisit the matter of his hesitancyin writing a generalizing equation. I presented Jim with yet another table (see Fig. 11). Thecolumn heading on the table werex, and a box figure, hereafter referred to asb. I offered noexplanation for the meanings ofx, b, or the table, except to indicate that the table was intendedto represent a function. At thex-value of 6, Jim filled in ab-value of 24. I then asked himto write a generalizing equation for the function represented in the table. I directed, “Write

Fig. 11. Final table presented to Jim.


Fig. 12. Jim corrects the reversal error.

a mathematical sentence for the relationship. Write a sentence that tells how to figure outb,if you know x.” Jim produced a sequence of equations (see Fig. 12). The first two equations(i.e.,x = 1, b = 4) serve as a point of initialization for Jim. The third equation is numeric,2 = 8, but Jim labeledx above the 2 andb above the 8. He then wrote, “x = b×4,” containingthe reversal error. After a moment of quiet thought, he added arrows indicating thatx shouldbe multiplied by 4, or, thatx should be placed in the place ofb. He finalized his equation bywriting “x × 4 = b.”

4.9. Analysis of addressing the reversal error

By writing the equation,x × 4 = b, Jim anchors and formalizes a relationship he hasobserved while reasoning about the numeric data in tables. In generating the equation, Jimengages in a process of notational variation to resolve discrepancies he notices when val-ues generated by an earlier version of the equation containing the reversal error did notcorrespond to values produced in operation of the spool system. As mention earlier, Izsák(2000) describes notational variation as “experimenting with and adjusting algebraic sym-bols patterns until computations on paper match measurements” on a physical system. Astudent uses this type of cognitive activity to makes adjustments to equations intended tomodel physical phenomena. Since Jim regards the table of numeric values he is using asrepresentation for a function such as that arising through operation of a spool system, hisdecisions regarding which variable is to be multiplied by the factor of elevation is an ex-ample of notational variation. Jim’s situation suggests that notational variation may be morethan trial-and-error adjustment of symbols within an equation to produce specific numericvalues. To engage in notational variation, Jim had to recognize a generalized systematic rela-tionship and anticipate a need for adjustment in the symbolic equation that he intends as itsrepresentation.


5. Discussion and conclusions

By building links between the spool elevating system and its representations includingtables, operations on the Etch-a-SketchTM, the graphs created on the Etch-a-SketchTM, andsymbolic equations, Jim developed a generalized view of multiplication that supported under-standing linear functions as processes involving systematic co-variation between two relatedvariables. His initial explorations with the spool elevating system enabled him to encounterand understand functions and their associated variables in a non-symbolic format. He devel-oped a sense of the change of one variable relative to another as he operated the system. Later,he began to focus on the importance of the slope factor in distinguishing between differentlinear relationships. This happened initially through graphs created on the Etch-a-SketchTM

and was formalized as he created equations for tables representing linear functions.Jim’s initial view of the function could be described as one of repeatable actions, where

although the procedures were consistent each time, the focus of his attention was not on thegeneral consistency of the actions, but on individual input and output values. This suggestedthat Jim did not view functions as generalized processes. However, when he began to exploreand reflect upon his use of the Etch-a-SketchTM to represent the relationship between thenumber of rotations of the handle on the spool system and the height of the object thatresulted, he deepened his understanding of the functions themselves.

Jim’s interpretation of the graphs on the Etch-a-SketchTM was rooted in translating theactions of the spool system to actions taken with the knobs of the Etch-a-SketchTM. Heshowed the strength of his incremental interpretation by indicating that the same graph couldbe used to show the relationship for two spools of different diameter. Later, when he facedtwo relationships that were alike except that the rate of elevation of the second was greaterthan the first, Jim developed mapping variations as described by Izsák (2000) while using theEtch-a-SketchTM to visually distinguish between the relationships. His thinking shifted to therates of elevation as slope parameters in function situations, and he distinguished betweenthe two relationships. Later, by recognizing similar generalized multiplicative relationshipsin tables of functions, he was able to develop awareness of systematic co-variation withinand across variables in the functions. His thinking evolved from that of accumulations madeby iterating a ratio to total accumulations in constant ratio. With that understanding in place,he was able to productively engage in the process of notational variation (Izsák, 2000) soas to create a symbolic equation that signifies the awareness of systematic co-variation ofvariables.

Consistent with the learning model proposed by Lesh et al. (1983), the findings from thisinvestigation suggest that students can use different representations to build a deep knowledgeof linear functions as generalized relationships involving systematic co-variation of variables.This investigation shows how explorations with a spool elevating system combined withthe opportunities to create and reason about Etch-a-SketchTM graphs and tables of functionsprovide the context for developing understanding by linking representations. In addition, thisresearch shows how Thompson’s (1994) model can be used to interpret the development ofknowledge of functions originating in dynamic physical models. Although the model was


originally developed to describe observations of learning using a computer micro-world tointerpret simulated motion, it is based upon the physical context of measurable quantities suchas those found in operating dynamic physical models. The development of Jim’s knowledgeof functions using the spool elevating system paralleled developmental patterns observed byThompson.

This investigation also uncovers details in the processes of linking representations by pro-viding additional examples of notational variations and mapping variations. Findings suggestthat the use of notational variations is effective in developing symbolic equations when abackground of understanding of functions has been established using other representations,and that mapping variations can be used to develop the background of understanding. Moreinvestigations are needed in order to fully understand how learners reason between action andprocess conceptions of functions and how learners use mapping and notational variations tobuild understanding of function.

Acknowledgments

The research reported here was conducted as part of the author’s doctoral dissertation com-pleted at Northern Illinois University. She gratefully acknowledges the support and guidanceof her dissertation co-directors, Frank Bazeli and Helen Khoury.

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developing the concept of linear function: one student’s experiences with dynamic physical models

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