calculus students' use & interpretation of variables

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Calculus Students' Use and Interpretation of Variables: Algebraic vs. ArithmeticThinkingSusan S. Gray a; Barbara J. Loud b; Carole P. Sokolowski c

a University of New England, Biddeford, Maine b Regis College, Weston, Massachusetts c Merrimack College,North Andover, Massachusetts

Online Publication Date: 01 April 2009

To cite this Article Gray, Susan S., Loud, Barbara J. and Sokolowski, Carole P.(2009)'Calculus Students' Use and Interpretation ofVariables: Algebraic vs. Arithmetic Thinking',Canadian Journal of Science, Mathematics and Technology Education,9:2,59 — 72

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CANADIAN JOURNAL OF SCIENCE, MATHEMATICSAND TECHNOLOGY EDUCATION, 9(2), 59–72, 2009Copyright C© OISEISSN: 1492-6156 print / 1942-4051 onlineDOI: 10.1080/14926150902873434

Calculus Students’ Use and Interpretation of Variables:Algebraic vs. Arithmetic Thinking

Susan S. GrayUniversity of New England, Biddeford, Maine

Barbara J. LoudRegis College, Weston, Massachusetts

Carole P. SokolowskiMerrimack College, North Andover, Massachusetts

Abstract: The ability to use and interpret algebraic variables as generalized numbers and changingquantities is fundamental to the learning of calculus. This study considers the use of variables inthese advanced ways as a component of algebraic thinking. College introductory calculus students’(n = 174) written responses to algebra problems requiring the use and interpretation of variables aschanging quantities were examined for evidence of algebraic and arithmetic thinking. A frameworkwas developed to describe and categorize examples of algebraic, transitional, and arithmetic thinkingreflected in these students’ uses of variables. The extent to which students’ responses showed evidenceof algebraic or arithmetic thinking was quantified and related to their course grades. Only one thirdof the responses of these entering calculus students were identified as representative of algebraicthinking. This study extends previous research by showing that evidence of algebraic thinking instudents’ work was positively related to successful performance in calculus.

Resume: La capacite d’utiliser et d’interpreter les variables algebriques comme des nombresgeneralises ou des quantites variables est fondamentale pour l’apprentissage du calcul differentielet integral. La presente etude considere une telle utilisation avancee des variables comme une com-posante de la pensee algebrique. Nous avons analyse les reponses ecrites de 174 etudiants dans uncours universitaire d’introduction au calcul differentiel et integral, a des problemes d’algebre quidemandaient d’utiliser et d’interpreter certaines variables comme des quantites variables, afin decerner les indices d’une pensee algebrique ou arithmetique. Un cadre a ete mis au point pour decrireet categoriser les exemples de pensee algebrique, transitionnelle et arithmetique, tels que refletes parl’utilisation des variables chez ces etudiants. Nous avons analyse dans quelle mesure les reponses desetudiants pouvaient etre considerees comme exemples de pensee algebrique ou arithmetique; cettemesure a ensuite ete quantifiee et mise en relation avec les notes obtenues dans les cours. Seulementun tiers des reponses des etudiants de premiere annee en calcul se sont averees des exemples depensee algebrique. Cette etude elargit le champ de certaines recherches precedentes, car elle montre

Address correspondence to Carole P. Sokolowski, Mathematics Department, Merrimack College, 315 Turnpike Street,North Andover, MA 01845. E-mail: [email protected]

Downloaded By: [de Villiers, Michael] At: 21:45 29 July 2009

60 GRAY ET AL.

que les indices de pensee algebrique dans le travail des etudiants sont effectivement lies a une bonneperformance en calcul.

INTRODUCTION

Algebraic variables can be used and interpreted in many ways. Variable is not a single, staticconcept that is easily understood. Although this concept is usually introduced to students bythe latter part of middle school, its many and complex uses are revealed only after years ofstudy. Ideally, the years of algebra instruction and practice in secondary school help studentsprogress from using variables as representatives of specific unknowns toward using variables inmore advanced ways. The study of calculus, with its fundamental concepts of limit, derivative,and integral, requires an ability to understand algebraic variables as generalized numbers and asfunctionally related varying quantities.

Past research indicates that college students who have difficulty using algebraic variablesas varying quantities often have difficulty in their performance in calculus (Carlson, Jacobs,Coe, Larson, & Hsu, 2002; Gray, Loud, & Sokolowski, 2005; White & Mitchelmore, 1996). Inan investigation of students’ understandings of the concept of derivative, White and Mitchel-more (1996) posited that their subjects’ difficulties with the derivative were attributable to an“under-developed concept of variable” (p. 91). Carlson et al. (2002) examined calculus students’covariational reasoning and concluded that such reasoning is fundamental to the understanding ofcalculus. We previously reported that calculus students who had shown an ability to use variablesas varying quantities achieved a mean final course grade of B− (reasonably good performance);by comparison, students who were categorized as having either a moderate or basic level of un-derstanding of variables only achieved a mean course grade of D+ (marginal performance; Grayet al., 2005). Mean grades were reported in that study. Marginal performance was not classifiedas successful because, for many majors, a grade of D+ was not sufficient for students to takethe next sequential course. There were students who had used variables at an advanced levelbut did not earn high grades in their calculus course. However, in that study, no students whoshowed only the most basic level of understanding of variables achieved even marginal grades incalculus. Although an advanced understanding of variables as varying quantities is not a suffi-cient condition for student success, it may be a necessary condition. In an effort to analyze whattypes of difficulties undergraduates exhibit in this regard, this article describes calculus students’approaches to algebra problems that require the use or interpretation of variables as generalizednumbers or varying quantities.

THEORETICAL FRAMEWORK

For the present study, an operational-structural (process-object) framework was used to categorizeand describe students’ responses to a selection of five problems from an algebra test (Sokolowski,1997) adapted from a large-scale British study (Hart, Brown, Kerslake, Kuchemann, & Ruddock,1985; Kuchemann, 1981). Many researchers have written about algebraic concepts from anoperational-structural perspective (Dubinsky, 1991; Dubinsky & Harel, 1992; Jacobs, 2002;Kieran, 1992; Sfard & Linchevski, 1994; Stacey & MacGregor, 2000; Tall et al., 2000; Trigueros


ALGEBRAIC VS. ARITHMETIC THINKING 61

& Ursini, 2003). Sfard and Linchevski (1994) explained the operational-structural framework bydescribing mathematical operations, or processes, as series of actions that are ultimately reifiedinto a cognitive structure, or object, upon which new operations can be performed in order toform even more complex structures.

In general, an operational viewpoint is characterized by an almost exclusive focus on ma-nipulating expressions or equations in search of numerical values. Sfard and Linchevski (1994)described an operational viewpoint as one which takes students from arithmetic to “algebra of afixed value” (p. 102), in which students solve an algebraic equation to find the unknown valueof the variable by performing a sequence of operations. Jacobs (2002) reported that secondaryschool students studying advanced calculus used a calculational approach when they treatedthe “variable as a tool for solving an equation or finding an unknown value” (p. 203). Staceyand MacGregor (2000) referred to this process of performing operations on equations to obtainnumerical answers as illustrative of arithmetic thinking.

By contrast, the ability to work with algebraic expressions or equations more generally andto interpret these as representations of mathematical relationships without needing to attachnumerical values to the variables is characteristic of a structural perspective. According to Sfardand Linchevski (1994), a structural viewpoint would be manifested in “functional algebra”(p. 108), in which students consider an algebraic function as a reified structure on which otheroperations can be performed. Similarly, Stacey and MacGregor (2000) referred to algebraicthinking as that which allows students to view variables, expressions, and equations as structuresof general representation. Students do not need to find numerical referents for the variablesin order to work with them as representations of generalized numbers or covarying quantities.Several researchers have developed models for describing students’ approaches to working withvariables based on these various interpretations of operational-structural cognitive development(Carlson et al., 2002; Jacobs, 2002; Trigueros & Ursini, 2003).

For this article, the terms algebraic thinking and arithmetic thinking most closely align withStacey and MacGregor’s (2000) descriptions given above. These terms are used to describethe characteristics of students’ responses to the algebra problems analyzed for this study. Toclarify this terminology more specifically with regard to students’ uses of algebraic variables,Kuchemann’s (1981) catagorizations of the Chelsea Diagnostic Algebra Test items (Hart et al.,1985) were utilized.

Kuchemann (1981) developed four hierarchical levels of students’ interpretations of algebraicvariables. Arithmetic thinking, as defined for the present study, can be viewed as roughly equivalentto Kuchemann’s Levels 1 and 2, which are very basic uses and interpretations of variables.According to Kuchemann, these two levels of using variables are characterized by evaluating,ignoring, or using variables as objects or labels as early learners of algebra tend to do. Forexample, in order to solve a simple equation such as x + 3 = 5, a student could easily find thatthe answer is x = 2 by replacing the x with values until the correct value is obtained; he/she couldsimply evaluate the variable. This tendency to evaluate variables is quite persistent. Even moreexperienced students sometimes use guess-and-check methods inappropriately to solve muchmore difficult equations. An example of a problem that allows a student to ignore the variables is:If q + r = 25, then q + r + 4 = ? Here, the student need only recognize that 4 must also be addedto the 25; the student need not actually work with the variables; they are essentially ignored. Theuse of variables as labels is illustrated in an expression such as 5a, for which the interpretation of“5 apples’ is common, rather than the intended interpretation of “5 times the number of apples.”


62 GRAY ET AL.

This interpretation of variables as labels or objects, rather than as indicators of quantities, hasproved to be very resilient. When students model word problems with algebraic equations, theyoften make errors based on an inappropriate use of variables as labels (Clement, Lochhead, &Monk, 1981; Gray et al., 2005).

Algebraic thinking as defined in this article is evidenced by the use of variables as specificunknowns, generalized numbers, and varying quantities, which are indicators of Kuchemann’s(1981) Levels 3 and 4, his more advanced levels of understanding. Generally, Level 3 refers tousing variables as specific unknowns. When one solves an equation such as 3x +5 = 4 − 2x,one is seeking the specific unknown, represented by the variable, that will make the equation atrue statement. The ability to solve equations with the variable appearing on both sides of theequal sign is an indication, according to some researchers, that students may have moved fromarithmetic to algebraic methods (Filloy & Rojano, 1989; Herscovics & Linchevski, 1994).

Most previous studies examining evidence of algebraic thinking in their subjects were con-ducted with young students who were just beginning to learn algebra (Kieran, 1992; Stacey &MacGregor, 2000). In the current study, we are concerned with undergraduate calculus studentswho have had a number of years of experience with the rules and syntax of algebra. For thesestudents, merely being able to solve an equation like 3x + 5 = 4 − 2x is not a sufficient indicatorof algebraic thinking, as it may be for younger students. For this reason, we are focusing on theuse and interpretation of variables as generalized numbers and varying quantities as more robustindicators of algebraic thinking. These two uses of variables are characteristics of Kuchemann’s(1981) Level 4. Variables are used as generalized numbers when they represent entire sets ofnumbers. For example, 5 + x = x + 5 represents the commutative property of addition forreal numbers. Students who understand variables as generalized numbers would recognize thatthe x in the equation is more than a place-holder for a specific unknown; rather, it implicitlyassumes infinitely many values. Finally, in a function such as y = 3x + 10, two variables, x andy, are covarying with one other and assume a range of related values. The focus of the presentstudy is on these advanced uses of variables that are at once the most crucially foundational forunderstanding the major concepts of calculus and the most difficult for students to grasp.

GOALS OF THE STUDY

There were three major goals of this study to examine entering calculus students’ responsesto problems that used variables as generalized numbers or varying quantities: first, to classifystudents’ responses to these problems as indicators of arithmetic or algebraic thinking; second,to determine their success rates on these problems with respect to algebraic thinking; and third,to determine the relationship between students’ course grades and their performance on theseproblems.

METHOD

An adaptation (Sokolowski, 1997) of the Chelsea Diagnostic Algebra Test (Hart et al., 1985)was administered over a period of four semesters on the first day of class to 174 introductorycalculus students at two private liberal arts colleges in New England. The test consisted of 23



TABLE 1Problems Analyzed

Problem

3 Which is larger, 2n or n+2? Explain.16 What can you say about c if c + d = 10 and c is less than d?19a a = b + 3. What happens to a if b is increased by 2?19b f = 3g + 1. What happens to f if g is increased by 2?21 If this equation (x + 1)3 + x = 349 is true when x = 6, then what value of x

will make this equation (5x + 1)3 + 5x = 349 true?

questions, with 15 of those having multiple parts. Students were given a maximum of 50 minutesto complete the test without the use of calculators. They received no directions other than the testquestions as stated, and all written work was recorded on the test paper.

For this study, a selection of five problems was analyzed because successful responses to theseproblems were likely to illustrate algebraic thinking. Four of the five problems incorporated theuse of variables as generalized numbers or varying quantities. All five problems were chosenbecause we expected that they could be easily answered from a structural perspective, thusindicating algebraic thinking, by students who have advanced to the study of calculus, and alsobecause evidence of arithmetic thinking would be observable in answers indicative of operationalor calculational approaches. The five problems analyzed are presented in Table 1, numbered asthey were on the algebra test (Hart et al., 1985; Sokolowski, 1997).

All responses were recorded and placed into one of three categories: arithmetic, algebraic,or transitional. Written characteristics of arithmetic responses included symbolic manipulation,evaluation of the variable as a single value, or numeric substitution. These responses wereconsidered to be operational approaches to the problems. Responses were categorized as algebraicwhen they showed evidence of using variables to represent multiple referents in generalizedexpressions and functional relationships and/or because they reflected a structural perspective insolving the problem. In addition, there were many responses that were deemed to be transitional.Responses in the transitional category included evidence of thinking about variables as takingon more than one value, but they were incomplete or not fully generalized. Consensus wasreached among the researchers regarding criteria and placement of responses into the appropriatecategories.

In addition to the qualitative analysis, responses were given a numeric score of 0, 1, or 2, basedon whether they were categorized as arithmetic, transitional, or algebraic, respectively. A scoreof 0 was also used for blanks or for the relatively few responses that could not be categorizedbecause their meanings could not be interpreted in this framework. The relationship betweenthese calculus students’ mean final course grades and their numeric scores on these five problemswas also examined.

RESULTS AND DISCUSSION

In this section, the responses to each problem are described and categorized according to thearithmetic/transitional/algebraic framework as previously defined. Percentages of responses in


64 GRAY ET AL.

each category and the relationship of numeric scores to course grades are also presented anddiscussed.

Problem 3

Which is larger, 2n or n + 2? Explain.Algebraic response: It depends. If n < 2, n + 2 is larger; if n = 2, 2n = n + 2; if n > 2, 2n is larger.

This problem involves the comparison of two expressions, both using the same variable. Thereis a need to think of the variable algebraically as taking on a range of values while making thiscomparison. Only 15% of the students responded to this problem by writing that it depends onthe value of n and included a good general explanation, either involving inequalities as shown inthe algebraic response above or clearly indicating an understanding of the structure of the answeras pivoting about the number 2.

An additional 36% of the responses were categorized as transitional because they indicateda recognition that the variable could represent more than one value. Although most of theseresponses gave two or three numerical examples to support their conclusion that it depends onthe value of n, they did not identify n = 2 as the structural pivot point of the comparison. Finally,a few other transitional responses stated that 2n is greater but then gave numerical exceptions.

Forty percent of the responses to Problem 3 were categorized as arithmetic. Almost all of thesestated that 2n is larger. These responses either gave one or more numerical examples to supportthis conclusion or gave an explanation stating that “multiplication makes numbers larger.” Thispersistent generalization that multiplication makes numbers larger than addition seems to indicatethat these students were thinking of natural numbers as the referents of the variable. They basedtheir conclusions on the results of testing only one or two values, thus suggesting an arithmeticapproach to this problem.

In general, the vast majority of students used natural numbers to test values to compare 2n andn + 2. This almost exclusive use of natural numbers may have been prompted by the use of theletter n as the variable in this problem.

Problem 16

What can you say about c if c + d = 10 and c is less than d?Algebraic response: c < 5.

In this problem, two variables are covarying and must be compared in the context of onefunction under the constraint of an inequality. Thirty-three percent of the students answered thisproblem algebraically with c < 5. This problem requires respondents to think of a set of valuescovarying with a second set of values. Therefore, the response c < 5 was the only one consideredto be indicative of algebraic thinking.

Twenty-four percent of the responses were classified as transitional. Approximately two thirdsof these were written in inequality form. For example, responses such as 0 ≤ c < 4 or 1 ≤ c ≤ 4were considered to be insufficient as algebraic responses because of the use of 0 or 1 as thelower bound and 4, rather than 5, as the upper bound of the inequality. Most of the remainingresponses consisted of a list of integers between 1 and 4, inclusive. Responses of these types were



categorized as transitional because they indicated that students considered a range of values butdid not fully generalize their answers.

Twenty-eight percent of the responses were categorized as arithmetic. These included solvingfor c (c = 10 − d or c < 10 − d) or giving a single number value for c. They reflected anarithmetic view of the problem because students simply performed a symbolic manipulation orgave a single numerical answer.

Problem 19a

a = b + 3. What happens to a if b is increased by 2?Algebraic response: a increases by 2.

Two variables are covarying in an additive functional relationship in this problem. Sixty-sevenpercent of the responses to this problem were categorized as algebraic. An algebraic responseindicates that a quantifiable change in a is the result of a change in b in the functional relationship.Although this problem has an algebraic response rate that is much higher than the others, it is soclosely aligned with Problem 19b that its results help to inform the interpretation of the resultsof the subsequent problem.

Eight percent of the responses stated that a increases. These responses were categorized astransitional, indicating that students recognized that there was a connection between increasingthe value of b and increasing the value of a; however, they did not specify the magnitude of theresulting increase. An alternative interpretation of this response is that students simply answeredmechanically, thinking that an increase in one variable automatically would produce an increasein the other.

Finally, there were nineteen percent of the responses that were reflective of arithmetic thinking.Typical responses in this category were a = b + 5 or a = (b + 2) + 3. Students were not increasingthe value of b by 2 but were representing their interpretation of the procedure of adding 2 to b.These responses and others, such as 2b + 3 or a = 5, suggest that students were using the 2to perform a computation instead of recognizing that the increase by 2 in one variable causes acorresponding change in the other variable.

Problem 19b

f = 3g + 1. What happens to f if g is increased by 2?Algebraic response: f increases by 6.

Here, two variables are covarying in a multiplicative and additive functional relationship.Although this problem is similar to the previous problem, the results were quite different. Only30% of the students gave an algebraic response to this problem. Most had only the answer andnothing else written. However, the written work accompanying 12 of these responses showed thatstudents tested a few values for g in the given function and then concluded that f increases by 6.The systematic testing of values indicates that students recognized the functional relationshipof the variables in this problem. The act of generalizing the results to produce the answer wasconsidered to be indicative of algebraic thinking.


66 GRAY ET AL.

Fourteen percent of the responses to this problem stated that f increases. As in Problem 19a,these responses were included in the transitional category because they illustrated a sense ofcovariation between the two increasing variables but did not specify the amount of the increase.

Another 40% of the responses to Problem 19b were categorized as arithmetic. Most of theresponses in this category stated that f doubles, increases by 2, or increases by some other number.Some students appeared to interpret the expression “increased by 2” as doubling. However, 13 ofthe 18 students who responded in this way did not state that a doubles in the previous problem.The multiplicative nature of Problem 19b may have changed their interpretation of “increased by2” into a doubling effect. There were other computational responses that indicated an arithmeticview, such as f = 7, f = 5g +1, or f = 6g + 1. Here, as in Problem 19a, students performed acalculation with the 2 or substituted the 2 for g. The relationship between two variables in whichone variable changes with respect to changes in the other variable was not apparent in theseresponses.

Problem 21

If this equation (x + 1)3 + x = 349 is true when x = 6, then what value of x will make this equation(5x + 1)3 + 5x = 349 true?Algebraic response: 6/5 or 1.2.

Only one variable is used in this problem. It is used as a specific unknown in both equations,but its value is different in the context of the second equation. Thus, it is necessary to see thestructural similarity between the two equations and to compare the value of the variable in oneequation with that in the other in order to answer this question from an algebraic perspective.

Only 27% of the students responded algebraically to this problem. Although a few studentswrote 5x = 6 on their papers, the majority showed just the answer, 6/5 or 1.2. To arrive at thisconclusion, it would be necessary to identify the structural similarity between the two equations,deduce that 5x = 6, and solve this equation for the new value of x. This approach is an indication ofalgebraic thinking. It is unlikely that students would arrive at this correct answer by an arithmeticmethod of testing values.

Thirty-one percent of the responses showed attempts to expand the cubic expression or toperform numeric substitutions or solve for x. Some other typical incorrect numerical answerswere: 30, 1, 2, 3, 4, or 6. Because none of these had anything else written on the test paper, theymay have been indications of attempts at numeric substitution or guess and check methods. Allof these indicate arithmetic or calculational approaches to the problem.

There were no responses to this problem that were categorized as transitional, but 42% of thestudents did not give a response. Although this problem was near the end of the test, most of thesestudents completed the following few problems, so the blanks do not necessarily signal a lack oftime. It is more likely that students saw this as a complex problem and decided to skip over it.

Summary of Response Categories

Table 2 summarizes the most common types of responses that were identified as indicators ofarithmetic, transitional, or algebraic thinking according to the categorization framework used inthis study. Examples of responses that reflected arithmetic thinking are those that showed evidenceof numeric substitution or rote symbolic manipulation. Transitional responses in general showed



TABLE 2Examples of Responses Indicative of Algebraic, Transitional, or Arithmetic Thinking

Problem Algebraic Transitional Arithmetic

Problem 3

Which is larger, 2n or

n + 2? Explain.

It depends.

If n < 2, n + 2 > 2n;

if n = 2, n + 2 = 2n;

if n > 2, n + 2 < 2n

It depends (with 2 or

3 numerical examples

for justification).

2n, with numerical

exceptions or

qualifications

2n (with numerical

examples)

2n, multiplication

makes it larger.

n + 2

Problem 16

What can you say

about c if c + d = 10

and c < d?

c < 5 0 < c < 4

0 ≤ c < 5

c ≤ 4

c = 1, 2, 3, 4

c = 10 - d

c < 10 - d

c equals a single

integer.

Problem 19a

a = b + 3. What

happens to a if b is

increased by 2?

a increases by 2. a increases. b + 5

(b + 2) + 3

2b + 3

a = 5

a + 2

Problem 19b

f = 3g + 1. What

happens to f if g is

increased by 2?

f increases by 6. f increases. Increases by 2, 7, or

some other number.

f doubles.

f = 7

f = 5g + 1

f = 6g + 1

Problem 21

If x +1( )3 + x = 349,

x = 6;

If 5x +1( )3 + 5x = 349,

x = ?

x = 6/5 or 1.2 No examples were

apparent in this

category.

Attempts to solve

for x or substitute

numbers

some indication of considering more than one value for the variables. Algebraic responses showedevidence of interpreting the variables as representative of covarying quantities or of taking astructural approach to solving the problem.

The percentages of responses in each category are given in Table 3. Responses identified inthe table as “Uncategorized” were incomplete answers or those whose intent or meaning could


68 GRAY ET AL.

TABLE 3Percentages of Responses in Each Category

Problem Number Algebraic Transitional Arithmetic Uncategorized Blank

3 15 36 40 9 116 33 24 28 5 1019a 67 8 19 3 219b 30 14 40 7 921 27 0 31 0 42

Note. Percentages may not sum to 100% horizontally due to round-off error.

not be determined within the context of the problem (e.g., for Problem 3, “n can be anything”;for Problem 16, “c does not equal 5”; for Problem 19a, “a becomes a − 2”; for Problem 19b,“subtracted by 2”).

Percentages of responses in each category varied among the five problems and were highlycontextual according to problem type. Of primary concern is the evidence that less than one thirdof these calculus students’ responses to four out of five of these relatively simple problems wereindicative of algebraic thinking as defined above.

Comparison of Mean Grades and Numeric Scores

Numeric scores ranged from 0 to 10 and were determined by assigning values of 0, 1, or 2to arithmetic, transitional, or algebraic responses, respectively. Course grades were representedwith a scale ranging from 0 to 4, indicating the following performance descriptors: 4.0, superior;3.0, good; 2.0, fair; 1.0, minimally passing; 0, fail. Students’ numeric scores were comparedto their mean final course grades in calculus. Figure 1 graphically illustrates the relationship,

FIGURE 1 Numeric Scores vs. Mean Course Grades in Calculus



TABLE 4Distribution of Numeric Scores and Mean Grades

Score 0, 1 2, 3 4, 5 6, 7 8, 9, 10

Mean grade 1.5 1.6 2.5 2.8 2.9n 27 41 54 30 22

showing that the group of students that was most successful with respect to algebraic think-ing (scores of 8, 9, or 10) earned more than a full grade point higher than those who wereleast successful (scores of 0, 1, 2, or 3). Moreover, these low scores represent, almost ex-clusively, responses that were categorized as arithmetic. Scores of 4 or greater were achievedprimarily by students whose responses to at least two of the problems were categorized as alge-braic. This break between scores of 2, 3 and 4, 5 is where the largest increase in mean gradesoccurs.

To make the comparison of numeric scores with mean grades, the scores 0 through 10 wereplaced into five groupings of two score points in each group, starting with zero. Because therewere only four scores of 10, these were combined with the group of scores including 8 and 9.The resulting frequencies formed an approximately normal distribution. These frequencies, alongwith the mean grades for each group, are recorded in Table 4. It is noteworthy that fewer than13% of these calculus students answered four or five problems algebraically (scores of 8, 9, or10), and 39% answered only one or no problems algebraically (scores of 0, 1, 2, or 3).

CONCLUSIONS AND IMPLICATIONS

The present study created a framework to describe some of the qualitative characteristics ofalgebraic and arithmetic thinking reflected in college students’ uses of variables. In addition, theextent to which students’ responses showed evidence of algebraic or arithmetic thinking wasquantified and related to their introductory calculus course grades.

The majority of students in this study had difficulty using variables as generalized numbersand varying quantities to answer these five problems algebraically. Although we thought theseproblems should have been relatively simple for entering calculus students, they proved to bequite challenging. Based on the arithmetic/transitional/algebraic framework of using variablesemployed in this article, only 34% of all the responses were categorized as representative ofalgebraic thinking, compared to the 31% that were indicative of arithmetic ways of thinking. Infact, only 4 of the 174 students answered all five problems algebraically. Many of the arithmeticresponses to Problems 16 and 21 showed the tendency to manipulate one of the variables in orderto “solve” for the other. Correctly performing operations with algebraic variables and symbols isnot in itself a reflection of algebraic thinking. Rather, algebraic thinking, as defined for this study,requires a structural perspective wherein variables are interpreted as representative of changingquantities.

The results of the present study align with the growing body of evidence documenting collegestudents’ difficulties using variables as varying quantities. Trigueros and Ursini (2003) found thatvery few first-year undergraduates who had not taken calculus were able to interpret variables


70 GRAY ET AL.

as varying quantities in relational situations. Although it may be assumed that calculus studentswould be able to interpret variables as varying quantities, previous studies report evidence to thecontrary and suggest that students’ abilities to interpret variables have an impact on their successin calculus. For example, Jacobs (2002) found that secondary school advanced calculus studentswho had difficulty interpreting variables as covarying quantities were also likely to have difficultyunderstanding the calculus concepts of limit and derivative. Moreover, completion of a semesterof calculus may not be sufficient to fully develop students’ understandings of variables as varyingquantities. Carlson et al. (2002) reported that undergraduates studying second-semester calculushad considerable difficulty with tasks that required reasoning with covarying quantities in relationto rates of change.

The present study extends previous research by documenting a relationship between under-graduate calculus students’ uses of variables as reflective of algebraic or arithmetic thinking andtheir performance in calculus. Students’ numeric scores on these five problems were positivelyrelated to their mean final grades in introductory calculus. There are a myriad of factors thatcontribute to course grades. The results of this study imply that facility in using variables alge-braically as varying quantities may be one of those factors for calculus. These findings, alongwith those of other researchers, indicate that there is a need to maintain an ongoing instructionalgoal in calculus and precalculus to develop an understanding of the concept of variable in all itscomplexities.

In addition to students’ strong tendencies to approach problems arithmetically, this study re-vealed numerous examples of questionable use of mathematical symbolism, particularly related toinequality notation. Many students gave responses suggesting that they were using the inequalitysymbol to refer to positive integers rather than to sets of real numbers. Evidence from responsesto Problems 3 and 16 indicate that, when working with situations of inequality, these studentsrarely appeared to consider real number domains for the variables. Students’ seemingly defaultthinking about referents of variables as being integers or natural numbers could be detrimentalto their understanding of the major concepts in calculus, in particular, the concepts of limit andcontinuity. However, it is possible that students were influenced by the use of the variable nin Problem 3 because the letter n is often used in mathematics to indicate an integer. It is alsopossible that students did not spontaneously think of using the real number domain in the contextof these problems without being prompted to do so.

The primary goals of this research were to document the ways that students responded toproblems requiring the use and interpretation of variables as changing quantities; to record thefrequencies and types of responses indicative of arithmetic, transitional, and algebraic thinking;and to determine whether success rates on these problems with respect to algebraic thinking wererelated to students’ grade performance in calculus. We decided to collect data from as large asample as was feasible in order to obtain a broad base for the range of possible responses fromundergraduates at the beginning of their study of calculus. Although the present study indicatesthat there was a strong tendency to use an arithmetic approach to answer these five problems, theresults are based on observations of students’ written responses and do not provide evidencethat students were unable to reason structurally. They may have been using numerical methodsthat were more familiar, intuitive, or convincing to them than were more generalized algebraicapproaches. The observations and conclusions reported in this study are based upon the authors’interpretations and could be further confirmed, extended, or modified by expanding the varietyof problems and/or conducting interviews with respondents.



The findings of this study support the viewpoint that instruction in courses prior to calculusshould include explicit attention to the many different uses of variables as a way to fosterdevelopment of algebraic thinking. Furthermore, the results of this study suggest that calculusinstruction should continue to emphasize the differing uses of variables in various contexts andstrive to develop students’ conceptions of variables as changing and covarying quantities. Whenstudents are attempting to learn the concepts of limit, derivative, integral, and the connectionsamong these concepts crystallized in the fundamental theorem of calculus, they must possess arobust and flexible view of all aspects of the changing quantities that form the foundation of theirmathematical studies.

ACKNOWLEDGEMENTS

Portions of this research were presented at the 2007 annual conference of Research in Undergradu-ate Mathematics Education (RUME) in Phoenix, Arizona. The authors express their appreciationto Professor Annie Selden for her helpful and insightful comments on an earlier draft of thisarticle.

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