Making the Connection: Research and Teaching in Undergraduate Mathematics

Part I

Student Thinking

a. Foundations for Beginning Calculus 1

On Developing a Rich Conception of Variable

María Trigueros, Instituto Tecnológico Autónomo de México

Sally Jacobs, Scottsdale Community College Introduction

Have you ever considered that what mathematicians call ‘variable' is not a mathematically well-defined concept? And that variable can have different meanings in different settings? Unlike the concept of function, for example, variable has no precise mathematical definition. It has come to be a "catch all" term to cover a variety of uses of letters in expressions and equations. As a result, students are often unclear about the different ways letters are used in mathematics.

Later in this chapter, we provide practical suggestions to help students develop a rich conception of variable as called for by the National Council of Teachers of Mathematics (NCTM) in Principles and Standards for School Mathematics (NCTM, 2000). For now, though, let's begin this chapter with some traditional problems that students typically encounter in their high school or college math courses. Work through each problem, and pay attention to the roles that your symbols play during the process of solving them.

Problem 1. Laureen trained for a bicycle race by repeatedly going up and down a hill near her house. Every time she went up the hill, she rode her bike at an average speed of 8 km/h and she rode back down the hill at an average speed of 17 km/h ending at the same spot where she started. One day, she went up and down the hill repeatedly for two and a half hours. How long did it take Laureen to goupthehill each time? What is the total distance she travelled that day?

Problem 2. Find the family of lines that pass through the point (–2, 3). What is the slope of the line that goes through (7, 4)?

Problem 3. Find the values of a so that the function given by is continuous over its entire domain.

As you solved these problems, did you notice the complexity of demands for thinking about variable? Did your symbols take on different roles during the problem solving process?

All of these problems are straightforward for most advanced students. But for many average students, it is difficult to understand the role of the symbols at each different step in the solution process. It is hard for them to consider symbols sometimes as variables related in a function, at other times as unknown numbers to be found, and at still other times as general numbers. (By "general number" we mean a symbol whose value is neither assigned throughout the solution process nor is to be determined at the final stage of the solution process.) Yet many teachers say ‘variable' for all these different instances of symbol. Using the same name for a symbol that plays a variety of roles is very confusing to students. Most of them think of symbols as unknowns that have to be found no matter where they appear; a few consider them as standing for any number and they know how to operate on them procedurally; but all students have difficulty integrating all the different meanings of what mathematicians call ‘variable.' Because the term ‘variable' covers a variety of uses of letters and lacks an analytic definition, it should be no surprise that attaining a well-developed robust variable conception is problematic for students.

To elaborate more fully, we turn now to a discussion of the complexity of the demands for thinking about variable in each of the problems above. Analysis of Problem 1

The solution to the first part of the problem requires dividing the distance travelled by Laureen into two parts: the way up and the way down the hill. It is necessary to consider the time Laureen took while going uphill as a general number. That is, it can be represented with a letter, say t, that can take any value in a set and on which one can operate. The time to go downhill can be symbolized as T – t, where T is another general number. In order to find the total distance, one has to consider the distance up the hill the same as the distance down, since Laureen ended at the same spot where she started. Using the fact that the distance travelled is proportional to the time spent travelling, and that the constant of proportionality is the average speed, it is necessary to introduce functional relationships between the distance travelled each part of the trip and time spent in each of them: v₁t = d₁ and v₂ (T – t) = d₂. In these relationships, it is important to note that t can be considered as the independent variable in the function, d₁ and d₂ as the dependent variables, and v and T as parameters. Using these two relationships, one gets a new functional relationship, v₁t + v₂ (T – t) = d₁ + d₂. Since the values of the parameters are given, d₁ = d₂ results in the equation, , where the symbol t is a specific unknown that must be determined. Once the value of t is obtained, it is necessary to go back to the function that relates time and distance, substitute the value for t and solve a new equation for d₁. The total distance can then be calculated by multiplying the uphill distance by 2.

Did you notice how the symbols take on different meanings at different times during the solution process? Initially, t and T were used as general numbers. Then t became an independent variable coordinating with the dependent variables d₁ and d₂. At that same time, T changed from a general number to a parameter. Near the end of the process, the roles of t and d₁ changed to unknown from independent variable and dependent variable, respectively. Analysis of Problem 2

The solution to this problem requires students to find an expression for all the possible lines that pass through the given point. They need to generalize what they know about a line to consider a family of lines each with different slopes. They must thus conceptualize slope as a varying quantity; but then they must introduce a parameter (a symbol used to generalize an algebraic statement so that the expression covers a family of cases) when they express the functional relationship between the independent variable x and the dependent variable y:. Students need to distinguish between the meaning of the symbol m, as a parameter, and the meaning of the symbols x and y, as covariates. This distinction is required in order for them to recognize that m (not x!) now becomes the unknown to be determined. Take note of the complexity of understanding required here: the student must move flexibly from variable as varying quantity to variable as parameter, as independent/dependent variable, and as unknown. Analysis of Problem 3

To solve this problem the student needs to understand the functional relationship between x and , where x can take any real value. It also helps if the student considers this relationship as a dynamic covariation between the independent and dependent variables. Furthermore, the ability to distinguish the different roles played by the variables becomes crucial when applying the definition of continuity for a real function. In the expression, , the student must realize that the independent variable x varies dynamically with the dependent variable f(x) as it nears the value of the parameter a. At the moment when x takes the value of a, the student must recognize that the role of changes from dependent variable to a specific unknown that must be found.

Following in the spirit of the analysis of these three problems, you may now have a more acute awareness of the different uses and the changes in meaning of the term ‘variable.' Several questions may come to mind: When is the last time you discussed the concept of variable with your students? How often do you point out the different ways in which variables are used in mathematics? Are you satisfied with the treatment given to variable in the textbooks? Do you assume, perhaps, that students just "know" variable and that it's not necessary to spend much class time talking about it?

Mainstream mathematics textbooks used in the high school, community college and first-year mathematics university programs (including the calculus sequence) generally give a cursory treatment to the concept of variable. The concept is not well developed and it is not given much attention. Thus it is not surprising that high school and college students possess variable conceptions that in some cases are narrow, limiting, and generally underdeveloped. In the next section, we present literature regarding the impoverished nature of some commonly held variable notions. Moreover, the research literature suggests that these limited conceptions of variable can pose obstacles for students when they advance to calculus and higher levels. This chapter discusses some of this research and provides practical suggestions. Research Literature About Students' Conceptions of Variable

Research on both high school and beginning college math students suggests that their conceptions of variable are often superficial and lacking in richness. A rich variable conception should include the notion of changing quantity and joint variation. Also, dynamic imagery plays a key role in a well-developed variable conception. In this section, we summarize several studies related to these aspects of a rich variable conception. Superficial Conceptions

In an investigation involving 167 beginning college students, Ursini and Trigueros (1997) concluded that these students had generally superficial conceptions of variable. The majority were able to interpret, symbolize and manipulate variables as specific unknowns only at a very elementary level. For example, they could interpret the symbol x in as representing an unknown value, and they knew that x in is not an unknown to be determined. Furthermore, they could symbolize the relationship from the data of a given table, and could correctly manipulate the symbols in expressions such as . They could recognize the presence of an unknown number in a problem; but their ability to process other given information was limited (e.g., the ability to use contextual data to symbolize an equation). In particular, they often had difficulty discriminating between variable as unknown and variable as general number in fairly simple expressions. For example, they had trouble seeing how variable is used differently in equations such as and . Further, they consistently avoided manipulation of variables. Additionally, situations involving related variables posed difficulties for these students. While they could adequately handle correspondence between specific numbers, they had trouble with the notion of related variation. Their difficulties seemed to stem from the absence of a conception of relation as a transformation process or a dynamical process of variation.

The researchers noticed, also, that while these students were capable of recognizing the role played by the variable in very simple expressions and problems, any small increase in complexity provoked inadequate generalizations. Overall, students' understanding of the concept of variable lacked the flexibility that is expected at the college level. In a later study, Trigueros and Ursini (2003) concluded that when students engaged in more complex problems in which different uses of variable are involved (for example, where they must pose and solve an equation or set up and work with a specific functional relationship), they were unable to differentiate among the different uses of variable and integrate them successfully. Varying Uses of Variable

Based on a seminal large scale British study, Küchemann (1980) developed a model for describing and classifying different ways secondary students use algebraic letters. This classification model delineates six levels of interpretation and use of letters: Letter Evaluated; Letter Ignored; Letter as Object; Letter as Specific Unknown; Letter as Generalized Number; and Letter as Variable. Building on the work of Küchemann and others, Trigueros and Ursini (1999, 2001, 2003) developed the "3 Uses of Variable Model" (3UV model) — not for classification purposes — but for the purpose of analyzing student difficulties, textbook treatment, and classroom observations. In addition, they use the 3UV model in the development and testing of instructional design. Their work analyzes the different uses of variable and the different aspects involved in its use when solving elementary algebra problems.

The 3UV model identifies variable as (a) unknown, sometimes called ‘indeterminate' in older textbooks, (b) general number, and (c) related variables, such as those found when working with functions or curves. In this model, the term general number refers to the meaning associated with symbols in general expressions in which it is necessary to perform algebraic operations; or when symbolizing generalizations (for example, those found in problems where it is needed to find the next term in a sequence of numbers, or the number of points or lines that a specific geometric pattern will have after a certain number of iterations). To be fluent with variables, students need to be able to interpret all three uses of variable in different parts of a multi-step problem. They also need to be able to symbolize a quantity with a variable and manipulate variables. In the case of functions and curves, they need to be able to construct graphs of related variables and interpret them. According to Trigueros and Ursini (1999, 2001, 2003), a well-developed understanding of algebra necessitates the ability to differentiate among the three uses of variable and to flexibly integrate their uses during the solution of any problem. Parameters

The 3UV model has also been used to analyze problems that include the use of parameters. Ursini and Trigueros (2004) consider parameters as a particular use of general numbers since they are needed to generalize expressions that already include variables. But in their study with 62 undergraduates, they found that students think of parameters as variables. During interviews, most students responded, "this letter stands for a constant that can change, it is another variable." These students interpreted parameters as general numbers and did not differentiate them from other variables unless the problem they were confronted with provided them with a concrete referent where the parameter acquires a specific meaning (for example, when parameters appear in the equation for a line or within a well-known formula such as the quadratic formula). Students showed difficulties manipulating the parameters, and on many occasions they tended to ignore them.

To illustrate, students solved the equation without taking the parameter into account when they were presented with the following problem: Given the equation , for which values of p does the equation have only one solution? What are the roles of p and x in this equation? When asked to explain their answer, they could not attach any meaning to p. They did not see p as the coefficient part of the linear term; they ignored it and were able to solve only a particular case. Most of the time, when these students were asked to symbolize a generalization, they either ignored the parameter or identified some general elements from the problem, but ultimately could not write an appropriate expression or equation. Changing Quantity, Dynamic versus Static Imagery, and Joint Variation

Using variables to represent changing quantities and express relationships is particularly problematic for students, as substantiated by numerous reports (e.g., Küchemann, 1980; Kieran, 1992; Ursini & Trigueros, 1997, 2001; Trigueros & Ursini, 1999, 2001, 2003; Jacobs, 2002). Studies show that high school and beginning college students can work appropriately with correspondence between numbers, but the idea of joint variation is not easy for them. They can plug in a value of one variable into a functional relationship, but they are unable to determine variation intervals (Kieran, 1992; English & Sharry, 1996) or think about this relationship in a more dynamic way (Ursini & Trigueros, 1997).

In exploring this difficulty, a few studies have revealed the role played by a covariational view of function that is supported by dynamic imagery in the formation of student conceptions of variables (e.g., Cottrill et al., 1996; Jacobs, 2002; see also Oehrtman, Carlson, and Thompson, this volume). In the context of limit, investigations of calculus students' views about variable have uncovered interesting results. Cottrill et al. (1996) found that, among first semester calculus students, mental construction of the domain process () was dynamic, whereas thinking about the range entailed the static image of considering only a single value, . A similar finding was reported by Jacobs (2002) in her exploratory study of Advanced Placement BC (AP/BC) calculus students' notions about variable. When these students were asked to discuss the meaning of ‘x' in the expression , they used dynamic imagery as they referred to x occurring in ; at the same time, however, they held a static view when they talked about x occurring in f(x) as a single value that is ‘plugged in.' In other words, they did not conceive of the two instances of x within the same equation in quite the same way (dynamic image in one instance, static in the other).

Moreover, Jacobs found that when these same students discussed derivative, the notion of continuous variation in a changing quantity seemed to be largely absent from their thinking. They gave no indication of holding an image of one variable changing in tandem with another variable (a dynamic covariational view) in the context of derivative. Also, they tended not to mention changing rate in their discussion of derivative. Jacobs concluded that, in general, the ability to view variable as capable of having changing values seems to play an important role in conceptualizing changing rate. Recognition that something is changing is an essential underpinning to understanding the key ideas in calculus such as derivative, changing rate, integration and the Fundamental Theorem.

White and Mitchelmore (1996) revealed serious deficiencies in first semester calculus students' conceptions of variable that affected their ability to represent changing quantities. Their study involved four versions of four problems (each problem having to do with application of the first derivative), where the most contextual presentation required more translation than the purely symbolic presentation. Most of the students had difficulty using variables to represent changing quantities; few were able to correctly symbolize. The investigators noted that "… defining and using new variables is qualitatively different from relating explicitly given variables in symbolic form… [it] involves forming relationships at a higher level of abstraction than relating those already given" (p. 91). In particular, students' written work and their follow-up interviews demonstrated that many of them were confused about how to define appropriate variables and whether letters represented changing or constant values. Moreover, they tended to think about two or more variables as things to be manipulated rather than as representations of quantities having a relationship. The researchers noted that these students seemed to view variables as literal symbols detached from any concrete meaning. That is, their conception of variable was limited to algebraic symbols. The study concluded that an underdeveloped conception of variable is a major obstacle to applying calculus successfully.

Examining variable understanding in the context of linear inequality, Sokolowski (2000) found that a conception of variable as a varying quantity is linked with the ability to model, solve, and interpret solutions to linear inequality problems. In addition, she concluded that most students lacked a deep and robust understanding of variable and the ability to use variables flexibly. Other authors have called attention to the role that covariational reasoning plays in students' understanding of variable quantities changing in tandem with each other (e.g., Carlson, Jacobs, Coe, Larsen, & Hsu, 2002). Possible Explanations

Why do secondary and post-secondary students have such difficulty with variable? The roots of the problem are indeed complex. Certainly over the past several centuries, variable has conveyed different meanings at different times. A retrospective look at these shifts in meaning provides some insight into why students may have difficulties. Historical Development: Shifts Over The Ages

Variable is a concept for which conventional meanings and published definitions have varied considerably since its early origins. The meaning of variable has changed in emphasis at different times. In the 16^th century, François Vieta (1540–1603) (or Viète) proposed a general method to solve problems. He referred to his method as the "analytical art" (Klein, 1968). Vieta's greatest innovation was his conception of (a) a general object which could be introduced in the general method and which could be operated on by well-defined rules and (b) a notational scheme: "a vowel to represent the quantity in algebra that was assumed to be unknown or undetermined and a consonant to represent a magnitude or number assumed to be known or given" (see Boyer & Merzbach, 1989, p. 341). He did not, however, apply his method to problems involving relationships between variables. The notion of related variables was advanced some years later. Working independently, both René Descartes (1596–1650) and Pierre Fermat (1601–1665) applied Vieta's ideas to geometry; Descartes, in particular, considered general objects as a useful tool to model and think about problems, to introduce dependence between symbols, and to calculate the value of one of the objects when the value of the other was known (Youschkevitch, 1976).

In the 18th century the mathematical term ‘variable' conveyed the meaning of something that actually varies, such as time that passes, temperature that oscillates, days that lengthen, mortality rate that decreases, etc. (see Freudenthal, 1983). The earlier meaning is imbued with a kinesthetic quality, as evidenced by mathematical expressions such as ‘e converges to 0,' ‘x runs through the set S,' or ‘n approaches infinity.' According to Hamley (1934), that kind of interpretation began to give way in the late 1800s to the idea of variable as an abstract concept relating to ‘pure' number values disassociated from any concrete embodiment in physical quantity. This latter notion of variable, which Freudenthal (1983) calls ‘polyvalent name,' has persisted throughout the last century. The modern mathematical practice, however, is to mix polyvalent names and variable mathematical objects into one term, ‘variable.'

A look at definitions used for the term ‘variable' over the last few centuries reveals qualitative shifts in emphasis. Schoenfeld and Arcavi (1988) cite 10 different definitions of variable in various technical publications printed between 1710 and 1984. These definitions differ according to the importance they attribute to notions such as domain (modern emphasis) or variable quantity (earlier emphasis). Mainstream mathematics curricula in several countries, for example México and the United States, tend to introduce variables as symbols (usually letters) that stand for numbers and whose value is changeable. Contrast this approach with the notion of changing amounts of some measurable quantity like amount of time. Appreciation of this distinction invites the question: Which is the variable, ‘time' or ‘t'? Adding to the complexity of variable for the uninitiated student, phrases such as ‘x varies,' ‘as x gets closer and closer to,' and ‘let number of hours be the independent variable' are often heard.

Some researchers have expressed their concern about the polyvalence of meaning of variable. Thompson (1994b) theorizes that Newton's insight leading him to the Fundamental Theorem of Calculus was supported by a mental image of dynamic quantities. Similarly, the development of an image of rate in 7th graders, he found, begins with an image of change in some quantity. Thus, according to Thompson (1994a), a conception of variable as changing magnitude is important for developing a mature image of rate. He asserts:

In today's K–14 mathematics curriculum there is no emphasis on function as covariation. In fact, there is no emphasis on variation. . . . This is in stark contrast to the Japanese elementary curriculum which repeatedly provokes students to conceptualize literal notations as representing a continuum of states in dynamic situations. . . . It seems, to me anyway, that a progressively more abstract notion of covariation rests upon a progressively more abstract image of variable magnitude. (p. 29)

Even earlier, Menger (1956) had expressed dismay over the blending of meanings into one term because it resulted in loss of preciseness in mathematical language. Freudenthal (1983) also disagreed with this convention on both pedagogical and mathematical grounds, since it obscures the important aspect of kinesthetics. Janvier (1996) notes the difference between magnitudes and numbers. He contends that the essence of the term ‘magnitude' captures the idea of a measuring number as opposed to a ‘pure' number. For him, this double meaning has serious implications for curriculum design, particularly in the areas of modeling and function approaches to algebra. Calculational Attitude

Another possible explanation for student difficulties may relate to the tendency for both teachers and students to focus more on calculations and less on concepts involving variable. The role of calculational versus conceptual orientation to variable surfaced in the Jacobs' (2002) study. Her investigation of AP/BC high school calculus students revealed that when they oriented themselves to a given task in a calculational manner (i.e., they talked about the task in terms of ‘find,' ‘solve,' ‘answer' or ‘plug in'), they viewed a variable as an unknown, a letter that stands for a number or general number, an input to a function, or as a receptacle that receives a number in specific cases or parameter. These views about variable contrasted sharply with the views of students who exhibited a conceptual orientation and who talked about the task in terms of dependency relationships and varying values. In the latter case, students tended to view variables as tools for expressing mathematical relationships; their variable conception was characterized by a concern for how a variable relates to its domain and how two or more variables relate to each other.

In the context of function, these AP/BC students tended to focus on the independent variable and the process of evaluating the function at a particular input value. The independent variable was in the foreground of their discussions while the dependent variable remained in the background. In other words, they were not mentally coordinating two changing quantities in a covarying relationship.

In the context of limit, students' calculational attitudes were associated with their tendency to view limit (or the variable L) as an unknown value to be procedurally determined. The limit L was never discussed as a variable. Also, a mental coordination of L with a (as in ‘x approaches a') was seldom observed in these students.

In the context of derivative, a calculational attitude was associated with students' inability to view variable as a changing quantity. Whether they talked about slope, difference quotient, rate, velocity or derivative, the matter of two simultaneously changing quantities was never mentioned. What they almost always alluded to, however, was the procedure for calculating slope/rate/velocity or finding the derivative function.

White and Mitchelmore (1996) reported similar findings regarding students' tendency to approach a derivative problem with a view toward using variables to perform a calculation or follow a procedure. As previously mentioned, they found that university calculus students' difficulties in applying calculus were directly related to their tendency to view variables merely as algebraic symbols that are to be manipulated. Lack of Rich Curricular Material

Current school mathematics curricula often fail to account for the conceptual complexity of variable. Not only do curriculum materials generally fall short of scaffolding a robust variable understanding, but instructional practices often neglect developing a rich conception. Textbook analysis and classroom observations indicate that curriculum and instruction at the secondary level concentrate mainly on the use of variable as unknown. Teachers seem to expect that as students encounter algebraic expressions, word problems, and problem-solving exercises, they will construct (all by themselves!) a robust, flexible and coherent conception of variable as a mathematical entity. However, the large body of research findings clearly indicates that this expectation has no grounding in empirical studies. Teachers need to become aware that something more explicit is needed to ensure that students develop the ability to differentiate general numbers from specific unknowns and work with variables as changing entities that can be related in a dynamic way. They need to concentrate on helping students work with the different uses of variable in a flexible way, to foster students' ability to overcome a merely computational way of thinking about symbols, and to ensure that students achieve a stronger conceptual understanding.

Simply having students take more math courses does not improve their variable understanding. Results from a study by Trigueros and Ursini (1999) suggest that, as students continue to take algebra courses through high school, their conception of variable shows little sign of substantial and sustained improvement between middle school and university levels. The researchers examined interpretation, symbolization and manipulation of variable in a variety of representations (verbal statements, tables, graphs, algebraic expressions). They found that even though students are exposed to different uses of variable in algebra courses, they nonetheless fail to comprehend variable as a multifaceted entity. Surprisingly, on tasks involving variables in a functional relationship or identification and interpretation of the unknown, students who had not yet taken algebra performed better than students who had already studied algebra. This finding indicates that student difficulties with variable conception are probably attributable to current didactical approaches. Interventions and Promising Results

Attempts to address the potential of students to understand the concept of variable after suitable intervention are reported by Ursini and Trigueros (2001). They designed a teaching approach that follows a spiral path where at each stage students are introduced first to a problem requiring only one use of variable at a time (variable as specific unknown, variable as general number, or variables in functional relationship) and then to a problem of the same level of difficulty but requiring integration of the different uses of variable. This procedure is repeated numerous times with ever increasing levels of task difficulty. This teaching design was tested in a study with 12- and 13-year-old public school students in México (Ursini, Trigueros, Escareño, & González, 2002). The analysis of these students' work showed that after 15 sessions they had acquired the ability to recognize the different uses of variable and use them appropriately when working on simple algebraic problems. Also, they were able to differentiate among the uses of variable and integrate the different uses throughout the process of solving the problem. Overall, they demonstrated gains in their ability to shift between the different uses of variable in a flexible way. The next section proposes specific suggestions for the practitioner. A Call To Action

The National Council of Teachers of Mathematics (NCTM) has designated specific goals relating to variable understanding in children from pre-K through grade 12. In the Principles and Standards for School Mathematics (NCTM, 2000), a comprehensive resource guide and set of recommendations for mathematics curriculum, strategies for developing student conceptions of variable are explicitly recommended for each grade band (Grades preK–2, Grades 3–5, Grades 6–8, Grades 9–12). One of the NCTM's guiding philosophies for developing variable conception is expressed in this document in a quote by Anna Sfard: "A thorough understanding of variable develops over a long time, and it needs to be grounded in extensive experience" (p. 39).

It is interesting to note that in the NCTM (2000) document, most of the curricular recommendations addressing variable concept development target the middle grades. There is very little discussion of variable at the secondary level. The same is true of the programs for the three compulsory mathematics courses for middle school education published by the Ministry of Education in México, as well as the most widespread programs for high school mathematics in México. A careful examination of all these curricula leaves us with the distinct impression that a robust variable understanding is tacitly assumed to be firmly established by the time students reach high school. The literature in mathematics education, however, strongly suggests otherwise. In fact, as we have documented throughout this chapter, high school algebra students tend to have under-developed conceptions of variable and this impoverishment holds true for college students as well. Practical Suggestions

The teaching of variable at all levels before calculus must develop the capacity of the learner to think of this concept as an entity and to understand its different uses as diverse facets of the same concept. Students need opportunities to discuss the differences in variables that appear in a problem so that they can become acquainted with different uses and can integrate and interiorize them into that which teachers and mathematicians call ‘variable.' Students also need opportunities to reflect on the roles that symbols play at different moments in the solution of specific problems, as demonstrated in the analysis of problem solutions discussed at the beginning of this chapter.

When students arrive at the calculus level, their conception of variable is far from complete. They still need to be given opportunities to revisit variables and think about them in the differentiated way presented in these pages. Students are capable of developing a rich conception of variable, but they need the full complexity of the concept to be addressed in a more explicit manner in their courses. And since many of the conceptual obstacles facing calculus students are related to variables playing a dynamic role in functional relationships, mathematics educators must give increased curricular attention to variables in such roles. We must be overt and intentional as we redirect our efforts toward fostering well-connected understandings.

Specifically, it is important that teachers realize the need (a) to become aware of students' difficulties with variables and (b) to address them explicitly in the classroom. In particular, teachers can incorporate into their classroom practice the following suggestions:

· Encourage students to choose meaningful symbols for variables in a contextual setting (not always x and y), and take care that they are able to change the name of the variable when needed to be sure they are not using the variable only as the replacement of the name of an object, but as a symbol to operate with. In expressions such as T – t in the bicycle problem presented at the beginning of this chapter, it is helpful to ask students to write 1–2 sentences to explain the difference in meaning between T and t.

· Ask students to identify all of their symbolic expressions. For example, have them write in words what each expression represents and discuss similarities and differences. Engage them in explaining the difference between equations and open expressions. Have them write in words what each open expression represents. For their equations, have them explain when the symbols stand for a number or numbers they need to find and when they represent any number in a certain domain. In the case where there is a relationship between variables, have them explain in words which variables are related and how.

· Discuss with students the different roles (unknown, general number, related variables) played by each of the symbols in specific problems. For instance, have students work in groups to construct a table that shows, for each successive step of the solution, what role each symbol plays in that particular step. Allow each group to present their table to the whole class and discuss the different roles played by their symbols with particular attention to when the roles change. Discuss similarities and differences between the tables presented.

· Request that students state the appropriate unit of measure that goes with each variable in a contextual setting. For example, when students identify variables and their relationship, have them write the unit of measure that goes with each variable. Also, when they identify their open expressions, have them write appropriate units.

· Provide opportunities for students to identify related variables and to explain how these quantities vary in different problems. Encourage them to imagine the variation and describe it in their own words. For example, when working with a problem such as the bicycle problem presented at the beginning of this chapter (e.g., number of times that Laureen rides up the hill, down the hill; distance travelled; duration of training that day, etc.), talk with students about all the quantities that vary. Ask students questions such as: What is the smallest and largest value that you can imagine for each quantity identified? What intermediate values do you imagine would occur between the smallest and largest values?

· Place increased emphasis on the covariation aspect of related variables. Have students work in groups to construct a 2-column table for each pair of variables that co-vary. Using their tables, have them construct graphs where they label and scale both axes. Have them select two points on their graph and discuss what each point represents (in terms of coordinated variables). Also have them explain what changes occur when moving from one point to the next. Then have them construct 3-column tables to show how the values of each variable change in tandem with the values of related variables.

· Help students develop dynamic imagery in connection with the concept of variable. Design classroom activities where students kinesthetically act out variation in some designated quantity over time (by walking, by hand movement along an axis on a graph, or by using physical models). Have them attend to aspects of the variation (for example, magnitude, amount of change, increasing/decreasing magnitudes, rate of change and changing rate). Then create for them a function setting where the varying quantities do not explicitly involve time and have them attend to aspects of variation in each quantity as it relates to the other.

· Reinforce all of these suggestions by including items on tests, quizzes, and other assessments that engage students in sense-making. For example, a problem like the bicycle problem (discussed earlier) lends itself to asking questions such as the following: What does the equation mean? Explain each letter. What is the unit of measure for ? What does the equation mean? What does mean?

Classroom time spent on developing a rich conception of variable is time well spent. With appropriate curriculum and instruction, students can achieve gains in their confidence and proficiency in working with variables. A better knowledge of variable can help students overcome a merely computational orientation and achieve rich conceptual thinking. When students possess rich variable conceptions, they are better prepared to advance to their next higher course in mathematics. Our goal for students is that they will no longer struggle with the complexities of variable in the ways that they are struggling today. Concluding Remarks

The research findings reported in this chapter make it clear that variable is a very complex concept and that students need help in developing a rich conception. The current body of research alerts us to specific difficulties that students face regarding the concept of variable and provides useful information to guide the design of interventions for helping students work more fluently with variables. The way is open for new curricular design; the way is open for new instructional methods. With research guiding the way, mathematics educators are now obligated to direct their efforts toward ensuring that students advance their conceptual understanding of variable. References

Boyer, C., & Merzbach, U. (1989). A history of mathematics (2^nd ed.). New York: Wiley.

Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352–378.

Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K., & Vidakovic, D. (1996). Understanding the limit concept: Beginning with a coordinated process schema. Journal of Mathematical Behavior, 15(2), 167–192.

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