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Chapter one: Introduction
Calculus is the study of variation, and it is also the central subject in mathematics (Boyer, 1959; Goldstine, 2012; Tall, 1990). The concept of calculus is applied in statistics, science, economics, and engineering to study the concepts of gravity, speed, velocity, variations, growth or decay function, and maximum or minimum of profit and cost function (Boyer, 1959, Goldstine, 2012; Tall, 1992). Many of scientific and technological growth since 1900s to the current time are linked with the idea of calculus (Boyer, 1959; Goldstine, 2012). However, the core concepts of calculus: infinitesimals, variations, rates of change, derivatives, and integrals are difficult for students to learn (Bezuidenhout, 1998; Burns, 2014; Carlson et al., 2002; Castillo-Garsow, 2012; Orhun, 2012, Park 2015; Tall, 1992; Thompson & Carlson, 2017). In this study, I seek to investigate the mathematical reasoning that undergird the concept of variation, rate of change, and derivative.
The concept of variation is fundamental for understanding the main idea of calculus and it has different mathematical meanings from elementary school to higher-level mathematics studies (Akgün & Özdemir, 2006; Gray, Loud, & Sokolowski, 2007; Jacobs, 2008; Kaput & Roschelle, 2013; National Governors Association (NGA), 2010; Philipp, 1992; Stephens, 2005). For example, within mathematics, variables have many meanings and it has a meaning of a fixed number or value, unknown values of an equation, an algebraic symbol, and varying quantities (Akgün & Özdemir, 2006; Ely & Ellis, 2018; Philipp, 1992; Stephens, 2005; Thompson & Carlson, 2017). The concepts of variation and infinitesimal variation are the foundation and main source of the development of the idea of constant rate of change, rate of change, function, derivative, and advanced concepts of calculus (Gray, Loud, & Sokolowski, 2007; Kaput, 1994; Meletiou, 2000; Philipp, 1992; Schoenfeld, Arcavi, 1988; Stephens, 2005). For instance, the concept of average rate of change is built on the quotient of two simultaneously varying quantities. However, the concept of variation is not well studied or it is often neglected in mathematics learning, despite its importance in the development of the fundamental concept of calculus (Akgün & Özdemir, 2006; Schoenfeld & Arcavi, 1988; Thompson & Carlson, 2017).
The concept of rate of change is first introduced in early middle school mathematics as connected to the concept of ratio, slope, and proportion (NGA, 2010). In elementary school, students discuss the concept of fractions, ratios, and proportions as they progressively learn the concept of whole numbers, integers, decimal numbers, and the rational number system in connection with the elementary mathematical operations (NGA, 2010). Similarly, the concept of rate of change is used to describe the relationship between two or more varying quantities in different fields, such as physics (e.g., speed and velocity which is used to describe the relationship between position and time), statistics (e.g. average and mean which is used to describe the distribution of data points), and economics (e.g., interest rate and percentage change which is used to describe the amount of money change in relation to a particular time interval). Within mathematics, the concept of rate of change has many meanings among high school, undergraduate, and graduate students (e.g., slope, rise over run, average rate of change, and derivative which is used to describe the relationship between two simultaneously changing quantities) (Bezuidenhout, 1998; Byerley, Hatfield, & Thompson, 2012; Ehlke & Hajra, 2017; Johnson, 2015; Musgrave & Carlson, 2016; Tyne, 2014, 2017; Weber, 2013). The fundamental concepts of calculus (e.g. functions, derivatives, integrals, and differential calculus) are founded on the conceptual understanding of the rate of change (Johnson, 2012; Thompson, 1994; Thompson & Carlson, 2017; Thompson & Thompson, 1992; Thompson & Thompson, 1994). However, many calculus students have weak or unproductive understanding towards the concept and idea of rate of change (Johnson, 2015; Musgrave & Carlson, 2016; Tyne, 2014, 2017; Thompson & Carlson, 2017).
The concept of derivative in calculus is deeply rooted in the ideas and concepts of variation and rate of change (Park, 2013; Thompson, 1994). The concept of derivative can be developed from the idea of constant rate of change or average rate of change to a continuously varying rate of change function, or an instantaneously varying rate of change function (Carlson et al., 2002; Thompson & Ashbrook, 2016; Thompson & Carlson, 2017). In calculus, the concept of derivative is typically introduced in a post-secondary school level or after a pre-calculus course using the concept of a limit of the average rate of change function (Hart, 2019; Stewart, 2015; Tan, 2018). Student’s conceptual understanding of integrals, differential equations, and advanced calculus concepts is founded on the idea and the concept of derivative function (Burns, 2014; Dufour, 2015; Firouzian, 2013; Habre & Abboud, 2006; Park, 2012; Park, 2013; Thompson & Carlson, 2017; Wagner, Roundy, Dray, Monogue & Weber, 2015). However, many students when joining their post-secondary school exhibited underdeveloped conceptual understanding to the concepts of derivative (Byerley, Hatfield & Thompson, 2012; Park, 2012; Park, 2013; Dorko & Weber, 2013).
The conceptual understanding of variation, rate of change, and the derivative are deeply founded on the students’ clear and sound mathematical reasoning abilities (Carlson et al., 2002; Gray, Loud, & Sokolowski, 2007; Moore, 2010; Thompson & Carlson, 2017). Mathematical reasoning is defined as “a capacity to think logically about the relationship among concepts and situations” (NRC, 2001, p.129). This idea of mathematical reasoning that focused on the students’ logical thinking ability about two or more covarying quantities is currently promoted in calculus learning (Carlson et al., 2002; Castillo-Garsow, 2012; Engelke, 2007; Gray, Loud, & Sokolowski, 2007; Moore, 2010; Thompson & Carlson, 2017). Researchers in calculus are currently investigating mathematical reasoning abilities like variational reasoning, quantitative reasoning, covariational reasoning, continuous covariational reasoning, and smooth continuous covariational reasoning, which are foundational to advanced calculus learning (Carlson, Larsen, & Jacobs, 2001; Carlson et al., 2002; Castillo-Garsow, 2012; Engelke, 2007; Gray, Loud, & Sokolowski, 2007; Moore, 2010; Moore& Paoletti, 2013; Thompson & Carlson, 2017; Weber & Carlson, 2010). Most study results in calculus showed that students’ learning of the foundational ideas of calculus like the rate of change, derivative, and integrals rely on their clear and explicit covariational reasoning capability (Carlson et al., 2002; Castillo-Garsow, 2012; Engelke, 2007; Gray, Loud, & Sokolowski, 2007; Moore, 2010; Moore & Paoletti, 2013; Moore & Paoletti, 2015; Thompson & Carlson, 2017; Weber & Carlson, 2010).
The notion of covariational reasoning means a student has an image of two quantities varying together within specified intervals (Carlson et al., 2002; Castillo-Garsow, 2012; Thompson & Carlson, 2017). According to Thompson (2011, p.46), “the mathematical reasoning of variation or covariation involves imagining a quantity whose value varies.” Additionally, Thompson (1996, p.1) indicated that “mathematical reasoning at all levels is firmly grounded in imagery.” Imagining for a student could mean someone visualizes mentally how “the volume and height of water are varying as the bottle is being filled with water without actually filling the bottle with the water” (Castillo-Garsow, Johnson & Moore, 2013; Thompson, 1994). Further discussion will be included about imagery and the idea of mathematical reasoning from the views of Thompson (1996) and Piaget (1967) in Chapter 2. The quote below gives an illustration to what it means to imagine how a quantity varies over certain given interval of time.
“An elementary school student might envision a situation in which one tree is three times as tall as another. What she will find difficult is to imagine this relationship holding over time as the trees grow—imagining that the taller tree is always three times as tall as its neighbor as each of them grow. The two instances reveal a stark contrast between two ways of thinking. In the first instance, “three times as tall” is a description of a static relationship. In the second instance, “three times as tall” is an invariant relationship between two quantities whose values vary. The second instance involves a way of thinking that is foundational for the concepts of variable and function in calculus” (Thompson, 2011, p.46).
Researchers in calculus are investigating numerous ways to improve students’ mathematical reasoning, for example, in several studies in calculus researchers are focused on developing and investigating students’ quantitative reasoning, continuous variational reasoning, and covariational reasoning. This study is similarly focused on investigating and understanding the students’ mathematical reasoning, especially students’ smooth continuous covariational reasoning ability, since covariational reasoning ability is viewed as a key mathematical reasoning ability that will help students to understand the central concept of calculus. Thompson and Carlson (2017) indicated that covariational reasoning is a foundational reasoning ability in mathematics, particularly in calculus, to advance students’ conceptual understanding of the concept of function, rate of change, and derivatives. There are two distinct views among researchers and educators regarding how students develop covariational reasoning. The two distinct views emerged from the continuous covariational reasoning view: (1) chunky continuous covariational reasoning view and (2) smooth continuous covariational view. More detailed discussion of these views will occur in the next section and chapter 2.
Background for the Study
Students’ variational and covariational reasoning have critical roles in conceptual understanding of the foundational concepts of rate of change, derivatives, integrals, and advanced calculus concepts (Castillo-Garsow, 2010, 2012; Castillo-Garsow et al., 2013; Ely & Ellis, 2018; Thompson & Carlson, 2017). Conflicted theoretical constructs on how students develop variational, covariational, and other related reasoning types exist among the major researchers like Saldanha and Thompson (1998), Carlson et al. (2002), Confrey and Smith (1994), and others. For example, for Confrey and Smith (1994) covariation means coordinating the change from y_m to y_(m+1) with the change from x_m to x_(m+1)without considering the change in-between the intervals. That is, covariation have a meaning of chunky by moving or adding values continuously between the start and the end values of one variable with successive values of change with the other variable. This equally means that chunky reasoning is the ability of coordinating a completed change in the first variable value with a completed change in the second variable (i.e., coordinating a complete ∆x with a complete ∆y)( Castillo-Garsow, 2012; Confrey & Smith, 1994; Saldanha &Thompson, 1998; Thompson & Carlson, 2017). On the other hand, Saldanha and Thompson (1998) viewed covariation in terms of imagining one quantity’s values varying continuously and then imagining the second quantity’s values varying simultaneously. For Saldanha and Thompson (1998) covariation has a meaning of continuous covariation where students imagining two quantities values moving together. That is, students imagining one quantity values varying by different values continuously as it passes all the way through the given interval and then imagining the second quantity simultaneously moving continuously at the beginning and end of the interval (Castillo-Garsow, 2012; Thompson, 2008; Thompson & Carlson, 2017). In continuous covariational reasoning there is an imagination of how quantities’ values vary within an interval (i.e., there is an imagination of change within ∆x). The continuous covariation reasoning proposed by Saldanha and Thompson (1998) could entail two reasonings chunky continuous covariation and smooth continuous covariation reasoning. For example, for the hypothetical problem “imagine a bottle filled with liquid. How do you think about the volume of the liquid as it changes relative to the height of the liquid in the bottle?” (Castillo-Garsow et al.,2013, p.31) For this kind of problem two ways of thinking are possible. One possible mathematical reasoning is filling the volume of the bottle smoothly and continuously by some amount of water. That is, it might have a mental image of change in progress (e.g., filling a bottle with a hose) which is a smooth reasoning. Another possible mathematical reasoning is filling the volume of the bottle in a discrete amount that is, it might entail images of change as occurring in discrete chunks (e.g., filling a bottle with successive cups of water) which is chunky reasoning. Therefore, chunky continuous covariation reasoning has a meaning of imagining a two covarying quantities changing their values continuously by adding the same amount of values at the end of the interval but not imagining variation within the interval. This type of reasoning can go with the covariational reasoning proposed by Confrey and Smith (1994). While smooth continuous covariation reasoning has a meaning of imagining two quantities covarying simultaneously; and at the same time thinking the two quantities varying continuously in a bit or vary smoothly and continuously in all in-between values over the interval, which is currently proposed by Carlso-Garwall (2012). The basic distinction between the two reasoning lays on their views of variation or change that is, in chunky reasoning there is starting and ending values (complete change) but in smooth reasoning there is starting but not ending values (keep moving or change in progress). When Thompson’s refuted Confrey’s idea (i.e. Confrey has chunky thinking towards two covarying quantities) he described the following statement which supported his idea of smooth continuous covariation. “Continuous variation in Confrey’s way of thinking about multiplicative change is very hard to imagine. If the underlying idea is that all multiplicative change happens by a split, then I do not know how to imagine the value of 2^x varying smoothly as I smoothly vary the value of x” (Thompson, 2008, p.39). The above quote shows that Thompson’s ideas of continuous covariation has smooth continuous reasoning but not chunky continuous reasoning. Though the two reasonings are co-related, since they emerge from continuous reasoning, nonetheless they are distinct in nature and they will produce two different mathematics. In the following section, I will discuss the two types of covariational reasoning.
Chunky Continuous Covariational Reasoning
Chunky continuous covariational reasoning is an imagination of change that focuses on how the change is happening at the start and end of the interval, or it is mathematical reasoning that happens by coordinating two end values (Carlso-Garwall, 2010, 2012; Confrey, 1991; Confrey and Smith, 1994; Thompson & Carlson, 2017). Figure 1 illustrates the nature of chunky continuous covariational reasoning.
Figure 1. Example of Chunky Continuous Covariational Reasoning
Figure 1 illustrates that changes in one variable x by 1 will coordinately increase the value of y by 2 (the left table of figure 1), or change by a factor of 3 (the right table of figure1). Confrey and Smith (1994) concluded that students who built their reasoning coordinately, as illustrated in figure 1, can produce a functional relationship between the two co-varying quantities. However, this type of reasoning was not helpful for students when students asked to think about what happens in between the endpoint of the interval (Carlso-Garwall, 2012; Thompson & Carlson, 2017).
For example, in Carlso-Garwall’s (2012, p.12) study for the per-capital policy problem, when two students were asked “to graph the first two seconds of a bank account under the given policy, beginning with an investment of $500”, the two students’ produced two different solutions to the problem. One of the subjects (i.e., Tiffany) produced an incorrect solution when she was asked what happened between the intervals because of her “completed chunks” reasoning. For instance, Tiffany had an imagination that the second value of the variable should be higher than the first variable value, but she didn’t think the variable values can continuously vary within the interval, due to this reasoning she produced incorrect curve or graph (see figure 2). “How do we fill in what goes on in-between points” (Carlso-Garwall, 2012, p.12), is what the researcher asks in order understand her reasoning for a very small chuck size value, and Tiffany repeated her chunk continuous process on a small scale and she described a process of finding the next value by using the first value (i.e., for any 1s chunk size length if she asked what goes on in-between 1s size length and she will definitely produce another 0.5s chunk size length interval within 1s chunk size length interval).
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