Chapter 4 DATA ANALYSIS AND RESULTS
4.1 Students’ personal definitions of parametric function
On question 3 of the final exam, students were asked to explain in their own words what is a parametric equation or parametric function (see Appendix B for details). In this report, it is considered a priori that a student with a strong understanding of parametric function should have a personal definition that relates, in some way, to the following operational definition1 of parametric function: At a value of the independent variable (or parameter) t, there is a value of the dependent variable x and a value of the dependent variable y, which together form a unique quantity (x, y). Based on this operational definition, each student’s definition of parametric function was analyzed according to whether it satisfied the following four criteria:
(i) Identify an independent variable (or parameter) t. (ii) Identify a dependent variable x.
(iii) Identify a dependent variable y.
(iv) Coordinate x and y to form an ordered pair (x, y).
Despite the fact that the parametric function concept was defined in class as a function that accepts a value oftas an input and returns an ordered pair (x, y) as an output, students rarely formulated their definition explicitly in such terms. Nevertheless, it was possible to relate aspects of students’ definitions to at least some of the four criteria of the above operational definition. Based on the analysis, students’ definitions were grouped into three categories: pre-action, action, and process.
1Since the focus of this study was on single-variable calculus students’ understanding of parametric
If a student possesses a personal definition of parametric function that satisfies criteria (i)–(iii) of the above operational definition, then he or she is considered to have at least a foundational understanding of the parametric function concept, even if his or her definition does not meet criterion (iv). On the other hand, if a student’s definition does not satisfy one or more of criteria (i)–(iii), then the student has little understanding of the parametric function concept. Such definitions are considered to be at the pre-action level of understand- ing of parametric function. Out of fifteen, two students did not give a response, and four students gave a response that was categorized as pre-action.
The following definition by Griffin is an example of a pre-action level definition: A function that takes into account time as a variable.
Griffin’s definition could possibly refer to a dependent variable as a function of time, but it is too vague. At most, his definition satisfies criteria (i) and (ii) of the operational definition of parametric function. Since there is no indication that he was thinking of two variables x and y that are depending on a parameter, sayt, that should be further coordinated to form an ordered pair, his definition was classified to be at the pre-action level.
Peggy also gave a definition that satisfied at most criteria (i) and (ii) and, thus, was classified as pre-action. She said wrote:
A 2-dimensional function with respect to t and x.
It is not clear what Peggy meant by her definition. She could be saying that y is a function of both t and x, but this is not necessarily true, since there is no requirement that y be a function of x on a curve defined parametrically. However, she does indicate that something is a function with respect tot. Therefore, her definition was considered to satisfy criteria (i) and (ii).
Some definitions appeared to have some merit on the surface, but a closer analysis found them to be pre-action, such as the following definition by Bailey:
A parametric equation is a representation of position vs time express[ed] on a radial graph.
The issue with Bailey’s definition is that it is not clear whether he meant “position” to be a one- or two-dimensional value. The textbook used for the calculus sequence at the university where this study was conducted initially introduces particle movement in the context of straight-line motion:
On the line of motion we choose a point of reference, a positive direction, a negative direction, and a unit distance. [. . .] There is no loss of generality in taking the line of motion as the x-axis. (Salas et al., 2007, p. 209)
With this in mind, Bailey’s definition is no more general than the one-dimensional case de- scribed above, which satisfies only criteria (i) and (ii) of the operational definition. Moreover, it is not clear what he meant by “radial graph.”
Unlike Griffin and Bailey’s definitions, which were categorized as pre-action because they did not satisfy criterion (iii) of the operational definition, Cindi’s definition was categorized as pre-action because it did not satisfy criterion (i):
A parametric equation is a representation of the path of a particle which forms a curve and can be graphed.
Cindi’s statement, “path of a particle which forms a curve,” seems to refer to two- dimensional motion, possibly satisfying criteria (ii) and (iii) of the operational definition. However, Cindi’s definition does not explain how the path of the particle relates to a param- eter t.
If a student’s definition, in some way, satisfies criteria (i)–(iii) or (i)–(iv) of the opera- tional definition of parametric function, then the definition was further analyzed to determine whether it qualifies as an action- or process-level definition. A definition was considered to be action-level if it expressed a reliance on superficial properties of parametric equations without expressing how parametric equations contribute to a more general process of trans- forming a value of tinto a value of xand y. Note that a definition would also be considered action-level if it relied on a particular example. However, none of the students in this study gave such a definition.
Only one student, Kevin, gave a parametric function or equation definition that ap- peared to be action-level:
When x & y are with respect to t.
Kevin’s definition satisfies criteria (i)–(iii) of the operational definition by identifying variables x, y, and t. However, it does not clearly utilize the function concept. By stating, “with respect to t,” it is possible that Kevin’s definition is in reference to a superficial property of parametric equations—the variables x and y are equal to expressions involving t. Therefore, with this in mind, Kevin’s definition was interpreted as indicating an action conception of parametric function.
A student’s definition was categorized as process-level if, in addition to satisfying criteria (i)–(iii), it described some sort of general process of x and y being determined by t. Seven students gave definitions that were categorized as process-level, two of which satisfied only criteria (i)–(iii). For example, Alex defined a parametric function as follows:
Parametric function is a function in which x and y [are] dependent on the pa- rametert, which is time.
Alex’s definition states thatxandyare dependent ont, satisfying criteria (i)–(iii) of the operational definition. Unlike Kevin’s “with respect to t” definition above, which suggests x and y are represented in terms of t, Alex’s “dependency” definition suggests functional relationships between tand x and betweent andy that are external to their representation. It is not surprising that Alex’s interpretation of the parameter t is limited to time, as that was the application of tthat was frequently emphasized in class. Viewing tas time does not make his definition any less of a process-level definition, since it still describes a process of an application of parametric functions.
Out of the seven students whose parametric function definitions were categorized as process-level, five gave a definition that satisfied all of criteria (i)–(iv) of the operational definition. Ron, for example, gave the following definition of parametric equation:
A parametric equation is one in which one measures not only the (x, y) coordi- nate points, but more importantly the specific time that the given graph passes through those points. It displays the time as each equation passes through its natural destination.
Ron’s definition refers in general to values of x, y, and t, satisfying criteria (i)–(iii) of the operational definition. Furthermore, it also coordinates the values of x and y to make an ordered pair (x, y), thereby satisfying criterion (iv). A process conception is indicated by the use of the phrase, “passes through,” which suggests that Ron has a dynamic perception of how an xy-curve is generated by a parametric equation.
Of the five students who gave definitions that satisfied all of criteria (i)–(iv), only one student, Lee, gave a close to formal definition of parametric function:
It [is] a new type of function where you have one input = t and your one out put is an ordered pair or a set of values, i.e. (x(t), y(t)) or (v(t), h(t)). It is determining the position, height, or volume of something based on the value of an implit varible [sic].
Lee’s definition is an explicit generalization of the definition of function. Her definition emphasizes that the output is an ordered pair instead of a single number. Furthermore, her definition is indicative of a process conception because the input, transformation, and output were general (Breidenbach et al., 1992).
One student, Whitney, gave a potentially correct response that did not fit the criteria used to discern students’ understanding. She wrote:
A parametric equation is a function in respect to time. It’s more 3-dimensional, while a regular y and x function will only be a 2-dimensional graph.
Whitney’s definition appears to satisfy criteria (i)–(iii) of the operational definition. When she states,“it’s more 3-dimensional,” she might be referring to a space curve. This is consistent with Euler’s use of parametric equations in Book II of his 1748 Introductio
Table 4.1 Students’ personal parametric function definition.
Definition criteria
Subject (i) (ii) (iii) (iv) APOS level
Alex * * * Process Bailey * * Pre-action Cindi * * Pre-action Griffin * * Pre-action Hannah * * * * Process Kevin * * * Action Lee * * * * Process Mary * * * * Process Nicole * * * * Process Peggy * * Pre-action Ron * * * * Process Sam * * * Process Whitney * * * 3-D
in Analysin Infinitorium. What is interesting is that space curves were never discussed in class, as the topic is reserved for Multivariable Calculus. Despite this fact, it seems that Whitney developed her own understanding of parametric function as a space curve. Although Whitney’s definition seems to be higher than a pre-action definition by satisfying criteria (i)–(iii), it is not clear how to classify her definition in terms of APOS. Therefore, it is categorized simply as 3-dimensional.
In summary, excluding Whitney’s definition, four out of twelve responses were consid- ered to be pre-action definitions, one was considered to be an action-level definition, and seven were considered to be process-level definitions. Table 4.1 summarizes the results of this section according to the criteria of the operational definition satisfied by each students’ personal definition of parametric function, as well as its corresponding APOS classification.
4.2 Students’ reasoning about real-valued functions and parametric functions