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Generalizing Actions of Forming: Identifying Patterns and Relationships Between Quantities
Halil Ibrahim Tasova Biyao Liang Kevin C. Moore
University of Georgia University of Georgia University of Georgia
In this paper, we illustrate and discuss two undergraduate students’ reasoning about quantities’
magnitudes. One student identified regularities regarding the relationship between two
quantities by focusing on successive amounts of change of one quantity (i.e., a pattern) while the
other attended to relative amounts of changes in both quantities (i.e., a relationship). We
illustrate that although reasoning about amounts of change is useful for making sense of the rate
of change in quantities, reasoning about relative changes in identifying a relationship between
quantities’ magnitudes is likely more productive in developing the concept of rate of change.
Keywords: Quantitative and covariational reasoning, Generalization, Rate of change.
Quantitative and covariational reasoning is critical to supporting students in understanding
major pre-calculus and calculus ideas (Ellis, 2007b; Confrey & Smith, 1995; Thompson, 1994,
2011; Thompson & Carlson, 2017). Moreover, Ellis (2007b) reported that quantitative reasoning
plays a significant role in students’ constructing productive generalizations. In this paper, we
characterize two undergraduate students’ generalizing actions during a teaching experiment
focused on modeling covariational relationships. We give specific attention to how the students’
engagement in covariational/quantitative reasoning differed and, in turn, how this difference led
them to generalize different regularities regarding a covariational relationship between two
quantities’ magnitudes. We report the generalizing actions of two students, with one student
operating with additive comparisons of amounts of change in one quantity, and the other student
operating with additive and multiplicative comparisons of amounts of change of two quantities
(i.e., relative changes and ratios). We also report the resulting identified regularities of these
ways of operating.
Background and Theoretical Framework
Quantitative and Covariational Reasoning
This study focuses on students’ generalizing actions involved in reasoning with relationships
between quantities in dynamic situations. We use quantity to refer to a conceptual entity an
individual construct as a measurable attribute of an object (Thompson, 2011). We also describe
students’ construction of quantitative structures by characterizing their quantitative operations
when determining a quantitative relationship. By quantitative operation, we mean the
conception of producing a new quantity from two others, and by a quantitative structure, we
mean a network of quantitative relationships (i.e., the conception of these three quantities;
Thompson, 1990, 2011). For example, someone can create a quantity as a result of additive
comparison of two quantities by answering the question, “How much more (less) of this is there
than that?”, whereas someone can create a quantity as a result of multiplicative comparison of
two quantities by answering the questions “‘How many times bigger is this than that?’ and ‘This
is (multiplicatively) what part of that?’” (Thompson, 1990, p. 11).
Furthermore, when students engage in a dynamic context that involve two quantities varying
simultaneously, they need to coordinate quantitative operations with covariational reasoning
(i.e., attending to how one quantity varies in relation to the other in tandem; Saldanha &
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Thompson, 1998; Carlson, Jacobs, Coe, Larsen, & Hsu, 2002). For example, in order to
determine a pattern of differences in a quantity’s variation in relation to the other, a student can
coordinate the variation of two quantities values or magnitudes and the variation of the resultant
difference quantity’s values or magnitudes (e.g., as two quantities increase, the difference of
these quantities decrease; see Mental Action 3 in Carlson et al., 2002).
Non-Ratio and Ratio-Based Reasoning
Many researchers have provided different ways of making sense of rate of change of one
quantity with respect to another. For example, some researchers (e.g., Carlson et al., 2002;
Confrey and Smith, 1994, 1995; Ellis, 2007b, 2011; Johnson, 2012, 2015b; Liang & Moore,
2017, 2018; Monk & Nemirovsky, 1994; Tasova & Moore, 2018) argued the importance of non-
ratio based reasoning, which is reasoning about amounts of change in one quantity in relation to
uniform changes in another quantity. For example, a constant rate of change in the perimeter of a
square with respect to changes in side length can be conceived by determining that amounts of
increase in the perimeter is “two centimeters” each time “if you increase both sides by point five
[centimeters]” (Johnson, 2012, p. 322).
There are also researchers (e.g., Confrey and Smith, 1994, 1995; Ellis, 2007b, 2007c, 2011;
Ellis, Özgür, Kulow, Williams, & Amidon, 2013, 2015; Johnson, 2015a) who have argued for
the importance of ratio-based reasoning (i.e., forming ratios of one quantity’s change to the other
quantity’s change) in making sense of the rate of change. For example, a constant speed of a
Clown can be conceived as a ratio of distance to time (i.e., “5cm:4s”; Ellis, 2007b, p. 472). We
note that these conceptualizations (mostly) included students’ reasoning with numbers. In this
paper, we expanded this body of literature by demonstrating ways in which students make sense
of rate of change in dynamic events and in graphs by reasoning with quantities’ magnitudes
independent of numerical values (see Liang, Stevens, Tasova, and Moore [2018] and Thompson,
Carlson, Byerley, and Hatfield [2014] for a detailed discussion on magnitude reasoning).
Because reasoning with quantities’ magnitudes does not necessitate reasoning with specified
values of the quantities, we conceptualize “ratio-based reasoning” as reasoning with a “quotient
[that] entails a multiplicative comparison of two quantities with the intention of determining their
relative size” (Byerley and Thompson, 2017, p. 173). We aim at demonstrating students’
generalizing actions by characterizing how they operate with magnitudes within a complex
quantitative structure.
Generalizing Framework
Building on Ellis’ (2007a) taxonomy of generalizations, Ellis, Tillema, Lockwood, and
Moore (submitted) introduced a generalization framework involving three major forms of
students’ generalizing—relating, forming, and extending. Students’ generalizing actions of
forming occur within one context, task, or situation. This type of generalizing action includes
students searching for similarity and regularity across cases, isolating constancy across varying
features by establishing a way of operating that has the potential to be repeated, and identifying a
regularity across cases, numbers, or figures. In this paper, we are using this framework to
illustrate two students’ generalizing actions of forming by focusing on their establishing ways of
operating and identifying regularities as they relate to covarying quantities.
Method
The data we present in this paper is from two semester-long teaching experiments (Steffe &
Thompson, 2000) conducted at a large public university in the southeastern U.S. A common goal
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of both teaching experiments was to investigate undergraduate students’ mental actions involved
in reasoning with dynamic situations, magnitudes, and graphs from a quantitative and
covariational reasoning perspective. In this paper, we focus on a student, Lydia, who at the time
of the study, was a pre-service secondary mathematics teacher in her first year in the program,
and another student, Caleb, who was a sophomore majoring in music education. Lydia
participated in 11 videotaped teaching experiment sessions and Caleb participated in 14, each of
which was approximately 1–2 hours long. We transcribed the video and digitized these students’
written work for both on-going and retrospective conceptual analyses (Thompson, 2008) to
analyze their observable and audible behaviors (e.g., talk, gestures, and task responses) and to
develop working models of their thinking. We choose to present these two cases here because the
students’ generalizing actions including their established ways of operating and identified
regularities are cognitively distinct, and thus are worth documenting and contrasting.
Analysis and Findings
In this paper, we illustrated two students’ generalizing actions—by focusing on their ways of
operating and identified regularity as they determined the covariational relationship between two
quantities.
Lydia’s Generalizing Actions
First, we characterize Lydia’s activities in Taking a Ride to discuss her generalizing actions
of establishing a way of operating (see Tasova & Moore [2018] for detailed account of her
generalizing activity). To start with, we presented Lydia an animation of a Ferris Wheel rider
that was indicated by a green bucket rotating counterclockwise from the 3:00 position (Desmos,
2014). Then, we asked her to describe how the height of the rider above the horizontal diameter
changes in relation to arc length it has traveled. After reasoning about directional change in
height in relation to arc length (i.e., height is increasing as the arc length increases in the first
quarter of rotation), she engaged in partitioning activity (Liang & Moore, 2017, 2018) in order to
investigate how height changes in relation to arc length. Namely, she used the spokes of the
Ferris wheel (i.e., each of the black bars [see Figure 1a] connecting the center of the wheel to its
edge) to partition the Ferris wheel into equal arc lengths, and then she drew corresponding
heights (see the green segments in Figure 1a and Figure 1b).
(a) (b) (c) (d)
Figure 1. Lydia engaging the Taking a Ride task. Figure 1b and 1d were designed for the reader.
With support from the teacher-researcher’s (TR) questioning, Lydia constructed successive
amounts of change in height (i.e., circled in blue seen in Figure 1c and blue segments in Figure
1d) that corresponded to successive uniform incremental changes in arc length. That is, Lydia
established a way of operating that involved the construction of a new quantity (i.e., amounts of
change in height) and associated partitioning activity. We inferred from her activity that Lydia
was constructing the difference of every two consecutive height magnitudes (i.e., D||H1||, D||H2||,
and D||H3||, see blue segments in Figure 1d) corresponding to the magnitude of arc length that
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accumulates in equal increments; Smith III & Thompson, 2008; Thompson, 1990). This served
as evidence that she was operating with additive comparisons among the accumulated height
magnitudes at successive states (i.e., ||H1||, ||H2||, and ||H3||, see green segments in Figure 1b)).
What’s more, she additively compared the amounts of change magnitudes in height. Namely, she
concluded D||H1|| > D||H2 > D||H3||.
We note that in her additive comparison, Lydia was not interested in measuring how much
one quantity’s magnitude exceeded (or fell short) of another quantity’s magnitude. Instead, her
quantitative operation included a gross additive comparison (Steffe, 1991) between the amounts
of change within a quantity (e.g., D||H3|| being “smaller” than D||H2||). From this activity,
therefore, we inferred that Lydia made a gross comparison of the differences, which is a more
complex quantitative reasoning because this requires relating results of quantitative operations
(i.e., an additive comparison of the results of two additive comparisons). After engaging in
repeated additive comparisons, Lydia was able to search for pattern in those quantities’
variation. With the recognition in the pattern of differences (i.e., decreasing change in height
along with those equal partitioning in arc length as shown in Figure 1c and 1d), Lydia had
identified the regularity in how height’s magnitude changes in relation to arc length in the first
quadrant, stating “as the arc length is increasing... [the] vertical distance from the center is
increasing ... but the value that we’re increasing by is decreasing.”
Caleb’s Generalizing Actions
We demonstrate Caleb’s generalizing actions when engaging in the Changing Bars Task,
which involved a simplified version of Ferris wheel situation (i.e., a circle) and six pairs of
orthogonally oriented bars (see Figure 2). On the circle, the red segment represents the
magnitude of the riders’ height above the horizontal diameter and the blue segment represents
the magnitude of the rider’s arc length traveled from the 3 o’clock position. Caleb was able to
move the end-point (i.e., the rider) along the circle between the 3:00 position to the 12:00
position. We asked Caleb to choose which, if any, of the orthogonal pairs accurately represents
the relationship between the height and the arc length of the rider as it travels.
Figure 2. Changing Bars Task (numbering and locations of the six pairs was edited for readers).
In this section, we report Caleb’s generalizing actions that involved him establishing ways of
operating that entailed additive and multiplicative comparisons. We note that identifying these
different operations does not imply that Caleb engaged in them in order. We believe that Caleb’s
reasoning involving additive and multiplicative comparisons was internally coherent and he
could make claims about either one depending on the TR’s questioning. Our goal of making such
distinction was to characterize his different ways of operating and contrast his ways of operating
with those of Lydia.
Additive comparison of amounts of change. Caleb started with comparing the amounts of
change in arc and amounts of change in height as the dynamic point traveled a small distance
from the 3:00 position. He stated that, “...at the very beginning, ... the height above the center
and the distance traveled from 3:00 position should be similar.” This way of operating was
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repeated several times during his generalizing actions with use of slightly different verbal
statements. For example, in a later conversation, he stated “at the beginning of the path [referring
to 3:00, see Figure 3a], ...the rate at which the height increases should be almost equal to the rate
at which the distance it's traveled.” We note that although he used the word of “rate,” we infer
that he meant amount of change in height and arc length. By repeating the same way of operating
in 12:00 position (i.e., new case in the first quarter of rotation), Caleb further stated that:
...from this point [pointing to the point denoted in orange in Figure 3b] ... to this point
[pointing to 12:00 position in Figure 3b], the height barely changes [green segment in Figure
3b and Figure 3c (i.e., D||H3||)], but you’re still traveling a fair distance around the circle
[blue annotation in Figure 3b and blue segment (i.e., D||A3||) in Figure 3c].
(a) (b) (c)
Figure 3. Recreation of Caleb’s activity in the Changing Bars task.
From his activity, we infer that Caleb’s established way of operating included an additive
comparison of D||H1|| with D||A1|| near the 3:00 position (i.e., D||H1|| is almost equal to D||A1||) and
of D||H3|| with D||A3|| near the 12:00 positon (i.e., D||H3|| is smaller than D||A3||). Similar to the
case of Lydia, we did not have evidence that Caleb constructed the difference between amounts
of change in two quantities (e.g., how much D||H3|| exceeded of D||A3||) beyond a gross additive
comparison between the amounts of change in each quantity (Steffe, 1991).
As the teaching experiment proceeded, he isolated a constant feature of the relationship
between the amounts of change in height’s magnitude and the amounts of change in arc length’s
magnitude across the first quarter of rotation. He stated that “from any point to any other point
along this stretch [referring to the first quarter of rotation], the amount that the red line [i.e.,
height’s magnitude] changes should always be smaller than the amount that the blue line [i.e., arc
length’s magnitude] changes.” Therefore, we infer that Caleb isolated a constant feature across
varying features of the relationship between D||H|| with D||A|| without reaching the final stage of
fully describing an identified regularity across the first quarter of rotation (e.g., D||H|| becomes
smaller relative to D||A|| as the rider travels from 3:00 positon to 12:00 position). It is important
to note that, however, Caleb knew that “when we’re looking down here [refers to 3:00 position]”
the relationship between D||H1|| and D||A1|| “should be vastly different from” the relationship
between D||H3|| and D||A3|| (see Figure 3c).
Eventually, Caleb identified a regularity regarding the relationship between D||H|| and D||A||
across the all cases in the first quarter of rotation. He stated that “the further you move away
from the 3:00 position, the more variance there would be between the red (i.e., D||H||) and the
blue lines (i.e., D||A||)” and by “variance” he meant that D||A|| became much bigger than D||H|| as
the dynamic point approached the 12:00 position.
Multiplicative comparison of amounts of change. Caleb also established a way of
operating that involved multiplicative comparisons between D||H|| and D||A||. He stated that “As
we approach this point right here [refers to 12:00 position], the ratio of the rate at which the
22nd Annual Conference on Research in Undergraduate Mathematics Education 606
height increases to the rate or to the distance we’ve traveled around...the circle, um, is at its
smallest...” This way of operating was also repeated several times during his generalizing
activity—both in the circle situation and in six pairs of bars. For example, near 12:00 position, he
established that there is a “...1 to 2.5 or 1 to 3 ratio in the amount that you change the red line’s
length [i.e., height’s magnitude] decreases to the blue line length [i.e., arc length’s magnitude]
decreasing.” We infer that Caleb constructed a quantity as ratios of D||H|| to D||A|| across the first
quarter rotation, and anticipated that the ratio gets smaller as the rider travels from 3:00 position
to 12:00 position. His way of operating that entailed multiplicative comparisons of quantities and
his identified regularity regarding the relationship between height and arc length became evident
in his graphing activity, which we report next.
(a) (b) (c)
Figure 4. (a) Caleb’s initial graph, (b) a resulting drawing of Caleb’s partitioning activity, and (c) a recreation of
Figure 4b for readers
Caleb’s graphing activity. The researchers then asked Caleb to produce a graph that
represents the relationship between height and arc length. He constructed the concave down
graph shown in Figure 4a. To interpret his displayed graph in terms of amounts of change in
height and arc length, Caleb engaged in partitioning activity (Liang & Moore, 2017, 2018) to
construct incremental changes that represented amounts of change in height (i.e., D||H1||, D||H2||,
and D||H3|| in Figure 4c; also see yellow vertical segments in Figure 4b) in relation to uniform
changes in arc length (i.e., D||A1||, D||A2||, and D||A3|| in Figure 4c; also see yellow horizontal
segments in Figure 4b). He then assigned estimated values for each segment to indicate its
magnitude (i.e., D||A1|| = D||A2|| = D||A3|| = 1; D||H1|| = .85 > D||H2|| = .5 > D||H3|| = .197 and
constructed ratios of each corresponding pairs, writing “.85/1”, “.5/1”, and “.197/1” (see Figure
4b). Caleb also operated on these ratios by additively comparing them, anticipating that these
ratios should decrease—“.85/1>.5/1>.197/1”. This suggested that Caleb continued and
generalized his ways of operating in the circle and bar situation to the graphical contexts and
identified a regularity that the ratios of successive pairs of amounts of change in height and arc
length should decrease as the rider travels in the first quarter of rotation. Caleb was also able to
extend his ways of operating to non-uniform intervals. Namely, he anticipated that when
increments of arc length are not equal (see his partitions in light blue in Figure 4b and his
estimated values for each increment), the same regularity should hold, writing “.8/1.9<.75/.82”
(see Figure 4b) without calculating the resulting value of the ratios.
Discussion
We focus on two students’ generalizing actions by giving attention to their ways of operating
and identified regularity. In establishing ways of operating, both Lydia and Caleb first
constructed differences (i.e., amounts of change) in magnitudes of height (i.e., D||H1||, D||H2||, and
22nd Annual Conference on Research in Undergraduate Mathematics Education 607
D||H3||) in relation to change of arc length’s magnitude (i.e., D||A1||, D||A2||, and D||A3||). However,
the way they operated on those differences differed. For example, they both engaged in
quantitative operation of additive comparisons; however, the operands that they considered in
their quantitative operations were different. That is, Lydia additively compared the successive
amounts of change in height’s magnitude, whereas Caleb additively compared amounts of
change in height’s magnitude with the corresponding amounts of change in arc length’s
magnitude. Therefore, the operands for Caleb were differences in height and differences in arc
length (i.e., D||Hn|| and D||An||, where n=1, 2, and 3), as opposed to Lydia whose operands were
differences of height in successive states (i.e., D||Hn|| and D||Hn+1||, where n=1 and 2). We
conjecture that the way of additive comparison in Caleb’s case might be more productive for
generalizing the rate of change in height with respect to arc length since such comparison
afforded him to anticipate a resultant ratio of differences in height and differences in arc length
(i.e., D||Hn||/D||An||).
We also find that these two students’ different ways of operating led them to identify
different regularities regarding the similar situations. Lydia searched for the pattern (Ellis,
2007b) by making within-measure additive comparisons among heights in different states. Thus,
she identified a pattern of how amounts of change in the height decrease as the arc length
increases. Caleb searched for the relationship (Ellis, 2007b) by making between-measure
multiplicative comparisons between height’s magnitudes and arc length’s magnitudes. Thus, he
identified a regularity of relative change of the height with respect to the arc length decreases as
the rider travels. We conjecture that this way of operating (i.e., multiplicative comparison
between changing quantities’ magnitudes) and the resultant identified regularity may afford
students to develop productive understandings of rate of change.
We want to point out that, when additively comparing the ratios in justifying his identified
regularity, Caleb’s engagement with numbers does not imply that he performed arithmetic
operations in a sense that he wanted to evaluate the quantities’ values. We infer that the reason
he assigned numbers to quantities’ magnitudes is that he needed to “propagate information”
(Thompson, 2011, p. 43) in order to deal with the complex quantitative situations. Thompson
(2011) claimed that propagation can be made under the conditions of being aware of (i)
quantitative structure and (ii) “numerical operations to perform to evaluate a quantity in that
structure” (p. 43). Even though Caleb did not perform numerical operations to evaluate
quantities, he satisfied the conditions of propagation. That is, he used numbers as intuitive
measurements of quantities’ magnitudes and he was aware of the quantitative structure.
Moreover, Caleb’s uses of numbers were necessary for him in order to compare the relative size
of two quantities’ magnitudes. Part of this necessity comes from the fact that there was no way
for him to visually represent the magnitude of a quantitative ratio. That is, Caleb used estimated
numbers to reason about the relationship between magnitudes of hard-to-visualize quantities, and
then re-interpreted this relationship between values in the context of quantitative structure in
order to propagate information about the relationship between quantities’ magnitudes. To
confirm if this is the case or to characterize the nature of this reasoning, we believe that future
research is necessary.
Acknowledgments
This paper is based upon work supported by the NSF under Grant No. DRL-1350342 and
DRL-1419973. Any opinions, findings, and conclusions or recommendations expressed in this
material are those of the authors and do not necessarily reflect the views of the NSF.
22nd Annual Conference on Research in Undergraduate Mathematics Education 608
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