Approaches to the Derivative in Korean and the U.S. Calculus Classrooms
Jungeun Park
University of Delaware
This study explored how one Korean and one U.S. calculus class defined the word
“derivative” as a point-specific object through the limit process on the difference quotient, and as
a function on its domain. The analysis using Commognitive approach showed that both class used
similar visual mediators for the limit process/object, but addressed different components of the
definitions; Discussion of the derivative as a function before it was defined were frequently found
in the U.S. class but rarely found in the Korean class; Words for the derivative at a point, and
words for the derivative as a function explicitly differed in the Korean class compared to the U.S.
class; and the derivative was first defined as a function through correspondence between x-value
and the derivative value in Korean class, but through expansion of x values from a number to
variable and corresponding changes in the U.S. class.
Keywords: commognition, calculus, derivative, language, teaching.
Introduction
Calculus is considered as a first college level mathematics course that students encounter in
the United States (U.S.). However, they often learn Calculus in high school as a form of
Advanced Placement (AP) or a regular course. Similarly, in South Korea, most high school
students who are on the college track learn basic calculus concepts because they are included in
the national curriculum. This study looks at calculus lessons about the derivative from South
Korea and the U.S. to explore how language specific terms for “the derivative at a point,” and
“the derivative of a function” are discussed in their lessons. In contrast to the English terms, both
of which include “derivative,” the corresponding terms in Korean, “Mi-bun-Gye-Sum”
(translated to “differential coefficient”) and “Do-ham-su” (translated to “leading function”), do
not include a common term. Specifically, this study addresses the following research question:
How is the derivative realized as an object at a point and an object on an interval in one high
school class in South Korea, and one AP calculus class in the United States?
This study adopted Sfard’s (2008) commognitive approach as an analytical tool. The purpose
for the analysis was not to compare the way the derivative is taught in two countries, but to apply
the same analytical lens in mathematical discourse involving two languages, focusing on how the
two terms were addressed as a number or a function in each language.
Theoretical Background
Various studies considered languages as a factor for students’ mathematical thinking. Some
studies explored the relationship between specific language features and students’ performance
in mathematics in Chinese (e.g., Wang & Lin, 2005), Korean (e.g., Kim, Ferrini-Mundy, &
Sfard, 2012), and Irish (e.g., Ni Riordain, 2013). Others considered cultural differences including
languages in teaching and learning such as indigenous non-native English speaking students’
learning, whose teachers were native English speakers (e.g., Favilli, Maffei, & Peroni, 2013;
Russell & Chernoff, 2013); they revealed that the language difference was a main factor of
teachers’ decision on the content and the level of difficulty for the class, and teachers’
knowledge about their students’ native language was a key component of the teacher’s
knowledge for teaching mathematics. The current study also explores teaching mathematics in
different languages, but also takes other means of communication into consideration such as use
of visuals while exploring discussions of “derivative” in Korean and English by adopting the
commognitive approach (Sfard, 2008) as a discourse analysis framework. It combines cognition
and communication, and explains mathematical thinking through one’s discourse through the
four characteristics: word use, visual mediators, routines, and endorsed narratives (Table 1).
Table 1.
Features of Mathematical Discourse in Commognitive Approach
Feature
Descriptions
Further Descriptions
Word use Use of words
Different speakers can use a word differently. It is an
signifying
"all-important matter" because "it is responsible to a great
mathematical objects extent for how the user sees the world" (Sfard, 2008, p. x).
Visual
Non-verbal means of Because ways people attend to visuals depend on contexts,
Mediators communication
mediators need to be viewed as part of the thinking process,
not auxiliary means of pre-existing thought.
Routines
Well-defined
Patterns can be found in speakers' use of words and visuals,
repetitive patterns
or in the process of creating and endorsing narratives.
Endorsed Utterances that
Students' endorsed narratives are often different from what
Narratives speakers endorse as the professional mathematics community endorses as true
true
(e.g., “Multiplication makes bigger” changes to
“Multiplication can make smaller.”).
Note. Adopted from Park (2015, p. 234)
The first two characteristics are also considered as realizations. Sfard (2008) defined
realizations of a signifier (either a word or visual mediator) as perceptually tangible entities that
share Endorsed narratives, and mathematical object as a collection of these realizations. For
example, a word “function” can be realized with a word, “mapping” or “relation,” graph,
equation, or gesture for a curve or straight line. With these four discursive characteristics, the
development of various mathematical objects has been examined (e.g., Kim et al., 2012; Sfard,
2008). The current study addresses the development of “the derivative” through its realizations –
words and visual mediators – in discourse from a Korean and an American calculus class.
The derivative, which is commonly mediated with symbols, is realized as a number (e.g.,
f '(a) = lim
h→0
f (a + h) − f (a)
) and as a function (e.g., f '(x) = lim f (x + h) − f (x) ). The realization of “the
h→0
h
h
derivative” includes several process and object transitions. First, for “the derivative at a point,”
the difference quotient (DQ) is considered as an initial object, and then the process of the limit
over smaller and smaller intervals is applied, and then the final object from this process is be the
derivative at a point. The limit process is often mediated with graphs of multiple secant lines;
symbols for the limit; words such as “as h approaches 0, the DQ approaches...”; numbers for DQ.
Through this process, the “derivative” is objectified as a number. Then, for “the derivative
function,” the derivative at a point is considered as an initial object, and the derivative process of
finding the derivative at every point is applied. This process can be mediated with several
tangent lines; several dots on an x-y plane; symbols including different letters (e.g., x); multiple
numbers for the derivative. The derivative of a function would be objectified from this process.
Method
The purpose of this study is to explore mathematical discourse about the derivative from one
Korean and one American calculus classroom. To this end, lessons for the derivative were
videotaped 7 times for the U.S. classroom and 10 times for the Korean classroom when the
teachers started the derivative unit. The video camera was located in the back of the classroom to
minimize the interruption, and field notes were taken. The two instructors also participated in a
30-minute interview about what they believe as important to teach in their class. The interviews
were also videotaped and used as a complementary data for the classroom data.
Participants
One Korean high school teacher, Mrs. Kim, and one U.S. AP calculus teacher, Mr. William
(pseudonyms) (Table 2) were recruited via email sent to the group of teachers recommended by
mathematics education professors at the researcher’s institution.
Table 2.
Teachers’ backgrounds and classes
First language
Degrees
Teaching experience
Calculus teaching
Number of Classes/week
Teaching method
Number of Students
Mrs. Kim
Korean
BS in Mathematics
13 years
3 times
Three 50-minute classes
Blackboard and chalk
34
Mr. William
English
BS and MS in Mathematics
10 years
7 times
Five 90 minute classes
SmartBoard, interactive graphs
32
Coding Scheme
Videos from each class were transcribed, and then the excerpts including realizations of the
derivative were selected. Excerpts for “the derivative at a point” were coded as a) Initial Object:
Location where the initial object was defined; b) Initial Object where the limit process was
applied; c) Limit Process:Location where the limit process was applied; d) Limit process:Change
in the object; e) Limit object from the limit process, and f) Limit Object:Location: where the limit
object was defined. Table 2 shows an example for such realizations. The first column shows
codes for the limit process/object. The first row shows Mrs Kim’s words or visual mediators.
Table 3.
Words and visual mediators in episode about limit process/object
Codes
Mrs. Kim’s Words
Initial Object
ARC [is] the slope of the line
Initial Object: location Passing through two points apart.
Limit Process: location How do we move the points? Closer, closer…
Limit Process: Change What happens to the slope? It’s getting smaller and
smaller (drawing three secant lines and arrows)
Limit Object
We see the slope, the differential coefficient, and
the instantaneous rate of change.
Final Object: location At one point
Graph [Fig. 2]
Top secant line
(a, f(a)), (b, f(b))
→a
Three secant
lines, arrow
Tangent line
A point a
Figure 1. Realization of the differential coefficient with graphs
The excerpts about the derivative as a function were coded as a) Limit Object: the initial
object where the derivative process was applied (e.g., the derivative at a point x=a); b)
Derivative Process-location: x on the derivative process (e.g., as a vary); c) Derivative
Process-change: change in the object (e.g., multiple numbers); d) Final object: the final object
from the derivative process (e.g., graph or equation). The derivative process was categorized by
five types: (a) Expansion, when excerpts expands from a number to a universal value on the
location (e.g., “a certain point” to “any points”) and/or the change (e.g., multiple tangent lines);
(b) Correspondence, when excerpts map x values to the derivative function; (c) Variation, when
excerpts include description of how the derivative varies on an interval (e.g., increasing); (d)
Universality, when excerpts include explicit realization of the derivative defined at every point
where the original function is differentiable; and (e) Specification, when excerpts include a
transition from the derivative of a function to the derivative at a point (e.g., substitution).
Results
The results present realizations of “differential coefficient” and “leading function” in Mrs.
Kim’s class, and “the derivative at a point” and “the derivative of a function” in Mr. William’s
class. Only one of the visualizations for these cases was presented here due to the limited space.
Limit Process and Object in Mrs. Kim’s Class
Among 44 episodes in Mrs. Kim’s lessons about the differential coefficient, 25 of them
included the limit process. The realizations for the limit process included words with symbols
(11 of 25), symbols (7), words with graphs (5), and words (2) (Figure 2). These realizations
highlighted the limited visual mediation of the limit process, but consistent use of words for the
initial and final objects. First, most episodes only included the realization of the locations of the
limit process with dynamic words or points on the curve without the change for the limit process.
The change was addressed only once with graphs of secant lines. Second, uses of different visual
mediators for the limit process were not consistent; two points on the curve were used for the
location of the limit process, which was not consistent with the two x values in the symbol ( lim ).
x→a
Third, in most cases involving symbols, both the location and change of the limit process were
implicit; in the episodes about evaluating “differential coefficient” using the definition (e.g.,
Let f (x + y) = f (x) + f (y) + 3xy, f '(0) = 3. Find f '(−4). ), she transitioned from the difference quotient
to the differential coefficient only with symbols ( lim ) without other explicit mediation of the
x→a
limit process. It should be also noted that, Mrs. Kim consistently used the term “the differential
coefficient” for the derivative as a point-specific object throughout the derivative lessons
including the ones about the derivative process (e.g., the slope of a linear function, and the slope
at any point on a graph). In most episodes, words mediating the initial and final objects were
consistent (e.g., “slope” for both, or “average rate of change” and “the rate of change”). The term
“differential coefficient” was defined synonymously with the terms for the final object.
Figure 2. Realization tree for limit process/object in Mrs. Kim’s class
Derivative Process and Object in Mrs. Kim’s Classroom
Among 20 episodes about the leading function in Mrs. Kim’s lesson, 17 addressed the
derivative process, consisting of correspondence and expansion (6), computation (6),
specification (4), and universality (1). Realizations of “the leading function” highlighted the
limited use of visual mediators for the derivative process, and explicit word use of “differential
coefficient” for the initial object, and “leading function” for the final object. First, the transition
from the differential coefficient to the leading function was explained with symbols and words
for the derivative process as correspondence; a diagram mapping a specific x value to “the
differential coefficient,” which was first realized as a “function” and then “the leading function.”
The two terms, “the differential coefficient” and “the leading function” were not directly related
until “the differential coefficient ” was computed as a value of “the leading function” with
symbols (i.e., specification). “The differential coefficient” and “the leading function” were
realized as the “same,” only through symbols. Graphs were not used in any types of derivative
process. Mrs. Kim mainly computed the leading function mainly following the limit
process/object without mentioning any types of the derivative process. Her word use separating
“the leading function” and “the differential coefficient” was also found. For example, she first
described “the differential coefficient” of a linear function as “same” “always,” and then used
“the leading function” for the constant function. Also, she used “the leading function” for the
equation, and “the differential coefficient” for its value at a point.
Limit Process and Object in Mr. William’s class
Among 17 episodes in Mr. William’s lessons about the derivative at a point, 11 of them
included the limit process. The limit process was mediated with words (3), graphs, gestures and
words (3), and symbols and words (5). Different components of the limit process/object was
included in the discussion when the different types of realizations were used. First, when words
and graphs were used, realizations included both the location and change for the limit process.
However, when the symbols were used, the realization mainly included the location for the limit
process besides one case where the words for the computation process mediated both the location
(e.g., “substitute,”) and the change (e.g., “zero over zero”). Second, the ways the limit process
was discussed also differed across different mediators. The location for the limit process was
mediated with words, gestures, and dynamic graphs for the values on the x-axis (e.g., “ two x
values,” horizontal hand gestures, and “h” with numbers approaching 0), which corresponded to
the limit notation in symbols (e. g. , lim!→! ). However, on the stationary graph, his gestures for
the location were mediating two points on the curve, not on the x-axis. Third, mismatches among
realizations with different visual mediators were observed. Specifically, his gesture mediating
the location for the limit process on a graph was not consistent with what is drawn on the graph.
Also, the letter for a moving point included in a dynamical graph (e.g., (x+h) approaching x)
differed from the letter included in symbols (e.g., x approaching a) that the graph mediated.
Regarding word use, words for the initial object, the process for the limit process, and the final
object were consistent in most cases (e.g., ARC and IRC; the slope for both the initial and final
objects). However, there were several cases, the word “slope” was inconsistently used to mediate
only either the initial object, the process for the limit process, or the final object. In some cases,
secant lines and tangent lines mediated the limit process/object without word “slope.”
Derivative Process and Object in Mr. William’s Classroom
Among 16 episodes about the leading function in Mr. William’s lesson, 15 addressed the
derivative process, consisting of correspondence and expansion (1), variation (2),
correspondence (2), universality (3), computation (4), and specification (3). He mainly used
gestures and graphs to transition from the derivative as a value to a function. First, the location
for the derivative process was mediated with his gesture of drawing and moving a tick mark on
the board horizontally, and the change was mediated with multiple tangent lines. Second, the
differentiability and velocity were addressed as specification after those words were realized as a
function with graphs. “Differentiability” was addressed as a specification of the derivative at
“every point” on its graph, and “velocity” at a point was realized as a specification of the
velocity over time (e.g., gestures for the tangent lines on a curve, and then a hitting gesture for
one point on the board). Similarly, the word “derivative” was used as a function after the graph
of the derivative was drawn, and compared to the use of the same word “derivative” as a number,
but the graphs for the “derivative” as a number (e.g., a tangent line, a point on the derivative
graph) were not directly compared to the graphs of the derivative function. Regarding word use,
transitions from the realization of the derivative as a point-specific object to a function were
mainly made with the word “slope.” The word “derivative” was not explicitly used in the
derivative process. Although Mr. William specified two uses of “derivative” as a number and as
a function several times while comparing the limit and derivative objects, he did not used the
word in the derivative process through which he objectified the “derivative” as a function.
Discussion and Conclusion
The analysis of realizations of “derivative,” in one Korean and one American calculus class
led to 4 observations regarding word use and visual mediation. First, similar visual mediators
were used in the realization of the limit process/object in both classes, but different components
were included in the realizations in each class. While using words and graphs for the limit
process, Mrs. Kim mainly mediated the location for the limit process, but Mr. William always
mediated both location and change. Mrs. Kim’s use for words and visuals for the initial object,
limit process, and final object were consistent (e.g., “slope,” “rate of change”), but Mr. William’s
words and visuals often varied (e.g., “the slope of a secant line” for the process and symbols for
the final object without “slope”). They used the terms (“the derivative at a point” and “the
differential coefficient”) only for the final object mediated with symbols. Second, the realizations
of the derivative as a function before it was defined were frequently found in Mr. William’s class
but rarely found in Mrs. Kim’s class. Mr. William used “velocity” “acceleration” and “force,”
with the phrase “always changing,” but Mrs. Kim never used “the leading function” before it was
defined, or the synonyms for the “derivative” (slope, rate of change) as a function. Similarly, the
word “differentiability” was realized as existence of “differential coefficient” at a point in Mrs.
Kim’s class, but was realized as a specific case of “the derivative function,” in Mr. William’s
class. Similarly, the notation 𝑓’(𝑎) was used both classes, but only Mr. William used the word
“any” for a before the derivative of a function was defined. Third, the words realizing the
derivative at a point and the words realizing the derivative as a function explicitly differed in
each class. The words differentiating the derivative as a point-specific object from the derivative
as a function were nouns in Mrs. Kim’s class. She explicitly used “the differential coefficient”
for the derivative as a number, and “the leading function” for the derivative as a function,
especially when discussing the relation between these two terms. In contrast, words realizing the
derivative as a number or a function were often attached phrases to the “derivative,” adjectives or
adverbs, in Mr. William’s class (e.g., “slope every point along x,” for the derivative function and
“individual slope” for the derivative at a point). Fourth, the types of the derivative process
through which the derivative was first defined as a function was different in each class: the
correspondence between x values and the differential coefficients in Mrs. Kim’s class, and the
expansion of the derivative at multiple points in Mr. William’s class. In Mrs. Kim’s class, there
was no direct transition from the term “the differential coefficient,” to “the leading function.”
Instead, the word “function” was used between the terms “the differential coefficient,” and “the
leading function.” In Mr. William’s class, the main visual mediator for the expansion was graphs
and the word “slope” throughout the initial object, derivative process, and the final object.
Although the differences in realizations between the two classes listed above seem related to
their different use of words, the analysis does not imply such differences are caused by the two
different words for the derivative as a point-specific object and the derivative as a function in
Korean and English. However, the realization of the derivative as a function or the transition
from the derivative at a point to the derivative function seems related to the key words mediating
those objects. Considering the initial motivation of this study, the differences in language related
terms, this study shows that Mr. William’s use of words were more consistent while realizing the
relation between the derivative at a point and the derivative of a function (e.g., use of “slope,” or
“velocity” throughout the initial object, derivative process, and the final object), than Mrs.
Kim’s. With the consistent use of words and corresponding visual mediators, the nouns realizing
the derivative at a point and the derivative of a function were used in a more coherent way (e.g.,
one tangent line, multiple gestures for several tangent line, and then the graph of the derivative)
in Mr. William’s class. Their uses of words and visuals were also different when they
transitioned from the derivative at a point to the derivative of a function, and when other related
terms were realized (e.g., “differentiability” in each class), when and how the terms synonymous
to “the derivative” was visually mediated (e.g., “rate of change” was realized as a function first,
and then realized at a point in the flea example). Mr. William’s consistent use of same words and
visuals mediating both the derivative at a point and the derivative function, and Mrs. Kim’s
separate use of words for “the differential coefficient,” and “the leading function” may not have
been affected only by the different terminology. However, the differences in realizations of the
derivative at a point and the derivative of a function seem consistent with the use of the word
“derivative” for both objects in English, and use of two different terms, which does not show the
relation between the two objects, in Korean.
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