Approaches to the Derivative in Korean and the U.S. Calculus Classrooms. Jungeun Park University of Delaware

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1 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

2 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 signifying mathematical objects Visual Mediators Routines Endorsed Narratives Non-verbal means of communication Well-defined repetitive patterns Utterances that speakers endorse as true Different speakers can use a word differently. It is an "all-important matter" because "it is responsible to a great extent for how the user sees the world" (Sfard, 2008, p. x). Because ways people attend to visuals depend on contexts, mediators need to be viewed as part of the thinking process, not auxiliary means of pre-existing thought. Patterns can be found in speakers' use of words and visuals, or in the process of creating and endorsing narratives. Students' endorsed narratives are often different from what the professional mathematics community endorses as 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 + h) f (a) f (x + h) f (x) f '(a) = lim ) and as a function (e.g., ). The realization of the h 0 f '(x) = lim h h 0 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.

3 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 sent to the group of teachers recommended by mathematics education professors at the researcher s institution. Table 2. Teachers backgrounds and classes Mrs. Kim Mr. William First language Korean English Degrees BS in Mathematics BS and MS in Mathematics Teaching experience 13 years 10 years Calculus teaching 3 times 7 times Number of Classes/week Three 50-minute classes Five 90 minute classes Teaching method Blackboard and chalk SmartBoard, interactive graphs Number of Students 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 Graph [Fig. 2] Initial Object ARC [is] the slope of the line Top secant line Initial Object: location Passing through two points apart. (a, f(a)), (b, f(b)) Limit Process: location How do we move the points? Closer, closer a Limit Process: Change What happens to the slope? It s getting smaller and smaller (drawing three secant lines and arrows) Three secant lines, arrow Limit Object We see the slope, the differential coefficient, and Tangent line the instantaneous rate of change. Final Object: location At one point A point a

4 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 x a ) without other explicit mediation of the 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.

5 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

6 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

7 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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