Student Gesture Use When Explaining the Second-Derivative Test and Optimization: Mimicking the Instructor?

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1 Student Gesture Use When Explaining the Second-Derivative Test and Optimization: Mimicking the Instructor? Tim McCarty West Virginia University Nicole Engelke Infante West Virginia University The Second-Derivative Test and optimization can naturally evoke gestures from an instructor while he or she is teaching. We wanted to establish how student learning might be affected by an instructor s use of gesture. Students viewed either a gesture-rich or gesture-free video of an instructor solving an optimization problem, and were interviewed a week later to assess both their understandings of optimization and how they used gesture to support their explanations. Very few gestures were used when the students explained how they solved the optimization problem. However, when describing the second derivative test separate from optimization, students used a number of gestures. We conclude that further study should be undertaken, but such study should be focused on the Second-Derivative Test without the context of optimization problems. Key words: Gesture, Calculus, Optimization, Instruction Background Gesture use in the classroom may be used to increase student interest and foster student success in calculus. Recent studies suggest that instructors use of gesture promotes student learning, and that instructors can intentionally alter their gesture production during instruction (Alibali, et al., 2013a; Alibali, et al., 2013b; Cook, Duffy, & Fenn, 2013; Hostetter, Bieda, Alibali, Nathan, & Knuth, 2006). As calculus can be considered the study of motion, it is a natural place to examine gesture. There are several types of calculus problems that require students to visualize or imagine situations involving changing rates, with optimization being one example. LaRue (2016) studied student responses and approaches to optimization problems, while classifying each action in a student s problem-solving process using Vinner s (1997) conceptual, analytical, pseudoconceptual and pseudo-analytical definitions. We use these definitions, along with Tall and Vinner s (1981) concept image, to describe student understanding. We seek to answer the following questions: Will students who view a gesture-enhanced lesson demonstrate a greater understanding of the presented concept than students who view a gesturefree lesson? How do students use gesture when explaining their work? Do they mimic any gestures used by their instructor? We present a pilot study that attempts to answer these questions. Methods Data collection was completed in two phases. The first phase included eleven students, each of whom were currently enrolled in a second-, third-, or fourth-semester Calculus class. The average amount of time it had been since each student had taken first-semester Calculus was about three-and-a-half years. These students were asked to take a short pretest, watch one of two possible videos, and then complete a short posttest. The second phase, which occurred a week later, involved semi-structured interviews in which students were asked questions to assess their understanding and recall of the first phase s material.

2 The six-question pretest had students determine if points on a function were maxima, minima, neither, or can t tell, using three of the standard function representations: algebraic, graphical, and verbal. The verbal problem was an optimization problem in which they were asked to maximize the area of a rectangle for a given perimeter. After completion of the pretest, students were shown one of two videos in which a similar perimeter/area optimization problem was presented, solved, and verified using the Second-Derivative Test. Both videos followed essentially the same script, but one video contained instruction that was given without any of the categories of gesture outlined in Engelke Infante (2016), while the second video contained instruction that utilized such gesture. After watching and taking notes on the video, the students were given a three-question posttest consisting of three optimization problems: the first was very similar to the one in the video, the second was an extension of the first, and the third was a different type of optimization problem. In the interest of space, we consider only the optimization problems that were similar to what was presented in the video. Six students viewed the gesture video, and the other five viewed the gesture-free video. Rubrics were created for each problem, and tests were graded and scored. For the second phase, we requested interviews with all participants. Three students agreed to participate and were asked to 1) tell us what you know about the Second-Derivative Test, and 2) solve a simple perimeter/area optimization problem. As students reached three specific points in their solution process, they were asked How do you know that the functions you have chosen represent area and perimeter?, How do you know to take the derivative at this step?, and How do you guarantee that the answer you obtained was a maximum? Follow-up questions were asked to clarify responses and ascertain that we had gained as much student knowledge as possible. The interviews were videotaped and an analysis of subject gesture use was begun. Data and Results To assess the extent that the gesture-rich video influenced student understanding, we compared the student scores of the last item on the pre-test (the only optimization problem) with the average scores of the two similar optimization problems in the posttest. The initial results suggest that the students who viewed the gesture-rich video improved their scores on the optimization problems more than the students who watched the gesture-free video, as shown in Table 1. Table 1 Scores of the Optimization Problems, Pre- and Post-video Students Pretest Problem Posttest Problems Improvement All Students 26% 52% 26% Gesture-rich video 22% 55% 33% Gesture-free video 32% 48% 16% During the interviews, we expected students to recall statements and solution methods that were shown to them in the video. However, of the three students interviewed, none could give an accurate description of the Second-Derivative Test, and while two students completed the initial optimization problem correctly, they did not verify their answers using the Second-Derivative Test, which was the method of verification in the video. The third student also obtained a correct

3 answer to the initial optimization problem, but he did so without using calculus. When given an additional optimization problem to solve, he was unable to solve it. Further discussion of the interview responses is given below. The Pretest and Posttest The pretest suggests that the students had poor mastery and recall of first-semester calculus objectives that required the use of more abstract algebraic representations of functions. The first question (Pre #1) asked students to determine extrema of a function, f, given its graph. The next four problems (Pre #2-5) asked students to determine extrema of f given either the graph of f or a list of algebraic information about f, f, and f (e.g., f(1) = 8, f (1) = 0, f (1) = 2.) The final pretest problem (Pre #6) asked students to solve an optimization problem requiring them to maximize the area of a rectangle for a given perimeter. Table 2 lists the scores for all problems on the pretest and posttest and shows this drop in scores after the pretest s first problem. We were not totally surprised by this score decrease, given that the literature indicates that the function concept takes a long time to develop, and that students have difficulty moving between different function representations (Carlson, 1998; Carlson, Oehrtman, & Engelke, 2010). We found a similar score decrease across the posttest optimization problems. The posttest problems asked the students to: 1) solve a problem similar to the example problem in the video, 2) solve a problem with a slight extension of the example problem, and 3) solve an optimization problem with a different context. It was our goal to determine the extent to which they could make extensions from the example that had been presented. As seen in Table 2, students scored well on the most similar problem (Post #1) and scores decreased as extensions needed to be made. Table 2 All Pre- and Posttest Scores Students Pre #1 Pre #2-5 Pre #6 Post #1 Post #2 Post #3 All Students 75% 24% 26% 67% 36% 20% Gesture-Rich Video 87% 25% 22% 70% 40% 12% Gesture-Free Video 60% 23% 32% 64% 32% 22% The Interviews During the interviews, no student accurately described the Second-Derivative Test. Each student gave responses that were a mixture of correct statements, pseudo-conceptual and pseudoanalytical ideas (Vinner, 1997), and statements that were incorrect or off-topic. Two of the three students were able to complete an optimization problem correctly, but neither of them used a version of the Second-Derivative Test to verify their answers, like had been done in the video example. We present case studies of each student s responses that focus on their gesture use and what they said about the Second-Derivative test. Students Who Watched the Gesture Video Student 105. Student 105 initially stated that the Second-Derivative Test was concavity. His first use of gesture occurred when he began discussing a sign chart which was part of his concept image of the test. While stating that one must find critical points, he used his hands to signal a

4 hierarchy of derivatives by moving them lower at each step while saying [given a] function, you find the first derivative, and then you find the second derivative. He seemed to show confusion between f, f, and f, stating that one must plug [critical number] back into the original function and that will tell you exactly where it changes it will change concavity and later compounding this error by drawing a sign chart that involved plugging critical values into f, claiming that the signs of those results determined decrease and increase, and stating incorrectly that from decreasing to increasing is concave down, and from increasing to decreasing is concave up. Thus, while his concept image includes this sign chart, his reasoning behind the creation of this chart is pseudo-conceptual. Nevertheless, he used many gestures during this explanation, indicating concavity by using five U -shaped hand positions and two U -shaped tracing motions within a span of about 40 seconds during the explanation of his sign chart. He was able to solve the optimization problem correctly, using the method shown in the video minus the verification step. However, when asked about why we take the derivative of our area function, he noted because The first derivative tells us the maximum and minimum, the second derivative tells you the concavity. While this statement is technically correct, it is pseudoanalytical as it is not an explicit reason as to why we take a derivative, and he did not include any of the U -shaped gestures he used before. When asked how he knew that his final answer led to a maximum, he did not invoke the Second-Derivative Test. Instead, he constructed a table by testing at least one value larger and one value smaller than his critical point; the areas resulting from those numbers were calculated and found to be less than the area resulting from his answer. Here, we see evidence of pseudo-analytical reasoning and the absence of gesture use. Aside from a few general hand motions given to emphasize certain words, he invoked minimal gesture during these particular explanations. Student 109. Student 109 mentioned that the Second-Derivative Test was to determine the point at which the (first) derivative changed direction, while positioning his index finger higher than his middle finger and then switching their positions to signify this change. When recalling the Second-Derivative Test, he invoked the image of a bell curve, and used gestures in his description, using two hand motions to trace, first upward, then downward, an imaginary bell curve in the space in front of him while saying, respectively, climbing up the bell curve, and it would not start to go negative, but start decreasing, and finishing with the same finger gesture as before while stating it s the change of the first derivative. Like Student 105, Student 109 was able to solve the optimization problem using the method shown in the video, and used a similar table instead of the Second-Derivative Test to show that his answer led to a maximum. Student 109 did not use much gesture during the explanation to his solution of the optimization problem; his gesture use consisted primarily of tracing elements of the many diagrams he drew to compliment his words. For example, after drawing a bell curve with a small horizontal tangent line at the top, he said at this point, slope should equal zero, and the tangent line would be flat, while tracing over the horizontal tangent line. Student Who Watched the Gesture-Free Video Student 106. Student 106 initially said that the Second-Derivative test told us rate of change and maximum and minimum. With that latter statement, he presented simultaneously a raised and lowered horizontal, flattened hand to delineate the ideas of maximum and minimum. He later said the test is where the first derivative is equal to zero, at which point he seemed to trace a small horizontal segment with his pencil, maybe to represent an image of a horizontal tangent line. He continued and it tells you the maximum and minimum, or maybe the local maximum and

5 minimum of the first derivative, while making the same flattened-hand gesture during both times he said maximum and minimum. Thus, it appears that Student 106 has at least a partial conceptual understanding of the second derivative. He answered the perimeter/area optimization question by quickly stating that the shape had to be a square. When asked how he knew this to be true, he stated that I know that circles have probably the largest area, then I think the square, in my mind is next so I just figured that I ll just make a square out of the rectangle. Since this answer did not take up much time, the interviewer gave him a second optimization problem to consider. He talked through several possible ways to approach the problem, but finally settled on constructing a graph. He drew two increasing segments followed by three decreasing segments and did not have an explanation as to why he drew this shape. While he was describing how he was interpreting his graph, he made a downward pointing motion when describing a function as decreasing, moved his hand in a precise straight line when saying that a function was linear, and held up his pencil horizontally then moved a finger down at the appropriate time while saying that the first derivative equaling zero can give us a maximum because the slope tells us where it levels off and starts to go back down. Ongoing Work The fact that students who viewed the gesture video showed more improvement when solving optimization problems on the posttest appears to be a promising area to explore. The interviews given a week later provided mixed results. While the students were describing their understanding of the Second-Derivative test in the interview, they used several gestures, some of which were similar to what they had seen in the video (Student 105) and some that were distinctly different (Student 106, who viewed the gesture-free video). However, students performance on similar optimization problems was not as strong as indicated on the posttest. Students did not verify that their answers were extrema. When asked how they knew their answer was a maximum, they did not invoke the second derivative as had been presented in the video. This leads us to believe that we were potentially asking them to recall too much mathematics. Analysis of the interviews is ongoing. Since the amount of mathematics in our script and tests seemed too much for these students, our next stage of research will be done on a smaller scale. We plan to devise a smaller, 4-5 minute script that only defines the Second-Derivative Test, eliminating the surrounding context of an optimization problem. This script will be filmed in four ways: with no gesture at all, with only tracing and pointing gestures at the board, with only hand gestures in the space between the instructor and the student, and with both of the preceding types of gesture. It is our hope that a simplified script, with more focused pre- and posttests, will allow for greater student success. With this success, we hope to not only have a greater understanding of how instructor use of no gesture, limited gestures, or many gestures might affect student learning, but also see students using more gesture to augment the explanation of their mathematical processes. Hence, we will be able to make suggestions for how instructors can incorporate gesture into their teaching to facilitate student understanding and success. As our work progresses over the next several months, we are certain several interesting questions will arise that we will bring to the conference.

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