Science, Mathematics, and Technology Learning

VOLUME 20 ISSUE 3

The International Journal of

Science, Mathematics, and Technology Learning __________________________________________________________________________

Triangle Based Scaffolding for Trigonometric Reasoning

ARTORN NOKKAEW, WANNAPONG TRIAMPO, NARIN NUTTAVUT, MEECHOKE CHUEDOUNG, DARAPOND TRIAMPO, AND CHARIN MODCHANG

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THE INTERNATIONAL JOURNAL OF SCIENCE, MATHEMATICS AND TECHNOLOGY LEARNING www.thelearner.com First published in 2014 in Champaign, Illinois, USA by Common Ground Publishing LLC www.commongroundpublishing.com ISSN: 2327-7971 © 2014 (individual papers), the author(s) © 2014 (selection and editorial matter) Common Ground All rights reserved. Apart from fair dealing for the purposes of study, research, criticism or review as permitted under the applicable copyright legislation, no part of this work may be reproduced by any process without written permission from the publisher. For permissions and other inquiries, please contact [email protected]. The International Journal of Science, Mathematics and Technology Learning is peer-reviewed, supported by rigorous processes of criterionreferenced article ranking and qualitative commentary, ensuring that only intellectual work of the greatest substance and highest significance is published.

Triangle Based Scaffolding for Trigonometric Reasoning Artorn Nokkaew, Mahidol University, Thailand Wannapong Triampo, Mahidol University, Thailand Narin Nuttavut, Mahidol University, Thailand Meechoke Chuedoung, Mahidol University, Thailand Darapond Triampo, Mahidol University, Thailand Charin Modchang, Mahidol University, Thailand Abstract: Trigonometry is an essential branch in mathematics education. It has been claimed that trigonometry students instructed through use of triangles will face difficulties in picturing the behavior of the trigonometric functions, and consequently, fail to apprehend them. This study elaborates on notions about the effects of dynamic images illustrating angle variation using an adjustable triangle tool to scaffold trigonometric reasoning. We demonstrate how the scaffolding tool can help students develop reasoning about trigonometric functions. Findings from interviews with non-formal education students showed that the adjustable triangle tool allowed students to better understand trigonometric ratios by means of visualization of the angle variation. As a consequence the students showed increased reasoning, leading to improvements in trigonometric understanding. Our findings suggest that students’ difficulties of relating the measured angle to a trigonometric function may be due to the use of a static triangle, which does not support imagination and does not provide students with an effective foundation for reasoning. In conclusion, we have reason to believe that our teaching tool may benefit trigonometry instructions. Keywords: Mathematics, Non-formal Education, Pedagogy, Trigonometry

Introduction

E

ven though Trigonometry can be represented in many forms and purposes (Byers 2010), in Thailand, triangle trigonometry is largely limited to elementary trigonometry class for the intention of promoting definitions of trigonometry and developing mathematical application ability. In triangle trigonometry, static triangle figures are normally presented to build a foundation of the ratio definitions of trigonometry. It is also common to use mnemonics, like ‘SOHCAHTOA’, to help remember trigonometric ratios. By considering special right triangles, namely 45-45-90 and 30-60-90 triangles, a table of trigonometric ratio values is given. The instruction emphasizes the lookup table approach. Based on this approach, students are expected to apply the trigonometric ratios and the obtained numerical values to solve a certain problem. This schema is useful and can be sufficiently utilized for solving certain types of word problems (Kendal and Stacey 1998). It has been found both in the mathematics curriculum of formal and non-formal education in Thailand. Researchers claim that triangle trigonometry has limitations (Kendal and Stacey 1998; Moore 2009; Moore 2010a, 2010b; Weber 2005; Weber 2008), such as the incapability of providing geometric objects that would allow students to develop a sense of changes of the trigonometric ratio as the angle measure varies. The common image of the angle variation based on static triangle setting is shown in Figure 1. It depicts how students think about the angle increasing in measure. The image of angle variation of a triangle, as shown in Figure1, is incoherent and inefficient for the reasoning about varying angle measures in relation to trigonometric ratio, which is crucial for the understanding of trigonometry (Moore 2010a; Thompson 2008). Notice from the representation in Figure 1 that to vary the angle students varied the length of the opposite side and the hypotenuse. Based on this perception of an angle’s variation, students face difficulties when attempting to envision the behavior of sin(x), opposite over hypotenuse, as the value of x varies (Thompson The International Journal of Science, Mathematics, and Technology Learning Volume 20, 2014, www.thelearner.com, ISSN 2327-7971 © Common Ground, Artorn Nokkaew, Wannapong Triampo, Narin Nuttavut, Meechoke Chuedoung, Darapond Triampo, Charin Modchang, All Rights Reserved Permissions: [email protected]

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2011). The changing hypotenuse becomes an obstacle in perceiving the hypotenuse as a unit of measurement which is a foundation of cosine and sine. Unfortunately, these two trigonometric ratios are normally presented first in classrooms. Consequently, holding images of static labeled triangles, students could not identify relationships between angle measures and the trigonometric ratios. In addition, they could not approximate values of trigonometric ratios for all angles (Weber 2005; Weber 2008).

Figure 1: A common way for students to imagine angle variation in a triangle (Thompson 2008, 47). Moore (2009) argued in favor for quantitative and covariation reasoning in constructing understanding of trigonometric functions. To do so, students should obtain potential images of an angle and the angle measure (Thompson 2008). Consequently, the radian as a unit of measurement has been taught to provide a foundation for reasoning. Unit circle trigonometry was firstly presented, then triangle trigonometry (Weber 2005; Weber 2008). As claimed by Weber (2005), introducing students to unit circle improved their performance. From a cognitive point of views, we argued that to present trigonometry in unit circle context using radians might increase cognitive load for non-formal education students. ‘Radian’ does not appear in the present non-formal education mathematics curriculum. In addition, a number of studies reported difficulties understanding the radian faced by students as well as teachers (Akkoc 2008; Fi 2006; Moore 2009; Topçu et al. 2006). Also, the radian is far removed from non-formal education students’ experience and they seem much more familiar and comfortable with degree measurement. Based on our direct experience, they could recognize images of 90, 180 and 360 degrees as right angle, straight line, and complete circle, even though they cannot explain what they are measuring. They also demonstrated applications of measuring angle in degree in their work, such as construction and mechanics. In addition, a triangle is one of the simplest geometrical figures with various applications. We believe that figures of right triangles and ‘degree’ as a unit of measurement are more appropriate for students to use as a foundation of reasoning. Based on this notion, we proposed a scaffolding tool for helping students’ trigonometric reasoning. This study was conducted to explore “How does the tool help promote covariation reasoning in a triangle trigonometric setting?”

Scaffolding Tool: Reducing Cognitive Load To understand trigonometry, students are expected to relate ratios of the lengths of two right triangle sides to an acute angle measurement. Unfortunately, the trigonometric ratios are relative quantities, measuring one side by using another side as a unit, which is not easy to directly measure and observe. The static triangle and angle variation image, shown in Figure 1, are insufficient to support reasoning. In this study, we provided a tool which was designed to relieve the cognitive

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NOKKAEW, ET AL.: TRIANGLE BASED SCAFFOLDING FOR TRIGONOMETRIC REASONING

load of observing trigonometric ratio relationships. The tool was designed and developed for teaching non-formal education students trigonometry. The instruction was purposely based on right triangle settings for the sake of minimizing cognitive load (Sweller 1988). However, due to drawbacks of static right triangle representation suggested in the previous research, adjustments were made. To promote an image of an angle as continuous quantities, the tool was designed as a frame of a quadrant which can form right triangles, see Figure 2A. In this case, the hypotenuse length is fixed. Right triangles can be constructed by moving the hypotenuse along vertical and horizontal tracks, namely varying the sizes of the two acute angles (Fig. 2B). In short, the tool provides an image of angle variations in which the length of the hypotenuse is fixed. In reasoning, it is necessary to know relevant quantities. Normally, a trigonometric ratio involves three quantities; two sides and one acute angle. To relieve cognitive load due to the ratio, the scale of the two arms is described as a fraction of the hypotenuse. In this sense, a ratio is perceived as one quantity. In other words, the horizontal arm represents a continuum of the cosine ratio and the vertical arm represents a continuum of the sine ratio. It may allow students to observe relationship between angle sizes and the ratios.

Figure 2: A tool for teaching trigonometry. Fig 2A (left) presents a right triangle (thick solid line) composing of the hypotenuse (shaded bar), vertical and horizontal hands (dash-line bar). Fig 2B (right) presents right triangles built by moving the hypotenuse along the track (dash-line bar). Based on the adjustable triangle tool, trigonometry was defined as depicted in Figure 3.

Figure 3: Instructional image based on adjustable triangle tool.

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The aim of providing the scaffolding tool is to promote Dynamic imagery (Presmeg 2006) that would be a foundation of trigonometric reasoning.

Methods In this study, 28 non-formal education students of Wat Thongsalagnam communal learning center, Bangkok, Thailand, participated in trigonometry classes using the adjustable triangle tool. The class was conducted once a week on Sunday. After two one-and-a-half hour classes, four students (S1-S4) participated voluntarily in group interviews. They are 19-20 years old. Two of them, S1 and S2, finished grade 6 and grade 8 respectively from formal education before enrolling in nonformal education. The other two, S3 and S4, finished compulsory education, grade 9. All of them left schools for a while before enrolling a non-formal education programme. The aim of the instruction was to develop covariation reasoning in a trigonometric context. To explore how students use the tool in trigonometric reasoning, a qualitative study was conducted. Although the qualitative approach which relies on interpretation has generality restriction, it could give us an insight into images constructed by students and their roles in trigonometric reasoning. With regard to interpretative objectivity, triangulation techniques were applied. Various forms of data were collected, namely verbal, gestures, drawing and writing from interviews. The interview questions were in the form “What happen to sine and cosine if the angle measure changes from … to …?” and “What is the approximated value of sine and cosine for …. degrees angle?” The students’ reasoning was recorded by audio and video recorders. Interpretations were drawn from the multiple representations for the purpose of ensuring internal consistency. Gestures and discourses were coded and analyzed to infer students’ images which were the base of reasoning (Bjuland, Cestari and Borgersen 2007). The analysis aimed to infer students’ mental images and conceptual development from their language and actions in argumentations. The collected data was examined iteratively to find counter examples of the interpretations. The interpretations were reviewed by a group of researchers and graduate students. The next section of this paper reports finding from analyzing students’ actions during the interviews. The reported results focus on the student’s languages and actions reflecting their internal images as engaging in trigonometric reasoning via the proposed tool.

Use of the Tool The tool was offered in order to scaffold students’ reasoning. The script in Table 1.1 is an instance of the use of the tool by a participant. Note that, considering an acute angle on the horizontal line; the student was introduced to sine and cosine as labels naming fractions of vertical and horizontal, respectively. The script presents a student’s argument about the trigonometric relationships. The student defined a sine as values lying on the vertical line. The sine values as well as angle measure are ordinal. It can be seen from words like ‘increase’ and ‘decrease’ which are regularly used to describe change of sine and angle measure. To make this argument, the student worked along with the tool. The relationship between angle measures and sine was illustrated from a sentence like: ‘if angle decrease, sine will decrease’. In brief, with the tool, arguments about relationships between angle measure and trigonometric ratio is likely viable. The tool provided a building block for trigonometric reasoning of angle variation. The script demonstrates a facilitating role of the tool in providing concrete experiences and evidences that support his reasoning about trigonometric function as the angle varies. The student reasoning shows the achievement of Moore’s (2010b) second level of covariation reasoning. Namely, the participating student perceived ‘the direction of change’.

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Table 1.1: An instance of a student’s argumentation. I S1 S1 I S1 I S1

S1 I S1

I S1

I S1

Verbal What happens to sine of an angle as the angle increase? If angle changes… sine will… Sine increases… Why do you think sine will increase? Sine will increase. Could you explain more? This is 30 degree angle. If the angle increases to 60 degree, sine will increase. Sine is here. It will increase. What happen to sine if the angle decreases? If the angle decreases, sine will decrease.

How about cosine? How does cosine vary according to the angle? Cos ….. If the angle increases, cosine decreases. If the angle decreases, what happen? If the angle decreases, cosine increases.

Actions

talked insinuatingly paused and thought took the tool and moved the hypotenuse spoke insinuatingly while working with the tool moved the hypotenuse in which the end at the horizontal hand moving toward the right angle. Pointed at the vertical hand.

moved the hypotenuse in which the end at the horizontal hand moving away horizontally from the right angle.

spoke insinuatingly while moving the hypotenuse.

moved the hypotenuse in which the end at the horizontal hand moving away horizontally from the right angle.

Students' Images in Trigonometric Reasoning Not only the tool supported the trigonometric reasoning of the participant, but it also allowed the students to construct images for the argumentation. According to the Pirie and Kieren’s (1994) growth of understanding, ones who are at the “image having“ stage do not need a scaffolding tool. Therefore, the emergence of images could be observed when the students started reasoning without the tool. They reasoned based on images that they had already constructed. In this study, various images could be observed after the students had had experience with the proposed tool. Some selected images are presented in this section.

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Image 1: ‘Plus Sign’ ‘Plus sign’ was named by a student, S1, who drew perpendicular lines to depict his image, see Figure 4. It might be worth pointing out that the participants had no any idea about a coordinate system. This image was constructed based on student’s prior image of quadrantal angle, including 0°, 90°, 180°, 270°, 360°. He defined the intersection as origin valued 0. Positive numbers lie on both sides of zero. The largest value is 1 placed at the end of each line. The horizontal line represents continuum of cosine for angle measures. At the right end of horizontal line, he defined as a cosine at 0 degree, where the value is 1. The cosine becomes 0 at the intersection where the angle measure is 90 degree. Then it becomes 1 as the angle measure increases to 180 degree. At this point, the teacher encouraged him to use a concept of a number line to represent the horizontal line. Namely, the numerical value at the left end will be negative. At 270 degree, cosine becomes 0 and places at the intersection again (see Fig. 5D). Figure 5 demonstrates a student’s image of cosine variation according to angles. He swept his finger along the horizontal line when explaining about cosine. Moreover, the word ‘increase’ and ‘decrease’ were often used in his explanation. It revealed that in his image cosine is continuous values along a horizontal straight line. Also, Figure 5D demonstrates an evidence of coordination between angle measures and cosine ratio by the student. Based on this image, a trigonometric relationship could be drawn as shown in Figure 6.

Figure 4: ‘Plus sign’ image (A) and (B) depicts changes of sine and cosine at angles

Figure 5. A student waves his finger along the horizontal line in order to describe relationships between angle and cosine. (A), (B), (C) and (D) presents cosines at 0, 90, 180 and 270 degree, respectively.

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Figure 6: S1’s pattern image

Image 2: ‘Clock’ A student (S2) drew a circle and a pair of perpendicular lines inside the circle (see Fig. 7). The circular trajectory represented an increase of angle measure. He, then, labeled 90, 180, 270 and 360 in each part of the circle. The number labels indicate upper limits in each quadrant. For example, the upper limit in the first quadrant is a 90 degree angle. Furthermore, in each quadrant, students described change of cosine and sine. In ‘90’, cosine decreases and sine increases. Decrease of both cosine and sine and increase of both cosine and sine were stated in ‘180’ and ‘360’, respectively. For ‘270’, increase of cosine and decrease of sine was described. The image was named as ‘Clock’ because the student explained that he was thinking of an angle variation as a running clock hand when he drew this image. The constructed images allowed the students to estimate values of trigonometric ratios (sine and cosine). Table 1.2 shows students’ estimations of trigonometric ratios in a group interview. The table presents the students’ estimation and also values of trigonometric ratios obtained from a calculator. It illustrated students’ an ability of trigonometric ratio value estimation. The approximated values were close to the value obtained from the calculator. Therefore, we can conclude that the scaffolding tool allowed the students to create an effective images which allowed them to roughly estimate the value of trigonometric ratios.

Figure 7: S2’s ‘Clock’ image 105


Table 1.2: Students’ estimation of trigonometric ratios (sine and cosine) from a group interview. Angle measures (degree) 100 180 200 270 340 360

Sine Student estimation 0 -0.2, -0.3 -1 -0.3 0

Cosine Calculator

0 -0.34 -1 -0.34 0

Student estimation -0.1 -1 -0.8, -0.9 0 0.8 1

Calculator -0.17 -1 -0.94 0 0.94 1

An example of how the student estimated is shown in Figure 8. The protocol of the estimation is a combination of the student’s previous image about the angle variation (Fig. 8A) and the instructional image obtained from the adjustable triangle tool (Fig. 8B). For instance, when the student was asked to estimate sine and cosine of 130 degree angle, he laid a pen as a clock hand on 12 o’clock where the pivot of the hand was positioned to the intersection. ‘This was 90 degrees’, said the student. He rotated the pen progressively counterclockwise and, simultaneously, said ‘100 120 130’, see Fig. 8A. Then, to estimate numerical value of sine and cosine of 130, the student changed to work on an image obtained from the adjustable triangle tool, see Fig. 8B. This method was consistent when he estimated sine and cosine of 200 degree angle. To do that, he started at 180 reference degree. His estimation for cosine 200 is around -0.8 to -0.9 and sine 200 is around 0.3. For estimation at 340, cosine and sine are around 0.8 and -0.3, respectively.

Figure 8: (A, right) an image of a clock that a student used to determine a quadrant that the assigned angle lie in. (B, left) an operation image that students used to estimate a numerical value of a trigonometric ratio.

Discussions As reported by the previous studies, the static image of the special triangles did not provide a sufficiently effective image for reasoning. This study was conducted to investigate the scaffolding roles of the adjustable triangle tool in trigonometric reasoning. The findings of this study demonstrated reasoning foundation based on triangle setting obtained from the adjustable triangle tool. The tool induced students’ constructions of visual-spatial imagery which allowed covariation reasoning in trigonometry. As a result, elementary understandings in trigonometry were developed based on a triangle setting. The findings suggested that it is possible to develop understanding of

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trigonometric ratios in a triangle setting. To do so, teachers have to provide a chance for students to construct an appropriate image of the trigonometric ratio. Unlike the static triangle, an adjustable image of a triangle can induce construction of a dynamic image that allows trigonometric reasoning as the results of in this study show. The constructed images can be various depending on students' previous knowledge and experience. Based on the finding, an emphasis on the relationships between angle measures and trigonometric ratios does not need to be delayed and reserved for advanced classes of trigonometry. Rather than emphasizing formal definitions at the beginning of the class, providing concrete experience (Kolb, Boyatzis and Mainemelis 1999) with the hands-on tool allowed non-formal education students who have weak mathematics backgrounds, to experience the behavior of the trigonometric objects and to construct some mathematical knowledge. Also, it allowed students to integrate their own existing images and newly constructed imagery. Engaging and accepting students' prior knowledge into classroom may make the learning meaningful for each student. Once an image is constructed, it is a chance for a teacher to encourage development of more advanced understanding (Pirie and Kieren 1994). From the constructs of the students in this study, we can see that the construction provided an image to which the teacher can add other concepts such as number line or Cartesian coordinates. Besides non-formal education, the tool may be used in regular formal and vocational education trigonometry classes. The tool can be presented in supplementary lesson which aim to promote conceptual understanding of trigonometric ratios. Because trigonometry is a procept (Gray and Tall 1992) which cover both process and concept, students need a chance to develop a concept about the trigonometric objects which is usually lacking in traditional classes of the triangle trigonometry. The tool can also be presented in a main lesson of the trigonometric ratio topic. However, to do so, it may require longer learning time than the traditional class.

Conclusions This paper presents a qualitative study about the effects of a scaffolding tool teaching mathematics in non-formal education programme. The tool is an adjustable triangle that supports visual-spatial learning in trigonometry and implies that trigonometric reasoning can be developed also by lowachieving students enrolled in non-formal education programmes. We demonstrated the scaffolding role of the tool on students’ improvement of visual-spatial thinking. However, this finding does not suffice generalization due to uniqueness of the participants, the nature of qualitative study, and the low number of the participants in this study. Future investigations may include a higher number of participants enrolled in formal education programmes to obtain statistical evidence. In addition, we intend to further develop the tool allowing the merging between triangle and unit circle trigonometry.

Acknowledgement We would like thank the institute for the promotion of teaching science and technology (IPST), and Institute for Innovative Learning, Faculty of Science, and Faculty of Graduate studies, Mahidol University. We thank Dr. Stefan Schreier for the final proofreading.

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REFERENCES Akkoc, Hatice. 2008. "Pre-service mathematics teachers’ concept images of radian." International Journal of Mathematical Education in Science and Technology 39: 857-878. Bjuland, Raymond, Maria Luiza Cestari, and Hans Erik Borgersen. 2007. "Pupils’ mathematical reasoning expressed through gesture and discourse: A case study from a sixth-grade lesson." Paper presented at the Fifth Congress of the European Society for Research in Mathematics Education, Larnaca, Cyprus, February 22–26. Byers, Patricia. 2010. "Investigating trigonometric representations in the transition to college mathematics." College Quarterly 13: 1-10. Fi, Cos. 2006. "Preservice secondary school mathematics teachers' knowledge of trigonometry: cofunctions." Paper presented at the 28th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Mérida, Yucatan, November 9-12. Gray, Eddie, and David Tall. 1992. "Success and failure in mathematics: Procept and procedure." Paper presented at Workshop on Mathematics Education and Computers, Taipei, April. Kendal, Margaret, and Kaye Stacey. 1998. "Teaching Trigonometry." Australian Mathematics Teacher 54: 34-39. Kolb, David A., Richard E. Boyatzis, and Charalampos Mainemelis. 2000. “Experiential Learning Theory: Previous Research and New Directions.” In Perspectives on cognitive, learning, and thinking styles, edited by Robert J. Sternberg and Li-Fang Zhang. NJ: Lawrence Erlbaum. Moore, Kevin C., 2009. "An investigation into precalculus students’ conceptions of angle measure." Paper presented at Twelfth Annual Special Interest Group of the Mathematical Association of America on Research in Undergraduate Mathematics Education (SIGMAA on RUME) Conference, Raleigh, NC, February 26-March 1. Moore, Kevin C. 2010a. "The role of quantitative and covariational reasoning in developing precalculus students’ images of angle measure and central concepts of trigonometry." Paper presented at the 13th Conference on Research in Undergraduate Mathematics Education, Raleigh, NC, February 25-28. Moore, K. C. 2010b. "The role of quantitative reasoning in precalculus students learning central concepts of trigonometry." PhD diss., Arizona State University Pirie, Susan, and Thomas Kieren. 1994. "Growth in mathematical understanding: how can we characterise it and how can we represent it?" Educational studies in Mathematics 2: 165190. Presmeg, Norma C. 2006. "Research on visualization in learning and teaching mathematics." In Handbook of research on the psychology of mathematics education, edited by Angel Gutiérrez and Paolo Boero, 205-235. Sense Publishers. Sweller, John. 1988. "Cognitive load during problem solving: Effects on learning." Cognitive science 12: 257-285. Thompson, Patrick W. 2008. "Conceptual analysis of mathematical ideas: Some spadework at the foundation of mathematics education." Paper presented at the Annual Meeting of the International Group for the Psychology of Mathematics Education, MorÈlia, Mexico. Thompson, Patrick W. 2011. Quantitative reasoning and mathematical modeling. Paper presented at the New perspectives and directions for collaborative research in mathematics education. WISDOMeMongraphs, Laramie, WY. Topçu, Tahsin, Mahmut Kertil, Hatice Akkoç, Kamil Yilmaz, and Osman Onder. 2006. "Preservice and in-service mathematics teachers' concept images of radian." Paper presented at 30th Conference of the International Group for the Psychology of Mathematics Education, Prague.

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Weber, Keith. 2005. "Students’ understanding of trigonometric functions." Mathematics Education Research Journal 17: 91-112. Weber, Keith. 2008. "Teaching trigonometric functions: Lessons learned from research." The Mathematics Teacher 102: 144-147.

ABOUT THE AUTHORS Dr. Artorn Nokkaew: Lecturer, Institute for Innovative Learning, Mahidol University, Nakhon Pathom, Thailand. Dr. Wannapong Triampo: Director and Associate Professor, Institute for Innovative Learning and Department of Physics, Faculty of Science, Mahidol University, Nakhon Pathom, Thailand. Dr. Narin Nuttavut: Assistant Professor, Department of Physics, Faculty of Science, Mahidol University, Bangkok, Thailand. Dr. Meechoke Chuedoung: Lecturer, Department of Mathematics, Faculty of Science, Mahidol University, Bangkok, Thailand. Dr. Darapond Triampo: Assistant Professor, Department of Chemistry, Faculty of Science, Mahidol University, Bangkok, Thailand. Dr. Charin Modchang: Assistant Professor, Department of Physics, Faculty of Science, Mahidol University, Bangkok, Thailand.

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