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Improving Calculus Learning Using a Scientific Calculator

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Improving Calculus Learning Using a Scientific Calculator Open Education Studies, 2020; 2: 220–227 Research Article Miriam Dagan*, Pavel Satianov, Mina Teicher Improving Calculus Learning Using a Scientific Calculator https://doi.org/10.1515/edu-2020-0125 received July 2, 2020; accepted September 28, 2020. made possible with calculators. This includes not only the basic computational tools, but also their potential use in Abstract: This article discusses the use of a scientific enhancing conceptual understanding of mathematical calculator in teaching calculus by using representations concepts and approaches, essential for developing of mathematics notions in different sub-languages research and critical thinking by the student. (analytical, graphical, symbolical, verbal, numerical This is very important nowadays, because academic and computer language). Our long-term experience institutions have been dealing with the failure of students shows that this may have a positive and significant effect in the first year of their studies in general and especially on the enhancement of conceptual understanding of in mathematical courses (Lowe and Cook, 2003; Yorke and mathematical concepts and approaches. This transcends Longden, 2004). With technological advances and a large the basic computational uses, and implies a potential flow of students to STEM professions at universities and for real improvement in the learning success, cognitive colleges, there is a constant decline in the mathematical motivation and problem solving skills of the student. basic knowledge of the novice academic students (Gueudet, We illustrate the steps we have taken towards doing this 2013; Bosch, Fonseca & Gascon, 2004). Undergraduate through some examples. students have difficulties in understanding definitions and various representation of mathematical concepts. It is Keywords: scientific calculator; calculus teaching; because teaching and learning in high school is still based different representations; conceptual understanding; more on algorithmic exercises and memorization, rather cognitive motivation. than on deeper understanding of mathematical language and using its computer applications in solving complex and interesting problems. Using the scientific calculator 1 Introduction may contribute to a reduction of this gap between enough developed algorithmic skills of novice engineering Despite the near half-century of their widespread students and insufficiency of research, creative and presence in schools, colleges and universities, educators critical mental skills, required in our current academic have yet to agree on the best way to use electronic pocket studies. calculators in teaching. For example, some claim that the use of electronic pocket calculators harms mathematics learning, while others are indifferent to their presence and are not fully aware of recent steps in their development. 2 Pedagogical strategy Indeed, many school and university teachers simply Our main pedagogical approach is based on the active ignore the possible benefits of calculator use in teaching use of different representations of mathematics notions mathematics, or other fields of STEM (Kissane and Kemp, and problems through diverse sub-languages (analytical, 2013). In academic studies, relatively few educators pay graphical, symbolical, verbal, numerical and computer enough attention to the broad educational opportunities language) of mathematics (Dagan et al., 2018, 2019b). For convenience, we will designate these basic forms of representations by means of the capital first letters of the *Corresponding author: Miriam Dagan, Bar-Ilan University, appropriate words: E-mail: [email protected] A – Analytical representation Pavel Satianov, Shamoon College of Engineering, Israel V – Verbal representation Mina Teicher, Bar-Ilan University Open Access. © 2020 Miriam Dagan, Pavel Satianov, Mina Teicher, published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. PLUS sub-language, in the context of this article) we must submit it in CN form: PLUS sub-language, in theImproving context Calculus of this Learningarticle) Usingwe must a Scientific submit Calculator it in CN form: 221 10 “Calculate X ”. 10 x1 “Calculate X ”. G – Graphical representation x1 N – Numerical representation C – Computer representation (Calculator depiction, in the context of this article) For example, the problem: “Calculate 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10” formulated in numerical form and required numerical answer is NN representation of the problem in this classification. If we want to present it in computer language (in CASIO fx-991ES PLUS sub-language, in the context of this article) we must submit it in CN form: 10 Figure 1. Graphical representation“Calculate of the sum X ”. x1 Verbal representation (VN) of the same problem will be “Calculate the sum of natural numbers from 1 to 10”. Graphical representation (GN) of the same problem will be “Calculate the area of the figure shown on figure 1”. Our hypothesis, based on long-term pedagogic 10 research and teaching experience, is: FigureFigure 1. Graphical 1: Graphical representation representation of of the the sum sum X . x1 “The permanent and systematic use of different sub-language representations, and various transitions among them, in mathematical disciplines). To correct this situation, we a calculus course, in addition to extensive use of scientific calculators and graphic applications, yields greater achievement organized special seminars dedicated to the use of the and a higher level of understanding, while enhancing cognitive fx-99IES PLUS CASIO calculator in teaching of mathematics motivation and creative thinking of students”. Such approach is for engineering students. We treat this calculator as an in accordance with the requirements of 21 century that needs interactive math dictionary and, as regards the calculus profound changes in the teaching of mathematics and other course, as a modern machine for evaluating elementary STEM disciplines for engineering students. functions, and, more generally, as a convenient device for storing and processing numerical information. This approach has changed the attitudes of students and teachers regarding the use of the scientific calculator, not 3 Using calculators in teaching only in the calculus course, but also in other math courses calculus and engineering disciplines. Note, that the skillful use of the calculator not only improves the student’s exam In most study programs, the calculus course plays a results, but also enhances the understanding of the key central role. After many years of experience in teaching ideas of calculus, as well as students’ motivation and calculus, we have found that the calculator presents interest in studying them. opportunities for significantly developing exploratory and critical thinking by the student. In this paper, we will describe some of our findings in this direction. Since most of our students have worked with the fx-99IES 4 Classroom teaching examples PLUS CASIO calculator, which allows for mid-term tests, final exams and high school exams, we have used this Consider an example involving simply the calculator in all our examples. As is the case for many skillful use of the scientific calculator. advanced calculators, the lecturer will play a key role in utilizing the full power of this device that transcends its Students are asked to evaluate the following expression: basic computational functions. 2.018 2.018× 2.019 In a survey of lecturers in our department, several 2.019 + − 2.018 2 2 years ago, we found that most lecturers did not use this 2.018 + 2.019 2.018 + 2.019 calculator in the teaching process and some of them were not even familiar with it and its potential for the teaching Although all students had the fx-991ES PLUS CASIO and learning of the calculus course (as well in others calculator, they did this task in a direct “arithmetic” 222 Miriam Dagan, Pavel Satianov, Mina Teicher Dear Dr. Beata Socha. After carefully reading the proof version of the paper several mistakes weremanner, found: step by step. This appears to be a routine We ask students to calculate the sum. calculation, requiring attention and numerous keystrokes, 1) Onand p.1 linesno more 6-7 ,at than the end that. of the Only article rarely title, theredid we is no observe letter R –a should be 13 + 23 + 33 + 43 + ... + 253 CALCULATOR and was written CALCULATO. student who might be familiar with the useful algebraic 2)Onoptions p.1 line of24 thein the calculator, right column before after the sign teacher’s & should beexplanation. Gascon, 2004 ).We instead ask Gasconstudents, to do this with their calculators and .(2004 This explanation begins by inviting the student We ask to studentsenter theto firstdo this one, with who their correctly calculators completes and thisthe firsttask, owillne, winwho correctly the following algebraic expression in the calculator:We ask students thisto do competition. this with their Our calculators experience and theshows first othatne, whonot correctlymany 3)On p.3 lines 9-10 (left column) in the formula shouldWecomplete beask ABstudentss insteadthis to task,do thisB will with win their this calculators competition and the. Ofirstur experienceone, who correctly shows that not many completes this task,students will win are this familiar competition with. O urthe experience summation shows operator that not many∑, completestudentss this are task, familiar will win with this the competition summation. O uroperator experience shows, which that isnot available
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