• Title/Summary/Keyword: Fundamental Theorem of Calculus

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A study on a genetic history of the fundamental theorem of calculus (미적분학의 기본정리에 대한 역사-발생적 고찰)

  • 한대희
    • Journal of Educational Research in Mathematics
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    • v.9 no.1
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    • pp.217-228
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    • 1999
  • The fundamental theorem of calculus is the most 'fundamental' content in teaching calculus. Since the aim of teaching the theorem goes beyond simple application of it, it is difficult to teach it meaningfully. Hence, for the meaningful teaching of the fundamental theorem of calculus, this article seeks to find the educational implication of the fundamental theorem of calculus through reviewing the genetic history of it. A genetic history of the fundamental theorem of calculus can be divided into the following five phases: 1. The deductive discovery of the fundamental theorem of calculus 2. Galileo's Law of falling body and the idea of the fundamental theorem of calculus 3. The discovery of the fundamental theorem of calculus and Barrow's proof 4. Newton's mensuration 5. the development of calculus in 19th century and the fundamental theorem of calculus The developmental phases of the fundamental theorem of calculus discussed above provides the three educational implications. first, we can rediscover this theorem through deductive methods and get the ideas of it in relation to kinetic problems. Second, the developmental phases of the fundamental theorem of calculus shows that the value of this theorem lies in the harmony of its theoretical beauty and practicality. Third, Newton's dynamic image of this theorem can be a typical way of understanding the theorem. We have different aims of teaching the fundamental theorem of calculus, according to which the teaching methods can be adopted. But it is self-evident that the simple application of the theorem is just a part of teaching the fundamental theorem of calculus. Hence we must try to put the educational implications reviewed above into practice.

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A Study on the Fundamental Theorem of Calculus : Focused on the Relation between the Area Under Time-velocity Graph and Distance (미적분의 기본정리에 대한 고찰 - 속도 그래프 아래의 넓이와 거리의 관계를 중심으로 -)

  • Joung, Youn-Joon;Lee, Kyung-Hwa
    • Journal of Educational Research in Mathematics
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    • v.19 no.1
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    • pp.123-142
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    • 2009
  • Dynamic context is considered as a source for intuitive understanding on the calculus. The relation between the area under time-velocity graph and distance is the base of the dynamic contexts which are treated in the integral calculus. The fundamental theorem of calculus has originated in dynamic contexts. This paper investigated the fundamental theorem of calculus via the relation between the area under time-velocity graph and distance. And we analyzed mathematics textbooks and the understanding of students. Finally we suggest some proposal for the teaching of the fundamental theorem of calculus.

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An exploration of alternative way of teaching the Fundamental Theorem of Calculus through a didactical analysis (미적분학의 기본정리의 교수학적 분석에 기반을 둔 지도방안의 탐색)

  • Kim, Sung-Ock;Chung, Soo-Young;Kwon, Oh-Nam
    • Communications of Mathematical Education
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    • v.24 no.4
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    • pp.891-907
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    • 2010
  • This study analyzed the Fundamental Theorem of Calculus from the historical, mathematical, and instructional perspectives. Based on the in-depth analysis, this study suggested an alternative way of teaching the Fundamental Theorem of Calculus.

Design of Teacher's Folding Back Model for Fundamental Theorem of Calculus (미적분학의 기본정리에 대한 교사의 Folding Back 사고 모형 제안)

  • Kim, Bu-Mi;Park, Ji-Hyun
    • School Mathematics
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    • v.13 no.1
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    • pp.65-88
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    • 2011
  • Epistemological development process of the Fundamental Theorem of Calculus is considered in a history of mathematical notions and the genetic process of the Fundamental Theorem is arranged by the order of geometric, algebraic and formalization steps. Based on this, we studied students' episte- mological obstacles and error and analyzed the content of textbooks related the Fundamental Theorem of Calculus. Then, We developed the "Folding Back Model" of the fundamental theorem of calculus for students to lead meaningful faithfully. The Folding Back Model consists of "the Framework of thou- ght"(figure V-1) and "the Model of genetic understanding of concept"(figure V-2). The framework of thought in the Folding Back Model is included steps of pedagogical intervention which is used "the Monitoring working questions"(table V-3) by the mathematics teacher. The Folding Back Model is applied the Pirie-Kieren Theory(1991), history of mathematical notions and students' epistemological obstacles to practical use of instructional design. The Folding Back Model will contribute the professional development of mathematics teachers and improvement of thinking skills of students when they learn the Fundamental Theorem of Calculus.

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RIEMANN-LIOUVILLE FRACTIONAL FUNDAMENTAL THEOREM OF CALCULUS AND RIEMANN-LIOUVILLE FRACTIONAL POLYA TYPE INTEGRAL INEQUALITY AND ITS EXTENSION TO CHOQUET INTEGRAL SETTING

  • Anastassiou, George A.
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.6
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    • pp.1423-1433
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    • 2019
  • Here we present the right and left Riemann-Liouville fractional fundamental theorems of fractional calculus without any initial conditions for the first time. Then we establish a Riemann-Liouville fractional Polya type integral inequality with the help of generalised right and left Riemann-Liouville fractional derivatives. The amazing fact here is that we do not need any boundary conditions as the classical Polya integral inequality requires. We extend our Polya inequality to Choquet integral setting.

A STUDY ON UNDERSTANDING OF DEFINITE INTEGRAL AND RIEMANN SUM

  • Oh, Hyeyoung
    • Korean Journal of Mathematics
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    • v.27 no.3
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    • pp.743-765
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    • 2019
  • Conceptual and procedural knowledge of integration is necessary not only in calculus but also in real analysis, complex analysis, and differential geometry. However, students show not only focused understanding of procedural knowledge but also limited understanding on conceptual knowledge of integration. So they are good at computation but don't recognize link between several concepts. In particular, Riemann sum is helpful in solving applied problem, but students are poor at understanding structure of Riemann sum. In this study, we try to investigate understanding on conceptual and procedural knowledge of integration and to analyze errors. Conducting experimental class of Riemann sum, we investigate the understanding of Riemann sum structure and so present the implications about improvement of integration teaching.

A study on the Relationship between Indefinite Integral and Definite Integral (부정적분과 정적분의 관계에 관한 고찰)

  • Joung, Youn-Joon;Lee, Kyeong-Hwa
    • School Mathematics
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    • v.11 no.2
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    • pp.301-316
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    • 2009
  • There are two distinct processes, definite integral and indefinite integral, in the integral calculus. And the term 'integral' has two meanings. Most students regard indefinite integrals as definite integrals with indefinite interval. One possible reason is that calculus textbooks do not concern the meaning in the relationship between definite integral and indefinite integral. In this paper we investigated the historical development of concepts of definite integral and indefinite integral, and the relationship between the two. We have drawn pedagogical implication from the result of analysis.

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Uniformity in Highschool Mathematics Textbooks in Definite Integral and its applications\ulcorner (정적분과 응용- 교과서 내용의 균일성\ulcorner)

  • 석용징
    • Journal of Educational Research in Mathematics
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    • v.11 no.2
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    • pp.307-320
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    • 2001
  • Traditionally, there are many inherent restrictions in highschool mathematics textbooks. They are restricted in its contents and inevitably resorted to reader's ability of intuition. So they are usually lacked logical precisions and have various differences in expressions. We are mainly concerned with the definite integral and its applications in current highschool mathematics II textbooks according to 6th curriculum. We choose 6 of them arbitrarily and survey by comparison to deduce some controversial topics among them as follows. 1) absurd metaphors in formula process 2) confusions in important notations and too much choices in terms and statements. 3) lack of precisions in - teaching hierarchy (between some contents of Physics and the applications of definite integral) - introducing a proof of theorem (fundamental theorem of Calculus I) - introducing the methods (integral substitutions 1, ll) 4) adopting small topics such as - mean value theorem of integral - integrals with variable limits. In coming 7th curriculum, highschool students in Korea are supposed to choose calculus as a whole, independent course. So we hope that the suggested controversial topics are to be referred by authors to improve the preceding Mathematics ll textbooks and for teachers to use them for better mathematics education.

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An Analysis of a Teacher's Decision Making in Mathematics Lesson: Focused on Calculus Class in Science Academy (수학 수업에서 교사의 의사결정 행동 분석 - 과학영재학교의 미적분학 수업 사례연구 -)

  • Oh, Taek-Keun;Kim, Jee-Ae;Lee, Kyeong-Hwa
    • School Mathematics
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    • v.16 no.3
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    • pp.585-611
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    • 2014
  • The purpose of this study is to understand the decision-making behavior of a mathematics teacher in science academy of Korea by applying the framework of class analysis through the theory of goal-oriented decision-making. To this end, we selected as the participant a mathematics teacher in charge of the class of basic calculus of science high school for the gifted in the metropolitan area, and observed the teacher's lesson. Based on a questionnaire derived from previous studies, we analyzed goals, orientations and resources of the teacher. Research results show that there are certain teaching routines by analyzing the behavior patterns that appear repeatedly in the teacher's lesson. Also we understand that it can be used on goals, orientations and resources of the teacher to adequately explain his teaching routine. In the present study, in particular, it was found to have a similar but partially different routines to the teaching routines shown in the study of Schoenfeld. From these findings, We can derive the implications that the theory of goal-oriented decision making can be suitably used as analytical tool for understanding the behavior of the teacher who pursue a productive interaction in mathematics lesson in Korea.

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