• Communications of the Korean Mathematical Society
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• v.31 no.1
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• pp.115-129
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• 2016

Fractional calculus operators have been investigated by many authors during the last four decades due to their importance and usefulness in many branches of science, engineering, technology, earth sciences and so on. Saigo et al. [9] evaluated the fractional integrals of the product of Appell function of the third kernel $F_3$ and multivariable H-function. In this sequel, we aim at deriving the generalized fractional differentiation of the product of Appell function $F_3$ and multivariable H-function. Since the results derived here are of general character, several known and (presumably) new results for the various operators of fractional differentiation, for example, Riemann-Liouville, $Erd\acute{e}lyi$-Kober and Saigo operators, associated with multivariable H-function and Appell function $F_3$ are shown to be deduced as special cases of our findings.

• Communications of Mathematical Education
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• v.24 no.1
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• pp.259-282
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• 2010

In this paper, we introduce a modified Moore Method designed for the multivariable calculus course, and discuss about the effective teaching and learning method by observing the changes in the understanding of students' learning and the effects on students' learning in the class implemented by this modified Moore Method. This teaching experiment research was conducted with the 15 students who took the multivariable calculus course offered as a 3 week summer session in 2008 at H University. To guide the students' active preparation, stepwise course materials structured in the form of questions on the important mathematical notions were provided to the students in advance. We observed the process of the students' small-group collaborative learning activities and their presentations in the class, and analysed the students' class journals collected at the end of every lecture and the survey carried out at the end of the course. The analysis of these results show that the students have come to recognize that a deeper understanding of the subjects are possible through their active process of search and discovery, and the discussion among the peers and teaching each other allowed a variety of learning experiences and reflective thinking.

• Research in Mathematical Education
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• v.16 no.3
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• pp.195-206
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• 2012

We adopt a spirit of Problem based learning to the class of Multivariable Calculus in a school of scientifically talented students and observed effects of our teaching-learning method in the Spring Semester of 2010. Twelve students who enrolled in this class participated in this research. We have proceeded with classroom experiment for the half of semester after midterm exam so that the students could compare our teaching-learning method with usual traditional one in the subject of multivariable calculus. Especially, we investigated changes in the learning attitude and cognitive development of the students toward definition and theorem of mathematics. Each group of 4 students worked on a sheet of our well-designed structured problems of several steps in each class and presented how they understood the way of constructing new definition and related theorems. Instructor's role in this research was to guide students' activities as questioner so that students could attain the clear meanings of definitions and theorems by themselves. We firstly analyzed students' process of mathematization of definition through observing their discussions and presentations as well as their achievements in the quizzes and final exams. Secondly, we analyzed students' class-diaries collected at the end of each class in addition to pre/post surveys.