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Teaching of Division of Fractions through Mathematical Thinking
( Cheng Chun Chor Litwin ) 한국수학교육학회 2013 수학교육 학술지 Vol.2013 No.1
Division of fractions is always a difficult topic for primary school students. Most of the presentations in teaching the topic in textbooks are procedural, asking students to invert the second fraction and multiply it with the first one, that is, (a/b)÷(c/d)=(a/b)×(d/c) Such procedural approach in teaching diminishes both the understanding of structure in mathematics and the interest in learning the subject. This paper discussed the formulation of teaching the division of fractions, which based on research lessons in some primary five classrooms. The formulated lessons started with an analogy to division of integers and working with division of fractions with equal denominators and then extended to division of fractions in general. It is found that the using of analogy helps students to invent their procedure in working the division problem. Some procedures found by students are discussed, with the focus on the development of their invention and mathematical thinking.
Mathematical Thinking through Different Representations and Analogy
( Cheng Chun Chor Litwin ) 한국수학교육학회 2011 수학교육연구 Vol.15 No.1
Mathematical thinking is a core element in mathematics education and classroom learning. This paper wish to investigate how primary four (grade 4) students develop their mathematical thinking through working on tasks in multiplication where greatest products of multiplication are required. The tasks include the format of many digit times one digit, 2 digits times 2 digits up to 3 digits times 3 digits. It is found that the process of mathematical thinking of students depends on their own representation in obtaining the product. And the solution is obtained through a pattern/analogy and "pattern plus analogy" process. This specific learning process provides data for understanding structure and mapping in problem solving. The result shows that analogy allows successful extension of solution structure in the tasks.
Mathematical Thinking through Problem Solving and Posing with Fractions
( Chun Chor Litwin Cheng ) 한국수학교육학회 2012 수학교육연구 Vol.16 No.1
One of the important aims in mathematics education is to enhance mathematical thinking for students. And students posing questions is a vital process in mathematical thinking as it is part of the reasoning and communication of their learning. This paper investigates how students develop their mathematical thinking through working on tasks in fractions and posing their own questions after successfully solved the problems. The teaching was conducted in primary five classes and the results showed that students` reasoning is related to their analogy with what previously learned. Also, posing their problems after solving the problem not only helps students to understand the structure of the problem, it also helps students to explore on different routes in solving the problem and extend their learning content.
A Theoretical Reflection of the Two-Basic approach in China -mathematics lessons in Sine Rule
( Cheng Chun Chor Litwin ) 한국수학교육학회 2011 수학교육 학술지 Vol.2011 No.-
This presentation aims to present a "Two Basic" model of teaching and learning mathematics in China. Two Basics refers to basic knowledge and basic skills. The "Two Basics Teaching Model" (Zhang 2006), which take the form of "Variation Methods" in Problem Solving teaching (Gu, 1994), and the application of "Mathematics Methodology" (Xue, 1983). Teacher intervention is important during the learning process. Such interventions include practice with variation, and working on group of selected related questions so that abstraction of learning is possible. The abstratction process allow students to link up mathematical expression and process. Mathematics knowledge and mathematical skills are intervened. Skills can be developed into knowledge and knowledge can induce skills. Sometimes, skills and knowledge may overlap at some instances. Two Basic approaches in teaching mathematics have its conceptual framework, that is a theory. This theory evolved from development and research in mathematics education in China. A course on teaching the Sine Rule is used as an example to analyze how the Two-Basic theory approach work in the process.
Ceyuan (測圓海鏡) and Jiuyong Yandai (九容演代)
Cheng, Chun Chor Litwin The Korean Society for History of Mathematics 2014 Journal for history of mathematics Vol.27 No.1
The book ${\ll}$Ceyuan Haijing${\gg}$ studies inscribed and circumscribed circles in a right triangle and shows equations that give the diameters of the circles. We discuss the development of mathematical contents written by the scholar Yang Zhaoyun in Qing dynasty on the contents of ${\ll}$Ceyuan Haijing${\gg}$ in his book ${\ll}$Jiuyong Yandai${\gg}$. He derived equations to find the diameters of the circles based on algebraic knowledge known in the Qing dynasty. In this paper, we conclude that Yang's methods in devising the equations include the Gou-Gu Theorem, mathematical expressions derived from Gou-Gu ratio table, and the technique of interchanging triangles and events. We conclude that the Gou-Gu ratio table was a very important tool when Yang devised the equations in ${\ll}$Ceyuan Haijing${\gg}$.