• School Mathematics
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• v.15 no.2
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• pp.243-262
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• 2013

This study attempts to create an analytical framework of the transformation and transmission of knowledge by teachers to students. I focuses on the assertion that the cognitive thinking of a teacher is reflected in his use of mathematical language. Mathematical language is one of the critical elements of communicating mathematical knowledge to students. I examined the cognitive teaching style of different teachers as expressed in their use of mathematical language. An analytical framework of Mathematics Teaching styles was created integrating thinking factors of each visual and analytic style into 5 categories. After that, I regarding the teaching style of mathmatics teachers places its significance not on which teaching style is right or wrong but on identifying the strong and weak points of the teaching styles through actual analysis. With the help of this analytical framework, I conducted an analysis on the videotaped classes and found that the teachers were not biased to one side but in fact there were teachers who demonstrated visual, analytic or mixed teaching style. Therefore, I concludes that math teachers can analyze their teaching styles and improve them through the analytical framework provided in these findings.

• Research in Mathematical Education
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• v.18 no.3
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• pp.171-185
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• 2014

Mathematically deductive reasoning skill is one of the major learning objectives stated in senior secondary curriculum (CDC & HKEAA, 2007, page 15). Ironically, student performance during routine assessments on geometric reasoning, such as proving geometric propositions and justifying geometric properties, is far below teacher expectations. One might argue that this is caused by teachers' lack of relevant subject content knowledge. However, recent research findings have revealed that teachers' knowledge of teaching (e.g., Ball et al., 2009) and their deductive reasoning skills also play a crucial role in student learning. Prior to a comprehensive investigation on teacher competency, we use a case study to investigate teachers' knowledge competency on how to teach their students to mathematically argue that, for example, two triangles are congruent. Deductive reasoning skill is essential to geometry. The initial findings indicate that both subject and pedagogical content knowledge are essential for effectively teaching this challenging topic. We conclude our study by suggesting a method that teachers can use to further improve their teaching effectiveness.

• Communications of Mathematical Education
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• v.26 no.1
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• pp.137-154
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• 2012

Approximation' is one of central conceptions in calculus. A basic conception for explaining 'approximation' is 'tangent', and 'tangent' is a 'line' with special condition. In this study, we will study pedagogically these mathematical knowledge on the ground of a viewpoint on the teaching of secondary geometry, and in connection with these we will suggest the teaching program and the chief end for the probable teaching. For this, we will examine point, line, circle, straight line, tangent line, approximation, and drive meaningfully mathematical knowledge for algebraic operation through the process translating from the above into analytic geometry. And we will construct the stream line of mathematical knowledge for approximation from a view of modern mathematics. This study help mathematics teachers to promote the pedagogical content knowledge, and to provide the basis for development of teaching model guiding the mathematical knowledge. Moreover, this study help students to recognize that mathematics is a systematic discipline and school mathematics are activities constructed under a fixed purpose.

• The Mathematical Education
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• v.45 no.2 s.113
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• pp.217-230
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• 2006

The purpose of this study is to inquire the building process of Pedagogical Content Knowledge of prospective mathematics teachers about the function concepts. For this purpose, We performed the following steps; First, we performed the survey relaying to the prospective mathematics teachers' teaching experiences, capabilities of their error evaluation of the students, and viewpoints about the function concepts. Second, we performed the survey on the subject-matter knowledge about the function concepts and the key items of designing teaching plans about the function concepts. And then, we interviewed the participants to check the results of the surveys and to supplement the necessary contents. The collected data was relatively correlative and analyzed in the process. As a result, we found the followings; First, subject-matter knowledge of prospective mathematics teachers about the function concepts is different depending on the grades. Second, prospective mathematics teachers are building more extended function concepts through the major subjects. Third, the major subjects are important to build the Pedagogical Content Knowledge of function concepts. Fourth, teaching experience plays an important role in transforming subject-matter knowledge of function concepts to Pedagogical Content Knowledge of it. Fifth, building the Pedagogical Content Knowledge means transferring the teacher's viewpoint from himself/herself to the learner.

• Research in Mathematical Education
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• v.6 no.1
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• pp.1-28
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• 2002

This study explored teachers (n = 219) from northern, central, southern and eastern Taiwan concerning their views about children's learning difficulties, mathematical instruction and school mathematics curricular. Results showed that teachers' mathematics knowledge or their instruction methods had no significant influence on their views of children's learning difficulties. Even though teachers indicated that understanding of abstract mathematical concepts was the most prominent difficulty for children, they tended to employ direct instruction rather than constructive and cooperative problem solving in their teaching. However, teachers' views of children's learning difficulties did influence their instructional practice. Results from in-dept interviews revealed that there were some obstacles that prevented teachers from putting constructiveism perspectives of instruction into teaching practice. Further investigation is needed to develop a better understanding of epistemology and teaming psychology as well as to help teachers create constructive learning situations.

• Education of Primary School Mathematics
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• v.8 no.2 s.16
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• pp.97-113
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• 2004

The purpose of this study is to develop instructional methods for the formalized algorithm through informal knowledge in teaching division of fractions. The following results have been drawn from this study: First, before students learn formal knowledge about division of fractions, they knowledge or strategies to solve problems such as direct modeling strategies, languages to reason mathematically, and using operational expressions. Second, students could solve problems using informal knowledge which is based on partitioning. But they could not solve problems as the numbers involved in problems became complex. In the beginning, they could not reinvent invert-and-multiply rule only by concrete models. However, with the researcher's guidance, they can understand the meaning of a reciprocal number by using concrete models. Moreover, they had an ability to apply the pattern of solving problems when dividend is 1 into division problems of fractions when dividend is fraction. Third, instructional activities were developed by using the results of the teaching experiment performed in the second research step. They consist of student's worksheets and teachers' guides. In conclusion, formalizing students' informal knowledge can make students understand formal knowledge meaningfully and it has a potential that promote mathematical thinking. The teaching-learning activities developed in this study can be an example to help teachers formalize students' informal knowledge.

• Journal of Elementary Mathematics Education in Korea
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• v.14 no.3
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• pp.745-768
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• 2010

The purpose of this study at gifted students' solving ability of the given study task by using all knowledge and tools which encompass mathematical contents and curriculums, and developing the teaching learning materials of gifted students in accordance with their level which tan enhance their mathematical thinking ability and develop creative idea. With these considerations in mind, this paper sought for the standard and procedures of teaching learning materials development according to the levels for the education of the mathematically gifted students. presented the procedure model of material development, produced teaching learning methods according to levels in the task of figurate number, and developed prototypes and examples of teaching learning materials for the mathematically gifted students. Based on the prototype of teaching learning materials for the gifted students in mathematics in accordance with their level, this research developed the materials for students and materials for teachers, and performed the modification and complement of material through the field application and verification. It confirmed various solving processes and mathematical thinking levels by analyzing the figurate number tasks. This result will contribute to solving the study task by using all knowledge and tools of mathematical contents and curriculums that encompass various mathematically gifted students, and provide the direction of the learning contents and teaching learning materials which can promote the development of mathematically gifted students.

• Communications of Mathematical Education
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• v.26 no.3
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• pp.301-316
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• 2012

Ball, Thames & Phelps(2008) introduced the idea of Mathematical Knowledge for Teaching(MKT) teacher. Specialized Content Knowledge(SCK) is one of six categories in MKT. SCK is a knowledge base, useful especially for math teachers to analyze errors, evaluate alternative ideas, give mathematical explanations and use mathematical representation. The purpose of this study is to analyze the elementary teacher's SCK. 29 six graders made word problems with respect to division fraction $9/10{\div}2/5$. These word problems were classified four sentence types based on Sinicrope, Mick & Kolb(2002) and then representative four sentence types were given to 10 teachers who have taught six graders. Data analysis was conducted through the teachers' evaluation of the answers(word problems) and revision of students' mathematical errors. This study showed how to know meanings of fraction division for effective teaching. Moreover, it suggested several implications to develop SCK for teaching and learning.

• School Mathematics
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• v.17 no.2
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• pp.179-202
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• 2015

In this research, I explored how to apply the history of mathematics in teacher education and investigated the applicability of Chosun Sanhak (mathematics of Chosun Dynasty) as the program that enriched the mathematical knowledge for teaching of prospective elementary school teachers. This program included not only mathematical knowledge but also socio-cultural knowledge and connection knowledge. Prospective teachers participated in various mathematical activities such as explaining, reasoning and problem solving in this program. The effects of this program are as follows. Prospective teachers learned the subject matter knowledge(SMK) which was helpful in teaching basic concepts and skills of elementary mathematics. Next, this program produced the pedagogical content knowledge(PCK) to prospective teachers by giving ideas how to teach.

• The Mathematical Education
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• v.55 no.3
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• pp.317-334
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• 2016

Knowledge and data interpretation on statistical estimation was important to have statistical literacy that current curriculum was said not to satisfy. The author investigated mathematics teachers' MKT on statistical estimation concerning interpretation of confidence interval by using questionnaire and interview. SMK of teachers' confidence was limited to the area of textbooks to be difficult to interpret data of real life context. Most of teachers wrongly understood SMK of interpretation of confidence interval to have influence upon PCK making correction of students' wrong concept. SMK of samples and sampling distribution that were basic concept of reliability and confidence interval cognized representation of samples rather exactly not to understand importance and value of not only variability but also size of the sample exactly, and not to cognize appropriateness and needs of each stage from sampling to confidence interval estimation to have great difficulty at proper teaching of statistical estimation. PCK that had teaching method had problem of a lot of misconception. MKT of sample and sampling distribution that interpreted confidence interval had almost no relation with teachers' experience to require opportunity for development of teacher professionalism. Therefore, teachers were asked to estimate statistic and to get confidence interval and to understand concept of the sample and think much of not only relationship of each concept but also validity of estimated values, and to have knowledge enough to interpret data of real life contexts, and to think and discuss students' concepts. So, textbooks should introduce actual concepts at real life context to make use of exact orthography and to let teachers be reeducated for development of professionalism.