Minggu, 28 Maret 2010

Mathematics Teacher Education in East Asian Countries - from the Perspective of Pedagogical Content Knowledge

Kyungmee Park (Hongik University, Seoul, Korea)


I. Introduction

In a new era of globalization and information technology, knowledge becomes the most important asset of a society and the driving force of nearly all types of economic, social and political developments. In this context, education is often expected to be the key for the future and can build up the necessary knowledge forces among young generations. Especially, as most East Asian countries lack in a sufficient supply of natural resources, the development of human capital is more crucial in East Asian countries. It is common knowledge that the quality of education highly depends on teachers' abilities. Teachers are the true driving force behind the improvement in educational quality, and effort to improve students' mathematical achievement cannot succeed without parallel attention to their teachers' competence. Thus many countries face challenges in preparing and maintaining a high-quality teaching force of professionals who can teach mathematics effectively, and who can help eventually prepare young people to contribute to the development of society.
Teacher education systems are built on features that are embedded in culture as well as the nature of schooling. The pre-service mathematics teacher preparation varies among East Asian countries both because of different educational environments and social and economic situations. However, East Asian countries share similarities in that there is a high educational zeal and a relatively high social status of teachers in comparison to western countries. In other words, while there are differences between East Asian countries, they still share several common aspects.
Moreover, East Asian countries including Chinese Taipei, Hong Kong, Japan, Korea, and Singapore have consistently shown outstanding performances in a series of international comparative studies, TIMSS 1995, 1999, 2003, and PISA 2000, 2003. Since one of the most important factors which influence student’s performance are the mathematics teachers and the mathematics lessons they receive in school. Therefore, it is significant that a comprehensive discussion is made on the issue of mathematics teacher education in East Asia.
There have been growing concerns about the Pedagogical Content Knowledge from the late 1980's(Shulman, 1987). More recently, a series of studies(Ma, 1999; Leung & Park, 2002; An, Kulm, & Wu, 2004) triggered the discussions about PCK in mathematics education in East Asian countries. This paper will give an overview the mathematics teacher education system in East Asian countries, bring up some issues of mathematics teacher education Korea from the angle of PCK, and then compare the courses offered for pre-service teacher education in East Asian countries also from the perspective of PCK.


II. Overview of Mathematics Teacher Education Programs in East Asian Countries

It is difficult to categorize the types of teacher education program in East Asian countries as prospective teachers are educated in different ways and several different programs exist within a country. It becomes more complicated to compare teacher education programs between primary and secondary school since the training each is usually different. However, for the sake of convenient discussion, teacher education program has been divided into the following two aspects (Leung, 2003). The first is whether the prospective teacher training is conducted in a mono-purpose university or institute (for example, normal universities in China and Taiwan, and Hong Kong Institute of Education in Hong Kong) or a comprehensive university. The second is whether the candidate has acquired a teacher certificate through usually a four year B.Ed.(Bachelor of Education) integrated program or has become a teacher after completing B.Sc.(Bachelor of Science or equivalent general degree) program where s/he has focused on mathematics alone and then taken one to two years of PGDE(Post Graduate Diploma in Education) or M.Ed.(Master of Education).
If the type of program is set up as the x axis (integrated program vs end-on program) and the type of university (mono-purpose vs comprehensive university) as the y axis, we come up with four quadrants. If we are to apply the countries China, Hong Kong, Japan, Korea, Singapore and Taiwan to this, it would look like the following.
Comprehensive university
Integrated program. Korea - secondary
Singapore – primary and secondary
Taiwan – secondary Hong Kong - secondary
Japan - secondary
Singapore - primary and secondary End-on program
China - primary and secondary
Hong Kong -primary
Japan - primary
Korea - primary
Taiwan – primary
Mono-purpose university(institute)


In the diagram above, each country was categorized according to the more dominant category of their teacher education programs. For instance, some of the primary school teachers in Japan and Korea are trained in comprehensive universities, and some secondary teachers in Korean are educated by B.Sc. followed by PGDE. In the case of Japan, half of the junior high school teachers are trained through an integrated in a mono-purpose university while the other half as well as senior high school teachers are trained through a B.Sc. in a comprehensive university plus a PGDE. By the same token, some of the secondary teachers in Hong Kong are trained through an integrated program at a mono-purpose university. Singaporean teacher education program has been included both in the integrated program and end-on program categories because no single method of training was dominant. NIE(National Institute of Education), established in 1991, takes charge of teacher education in Singapore. NIE is part of the Nanyang Technological University and thus viewed as a comprehensive university but it can also be seen as an independent body that exists for a single purpose.

Integrated program vs. end-on program
While we cannot judge whether a specific program of teacher education is absolutely superior or more efficient in comparison to another, we can discuss the strengths and weaknesses of each. First of all, the end-on program usually takes longer consuming five to six years compared to four years for the B.Ed.(or B.Sc. with diploma in Education). As PGDE or M.Ed. is taken as a professional graduate school(similar as law school) that one goes to after undergraduate studies, it is easier to instill professional qualities to the prospective teachers. Another strength of the PGDE or M.Ed. is that teachers trained through the programs think of teaching as a true vocation as they spent their undergraduate years giving sincere consideration to becoming teachers. However, in the B.Sc. course students are taught by mathematicians who make less consideration towards school mathematics and its education, leading to a possible discontinuity between the mathematics learned during four years as undergraduate and that learned during one or two years at PGDE or M.Ed. There is a possibility that the pedagogy dealt with in PGDE or M.Ed does not go beyond a superficial level since student teachers should learn general education courses as well as mathematics education courses within 1-2 years. Thus the link between the subject matter knowledge learned in undergraduate and the subject matter teaching that the student teachers will be facing in the future might not be fully implemented in the graduate courses.
The duration of integrated program is relatively shorter than that of end-on program, yet there is a better chance that students receive education which combines mathematics and pedagogy under the clear purpose to train teachers. While learning about linear algebra, students do not just learn matrix, but probably focus on how to teach the topic as well. During their courses on complex analysis, prospective teachers have chance to discuss the meaningful ways to introduce complex numbers to their future classes. Learning Peano's axiom, prospective teachers have opportunity to envisage how they will explain the properties of natural numbers. Such discussions can lead prospective teachers to true acquisition of PCK.
Of course, the integrated or end-on teacher education program themselves do not guarantee absolute efficiency. For instance, the student-teacher may effectively utilize the mathematical knowledge s/he acquired as an undergraduate and learn the related pedagogy during the graduate course under the B.Sc+PGCE system. Also, since most courses in the B.Ed are taught by mathematicians who rarely link the mathematical topics to pedagogy, the mathematics and the related pedagogy may be separated.
Meanwhile, one factor we must take note of is that the situation differs in the case of primary school teachers depending on whether the teacher is a specialized teacher exclusively instructing math (such as primary teachers in China and Hong Kong) or a general instructor of all subjects (such as primary teachers in Korea and Japan). In the case of the former, the same logic that applies to secondary teachers can be applied. However, in the case of the general instructor, it would be virtually impossible to major in one subject as an undergraduate and then acquire within one to two years of a graduate program the comprehensive attainments needed to become a teacher including native language education, science education and social studies education. Thus, in the case of the comprehensive primary teacher, training through the integrated program would be more adequate.

Mono-purpose university(institute) vs. comprehensive university
While mono-purpose universities(institutes) have a merit in that their primary concern is to nurture teachers, they are limited in their ability to provide a variety of courses due to their small size. They also experience limitation due to the fact that they are composed of students that share the same future career. As education is ultimately an understanding of people, it is better if students are able to meet various students different from themselves and widen their understanding of others. Mono-purpose universities are also outdone by comprehensive universities in terms of facilities such as libraries.
The issue related to this mono-purpose and comprehensive university is whether the elementary and secondary teacher education take place in the same university(institute) or not. Although the elementary and secondary teacher education universities are segregated in many East Asian countries, integration of these two seems to be a trend recently. This is the direction taken by western countries such as the US, UK and France. In case of France, for example, the government runs a unified graduate school IUFM(Instituts Universitaires de Formation des Maitre) which was established in 1990 based on the new Education Law in 1989 and covers pre-school to secondary school teacher education (Comiti & Ball, 1996). In fact, similar integrated teacher education institutions are also found in westernized East Asian countries such as Hong Kong(Hong Kong Institute of Education) and Singapore(NIE).
Such integrated teacher education has advantages in the aspect of continuity and PCK. Even if primary and secondary teachers are not being trained in exactly the same program, there is a higher possibility that primary teachers would be exposed to secondary school math and secondary teachers to elementary school math if they are both being educated in the same institution. If a secondary school teacher knows what math lesson is provided to students at primary school and when, s/he can have a better understanding of the learner who has completed primary school and come to secondary school. In the same light, it is important that primary school teachers know what math lessons their students will receive in the future. Integrated graduate programs are more likely to provide the courses from the perspective of such continuity. Also such integrated teacher education programs hold a merit in that it can enable comprehensive PCK as oppose to fragmented PCK. For example, the numbers that are taught from primary to high school start from natural number and progress through integer, rational number, real number and complex number. It is important that teachers have an understanding of how the number concepts are expanded rather than knowing just a section as well as understanding the closedness and operation within each scope of numbers.

III. Mathematics Teacher Education in Korea and Pedagogical Content Knowledge

Korea has a teacher education system designed and closely controlled by the government, with the Ministry of Education(MOE) and the 16 boards of education in each region. An important characteristic of teacher education in Korea is that the government plays a critical role in pre-service education, certification, teacher selection, and in-service training. Teacher education is both prescriptive and standardized, and can only be attained at accredited institutes. Further, the government determines the admission quota of every institute on an annual base (Kim, 2001).
Most elementary school teachers in Korea are trained at one of the 11 national universities of education (mono-purpose university) including Seoul National University of Education, which offer a bachelor's degree to students after their completing a four-year training. In addition, the Korea National University of Education(KNUE) and Ewha Woman’s University also produce teacher candidates for elementary teaching. In turn, all the elementary school teachers, except teachers from one private university, are trained at national institutions. Under this ‘objective-oriented system’ or ‘closed system’, the government gains greater influence over the quality and form of elementary school education.
Training for secondary mathematics teacher is relatively open, it is offered at comprehensive universities (both national and private) and can follow three tracks: 1) dept of mathematics education in the college of education, 2) B.Sc. parallel enrollment in teacher certification programs, and 3) graduate schools of education. Among these programs, department of mathematics education in the college of education plays a primary role in training secondary mathematics teachers. Graduate schools of education aim mainly at in-service teacher training, and thus the pre-service training is only the minor function of the university.

1. Mathematics Education: mixture vs. compound?

As mentioned in the previous section, secondary mathematics teacher education in Korea mostly takes place in the department of mathematics education in the college of education in comprehensive universities. However, up until a decade or so ago, the curriculum of mathematics education department did not show a significant difference with that adopted by mathematics department in the college of science. That is to say that the pre-service teacher education program conducted by the department of mathematics education was composed of advanced mathematics found in the department of mathematics and general pedagogy found in the department of education. Therefore there has been criticism that Korean teachers experience "double discontinuity" as a result. The first discontinuity comes from the transition from senior high school to university as mostly advance pure mathematics is taught in university. The second comes when the teachers complete pre-service teacher education and return to the classroom where they have to put aside the advanced mathematics they studied in college and teach secondary school mathematics by recalling the lessons they received as high-school students.

secondary school → advanced pure → secondary school
mathematics mathematics mathematics
(as a student) (as a student) (as a teacher)

In order for the pedagogy taught in university to provide substantial suggestions in teaching school mathematics and not fall short merely as general consideration, mathematics and pedagogy have to be "compounded" as mathematics education rather than a mere "mixture" of mathematics and pedagogy. Just as A and B lose their properties and become a different compound after going through a chemical reaction, mathematics and pedagogy need to meet and become a new discipline, mathematics education.

mathematics + pedagogy → mixture
mathematics + pedagogy → compound

chemical reaction

The identity of mathematics education might be illuminated from the lens of Pedagogical Content Knowledge. The discussions of mathematics education-related knowledge such as Content Knowledge, Pedagogical Knowledge and Pedagogical Content Knowledge(which is a special amalgam of content and pedagogy, and teachers' special form of professional understanding) were initiated by Shulman(1987). There have been many interpretations about the nature and scope of PCK, yet it appears that there is no clear definition that has been agreed upon. Shulman(1987) stated that PCK is the ability of the teacher to transform mathematics into forms that are "pedagogically powerful and yet adaptive to the variations in ability and background presented by the students" (p. 15). Also PCK is defined as the blending of content and pedagogy into an understanding of how particular topics, problems or issues are organized and adapted to the diverse interests and abilities of learners, and presented for instruction. In short, PCK is content-specific since it is an interface of content and pedagogy and an understanding of how topics and skills can be organized and taught to pupils.
PCK made another comeback after Ma published the landmark book in 1999. More recently, An, Kulm, and Wu(2004) broadened the scope of PCK and made a model of PCK which includes three components; knowledge of content , knowledge of curriculum and knowledge of teaching . Taking the risk of simplification, mathematics and education correspond to CK and PK respectively, and mathematics education as the integration of pedagogy and content is referred to as PCK.
To discuss the PCK in more detail, let’s consider the concept of the ‘convergence of sequence’. The convergence of sequence is defined on the following two levels.

【 high school level 】
If n becomes larger and larger, then {pn} goes closer and closer to p. In this case we say that {pn} converges to p.

【 college level 】
For every ε>0, there is an integer N such that n≥N implies that d(pn, p)<ε . In this case we say that {pn} converges to p.

Mathematics teachers are first required to understand the convergence as rigorous mathematics in college level, and transpose this to the high-school level which provides ease in understanding by intuition. At the same time, mathematics teachers need to contemplate the method on how to facilitate students’ understanding of the concept of the convergence of sequence. In addition, it would be helpful if mathematics teachers have the historical knowledge that the epsilon-delta method was established in the 19th century in the process of the theorization of the limit concept of functions by Cauchy, Riemann, Dirichlet etc. All this knowledge encompasses the concept of convergence can be called as PCK. Thus PCK as well as CK and PK should be emphasized in the teacher education program.

2. PCK and Teacher Employment Test in Korea

The teacher education program provided by the department of mathematics education in Korea has evolved with an emphasis on PCK. One of the reasons behind this development is the Teacher Employment Test. The existing overcapacity of secondary school teachers in Korea became more severe in the late 1990's after the Asian financial crisis. In Korea, completing the four-year pre-service teacher education at a college of education does not in itself qualify the graduates to teach in public schools. Completion of the four-year training awards the graduates with a teacher's certificate which makes them eligible to teach in private schools. To qualify to teach in public schools, however, certificate holders are required to pass a very demanding national examination, the Teachers Employment Test (TET). The TET for elementary school teachers is not competitive, usually the success rate is lower than 2:1. However, there is a tough competition for secondary school teachers. The low success rate of the TET for secondary school teachers has earned it the nickname the 'bar exam' of the college of education. As PCK as well and CK and PK is addressed with importance in the TET which has a big influence over pre-service teacher education in Korea, increasing interest has been shown to PCK accordingly (Kwon, 2004; Park & Leung, 2003).
The preliminary TET consists of three parts, encompassing mathematics (content knowledge or subject matter knowledge), general educational theory(pedagogical knowledge), and mathematics education(pedagogical content knowledge). The student teachers who pass the preliminary TET are required to take further essay test (making a teaching plan), interview, and teaching demonstration. The proportion of preliminary TET in mathematics, education, and mathematics education is as follows.

Content Percent of
items Item type Relevant knowledge
Education in general 20% Multiple choice items Pedagogical knowledge
Mathematics Linear Algebra
Abstract Algebra
Complex Analysis
Topology
Real Analysis
Differential Geometry
Number Theory
Probability and Statistics
Discrete Mathematics 54% Open-ended items Content knowledge
or
Subject matter
knowledge
Mathematic Education 26% Open-ended items Pedagogical content
Knowledge
[Table 1] Content components of preliminary Teacher Employment Test for secondary school teachers

The following is part of the mathematics education questions included in the TET held in December 2004. Items on mathematics and education in general are provided in Appendix 1.

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In order to find out if the students understand the geometrical aspect of derivatives, the teacher presented the students with several graphs and asked them to draw a tangent line from point P. The most common mistake the students made was as follows.



1. Explain the reason behind such a mistake from the perspective of “epistemological obstacle”

【Model Answer】
“Epistemological obstacle” refers to a case where knowledge that was successful and useful in a special situation is found to be inadequate in a new or more comprehensive situation. The error made by the students seems to be one stemming from the tangent line to the circle which was dealt in the 7th grade mathematics. Such an error was made when the lesson that a circle and line meet at a point the line becomes the tangent line of the circle is simply expanded and applied to curved lines by the students.

2. Based on Freudenthal's histo-genetic principle of mathematization and differentiation, state three factors, in order, that are needed in teaching derivatives.

【Model Answer】
Historically, the differentiation concept was created in the process of mathematizing the tangent line problem and velocity problem. Therefore, the realistic context has to be presented first before making formal definitions of derivatives. The three factors are 1) tangent line as a limit of secants, 2) instantaneous velocity as a limit of average velocity, 3) definition of derivative
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To solve the above questions, the teacher-to-be needs to be armed not only with knowledge on pure mathematics or general pedagogy but also with PCK in order to interpret the topic of mathematics from an educational perspective.

3. PCK embedded in pre-service teacher education program in Korea

Aside from the TET, another reason that PCK is being addressed with importance in pre-service teacher education programs is the periodic evaluations conducted on colleges of education since the end of the 90's. One important criteria of this evaluation is the ratio of mathematics education courses that emphasize PCK. As a result, more colleges of education are establishing courses that address PCK such as Didactics of Mathematics, History of Mathematics, Problem Solving, Teaching Algebra and Teaching Geometry. Just as tests are the baton that orchestrates the education in Korean elementary, middle and high schools, the TET and the evaluations of colleges of education are becoming important factors in determining pre-service teacher education. The following are examples of how PCK is dealt with in PCK courses.

PCK in function (correspondence vs. dependency)
In order to instruct students efficiently on function, the teacher needs to understand the historic background behind the concept of function. In the 17th century, Leibniz defined function from the perspective of dependency that variable y was determined as variable x changed. In the 18th century, Euler viewed function as something that could be presented in analytic formula. However, with the discovery that a single function can be expressed in numerous formulae using Fourier series and Taylor series, the perspective that function was a formula(expression) changed. Consequently, in the 19th century, Dirichlet considered function as arbitrary correspondence. With the establishment of the set theory in the early 20th century, it was defined as a Cartesian product and a relation.
In the previous school curriculum in Korea, function used to be introduced from the “correspondence” perspective in the 7th grade. However, this was inadequate in light of the histo-genetic principle that students should be instructed according to the historic phases of mathematics. With the revision in the curriculum, function taught in the 7th grade is now defined from the “dependency” perspective following topics on proportional and inverse proportional. After a three year interval, students are taught in the 10th grade a new definition of function from the perspective of correspondence, which reflects more proactively the histo-genetic principle.

PCK in geometric proof (synthetic method vs. analytic method)
The two methods in geometric proof are the "synthetic method" and "analytic method". The analytic method which can be traced back to Pappus searches for a method where the proposition needed proving is assumed to be true (or the answer needed finding is assumed) from which various propositions are induced. On the other hand, the synthetic method which has its origin in Euclid's Elements, follows a deductive path which starts from a proposition which is already known or assumed to be true and follows the process to reach the proposition which needs to be proved in the final stage.
For example, the first step to prove that there is one inscribed circle in a triangle using the synthetic method is to draw the angle bisector. However, it is not easy for students to think of an angle bisector to show the existence of inscribed circle because the concept of inscribed circle and that of angle bisector are quite remote. Conversely, the analytic method is one that students can feel more comfortable with. If it is assumed that an inscribed circle does exist, there must be an inner center which has to be the same distance away from the three sides. Such deduction can lead the student to the idea of drawing an angle bisector. The analytic method of thinking is much more natural and the various stages of thinking create a logical causality.
It is highly probable that the early mathematicians who proved the theories of geometry utilized the analytic method of thinking. However, analysis as a method of discovery is commonly hidden in completed ready-made mathematics, and textbooks tend to present deductive proof in sophisticated manner. If teachers use ideas that utilize the analytic method when instructing students of proof, the students would face less difficulty in learning proof.

IV. Mathematics Teacher Education Courses in East Asian Countries

Pre-service mathematics teacher education programs consist of a variety of courses including liberal arts courses and even sports courses as well as mathematics, education and mathematics education courses. In other words, teacher education programs are comprised of more than mathematics(CK), general education(PK) and mathematics education courses(PCK). Here, only the courses corresponding to CK, PK, PCK were included.


When categorizing the courses, you can judge whether the course falls into the categories of CK, PK or PCK by looking at its title. For example, ‘number theory' and 'complex variables' are definitely included in CK, 'educational sociology' is surely PK, and 'mathematical problem solving' and ‘computers in mathematics education' are PCK courses. However, the title of the course itself is not an absolute reflection of the content that is covered. For instance, the course 'psychology of mathematics learning' is clearly included in PCK, but if the part on Piaget's developmental stage, Bruner's structure of knowledge, Ausubel's meaningful learning are taught as a general theory out of reference from school mathematics, it would be closer to PK. On the other hand, even a theory stemming from general education such as that of Piaget, Bruner, and Ausubel would become a PCK if it is taught in close linkage to school mathematics or if it covers the generic theories of mathematics education such Dienes’ mathematical variability theory and Skemp’s schematic learning, as it is implied in its title. The opposite could also be found. Even if the course is named ‘educational psychology’, it would have strong features of a PCK course if it was a course prepared exclusively for prospective mathematics teachers and taught with consideration to school mathematics. As such, it may be irrational to categorize the courses according to their names. However, judgments were made inevitably on the titles of the courses in order to review the trend of teacher education programs in various countries although such a distinction may be irrational.


1. Curricula for Teacher Education in Korea
The distribution of mathematics, education, and mathematics education courses slightly differs from one university to other; nevertheless the curricula of pre-service teacher education in Korea are quite similar. As representative teacher education curricula in Korea, the curricula of Gyeongin National University of Education and Seoul National Education were selected.

Curricula for Elementary Teacher Education (B.Ed. in mono-purpose university)
Gyeongin National University of Education - Department of Mathematics Education
Category Titles of courses Units
Content knowledge Introduction to algebra (3)
Introduction to geometry (3)
Probability and statistics (3) Required
26-29
Introduction to analysis (3)
Set theory and topology (3) Elective
Pedagogical content knowledge Mathematics education 1 (2)
Mathematics education 2 (3)
Didactics of elementary mathematics education (3)
Required
Guidance for mathematics problem solving (3)
Psychology of mathematics for instruction (3) Elective
Pedagogical knowledge Education in general courses 20
Other subject matter education Korean language education, Moral education, Social studies education, Physical education, Science education, English education, Music education, Art education etc 42-44
Practicum 5
One unit means that one hour per week during one semester consists of 15 weeks.

In order to get elementary school teacher certificates with mathematics education as the declared major, pre-service elementary school teachers need to take at least 26 (up to 29) credits from mathematics courses and mathematics education courses. Korean pre-service primary teachers are required to take relatively small number units in mathematics and mathematics courses because Korean elementary teachers are comprehensive teachers and they need to take other subject matter education courses as well as mathematics education courses.

Curricula for Secondary Teacher Education (B.Ed. in comprehensive university)
Seoul National University – College of Education – Dept. of Mathematics Education
Category Titles of courses Units
Content knowledge Linear algebra 1(3)
Linear algebra 2(3)
Number theory (3)
Differential equations (3)
Real analysis (3)
Advanced analysis 1(3)
Multiple variable functions (3)
Complex analysis (3)
Abstract algebra 1(3)
Abstract algebra 2(3)
Geometry (3)
Introduction to differential geometry (3)
Combinatorics (3)
Topology 1(3)
Topology 2(3)
Statistics (3)
Numerical analysis (3) 33 units
(11 out of 17 courses)
Pedagogical content knowledge Teaching secondary school mathematics (3)
Theory of mathematics education (3)
Computer and mathematics education (3)
Mathematics and education (3) 9 units
(3 out of 4 courses)
Pedagogical knowledge Education in general courses 9 units
Practicum 3 units
In comparison with the curricula of pre-service elementary school teacher program, subject matter knowledge(mathematics) is much more emphasized in the curricula of secondary mathematics teacher institutions. This is a natural phenomenon considering the fact that more mathematical proficiency is required for secondary school teachers.
2. Curricula for Teacher Education in China
Curricula for Secondary Teacher Education (B.Ed. in mono-purpose university)
East China Normal University – Dept. of Mathematics Education
Category Titles of courses Units
Content knowledge Analytic geometry and higher algebra 1 (5)
Analytic geometry and higher algebra 2 (5)
Calculus (5)
Multivariate calculus (5)
Mathematical analysis (5)
Ordinary differential equations (3)
Complex analysis (3)
Probability and statistics (4)
Abstract algebra (3)
Differential geometry (3)
Mathematical equation (3) 44 units
Pedagogical content knowledge Mathematical software and mathematical experiments (3)
Mathematical modeling (3.5) 6.5 units
Pedagogical knowledge Classroom language (1)
Education theory (2)
Education technology (2)
Psychology (2)
Teaching methods in mathematics (3) 10 units
Practicum 6 units

3. Curricula for Teacher Education in Taiwan

Curricula for Elementary Teacher Education (B.Ed. in mono-purpose university)
National Taipei Teacher College - Department of Mathematics Education
Category Titles of courses Units
Content knowledge Information science (2)
Calculus1 (3)
Calculus2 (3)
Linear algebra 1 (3)
Linear algebra 2 (3)
Advanced calculus 1 (3)
Advanced calculus 2 (3)
Algebra (3)
Discrete mathematics (3)
Probability and statistics 1 (3)
Probability and statistics 1 (3)




Required 59-80
Differential equation
Real analysis
Complex analysis
Geometry
Topology
Advanced algebra

Elective
Pedagogical content knowledge Introduction to mathematics education(2)
Conceptual development in mathematics(3)
Research in mathematics curriculum (3)
Required
Study of problem solving in mathematics
Teaching aid for mathematics
History of mathematics
Studies in mathematics education
Elective
Pedagogical knowledge Education in general courses 65
Other subject matter education Madarin education, Moral education, Social studies education, Physical education, Science education, English education, Music education, Art education
Practicum 6

Curricula for Secondary Teacher Education (B.Ed. in comprehensive university)
National Taiwan Normal University– College of Science – Dept. of Mathematics Education
Category Title of Courses Units
Content knowledge Calculus1 (4)
Calculus2 (4)
Linear algebra 1(3)
Linear algebra 2(3)
Number theory (3)
Advanced calculus 1 (4)
Advanced calculus 2 (4)
Algebra 1 (3)
Algebra 2 (3)
Differential equations (3)
Complex analysis (3)
Advanced linear algebra 1(3)
Advanced linear algebra 2(3)
Differential geometry 1 (3)
Differential geometry 2 (3)






Required 51
Set theory (3)
Analytic geometry (3)
Advanced geometry1 (3)
Advanced geometry2 (3)
Topology 1 (3)
Topology 2 (3)
Real analysis (3)
Introduction to Fourier series (3)
Functional Analysis (3)



Elective 24
Pedagogical content knowledge Mathematics problem solving (2)
Mathematics learning 1 (2)
Mathematics learning 2 (2)
Mathematics teaching and assessment (2)
Computer and mathematics (2)
Study in mathematics teaching materials and methods(2)


Required 12
Pedagogical knowledge Education in general courses 10
Practicum 4

4. Curricula for Teacher Education in Hong Kong

Curricula for Elementary Teacher Education (B.Ed. in mono-purpose university)
Hong Kong Institute of Education – Major in Mathematics Education
Category Titles of courses Credit points
Content knowledge Geometry and measurement (3)
Elementary number theory (3)
Elementary linear algebra (3)
Elements of mathematical analysis (3)
Discrete mathematics (3)
Introduction to probability (3)

Required
18
Pedagogical content knowledge Understanding numbers (3)
Foundation of mathematics (3)
History of mathematics (3)
Problem solving in mathematics (3)
Learning, teaching and assessment in primary mathematics (3)
Issues in primary mathematics education (3)
Technology assisted learning in primary mathematics (3)


Required 21
Pedagogical knowledge Education in general courses 33
Practicum 16

Curricula for Secondary Teacher Education (PGDE in comprehensive university)
The University of Hong Kong– Faculty of Science
Category Title of Courses hours
Content knowledge Done in the undergraduate program
3 years full-time study
Pedagogical content knowledge Major methods course

Strand 1: Generic Issues
The Context of School Mathematics
Aims and objectives
The mathematics curriculum
Assessment
The Nature of Mathematics and its Learning
Re-thinking mathematical concepts
Historical development of mathematics
Problem-solving and investigating
Psychological theories of learning
mathematics
Individual differences in learning mathematics
Becoming a Mathematics Teacher
Planning
Effective teaching
Explaining and questioning
Classroom organization
Use of IT and other resources
Reflection and professional development
Strand 2: Specific Applications
Presentation of ideas related to the teaching and learning of specific topics in the mathematics curriculum









Required 84 hours
Pedagogical knowledge Education in general courses Required 48 hours
Elective 18 hours
Practicum 14 weeks

5. Curricula for Teacher Education in Singapore

Curricula for Elementary Teacher Education (PGDE in comprehensive university)
National Institute of Education
Category Title of courses Units
Content knowledge Done in the undergraduate program
Pedagogical content knowledge Art education (8)
English language education (8)
Mathematics education (8)
Music education (8) Science education (8)
Social Studies education (8)
* Student-teachers who major in mathematics are expected to take 3 curriculum studies areas including mathematics. Another option is taking 2 curriculum studies areas and take additional 8 units in subject matter courses Elective 16-24
Pedagogical knowledge Educational psychology 1 (2)
Educational psychology 2 (2)
ICT for engaged learning (2)
The social context of teaching and learning (2)

Required 8
Communication skills for teachers (2) 2
Practicum 10

Curricula for Secondary Teacher Education (PGDE in comprehensive university)
Category Title of courses Units
Content knowledge Done in the undergraduate program
Pedagogical content knowledge The teaching of mathematics (9)
The teaching of lower secondary mathematics (9)
* Prospective teachers of various subjects are trained in the same program. Student-teachers in major in mathematics are expected to take the above two courses. 18
Pedagogical knowledge Educational psychology 1 (2)
Educational psychology 2 (2)
ICT for engaged learning (2)
The social context of teaching and learning (2)

Required 10
Select 1 course among 23 courses Elective
Communication skills for teachers (2) 2
Practicum 10

As we have seen, East Asian countries have a variety of teacher education programs and accordingly a variety of curricula. As the definition of a unit within a curriculum differs depending on the program, it is more appropriate to compare the proportion of the program that is taken up by PK, CK and PCK rather than comparing the number of units. However, this also entails many difficulties. For example, in the case of Korea and Taiwan, the primary teacher education programs only show the total units of PCK and CK programs and therefore one cannot know the individual proportion. Also, prospective teachers who choose PGDE acquire Content Knowledge through B.Sc. which makes it equally difficult to know the proportion of each. Taking these few factors into account, the following tendencies can be found. First, the proportion of PK courses as well as courses covering other subject matter education is high in elementary teacher education programs. This seems to be the case as primary teachers tend to be comprehensive instructors teaching all subjects, and the level of CK demanded for in primary school mathematics is not so high. Second, the curriculum of secondary teacher education programs is composed around CK, especially in the case of B.Ed. courses, so that PCK plays a secondary role.


V. Discussion
Zero-sum game
Teacher preparation programs tend to work in a zero-sum game where additional preparation in one area will result in decreased preparation in some other area. In other words, emphasis on CK within a teacher education program results in decrease in PK or PCK (Jeremy, Duane, & Kimberly, 2003). Therefore, one of the challenging issue in teacher education program is the need to find a proper balance between the knowledge and understanding of the discipline(mathematics) and the pedagogy needed to transform it in way by which learning among students may be fostered (Gopinathan, Ho, & Tan, 2001).
Teachers' mathematical knowledge is a critical foundation in the practice of teaching, and instruction is shaped by what teachers know and do not know about the subject they teach. As students would learn more mathematics if their teacher knows more mathematics, no one would disagree that the teacher needs to take many math-related courses and learn about mathematics on an advanced level. In order to teach school mathematics, the teacher must have a complete mastery of it which can be acquired by being proficient in mathematics of a much higher level than that which is taught in the classroom. For instance, a teacher who has a good knowledge of ‘Analysis’ ‘will be able to teach continuity and differentiation of function with a sure focus on their meanings. Even in cases where the themes of advanced mathematics and school mathematics are not directly connected, a teacher who has experienced the thought process required in advanced-level math will have a better understanding of the essence of school mathematics. In this regards, the Glenn commission(U.S. Department of Education, 2000) stated that "High quality teaching requires that teachers have a deep knowledge subject matter. For this there is no substitute" (p. 22).
However, as some studies have shown, the number of mathematics courses taken by the teacher during his/her training has a positive relationship with students' achievement but tends to show a weaker correlation after a certain level (Monk, 1994; Darling-Hammond, 2000). Askew(1999) noticed that for British elementary teachers “being highly effective was not positively associated with higher levels of qualification in mathematics. The amount of continuing professional development in mathematics education that teachers had undertaken was a better predictor of their effectiveness than the level to which they had formally studied mathematics" (p. 96). Given the results of such studies, there is a need to put priority on PCK in teacher preparation programs.

Toward more emphasis on PCK
Teaching was considered to have a personal and artistic nature, and not perceived as something to be developed systematically. Pedagogy was also seen as generic, not as something that was deeply interwoven with the particular content(Comti & Ball, 1996). Pedagogy should be topic (within the subject) specific as well as subject specific. During lessons, the approach towards probability should definitely be different from the approach taken to teach quadric equations. In this light, there should be a stronger emphasis on PCK, which is an interface of content and pedagogy, an understanding of how topics and skills can be organized and taught to pupils. That is because, to work with the words of Kant, ‘pedagogy without mathematics is empty, mathematics without pedagogy is blind.’
One of the main components of PCK is didactic transposition. Didactic transposition delves into the multi-dimensioned variation process that begins with the 'advanced mathematics as a discipline' to 'mathematics as a school subject' or 'mathematics for mathematicians' to 'mathematics for students'. In this process, one of major issue is that how to transform and simplify the topics of mathematics into something that can easily be understood by students, without distorting the nature of content.
As stated earlier, while it may be difficult to find the optimal composition of courses in a teacher education curriculum, we can at least find a direction that PCK should be stressed more than it currently is. PCK gives identity to mathematics education as a discipline. Also, PCK is not simply mathematics, the field of the mathematician, or pedagogy, the field of the education in general scholar, but a third ability all together which can guarantee the expertise of the mathematics teacher.

References

An, S., Kulm, G., and Wu, Z. (2004). The pedagogical content knowledge of middle school mathematics teachers in China and the U.S. Journal of Mathematics Teacher Education, 7. 145-172.
Askew, M. (1999). It ain’t (just) what you do: Effective teachers of numeracy. In I. Thompson (ed.), Issues in Teaching Numeracy in Primary School. pp. 91-102. Buckingham, England: Open University Press.
Comiti, C., & Ball, D. L.(1996). Preparing teachers to teach mathematics: A comparative perspective. In A.J. Bishop, K. Clements, C. Keitel, J. Kiplatrick & C. Laborde. (eds.) International Handbook of Mathematics Education. pp. 1123-1154.
Darling-Hammond, L. (2000). Teacher quality and student achievement: A review of state policy evidence. Educational Policy Analysis Archives, 8(1).
Gopinathan, S., Ho, W.K., & Tan, J. (2001). Teacher education and teaching in Singapore: Into the new century. In Y.C. Cheng, M.M.C. Mok, K.T. Trui (eds.) Teaching Effectiveness and Teacher Development. Towards a New Knowledge Base. pp. 407-430. The Hong Kong Institute of Education. Hong Kong: Kluwer Academic Publishers.
Jeremy, A.. K., Duane, A. C., & Kimberly, A. B. (2003). The role of mathematics teachers’ content knowledge in their teaching: A Framework for research applied to a study of student teachers. Journal of Mathematics Teacher Education, 6. 223-252.
Kim, H. S. (2001). Towards achieving high quality pre-service teacher training in Korea. In Y.C. Cheng, M.M.C. Mok, & K.T. Tsui (eds.), Teaching Effectiveness and Teacher Development. pp. 431-452. The Hong Kong Institute of Education. Hong Kong: Kluwer Academic Publishers.
Kwon, O. (2004). Mathematics Teacher Education in Korea. The Korean Presentation at ICME-10. Copenhage, Denmark, July 6, 2004.
Park, K.M. & Leung, F.K.S. (2003). Factors contributing to East Asian students' high achievement in mathematics: the case Korea. The Mathematics Educator, 1. 7-19.
Leung, F.K.S. and Park, K.M. (2002). “Competent students, competent teachers?”, International Journal of Educational Research, 37(2), 113-129.
Leung, Frederick, K. S. (2003). Issues concerning teacher education in the East Asian region. Asia-Pacific Journal of Teacher Education Development, 6(2). pp.
Ma, L. (1999). Knowing and Teaching Elementary Mathematics. Mahwah, N.J.: Lawrence Erlbaum Associates, Publishers.
Monk, D. (1994). Subject area preparation of secondary mathematics and science teachers and student achievement. Economics and Education Review, 13, 125-145.
Shulman, L. (1987). Knowledge and Teaching: Foundations of the New Reform. Harvard Educational Review, 57(1). 1-22.
United States Department of Education (2000). Before it’s Too Late: A Report to the Nation from the National commission on Mathematics and Science Teaching for the 21st Century. Washington, DC: Author.
Appendix 1. Sample items of TET (2004)

Education (Pedagogical Knowledge)

Which theory encompasses all of the following?

∙ The meaning of growth and habit have to be interpreted through an educational process.
∙ The logic of the subject and the psychology of the learner both have to be considered.
∙ The principles of continuity and interaction are emphasized.

① J. Locke - Empiricism
② J. Dewey - Experience theory
③ H-G. Gadamer - Hermeneutic Experience theory
④ Oakeshott- Experience and Its Modes theory


Mathematics (Content Knowledge)

Abstract Algebra
Fix a field F, and let K be an algebraic extension field of F with [K:F]=10, i.e. dimFK = 10. If f(x)∈F[x] is an irreducible polynomial of degree 3, prove that K does not contain a root of f(x).

Real Analysis
Suppose D is an open subset of R2, and f(x,y) : D→R is a differentiable function. Let G be the surface defined by the graph z=f(x,y), and for each point p of G, define θ as the angle between the normal vector of G at p and the positive z-axis direction. Show that ∫G cos2(θ/2)dS = 1/2 S(G) + 1/2 A(D) where S(G) is the area of G, A(D) is the area of D, both of which are assumed finite, and dS = sec θ dA.

Complex Analysis
Suppose f: D→C is an analytic function defined on an open subset D⊂C(complex plane). If Im f(z) = 2 Re f(z) for all z in D, prove that f is a constant function on D.

Differential Geometry
Suppose α: [a,b]→R3 is a curve parametrized by arc length. Let k(s) be the curvature of α(s), and let N(s) be the principal normal vector of α (s). If α ''(s)≠0 and α (s)+ 1/k(s) N(s) is a fixed point, prove that α lies on a circle.

Topology
Given a topological space X and a surjective map g: X→Y, one may define the quotient topology on Y by the following condition: O is open in Y if and only if g-1 (O) is open in X. When X=R, the set of real numbers with its usual topology, and Y=Z, the set of integers, and f(x)=[x], the largest integer not exceeding x, determine the quotient topology of Z.

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