Effective Teaching: Pedagogical Content Knowledge

Proceeding of International Joint Seminar Garut, Garut, Indonesia, 21 September 2014

Effective Teaching: Pedagogical Content Knowledge Palanisamy Kathirveloo Marzita Puteh Fakulti Sains Matematik Universiti Pendidikan Sultan Idris Corresponding Author: [email protected]

ABSTRACT

Pedagogical Content Knowledge (PCK) plays an important role in classroom instructions. In the teaching and learning process, a PCK involves teachers’ competence in delivering the conceptual approach, relational understanding and adaptive reasoning of the subject matter. The aim of this study is to explore the PCK of mathematics teachers in teaching fractions. Students are facing difficulties in the conceptual understanding of fractions due to teachers’ ignorant and lack of PCK in teaching fractions. Evidence provided in this paper is basically based on the past literature. Without full grasp of PCK, teachers may face difficulty in teaching the subject effectively.

Keyword: Pedagogy Content Knowledge, Conceptual Approach, Relational Understanding, mathematics

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Introduction

In the era of globalization, education needs more real and practical instruction. Not only students have misconceptions of the subject matters, teachers may teach wrong concepts especially in mathematics. Thus, Shulman (1987) proposed PCK to reduce teachers’ misconceptions. PCK is a blending of content and pedagogy in order to enhance the understanding of how particular topics, problems, or issues are organized, represented, and adapted to the diverse interests and various levels of abilities of learners (Shulman, 1987). The important skill that a teacher should process is the capacity to transform the knowledge to be taught to the students in a way that could be easily understood. An actual teaching should not only contain the teacher’s skillful demonstration of his knowledge but should also include the ability to guide the students to understand meaningfully the content of the knowledge (Hansen, 1995). This shows the importance of PCK in instruction of any classroom. Shulman and Grossman (1988) divided knowledge into two components;

substantive knowledge - the key facts, concepts, principles and explanatory frameworks in a discipline and syntactic knowledge - the rules of evidence and proof within a discipline. Most studies have shown that PCK is important to the educators and teachers in the classroom instructions. We can summarize what most studies founds in their findingsproper and structured planning of lesson plans would contribute to better understanding and would cater for various levels of abilities. Issues on Pedagogical Content Knowledge (PCK) Although the academics give varied opinions to the various components of teachers’ knowledge but they agreed about the importance of teachers’ proper in teaching the

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students. Shulman defined PCK as teachers’ interpretations of subject matter knowledge in the context of helping the students learning. We can said that teachers’ understanding of subject matter itself will influence children’s learning. In PCK, students will only response if they are confident with their understanding and will communicate to the teachers. This will help the teachers to ‘catch’ when their misconceptions and errors occur. By this way, teachers may be able to overcome their misconceptions and errors immediately. Thus teachers need to have proper understanding of the subject they teach. Shulman and other researchers have contributed to our understanding of particular aspects of PCK. Shulman (1986) also stressed the need for teachers to know their students thinking. The transformations of subject matter knowledge (SMK) into pedagogical content knowledge is a significant focus in teacher education. Educators and researchers (e.g., Graeber, 1999; Leinhardt, Putnam, Stein, & Baxter, 1991) have stressed that teachers need to alleviate their misconceptions about their subject matter. In New Zealand, a recent report from the Education Review Office claims that 23% of the teachers have low pedagogical content knowledge in mathematics (Education Review Office, 2006). The finding shows the importance and the needs of PCK among teachers. Lack of PCK will influence teacher’s effective teaching and learning process in the classroom. Teachers is a person who is entrusted to enhance students’ achievement. Thus, the teachers should be free of misconceptions and errors. In order to carry out their duties well they need particular and special knowledge that PCK suggests.

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Mathematics Pedagogical Content Knowledge (MPCK) In general, pedagogical content knowledge affects how teachers think about their subject matter knowledge. A skilful and very knowledgeable teacher has the potential to make the learning of mathematics more meaningful to the students (Public Agenda, 2000 in Graham & Fennell, (2001)). Teaching or instructions without deep understanding is meaningless. Effective instruction should involve PCK. PCK covers conceptual and procedural knowledge, and the stages of understanding that they are likely to pass through in moving from a state of having little understanding to the mastery of it. As such, teachers should have knowledge on how to teach, their students effectively. The National Council of Teachers of Mathematics (2000) pointed out that effective teaching requires knowing and understanding mathematics, students as learners, and pedagogical strategies (p.17).

MPCK and Fractions

Learning and mastering fractions is still a major issue for students in both primary and middle schools (Saxe, Taylor, Mcintosh, & Gearhart, 2005). Fractions is considered as one of the problematic topics to teach and to understand especially at elementary or primary level. According to Smith (2002), difficulty with fractions among teachers is well documented in many countries, and many authors consider fractions to be the most difficult area of mathematics covered in primary schools. Studies have shown that teachers and students, have difficulties on fraction concept and division in fractions (Ball, D.L., 1990; Redmond, 2009). Shweta Naik in her findings said that confusion arise

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among teachers, to determine which answer is right or wrong provided by their students for the given questions. 7 2/5 – 7 × 2/5 a) 0

b) 2/5

c) 4 3/5

d) 23/5

The above question with multiple responses is a challenge for the teachers’ knowledge about the concepts. The conversion of mixed number into fractions are learned as multiplication of the whole number and the denominator followed by addition of the numerator. This procedural understanding develops a belief of the existence of a multiplication sign between 7 and 2/5. The existence is also supported by the rules from algebra as it is often said that if there is no sign between two letters (or a letter and a number) then there is a multiplication sign. So xy indicates x x y. Similarly, 7 2/5 indicates 7 × 2/5. These lead teachers towards wrong answer i. e., option (a) 0. But interestingly when these teachers are interviewed and asked about the reasons for the rest of the answers, they started thinking about students reasoning. This unpacking of what students thought, gave them insight about the structure of fraction representation itself. Thus, we can determine the level of teacher’s conceptual understanding and relational connectedness in teaching of fractions. PCK is assumed to help teachers to sequence their instructions in a workable way. Conceptual and relational understanding will help teachers to understand their students thinking or reasoning. In order to become a better teachers of mathematics they need to deepen their understanding of the mathematics. They are also expected to apply PCK whenever they teach subject matter for the various levels of abilities of their students.

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Conclusion In the context of Malaysia, there are few studies on PCK.

As educators, we are

concerned about the effectiveness of, PCK. Still others claim that we are “deskilling” or “deprofessionalizing” teachers by “testing” them. This is counterproductive. In conclusion, there is a powerful relationship between what a teacher knows, how she knows it, and what she can do in the context of instruction. Mathematics teacher educators assume a significant responsibility for enhancing the content and pedagogical knowledge of prospective teachers as well as influencing their belief regarding mathematics teaching and learning and their awareness of their students’ mathematical dispositions. In our context, we need to assess the teacher education system in Malaysia for better and quality instructions in the future. Most of the studies on PCK focus on preservice teachers, novice teachers, and prospective teachers but very few on in-service teachers or experience teacher or expert teachers.

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Reference Ball, D.L. (1990). Pre-service elementary and secondary teachers’ understanding of division. Journal for Research in Mathematics Education,, 21(2), 132–144. Education Review Office. (2006). The Quality of Teaching in Years 4 and 8 : Social Studies June 2006. Graeber, A. (1999). Forms of knowing mathematics: What pre-service teachers should learn. Educational Studies in Mathematics, 38(1-3), 189-208. Graham, K. J., & Fennell, F. S. (2001). Principles and standards for school mathematics and teacher education: Preparing and empowering teachers. School Science and Mathematics, 101(6), 319-327. Grossman, P. L. (2012). Why Models Matter : An Alternate View on Professional Growth in Teaching, 62(2), 171–179. Hansen, D. . (1995). The call to teach. Teachers College Press. Leinhardt, G., Putnam, R. T., Stein, M. K., & Baxter, J. (1991). Where subject knowledge matters. Advances in research on teaching, 2, 87-113. Naik, S., Bhabha, H., & Education, S. (n.d.). The measures for understanding teachers ’ mathematical knowledge for teaching fractions – how do they really work ? National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: Author. Redmond, A. (2009). Prospective Elementary Teachers’ division of Fractions Understanding. Oklahoma State University. S.Shulman, L. (1987). Knowledge and teaching. Harvard Educational Review, 57(1), 1– 21. Saxe, G. B., Taylor, E. V, Mcintosh, C., & Gearhart, M. (2005). Representing Fractions with Standard Notation : A Developmental Analysis, 36(2), 137–157.

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BIODATA AND CONTACT ADDRESSES OF AUTHORS

Palanisamy Kathir Veloo is currently a PhD Scholar of the Ministry of Education, pursuing his PhD in the Faculty of Science and Mathematics, Sultan Idris Education University, Malaysia. He holds a Master Degree in Teaching of Mathematics from University of Science Malaysia, He obtained a Bachelor Degree from Sultan Idris Education University and a Diploma in Teaching Tamil Language from Teachers Training College Sultan Abdul Halim, Malaysia. Palanisamy Kathir Veloo Sultan Idris Education University Faculty of Science and Mathematics Tanjung Malim 35900 Perak MALAYSIA [email protected]