The aim of this study is to explore high-school school mathematics teachers' topic-specific pedagogical content knowledge. First, 639 high-school students were asked to give explanations about "a0 = 1, 0! = 1" and "a ÷ 0" where a ≠ 0. Weak explanations by the students led to a detailed research on teachers. Fifty-eight high school mathematics teachers in Northern Cyprus were the participants. They were asked to write how they teach the above topics to high school students. The researcher determined some categories and themes from the written explanations of the teachers using inductive content analysis. Then, three pre-service mathematics teachers had a four-hour training on deductive content analysis. The pre-service teachers went over the written explanations independently and used the pre-determined categories and themes to code the explanations in a deductive way. The results indicate (χ2) that experienced teachers propose more conceptually based instructional strategies than novice teachers. The results also indicate that strategies proposed by all the participants were mainly procedural, fostering memorization. Hence, giving teachers the opportunity to view and teach mathematics in a more constructivist way and more emphasis on topic-specific pedagogical content knowledge in teacher training programs are recommended.
Mathematics Teachers, Topic-Specific Pedagogical Content Knowledge, Teaching Zero.
Rapid developments in science and technology demand for better educated students who can solve problems creatively, learn how to learn, and think critically. For these reasons, in the last two decades a number of studies have attempted to define the nature and the components of teacher knowledge that are necessary especially for mathematics teaching in line with the above skills (Adey & Shayer, 1994; Halpern, 1992; Hill, Ball & Schilling, 2008; Neubrand, 2008; Nickerson, Perkins, & Smith, 1985; Schoenfeld, 1992; Sternberg, 1994; Swartz & Parks, 1994; Tishman, Perkins, & Jay, 1995; Zohar, 2004). Since many research findings emphasize the effects of teachers' knowledge on students' achievement (Brickhouse, 1990; Clark & Peterson, 1986; Hanushek, 1971; Nespor, 1987; Neubrand, 2008; Strauss & Sawyer, 1986; Tobin & Fraser, 1989; Wilson & Floden, 2003) and students' weak knowledge in zero concept (Ball, 1990; Henry, 1969; Ma, 1999; Reys, 1974; Tsamir, Sheffer, & Tirosh, 2000; Quinn, Lamberg & Perrin, 2008), in this study, we aimed to explore the topic-specific pedagogical content knowledge of mathematics teachers in the context of teaching a0, 0! and a ÷ 0. Many research findings about the differences of novice and experienced teachers' pedagogical content knowledge (Ball & Bass, 2000; Cooney & Wiegel, 2003; Çakmak, 1999; Hill et al., 2008; Neubrand, 2008) led us to emphasize experience as a categorical variable in this study. For this reason, answers were sought for the following questions:
1) What are the instructional strategies proposed by mathematics teachers for teaching of a0 = 1?
2) Are the instructional strategies proposed for teaching of a0 = 1, teaching experience dependent?
3) What are the instructional strategies proposed by mathematics teachers for teaching of 0! = 1?
4) Are the instructional strategies proposed for teaching of 0! = 1, teaching experience dependent?
5) What are the instructional strategies proposed by mathematics teachers for teaching of a ÷ 0?
6) Are the instructional strategies proposed for teaching of a ÷ 0, teaching experience dependent?
Pedagogical Content Knowledge and Topic-Specific Mathematics Pedagogical Content Knowledge
In the last two decades, a great deal of studies have been attempted to investigate the nature and the components of teacher knowledge that is necessary for teaching mathematics. Many researchers specify that actual teaching practice might differ from the knowledge, which a teacher has acquired in the formal education (Ball & Bass, 2000; Cooney & Wiegel, 2003; Çakmak, 1999; Hill et al. …