Analyzing Students’ Works: An Opportunity for Pre-Service Teachers to Learn About Teaching Mathematics

Author(s):

Makbule Gozde Didis (presenting / submitting) Sinem Bas Ayhan Kursat Erbas Bulent Cetinkaya Erdinc Cakiroglu

Conference:

ECER 2013

Network:

24. Mathematics Education Research

Format:

Paper

Session Information

24 SES 02, Pre-Service Mathematics Teacher Education

Paper Session

Time:

2013-09-10

15:15-16:45

Room:

D-405

Chair:

Betül Yeniterzi

Contribution

In order to become a good mathematics teacher, pre-service teachers need to learn both mathematics and how to teach it (Olivera & Hannula, 2007). Knowing mathematics (subject matter knowledge), knowing teaching (pedagogic knowledge) and knowing how to teach mathematics (pedagogical content knowledge) are three fundamental constructs of teachers’ knowledge for teaching mathematics (Shulman, 1986). Subject matter knowledge is a cornerstone of teaching which includes knowledge of facts, rules and concepts in a certain domain that are to be learned by students. However, teachers need more. Teachers need to know about students’ experiences, backgrounds and understanding in order to help students to access subject matter (Ball, 1997). Many researches indicate that pre-service teachers’ ideas and knowledge about students are weak. They are not familiar with students’ common conceptions, errors, difficulties or misconceptions (e.g., Ball, 1988; Tirosh, 2000) and they had poor knowledge to understand and interpret what students know and think (e.g., Akkoç, Yeşildere, & Özmantar, 2007; Tirosh, 2000). For example, Tirosh (2000) aimed to enhance pre-service teachers’ subject matter knowledge of division of fractions and their awareness of the misconceptions held by students in the context of a method course. The result of the study showed that pre-service teachers were not aware of the students’ misconceptions in this domain before the instruction and they mostly mentioned only algorithmically based mistakes. On the other hand, most of pre-service teachers became aware of the intuitive based mistakes and were familiar with the sources of incorrect responses at the end of the course.

In the last decade, mathematics education researchers focused on investigating teachers’ knowledge of students’ thinking, a fundamental component of pedagogical content knowledge. Researchers emphasized that pre-service teachers should be aware of topics that students often had difficulty or misunderstood; in addition, they should also know ways to look into the subject matter through the eyes and minds of the learner (e.g., Ball, 1997; Tirosh, 2000). One of the strongly suggested ways of developing this knowledge is by organizing pre-service teachers learning around real students’ works (e.g., copies of students’ works, videotapes of classroom lessons, etc.) because these artifacts are based on practice (Ball &Cohen, 1999).

The purpose of this study is to investigate the contributions of analysis of students’ works to the pre-service secondary teachers’ knowledge for teaching mathematics through their self-reports. The following research question guided this study:

What are the pre-service mathematics teachers’ perceptions about the contributions of examination of students’ works for their knowledge for teaching mathematics?

Method

Twenty five pre-service secondary mathematics teachers who were enrolled in a course on mathematical modeling for teachers participated in this study. During the semester, as part of a coursework, pre-service teachers worked on thought revealing non-routine tasks in groups (3 or 4); developed solutions to the tasks; presented their solutions to the other students; and responded to the comments and questions related to their solutions. For each task, a week after pre-service teachers worked on a task, they were given high school students’ written works on this task and classroom videos of different groups of students’ discussion on this task. Pre-service teachers examined the students’ works and discussed various aspects of these works, such as “what do the students think, and what is that the students understand, what might be the sources of students’ mistakes?” Qualitative study was used as method-of-inquiry. The data on pre-service mathematics teachers’ perceptions about what they think they learned about teaching mathematics were collected through individual reflection papers, interviews conducted with seven voluntary pre-service teachers from each group, and pre/post self reports. In analyzing data, initially, audio-taped interviews were transcribed. Then, all written data sources were carefully organized and analyzed by the help of qualitative data analysis software (NVIVO).

Expected Outcomes

The analysis of the data indicated that pre-service mathematics teachers’ examinations of students’ works contributed to their knowledge for teaching mathematics in the dimensions of knowledge of students, knowledge of teacher/teacher role, and knowledge of assessment. According to pre-service teachers’ self-reports, development of their understanding of students’ thinking was the most important contributions to their knowledge for teaching mathematics. They stated that in examining students’ works, they learnt about how students think and what difficulties they have on a particular mathematics topic, which they were unaware before the course. Secondly, they reported that until this course, they have never examined students’ works and analyzed their thinking like a teacher. Therefore, they stated that they started to feel that like a real teacher and developed ideas on teaching of a particular mathematics topic and different roles of a teacher. Lastly, they indicated that they learnt to evaluate students’ work beyond determining the accuracy of the responses. They reported that at the end of the course, the aspect of examining and interpreting details of students’ thinking were in their focus of attention.

References

Akkoç, H., Yeşildere, S., & Özmantar, F. (2007). Prospective mathematics teachers’ pedagogical content knowledge of definite integral: The problem of limit process. In D. Küchemann (Ed.) Proceedings of the British Society for Research into Learning Mathematics, 27(3), 7-12. Ball, D. L. (1988). Knowledge and reasoning in mathematical pedagogy: Examining what prospective teachers bring to teacher education. Unpublished doctoral dissertation, Michigan State University, USA. Ball, D. L. (1997). What do students know? Facing challenges of distance, context, and desire in trying to hear children. In B. Biddle, T. Good, & I. Goodson (Eds.), International handbook on teachers and teaching (Vol. II), (pp. 679-718). Dordrecht, Netherlands: Kluwer Press. Ball, D. L., & Cohen, D. K. (1999). Developing practice, developing practitioners: Toward a practice-based theory of professional education.Teaching as the learning profession: Handbook of policy and practice, 1, 3-22. Oliveira, H., & Hannula, M. S (2008). Individual prospective mathematics teachers. In K. Krainer, & T. Wood (Eds.), The International Handbook of Mathematics Teacher Education : Vol. 3. Participants in mathematics teaching education (pp. 13-34). Rotterdam: Sense Publisher. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14. Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children's conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31(1), 5-25.