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Mathematics Teachers' Use of Mathematical Descriptions, Explanations and Justifications: The Case of Samet

Year 2019, Volume: 12 Issue: 4, 1284 - 1305, 03.10.2019
https://doi.org/10.30831/akukeg.478101

Abstract

It is important for teachers to design and use mathematically accurate descriptions, explanations and justifications that are comprehensible and useful for students in the context of reflecting their mathematical knowledge for teaching. The purpose of this study is to examine a mathematics teacher’s mathematical knowledge for teaching function concept by investigating his use of mathematical descriptions, explanations and justifications. The study was conducted as a descriptive case study. The participant in the study was one mathematics teacher (Samet) who volunteered to join the research. Data was collected via observing and recording the teacher’s teaching of the function concept and survey of function concept. The results of the study revealed that the teacher mostly used mathematical descriptions in his teaching. This was followed by mathematical explanations, and mathematical justifications. The teacher’s use of mathematical descriptions, explanations and justifications and his sufficiency at using these varied according to cases. Results indicated some deficiencies in the teacher’s mathematical knowledge for teaching.

References

  • Aksu, Z., & Kul, Ü. (2016). Exploring mathematics teachers’ pedagogical content knowledge in the context of knowledge of students. Journal of Education and Practice, 7(30), 35-42.
  • An, S., Kulm, G., & 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.
  • Ball, D. L. (1990). Reflections and deflections of policy: The case of Carol Turner. Educational Evaluation and Policy Analysis, 12(3), 247-259.
  • Ball, D. L., & Bass, H. (2002). Toward a practice-based theory of mathematical knowledge for teaching. In Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group (pp. 3-14).
  • Ball, D. L., Thames, M. H. & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389- 407.
  • Cai, J. (2003). Singaporean students' mathematical thinking in problem solving and problem posing: an exploratory study. International Journal of Mathematical Education in Science and Technology, 34(5), 719-737.
  • Charalambous, C. Y. (2010). Mathematical knowledge for teaching and task unfolding: An exploratory study. The Elementary School Journal, 110(3), 247-278.
  • Charalambous, C. Y., & Hill, H. C. (2012). Teacher knowledge, curriculum materials, and quality of instruction: Unpacking a complex relationship. Journal of Curriculum Studies, 44(4), 443-466.
  • Chazan, D. (1993). High school geometry students' justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24(4), 359-387.
  • Cohen, D. K. (1990). A revolution in one classroom: The case of Mrs. Oublier. Educational Evaluation and Policy Analysis, 12(3), 311-329.
  • Creswell, J. W. (2003). Research design: Qualitative, quantitative, and mixed method approaches. Second Edition. Thousand Oaks, CA: Sage Publications.
  • Erlandson, D. A., Harris, E. L., Skipper, B. L., & Allen, S. D. (1993). Doing naturalistic inquiry: A guide to methods. Newbury Park, CA: Sage Publications.
  • Even, R. (1990). Subject matter knowledge for teaching and the case of functions. Educational Studies in Mathematics, 21(6), 521-544.
  • Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept. Journal for research in mathematics education, 24(2), 94-116.
  • Even, R., & Tirosh, D. (1995). Subject-matter knowledge and knowledge about students as sources of teacher presentations of the subject-matter. Educational studies in mathematics, 29(1), 1-20.
  • Fennema, E., & Franke, M. (1992). Teachers’ knowledge and its impact. In D. Grouws (Ed.), Handbook of Research on Mathematical Teaching and Learning (pp. 575-596). New York: Macmillan.
  • Forman, E. A., McCormick, D. E., & Donato, R. (1997). Learning what counts as a mathematical explanation. Linguistics and Education, 9(4), 313-339.
  • Fraivillig, J. L., Murphy, L. A., & Fuson, K. C. (1999). Advancing children’s mathematical thinking in everyday mathematics classrooms. Journal for Research in Mathematics Education, 30(2), 148-170.
  • Grossman, P. L. (1990). The making of a teacher: Teacher knowledge and teacher education. New York: Teachers College Press.
  • Haciomeroglu, G. (2006). Prospective secondary teachers' subject matter knowledge and pedagogical content knowledge of the concept of function (Unpublished Doctoral Dissertation). Florida State University, Tallahassee, FL.
  • Hatisaru, V., & Erbas, A. K. (2017). Mathematical knowledge for teaching the function concept and student learning outcomes. International Journal of Science and Mathematics Education, 15(4), 703-722.
  • Heaton, R. M. (1992). Who is minding the mathematics content? A case study of a fifth-grade teacher. The Elementary School Journal, 93(2), 153-162.
  • Hill, H. C. (2010). The nature and predictors of elementary teachers' mathematical knowledge for teaching. Journal for Research in Mathematics Education, 41(5), 513-545.
  • Hill, H. C., Blunk, M. L., Charalambous, C. Y., Lewis, J. M., Phelps, G. C., Sleep, L., & Ball, D. L. (2008). Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study. Cognition and Instruction, 26(4), 430-511.
  • Karahasan, B. (2010). Preservice secondary mathematics teachers‟ pedagogical content knowledge of composite and inverse function (Unpublished doctoral dissertation). Middle East Technical University, Ankara, Turkey.
  • Kersting, N. B., Givvin, K. B., Thompson, B. J., Santagata, R., & Stigler, J. W. (2012). Measuring usable knowledge: Teachers’ analyses of mathematics classroom videos predict teaching quality and student learning. American Educational Research Journal, 49(3), 568-589.
  • Krippendorff, K. (1980). Content Analysis: An introduction to its methodology. Beverly Hills, CA: Sage Publications.
  • Lachner, A., & Nückles, M. (2016). Tell me why! Content knowledge predicts process-orientation of math researchers’ and math teachers’ explanations. Instructional Science, 44(3), 221-242.
  • Learning Mathematics for Teaching (2006). A Coding rubric for Measuring the Quality of Mathematics in Instruction (Technical Report LMT1.06). Ann Arbor, MI: University of Michigan, School of Education.
  • Learning Mathematics for Teaching Project. (2011). Measuring the mathematical quality of instruction. Journal of Mathematics Teacher Education, 14(1), 25-47.
  • Leinhardt, G. (2001). Instructional explanations: A commonplace for teaching and location of contrast. In V. Richardson (Eds.), Handbook of research on teaching [Fourth Edition]. Washington, DC: American Educational Research Association.
  • Leinhardt, G. (2010). Introduction: Explaining instructional explanations. In M. K. Stein, & L. Kucan (Eds.), Instructional explanations in the discipline (pp.1-5). New York: Springer.
  • Llinares, S. (2000). Secondary School Mathematics Teacher's Professional Knowledge: A case from the teaching of the concept of function, Teachers and Teaching, 6(1), 41-62, DOI: 10.1080/135406000114744
  • Lloyd, G. M., & Wilson, M. R. (1998). Supporting innovation: The impact of a teacher's conceptions of functions on his implementation of a reform curriculum. Journal for Research in Mathematics Education, 29(3), 248-274.
  • Magnusson, S., Krajcik, J., & Borko, H. (1999). Nature, sources and development of pedagogical content knowledge for science teaching. In J. Gess Newsome and N.G. Lederman (Eds.), Examining pedagogical content knowledge. (95–132). Dordrecht, The Netherlands: Kluwer Academic Publishers.
  • Marshall, C., & Rossman, G. B. (1999). Designing qualitative research (3rd ed.). Thousand Oaks, CA: Sage.
  • Miles, M. B., & Huberman, M. A. (1994). Qualitative analysis: An expanded sourcebook. Thousand Oaks, CA: Sage.
  • National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
  • Nyikahadzoyi, M. (2015). Teachers' knowledge of the concept of a function: A theoretical framework. International Journal of Science & Mathematics Education, 13, 261-283.
  • Ronau, R. N., Meyer, D., Crites, T., & Dougherty, B. J. (2014). Putting essential understanding of functions into practice in grades 9–12. Reston, VA: National Council of Teachers of Mathematics.
  • Shulman, L. S. (1986). Those who understand; knowledge growth in teaching. Educational Researcher, 15(2), 4-14.
  • Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1-22.
  • Smith, H. W. (1981). Strategies of social research: The methodological imagination, 2nd Edition, Englewood Cliffs, NJ: Prentice-Hall.
  • Snider, R. B. (2016). How mathematical knowledge for teaching intersects with teaching practices: the knowledge and reasoning entailed in selecting examples and giving explanations in secondary mathematics. (Unpublished doctoral dissertation). United States of America: University of Michigan.
  • Staples, M. E., Bartlo, J., & Thanheiser, E. (2012). Justification as a teaching and learning practice: Its (potential) multifaceted role in middle grades mathematics classrooms. The Journal of Mathematical Behavior, 31(4), 447-462.
  • Steele, M. D., & Rogers, K. C. (2012). Relationships between mathematical knowledge for teaching and teaching practice: the case of proof. Journal of Mathematics Teacher Education, 15(2), 159-180.
  • Steele, M. D., Hillen, A. F., & Smith, M. S. (2013). Developing mathematical knowledge for teaching in a methods course: the case of function. Journal of Mathematics Teacher Education, 16(6), 451-482.
  • Stein, M. K., Baxter, J. A., & Leinhardt, G. (1990). Subject-matter knowledge and elementary instruction: A case from functions and graphing. American Educational Research Journal, 27(4), 639-663.
  • Author (2017)
  • Weber, R. P. (1985). Basic content analysis, quantitative applications in the social sciences. Beverly Hills, CA: Sage Publications.
  • Wilson, S. M. (1990). A conflict of interests: The case of Mark Black. Educational Evaluation and Policy Analysis, 12(3), 293-310.
  • Wilson, M. R. (1992). A study of three preservice secondary mathematics teachers’ knowledge and beliefs about functions (Unpublished doctoral dissertation). University of Georgia, Athens, GA.
  • Xenofontos, C., & Andrews, P. (2017). Explanations as tools for evaluating content knowledge for teaching: A cross-national pilot study in Cyprus and Greece. In Dooley, T. & Gueudet, G. (Eds.), Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (CERME10, February 1 – 5, 2017). Dublin, Ireland: DCU Institute of Education & ERME.
  • Yackel, E. (2001). Explanation, justification and argumentation in mathematics classrooms. In M. v. d. Heuvel-Panhuizen (Eds.), Proceedings of the 25th conference of the International Group for the Psychology of Mathematics Education Vol. 1 (pp. 9 - 24). Utrecht, The Netherlands: PME.
  • Yin, R. (2009). Case study research: Design and methods, Fourth edition. Thousand Oaks, CA: Sage Publications.

Matematik Öğretmenlerinin Matematiksel Tanımlamaları, Açıklamaları ve Doğrulamaları Kullanımı: Samet Örneği

Year 2019, Volume: 12 Issue: 4, 1284 - 1305, 03.10.2019
https://doi.org/10.30831/akukeg.478101

Abstract

Öğretmenlerin, öğretmek için matematik bilgilerini yansıtmaları bağlamında, öğrenciler için anlaşılır ve kullanışlı ve matematiksel olarak doğru tanımlamaları, açıklamaları ve doğrulamaları tasarlamaları ve kullanmaları önemlidir. Bu çalışmanın amacı, bir matematik öğretmeninin fonksiyon kavramının öğretiminde matematiksel tanımlamaları, açıklamaları ve doğrulamaları kullanımını araştırarak öğretmek için matematik bilgisini incelemektir. Çalışma, tanımlayıcı bir durum çalışması olarak gerçekleştirilmiştir. Araştırmanın katılımcısı araştırmaya katılmak için gönüllü olan bir matematik öğretmenidir (Samet). Veriler, öğretmenin fonksiyon kavramı öğretiminin gözlemlenmesi ve kaydedilmesi ve fonksiyon kavramı anketi ile toplanmıştır. Araştırmanın sonuçları, öğretmenin öğretiminde çoğunlukla matematiksel tanımlamaları kullandığını ortaya koymuştur. Bunu matematiksel açıklamalar ve matematiksel doğrulamalar takip etmiştir. Öğretmenin matematiksel tanımlamaları, açıklamaları ve doğrulamaları kullanımı ve bunları kullanmadaki yeterliliği farklı durumlara göre değişiklik göstermiştir. Sonuçlar, öğretmenin öğretmek için matematik bilgisindeki bazı eksiklikleri olduğunu göstermiştir.

References

  • Aksu, Z., & Kul, Ü. (2016). Exploring mathematics teachers’ pedagogical content knowledge in the context of knowledge of students. Journal of Education and Practice, 7(30), 35-42.
  • An, S., Kulm, G., & 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.
  • Ball, D. L. (1990). Reflections and deflections of policy: The case of Carol Turner. Educational Evaluation and Policy Analysis, 12(3), 247-259.
  • Ball, D. L., & Bass, H. (2002). Toward a practice-based theory of mathematical knowledge for teaching. In Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group (pp. 3-14).
  • Ball, D. L., Thames, M. H. & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389- 407.
  • Cai, J. (2003). Singaporean students' mathematical thinking in problem solving and problem posing: an exploratory study. International Journal of Mathematical Education in Science and Technology, 34(5), 719-737.
  • Charalambous, C. Y. (2010). Mathematical knowledge for teaching and task unfolding: An exploratory study. The Elementary School Journal, 110(3), 247-278.
  • Charalambous, C. Y., & Hill, H. C. (2012). Teacher knowledge, curriculum materials, and quality of instruction: Unpacking a complex relationship. Journal of Curriculum Studies, 44(4), 443-466.
  • Chazan, D. (1993). High school geometry students' justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24(4), 359-387.
  • Cohen, D. K. (1990). A revolution in one classroom: The case of Mrs. Oublier. Educational Evaluation and Policy Analysis, 12(3), 311-329.
  • Creswell, J. W. (2003). Research design: Qualitative, quantitative, and mixed method approaches. Second Edition. Thousand Oaks, CA: Sage Publications.
  • Erlandson, D. A., Harris, E. L., Skipper, B. L., & Allen, S. D. (1993). Doing naturalistic inquiry: A guide to methods. Newbury Park, CA: Sage Publications.
  • Even, R. (1990). Subject matter knowledge for teaching and the case of functions. Educational Studies in Mathematics, 21(6), 521-544.
  • Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept. Journal for research in mathematics education, 24(2), 94-116.
  • Even, R., & Tirosh, D. (1995). Subject-matter knowledge and knowledge about students as sources of teacher presentations of the subject-matter. Educational studies in mathematics, 29(1), 1-20.
  • Fennema, E., & Franke, M. (1992). Teachers’ knowledge and its impact. In D. Grouws (Ed.), Handbook of Research on Mathematical Teaching and Learning (pp. 575-596). New York: Macmillan.
  • Forman, E. A., McCormick, D. E., & Donato, R. (1997). Learning what counts as a mathematical explanation. Linguistics and Education, 9(4), 313-339.
  • Fraivillig, J. L., Murphy, L. A., & Fuson, K. C. (1999). Advancing children’s mathematical thinking in everyday mathematics classrooms. Journal for Research in Mathematics Education, 30(2), 148-170.
  • Grossman, P. L. (1990). The making of a teacher: Teacher knowledge and teacher education. New York: Teachers College Press.
  • Haciomeroglu, G. (2006). Prospective secondary teachers' subject matter knowledge and pedagogical content knowledge of the concept of function (Unpublished Doctoral Dissertation). Florida State University, Tallahassee, FL.
  • Hatisaru, V., & Erbas, A. K. (2017). Mathematical knowledge for teaching the function concept and student learning outcomes. International Journal of Science and Mathematics Education, 15(4), 703-722.
  • Heaton, R. M. (1992). Who is minding the mathematics content? A case study of a fifth-grade teacher. The Elementary School Journal, 93(2), 153-162.
  • Hill, H. C. (2010). The nature and predictors of elementary teachers' mathematical knowledge for teaching. Journal for Research in Mathematics Education, 41(5), 513-545.
  • Hill, H. C., Blunk, M. L., Charalambous, C. Y., Lewis, J. M., Phelps, G. C., Sleep, L., & Ball, D. L. (2008). Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study. Cognition and Instruction, 26(4), 430-511.
  • Karahasan, B. (2010). Preservice secondary mathematics teachers‟ pedagogical content knowledge of composite and inverse function (Unpublished doctoral dissertation). Middle East Technical University, Ankara, Turkey.
  • Kersting, N. B., Givvin, K. B., Thompson, B. J., Santagata, R., & Stigler, J. W. (2012). Measuring usable knowledge: Teachers’ analyses of mathematics classroom videos predict teaching quality and student learning. American Educational Research Journal, 49(3), 568-589.
  • Krippendorff, K. (1980). Content Analysis: An introduction to its methodology. Beverly Hills, CA: Sage Publications.
  • Lachner, A., & Nückles, M. (2016). Tell me why! Content knowledge predicts process-orientation of math researchers’ and math teachers’ explanations. Instructional Science, 44(3), 221-242.
  • Learning Mathematics for Teaching (2006). A Coding rubric for Measuring the Quality of Mathematics in Instruction (Technical Report LMT1.06). Ann Arbor, MI: University of Michigan, School of Education.
  • Learning Mathematics for Teaching Project. (2011). Measuring the mathematical quality of instruction. Journal of Mathematics Teacher Education, 14(1), 25-47.
  • Leinhardt, G. (2001). Instructional explanations: A commonplace for teaching and location of contrast. In V. Richardson (Eds.), Handbook of research on teaching [Fourth Edition]. Washington, DC: American Educational Research Association.
  • Leinhardt, G. (2010). Introduction: Explaining instructional explanations. In M. K. Stein, & L. Kucan (Eds.), Instructional explanations in the discipline (pp.1-5). New York: Springer.
  • Llinares, S. (2000). Secondary School Mathematics Teacher's Professional Knowledge: A case from the teaching of the concept of function, Teachers and Teaching, 6(1), 41-62, DOI: 10.1080/135406000114744
  • Lloyd, G. M., & Wilson, M. R. (1998). Supporting innovation: The impact of a teacher's conceptions of functions on his implementation of a reform curriculum. Journal for Research in Mathematics Education, 29(3), 248-274.
  • Magnusson, S., Krajcik, J., & Borko, H. (1999). Nature, sources and development of pedagogical content knowledge for science teaching. In J. Gess Newsome and N.G. Lederman (Eds.), Examining pedagogical content knowledge. (95–132). Dordrecht, The Netherlands: Kluwer Academic Publishers.
  • Marshall, C., & Rossman, G. B. (1999). Designing qualitative research (3rd ed.). Thousand Oaks, CA: Sage.
  • Miles, M. B., & Huberman, M. A. (1994). Qualitative analysis: An expanded sourcebook. Thousand Oaks, CA: Sage.
  • National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
  • Nyikahadzoyi, M. (2015). Teachers' knowledge of the concept of a function: A theoretical framework. International Journal of Science & Mathematics Education, 13, 261-283.
  • Ronau, R. N., Meyer, D., Crites, T., & Dougherty, B. J. (2014). Putting essential understanding of functions into practice in grades 9–12. Reston, VA: National Council of Teachers of Mathematics.
  • Shulman, L. S. (1986). Those who understand; knowledge growth in teaching. Educational Researcher, 15(2), 4-14.
  • Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1-22.
  • Smith, H. W. (1981). Strategies of social research: The methodological imagination, 2nd Edition, Englewood Cliffs, NJ: Prentice-Hall.
  • Snider, R. B. (2016). How mathematical knowledge for teaching intersects with teaching practices: the knowledge and reasoning entailed in selecting examples and giving explanations in secondary mathematics. (Unpublished doctoral dissertation). United States of America: University of Michigan.
  • Staples, M. E., Bartlo, J., & Thanheiser, E. (2012). Justification as a teaching and learning practice: Its (potential) multifaceted role in middle grades mathematics classrooms. The Journal of Mathematical Behavior, 31(4), 447-462.
  • Steele, M. D., & Rogers, K. C. (2012). Relationships between mathematical knowledge for teaching and teaching practice: the case of proof. Journal of Mathematics Teacher Education, 15(2), 159-180.
  • Steele, M. D., Hillen, A. F., & Smith, M. S. (2013). Developing mathematical knowledge for teaching in a methods course: the case of function. Journal of Mathematics Teacher Education, 16(6), 451-482.
  • Stein, M. K., Baxter, J. A., & Leinhardt, G. (1990). Subject-matter knowledge and elementary instruction: A case from functions and graphing. American Educational Research Journal, 27(4), 639-663.
  • Author (2017)
  • Weber, R. P. (1985). Basic content analysis, quantitative applications in the social sciences. Beverly Hills, CA: Sage Publications.
  • Wilson, S. M. (1990). A conflict of interests: The case of Mark Black. Educational Evaluation and Policy Analysis, 12(3), 293-310.
  • Wilson, M. R. (1992). A study of three preservice secondary mathematics teachers’ knowledge and beliefs about functions (Unpublished doctoral dissertation). University of Georgia, Athens, GA.
  • Xenofontos, C., & Andrews, P. (2017). Explanations as tools for evaluating content knowledge for teaching: A cross-national pilot study in Cyprus and Greece. In Dooley, T. & Gueudet, G. (Eds.), Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (CERME10, February 1 – 5, 2017). Dublin, Ireland: DCU Institute of Education & ERME.
  • Yackel, E. (2001). Explanation, justification and argumentation in mathematics classrooms. In M. v. d. Heuvel-Panhuizen (Eds.), Proceedings of the 25th conference of the International Group for the Psychology of Mathematics Education Vol. 1 (pp. 9 - 24). Utrecht, The Netherlands: PME.
  • Yin, R. (2009). Case study research: Design and methods, Fourth edition. Thousand Oaks, CA: Sage Publications.
There are 55 citations in total.

Details

Primary Language English
Subjects Other Fields of Education
Journal Section Articles
Authors

Berna Tataroğlu-taşdan

Publication Date October 3, 2019
Submission Date November 2, 2018
Published in Issue Year 2019 Volume: 12 Issue: 4

Cite

APA Tataroğlu-taşdan, B. (2019). Mathematics Teachers’ Use of Mathematical Descriptions, Explanations and Justifications: The Case of Samet. Journal of Theoretical Educational Science, 12(4), 1284-1305. https://doi.org/10.30831/akukeg.478101