An Investigation of Prospective Mathematics Teachers’ Knowledge of Basic Algorithms with Whole Numbers: A Case of Turkey

Abstract

The aim of this qualitative case study is to investigate prospective mathematics teachers’ subject matter knowledge of the underlying concepts of standard and nonstandard algorithms used to solve the problems with whole numbers. Twenty three prospective mathematics teachers enrolled in the Elementary Mathematics Education Program of one of the most successful universities in Turkey were the participants of the study. The data was collected through four tasks containing basic algorithms. More specifically, the Ones Task assessed participants’ understanding of the underlying place value concepts of standard algorithms. The Andrew Task and the Doubling Task required participants to conceptualize and interpret nonstandard strategies. In the Division Task, participants were expected to provide in-depth explanation for the difference between multiplication and division and between partitive division and measurement division. The content analysis method was used to analyze the data. The results of the study revealed that more than half of the prospective mathematics teachers had knowledge about the place value of 1 in addition and subtraction, and also multiplication. However, most of the prospective teachers could not explain the underlying principle and the meaning of the nonstandard algorithm in subtraction. Similar to their knowledge on subtraction, prospective teachers’ knowledge on division was limited.

Keywords

• Content knowledge,basic algorithms,whole numbers,prospective mathematics teacher

Kaynakça

1. Ball, D. L. (1990). Prospective elementary and secondary teachers’ understanding of division. Journal of Research in Mathematics Education, 21, 132-144.
2. 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.
3. Creswell, J. W. (2007). Qualitative inquiry and research design: Choosing among five approaches (2nd ed.). Thousand Oaks, California: Sage Publications, Inc.
4. Engelbrecht, J., Harding, A., & Potgieter, M. (2005). Undergraduate students’ performance and confidence in procedural and conceptual mathematics. International Journal of Mathematical Education in Science and Technology, 36(7), 701-712.
5. 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.
6. Grossman, P. L. (1990). The making of a teacher: Teacher knowledge and teacher education. Teachers College Press, Teachers College, Columbia University.
7. Hiebert, J., & Lefevre P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: the case of mathematics, (pp.1- 27). Hillsdale, NJ: Lawrence Erlbaum Associates.
8. Isiksal, M. (2006). A study on pre-service elementary mathematics teachers‟ subject matter knowledge and pedagogical content knowledge regarding the multiplication and division of fractions. Unpublished doctoral dissertation, Middle East Technical University, Turkey.

9. Lucus, C. A. (2006). Is subject matter knowledge affected by experience? The case of composition of functions. In J. Novotna, H. Moraova, M. Kraka & N. Stehlikova (Eds.) Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 97-104). Prague: PME
10. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.
11. Merriam, S. B. (2009). Qualitative research: A guide to design and implementation. San Francisco: Jossey-Bass.
12. National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
13. Nilsson, P., & Lindstrom, T. (2012, July). Connecting Swedish compulsory school teachers' content knowledge of probability to their level of education, teaching years and self-assessments of probability concepts. Paper presented at the 12th International Congress on Mathematical Education, Seoul, Korea.
14. Philipp, R., Schappelle, B., Siegfried, J., Jacobs, V., & Lamb, L. (2008). The effects of professional development on the mathematical content knowledge of K-3 teachers. In annual meeting of the American Educational Research Association, New York.
15. Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346.
16. Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers’ mathematics subject knowledge: The Knowledge Quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8(3), 255-281.
17. Siegfried, J. M., Gordon, J. M., & Garcia, J. R. (2007). Barely in STEP: How professional development affects teachers’ perspectives on and analysis of student work. In T. Lambert & L. R. Wiest (Eds.), Proceedings of the 29th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. University of Nevada: Reno.
18. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.
19. Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harward Educational Review, 57(1), 1-22.
20. Strauss, A., & Corbin, J. M. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Newbury Park, California: Sage Publications, Inc.
21. Thanheiser, E. (2009). Preservice elementary school teachers' conceptions of multidigit whole numbers. Journal for Research in Mathematics Education, 40(3), 251-281.