Understanding Pre-service Teachers Mathematical Concepts Representation for Problem Posing

Adriano V. Patac, Jr., Fernando T. Herrera

Abstract


The purpose of the study is to apply the structure of problem posing as an instructional scaffold in posing problems. The conditions and explorations used by the pre-service teachers during the process of posing reflects how they manage previously learned concepts in mathematics. Results showed how participants posed textbook types of problems, utilized implicit assumptions, compare unrelated concepts, and insufficient information about the problem goal or the sentence structure. The implementation of the structured scheme using Pythagorean Theorem revealed that the instructional scaffold only serves as a “map” rather than a reflective guide so that posing of problems can be systematic and meaningful. Compartmentalization of concepts was a dominant conditions and actions used during the exploration and mainly focused on processing within the same area of representation. Integration of different concepts used during the manipulations of the Pythagorean Theorem reflects a lack of interrelatedness among the elements within the larger structure. Thus, pre-service teachers concerns mostly on how to complete the scheme. The algebraic and geometric representations of Pythagorean Theorem were treated independently rather than treating how the elements interrelates to enable them to function together. Hence, challenging a new concept to replace the existing concept was not successful. Thus, the structured scheme of problem posing suggests an important pedagogical role in understanding the mathematical knowledge of the pre-service teachers.  

 


Keywords


problem posing; structured scheme; mathematical explorations; meaningful relations

Full Text:

PDF

References


Australian Association of Mathematics Teacher (AAMT). (2002). Standards for excellence in teaching mathematics in Australian schools. Available from: http://www.aamt.edu.au/standards/.

Ball, D. L., Lubienski, S. T., & Mewborn, D. S. (2001). Research on teaching mathematics: The unsolved problem of teachers’ mathematical knowledge. In V. Richardson (Ed.), Handbook of research on teaching (pp. 433–456). Washington, DC: American Educational Research Association.

Borko, H., & Putnam, R. (1996). Learning to teach. In D. Berliner, & R. Calfee (Eds.), Handbook of educational psychology (pp. 673–708). New York: Macmillan.

Brown, S. I., & Walter, M. I. (2005). The art of problem posing (3rd ed.). Hillsdale, NJ: Erlbaum.

Brown, S. I., & Walter, M. I. (1990). The art of problem posing (2nd ed.). Hillsdale, NJ: Erlbaum.

Brown, S. I., & Walter, M. I. (1983). The art of problem posing. Hillsdale, NJ: Lawrence Erlbaum.

Cai, J., Hwang, S., Chunlian Jiang, C., and Silber,S. (2015) Problem-posing research in mathematics education: Some answered and unanswered. In F. M. Singer, N. F. Ellerton, J. Cai (Eds.) Mathematical Problem Posing: From Research to Effective Practice (pp. 3-34). New York: Springer.

Crespo, S., & Sinclair, N. (2008). What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems. Journal of Mathematics Teacher Education, 11 ,395–415.

Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht, The Netherlands: Kluwer.

Hall, R., Kibler, D., Wenger, E., & Truxaw, C. (1989). Exploring the episodic structure of algebra story problem solving. Cognition and Instruction, 6 (3), 223–283.

Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning.Journal for Research in Mathematics Education, 28, 524–549.

Lavy, I., & Bershadsky, I. (2003). Problem posing via “What if not?” strategy in solid geometry— A case study. The Journal of Mathematical Behavior, 22(4), 369–387.

Livy, S. L., Vale, C., & Herbert, S. (2016). Developing Primary Pre-service Teachers' Mathematical Content Knowledge During Practicum Teaching. Australian Journal of Teacher Education, 41(2), 152-173.

Ministry of Education of China. (2011). Mathematics curriculum standard of compulsory education (2011 version) [in Chinese]. Beijing, China: Beijing Normal University Press. Retrieved from http://www.moe.gov.cn/publicfiles/business/htmlfiles/moe/moe_711/201201/xxgk_129268.html

National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (NCTM). (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

Polya, G. (1957). How to solve it. Princeton, NJ: Princeton University Press.

Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14 (1),19–28.

Silver, E. A., & Cai, J. (1996). An analysis of arithmetic problem posing by middle school students. Journal for Research in Mathematics Education, 27 (5), 521–539.

Silver, E. A., Mamona-Downs, J., Leung, S., & Kenney, P. A. (1996). Posing mathematical problems in a complex task environment: An exploratory study. Journal for Research in Mathematics Education, 27 (3), 233–309.

Stoyanova, E. (1998). Problem posing in mathematics classrooms. In A. McIntosh & N. F. Ellerton (Eds.), Research in mathematics education: A contemporary perspective (pp. 164–185). Perth, Australia, Australia: MASTEC.

Tutak, F. A. (2009). A study of geometry content knowledge of elementary preservice teachers: The case of quadrilaterals (Doctoral dissertation. University of Florida).


Refbacks

  • There are currently no refbacks.