# Strategy ranges: describing change in prospective elementary teachers’ approaches to mental computation of sums and differences

- 818 Downloads
- 2 Citations

## Abstract

This study investigated the sets of mental computation strategies used by prospective elementary teachers to compute sums and differences of whole numbers. In the context of an intervention designed to improve the number sense of prospective elementary teachers, participants were interviewed pre/post, and their mental computation strategies were analyzed. The analysis led to the identification of the *strategy ranges* used by the participants, as well as descriptions of changes pre/post in those strategy ranges. This article illustrates how strategy ranges, as an analytic tool, afford useful descriptions of the repertoires of mental computation strategies that individuals use.

## Keywords

Prospective elementary teachers Mental computation Flexibility Strategy ranges## Notes

### Acknowledgments

I am grateful to the editor and anonymous reviewers for their very thoughtful reading and helpful feedback. I thank Dr. Susan Nickerson for her efforts as the instructor of the course and for her mentorship. Finally, I thank the interview participants for their time and willingness to share their thinking.

## References

- Anghileri, J. (2000).
*Teaching number sense*. London: Continuum.Google Scholar - Baek, J.-M. (1998). Children’s invented algorithms for multidigit multiplication problems.
*NCTM Yearbook,**1998*, 151–160.Google Scholar - Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education.
*The Elementary School Journal,**90*, 449–466.CrossRefGoogle Scholar - 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.Google Scholar - Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999).
*Children’s mathematics: Cognitively guided instruction*. Portsmouth, NH: Heinemann.Google Scholar - Carraher, T. N., Carraher, D. W., & Schliemann, A. D. (1987). Written and oral mathematics.
*Journal for Research in Mathematics Education, 18,*83–97.Google Scholar - Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research.
*Educational Psychologist,**31*, 175–190.CrossRefGoogle Scholar - Common Core State Standards Initiative (CCSSI). (2010).
*Common core state standards for mathematics*. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.Google Scholar - Conference Board of the Mathematical Sciences. (2012).
*The mathematical education of teachers II*. Providence, RI and Washington, DC: American Mathematical Society and Mathematical Association of America.Google Scholar - Fuson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., Olivier, A. I., et al. (1997). Children’s conceptual structures for mulitidigit numbers and methods of multidigit addition and subtraction.
*Journal for Research in Mathematics Education,**28*, 130–162.CrossRefGoogle Scholar - Greeno, J. (1991). Number sense as situated knowing in a conceptual domain.
*Journal for Research in Mathematics Education,**22*, 170–218.CrossRefGoogle Scholar - Harkness, S. S., & Thomas, J. (2008). Reflections on “multiplication as original sin”: The implications of using a case to help preservice teachers understand invented algorithms.
*Journal of Mathematical Behavior,**27*, 128–137.CrossRefGoogle Scholar - Heirdsfield, A. M., & Cooper, T. J. (2004). Factors affecting the process of proficient mental addition and subtraction: Case studies of flexible and inflexible computers.
*Journal of Mathematical Behavior,**23*, 443–463.CrossRefGoogle Scholar - Hope, J. A., & Sherrill, J. M. (1987). Characteristics of unskilled and skilled mental calculators.
*Journal for Research in Mathematics Education, 18*, 98–111.Google Scholar - Jacobs, V. R., Lamb, L. L. C., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking.
*Journal for Research in Mathematics Education, 41*(2), 169–202.Google Scholar - Ma, L. (1999).
*Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States*. New Jersey: Erlbaum.Google Scholar - Markovits, Z., & Sowder, J. (1994). Developing number sense: An intervention study in grade 7.
*Journal for Research in Mathematics Education,**25*, 4–29.CrossRefGoogle Scholar - McIntosh, A. (1998). Teaching mental algorithms constructively. In L. J. Morrow & M. J. Kenney (Eds.),
*The teaching and learning of algorithms in school mathematics, 1998 yearbook*(pp. 44–48). Reston, VA: NCTM.Google Scholar - Menon, R. (2003). Exploring preservice teachers understanding of two-digit multiplication.
*The International Journal for Mathematics Teaching and Learning*. Retrieved from http://www.cimt.plymouth.ac.uk/journal/ramakrishnanmenon.pdf - Menon, R. (2004). Preservice teachers’ number sense.
*Focus on Learning Problems in Mathematics,**26*(2), 49–61.Google Scholar - Menon, R. (2009). Preservice teachers’ subject matter knowledge of mathematics.
*The International Journal for Mathematics Teaching and Learning*. Retrieved from http://www.cimtplymouth.ac.uk/jjournal/menon.pdf. - National Council of Teachers of Mathematics. (2000).
*Principles and standards for school mathematics*. Reston, VA: Author.Google Scholar - National Research Council. (2001). Adding it up: Helping children learn mathematics. In J. Kilpatrick, J. Swafford, & B. Findel (Eds.),
*Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education*. Washington, DC: National Academy Press.Google Scholar - Newton, K. J. (2008). An extensive analysis of preservice elementary teachers’ knowledge of fractions.
*American Educational Research Journal, 45,*1080–1110.Google Scholar - Reys, R. E., & Yang, D. C. (1998). Relationship between computational performance and number sense among sixth- and eighth-grade students in Taiwan.
*Journal for Research in Mathematics Education,**29*, 225–237.CrossRefGoogle Scholar - Reys, R. E, Reys, B. J., Nohda, N., & Emori, H (1995). Mental computation performance and strategy use of Japanese students in grades 2, 4, 6, and 8.
*Journal for Research in Mathematics Education, 26*(4), 304–326.Google Scholar - Reys, R., Rybolt, J., Bestgen, B., & Wyatt, J. (1982). Processes used by good computational estimators.
*Journal for Research in Mathematics Education, 13,*183–201.Google Scholar - Richards, J. (1991). Mathematical discussions. In E. von Glaserfeld (Ed.),
*Radical constructivism in mathematics education*(pp. 13–51). Dordrecht, The Netherlands: KluwerGoogle Scholar - Simon, M. (1993). Prospective elementary teachers’ knowledge of division.
*Journal for Research in Mathematics Education,**24*(3), 233–254.CrossRefGoogle Scholar - Smith, J. P., III. (1995). Competent reasoning with rational numbers.
*Cognition and Instruction, 13,*3–50.Google Scholar - Sowder, J. (1992). Estimation and number sense. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 371–389). New York: Macmillan.Google Scholar - Sowder, J., Sowder, L., & Nickerson, S. (2010).
*Reconceptualizing mathematics for elementary school teachers*. New York: W. H. Freeman.Google Scholar - Strauss, A., & Corbin, J. (1998).
*Basics of qualitative research: Techniques and procedures for developing grounded theory*(2nd ed.). Thousand Oaks, CA: Sage.Google Scholar - Thanheiser, E. (2009). Preservice elementary teachers’ conceptions of multidigit whole numbers.
*Journal for Research in Mathematics Education,**40*, 251–281.Google Scholar - Thanheiser, E. (2010). Investigating further preservice teachers’ conceptions of multidigit whole numbers: refining a framework.
*Educational Studies in Mathematics,**75*, 241–251.CrossRefGoogle Scholar - Tirosh, D., & Graeber, A. O. (1991). The effect of problem type and common misconceptions on preservice elementary teachers’ thinking about division.
*School Science and Mathematics, 91*(4), 157–163.Google Scholar - Tsao, Y.-L. (2005). The number sense of preservice elementary school teachers.
*College Student Journal,**39*, 647–679.Google Scholar - Whitacre, I. (2007). Preservice teachers’ number sensible mental computation strategies. In
*Proceedings of the tenth special interest group of the mathematical association of America on research in undergraduate mathematics education*. San Diego, CA. Retrieved from http://sigmaa.maa.org/rume/crume2007/papers/whitacre.pdf. - Whitacre, I. (2012).
*Investigating number sense development in a mathematics content course for prospective elementary teachers*. Unpublished doctoral dissertation, University of California, San Diego, and San Diego State University.Google Scholar - Yang, D. C. (2003). Teaching and learning number sense – an intervention study of fifth grade students in Taiwan.
*International Journal of Science and Mathematics Education, 1*(1), 115–134.Google Scholar - Yang, D. C. (2007). Investigating the strategies used by preservice teachers in Taiwan when responding to number sense questions.
*School Science and Mathematics,**107*, 293–301.CrossRefGoogle Scholar - Yang, D. C., Reys, R. E., & Reys, B. J. (2009). Number sense strategies used by pre-service teachers in Taiwan.
*International Journal of Science and Mathematics Education,**7*, 383–403.CrossRefGoogle Scholar - Zazkis, R. (2005). Representing numbers: Primes and irrational.
*International Journal of Mathematical Education in Science and Technology,**36*, 207–218.CrossRefGoogle Scholar - Zazkis, R., & Campbell, S. (1996). Divisibility and multiplicative structure of natural numbers: Preservice teachers’ understanding.
*Journal for Research in Mathematics Education,**27*, 540–563.CrossRefGoogle Scholar