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.
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Notes
It is beyond the scope of this paper to attempt to account for the differential changes in PTs’ strategy ranges, e.g., to explain why Zelda’s addition strategy range changed substantially, while her subtraction strategy range did not. The author’s dissertation includes two case studies of the number sense development of individual PTs. These case studies attempt to answer such finegrained questions (Whitacre 2012).
Note that, in theory, Inflexible need not imply MASAbound. However, individuals who use only one strategy to perform an operation mentally do tend to be MASAbound. Likewise, Heirdsfield and Cooper (2004) described accurate, inflexible mental calculators as relying on the MASAs specifically. It is not clear how to integrate invalid strategies into the picture of strategy ranges. This matter will require further attention.
References
Anghileri, J. (2000). Teaching number sense. London: Continuum.
Baek, J.M. (1998). Children’s invented algorithms for multidigit multiplication problems. NCTM Yearbook, 1998, 151–160.
Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education. The Elementary School Journal, 90, 449–466.
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.
Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.
Carraher, T. N., Carraher, D. W., & Schliemann, A. D. (1987). Written and oral mathematics. Journal for Research in Mathematics Education, 18, 83–97.
Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist, 31, 175–190.
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.
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.
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.
Greeno, J. (1991). Number sense as situated knowing in a conceptual domain. Journal for Research in Mathematics Education, 22, 170–218.
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.
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.
Hope, J. A., & Sherrill, J. M. (1987). Characteristics of unskilled and skilled mental calculators. Journal for Research in Mathematics Education, 18, 98–111.
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.
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. New Jersey: Erlbaum.
Markovits, Z., & Sowder, J. (1994). Developing number sense: An intervention study in grade 7. Journal for Research in Mathematics Education, 25, 4–29.
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.
Menon, R. (2003). Exploring preservice teachers understanding of twodigit 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.
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.
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.
Newton, K. J. (2008). An extensive analysis of preservice elementary teachers’ knowledge of fractions. American Educational Research Journal, 45, 1080–1110.
Reys, R. E., & Yang, D. C. (1998). Relationship between computational performance and number sense among sixth and eighthgrade students in Taiwan. Journal for Research in Mathematics Education, 29, 225–237.
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.
Reys, R., Rybolt, J., Bestgen, B., & Wyatt, J. (1982). Processes used by good computational estimators. Journal for Research in Mathematics Education, 13, 183–201.
Richards, J. (1991). Mathematical discussions. In E. von Glaserfeld (Ed.), Radical constructivism in mathematics education (pp. 13–51). Dordrecht, The Netherlands: Kluwer
Simon, M. (1993). Prospective elementary teachers’ knowledge of division. Journal for Research in Mathematics Education, 24(3), 233–254.
Smith, J. P., III. (1995). Competent reasoning with rational numbers. Cognition and Instruction, 13, 3–50.
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.
Sowder, J., Sowder, L., & Nickerson, S. (2010). Reconceptualizing mathematics for elementary school teachers. New York: W. H. Freeman.
Strauss, A., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory (2nd ed.). Thousand Oaks, CA: Sage.
Thanheiser, E. (2009). Preservice elementary teachers’ conceptions of multidigit whole numbers. Journal for Research in Mathematics Education, 40, 251–281.
Thanheiser, E. (2010). Investigating further preservice teachers’ conceptions of multidigit whole numbers: refining a framework. Educational Studies in Mathematics, 75, 241–251.
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.
Tsao, Y.L. (2005). The number sense of preservice elementary school teachers. College Student Journal, 39, 647–679.
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.
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.
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.
Yang, D. C., Reys, R. E., & Reys, B. J. (2009). Number sense strategies used by preservice teachers in Taiwan. International Journal of Science and Mathematics Education, 7, 383–403.
Zazkis, R. (2005). Representing numbers: Primes and irrational. International Journal of Mathematical Education in Science and Technology, 36, 207–218.
Zazkis, R., & Campbell, S. (1996). Divisibility and multiplicative structure of natural numbers: Preservice teachers’ understanding. Journal for Research in Mathematics Education, 27, 540–563.
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.
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Appendices
Appendix 1: Mental computation interview tasks
Instructions
For the following tasks, please find an exact answer mentally and explain your thinking. Each task involves a story about Bobo, who sells oboes.
(Addition Story)
Bobo’s oboe business is booming! However, he could use some help keeping track of his sales.

A1. Bobo sells 37 oboes 1 month and 52 the next month. How many oboes did he sell in those 2 months?

A2. Bobo sells 64 oboes 1 month and 87 the next month. How many oboes did he sell in those 2 months?

A3. Bobo sells 96 oboes 1 month and 157 the next month. How many oboes did he sell in those 2 months?

A4. Bobo sells 38 oboes 1 month and 99 the next month. How many oboes did he sell in those 2 months?
(Subtraction Story)
As everyone knows, oboes don’t grow on trees. Bobo has to spend money to make money.

S1. If Bobo buys an oboe for $34 and then sells it for $78, how much money does he make?

S2. If Bobo buys an oboe for $52 and then sells it for $178, how much money does he make?

S3. If Bobo buys an oboe for $45 and then sells it for $82, how much money does he make?

S4. If Bobo buys an oboe for $49 and then sells it for $125, how much money does he make?
Appendix 2
Coding scheme for mental addition strategies
Strategy  Description 

MASA  The student used the mental analogue of the standard (USA) addition algorithm. Language such as “carry the one” often accompanied the use of this strategy. Students generally used nonplacevalue language 
Right to Left  The student added placevaluewise from right to left but did not necessarily picture the digits aligned, as in the standard algorithm. In contrast to the MASA, the student used placevalue language 
Left to Right  The student added placevaluewise from left to right. Typically, she used placevalue language 
Aggregation  The student began with one of the two addends and added the other one on in convenient chunks, generally working from big to small and keeping a running subtotal 
Giving  The student altered the problem such that part of one addend (usually a small number of ones) was added (“given”) to the other prior to finding their sum 
Single Compensation  The student altered one of the two addends (usually rounding up or down to the nearest multiple of ten) prior to performing the addition. The student added the rounded numbers and then compensated for rounding 
Double Compensation  The student altered both addends (usually rounding them up or down to the nearest multiple of ten) prior to performing the addition. The student added the rounded numbers and then compensated for rounding 
Appendix 3
Coding scheme for mental subtraction strategies
Strategy  Description 

MASA  The student used the mental analogue of the standard (USA) subtraction algorithm. Language such as “borrowing” often accompanied the use of this strategy 
Right to le  The student subtracted placevaluewise from right to left but without visualizing the numbers aligned as in the standard algorithm 
Left to Right  The student subtracted placevaluewise from left to right 
Aggregation  The student either (a) began with the subtrahend and added onto it in convenient chunks until the minuend was reached or (b) began with the minuend and subtracted off the subtrahend in convenient chunks. The student kept a cumulative mental record of the amount added or subtracted. In the case of adding on to the subtrahend, this amount gave the difference. In the case of subtracting from the minuend, the result gave the difference 
Minuend Compensation  The student altered the minuend (often rounding up or down to a multiple of ten) prior to performing the subtraction. The student found the difference between the subtrahend and rounded minuend and then compensated appropriately for rounding 
Subtrahend Compensation  The student altered the subtrahend (often rounding up or down to the nearest multiple of ten) prior to performing the subtraction. The student found the difference between the minuend and rounded subtrahend and then compensated for rounding. Specifically, she compensated correctly: She added to the difference to compensate for having added to the subtrahend, or she subtracted from the difference to compensate for having subtracted from the subtrahend 
Shifting the difference  The student added the same amount to, or subtracted the same amount from, both the minuend and subtrahend. She then found the difference between the rounded numbers 
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Whitacre, I. Strategy ranges: describing change in prospective elementary teachers’ approaches to mental computation of sums and differences. J Math Teacher Educ 18, 353–373 (2015). https://doi.org/10.1007/s1085701492818
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DOI: https://doi.org/10.1007/s1085701492818
Keywords
 Prospective elementary teachers
 Mental computation
 Flexibility
 Strategy ranges