Documenting the Process of a Prospective Elementary Teacher’s Flexibility Development: Scaffolded Strategy Ranges and Sociomathematical Norms for Mental Computation

References

• Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education. The Elementary School Journal, 90 (4), 449–466.
   Web of Science ®Google 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. doi:10.1177/0022487108324554
   Web of Science ®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
• Chaiklin, S. (2003). The zone of proximal development in Vygotsky’s analysis of learning and instruction. In A. Kozulin, B. Gindis, V. S. Ageyev, & S. M. Miller (Eds.), Vygotsky’s educational theory in cultural context (pp. 39–64). Cambridge, UK: Cambridge University Press.
   Google Scholar
• Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist, 31, 175–190.
   Web of Science ®Google Scholar
• Cobb, P., Boufi, A., McClain, K., & Whitenack, J. (1997). Reflective discourse and collective reflection. Journal for Research in Mathematics Education, 28 (3), 258–277.
   Web of Science ®Google Scholar
• Cobb, P., Stephan, M., McClain, K., & Gravemeijer, K. (2001). Participating in Classroom Mathematical Practices. Journal of the Learning Sciences, 10 (1/2), 113–163.
   Web of Science ®Google Scholar
• Cole, R., Becker, N., Towns, M., Sweeney, G., Wawro, M., & Rasmussen, C. (2012). Adapting a methodology from mathematics education research to chemistry education research: Documenting collective activity. International Journal of Science and Mathematics Education, 10 (1), 193–211. doi:10.1007/s10763-011-9284-1
   Web of Science ®Google Scholar
• Corbin, J., & Strauss, A. (2015). Basics of qualitative research: Techniques and procedures for developing grounded theory (4th ed.). Los Angeles, CA: Sage publications.
   Google Scholar
• Fukawa-Connelly, T. (2012). Classroom sociomathematical norms for proof presentation in undergraduate in abstract algebra. Journal of Mathematical Behavior, 31, 401–416.
   Google Scholar
• Fuson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., Olivier, A. I., … Fennema, E. (1997). Children’s conceptual structures for mulitdigit numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics Education, 28 (2), 130–162.
   Web of Science ®Google Scholar
• Hammer, D. (1996). Misconception or p-prims: How may alternative perspectives of cognitive structure influence instructional perceptions and intentions? The Journal of the Learning Sciences, 5 (2), 97–127.
   Web of Science ®Google Scholar
• Hanson, S. A., & Hogan, T. P. (2000). Computational estimation skill of college students. Journal for Research in Mathematics Education, 31 (4), 483–499.
   Web of Science ®Google 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 (2), 128–137. doi:10.1016/j.jmathb.2008.07.004
   Google 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 (4), 443–463.
   Google Scholar
• Hope, J. A., & Sherrill, J. M. (1987). Characteristics of unskilled and skilled mental calculators. Journal for Research in Mathematics Education, 18 (2), 98–111.
   Web of Science ®Google Scholar
• Horn, I. S. (2008). Accountable argumentation as a participation structure to support learning through disagreement. Journal for Research in Mathematics Education, Monograph, 14, 97–126.
   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.
   Web of Science ®Google Scholar
• Kazemi, E., Elliott, R., Mumme, J., Carroll, C., Lesseig, K., & Kelley-Petersen, M. (2011). Noticing leaders’ interactions with videocases of teachers engaged in mathematics tasks in professional development. In M. Sherin, V. Jacobs, & R. Phillipp (Eds.), Mathematics teacher noticing: Seeing through teachers' eyes (pp. 188–203). New York: Routledge.
   Google Scholar
• Markovits, Z., & Sowder, J. (1994). Developing number sense: An intervention study in grade 7. Journal for Research in Mathematics Education, 25 (1), 4–29.
   Web of Science ®Google 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
• McIntosh, A., Nohda, N., Reys, B. J., & Reys, R. E. (1995). Mental computation performance in Australia, Japan and the United States. Educational Studies in Mathematics, 29 (3), 237–258.
   Google Scholar
• McIntosh, A., Reys, B. J., & Reys, R. E. (1992). A proposed framework for examining basic number sense. For the Learning of Mathematics, 12 (3), 2–8, 44.
   Google Scholar
• Menon, R. (2003). Exploring preservice teachers? understanding of two-digit multiplication. International Journal for Mathematics Teaching and Learning. Retrieved from http://www.cimt.plymouth.ac.uk/journal/ramakrishnanmenon.pdf
   Google Scholar
• 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
   Google Scholar
• Mercer, N. (1995). The guided construction of knowledge: Talk amongst teachers and learners. Clevedon: Multilingual Matters.
   Google Scholar
• Merriam, S. B. (1998). Qualitative research and case study applications in education. San Francisco: Josey-Bass.
   Google Scholar
• Mewborn, D. (2001). Teachers content knowledge, teacher education, and their effects on the preparation of elementary teachers in the United States. Mathematics Teacher Education and Development, 3, 28–36.
   Google Scholar
• Moll, L. C., Amanti, C., Neff, D., & Gonzalez, N. (1992). Funds of knowledge for teaching: Using a qualitative approach to connect homes and classrooms. Theory into Practice, 31 (2), 132–141.
   Google Scholar
• National Council of Teachers of Mathematics (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: NCTM.
   Google Scholar
• National Research Council (2001). Adding it up: Helping children learn mathematics. Washington, DC: The National Academies Press. doi:10.17226/9822
   Google Scholar
• Nickerson, S. D., & Whitacre, I. M. (2010). A local instruction theory for the development of number sense. Mathematical Thinking and Learning, 12, 227–252.
   Web of Science ®Google Scholar
• Nunes, T., Schliemann, A. D., & Carraher, D. W. (1993). Street mathematics and school mathematics. Cambridge, MA: Cambridge University Press.
   Google Scholar
• Philipp, R. A., Cabral, C., & Schappelle, B. P. (2005). IMAP CD-ROM: Integrating mathematics and pedagogy to illustrate children’s reasoning [Computer software]. Upper Saddle River, NJ: Pearson Education.
   Google Scholar
• Rasmussen, C., & Stephan, M. (2008). A methodology for documenting collective activity. In A. E. Kelly, R. A. Lesh, & J. Y. Baek (Eds.), Handbook of design research methods in education: Innovations in science, technology, engineering, and mathematics learning and teaching (pp. 195–215). New York, NY: Routledge.
   Google Scholar
• Russell, S. J. (2000). Developing computational fluency with whole numbers. Teaching Children Mathematics, 7 (3), 154–158.
   Google Scholar
• Schoenfeld, A. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics . In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). New York, NY: MacMillan.
   Google Scholar
• Simon, M. (1993). Prospective elementary teachers? knowledge of division. Journal for Research in Mathematics Education, 24 (3), 233–254.
   Web of Science ®Google Scholar
• Smith, J. P. III. (1995). Competent reasoning with rational numbers. Cognition and Instruction, 13 (1), 3–50.
   Web of Science ®Google Scholar
• Smith, J. P. III, diSessa, A. A., & Roschelle, J. (1994). Misconceptions reconceived: A constructivist analysis of knowledge in transition. Journal of the Learning Sciences, 3 (2), 115–163.
   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
• Star, J. R., & Newton, K. J. (2009). The nature and development of experts’ strategy flexibility for solving equations. ZDM Mathematics ZDM, 41 (5), 557–567.
   Google Scholar
• Star, J. R., & Seifert, C. (2006). The development of flexibility in equation solving. Contemporary Educational Psychology, 31 (3), 280–300.
   Web of Science ®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. (2015). Developing prospective teachers’ conceptions with well-designed tasks: Explaining successes and analyzing conceptual difficulties. Journal of Mathematics Teacher Education, 18 (2), 141–172.
   Web of Science ®Google Scholar
• Thanheiser, E., Browning, C., Edson, A. J., Lo, J.-J., Whitacre, I., Olanoff, D., & Morton, C. (2014). Prospective elementary mathematics teacher content knowledge: What do we know, what do we not know. And Where Do We Go? The Mathematics Enthusiast, 11 (2), 433–448.
   Google Scholar
• Thanheiser, E., Whitacre, I., & Roy, G. (2014). Mathematical content knowledge for teaching elementary mathematics: A focus on whole-number concepts and operations. The Mathematics Enthusiast, 11 (2), 217–266.
   Google Scholar
• Toulmin, S. E. (2003). The uses of argument. Cambridge, UK: Cambridge University Press. doi:10.1017/CBO9780511840005
   Google Scholar
• Tsao, Y.-L. (2005). The number sense of preservice elementary school teachers. College Student Journal, 39 (4), 647–679.
   Google Scholar
• Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. (M. Cole, V. John-Steiner, S. Scribner, & E. Souberman, Eds.). Cambridge, MA: Harvard University Press.
   Google Scholar
• Whitacre, I. M. (2012). Investigating number sense development in a mathematics content course for prospective elementary teachers (Doctoral dissertation). Retrieved from http://escholarship.org/uc/item/3q58m77r
   Google Scholar
• Whitacre, I. (2013). Viewing prospective elementary teachers' prior knowledge as a resource in their number sense development. In M. Martinez & A. Castro Superfine (Ed.), 35th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 901–904). Chicago, IL: University of Illinois at Chicago.
   Google Scholar
• Whitacre, I. (2015). Strategy ranges: Describing change in prospective elementary teachers' approaches to mental computation of sums and differences. Journal of Mathematics Teacher Education, 18 (4), 353–373. doi:10.1007/s10857-014-9281-8
   Web of Science ®Google Scholar
• Whitacre, I. (2017). Prospective elementary teachers learning to reason flexibly with sums and differences: Number sense development viewed through the lens of collective activity. Cognition and Instruction, 36 (1), 56–82. doi:10.1080/07370008.2017.1394303
   Web of Science ®Google Scholar
• Whitacre, I., Atabaş, S., Findley, K. (2018). Exploring unfamiliar paths through familiar mathematical territory: Constraints and affordances in a preservice teacher’s reasoning about fraction comparisons. Journal of Mathematical Behavior. doi:10.1016/j.jmathb.2018.06.006
   Google Scholar
• Whitacre, I., & Nickerson, S. D. (2012). Prospective elementary teachers' justifications for their nonstandard mental computation strategies. In L. R. Van Zoest, J.-J. Lo, & J. L. Kratsky (Ed.), 34th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 777–780). Reno, NV: University of Reno.
   Google Scholar
• Whitacre, I., & Nickerson, S. D. (2016a). Investigating the improvement of prospective elementary teachers' number sense in reasoning about fraction magnitude. Journal of Mathematics Teacher Education, 19 (1), 57–77. doi:10.1007/s10857-014-9295-2
   Web of Science ®Google Scholar
• Whitacre, I., & Nickerson, S. D. (2016b). Prospective elementary teachers making sense of multi-digit multiplication: Leveraging resources. Journal for Research in Mathematics Education, 47, 270–307.
   Web of Science ®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 (7), 293–301.
   Google 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 (2), 383–403.
   Web of Science ®Google Scholar
• Yin, R. K. (1994). Case study research: Design and methods. Thousand Oaks, CA: Sage.
   Google Scholar
• Zazkis, R. (2005). Representing numbers: Primes and irrational. International Journal of Mathematical Education in Science and Technology, 36, 207–218.
   Google 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.
   Web of Science ®Google Scholar