References

Behr, M., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio and proportion. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 296–333). New York: Macmillan.

Behr, M., Harel, G., Post, T., & Lesh, R. (1994). Units of quantity: A conceptual basis common to additive and multiplicative structures. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 123–180). Albany, NY: SUNY Press.

Behr, M., Lesh, R., Post, T., & Silver, E. A. (1983). Rational number concepts. In R. Lesh and M. Landau (Eds.), Acquisition of mathematical concepts and processes (pp. 91–126). New York: Academic Press.

Behr, M., & Post, T. (1992). Teaching rational number and decimal concepts. In T. Post (Ed.), Teaching mathematics in grades K–8: Research-based methods (2nd ed.) (pp. 201–248). Boston: Allyn and Bacon.

Bezuk, N., & Cramer, K. (1989). Teaching about fractions: What, when, and how? In P. Trafton (Ed.), National Council of Teachers of Mathematics 1989 yearbook: New directions for elementary school mathematics (pp. 156–167). Reston, VA: National Council of Teachers of Mathematics.

Blanton, M. L. (2008). Algebra in the elementary classroom: Transforming thinking, transforming practice. Portsmouth, NH: Heinemann.

Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic & algebra in elementary school. Portsmouth, NH: Heinemann. Clements, D. H. (1999). Concrete manipulatives, concrete ideas. Contemporary Issues in Early Childhood, 1(1), 45–60.

Collins, A., & Dacey, L. (2010). Zeroing in on number and operations: Key ideas and common misconceptions, Grades 5–6 . Portland, ME. Stenhouse.

Common Core State Standards Initiative. (2010). Common Core State Standards for mathematics. Washington, DC: CCSSO & National Governors Association. Retrieved from www.corestandards.org/math

Cramer, K., & Whitney, S. (2010). Learning rational number concepts and skills in elementary school classrooms. In D. V. Lambdin & F. K. Lester (Eds.), Teaching and learning mathematics: Translating research for elementary school teachers (pp. 15–22). Reston, VA: National Council of Teachers of Mathematics.

Cuoco, A., Goldenberg, E. P., & Mark, J. (1996). Habits of mind: An organizing principle for a mathematics curriculum. Journal of Mathematical Behavior, 14(4), 375–402.

Empson, S. B., & Levi, L. (2011). Extending children's mathematics: Fractions and decimals. Portsmouth, NH: Heinemann.

Enyedy, N., Rubel, L., Castellón, V., Mukhopadhyay, S., Esmonde, I., & Secada, W. (2008). Revoicing in a multilingual classroom. Mathematical Thinking and Learning, 10(2), 134–162.

Falkner, K. P., Levi, L., & Carpenter, T. P. (1999). Children's understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6(4), 232–236.

Fosnot, C. T., & Dolk, M. (2002). Young mathematicians at work: Constructing fractions, decimals, and percents. Portsmouth, NH: Heinemann.

Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Boston: D. Reidel.

Huinker, D. (2002). Examining dimensions of fraction operation sense. In B. Litwiller & G. Bright (Eds.), Making sense of fractions, ratios, and proportions: 2002 yearbook of the National Council of Teachers of Mathematics (pp. 72–78). Reston, VA: National Council of Teachers of Mathematics.

Kamii, C., & Dominick, A. (1998). The harmful effects of algorithms in grades 1–4. In L. J. Morrow & M. J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics (pp. 130–140). Reston, VA: National Council of Teachers of Mathematics.

Kieren, T. (1976). On the mathematical, cognitive, and instructional foundations of rational numbers. In R. Lesh (Ed.), Number and measurement: papers from a research workshop (pp. 101–144). Columbus, OH: ERIC/SMEAC.

Lamon, S. J. (1996). The development of unitizing: Its role in children's partitioning strategies. Journal for Research in Mathematics Education, 27(2), 170–193.

Lamon, S. J. (1999). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. Mahwah, NJ: Lawrence Erlbaum.

Lamon, S. J. (2005). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (2nd ed.). Mahwah, NJ: Lawrence Erlbaum.

Lamon, S. J. (2012). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. New York: Routledge.

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

National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.

National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.

Neagoy, M. (2012). Planting the seeds of algebra, PreK–2: Explorations for the early grades. Thousand Oaks, CA: Corwin.

Neagoy, M. (2014). Planting the seeds of algebra, 3–5: Explorations for the upper elementary grades. Thousand Oaks, CA: Corwin.

O'Connor, M. C., & Michaels, S. (1996). Shifting participant frameworks: Orchestrating thinking practices in group discussion. In D. Hicks (Ed.), Discourse, learning, and schooling (pp. 63–103). New York: Cambridge University Press.

Otto, A. D., Caldwell, J. H., Hancock, S. W., & Zbiek, R. M. (2011). Developing essential understanding of multiplication and division for teaching mathematics in grades 3–5. Reston, VA: National Council of Teachers of Mathematics.

Seeley, C. L. (2015). Faster isn't smarter (2nd ed.). Sausalito, CA: Math Solutions.

Smith III, J. P. (2002). The development of students' knowledge of fractions and ratios. In B. Litwiller & G. Bright (Eds.), Making sense of fractions, ratios, and proportions: 2002 yearbook of the National Council of Teachers of Mathematics (pp. 3–17). Reston, VA: National Council of Teachers of Mathematics.

Sowder, J. (2002). Place value as the key to teaching decimal operations. In D. L. Chambers & National Council of Teachers of Mathematics, Putting research into practice in the elementary grades: Readings from journals of the National Council of Teachers of Mathematics (pp. 113–118). Madison: Wisconsin Center for Education Research, University of Wisconsin-Madison.

Tirosh, D., & Graeber, A. O. (1989). Preservice elementary teachers' explicit beliefs about multiplication and division. Educational Studies in Mathematics , 20(1), 79–96.

Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (8th ed.). Boston: Pearson.

Van de Walle, J. A., & Lovin, L. A. H. (2006). Teaching student-centered mathematics: Grades 3–5. Boston: Pearson.

Van de Walle, J. A., & Lovin, L.A.H. (2006). Teaching student-centered mathematics: Grades K–2. Boston: Pearson.

Vergnaud, G. (1979). The acquisition of arithmetic concepts. Educational Studies in Mathematics, 10, 263–274.