Efficacy of Different Concrete Models for Teaching the Part-Whole Construct for FractionsKathleen A Cramer and Terry Wyberg
Ashlock, R. B. (2010) Error patterns in computation: Using error patterns to help each student learn. Boston: Allyn & Bacon.
Ball, D. (1992). Magical hopes: Manipulatives and reform in math education. American Federation of Teachers, summer, 14-18; 46-47.
Behr, M. LEsh, R., Post, T. & Silver, E. (1983). Rational number concepts. In R. Lesh & M. Landau (Eds.), Acquisitions of mathematics concepts and processes (pp. 92-127). New York: Academic Press.
Behr, Wachsmuth & Post (1988). Rational number learning aids: Transfer from continuous models to discrete models. Focus on Learning Problems in Mathematics, 10(4), 1-17.
Bright, George, Merlyn Behr, Thomas Post, and Ipke Wachsmuth. 1988. “Identifying Fractions on Number Lines.” Journal for Research in Mathematics Education 19 (3): 215-232.
Bruner, J. (1960). The process of education. Cambridge, MA: Harvard University Press.
Cramer, K. (2003). Using a translation model for curriculum development and classroom instruction. In R. Lesh & H. Doerr (Eds.), Beyond Constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp.449 - 464 ). Mahwah, N.J.: Lawrence Erlbaum Associates.
Cramer, K. & Henry, A. (2002) Using manipulative models to build number sense for addition of fractions. In B.Litwiller & G. Bright (Eds.), Making sense of fractions, ratios, and proportions: 2002 yearbook (41-48). Reston, VA: National Council of Teachers of Mathematics.
Cramer, K., Post, T., & delMas, R. (2002). Initial fraction learning by fourth- and fifth-grade students: A comparison of the effects of using commercial curricula with the effects of using the Rational Project curriculum. Journal for Research in Mathematics Education, 33 (2), 111-144.
English, L. D. (1997). Analogs, metaphors, and images: Vehicles for mathematical reasoning. In L. English (Ed.) Mathemaical Reasoning: Analogs, metaphors, and images. Mahwah, N.J.:Lawrence Erlbaum Associates.
English, L.D., & Halford, G. S. (1995). Mathematics education: Models and processes. Mahwah, N.J.:Lawrence Erlbaum Associates.
Ginsberg, H., & Opper, S. (1969). Piaget’s theory of intellectual development: An introduction. Englewood Cliffs, NJ: Prentice-Hall.
Hannula, M. (2003). Location fraction on a number line. In N. Pateman, B. J. Dougherty, & J. T. Zillox (Eds.), Proceedings of the 2003 joint meeting of PME and PMNA (Vol. 3, pp3-17). Honolulu, HI: College of Education, University of Hawaii.
Kendall-Hunt Publishing Company (2003). Math trailblazers- Grade 4 (second edition). Dubuque, Iowa: Author.
Kendall-Hunt Publishing Company (2003). Math trailblazers- Grade 5 (second edition). Dubuque, Iowa: Author.
Keijzer, R., & Terwel, J. (2003). Learning for mathematical insight: A longitudinal comparative study on modeling. Learning and Instruction, 13, 285-304.
Koupa, V., Zawojewski, J., & Strutchen, M. (1997). What do students know about number and operations? In P. A. Kenny & E. A. Silver (Eds.), Results for the sixth mathematics assessment of the National Assessment of Educational Progress (pp.87-140). Reston, VA: National Council of Teachers of Mathematics.
Moss, J. (2005). Pipes, tubes, and beakers: New approaches to teaching rational-number system. In M. Donovan & J.D. Bransford (Eds.), How students learn: Mathematics in the classroom (pp.309-350). Washington, D.C.: The National Academies Press.
Moss, J., & Case, R. (1999). Developing children’s understanding of rational numbers: A new model and experimental curriculum. Journal for Research in Mathematics Education, 30(2), 119, 122-147, 1999
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. J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.
Post, T., & Cramer, K. (1987). Children's strategies in ordering rational numbers. Arithmetic Teacher, 35 (2), 33-35.
Post, T. (1992). Some notes on the nature of mathematics learning. In T. Post (Ed.) Teaching mathematics in grades K- 8: Research-based methods, pp. 1 – 22. Needham, MA: Allyn and Bacon.
Post, T., Cramer, K., Behr, M., Lesh, R., & Harel, G. (1993). Curriculum implications of research on the learning, teaching and assessing of rational number concepts. In T. Carpenter, E. Fennema & T. Rhomberg (Eds.), Rational Number: An integration of research (pp. 327 - 362). Hillsdale, NJ: Lawrence Erlbaum Associates.
Kathleen Cramer - Associate Professor in Mathematics Education at the University of Minnesota
Terry Wyberg - Senior Lecturer in Mathematics Education at the University of Minnesota