U.S. grade 4–6 students’ rational-number substitutions for odd-sum unknown addend tasks

References

• Asquith, P., Stephens, A. C., Knuth, E. J., & Alibali, M. W. (2007). Middle school mathematics teachers’ knowledge of students’ understanding of core algebraic concepts: Equal sign and variable. Mathematical Thinking Amd Learning, 9(3), 249–272. doi:10.1080/10986060701360910
   Google Scholar
• Blanton, M., & Knuth, E. (2013). Progression for developing understanding of variable across grades 3-7. Project LEAP: Learning through an Early ALgebra Progression. Retrieved from http://algebra.wceruw.org/documents/GA_LP_web.pdf
   Google Scholar
• Blanton, M., Levi, L., Crites, T., & Dougherty, B. J. (2011). Developing essential understanding of algebraic thinking for teaching mathematics in grades 3-5 (B. J. Dougherty Ed.). Reston, VA: NCTM.
   Google Scholar
• Booth, L. R. (1984). Algebra: Children’s strategies and errors. Windsor, Berkshire: The NFER-NELSON Publishing Company.
   Google Scholar
• Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). (2000). How people learn: Brain, mind, experience, and school (Expanded ed.). Washington, DC: National Academy Press.
   Google Scholar
• Brizuela, B., Blanton, M., Gardiner, A., Gravel, B., & Newman-Owens, A. (2013). Bringing variable notation to the forefront of early mathematics education. Paper presented at the National Council of Teachers of Mathematics Research Pre-Session, Denver, CO.
   Google Scholar
• Brizuela, B., Gardiner, A., Sawrey, K. B., Newman-Owens, A., & Blanton, M. (2015). First grade students’ uses of tables as they explore functional relations. Paper presented at the National Council of Teachers of Mathematics Research Conference, Boston, MA.
   Google Scholar
• Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth, NH: Hienemann.
   Google Scholar
• Carraher, D. W., Brizuela, B., & Schliemann, A. (2000). Bringing out the algebraic character of arithmetic: Instantiating variables in addition and subtraction. Paper presented at the Psychology of Mathematics Education (PME), Hiroshima, Japan.
   Google Scholar
• Carraher, D. W., & Schliemann, A. D. (2007). Early algebra and algebraic reasoning: Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 669–705). Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Carraher, D. W., Schliemann, A. D., & Schwartz, J. (2007). Early algebra is not the same as algebra early. In J. Kaput, D. W. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 235–272). New York, NY: Lawrence Erlbaum Associates.
   Google Scholar
• Charles, R. I., Fennell, F. S., Caldwell, J. H., Ramirez, A. B., Cavanagh, M., Sammons, K. B.,… Van de Walle, J. A. (2009). enVisionMATH. Glenview, IL: Pearson Education.
   Google Scholar
• Christou, K. P., & Vosniadou, S. (2012). What kinds of numbers do students assign to literal symbols? Aspects of the transition from arithmetic to algebra. Mathematical Thinking and Learning, 14(1), 1–27. doi:10.1080/10986065.2012.625074
   Web of Science ®Google Scholar
• Christou, K. P., Vosniadou, S., & Vamvakoussi, X. (2007). Students’ interpretations of literal symbols in algebra: Reframing the conceptual change approach in learning and instruction (pp. 283–297). Emerald Group Publishing:New Milford: Turpin Distribution, Incorporated [Distributer], Bingley.
   Google Scholar
• Creswell, J. W. (2003). Research design: Qualitative, quantitative, and mixed methods approaches (2nd ed.). Thousand Oaks, CA: Sage Publishing.
   Google Scholar
• Dougherty, B. (2004). Early algebra: Perspectives and assumptions. For the Learning of Mathematics, 24(3), 28–30. Retrieved from http://www.jstor.org/stable/40248469
   Google Scholar
• Dougherty, B. (2008). Measure Up: A quantitative view of early algebra. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 389–412). New York, NY: Lawrence Erlbaum Associates.
   Google Scholar
• Dougherty, B. J., & Venenciano, L. C. H. (2007). Measure Up for understanding. Teaching Children Mathematics, 13(9), 452–456.
   Google Scholar
• Dougherty, B. J., & Zillox, J. (2003). Voyaging from theory to practice in teaching and learning mathematics: A view from Hawai’i. Paper presented at the International Group for the Psychology of Mathematics Education, Honolulu, HI.
   Google Scholar
• Ellis, A. B. (2007). Connections between generalizing and justifying: Students’ reasoning with linear relationships. Journal for Research in Mathematics Education, 38(3), 194–229.
   Web of Science ®Google Scholar
• Fennell, F. M., Altieri, M. B., Silver Burderr Ginn (Firm), & Krulik, S. (1998). Mathematics: The Path to Math Success! Parsippany, NJ: Silver Burdett Ginn.
   Google Scholar
• Fujii, T. (2003). Probing students’ understanding of variables through cognitive conflict problems: Is the concept of a variable so difficult for students to understand? Paper presented at the 27th annual conference of the International Group for the Psychology of Mathematics Education, Honolulu, Hawaii.
   Google Scholar
• Fujii, T., & Stephens, M. (2008). Using number sentences to introduce the idea of variable. In C. E. Greenes & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics: Seventieth yearbook (pp. 127–149). Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Gee, J. P. (2008). A Sociocultural Perspective on Opportunity to Learn. Assessment, Equity, and Opportunity to Learn. doi: 10.1017/Cbo9780511802157
   Google Scholar
• Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65–100). Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Hirsch, C. R. (Ed.). (2007). Perspectives on the design and development of school mathematics. Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Kaput, J. J. (2008). What is algebra? What is algebraic reasoning? In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 5–17). New York, NY: Taylor & Francis Group, LLC.
   Google Scholar
• Kaput, J., & Blanton, M. (2001). Algebrafying the elementary mathematics experience. Part 1: Tranforming task structures. Paper presented at the 12th ICMI Study Conference: The Future of the Teaching and Learning of Algebra, Melbourne, VC, Australia.
   Google Scholar
• Kaput, J. J., Blanton, M. L., & Moreno, L. (2008). Algebra from a symbolization point of view. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 19–55). New York, NY: Taylor & Francis.
   Google Scholar
• Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A., & Stephens, A. C. (2005). Middle school students’ understanding of core algebraic concepts: Equivalence & variable. ZDM: The International Journal on Mathematics Education, 37(1), 68–76.
   Google Scholar
• Koedinger, K. R., & Nathan, M. J. (2004). The real story behind story problems: Effects of representations on quantitative reasoning. Journal of the Learning Sciences, 13(2), 129–164. doi:10.1207/s15327809jls1302_1
   Web of Science ®Google Scholar
• Küchemann, D. (1981). Children’s understanding of mathematics. In K. Hart (Ed.), Algebra (pp. 11–16). London, UK: Murray.
   Google Scholar
• Lobato, J., Burns Ellis, A., & Muñoz, R. (2003). How “focusing phenomena” in the instructional environment support individual students’ generalizations. Mathematical Thinking and Learning, 5(1), 1–36. doi:10.1207/S15327833MTL0501_01
   Google Scholar
• Lucariello, J., Tine, M. T., & Ganley, C. M. (2014). A formative assessment of students’ algebraic variable misconceptions. The Journal of Mathematical Behavior, 33, 30–41. doi:10.1016/j.jmathb.2013.09.001
   Google Scholar
• MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation: 11-15. Educational Studies in Mathematics, 33, 1–19. doi:10.1023/A:1002970913563
   Google Scholar
• Marum, T., Isler, I., Stephens, A. C., Gardiner, A., Blanton, M., & Knuth, E. (2011). From specific value to variable: Developing students’ avilities to represent unknowns. Paper presented at the 33rd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Reno, NV.
   Google Scholar
• McNeil, N. M., Grandau, L., Knuth, E. J., Alibali, M. W., Stephens, A. C., Hattikudur, S., & Krill, D. E. (2006). Middle-school students’ understanding of the equal sign: The books they read can’t help. Cognition and Instruction, 24(3), 367–385. doi:10.1207/s1532690xci2403_3
   Web of Science ®Google Scholar
• Powell, S. R. (2012). Equations and the equal sign in elementary mathematics textbooks. The Elementary School Journal, 112(4), 627–648. doi:10.1086/665009
   PubMed Web of Science ®Google Scholar
• Radford, L. (2000). Signs and meanings in students’ emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics, 42, 237–268. doi:10.1023/A:1017530828058
   Google Scholar
• Russell, S. J., Economopoulos, K., Cochran, K., Murray, M., Hollister, A., Batable, V.,… Wittenberg, L. (2008). Investigations in Number, Data, and Space (2 ed.). Glenview, IL: Pearson/Scott Foresman.
   Google Scholar
• Sfard, A., & Linchevski, L. (1994). The gains and the pitfalls of reification: The case of algebra. Educational Studies in Mathematics, 26(2/3), 191–228. doi:10.1007/BF01273663
   Google Scholar
• Skemp, R. R. (1982). Communicating mathematics: Surface structures and deep structures. Visible Language, 16(3), 281–288. Retrieved from < Go to ISI >://WOS:A1982PR32300008
   Google Scholar
• Stacey, K. (1989). Finding and using patterns in linear generalising problems. Educational Studies in Mathematics, 20(2), 147–164. doi:10.1007/BF00579460
   Google Scholar
• Swafford, J. O., & Langrall, C. W. (2000). Grade 6 students’ preinstructional use of equations to describe and represent problem situations. Journal for Research in Mathematics Education, 31(1), 89–112. doi:10.2307/749821
   Web of Science ®Google Scholar
• Switzer, J. M. (2016, May). What conceptions have US Grade 4-6 students’ generalized for formal and informal common representations of unknown addends? International Journal of Mathematical Teaching and Learning, 1–43. Retrieved from International Journal for Mathemtics Teaching and Learning website http://www.cimt.org.uk/journal/switzer.pdf
   Google Scholar
• Switzer, J. M., Buchheister, K., & Dougherty, B. (2014). Examining number: A soft idea. [News & Views]. Teaching Children Mathematics, 21(2), 69–73. doi:10.5951/teacchilmath.21.2.0069
   Google Scholar
• Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169. doi:10.1007/BF00305619
   Google Scholar
• Usiskin, Z. (1999). Conceptions of school algebra and uses of variables. In B. Moses (Ed.), Algebraic thinking, grades K-12: Readings from NCTM’s school-based journals and other publications (pp. 7–13). Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Warren, E., & Cooper, T. (2007). Generalising the pattern rule for visual growth patterns: Actions that support 8 year olds’ thinking. Educational Studies in Mathematics, 67(2), 171–185. doi:10.1007/s10649-007-9092-2
   Google Scholar