Early algebraic thinking and the US mathematics standards for grades K to 5 / El pensamiento algebraico temprano y los estándares matemáticos en la Educación Primaria (6–12 años) en Estados Unidos

References / Referencias

• Balacheff, N. (2001). Symbolic arithmetic vs. algebra the core of a didactical dilemma. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives on school algebra (pp. 249–260). Dordrecht: Springer.
   Google Scholar
• Ball, D. L., Ferrini-Mundy, J., Kilpatrick, J., Milgram, R. J., Schmid, W., & Schaar, R. (2005). Reaching for common ground in K–12 mathematics education. Notices of the AMS, 52, 1055–1058.
   Google Scholar
• Beckmann, S. (2004). Solving algebra and other story problems with simple diagrams: A method demonstrated in grade 4–6 texts used in Singapore. The Mathematics Educator, 14, 42–46.
   Google Scholar
• Blanton, M. (2008). Algebra and the elementary classroom: Transforming thinking, transforming practice. New York, NY: Pearson Education.
   Google Scholar
• Blanton, M., Brizuela, B., Gardiner, A. M., Sawrey, K., & Newman-Owens, A. (2015). A learning trajectory in 6-year-olds’ thinking about generalizing functional relationships. Journal for Research in Mathematics Education, 46, 511–558.
   Web of Science ®Google Scholar
• Blanton, M., Brizuela, B. M., Gardiner, A. M., Sawrey, K., & Newman-Owens, A. (2017). A progression in first-grade children’s thinking about variable and variable notation in functional relationships. Educational Studies in Mathematics, 95, 181–202. Retrieved from http://rdcu.be/osuZ
   Web of Science ®Google Scholar
• Blanton, M., Levi, L., Crites, T., & Dougherty, B. (2011). Developing essential understanding of algebraic thinking for teaching mathematics in grades 3–5. Essential understanding series. Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Blanton, M., Stephens, A., Knuth, E., Gardiner, A. M., Isler, I., & Kim, J.-S. (2015). The development of children’s algebraic thinking: The impact of a comprehensive early algebra intervention in third grade. Journal for Research in Mathematics Education, 46, 39–87.
   Web of Science ®Google Scholar
• Bodanskii, F. (1969/1991). The formation of an algebraic method of problem solving in primary school. In V. V. Davydov (Ed.), Psychological abilities of primary school children in learning mathematics (Vol. 6, pp. 275–338). Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Boyer, C. B. (1968). A history of mathematics. New York, NY. John Wiley & Sons, Inc.
   Google Scholar
• Brizuela, B. (2016). Variables in elementary mathematics education. The Elementary School Journal, 117, 46–71.
   Web of Science ®Google Scholar
• Brizuela, B., & Lara-Roth, S. (2002). Additive relations and function tables. Journal of Mathematical Behavior, 20, 309–319.
   Google Scholar
• Brizuela, B. M., Blanton, M., Sawrey, K., Newman-Owens, A., & Murphy Gardiner, A. (2015). Children’s use of variables and variable notation to represent their algebraic ideas. Mathematical Thinking and Learning, 17, 34–63.
   Web of Science ®Google Scholar
• Brizuela, B. M., & Earnest, D. (2008). Multiple notational systems and algebraic understanding: The case of the “best deal” problem. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 373–402). Mahwah, NJ: Erlbaum & NCTM.
   Google Scholar
• Brizuela, B. M., & Schliemann, A. D. (2004). Ten-year-old students solving linear equations. For the Learning of Mathematics, 24(2), 33–40.
   Google Scholar
• Cai, J., & Knuth, E. (Eds.). (2011). Early algebraization. A Global dialogue from multiple perspectives. Berlin: Springer.
   Google Scholar
• Cañadas, M. C., Brizuela, B. M., & Blanton, M. (2016). Second graders articulating ideas about linear functional relationships. Journal of Mathematical Behavior, 41, 97–103.
   Web of Science ®Google Scholar
• Carpenter, T., & Levi, L. (1999). Developing conceptions of algebraic reasoning in primary grades. Paper presented at the Annual Meeting of the American Educational Research Association, Montreal, Canada.
   Google Scholar
• Carraher, D. W., Martinez, M., & Schliemann, A. D. (2008). Early algebra and mathematical generalization. ZDM – The International Journal on Mathematics Education (formerly Zentralblatt fur Didaktik der Mathematik), 40, 3–22.
   Google Scholar
• Carraher, D. W., & Schliemann, A. D. (2007). Early algebra and algebraic reasoning. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 669–705). Charlotte, NC: Information Age Publishing.
   Google Scholar
• Carraher, D. W., & Schliemann, A. D. (2016). Powerful ideas in elementary school mathematics. In L. English, & D. Kirshner (Eds.), Handbook of international research in mathematics education (pp. 191–218). New York, NY: Taylor and Francis.
   Google Scholar
• Carraher, D. W., & Schliemann, A. D. (2018). Cultivating early algebraic reasoning. In C. Kieran (Ed.), Teaching and learning algebraic thinking with 5-12- year-olds (pp. 107–138). The Global Evolution of an Emerging Field of Research and Practice. ICME-13 Monographs. Chaim: Springer International Publishing.
   Google Scholar
• Carraher, D. W., Schliemann, A. D., & Brizuela, B. (2000, October). Early algebra, early arithmetic: Treating operations as functions. [Medium format: CD] Plenary presentation at XXII Meeting of the North American chapter of the International Group for the Psychology of Mathematics Education, Tucson, AZ.
   Google Scholar
• Carraher, D. W., Schliemann, A. D., & Brizuela, B. (2005). Treating operations as functions. In D. W. Carraher & R. Nemirovsky (Eds.), Media and meaning. [Medium format: CD]. Monographs of the Journal for Research in Mathematics Education.
   Google Scholar
• Carraher, D. W., Schliemann, A. D., Brizuela, B., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics Education, 37, 87–115.
   Web of Science ®Google Scholar
• Carraher, D. W., Schliemann, A. D., & Schwartz, J. (2008). Early algebra is not the same as algebra early. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 235–272). Mahwah, NJ: Erlbaum & NCTM.
   Google Scholar
• CCSSI [Common Core State Standards Initiative]. (2010). Common core state standards for mathematics. Retrieved from http://www.corestandards.org/wp-content/uploads/Math_Standards1.pdf.
   Google Scholar
• Davydov, V. V. (1969/1991). Psychological abilities of primary school children in learning mathematics (Vol. 6). Reston, VA: National Council of Teachers of Mathematics.
   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). Mahwah, NJ: Erlbaum & NCTM.
   Google Scholar
• Earnest, D. (2014). Exploring functions in elementary school: Leveraging the representational context. In K. Karp (Ed.), Annual perspectives in mathematics education: Using research to improve instruction (pp. 171–179). Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Ferrara, F., & Sinclair, N. (2016). An early algebra approach to pattern generalisation: Actualising the virtual through words, gestures and toilet paper. Educational Studies in Mathematics, 92(1), 1–19.
   Web of Science ®Google Scholar
• Ferrini-Mundy, J., Lappan, G., & Phillips, E. (1997). Experiences with patterning. Teaching Children Mathematics, 3, 282–289.
   Google Scholar
• Fujii, T., & Stephens, M. (2001). Fostering understanding of algebraic generalisation through numerical expressions: The role of the quasi-variables. In H. Chick, K. Stacey, J. Vincent, J. Cooper, & E. Warren (Eds.), Proceedings of the 12th ICMI study Conference: The future of the teaching and learning of algebra (Vol 1, pp. 258–264). Melbourne: The University of Melbourne. Retrieved from https://minerva-access.unimelb.edu.au/handle/11343/35000
   Google Scholar
• Gowers, T. (Ed.). (2008). The Princeton companion to mathematics. Princeton, NJ: Princeton University Press.
   Google Scholar
• Greenberg, J., & Walsh, K. (2008). No common denominator: The preparation of elementary teachers in mathematics by America’s Education Schools. [Mathematics advisory group: R. Askey, A. Chen, M. Goldenberg, R. Howe, J. Kamras, J. Milgram, R. Ramos & Y. Sagher] National Council on Teacher Quality. Retrieved from www.nctq.org
   Google Scholar
• Hobgood, K., & Kitchings, C. (2019). Mathematical essay: Investigating figurate numbers with technology. Retrieved from http://jwilson.coe.uga.edu/EMAT6680Su11/Kitchings/CK6690/Figurate%20Numbers.html
   Google Scholar
• Hotomski, M., Schliemann, A. D., Carraher, D. W., & Teixidor-i-Bigas, M. (2018). Poincaré Institute: Impact on Mathematics Achievement of Ethnic Groups (Unpublished research report). Tufts University, Medford, MA: The Poincaré Institute ( retrieved 15 July 2012 from: https://sites.tufts.edu/poincare/files/2018/07/Poicaré-Institute-Impact-on-Mathematics-Achievement-of-Ethnic-Groups.pdf)
   Google Scholar
• Kaput, J. J. (1995). Long-term algebra reform: Democratizing access to big ideas. In B. LaCampagne, & Kaput (Eds.), The algebra initiative colloquium (pp. 37–53). Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement.
   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: NCTM & Lawrence Erlbaum Associates.
   Google Scholar
• Kaput, J. J., Carraher, D. W., & Blanton, M. L. (Eds.). (2008). Algebra in the early grades. Mahwah, NJ: Erlbaum & NCTM.
   Google Scholar
• Katz, V. J. (1995). The development of algebra and algebra education. In C. B. LaCampagne, W. Blair, & J. J. Kaput (Eds.), The algebra initiative colloquium (Vol. 1, pp. 19–36). Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement. Retrieved from https://files.eric.ed.gov/fulltext/ED385436.pdf
   Google Scholar
• Kieran, C. (2004). Algebraic thinking in the early grades: What is it. The Mathematics Educator, 8, 139–151.
   Google Scholar
• Kieran, C. (Ed.). (2018). Teaching and learning algebraic thinking with 5-12- year-olds. The global evolution of an emerging field of research and practice. In ICME-13 monographs, New York, NY: Chaim: Springer.
   Google Scholar
• Kleiner, I. (1989). Evolution of the function concept: A brief survey. The College Mathematics Journal, 20, 282–300.
   Google Scholar
• LaCampagne, C. B., Blair, W., & Kaput, J. J., (Eds.). (1995). The algebra initiative colloquium: Vol 1: Plenary and reactor papers. Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement. Vol 2: Working group papers ED385437.pdf Retrieved from https://files.eric.ed.gov/fulltext/ED385436.pdf
   Google Scholar
• Loveless, T. (2001). A tale of two math reforms: The politics of the new math and the NCTM standards. In T. Loveless (Ed.), The great curriculum debate: How should we teach reading and math (pp. 184–209). Washington, DC: Brookings Institution.
   Google Scholar
• Martinez, M. V., & Brizuela, B. M. (2006). A third grader’s way of thinking about linear function tables. Journal of Mathematical Behavior, 25, 285–298.
   Google Scholar
• National Commission on Teaching and America’s Future (NCTAF). (1996). What matters most: Teaching for America’s future. New York, NY: NCTAF. Retrieved from: https://eric.ed.gov/?id=ED572506
   Google Scholar
• National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Author. Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
   Google Scholar
• NCTM [National Council of Teachers of Mathematics]. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics. Retrieved from https://www.nctm.org/Standards-and-Positions/Principles-and-Standards/Algebra/
   Google Scholar
• NCTM [National Council of Teachers of Mathematics]. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: Author.
   Google Scholar
• NCTM [National Council of Teachers of Mathematics] Algebra Working Group. (1995). Algebra in the K–12 Curriculum: Dilemmas and Possibilities (Final Report to the Board of Directors). East Lansing, MI: Michigan State University.
   Google Scholar
• Quine, W. V. (1987). Quiddities: An intermittently philosophical dictionary. Cambridge, MA: Harvard University Press.
   Google Scholar
• Radford, L. (2018). The emergence of symbolic algebraic thinking in primary school. In C. Kieran (Ed.), Teaching and learning algebraic thinking with 5-to 12-year-olds (pp. 3–25). Chaim: Springer.
   Google Scholar
• Resnick, M. D. (1981). Mathematics as a science of patterns: Ontology and reference. Noûs, 15, 529–550. Special Issue on Philosophy of Mathematics. Retrieved from https://www.jstor.org/stable/2214851
   Web of Science ®Google Scholar
• Schliemann, A. D., Carraher, D. W., & Brizuela, B. M. (2001). When tables become function tables. In M. V. D. Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 145–152). Utrecht: Freudenthal Institute.
   Google Scholar
• Schliemann, A. D., Carraher, D. W., & Brizuela, B. M. (2007). Bringing out the algebraic character of arithmetic: From children’s ideas to classroom practice. Hillsdale, NJ: Lawrence Erlbaum.
   Google Scholar
• Schliemann, A. D., Carraher, D. W., & Brizuela, B. M. (2012). Algebra in elementary school. In L. Coulange, & J.-P. Drouhard (Eds.), Enseignement de l’algèbre élémentaire: Bilan et perspectives (pp. 109–124). (Special Issue of Recherches en Didactique des Mathématiques). Grenoble: La Pensée Sauvage, Editions.
   Google Scholar
• Schliemann, A. D., Carraher, D. W., & Teixidor-i-Bigas, M. (2013). Teacher development and student learning. Invited Presentation. 13th International Congress on Mathematical Education. Hamburg, Germany (July 25–30). Retrieved 17 May 2018 from: https://sites.tufts.edu/poincare/files/2016/10/Schliemann-Carraher-Teixidor-2016-Teacher-Development-and-Student-Learning-ICME-13-1.pdf.
   Google Scholar
• Schmittau, J. (2005). The development of algebraic thinking. ZDM, 37, 16–22.
   Google Scholar
• Schoenfeld, A. (1995). Report of working group 1. In C. B. LaCampagne, W. Blair, & J. J. Kaput (Eds.), The algebra initiative colloquium: Vol 2 ED385437.pdf#page=11 (pp. 11–18). Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement.
   Google Scholar
• Schoenfeld, A. H., & Arcavi, A. (1988). On the meaning of variable. The Mathematics Teacher, 81, 420–427.
   Google Scholar
• Schwartz, J., & Yerushalmy, M. (1992). Getting students to function in and with algebra. The Concept of Function: Aspects of Epistemology and Pedagogy, 25, 261–289.
   Google Scholar
• Steen, L. A. (1988). The science of patterns. Science, 240, 611–616. doi:10.1126/science.240.4852.611
   PubMed Web of Science ®Google Scholar
• Steen, L. A. (1990). Pattern. In L. A. Steen (Ed.), On the shoulders of giants: New approaches to numeracy (pp. 1–10). Mathematical Sciences Education Board & National Research Council. Washington, DC: National Academy Press.
   Google Scholar
• Teixidor-i-Bigas, M., Schliemann, A. D., & Carraher, D. W. (2013). Integrating disciplinary perspectives: The Poincaré Institute for Mathematics Education. The Mathematics Enthusiast, 10, 519–562.
   Google Scholar
• Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. F. Coxford, & A. P. Shulte (Eds.), The Ideas of algebra, K–12. 1988 Yearbook (pp. 7–13). Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar