Fourth-graders’ justifications in early algebra tasks involving a functional relationship

Referencias

Ayala-Altamirano, C., & Molina, M. (2020). Meanings attributed to letters in functional contexts by primary school students. International Journal of Science and Mathematics Education, 18(7), 1271–1291. https://doi.org/10.1007/s10763-019-10012-5. Ayalon, M., & Hershkowitz, R. (2018). Mathematics teachers’ attention to potential classroom situations of argumentation. Journal of Mathematical Behavior , 49, 163–173. https://doi.org/10.1016/j.jmathb.2017.11.010 Banes, L. C., López, G., Skubal, M., & Perfecto, L. (2017). Co-constructing written explanations. Mathematics Teaching in the Middle School, 23(1), 30–38. https://doi.org/10.5951/mathteacmiddscho.23.1.0030 Blanton, M. L. (2008). Algebra and the elementary classroom: Transforming thinking, transforming practice. Portsmouth, NA: Heinemann. Blanton, M.L. (2017). Algebraic reasoning in grades 3-5. In M. Battista (Ed.), Reasoning and sense making in grades 3-5 (pp. 67–102). Reston, VA: NCTM. Blanton, M. L., & Kaput, J. J. (2011). Functional thinking as a route into algebra in the elementary grades. In J. Cai, & E. Knuth (Eds.), Early algebraization. advances in mathematics education (pp. 5–23). Heidelberg, Germany: Springer. Blanton, M. L., Brizuela, B., 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(2), 181–202. https://doi.org/10.1007/s10649-016-9745-0 Blanton, M. L., Brizuela, B., Stephens, A., Knuth, E., Isler, I., Gardiner, A., Stroud, R., Fonger, N., & Stylianou, D. (2018). Implementing a framework for early algebra. In C. Kieran (Ed.), Teaching and learning algebraic thinking with 5-to 12-year-olds: The global evolution of an emerging field of research and practice (pp. 27–49). Hamburg, Germany: Springer. Blanton, M. L., Levi, L., Crites, T., & Dougherty, B. J. (2011). Developing essential understanding of algebraic thinking for teaching mathematics in grades 3-5. Reston, VA: NCTM. Booth, L. R. (1988). Children’s difficulties in beginning algebra. In A. Coxforf & A. Schulte (Eds.), The ideas of algebra, K-12 (pp. 20–32). Reston, VA: NCTM Brizuela, B., Blanton, M. L., Gardiner, A. M., Newman-Owens, A., & Sawrey, K. (2015). A first grade student’s exploration of variable and variable notation/una alumna de primer grado explora las variables y su notación. Studies in Psychology/ Estudios De Psicología, 36(1), 138–165. https://doi.org/10.1080/02109395.2014.1000027 Cañadas, M. C. & Molina, M. (2016). Una aproximación al marco conceptual y principales antecedentes del pensamiento funcional en las primeras edades [An approach to the conceptual framework and background of functional thinking in early ages]. In E. Castro, E. Castro, J. L. Lupiáñez, J. F. Ruiz-Hidalgo, & M. Torralbo (Eds.), Investigación en Educación Matemática. Homenaje a Luis Rico (pp. 209–218). Granada, España: Comares. Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically. Integrating arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann. Carraher, D. W., & Schliemann, A. D. (2007). Early algebra and algebraic reasoning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 669– 705). Reston, VA: NCTM. Carraher, D. W., & Schliemann, A. D. (2015). Powerful ideas in elementary school mathematics. In L. D. English, & D. Kirshner (Eds.), Handbook of international research in mathematics education (pp. 191–208). New York, NY: Routledge. Carraher, D. W., & Schliemann, A. D. (2018). Cultivating early algebraic thinking. In C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12-Year-Olds. ICME-13 Monographs (pp. 107–138). Cham, Germany: Springer. https://doi.org/10.1007/978-3-319- 68351-5_5 Chua, B. L. (2016). Justification in Singapore secondary mathematics. In P. C. Toh, & B. Kaur (Eds.), Developing 21st century competencies in the mathematics classroom (pp. 165–188). Singapore: World Scientific. https://doi.org/10.1142/9789813143623_0010 Cobb, P., & Gravemeijer, K. (2008). Experimenting to support and understand learning processes. 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. 68–95). Mahwah, NJ: LEA. Ellis, A. B. (2007). A taxonomy for categorizing generalizations: Generalizing actions and reflection generalizations. Journal of the Learning Sciences, 16(2), 221–262. https://doi.org/10.1080/10508 400701193705 Ingram, J., Andrews, N., & Pitt, A. (2019). When students offer explanations without the teacher explicitly asking them to. Educational Studies in Mathematics, 101(1), 51–66. https://doi.org/10.10 07/s10649-018-9873-9 Kaput, J. J. (2008). What is algebra? What is the algebraic reasoning? In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 5–17). New York, NY: Lawrence Erlbaum Associates. Kaput, J. (2009). Building intellectual infrastructure to expose and understand ever-increasing complexity. Educational Studies in Mathematics, 70(2), 211–215. https://doi.org/10.1007/s10649-008-9169-6 Kelly, A. E., & Lesh, R. A. (2000). Handbook of research design in mathematics and science education. New Jersey: NJ: LEA. Kieran, C. (1989). The early learning of algebra: A structural perspective. In S. Wagner, & C. Kieran (Eds.) Research Issues in the Learning and Teaching of Algebra (pp. 33–56). Reston, VA: NCTM Kieran, C. (2014). Algebra teaching and learning. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 27–32). Dordrecht: Springer Netherlands. https://doi.org/10.1007/978-3-319-77487-9 Kieran, C. (2018). Seeking, using, and expressing structure in numbers and numerical operations: A fundamental path to developing early algebraic thinking. In C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12-year-olds: The global evolution of an emerging field of research and practice (pp. 79-105). New York: Springer. Knuth, E., Alibali, M.W., McNeil, N.M., Weinberg, A., & Stephens, A.C. (2005). Middle School Students’ understanding of core algebraic concepts: equivalence & variable. Zentralblatt für Didaktik der Mathematik, 37(1), 68-76 Knuth, E., Choppin, J., & Bieda, K. (2009). Middle school students' production of mathematical justifications. In M. L. Blanton, D. Stylianou, & E. J. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (pp. 153–170). New York, NY: Routledge. Krummheuer, G. (2013) The relationship between diagrammatic argumentation and narrative argumentation in the contex of the development of mathematical thinking in the early years. Educational Studies in Mathematics, 84, 249–265. https://doi.org/10.1007/s10649-013-9471- 9 Küchemann, D. (1981). Algebra. In K. Hart (Ed.), Children’s understanding of mathematics (pp. 11–16). London, UK: Murray. Lannin, J. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231–258. https://doi.org/10.1207/s15327833mtl0703_3 Molina, M., & Mason, J., (2009) Justifications-on-Demand as a Device to Promote Shifts of Attention Associated with Relational Thinking in Elementary Arithmetic. Canadian Journal of Science, Mathematics and Technology Education, 9(4), 224–242. https://doi.org/10.1080/14926150903191885 Mason, J., Grahamn, A., Pimm, D., & Gowar, N. (1985). Routes to/Roots of algebra. London, UK: Center for Mathematics Education, The Open University. Mason, J. (1996). Expressing generality and roots of algebra. In N. Bernarz, C. Kieran, & L. Lee (Eds.) Approaches to algebra (pp. 65–86). Springer, Dordrecht. https://doi.org/10.1007/978- 94-009-1732-3_5 Mason, J. (2017). Overcoming the algebra barrier: Being particular about the general, and generally looking beyond the particular, in homage to Mary Boole. In S. Stewart (Ed.), And the rest is just algebra (pp. 97–117). Cham, Germany: Springer International Publishing. https://doi.org/10.1007/978-3-319-45053-7_6 Ministerio de Educación, Cultura y Deporte (2014). Real Decreto 126/2014 de 28 de febrero, por el que se establece el currículo básico de la Educación Primaria [Royal Decree 126/2014 of February 28, which establishes the basic curriculum of Primary Education]. BOE, 52, 19349– 19420. Morgan, C., Craig, T., Schuette, M., & Wagner, D. (2014). Language and communication in mathematics education: An overview of research in the field. ZDM – The international Journal on Mathematics Education, 46(6), 843–853. https://doi.org/10.1007/s11858-014-0624-9 Presmeg, N., Radford, L., Roth, W., & Kadunz, G. (2016). Semiotics in mathematics education. ICME-13 Topical Surveys. Berlin, Germany: Springer. https://doi.org/10.1007/978-3-319- 31370-2_1. Radford, L. (2002). The seen, the spoken and the written: A semiotic approach to the problem of objectification of mathematical knowledge. For the Learning of Mathematics, 22(2), 14–23. Radford, L. (2009). Why do gestures matter? sensuous cognition and the palpability of mathematical meanings. Educational Studies in Mathematics, 70(2), 111–126. https://doi.org/10.1007/s10649-008-9127-3 Radford, L. (2010). Layers of generality and types of generalization in pattern activities. PNA, 4(2), 37–62. 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. ICME-13 Monographs (pp. 3–25). Cham, Germany: Springer. https://doi.org/10.1007/978-3-319- 68351-5_1 Radford, L., & Sabena, C. (2015). The question of method in a Vygotskyan semiotic approach. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Approaches to qualitative research in mathematics education: Examples of methodology and methods (pp. 157–182). New York, NY: Springer. https://doi.org/10.1007/978-94-017-9181-6_7 Simon, M. A., & Blume, G. W. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. Journal of Mathematical Behavior, 15(1), 3–31 Smith, E. (2008). Representational thinking as a framework for introducing functions in the elementary curriculum. In J. J. Kaput, D. W. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 133–163). New York, NY: LEA. Staples, M. E., Bartlo, J., & Thanheiser, E. (2012). Justification as a teaching and learning practice: Its (potential) multifaceted role in middle grades mathematics classrooms. The Journal of Mathematical Behavior , 31(4), 447–462. https://doi.org/10.1016/j.jmathb.2012.07.001 Stephens, A., Ellis, A., Blanton, M., & Brizuela, B. (2017). Algebraic thinking in the elementary and middle grades. In J. Cai (Ed.), Compendium for research in mathematics education. third handbook of research in mathematics education. (pp. 386–420). Reston, VA: NCTM. Strachota, S., Knuth, E., & Blanton, M. (2018). Cycles of generalizing activities in the classroom. In C. Kieran (Ed.), Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds: The global evolution of an emerging field of research and practice (pp. 351–378). Cham, Germany: Springer. https://doi.org/10.1007/978-3-319-68351-5_15 Stylianides, A. (2015). The role of mode of representation in students' argument constructions. In K. Krainer, & N. Vondrová (Eds.), Proceedings of the 9th Congress of the European Society for Research in Mathematics Education, (pp. 213–220). Prague, Czech Republic. Vergel, R. (2014). El signo en Vygotsky y su vínculo con el desarrollo de los procesos psicológicos superiores [the sign for Vygotsky and its connection with the development of superior psychological processes]. Folios, 39(1), 65-76 https://doi.org/10.17227/01234870.39folios65.76 Vygotsky, L. S. (1995). Thought and Language (J.P. Tousaus, Trans.). Barcelona, Spain: Editorial Planeta. (Original work published 1934) Wertsch, J. V. (1995). Vygotsky and the social formation of mind (J. Zanón & M. Cortés, Trans.; 2nd ed.). Barcelona, Spain: Ediciones Paidós. (Original work published 1985). Yakubinskii, L.P. (1923). O dialogicheskoi rechi [On Dialogic Speech]. Petrogrado: Trudy Foneticheskogo Instituta Prakticheskogo Izucheniya