Expanded model for elementary algebraic reasoning levels

María Burgos; Nicolás Tizón-Escamilla; Juan Díaz Godino

doi:10.29333/ejmste/14753

Full Text (PDF)

María Burgos ¹ , Nicolás Tizón-Escamilla ¹ ^* , Juan Díaz Godino ¹

More Detail

¹ University of Granada, Granada, SPAIN^* Corresponding Author

Abstract

The development of algebraic reasoning from the earliest educational levels is an objective that has solid support both from the point of view of research and curricular development. Effectively incorporating algebraic content to enrich mathematical activity in schools requires considering the different degrees of generality of the objects and processes involved in algebraic practices. In this article, we present an expanded version of the model of levels of algebraization proposed within the framework of the onto-semiotic approach, establishing sublevels that provide a more microscopic view of the structures involved and the processes of generalization, representation, and analytical calculation at stake. We exemplify the model with mathematical activities that can be approached from primary education, classified according to the different sublevels of algebraization. The use of this expanded model can facilitate the development of didactic-mathematical knowledge of teachers in training on algebraic reasoning and its teaching.

Keywords

algebraic reasoning
primary education
onto-semiotic approach
teacher training

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, Volume 20, Issue 7, July 2024, Article No: em2475

https://doi.org/10.29333/ejmste/14753

Publication date: 01 Jul 2024

Online publication date: 20 Jun 2024

Article Views: 560

Article Downloads: 108

Open Access References How to cite this article

References

Aké, L. Godino, J. D., Gonzato, M. & Wilhelmi, M. R. (2013). Proto-algebraic levels of mathematical thinking. In A. M. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 1-8). IGPME.
Arzarello, F. (2006). Semiosis as a multimodal process. Revista Latinoamericana de Investigación en Matemática Educativa [Latin American Journal of Research in Educational Mathematics], 9(1), 267-300.
Ayala-Altamirano, C., & Molina, M. (2021). El proceso de generalización y la generalización en acto: Un estudio de casos [The generalization process and generalization in action: A case study]. PNA, 15(3), 211-241. https://doi.org/10.30827/pna.v15i3.18109
Blanton, M. (2008). Algebra and the elementary classroom: Transforming thinking, transforming practice. Heinemann.
Blanton, M. Brizuela, B., Gardiner, A., 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., Levi, L., Crites, T., & Dougherty, B. J. (2011). Developing essential understanding of algebraic thinking for teaching mathematics in grades 3-5. NCTM.
Blanton, M., & Kaput, J. (2004). Elementary grades students’ capacity for functional thinking. In M. Johnsen, & A. Berit (Eds.), Proceedings of the 28^th International Group of the Psychology of Mathematics Education (pp. 135-142). Bergen University College.
Blanton, M., Brizuela, B. M., Murphy, A., Sawrey, K., & Newman-Owens, A. (2015). A learning trajectory in 6-year-olds´s thinking about generalizing functional relationship. Journal for Research in Mathematics Education, 46(5), 511-559. https://doi.org/10.5951/jresematheduc.46.5.0511
Burgos, M., & Godino, J.D. (2018). Recognizing algebrization levels in an inverse proportionality task by prospective secondary school mathematics teachers. In L. Gómez Chova, A. López Martínez, & I. Candel Torres (Eds.), Proceedings of EDULEARN18 Conference (pp. 2483–2491). IATED Academy. https://doi.org/10.21125/edulearn.2018.0672
Burgos, M., & Godino, J. D. (2022). Assessing the Epistemic Analysis Competence of Prospective Primary School Teachers on Proportionality Tasks. International Journal of Science and Mathematics Education, 20, 367-389. https://doi.org/10.1007/s10763-020-10143-0
Campbell-Kelly, M., Croarken, M., Flood, R., & Robson, E. (2003). The history of mathematical tables: From summer to spreadsheets. Oxford University Press. https://doi.org/10.1093/acprof:oso/9780198508410.001.0001
Carraher, D., & Schliemann, A. (2018). Cultivating 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. 107-138). Springer. https://doi.org/10.1007/978-3-319-68351-5_5
Cooper, T. J., & Warren, E. (2011). Years 2 to 6 students’ ability to generalize: Models, representations and theory for teaching and learning. In J. Cai, & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 187-214). Springer. https://doi.org/10.1007/978-3-642-17735-4_12
Drijvers, P., Goddijn, A., & Kindt, M. (2011). Algebra education: Exploring topics and themes. In P. Drijvers (Ed.), Secondary algebra education (pp. 5-26). Sense Publishers. https://doi.org/10.1007/978-94-6091-334-1_1
Duval, R. (2006). Cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103-131. https://doi.org/10.1007/s10649-006-0400-z
Duval, R. (2017). Understanding the mathematical way of thinking–The registers of semiotic representations. Springer. https://doi.org/10.1007/978-3-319-56910-9
Ellis, A. B. (2007). A taxonomy for categorizing generalizations. The Journal of the Learning Sciences, 16(2), 221-262. https://doi.org/10.1080/10508400701193705
Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19-25.
Filloy, E., Puig, L., & Rojano, T. (2008). Educational algebra. A theoretical and empirical approach. Springer. https://doi.org/10.1007/978-0-387-71254-3
Font, V., Godino, J. D. & Gallardo, J. (2013). The emergence of objects from mathematical practices. Educational Studies in Mathematics, 82, 97-124. https://doi.org/10.1007/s10649-012-9411-0
Font, V., Planas, N. & Godino, J. D. (2010). Modelo para el análisis didáctico en educación matemática [Model for didactic analysis in mathematics education]. Infancia y Aprendizaje, 33(2), 89-105. https://doi.org/10.1174/021037010790317243
Fujii, T. (2003). Probing students’ understanding of variables through cognitive conflict problems: Is the concept of a variable so difficult for students to understand? In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of the 27^th Conference of the International Group for the Psychology of Mathematics Education (pp. 49-65). PME.
Giacomone, B., Godino, J. D., Blanco, T. F., & Wilhelmi, M. R. (2022). Onto-semiotic analysis of diagrammatic reasoning. International Journal of Science and Mathematics Education, 21, 1495-1520. https://doi.org/10.1007/s10763-022-10316-z
Godino, J. D., Aké, L., Gonzato, M., & Wilhelmi, M. R. (2014). Niveles de algebrización de la actividad matemática escolar. Implicaciones para la formación de maestros [Levels of algebrization of school mathematical activity. Implications for teacher training]. Enseñanza de las Ciencias, 32(1), 199-219. https://doi.org/10.5565/rev/ensciencias.965
Godino, J. D. Batanero, C., & Font, V. (2019). The onto-semiotic approach: Implications for the prescriptive character of didactic. For the Learning of Mathematics, 39(1), 37-42.
Godino, J. D., Castro, W., Aké, L., & Wilhelmi, M. R. (2012). Naturaleza del razonamiento algebraico elemental [Nature of elementary algebraic reasoning]. BOLEMA, 26(42B), 483-511. https:// doi.org/10.1590/S0103-636X2012000200005
Hewitt, D. (2019). “Never carry out any arithmetic”: The importance of structure in developing algebraic thinking. In U.T. Jankvist, M. Van den Heuvel-Panhuizen, & M. Veldhuis (Eds.), Proceedings of the 11^th Congress of the European Society for Research in Mathematics Education (pp. 558-565). Freudenthal Group & Freudenthal Institute, Utrecht University and ERME.
Hohensee, C. (2017). Preparing elementary prospective teachers to teach early algebra. Journal of Mathematics Teacher Education, 20, 231-257. https://doi.org/10.1007/s10857-015-9324-9
Johnson, H. L. (2022). Task design for graphs: Rethink multiple representations with variation theory, Mathematical Thinking and Learning, 24(2), 91-98. https://doi.org/10.1080/10986065.2020.1824056
Kaput, J. J. (1993). The urgent need for proleptic research in the representation of quantitative relationships. In T. A. Romberg, T. P. Carpenter, & E. Fennema (Eds.), Integrating research on the graphical representation of functions (pp. 279-312). Lawrence Erlbaum Associates.
Kaput, J. J., Blanton, M., & 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). Routledge. https://doi.org/10.4324/9781315097435-3
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). NCTM. https://doi.org/10.4324/9781315044378-4
Kieran, C. (2022) The multi-dimensionality of early algebraic thinking: Background, overarching dimensions, and new directions. ZDM Mathematics Education, 54, 1131-1150. https://doi.org/10.1007/s11858-022-01435-6
Kieran, C., Pang, J. S., Ng, S. F., Schifter, D., & Steinweg, A. S. (2017). Topic study group 10: Teaching and learning of early algebra. In G. Kaiser (Ed.), The Proceedings of the 13^th International Congress on Mathematical Education. Springer. https://doi.org/10.1007/978-3-319-62597-3_37
Kilhamn, C., Röj-Lindberg, A. S., & Björkqvist, O. (2019). School algebra. In C. Kilhamn, & R. Säljö (Eds.), Encountering algebra (pp. 1-11). Springer. https://doi.org/10.1007/978-3-030-17577-1_1
Kop, P. M., Janssen, F. J., Drijvers, P. H., & van Driel, J. H. (2020b). The relation between graphing formulas by hand and students’ symbol sense. Educational Studies in Mathematics, 105, 137-161. https://doi.org/10.1007/s10649-020-09970-3
Kop, P. M., Janssen, F. J., Drijvers, P. H., & van Driel, J. H. (2020a). Promoting insight into algebraic formulas through graphing by hand. Mathematical Thinking and Learning, 25(2), 125-144. https://doi.org/10.1080/10986065.2020.1765078
Krutetskii, V. A. (1976). The psychology of mathematical abilities in schoolchildren. University of Chicago Press.
Linchevski, L., & Herscovics, N. (1996). Crossing the cognitive gap between arithmetic and algebra: Operating on the unknown in the context of equations. Educational Studies in Mathematics, 30, 39-65. https://doi.org/10.1007/BF00163752
Malara, N. A., & Navarra G., (2018). New words and concepts for early algebra teaching: Sharing with teachers epistemological issues in early algebra to develop students’ 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. 27-50). Springer. https://doi.org/10.1007/978-3-319-68351-5_3
Mason, J., Stephens, M., & Watson, A. (2009). Appreciating mathematical structure for all. Mathematics Education Research Journal, 21(2), 10-32. https://doi.org/10.1007/BF03217543
Pittalis, M. (2023). Young students’ arithmetic-algebraic structure sense: An empirical model and profiles of students. International Journal of Sciences and Mathematics Education, 21, 1865-1887. https://doi.org/10.1007/s10763-022-10333-y
Pittalis, M., Pitta-Pantazi, D., & Christou, C. (2020). Young students’ functional thinking modes: The relation between recursive patterning, covariational thinking, and correspondence relations. Journal for Research in Mathematics Education, 51(5), 631-674. https://doi.org/10.5951/jresematheduc-2020-0164
Radford, L. (2010). Layers of generality and types of generalization in pattern activities. PNA, 4(2), 37-62. https://doi.org/10.30827/pna.v4i2.6169
Radford, L. (2011). Grade 2 students’ non-symbolic algebraic thinking. In J. Cai, & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 303-322). Springer. https://doi.org/10.1007/978-3-642-17735-4_17
Radford, L. (2013). Three key concepts of the theory of objectification: Knowledge, knowing, and learning. Journal of Research in Mathematics Education, 2(1), 7-44. https://doi.org/10.4471/redimat.2013.19
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: The global evolution of an emerging field of research and practice (pp. 3-25). Springer. https://doi.org/10.1007/978-3-319-68351-5_1
Radford, L. (2021). O ensino-aprendizagem da ágebra na teoria da objetivação [The teaching-learning of agebra in the theory of objectification]. In V. Moretti, & L. Radford (Eds.), Pensamento algébrico nos anos iniciais: Diálogos e complementaridades entre a teoriada objetivação e a teoria histórico-cultural [Algebraic thinking in the early years: Dialogues and complementarities between objectification theory and historical-cultural theory] (pp. 171-195). Livraria da Física [Physics Bookstore].
Radford, L., Miranda, I., & Guzmán, J. (2008). Relative motion, graphs and the heteroglossic transformation of meanings: A semiotic analysis. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda (Eds.), Proceedings of the Joint 32^nd Conference of the International Group for the Psychology of Mathematics Education and the 30^th North American Chapter (pp. 161-168). The Ohio State University.
Sfard, A. (2020). Commognition. In S. Lerman (Ed.), Encyclopedia of mathematics education. Springer. https://doi.org/10.1007/978-3-030-15789-0_100031
Sfard, A., & Linchevski, L. (1994). The gains and the pitfalls of reification–The case of algebra. Educational Studies in Mathematics, 26, 191-228. https://doi.org/10.1007/BF01273663
Stephens, A. C., Fonger, N., Strachota, S., Isler, I., Blanton, M. L., Knuth, E., & Gardiner, A. M. (2017). A learning progression for elementary students’ functional thinking a learning progression for elementary students’ functional. Mathematical Thinking and Learning, 19(3), 143-166. https://doi.org/10.1080/10986065.2017.1328636
Torres, M. D., Brizuela, B. M., Cañadas, M. C., & Moreno, A. (2022). Introducing tables to second-grade elementary students in an algebraic thinking context. Mathematics, 10(1), 56. https://doi.org/10.3390/math10010056
Venkat, H., Askew, M., Watson, A., & Mason, J. (2019). Architecture of mathematical structure. For the Learning of Mathematics, 39(1), 13-17.
Vergel, R., Godino, J. D., Font, V., & Pantano, O. L. (2023). Comparing the views of the theory of objectification and the onto-semiotic approach on the school algebra nature and learning. Mathematics Education Research Journal, 35, 475-496 https://doi.org/10.1007/s13394-021-00400-y
Vergel, R., Radford, L., & Rojas, P. J. (2022). Zona conceptual de formas de pensamiento aritmético “sofisticado” y proto-formas de pensamiento algebraico: Una contribución a la noción de zona de emergencia del pensamiento algebraico [Conceptual zone of “sophisticated” arithmetic thought forms and proto-forms of algebraic thought: A contribution to the notion of emergency zone of algebraic thought]. Bolema, 36(74), 1174-1192. https://doi.org/10.1590/1980-4415v36n74a11
Warren, E., Trigueros, M., & Ursini, S. (2016). Research on the learning ad teaching of algebra. In Á. Gutierrez, G. C. Leder, & P. Boero (Eds.), The second handbook of research on the psychology of mathematics education (pp. 73-108). Sense Publishers. https://doi.org/10.1007/978-94-6300-561-6_3
Zapatera, A., & Quevedo, E. (2021). The initial algebraic knowledge of preservice teachers. Mathematics, 9, 2117. https://doi.org/10.3390/math9172117
Zeljić, M. (2015). Modelling the relationships between quantities: Meaning in literal expressions. EURASIA Journal of Mathematics, Science and Technology Education, 11(2), 431-442. https://doi.org/10.12973/eurasia.2015.1362a

How to cite this article

APA

Burgos, M., Tizón-Escamilla, N., & Godino, J. D. (2024). Expanded model for elementary algebraic reasoning levels. Eurasia Journal of Mathematics, Science and Technology Education, 20(7), em2475. https://doi.org/10.29333/ejmste/14753

Vancouver

Burgos M, Tizón-Escamilla N, Godino JD. Expanded model for elementary algebraic reasoning levels. EURASIA J Math Sci Tech Ed. 2024;20(7):em2475. https://doi.org/10.29333/ejmste/14753

AMA

Burgos M, Tizón-Escamilla N, Godino JD. Expanded model for elementary algebraic reasoning levels. EURASIA J Math Sci Tech Ed. 2024;20(7), em2475. https://doi.org/10.29333/ejmste/14753

Chicago

Burgos, María, Nicolás Tizón-Escamilla, and Juan Díaz Godino. "Expanded model for elementary algebraic reasoning levels". Eurasia Journal of Mathematics, Science and Technology Education 2024 20 no. 7 (2024): em2475. https://doi.org/10.29333/ejmste/14753

Harvard

Burgos, M., Tizón-Escamilla, N., and Godino, J. D. (2024). Expanded model for elementary algebraic reasoning levels. Eurasia Journal of Mathematics, Science and Technology Education, 20(7), em2475. https://doi.org/10.29333/ejmste/14753

MLA

Burgos, María et al. "Expanded model for elementary algebraic reasoning levels". Eurasia Journal of Mathematics, Science and Technology Education, vol. 20, no. 7, 2024, em2475. https://doi.org/10.29333/ejmste/14753