Introducing Equations and Inequations throught Arithmetic Problems with Visual Support
Abstract This paper presents the findings of a teaching experiment conducted within the design research framework to promote algebraic thinking in primary school students. The aim of this paper is to identify the structures they recognize in problems involving the visual representation of equations and inequalities. The written and oral responses of two groups of students, comprising third and fourth graders from primary school, were subjected to analysis, with particular attention paid to the processes of translating problems into algebraic alphanumeric language. The findings indicate that the students were able to comprehend various structures and establish the equivalence between their components and mathematical operations. However, the expression of inequalities proved to be more complex in comparison to equations. It can be concluded that the visual representation in the problems facilitates flexible thinking, enables the perception of mathematical structures and justifies their equivalence, thus proving to be an effective and accessible tool for the students.
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Ayala-Altamirano, C., Pinto, E., Molina, M., & Cañadas, M. C. (2022). Interacting with Indeterminate Quantities through Arithmetic Word Problems: Tasks to Promote Algebraic Thinking at Elementary School. Mathematics, 10(13), 2229. https://doi.org/10.3390/math10132229
Blanton, M. L., Brizuela, B. M., Stephens, A., Knuth, E., Isler, I., Gardiner, A. M., Stroud, R., Fonger, N. L., & Stylianou, D. (2018). Implementing a framework for early algebra. En C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12-year-olds (pp. 27-49). Springer.
Blanton, M. L., & Kaput, J. J. (2003). Developing elementary teachers’ algebra eyes and ears. Teaching Children Mathematics, 10(2), 70-77. https://doi.org/10.5951/TCM.10.2.0070
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.
Carraher, D. W., Schliemann, A. D., Brizuela, B. M., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics education, 37(2), 87-115.
Cobb, P., & Gravemeijer, K. (2008). Experimenting to support and understand learning processes. En 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). Lawrence Erlbaum Associates.
Coquin-Viennot, D., & Moreau, S. (2003). Highlighting the role of the episodic situation model in the solving of arithmetical problems. European Journal of Psychology of Education, 18(3), 267-279. https://doi.org/10.1007/BF03173248
Diezmann, C. M., & English, L. D. (2001). Promoting the use of diagrams as tools for thinking. En A. A. Cuoco y F. R. Curcio (Eds.), The roles of representation in school mathematics: 2001 yearbook (pp. 77-98). NCTM.
Goldin, G. A. (2002). Representation in mathematical learning and problem solving. En L. D. English (Ed.), Handbook of international research in mathematics education, (pp. 196-218). Lawrence Erlbaum Associates.
Hunter, J., & Miller, J. (2022). The use of cultural contexts for patterning tasks: Supporting young diverse students to identify structures and generalise. ZDM - Mathematics Education, 54(6), 1349-1362. https://doi.org/10.1007/s11858-022-01386-y
Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. Building meaning for symbols and their manipulation. En F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (vol. 2, pp. 707-762). Information Age Publishing, Inc. y NCTM.
Kieran, C. (2011). Overall commentary on early algebraization: Perspectives for research and teaching. En J. Cai y E. J. Knuth (Eds.), Early algebraization (pp. 579-593). Springer.
Kieran, C. (2019). Task Design Frameworks in Mathematics Education Research: An Example of a Domain-Specific Frame for Algebra Learning with Technological Tools. En G. Kaiser y N. Presmeg (Eds.), Compendium for early career researchers in mathematics education (pp. 265-287). Springer.
Kieran, C. (2022). The multi-dimensionality of early algebraic thinking: Background, overarching dimensions, and new directions. ZDM - Mathematics Education, 54(6), 1131-1150. https://doi.org/10.1007/s11858-022-01435-6
Kieran, C., & Martínez-Hernández, C. (2022). Structure sense at early ages: The case of equivalence of numerical expressions and equalities. En T. Rojano (Ed.), Algebra structure sense development amongst diverse learners (pp. 35-66). Routledge.
Lloyd, G. M., Herbel-Eisenmann, B. A., & Star, J. R. (2011). Developing essential understanding of expressions, equations, and functions for teaching mathematics in grades 6-8. NCTM.
Marghetis, T., Landy, D., & Goldstone, R. L. (2016). Mastering algebra retrains the visual system to perceive hierarchical structure in equations. Cognitive Research: Principles and Implications, 1(1), 25. https://doi.org/10.1186/s41235-016-0020-9
Ministerio de Educación de Chile (2012). Bases Curriculares Educación Básica. Unidad de Currículo y Evaluación.
Ministerio de Educación y Formación Profesional (MEFP) (2022). Real Decreto 157/2022, de 01 de marzo, por el que se establecen la ordenación y enseñanzas mínimas de la Educación Primaria. BOE, 52, 24386-24504.
National Council of Teacher of Mathematics (NCTM) (2003). Principios y estándares para la educación matemática. SAEM Thales.
Paoletti, T., Stevens, I., & Vishnubhotla, M. (2021). Comparative and restrictive inequalities. The Journal of Mathematical Behavior, 63, 100895. https://doi.org/10.1016/j.jmathb.2021.100895
Pinto, E., Ayala-Altamirano, C., Molina, M., & Cañadas, M. C. (2023). Desarrollo del pensamiento algebraico a través de la justificación en educación primaria. Enseñanza de las Ciencias, 41(1), 149-173. https://doi.org/10.5565/rev/ensciencias.5835
Radford, L. (2011). Grade 2 Students’ Non-Symbolic Algebraic Thinking. En J. Cai y E. J. Knuth (Eds.), Early algebraization (pp. 303-322). Springer.
Radford, L. (2018). The emergence of symbolic algebraic thinking in primary school. En C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12-year-olds (pp. 3-25). Springer.
Radford, L. (2022). Introducing equations in early algebra. ZDM Mathematics Education, 54(6), 1151-1167. https://doi.org/10.1007/s11858-022-01422-x
Rojano, T. (2022). Algebra structure sense: Conceptual approaches and elements for its development. En T. Rojano (Ed.), Algebra structure sense development amongst diverse learners (pp. 1-19). Routledge.
Tsamir, P., & Almog, N. (2001). Students’ strategies and difficulties: The case of algebraic inequalities. International Journal of Mathematical Education in Science and Technology, 32(4), 513-524. https://doi.org/10.1080/00207390110038277
Blanton, M. L., Brizuela, B. M., Stephens, A., Knuth, E., Isler, I., Gardiner, A. M., Stroud, R., Fonger, N. L., & Stylianou, D. (2018). Implementing a framework for early algebra. En C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12-year-olds (pp. 27-49). Springer.
Blanton, M. L., & Kaput, J. J. (2003). Developing elementary teachers’ algebra eyes and ears. Teaching Children Mathematics, 10(2), 70-77. https://doi.org/10.5951/TCM.10.2.0070
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.
Carraher, D. W., Schliemann, A. D., Brizuela, B. M., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics education, 37(2), 87-115.
Cobb, P., & Gravemeijer, K. (2008). Experimenting to support and understand learning processes. En 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). Lawrence Erlbaum Associates.
Coquin-Viennot, D., & Moreau, S. (2003). Highlighting the role of the episodic situation model in the solving of arithmetical problems. European Journal of Psychology of Education, 18(3), 267-279. https://doi.org/10.1007/BF03173248
Diezmann, C. M., & English, L. D. (2001). Promoting the use of diagrams as tools for thinking. En A. A. Cuoco y F. R. Curcio (Eds.), The roles of representation in school mathematics: 2001 yearbook (pp. 77-98). NCTM.
Goldin, G. A. (2002). Representation in mathematical learning and problem solving. En L. D. English (Ed.), Handbook of international research in mathematics education, (pp. 196-218). Lawrence Erlbaum Associates.
Hunter, J., & Miller, J. (2022). The use of cultural contexts for patterning tasks: Supporting young diverse students to identify structures and generalise. ZDM - Mathematics Education, 54(6), 1349-1362. https://doi.org/10.1007/s11858-022-01386-y
Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. Building meaning for symbols and their manipulation. En F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (vol. 2, pp. 707-762). Information Age Publishing, Inc. y NCTM.
Kieran, C. (2011). Overall commentary on early algebraization: Perspectives for research and teaching. En J. Cai y E. J. Knuth (Eds.), Early algebraization (pp. 579-593). Springer.
Kieran, C. (2019). Task Design Frameworks in Mathematics Education Research: An Example of a Domain-Specific Frame for Algebra Learning with Technological Tools. En G. Kaiser y N. Presmeg (Eds.), Compendium for early career researchers in mathematics education (pp. 265-287). Springer.
Kieran, C. (2022). The multi-dimensionality of early algebraic thinking: Background, overarching dimensions, and new directions. ZDM - Mathematics Education, 54(6), 1131-1150. https://doi.org/10.1007/s11858-022-01435-6
Kieran, C., & Martínez-Hernández, C. (2022). Structure sense at early ages: The case of equivalence of numerical expressions and equalities. En T. Rojano (Ed.), Algebra structure sense development amongst diverse learners (pp. 35-66). Routledge.
Lloyd, G. M., Herbel-Eisenmann, B. A., & Star, J. R. (2011). Developing essential understanding of expressions, equations, and functions for teaching mathematics in grades 6-8. NCTM.
Marghetis, T., Landy, D., & Goldstone, R. L. (2016). Mastering algebra retrains the visual system to perceive hierarchical structure in equations. Cognitive Research: Principles and Implications, 1(1), 25. https://doi.org/10.1186/s41235-016-0020-9
Ministerio de Educación de Chile (2012). Bases Curriculares Educación Básica. Unidad de Currículo y Evaluación.
Ministerio de Educación y Formación Profesional (MEFP) (2022). Real Decreto 157/2022, de 01 de marzo, por el que se establecen la ordenación y enseñanzas mínimas de la Educación Primaria. BOE, 52, 24386-24504.
National Council of Teacher of Mathematics (NCTM) (2003). Principios y estándares para la educación matemática. SAEM Thales.
Paoletti, T., Stevens, I., & Vishnubhotla, M. (2021). Comparative and restrictive inequalities. The Journal of Mathematical Behavior, 63, 100895. https://doi.org/10.1016/j.jmathb.2021.100895
Pinto, E., Ayala-Altamirano, C., Molina, M., & Cañadas, M. C. (2023). Desarrollo del pensamiento algebraico a través de la justificación en educación primaria. Enseñanza de las Ciencias, 41(1), 149-173. https://doi.org/10.5565/rev/ensciencias.5835
Radford, L. (2011). Grade 2 Students’ Non-Symbolic Algebraic Thinking. En J. Cai y E. J. Knuth (Eds.), Early algebraization (pp. 303-322). Springer.
Radford, L. (2018). The emergence of symbolic algebraic thinking in primary school. En C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12-year-olds (pp. 3-25). Springer.
Radford, L. (2022). Introducing equations in early algebra. ZDM Mathematics Education, 54(6), 1151-1167. https://doi.org/10.1007/s11858-022-01422-x
Rojano, T. (2022). Algebra structure sense: Conceptual approaches and elements for its development. En T. Rojano (Ed.), Algebra structure sense development amongst diverse learners (pp. 1-19). Routledge.
Tsamir, P., & Almog, N. (2001). Students’ strategies and difficulties: The case of algebraic inequalities. International Journal of Mathematical Education in Science and Technology, 32(4), 513-524. https://doi.org/10.1080/00207390110038277
Pacheco-Cáceres, E., Pérez-Martos, M. D. C., & Ayala-Altamirano, C. (2025). Introducing Equations and Inequations throught Arithmetic Problems with Visual Support. Aula, 31, e32206. https://doi.org/10.14201/aula2025.32206
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