Prospective Mathematics Teachers’ Profiles When Noticing Proportional Reasoning

Authors

Abstract

The study characterizes profiles of professional noticing proportional reasoning of fourteen prospective secondary mathematics teachers as they attend to and interpret characteristics of proportional reasoning to decide how to proceed with teaching. We characterize three prospective secondary teachers’ profiles regarding the development of the three skills (attend to, interpret and decide). The notable profile was characterized by high performance in the three skills; the mixed profile was characterized by uneven performance across the skills; and the emerging profile when prospective teachers had difficulties in the three skills. These profiles reveal two features of the learning of professional noticing: (i) the difficulty in simultaneously considering multiple mathematically relevant elements, and (ii) the non-linearity between the skills.

Keywords

Professional noticing, Proportional reasoning, Prospective teachers, Development of teaching competences, Mathematics teacher education

References

Amador, J. M., Glassmeyer, D., & Brakoniecki, A. (2025). Teachers’ noticing of proportional reasoning. Journal of Mathematics Teacher Education, 28(4), 879-907. https://doi.org/10.1007/s10857-024-09625-7

Barnhart, T. & van Es, E. (2015). Studying teacher noticing: Examining the relationship among pre-service science teachers’ ability to attend, analyze and respond to student thinking. Teaching and Teacher Education, 45, 83-93. https://doi.org/10.1016/j.tate.2014.09.005

Buforn, À., Llinares, S., Coles, A., & Brown, L. C. (2022). Pre-service teachers’ knowledge of the unitizing process in recognizing students’ reasoning to propose teaching decisions. International Journal of Mathematical Education in Science and Technology, 53(2), 425-443. https://doi.org/10.1080/0020739x.2020.1777333

Buforn, À., Llinares, S., & Fernández, C. (2018). Características del conocimiento de los estudiantes para maestro españoles en relación con la fracción, razón y proporción. Revista Mexicana de Investigación Educativa, 23(76), 229-251.

Burgos, M. & Godino, J. (2022). Prospective primary school teachers’ competence for the cognitive analysis of students’ solutions to proportionality tasks. Journal of Mathematik-Didactik, 43, 347-376. https://doi.org/10.1007/s13138-021-00193-4

Copur-Gencturk, Y., Baek, C., & Doleck, T. (2023). A closer look at teachers’ proportional reasoning. International Journal of Science and Mathematics Education, 21(1), 113-129. https://doi.org/10.1007/s10763-022-10249-7

Cramer, K. & Post, T. (1993). Proportional reasoning. The Mathematics Teacher, 86, 404-407.

Dindyal, J., Schack, E., Choy, B. H., & Gamoran, M. (2021). Exploring the terrains of mathematics teacher noticing. ZDM Mathematics Education, 53(1), 1-16. https://doi.org/10.1007/s11858-021-01249-y

Espinoza-González, J., Buforn, À., & Llinares, S. (2025). Anexo 1. Diagramas con las puntuaciones asignadas a cada grupo de estudiantes para profesor en las tres destrezas según los tres niveles de razonamiento proporcional. [Archivo de datos]. Figshare. https://doi.org/10.6084/m9.figshare.29803679.v3

Fernández, C., Valls, J., & Llinares, S. (2011). El desarrollo de un esquema para caracterizar la competencia docente «mirar con sentido» el pensamiento matemático de los estudiantes. En M. Marín et al. (Eds.), Investigación en Educación Matemática XV (pp. 351-360). SEIEM.

Fernández, C. & Llinares, S. (2012). Características del desarrollo del razonamiento proporcional en la educación primaria y secundaria. Enseñanza de las Ciencias, 30(1), 129-142. https://doi.org/10.5565/rev/ec/v30n1.596

Jacobs, V. R., Empson, S. B., Jessup, N., Dunning, A., Pynes, D., Krause, G., & Franke, T. M. (2024). Profiles of teachers’ expertise in professional noticing of children’s mathematical thinking. Journal of Mathematics Teacher Education, 27(3), 295-324. https://doi.org/10.1007/s10857-022-09558-z

Jacobs, V. R., Lamb, L. C., & Philipp, R.A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169-202. https://doi.org/10.5951/jresematheduc.41.2.0169

Kaput, J. & West, M. M. (1994). Missing-value proportional reasoning problems: Factors affecting informal reasoning patterns. En G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 235-287). State University of New York Press.

Lamon, S. (1993). Ratio and proportion: Children’s cognitive and metacognitive processes. En T. Carpenter, E. Fennema, & T. Romberg (Eds.), Rational Numbers: An Integration of Research (pp. 131-156). Lawrence Erlbaum Associates.

Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework. En F. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 629-669). Information Age Publishing.

Lobato, J., Ellis, A. B., & Zbiek, R. M. (2010). Developing essential understanding of ratios, proportions, and proportional reasoning for teaching mathematics: Grades 6-8. National Council of Teachers of Mathematics.

Mason, J. (2002). Researching your own practices: The discipline of noticing. Routledge.

Moreno, M., Valls, J., Sánchez-Matamoros, G., & Aráuz, D. F. (2025). Desarrollo profesional de futuros docentes de matemáticas de educación secundaria. Enseñanza de las Ciencias, 43(2), 61-80. https://doi.org/10.5565/rev/ensciencias.6320

Moreno, M., Sánchez-Matamoros, G., & Valls, J. (2025). Influence of professional materials on the decision-making of preservice secondary teachers when noticing students’ mathematical thinking. Education Sciences, 15(4), 418. https://doi.org/10.3390/educsci15040418

Noelting, G. (1980a). The development of proportional reasoning and the ratio concept. Part I— Differentiation of stages. Educational Studies in Mathematics, 11(2), 217-253. https://doi.org/10.1007/BF00304357

Noelting, G. (1980b). The development of proportional reasoning and the ratio concept. Part II— Problem-structure at successive stages—Problem-solving strategies and the mechanism of adaptive restructuring. Educational Studies in Mathematics, 11(3), 331-363. https://doi.org/10.1007/BF00697744

Sánchez-Matamoros, G., Fernández, C., & Llinares, S. (2019). Relationships among prospective secondary mathematic teachers’ skills of attending, interpreting and responding to students’ understanding. Educational Studies in Mathematics, 100(1), 83-99. https://doi.org/10.1007/s10649-018-9855-y

Scheiner, T. (2021). Towards a more comprehensive model of teacher noticing. ZDM-Mathematics Education, 53(1), 85-94. https://doi.org/10.1007/s11858-020-01202-5

Scheiner, T. (2023). Shifting the ways prospective teachers frame and notice student mathematical thinking: from deficits to strengths. Educational Studies in Mathematics, 114(1), 35-61. https://doi.org/10.1007/s10649-023-10235-y

Tjoe, H. & De la Torre, J. (2014). On recognizing proportionality: Does the ability to solve missing value proportional problems presuppose the conception of proportional reasoning? Journal of Mathematical Behavior, 33, 1-7. https://doi.org/10.1016/j.jmathb.2013.09.002

Tyminsky, A. M., Simpson, A. J., Land, T. J., Drake, C., & Dede, E. (2021). Prospective elementary mathematics teachers’ noticing of children’s mathematics: A focus on extending moves. Journal of Mathematics Teacher Education, 24(6), 533-561. https://doi.org/10.1007/s10857-020-09472-2

Valverde, G. & Castro, E. (2012). Prospective elementary school teachers’ proportional reasoning. PNA, 7, 1-19. https://doi.org/10.30827/pna.v7i1.6134

Van Dooren, W., De Bock, D., Evers, M., & Verschaffel, L. (2009). Students’ overuse of proportionality on missing-value problems: How numbers may change solutions. Journal for Research in Mathematics Education, 40(2), 187-211. https://doi.org/10.2307/40539331

Van Dooren, W., De Bock, D., Hessels, A., Janssens, D., & Verschaffel, L. (2005). Not everything is proportional: Effects of age and problem type on propensities of overgeneralization. Cognition and Instruction, 23(1), 57-86. https://doi.org/10.1207/s1532690xci2301_3

Weyers, J., König, J., Scheiner, T., Santagata, R., & Kaiser, G. (2024). Teacher noticing in mathematics education: A review of recent developments. ZDM Mathematics Education, 56(2), 249-264. https://doi.org/10.1007/s11858-023-01527-x

Published

2025-11-03

How to Cite

Espinoza-González, J., Buforn, Àngela, & Llinares, S. (2025). Prospective Mathematics Teachers’ Profiles When Noticing Proportional Reasoning. Enseñanza De Las Ciencias. Revista De investigación Y Experiencias didácticas, 43(3), 31–50. https://doi.org/10.5565/rev/ensciencias.6472

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