Cognitive analysis of probability comparison tasks by preservice primary school teachers




Probability, proportional reasoning, teachers’ education, onto-semiotic approach, cognitive analysis


To promote mathematical learning, teachers must be able to analyze, interpret and assess students’ mathematical activity, making decisions about their understanding of, and difficulties in, solving mathematical tasks. This cognitive analysis skill enables teachers to understand mathematical learning processes and to establish different ways of institutionalizing the mathematical knowledge involved. [Objective] This work is intended to present the results of an assessment of preservice Primary Education teachers’ knowledge and abilities to interpret students’ responses to probability comparison tasks, identify incorrect strategies and recognize proportional reasoning in mathematical activity. Furthermore, the strategies proposed by preservice teachers to help students overcome the difficulties that led them to obtain inadequate solutions are analyzed. [Methodology] Descriptive and qualitative research was carried out with the collaboration of 116 preservice Primary Education teachers from the University of Almería (Spain). The investigation was carried out once the process of training in the mathematical contents of Statistics and Probability had been completed. [Results] One of the most important results obtained is a didactic-mathematical knowledge of the type of proportional and probabilistic reasoning that impedes preservice teachers in their interpretation and decision-making about student responses. [Conclusions] These results highlight the need to implement educational solutions to adequately resolve these common situations in schools.


Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407. 10.1177/0022487108324554

Bartell, T. G., Webel, C., Bowen, B., & Dyson, N. (2013). Prospective teacher learning: recognizing evidence of conceptual understanding. Journal of Mathematics Teacher Education, 16(1), 57-79.

Batanero, C., Godino, J. D., & Roa, R. (2004). Training teachers to teach probability. Journal of statistics Education, 12(1).

Batanero, C., Gómez, E., Contreras, J. M., & Díaz, C. (2015). Conocimiento matemático de profesores de primaria en formación para la enseñanza de la probabilidad: Un estudio exploratorio. Práxis Educativa, 10(1), 11-34.

Batanero, C., Gómez, E., Serrano, L., & Contreras, J. M. (2012). Comprensión de la aleatoriedad por futuros profesores de educación primaria. Redimat, 1(3), 222-245.

Begolli, K. N., Dai, T., McGinn, K. M., & Booth, J. L. (2021). Could probability be out of proportion? Self-explanation and example-based practice help students with lower proportional reasoning skills learn probability. Instructional Science 49, 441–473.

Ben-Chaim, D., Keret, Y., & Ilany, B. (2012). Ratio and proportion: Research and teaching in mathematics teachers’ education. Sense Publisher.

Berk, D., Taber, S. B., Gorowara, C. C., & Petzl, C. (2009). Developing prospective elementary teachers’ flexibility in the domain of proportional reasoning. Mathematical Thinking and Learning, 11(3), 113-135. 0.1080/10986060903022714

Bisquerra, R., & Alzina, R. B. (2004). Metodología de la investigación educativa. Editorial La Muralla.

Biza, I., Nardi, E., & Zhachariades, T. (2007). Using tasks to explore teacher knowledge in situation-specific contexts. Journal of Mathematics Teacher Education, 10, 301–309.

Boyer, T. W., & Levine, S. C. (2015). Prompting Children to Reason Proportionally: Processing Discrete Units as Continuous Amounts. Developmental Psychology, 51(5), 615–620.

Bryant, P., & Nunes, T. (2012). Children’s understanding of probability: A literature review (full report). The Nuffield Foundation.

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

Burgos, M., & Godino, J. D. (2020). Modelo ontosemiótico de referencia de la proporcionalidad: Implicaciones para la planificación curricular en primaria y secundaria. AIEM Avances de Investigación en Educación Matemática, 18, 1–20.

Chapman, O. (2014). Overall commentary: understanding and changing mathematics teachers. In: J. J. Lo; K. R. Leatham; L. R. Van Zoest (Eds.) Research Trends in Mathematics Teacher Education (pp. 295-309). Springer International Publishing.

Choy, B. H. (2016). Snapshots of mathematics teacher noticing during task design. Mathematics Education Research Journal, 28(3), 421-440.

English, L. D. (2008). Setting an agenda for international research in mathematics education. In Handbook of international research in mathematics education, 2nd Edition, p 3-19. New York y London: Taylor and Francis (Routledge).

Fernández, C., Llinares, S., & Valls, J. (2011). Características del desarrollo de una mirada profesional en estudiantes para profesor de matemáticas en un contexto b-learning. Acta Scientiae, 13(1), 10-28.

Fernández, C., Llinares, C., & Valls, J. (2012). Learning to notice students' mathematical thinking through online discussions. ZDM. Mathematics Education. 10.1007/s11858-012-0425-y

Gea, M.M., Parraguez, R., & Batanero, C. (2017). Comprensión de la probabilidad clásica y frecuencial por futuros profesores. En J. M. Muñoz-Escolano; A. Arnal-Bailera; P. Beltrán-Pellicer; M. L. Callejo; J. Carrillo (Eds.), Investigación en educación matemática XXI (pp. 267-276). SEIEM.

Godino, J. D. (2009). Categorías de análisis de los conocimientos del profesor de matemáticas. Unión, Revista Iberoamericana de Educación Matemática, 20, 13-31.

Godino, J. D., Batanero, C., & Font, V. (2007). The onto-semiotic approach to research in mathematics education. ZDM. The International Journal on Mathematics Education, 39(1-2), 127-135.

Godino, J. D., Giacomone, B., Batanero, C., & Font, V. (2017). Enfoque ontosemiótico de los conocimientos y competencias del profesor de matemáticas. Bolema, 31(57), 90-113.

Gómez, E., Batanero, C., & Contreras, C. (2013). Conocimiento matemático de futuros profesores para la enseñanza de la probabilidad desde el enfoque frecuencial. Bolema, 28(48), 209-229.

Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39, 372-400.

Jacobs, V. R., Lamb, L. C. & Philipp, R. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169-202.

Jakobsen, A., Ribeiro, C. M., & Mellone, M. (2014). Norwegian prospective teachers’ MKT when interpreting pupils’ productions on a fraction task. Nordic Studies in Mathematics Education, 19(3-4), 135-150.

Lamon, S. (2007). Rational number and proportional reasoning: toward a theoretical framework for research. In: F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629-667). NCTM.

Langrall, C. W., & Mooney, E. S. (2005). Characteristics of elementary school students’ probabilistic reasoning. In G. Jones (Ed.), Exploring probability in school (pp. 95–119). Springer.

Lesh, R., Post, T., & Behr, M. (1988). Proportional reasoning. In: J. Hiebert; M. Behr (Eds.), Number concepts and operations for the middle grades (pp. 93-118). Reston, VA: NCTM.

Llinares, S. (2013). Professional Noticing: A component of the mathematics teacher’s professional practice. Sisyphus, Journal of Education, 1(3), 76-93.

Mason, J. (2016). Perception, interpretation and decision making: understanding gaps between competence and performance-a commentary. ZDM, 48(1-2), 219-226.

Mohamed, N. (2012). Evaluación del conocimiento de los futuros profesores de educación primaria sobre probabilidad (Doctoral dissertation). Universidad de Granada.

Pereira-Mendoza, L. (2002). Would you allow your accountant to perform surgery? Implications for the education of primary teachers. In B. Phillips (Ed.), Proceedings of the Sixth International Conference on the Teaching of Statistics. Hawthorn, VIC: International Statistical Institute.

Pino-Fan, L., Assis, A., & Castro, W. F. (2015). Towards a methodology for the characterization of teachers' didactic-mathematical knowledge. EURASIA Journal of Mathematics, Science and Technology Education, 11(6), 1429-1456.

Pino-Fan, L., & Godino, A. (2015). Perspectiva ampliada del conocimiento didáctico-matemático del profesor. PARADIGMA, 36(1), 87-109.

Simpson, A., & Haltiwanger, L. (2017). This is the first time I’ve done this: Exploring secondary prospective mathematics teachers’ noticing of students’ mathematical thinking. Journal of Mathematics Teacher Education, 20(4), 335-355.

Son, J. (2013). How preservice teachers interpret and respond to student errors: Ratio and proportion in similar rectangles. Educational Studies in Mathematics, 84, 49–70.

Stahnke, R., Schueler, S., & Roesken-Winter, B. (2016). Teachers’ perception, interpretation, and decision making: a systematic review of empirical mathematics education research. ZDM. Mathematics Education, 48(1-2), 1-27.

Tourniaire, F., & Pulos, S. (1985). Proportional reasoning: A review of the literature. Educational Studies in Mathematics, 16, 181-204.

Van Dooren, W. (2014). Probabilistic thinking: analyses from a psychological perspective. In Chernoff E., Sriraman B. (Eds.), Probabilistic Thinking. Advances in Mathematics Education (pp. 123-126). Springer.

Vásquez, C., & Alsina, Á. (2015a). Conocimiento didáctico-matemático del profesorado de educación primaria sobre probabilidad: Diseño, construcción y validación de un instrumento de evaluación. BOLEMA, 29 (52), 681-703.

Vásquez, C., & Alsina, A. (2015b). El conocimiento del profesorado para enseñar probabilidad: Un análisis global desde el modelo del conocimiento didáctico-matemático. Avances de Investigación en Educación Matemática,7, 27-48.

Watson, J. (2005). The probabilistic reasoning of middle school students. In G. A. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 145-169). Springer.

Watson, J. M., Collis, K. F., & Moritz, J. B. (2007). The development of chance measurement. In: Stepping Stones for the 21st Century (pp. 113-138). Brill Sense.



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Cognitive analysis of probability comparison tasks by preservice primary school teachers. (2022). Uniciencia, 36(1), 1-24.



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How to Cite

Cognitive analysis of probability comparison tasks by preservice primary school teachers. (2022). Uniciencia, 36(1), 1-24.

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