Workshop on non-routine problem solving for students aged 8 to 9: A case study

Keywords: Heuristics, arithmetic thinking, non-routine problems, implicative analysis


[Objective] This article describes heuristics used by 8 and 9 year-old students to solve four non-routine problems that promote the development of arithmetic thinking from two perspectives: number distribution under a graphical representation, and use of arithmetic operations in the positional decimal system. [Methodology] The analysis included the formulation of categories that made it possible to characterize a priori the heuristics that could arise when solving each problem. These categories were used in the implementation of a mixed methodological approach with an exploratory and descriptive orientation. Qualitative analysis was carried out through a case study that helped to identify key actions based on written results provided by students. Quantitative analysis was carried out using implicative analysis, which includes a similarity tree and identification of significant classes. [Results] The results point to the importance of using simple heuristics in finding solutions to non-routine arithmetic problems, and confirm the presence of some characteristics of heuristic reasoning, such as attention, reduction, and change of assumptions. Likewise, implicative relationships were identified among some heuristics that share common characteristics, depending on the type of problem. [Conclusions] The results of this study show that, even in incorrect or incomplete answers, it is possible to recognize logical processes for the elaboration of partial answers and intuitive approaches, which are consistent with the action of simplifying or facilitating the search for a solution.


Berenger, A. (2018). Pre-service teachers’ difficulty with problem solving. In J. Hunter, P. Perger, & L. Darragh (Eds.), Proceedings of the 41st Annual Conference of the Mathematics Education Research Group of Australasia (pp. 162-169). Auckland: MERGA.
Bergman, M. M. (2008). The straw men of the qualitative-quantitative divide and their influence on mixed methods research. In Author (Ed.), Advances in mixed methods research: Theories and applications, 6, 11-21.
Bruder, R. (2016). Role of heuristics for problem solving. In G. Kaiser (Ed.). Problem Solving in Mathematics Education, ICME-13 Topical Surveys (pp. 2-6). Hamburg: Springer.
Casetta, I., González, V., & Rodríguez, M. (2017). Un procedimiento para diseñar entrevistas personalizadas que permiten identificar heurísticas matemáticas. Paradigma, 38(1), 235-258.
Clements, D. H., & Sarama, J. (2015). El aprendizaje y la enseñanza de las matemáticas a temprana edad: El enfoque de las trayectorias de aprendizaje. Gran Bretaña: Learning Tools.
Cui, Z., & Ng, O. L. (2021). The Interplay Between Mathematical and Computational Thinking in Primary School Students’ Mathematical Problem-Solving Within a Programming Environment. Journal of Educational Computing Research, 59(5), 988-1012.
Dunphy, E., Dooley, T., & Shiel, G. (2014). Mathematics in early childhood and primary education (3-8 years). Volume 1: Definitions, theories, development and progression. Dublin: NCCA.
Elia, I., van den Heuvel-Panhuizen, M. & Kolovou, A. (2009). Exploring strategy use and strategy flexibility in non-routine problem solving by primary school high achievers in mathematics. ZDM Mathematics Education 41(5), 605-618.
Felmer, P., Liljedahl, P., & Koichu, B. (Eds.). (2019). Problem solving in mathematics instruction and teacher professional development. Springer International Publishing.
Gigerenzer, G. (2008). Why heuristics work. Perspectives on psychological science, 3(1), 20-29.
Gigerenzer, G., & Todd, P. M. (1999). Simple heuristics that make us smart. New York: Oxford University Press.
Gras, R., Suzuki, E., Guillet, F., & Spagnolo, F. (2008). Statistical implicative analysis. New York: Springer.
Kahneman, D. (2012). Pensar rápido, pensar despacio. Barcelona: Penguin Random House Grupo Editorial.
Koichu, B. (2019). A discursively oriented conceptualization of mathematical problem solving. In P. Felmer, P. Liljedahl, & B. Koichu (Eds.). Problem solving in mathematics instruction and teacher professional development (pp. 43-66). Cham: Springer.
Kruglanski, A. W., & Gigerenzer, G. (2011). Intuitive and deliberate judgments are based on common principles. Psychological review, 118(1), 97-109.
Kuzle, A. (2019). Design and evaluation of practice-oriented materials fostering students’ development of problema-solving competence: The case of working backward strategy. LUMAT: International Journal on Math, Science and Technology Education, 7(3), 28-54.
Lester Jr, F. K. (2013). Thoughts about research on mathematical problem-solving instruction. The mathematics enthusiast, 10(1), 245-278.
Liang, W., & Toh, T. L. (2018). Mathematical problem solving on numbers and arithmetic in upper primary mathematics classroom. Journal of Science and Mathematics Education in South-east Asia, 41 (1), 1-24.
Liljedahl, P., Santos-Trigo, M., Malaspina, U., & Bruder, R. (2016). Problem solving in mathematics education, ICME-13 Topical Surveys. Cham: Springer Nature.
Lompscher, J. (1975). The conditions for increasing the efficiency of learning activity. Voprosy Psychologii, 6, 8-74.
Lubis, A. B., Miaz, Y., & Putri, I. E. (2019). Influence of the Guided Discovery Learning Model on Primary School Students' Mathematical Problem-Solving Skills. Jurnal Mimbar Sekolah Dasar, 6(2), 253-266.
Ministerio de Educación de Chile. (2018). Matemática. Bases Curriculares Primero a Sexto básico (pp. 213-261). Recuperado de
NCTM. (2000). Principles and standards for school mathematics. Reston. VA: National Council of Teachers of Mathematics.
Ng, O. L., Liu, M., & Cui, Z. (2021). Students’ in-moment challenges and developing maker perspectives during problem-based digital making. Journal of Research on Technology in Education, 1-15.
Novotná, J., Eisenmann, P., Přibyl, J., Ondrušová, J., & Břehovský, J. (2014). Problem solving in school mathematics based on heuristic strategies. Journal on Efficiency and Responsibility in Education and Science, 7(1), 1-6.
Olivares, D., Lupiáñez, J. L., & Segovia, I. (2021). Roles and characteristics of problem solving in the mathematics curriculum: a review. International Journal of Mathematical Education in Science and Technology, 52(7), 1079-1096.
Organization for Economic Cooperation and Development [OECD]. (2019). Mathematics performance (PISA indicator). New York: OECD Publishing. (Accessed on December 2021).
Orús, P.; Zamora, L. y Gregori, P. (2009). Teoría y aplicaciones del análisis estadístico implicativo: primera aproximación en lengua hispana. Departamento de Matemáticas, Universitat Jaume I; Facultad de Matemática y Computación, Universidad de Oriente de Santiago de Cuba.
Pehkonen, E. (1999). Beliefs as obstacles for implementing an educational change in problem solving. In E. Pehkonen, & G. Törner (Eds.), Mathematical beliefs and their impact on teaching and learning of Mathematics (pp. 109-117). Duisburg: Springer Spektrum.
Phonapichat, P., Wongwanich, S., & Sujiva, S. (2014). An analysis of elementary school students’ difficulties in mathematical problem solving. Procedia-Social and Behavioral Sciences, 116, 3169-3174.
Podkhodova, N., Snegurova, V., Stefanova, N., Triapitsyna, A., & Pisareva, S. (2020). Assessment of Mathematics Teachers' Professional Competence. Journal on Mathematics Education, 11(3), 477-500.
Pólya, G. (1945). How to solve it. Princeton University.
Pólya, G. (1964). Die Heuristik. Versuch einer vernünftigen Zielstellung. Der Mathematikunterricht, 10, 5-15.
Reiss, K. M., Heinze, A., Renkl, A., & Groß, C. (2008). Reasoning and proof in geometry: Effects of a learning environment based on heuristic worked-out examples. ZDM, 40(3), 455-467.
Rodríguez, M. y Parraguez M. (2014). Interpretando estrategias en resolución de problemas desde dos constructos teóricos: Un estudio de caso. Revista electrónica de investigación en educación en ciencias, 9(2), 1-12.
Rodríguez, M.; Gregori, P.; Riveros, A.; Aceituno, D. (2017). Análisis de las estrategias de resolución de problemas en matemática utilizadas por estudiantes talentosos de 12 a 14 años. Revista Enseñanza de las Ciencias, 29(2), 159-186.
Rodríguez, M.; Oyarce, E.; Lara, M. y Celis, P. (2020). Fomentando la indagación en estudiantes de secundaria mediante la resolución de problemas, una estrategia para articular matemática y ciencias: Un estudio de caso. Revista electrónica de investigación en educación en ciencias, 15(1), 60-71.
Rott, B. (2014). Rethinking Heuristics–Characterizations and Examples. In A. Ambrus, & É. Vásárhelyi (Eds.) Problem Solving in Mathematics Education, Proceedings of the 15th ProMath Conference, 30 August – 1 September, 2013 (pp. 176-192). Eötvös Loránd University, Faculty of Science, Institute of Mathematics Mathematics Teaching and Education Center and Eszterházi Károly College, Institute of Mathematics and Informatics.
Schoenfeld, A. H. (1979). Explicit heuristic training as a variable in problem-solving performance. Journal for Research in Mathematics Education, 10(3), 173-187.
Schoenfeld, A. H. (1985). Making sense of “out loud” problem-solving protocols. The Journal of Mathematical Behavior, 4(2), 171-191.
Singh, P., Teoh, S. H., Cheong, T. H., Rasid, N. S. M., Kor, L. K., & Nasir, N. A. M. (2018). The use of problem-solving heuristics approach in enhancing STEM students development of mathematical thinking. International Electronic Journal of Mathematics Education, 13(3), 289-303.
Stake, R. E. (2011). Qualitative research and case study. Silpakorn Educational Research Journal, 3(1-2), 7-13.
Stanic, G., & Kilpatrick, J. (1989). Historical perspectives on problem solving in the mathematics curriculum. In R. I. Charles, & E. A. Silver (Eds.), The Teaching and Assessing of Mathematical Problem Solving (pp. 1-22). Reston, VA: NCTM/Lawerance Erlbaum Associates.
Susperreguy, M. I., Peake, C., & Gómez, D. M. (2020). Research on numerical cognition in Chile: current status, links to education and challenges (Investigación en cognición numérica en Chile: estado actual, vínculos con la educación y desafíos). Studies in Psychology, 41(2), 404-438.
Törner, G., Schoenfeld, A. H., & Reiss, K. M. (2007). Problem solving around the world: Summing up the state of the art. ZDM-The International Journal on Mathematics Education, 39(1), 5-6.
Ukobizaba, F., Nizeyimana, G., & Mukuka, A. (2021). Assessment Strategies for Enhancing Students' Mathematical Problem-Solving Skills: A Review of Literature. Eurasia Journal of Mathematics, Science and Technology Education, 17(3), em1945.
How to Cite
Rodríguez Jara, M., Vergara-Gómez, A., Mondaca-Saavedra, A., & Gregori Huerta, P. (2023). Workshop on non-routine problem solving for students aged 8 to 9: A case study. Uniciencia, 37(1), 1-23.
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