Learning trajectory of narrative arithmetic problems: numerical concepts and semantic structure
DOI:
https://doi.org/10.34019/1982-1247.2026.v20.45151Keywords:
Problemas aritméticos narrativos, Cognição numérica, Aprendizagem MatemáticaAbstract
Narrative arithmetic problems are an important tool for encouraging children to mathematize the world. On one hand, the semantic structure is the cognitive schema underlying arithmetic problems that most influences their resolution. On the other hand, the acquisition of the numerical concepts underlying mathematical reasoning follows a well-defined trajectory. This article aims to perform a conceptual mapping between the semantic structure of additive arithmetic problems and the trajectory of development of the concept of number, uniting two different theoretical paradigms. The article brings contributions to the pedagogical and research areas.
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Balt, M., Ehlert, A., & Fritz, A. (2019). Assessment in inclusive mathematics education. Approaches to designing progress assessments for numeracy learning. In D. Kollosche, R. M. J. Souza, M. Knigge, M. G. Penteado, & O. Skovsmose (Eds.), Inclusive mathematics education (pp. 197–216). Springer. https://doi.org/10.1007/978-3-030-11518-0_14
Balt, M., Fritz, A., & Ehlert, A. (2020). Insights into first grade students’ development of conceptual numerical understanding as drawn from progression-based assessments. Frontiers in Education, 5, Article 80. https://doi.org/10.3389/feduc.2020.00080
Carpenter, T. P., & Moser, J. M. (1979). An investigation of the learning of addition and subtraction. (Theoretical Paper No. 79, pp. 10–53). Wisconsin Research and Development Center for Individualized Schooling. https://eric.ed.gov/?id=ED188892
Carpenter, T. P., & Moser, J. M. (1982). The development of addition and subtraction problem-solving skills. In T. P. Carpenter, J. M. Moser & T. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 9–24). Taylor & Francis. https://doi.org/10.4324/9781003046585
Carpenter, T. P., Hiebert, J., & Moser, J. M. (1983). The effect of instruction on children's solutions of addition and subtraction word problems. Educational Studies in Mathematics, 14(1), 55–72. https://www.jstor.org/stable/3482306
Chan, W. W. L., & Kwan, J. L. Y. (2021). Pathways to word problem solving: The mediating roles of schema construction and mathematical vocabulary. Contemporary Educational Psychology, 65, Article 101963. https://psycnet.apa.org/doi/10.1016/j.cedpsych.2021.101963
Chapman, O. (2006). Classroom practices for context of mathematics word problems. Educational Studies in Mathematics, 62, 211–230. https://doi.org/10.1007/s10649-006-7834-1
Ching, B. H. H., & Nunes, T. (2017). The importance of additive reasoning in children’s mathematical achievement: A longitudinal study. Journal of Educational Psychology, 109(4), 477–508. https://psycnet.apa.org/doi/10.1037/edu0000154
Clements, D. H. & Sarama, J. (2021). Learning and teaching early math. The learning trajectories approach (3rd. ed.). Routledge.
Clements, D. H., & Sarama, J. (2014). Learning trajectories. In A. P. Maloney, J. Confrey, & K. H. Nguyen (Eds.), Learning over time: Learning trajectories in mathematics education (pp. 1–30). IAP Information Age Publishing.
Daroczy, G., Wolska, M., Meurers, W. D., & Nuerk, H. C. (2015). Word problems: A review of linguistic and numerical factors contributing to their difficulty. Frontiers in Psychology, 6, Article 129726. https://doi.org/10.3389/fpsyg.2015.00348
Freitas, F. R., Herzog, M., Haase, V. G., & Fritz, A. (2022). Compreensão conceitual do número no diagnóstico e intervenção para as dificuldades de aprendizagem na aritmética. In: V. G. Haase, H. A. T. Simplício e K. S. Benedetti (Orgs.). Pedagogia do Sucesso (Vol. 1, pp. 289–312). Ampla Editora.
Fritz, A., Ehlert, A., & Balzer, L. (2013). Development of mathematical concepts as basis for an elaborated mathematical understanding. South African Journal of Childhood Education, 3(1), 38–67. https://doi.org/10.4102/sajce.v3i1.31
Fritz, A., Haase, V. G., & Rasanen, P. (2019). International handbook of mathematical learning difficulties. Springer.
Fuson, K. C. (2009). Avoiding misinterpretations of Piaget and Vygotsky: Mathematical teaching without learning, learning without teaching, or helpful learning-path teaching?. Cognitive Development, 24(4), 343–361. http://dx.doi.org/10.1016/j.cogdev.2009.09.009
Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Harvard University Press.
Gomides, M. R. A., Lopes-Silva, J. B., Moura, R., Salles, J. F. & Haase, V. G. (2021). PRONUM3RO: Bateria de avaliação do processamento numérico e cálculo. Vetor.
González, J. E. J., & Espinel, A. I. G. (2002). Strategy choice in solving arithmetic word problems: Are there differences between students with learning disabilities, G-V poor performance and typical achievement students? Learning Disability Quarterly, 25(2), 113–122. https://psycnet.apa.org/doi/10.2307/1511278
Gotwals, A. W. (2018). Where are we now? Learning progressions and formative assessment. Applied Measurement in Education, 31, 157–164. https://doi.org/10.1080/08957347.2017.1408626
Jitendra, A. K., Dupuis, D. N., Rodriguez, M. C., Zaslofsky, A. F., Slater, S., Cozine-Corroy, K., & Church, C. (2013). A randomized controlled trial of the impact of schema-based instruction on mathematical outcomes for third-grade students with mathematics difficulties. The Elementary School Journal, 114(2), 252–276. https://psycnet.apa.org/doi/10.1086/673199
Jitendra, A. K., Harwell, M. R., Dupuis, D. N., & Karl, S. R. (2017). A randomized trial of the effects of schema-based instruction on proportional problem-solving for students with mathematics problem-solving difficulties. Journal of Learning Disabilities, 50(3), 322–336. https://doi.org/10.1177/0022219416629646
Krajewski, K. (2008). Pra ̈vention der Rechenschwa ̈che [The early prevention of math problems]. In W. Schneider, & M. Hasselhorn (Eds.), Handbuch der Pädagogischen Psychologie (pp. 360–370). Hogrefe.
Kutaka, T. S., Chernyavskiy, P., Cong, M., McCreadie, K., Sarama, J., & Clements, D. H. (2023). How arithmetic problem-solving strategy sophistication evolves over time in a kindergarten sample [Preprint]. https://doi.org/10.31234/osf.io/hjbzg
Langhorst, P., Ehlert, A., & Fritz, A. (2012). Non-numerical and numerical understanding of the part-whole concept of children aged 4 to 8 in word problems. Journal für Mathematik-Didaktik, 2(33), 233–262. https://doi.org/10.1007/S13138-012-0039-5
Le Corre, M., & Carey, S. (2007). One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles. Cognition, 105(2), 395–438. https://doi.org/10.1016/j.cognition.2006.10.005
Le Corre, M., Van de Walle, G., Brannon, E. M. & Carey, S. (2006). Re-visiting the competence/performance debate in the acquisition of the counting principles. Cognitive Psychology, 52(2), 130–169. https://doi.org/10.1016/j.cogpsych.2005.07.002
Morales, R. V., Shute, V. J., & Pellegrino, J. W. (1985). Developmental differences in understanding and solving simple mathematics word problems. Cognition and Instruction, 2(1), 41–57. https://doi.org/10.1207/s1532690xci0201_2
Moretti, M. T., & Brandt, C. F. (2014). Dificuldades na resolução de problemas aditivos a uma operação: Ponto de encontro esclarecedor à luz da noção de congruência semântica. Acta Scientiae, 16(3). http://www.periodicos.ulbra.br/index.php/acta/article/view/1173
Munez, D., Orrantia, J., & Rosales, J. (2013). The effect of external representations on compare word problems: Supporting mental model construction. The Journal of Experimental Education, 81(3), 337–355. https://psycnet.apa.org/doi/10.1080/00220973.2012.715095
Nuerk, H. C., Moeller, K., Klein, E., Willmes, K., & Fischer, M. H. (2015). Extending the mental number line. Zeitschrift Für Psychologie, 219(1), 3–22. https://doi.org/10.1027/2151-2604/a000041
Nunes, T., & Bryant, P. (1996). Children doing mathematics. Wiley-Blackwell.
Nunes, T., & Bryant, P. (2015). The development of mathematical reasoning. In R. M. Lerner (Ed.), Handbook of child psychology and developmental science (pp. 1–48). Wiley.
Nunes, T., & Bryant, P. (2021). Using mathematics to understand the world: How culture promotes children's mathematics. Routledge.
Nunes, T., Bryant, P., Barros, R., & Sylva, K. (2012). The relative importance of two different mathematical abilities to mathematical achievement. British Journal of Educational Psychology, 82(1), 136–156. https://psycnet.apa.org/doi/10.1111/j.2044-8279.2011.02033.x
Nunes, T., Dorneles, B. V., Lin, P. J., & Rathgeb-Schnierer, E. (2016). Teaching and learning about whole numbers in primary school. Springer Nature.
Oswald, T. M., Beck, J. S., Iosif, A. M., McCauley, J. B., Gilhooly, L. J., Matter, J. C., & Solomon, M. (2016). Clinical and cognitive characteristics associated with mathematics problem solving in adolescents with autism spectrum disorder. Autism Research, 9(4), 480–490. https://doi.org/10.1002/aur.1524
Peake, C., Jiménez, J. E., Rodriguez, C., Bisschop, E., & Villarroel, R. (2015). Syntactic awareness and arithmetic word problem solving in children with and without learning disabilities. Journal of Learning Disabilities, 48(6), 593–601. https://doi.org/10.1177/0022219413520183
Pongsakdi, N., Kajamies, A., & Veermans, K. (2020). What makes mathematical word problem solving challenging? Exploring the roles of word problem characteristics, text comprehension, and arithmetic skills. ZDM Mathematics Education, 52, 33–44. https://doi.org/10.1007/s11858-019-01118-9
Powell, S. R., & Fuchs, L. S. (2014). Does early algebraic reasoning differ as a function of students’ difficulty with calculations versus word problems? Learning Disabilities Research & Practice, 29(3), 106–116. https://psycnet.apa.org/doi/10.1111/ldrp.12037
Riley, M. S., Greeno, J. G., & Heller, J. I. (1983). Development of children’s problem solving ability in arithmetic. In H. P. Ginsburg (Ed.), The development of mathematical thinking (pp. 153–196). Academic Press.
Robinson, K. M. (2019). Arithmetic concepts in the early school years. In K. M. Robinson, H. P. Osana & D. Kotsopoulos (Eds.) Mathematical learning and cognition in early childhood. Integrating interdisciplinary research into practice (pp. 165–185). Springer.
Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. Routledge.
Selter, C., Prediger, S., Nührenbürger, M., & Hußmann, S. (2012). Taking away and determining the difference: A longitudinal perspective on two models of subtraction and the inverse relation to addition. Educational Studies on Mathematics, 79, 389–408. https://doi.org/10.1007/s10649-011-9305-6
Siegler, R. S., & Braithwaite, D. W. (2017). Numerical development. Annual Review of Psychology, 68, 187–213. https://doi.org/10.1146/annurev-psych-010416-044101
Swanson, H. L. (2014). Does cognitive strategy training on word problems compensate for working memory capacity in children with math difficulties? Journal of Educational Psychology, 106(3), 831–848. https://psycnet.apa.org/doi/10.1037/a0035838
Ta'ir, J., Brezner, A., & Ariel, R. (1997). Profound developmental dyscalculia: Evidence for a cardinal/ordinal skills acquisition device. Brain and Cognition, 35(2), 184–206. https://doi.org/10.1006/brcg.1997.0937
Thevenot, C., & Barrouillet, P. (2015). Arithmetic word problem solving and mental representations. In R. Kadosh & A. Dowker (Eds.), The Oxford handbook of numerical cognition (pp. 158–179). Oxford University Press.
Vergnaud, G. (1982). Cognitive and developmental psychology and research in mathematics education: Some theoretical and methodological issues. For the Learning of Mathematics, 3(2), 31–41. http://www.jstor.org/stable/40248130
Vergnaud, G. (1997). The nature of mathematical concepts. In T. Nunes & P. Bryant (Eds.), Learning and teaching mathematics: An international perspective (pp. 5–28). Psychology Press/Erlbaum (UK) Taylor & Francis.
Vergnaud, G. (2009). The theory of conceptual fields. Human Development, 52(2), 83–94. https://psycnet.apa.org/doi/10.1159/000202727
Verschaffel, L., & Corte, E. (1997). Word problems: A vehicle for promoting authentic mathematical understanding and problem solving in the primary school. In T. Nunes & P. Bryant (Eds), Learning and teaching mathematics: An international perspective (pp. 69–97). Psychology Press.
Verschaffel, L., Schukajlow, S., Star, J., & Van Dooren, W. (2020). Word problems in mathematics education: A survey. ZDM – Mathematics Education, 52, 1–16. https://eric.ed.gov/?id=EJ1243930
Vicente, S., Orrantia, J., & Verschaffel, L. (2008). Influencia del conocimiento matemático y situacional en la resolución de problemas aritméticos verbales: Ayudas textuales y gráficas. Journal for the Study of Education and Development, 31(4), 463–483. https://psycnet.apa.org/doi/10.1174/021037008786140959
Von Aster, M. G., & Shalev, R. S. (2007). Number development and developmental dyscalculia. Developmental Medicine & Child Neurology, 49(11), 868–873. https://doi.org/10.1111/j.1469-8749.2007.00868.x
Zheng, X., Swanson, H. L., & Marcoulides, G. A. (2011). Working memory components as predictors of children’s mathematical word problem solving. Journal of Experimental Child Psychology, 110(4), 481–498. https://psycnet.apa.org/doi/10.1016/j.jecp.2011.06.001
