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27 - Reasoning in Mathematical Development

Neurocognitive Foundations and Their Implications for the Classroom

from Part III - Education and School-Learning Domains

Published online by Cambridge University Press:  24 February 2022

By
Jérôme Prado  and
Marie-Line Gardes
Edited by
Olivier Houdé  and
Grégoire Borst
Olivier Houdé
Affiliation:
Université de Paris V
Grégoire Borst
Affiliation:
Université de Paris V
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Summary

Relations are ubiquitous in mathematics, from the understanding of measurement and patterns to the acquisition of algebra and fractions. In line with this observation, a growing body of literature indicate that individual differences in mathematical skills are associated with individual differences in the ability to reason about relations. In the present chapter, we review these studies and discuss what is known about the neural and behavioral development of two major forms of relational reasoning (i.e., transitive reasoning and analogical reasoning). We argue that relational reasoning may not only relate to mathematical skills because both place demands on common general cognitive resources, but also because relational reasoning and numerical skills share some underlying neurocognitive representations. Finally, the educational implications of these studies are discussed. Notably, we suggest that teachers may help scaffold the development of relational reasoning skills in the classroom by promoting situations in which children are engaged in problem-solving.

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The Cambridge Handbook of Cognitive Development , pp. 566 - 584
Publisher: Cambridge University Press
Print publication year: 2022

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References

Aldon, G., Durand-Guerrier, V., & Ray, B. (2017). Problems promoting the devolution of the process of mathematisation: An example in number theory and a realistic fiction. In Aldon, G., Hitt, F., Bazzini, L., & Gellert, U. (eds.), Mathematics and Technology (pp. 411–429). Cham, Switzerland: Springer.Google Scholar
Ansari, D. (2008). Effects of development and enculturation on number representation in the brain. Nature Reviews Neuroscience, 9, 278–291.Google Scholar
Arsac, G., & Mante, M. (2007). Les pratiques du problème ouvert. SCEREN-CRDP de l’Académie de Lyon.Google Scholar
Artigue, M., & Houdement, C. (2007). Problem solving in France: Research and curricular perspectives. Zentral Blatt Für Didaktik Der Mathematik, 39, 365–382.Google Scholar
Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In Pimm, D. (ed.), Mathematics, Teachers and Children (pp. 216–235). London: Hodder and Stoughton.Google Scholar
Bartha, P. (2013). Analogy and analogical reasoning. In Zalta, E. N. (ed.), The Stanford Encyclopedia of Philosophy. Available from https://plato.stanford.edu/cite.html.Google Scholar
Brannon, E. M., & Terrace, H. S. (1998). Ordering of the numerosities 1 to 9 by monkeys. Science, 282, 746–749.Google Scholar
Brousseau, G. (1997). Theory of Didactical Situations in Mathematics, trans by N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield. Dordrecht, NL: Kluwer.Google Scholar
Bryant, P. E., & Kopytynska, H. (1976). Spontaneous measurement by young children. Nature, 260, 773.Google Scholar
Bryant, P. E., & Trabasso, T. (1971). Transitive inferences and memory in young children. Nature, 232, 458–465.Google Scholar
Bunge, S., Helskog, E., & Wendelken, C. (2009). Left, but not right, rostrolateral prefrontal cortex meets a stringent test of the relational integration hypothesis. Neuroimage, 46, 338–342.CrossRefGoogle Scholar
Bunge, S., & Wendelken, C. (2009). Comparing the bird in the hand with the ones in the bush. Neuron, 62, 609–611.CrossRefGoogle Scholar
Bunge, S., Wendelken, C., Badre, D., & Wagner, A. (2005). Analogical reasoning and prefrontal cortex: Evidence for separable retrieval and integration mechanisms. Cerebral Cortex, 15, 239–249.Google Scholar
Burgoyne, K., Witteveen, K., Tolan, A., Malone, S., & Hulme, C. (2017). Pattern understanding: Relationships with arithmetic and reading development. Child Development Perspectives, 11, 239–244.Google Scholar
Crone, E., Wendelken, C., van Leijenhorst, L., Honomichl, R., Christoff, K., & Bunge, S. (2009). Neurocognitive development of relational reasoning. Developmental Science, 12, 55–66.Google Scholar
Delius, J. D., & Siemann, M. (1998). Transitive responding in animals and humans: exaption rather than adaption? Behavioural Processes, 42, 107–137.Google Scholar
Dias, T., & Durand-Guerrier, V. (2005). Expérimenter pour apprendre en mathématiques. Repères IREM, 60, 61–78.Google Scholar
Douek, N. (2010). Approaching proof in school: From guided conjecturing and proving to a story of proof construction. Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education, January 28–February 1, 2009. Lyon, France.Google Scholar
Dumontheil, I., Burgess, P. W., & Blakemore, S. J. (2008). Development of rostral prefrontal cortex and cognitive and behavioural disorders. Developmental Medicine & Child Neurology, 50, 168–181.Google Scholar
Dumontheil, I., Houlton, R., Christoff, K., & Blakemore, S.-J. (2010). Development of relational reasoning during adolescence. Developmental Science, 13, 24.Google Scholar
English, L. D. (2004). Mathematical and Analogical Reasoning of Young Learners. Mahwah, NJ: Lawrence ErlbaumGoogle Scholar
Evans, J. S. (2003). In two minds: Dual-process accounts of reasoning. Trends in Cognitive Science, 7, 454–459.CrossRefGoogle ScholarPubMed
Feigenson, L., Libertus, M. E., & Halberda, J. (2013). Links between the intuitive sense of number and formal mathematics ability. Child Development Perspectives, 7, 74–79.Google Scholar
Fias, W., Menon, V., & Szucs, D. (2013). Multiple components of developmental dyscalculia. Trends in Neuroscience and Education, 2, 43–47.Google Scholar
Frank, M. J. (2005). Dynamic dopamine modulation in the basal ganglia: A neurocomputational account of cognitive deficits in medicated and nonmedicated Parkinsonism. Journal of Cognitive Neuroscience, 17, 51–72.Google Scholar
Frank, M. J., Rudy, J. W., & O’Reilly, R. C. (2003). Transitivity, flexibility, conjunctive representations, and the hippocampus. II. A computational analysis. Hippocampus, 13, 341–354.Google Scholar
Fuchs, L. S., Compton, D. L., Fuchs, D., Paulsen, K., Bryant, J. D., & Hamlett, C. L. (2005). The prevention, identification, and cognitive determinants of math difficulty. Journal of Educational Psychology, 97, 493–513.Google Scholar
Gardes, M. L. (2018). Démarches d’investigation et recherche de problèmes. In Aldon, G. (ed.), Le Rallye mathématique, un jeu très sérieux! (pp. 73–96). Poitiers: Canopée.Google Scholar
Gardes, M. L. & Durand-Guerrier, V. (2016). Designation at the core of the dialectic between experimentation and proving: A study in number theory. Paper presented at the First Conference of International Network for Didactic Research in University Mathematics. March 31–April 2, 2016, Montpellier, France.Google Scholar
Gibel, P. (2013). The presentation and setting up of a model of analysis of reasoning processes in mathematics lessons in primary schools. Paper presented at the Proceedings of the Eighth Congress of the European Society for Research in Mathematics Education. February 6–10, 2013, Manavgat-Side, Antalya, Turkey.Google Scholar
Goswami, U., & Brown, A. L. (1989). Melting chocolate and melting snowmen: Analogical reasoning and causal relations. Cognition, 35, 69–95.CrossRefGoogle Scholar
Green, C. T., Bunge, S. A., Chiongbian, V. B., Barrow, M., & Ferrer, E. (2017). Fluid reasoning predicts future mathematical performance among children and adolescents. Journal of Experimental Child Psychology, 157, 125–143.Google Scholar
Grenier, D. (2013). Research situations to learn logic and various types of mathematical reasonings and proofs. Paper presented at the Proceedings of the Eighth Congress of the European Society for Research in Mathematics Education. February 6–10, 2013, Manavgat-Side, Antalya, Turkey.Google Scholar
Grosenick, L., Clement, T. S., & Fernald, R. D. (2007). Fish can infer social rank by observation alone. Nature, 445, 429–432.CrossRefGoogle ScholarPubMed
Halford, G. (1992). Analogical reasoning and conceptual complexity in cognitive development. Human Development, 35, 193–217.Google Scholar
Halford, G., Wilson, W. H., & Phillips, S. (2010). Relational knowledge: The foundation of higher cognition. Trends in Cognitive Sciences, 14, 497–505.Google Scholar
Handley, S., Capon, A., Beveridge, M., Dennis, I., & Evans, J. S. B. T. (2004). Working memory, inhibitory control, and the development of children’s reasoning. Thinking & Reasoning, 10, 175–195.CrossRefGoogle Scholar
Hinton, E. C., Dymond, S., von Hecker, U., & Evans, C. J. (2010). Neural correlates of relational reasoning and the symbolic distance effect: Involvement of parietal cortex. Neuroscience, 168, 138–148.Google Scholar
Holyoak, K. J., & Thagard, P. (1995). Mental Leaps: Analogy in Creative Thought. Cambridge, MA: MIT Press.Google Scholar
Houdé, O. (2000). Inhibition and cognitive development: Object, number, categorization, and reasoning. Cognitive Development, 15, 63–73.Google Scholar
Houdé, O. (2019). 3-System Theory of the Cognitive Brain: A Post-Piagetian Approach. New York: Routledge.CrossRefGoogle Scholar
Houdé, O., Pineau, A., Leroux, G., Poirel, N., Perchey, G., Lanoë, C., … Mazoyer, B. (2011). Functional MRI study of Piaget’s conservation-of-number task in preschool and school-age children: A neo-Piagetian approach. Journal of Experimental Child Psychology, 110, 332–346.Google Scholar
Hubbard, E., Piazza, M., Pinel, P., & Dehaene, S. (2005). Interactions between number and space in parietal cortex. Nature Reviews Neuroscience, 6, 435–448.Google Scholar
Inglis, M., & Attridge, N. (2017). Does Mathematical Study Develop Logical Thinking?: Testing the Theory of Formal Discipline. Singapore: World Scientific.Google Scholar
Inhelder, B., & Piaget, J. (1958). The Growth of Logical Thinking from Childhood to Adolescence: An Essay on the Construction of Formal Operational Structures. New York: Basic Books.Google Scholar
Ischebeck, A., Schocke, M., & Delazer, M. (2009). The processing and representation of fractions within the brain: An fMRI investigation. Neuroimage, 47, 403–413.Google Scholar
Jacob, S. N., & Nieder, A. (2009). Notation-independent representation of fractions in the human parietal cortex. Journal of Neuroscience, 29, 4652–4657.Google Scholar
Kallio, K. D. (1988). Developmental difference in the comprehension of simple and compound comparative relations. Child Development, 59, 397–410.Google Scholar
Kuhn, D., & Crowell, A. (2011). Dialogic argumentation as a vehicle for developing young adolescents’ thinking. Psychological Science, 22, 545–552.Google Scholar
Lakatos, I. (1976). Proofs and Refutations. Cambridge: Cambridge University Press.Google Scholar
Lee, K., Ng, S. F., Bull, R., Pe, M. L., & Ho, R. (2011). Are patterns important? An investigation of the relationships between proficiencies in patterns, computation, executive functioning and algebraic word problems. Journal of Educational Psychology, 103, 269–281.Google Scholar
Lyons, I. M., Vogel, S. E., & Ansari, D. (2016). On the ordinality of numbers: A review of neural and behavioral studies. Progress in Brain Research, 227, 187–221.CrossRefGoogle Scholar
Mayer, R. E., & Wittrock, M. C. (2006). Problem solving. In Alexander, P. A., & Winne, P. H. (eds.), Handbook of Educational Psychology (2nd ed., pp. 287–304). Mahwah, NJ: Erlbaum.Google Scholar
Mercier, H., Boudry, M., Paglieri, F., & Trouche, E. (2017). Natural-born arguers: Teaching how to make the best of our reasoning abilities. Educational Psychologist, 52, 1–16.CrossRefGoogle Scholar
Mercier, H., & Sperber, D. (2011). Why do humans reason? Arguments for an argumentative theory. Behavioral and Brain Sciences, 34, 57–74.Google Scholar
Morsanyi, K., Devine, A., Nobes, A., & Szűcs, D. (2013). The link between logic, mathematics and imagination: Evidence from children with developmental dyscalculia and mathematically gifted children. Developmental Science, 16, 542–553.Google Scholar
Morsanyi, K., Kahl, T., & Rooney, R. (2017a). The link between math and logic in adolescence: The effect of argument form. In Toplak, M. E., & Weller, J. (eds.), Individual Differences in Judgment and Decision Making from a Developmental Context (pp. 166–185). Hove: Psychology Press.Google Scholar
Morsanyi, K., McCormack, T., & O’Mahony, E. (2017b). The link between deductive reasoning and mathematics. Thinking & Reasoning, 24, 1–24.Google Scholar
Morsanyi, K., & Szücs, D. (2014). The link between mathematics and logical reasoning. In Chinn, S. (ed.), The Routledge International Handbook of Dyscalculia and Mathematical Learning Difficulties (pp. 101–114). London: Routledge.Google Scholar
Moutier, S., Plagne, S., Melot, A.-M., & Houdé, O. (2006). Syllogistic reasoning and belief-bias inhibition in school children. Developmental Science, 9, 166–172.Google Scholar
Moyer, R. S., & Bayer, R. H. (1976). Mental comparison and the symbolic distance effect. Cognitive Psychology, 8, 228–246.Google Scholar
Newstead, S., Keeble, S., & Manktelow, K. (1985). Children’s performance on set inclusion and linear ordering relationships. Bulletin of the Psychonomic Society, 23, 105–108.Google Scholar
Pasnak, R., Schmerold, K. L., Robinson, M. F., Gadzichowski, K. M., Bock, A. M., O’Brien, S. E., … Gallington, D. A. (2016). Understanding number sequences leads to understanding mathematics concepts. The Journal of Educational Research, 109, 640–646.Google Scholar
Paz, Y. M. C. G., Bond, A. B., Kamil, A. C., & Balda, R. P. (2004). Pinyon jays use transitive inference to predict social dominance. Nature, 430, 778–781.Google Scholar
Peters, L., & De Smedt, B. (2017). Arithmetic in the developing brain: A review of brain imaging studies. Developmental Cognitive Neuroscience, 30, 265–279.Google Scholar
Piaget, J. (1952). The Child’s Conception of Number. London: Routledge & Kegan Paul.Google Scholar
Piaget, J., & Inhelder, B. (1967). The Child’s Conception of Space. London: Routledge & Kegan Paul.Google Scholar
Polya, G. (1945). How to Solve It. Princeton, NJ: Princeton University Press.Google Scholar
Polya, G. (1954). Mathematics and Plausible Reasoning. Princeton, NJ: Princeton University Press.Google Scholar
Potts, G. (1972). Information processing stragies used in the encoding of linear ordering. Journal of Verbal Learning and Verbal Behavior, 11, 727–740.Google Scholar
Potts, G. (1974). Storing and retrieving information about ordered relationship. Journal of Experimental Psychology: General, 103, 431–439.Google Scholar
Prado, J., Chadha, A., & Booth, J. (2011). The brain network for deductive reasoning: A quantitative meta-analysis of 28 neuroimaging studies. Journal of Cognitive Neuroscience, 23, 3483–3497.Google Scholar
Prado, J., Noveck, I. A., & Van Der Henst, J.-B. (2010). Overlapping and distinct neural representations of numbers and verbal transitive series. Cerebral Cortex, 20, 720–729.Google Scholar
Prado, J., Van der Henst, J.-B., & Noveck, I. A. (2008). Spatial associations in relational reasoning: evidence for a SNARC-like effect. Quarterly Journal of Experimental Psychology (Colchester), 61, 1143–1150.CrossRefGoogle ScholarPubMed
Primi, R., Ferrão, M. E., & Almeida, L. S. (2010). Fluid intelligence as a predictor of learning: A longitudinal multilevel approach applied to math. Learning and Individual Differences, 20, 446–451.Google Scholar
Rabinowitz, F. M., & Howe, M. L. (1994). Development of the middle concept. Journal of Experimental Child Psychology, 57, 418–449.Google Scholar
Rattermann, M. J., & Gentner, D. (1998). More evidence for a relational shift in the development of analogy: Children’s performance on a causal-mapping task. Cognitive Development, 13, 453–478.Google Scholar
Richland, L. E., Holyoak, K. J., & Stigler, J. W. (2004). Analogy use in eighth-grade mathematics classrooms. Cognition and Instruction, 22, 37–60.CrossRefGoogle Scholar
Richland, L. E., Morrison, R. G., & Holyoak, K. J. (2006). Children’s development of analogical reasoning: insights from scene analogy problems. Journal of Experimental Child Psychology, 94, 249–273.Google Scholar
Richland, L. E., & Simms, N. (2015). Analogy, higher order thinking, and education. Wiley Interdisciplinary Reviews: Cognitive Science, 6, 177–192.Google Scholar
Richland, L. E., Zur, O., & Holyoak, K. J. (2007). Mathematics. Cognitive supports for analogies in the mathematics classroom. Science, 316, 1128–1129.Google Scholar
Rittle-Johnson, B., Fyfe, E. R., Hofer, K. G., & Farran, D. C. (2016). Early math trajectories: Low-income children’s mathematics knowledge from age 4 to 11. Child Development, 88, 1727–1742.CrossRefGoogle Scholar
Rittle-Johnson, B., Zippert, E. L., & Boice, K. L. (2018). The roles of patterning and spatial skills in early mathematics development. Early Childhood Research Quarterly, 46, 166–178.Google Scholar
Russell, J., McCormack, T., Robinson, J., & Lillis, G. (1996). Logical (versus associative) performance on transitive inference tasks by children: Implications for the status of animals’ performance. The Quarterly Journal of Experimental Psychology, 49B, 231–244.Google Scholar
Schoenfeld, A. (1985). Mathematical Problem Solving. Cambridge, MA: Academic press.Google Scholar
Schwartz, F., Epinat-Duclos, J., Léone, J., Poisson, A., & Prado, J. (2018). Impaired neural processing of transitive relations in children with math learning disability. NeuroImage: Clinical, 20, 1255–1265.Google Scholar
Schwartz, F., Epinat-Duclos, J., Léone, J., Poisson, A., & Prado, J. (2020). Neural representations of transitive relations predict current and future math calculation skills in children. Neuropsychologia, 141, 107410.Google Scholar
Singer-Freeman, K. E., & Goswami, U. (2001). Does half a pizza equal half a box of chocolates? Proportional matching in an analogy task. Cognitive Development, 16, 811–829.Google Scholar
Singley, A. T. M., & Bunge, S. A. (2014). Neurodevelopment of relational reasoning: Implications for mathematical pedagogy. Trends in Neuroscience and Education, 3, 33–37.Google Scholar
Sriraman, B. (2005). Mathematical and analogical reasoning of young learners. ZDM – Mathematics Education, 37, 506–509.Google Scholar
Taub, G. E., Floyd, R. G., Keith, T. Z., & McGrew, K. S. (2008). Effects of general and broad cognitive abilities on mathematics achievement. School Psychology Quarterly, 23, 187–198.Google Scholar
Törner, G., Schoenfeld, A. H., & Reiss, K. M. (2007). Problem solving around the world: Summing up the state of the art. Zentral Blatt Für Didaktik Der Mathematik, 39, 353.Google Scholar
Tunteler, E., & Resing, W. C. (2002). Spontaneous analogical transfer in 4-year-olds: A microgenetic study. Journal of Experimental Child Psychology, 83, 149–166.Google Scholar
Vanderheyden, A. M., Broussard, C., Snyder, P., George, J., Meche Lafleur, S., & Williams, C. (2011). Measurement of kindergartners’ understanding of early mathematical concepts. School Psychology Review, 40, 296–306.Google Scholar
Vasconcelos, M. (2008). Transitive inference in non-human animals: An empirical and theoretical analysis. Behavioral Processes, 78, 313–334.Google Scholar
Vendetti, M. S., Matlen, B. J., Richland, L. E., & Bunge, S. A. (2015). Analogical reasoning in the classroom: Insights from cognitive science. Mind, Brain, and Education, 9, 100–106.Google Scholar
von Fersen, L., Wynne, C. D., Delius, J. D., & Staddon, J. (1991). Transitive inference formation in pigeons. Journal of Experimental Psychology Animal Behavior Processes, 17, 334–341.Google Scholar
Waltz, J. A., Lau, A., Grewal, S. K., & Holyoak, K. J. (2000). The role of working memory in analogical mapping. Memory & Cognition, 28, 1205–1212.Google Scholar
Wendelken, C. (2015). Meta-analysis: How does posterior parietal cortex contribute to reasoning? Frontiers in Human Neuroscience, 8, 1042.Google Scholar
Wendelken, C., Nakhabenko, D., Donohue, S., Carter, C., & Bunge, S. (2008). “Brain is to thought as stomach is to??”: Investigating the role of rostrolateral prefrontal cortex in relational reasoning. Journal of Cognitive Neuroscience, 20, 682–693.Google Scholar
Wendelken, C., O’Hare, E., Whitaker, K., Ferrer, E., & Bunge, S. (2011). Increased functional selectivity over development in rostrolateral prefrontal cortex. Journal of Neuroscience, 31, 17260–17268.Google Scholar
Wright, B. C. (2001). Reconceptualizing the transitive inference ability: A framework for existing and future research. Developmental Review, 21, 375–422.Google Scholar
Wright, B. C. (2012). The case for a dual-process theory of transitive reasoning. Developmental Review, 32, 89–124.Google Scholar
Wright, S., Matlen, B., Baym, C., Ferrer, E., & Bunge, S. (2007). Neural correlates of fluid reasoning in children and adults. Frontiers in Human Neuroscience, 1, 8.Google Scholar

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