Skip to main content
SpringerLink
Log in

Playing on patterns: is it a case of analogical transfer?

  • Original Paper
  • Published:
ZDM – Mathematics Education Aims and scope Submit manuscript

Abstract

While patterning was commonly seen as evidence of mathematical thinking, interdisciplinary interest has recently increased due to pattern-recognition applications in artificial intelligence. Within two empirical studies, we analyze the analogical-transfer capability of primary school students when completing three types of bi-dimensional patterns, namely, numerical, discrete geometric, and continuous geometric. We found that the mechanisms involved in analogical transfer for continuing sequential patterns are based on two complementary cognitive processes: decoding and adapting. In addition, at a basic level of processing, students activate one of the operational tools of shape recognition or counting, and based on it, they find a surface analogy that leads them to use isometric transformations or one-dimensional development for continuing the given pattern. At a more complex level of processing, students activate both shape recognition and counting and are thus able to apply a filter of processing that uncovers a deep-structure analogy, which allows cognitive framing of the problem and leads to coherent 2D developments within the understood conceptual frame. At a more advanced level of processing, students can use a refined filter not only to uncover a deep-structure analogy but also to use an external language to verbalize that analogy, and consequently, to find 2D developments that trigger changes in cognitive framing, showing that pattern generation is a creative activity. Teaching and learning implications are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig.5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  • Baer, J. (2015). The importance of domain-specific expertise in creativity. Roeper Review, 37(3), 165–178.

    Article  Google Scholar 

  • Barbosa, A., & Vale, I. (2015). Visualization in pattern generalization: potential and challenges. Journal of the European Teacher Education Network, 10, 57–70.

    Google Scholar 

  • Bartha, P. (2013). Analogy and analogical reasoning. In E. N. Zalta (Ed.), The Stanford encyclopedia. of philosophy (Fall 2013 ed.)

  • Bassok, M. (2003). Analogical transfer in problem solving. The Psychology of Problem Solving, 343–369

  • Bates, K. E., Gilligan-Lee, K., & Farran, E. K. (2021). Reimagining mathematics: the role of mental imagery in explaining mathematical calculation skills in childhood. Mind, Brain, and Education, 15(2), 189–198.

    Article  Google Scholar 

  • Baylor, G. W., & Lemoyne, G. (1975). Experiments in seriation with children: towards an information processing explanation of the horizontal decalage. Canadian Journal of Behavioral Science, 7(1), 4–29.

    Article  Google Scholar 

  • Chomsky, N. (1980). Rules and representations. Columbia U.P.

    Book  Google Scholar 

  • Conway, K. D. (1999). Assessing open-ended problems. Math Teaching in the Middle School, 4(8), 510–514.

    Article  Google Scholar 

  • Copi, I. M., & Cohen, C. (2005). Introduction to logic. Prentice-Hall.

    Google Scholar 

  • Csikszentmihalyi, M. (1999). Creativity. In R. A. Wilson & F. C. Keil (Eds.), The MIT encyclopedia of the cognitive sciences. MIT Press.

    Google Scholar 

  • Fischer, K. W., & Bidell, T. R. (1998). Dynamic development of psychological structures in action and thought. In R. M. Lerner (Ed.), Handbook of child psychology (pp. 467–561). Wiley.

    Google Scholar 

  • Gentner, D. (1983). Structure-mapping: a theoretical framework for analogy. Cognitive Science, 7, 155–170.

    Article  Google Scholar 

  • Gick, M. L., & Holyoak, K. J. (1980). Analogical problem solving. Cognitive Psychology, 12, 306–355.

    Article  Google Scholar 

  • Gick, M. L., & Holyoak, K. J. (1983). Schema induction and analogical transfer. Cognitive Psychology, 15(1), 1–38.

    Article  Google Scholar 

  • Goncalo, J. A., Vincent, L., & Audia, P. G. (2010). Early creativity as a constraint on future achievement. In D. Cropley, J. Kaufman, A. Cropley, & M. Runco (Eds.), The dark side of creativity (pp. 14–133). New York: Cambridge University Press.

    Google Scholar 

  • Gray, M. E., & Holyoak, K. J. (2021). Teaching by analogy: from theory to practice. Mind, Brain, and Education, 15(3), 250–263.

    Article  Google Scholar 

  • Gust, H., Krumnack, U., Kühnberger, K. U., & Schwering, A. (2008). Analogical reasoning: a core of cognition. Künstliche Intelligenz, 22(1), 8–12.

    Google Scholar 

  • Hauser, M. D., Chomsky, N., & Fitch, W. T. (2002). The faculty of language: what is it, who has it, and how did it evolve? Science, 298, 1569–1579.

    Article  Google Scholar 

  • Haverty, L. A., Koedinger, K. R., Klahr, D., & Alibali, M. W. (2000). Solving inductive reasoning problems in mathematics: not-so-trivial pursuit. Cognitive Science, 24(2), 249–298.

    Article  Google Scholar 

  • Hoth, J., Kaiser, G., Busse, A., Doehrmann, M., Koenig, J., & Blömeke, S. (2017). Professional competences of teachers for fostering creativity and supporting high-achieving students. ZDM Mathematics Education, 49(1), 107–120.

    Article  Google Scholar 

  • Kaniel, S. (2013). When creativity met transfer: increasing creativity and transfer by controlling the styles of processing. Gifted Education International, 29(1), 13–27.

    Article  Google Scholar 

  • Lakoff, G. (2010). Why it matters how we frame the environment. Environmental Communication, 4(1), 70–81.

    Article  Google Scholar 

  • Lassig, C. J. (2013). Approaches to creativity: how adolescents engage in the creative process. Thinking Skills and Creativity, 10, 3–12.

    Article  Google Scholar 

  • Maguire, M., & Delahunt, B. (2017). Doing a thematic analysis: a practical, step-by-step guide for learning and teaching scholars. All Ireland Journal of Higher Education, 9(3).

  • Mattson, M. P. (2014). Superior pattern processing is the essence of the evolved human brain. Frontiers in Neuroscience, 8, 265. https://doi.org/10.3389/fnins.2014.00265

    Article  Google Scholar 

  • Piaget, J. (1957). Logic and psychology. Basic Books.

    Google Scholar 

  • Rivera, F. D. (2013). Teaching and learning patterns in school mathematics: psychological and pedagogical considerations. Springer.

    Book  Google Scholar 

  • Rivera, F. D. (2018). Pattern generalization processing of elementary students: cognitive factors affecting the development of exact mathematical structures. Eurasia Journal of Mathematics, Science and Technology Education, 14(9), em158.

    Article  Google Scholar 

  • Rivera, F. D., & Becker, J. R. (2008). Middle school children’s cognitive perceptions of constructive and deconstructive generalizations involving linear figural patterns. ZDM Mathematics Education, 40, 65–82.

    Article  Google Scholar 

  • Simms, N. K., & Richland, L. E. (2019). Generating relations elicits a relational mindset in children. Cognitive Science, 43(10).

  • Singer, F. M. (2001). Structuring the information—a new way of perceiving the content of learning. ZDM Mathematics Education, 6, 204–217.

    Article  Google Scholar 

  • Singer, F. M. (2003). From cognitive science to school practice: building the ridge. PME, 4, 207–214.

    Google Scholar 

  • Singer, F. M. (2007a). Beyond conceptual change: Using representations to integrate domain-specific structural models in learning mathematics. Mind, Brain, and Education, 1(2), 84–97.

    Article  Google Scholar 

  • Singer, F. M. (2007b). Modelling both complexity and abstraction: a paradox? In W. Blum, P. Galbraith, H. W. Henn, & N. Mogens (Eds.), Applications and modelling in mathematics education (pp. 233–240). Springer.

    Chapter  Google Scholar 

  • Singer, F. M. (2009). The dynamic infrastructure of mind—a hypothesis and some of its applications. New Ideas in Psychology, 27(1), 48–74.

    Article  Google Scholar 

  • Singer, F. M., & Voica, C. (2017). When mathematics meets real objects: How does creativity interact with expertise in problem solving and posing? In Creativity and Giftedness (pp. 75–103). Springer

  • Singer, F. M., & Voica, C. (2008a). Extrapolating rules: How do children develop sequences? In O. Figueras, J. L. Cortina, S. Alatorr & A. Mepúlveda (Eds.), Proceedings of PME 32 (Vol. 4, pp. 256–263).

  • Singer, F. M., & Voica, C. (2008b). Between perception and intuition: learning about infinity. The Journal of Mathematical Behavior, 27(3), 188–205.

    Article  Google Scholar 

  • Singer, F. M., & Voica, C. (2013). A problem-solving conceptual framework and its implications in designing problem-posing tasks. Educational Studies in Mathematics, 83(1), 9–26.

    Article  Google Scholar 

  • Singer, F. M., & Voica, C. (2015). Is problem posing a tool for identifying and developing mathematical creativity? In F. M. Singer, N. Ellerton, & J. Kai (Eds.), Mathematics problem posing (pp. 141–174). Springer.

    Chapter  Google Scholar 

  • Terwel, J., van Oers, B., van Dijk, I., & van den Eeden, P. (2009). Are representations to be provided or generated in primary mathematics education? Effects on transfer. Educational Research and Evaluation, 15(1), 25–44.

    Article  Google Scholar 

  • Torrance, E. P. (1965). Rewarding creative behavior. Prentice-Hall.

    Google Scholar 

  • Vale, I., Pimentel, T., Cabrita, I., Barbosa, A., & Fonseca, L. (2012). Pattern problem solving tasks as a mean to foster creativity in mathematics. Proceedings of PME 36, 4, 171–178.

    Google Scholar 

  • Voica, C., & Singer, F. M. (2012). Creative contexts as ways to strengthen mathematics learning. Procedia SBS, 33, 538–542.

    Google Scholar 

  • Voica, C., & Singer, F. M. (2013). Problem modification as a tool for detecting cognitive flexibility in school children. ZDM Mathematics Education, 45(2), 267–279.

    Article  Google Scholar 

  • Voica, C., Singer, F. M., & Stan, E. (2020). How are motivation and self-efficacy interacting in problem-solving and problem-posing? Educational Studies in Mathematics, 105(3), 487–517.

    Article  Google Scholar 

Download references

Acknowledgements

We thank the Editor-in-Chief and three anonymous reviewers for their suggestions.

Author information

Authors and Affiliations

  1. UPG University of Ploiesti, Ploiești, Romania

    Florence Mihaela Singer

  2. University of Bucharest, Bucharest, Romania

    Cristian Voica

Authors
  1. Florence Mihaela Singer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cristian Voica
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Florence Mihaela Singer.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix 1. Examples of items in performance tests addressed to the first sample

figure a

Appendix 2. The patterns proposed to students from the second sample

figure b

Appendix 3. Two performance tests filled in by students in grade 2, from the first sample

figure dfigure d

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Singer, F.M., Voica, C. Playing on patterns: is it a case of analogical transfer?. ZDM Mathematics Education 54, 211–229 (2022). https://doi.org/10.1007/s11858-022-01334-w

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11858-022-01334-w

Keywords

Search

Navigation