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“Where does the square go?”: reinterpreting shapes when solving a tangram puzzle

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Abstract

Thirty-seven third graders and thirty-two first graders engaged in solving a tangram puzzle in the shape of a fox. They had five minutes to solve the puzzle, and after this time, they received guidance on the particular piece they had difficulties with. Through the lenses of navigating flexible abstraction, reinterpretation, combinations, and borrowing structure to expand upon the existing 2D shape composition and decomposition learning trajectory, we examined ways in which students’ puzzle-solving processes and their challenges related to the fox puzzle’s features. Students’ initial shape placements suggest that parts of the fox puzzle primed the use of particular pieces, which reduced the abstraction of the puzzle. The most challenging part of the puzzle for students was navigating reinterpretation to place the square and two small triangles on the fox’s head in nonstandard orientations. Even though students faced challenges at different steps, they overcame them similarly by trying new combinations and by borrowing structure. Some students did not complete the puzzle even though they used flips and turns (reinterpretation) strategically. The results suggest potential modifications of the current learning trajectory to account for differences between tangram and pattern block puzzles and differences due to tangram puzzles’ features. Because the puzzle’s features played a role in students’ challenges, future work needs to focus on the interaction between students’ puzzle-solving processes and puzzles’ features for a variety of tangram puzzles.

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References

  • Baran, B., Dogusoy, B., & Cagiltay, K. (2007). How do adults solve digital tangram problems? Analyzing cognitive strategies through eye tracking approach. In J. A. Jacko (Ed.), Human-Computer interaction. HCI intelligent multimodal interaction environments. Lecture notes in computer science: Vol. 4552 (Part III, pp. 555–563). Springer-Verlag Berlin Heidelberg. https://doi.org/10.1007/978-3-540-73110-8_60

  • Bofferding, L., & Foster, A. (2021). A theoretical analysis of tangram puzzles. In D. Olanoff, K. Johnson, & S. Spitzer (Eds.), Proceedings of the 43rd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 83–97).

  • Casey, B., Erkut, S., Ceder, I., & Young, J. (2008). Use of a storytelling context to improve girls’ and boys’ geometry skills in kindergarten. Journal of Applied Developmental Psychology, 29, 29–48. https://doi.org/10.1016/j.appdev.2007.10.005

    Article  Google Scholar 

  • Chou, C., Yang, C., & Chen, Z. (2013). Applying augmented reality to assisting children in solving tangram puzzles. In L.-H. Wong et al. (Eds.), Proceedings of the 21st International Conference on Computers in Education (pp. 139–144). Asia-Pacific Society for Computers in Education.

  • Clements, D. H. (2004). Major themes and recommendations. In D. H. Clements, J. Sarama, & A. DiBiase (Eds.), Engaging young children in mathematics: Standards for early childhood mathematics education (pp. 7–72). Lawrence Erlbaum Associates.

    Google Scholar 

  • Clements, D. H., & Sarama, J. (2000). Young children’s ideas about geometric shapes. Teaching Children Mathematics, 6(8), 482–488.

    Article  Google Scholar 

  • Clements, D. H., & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach (Studies in Mathematical Thinking and Learning Series). Routledge.

  • Clements, D. H., & Sarama, J. (2014). Learning and teaching early math: The learning trajectories approach (Studies in Mathematical Thinking and Learning Series, 2nd Edition). Routledge.

  • Clements, D. H., & Sarama, J. (2017/2019). Learning and teaching with learning trajectories [LT]2. Retrieved November 2020, from Marsico Institute, Morgridge College of Education, University of Denver. https://learningtrajectories.org/

  • Clements, D. H., Wilson, D. C., & Sarama, J. (2004). Young children’s composition of geometric figures: A learning trajectory. Mathematical Thinking and Learning, 6(2), 163–184. https://doi.org/10.1207/s15327833mtl0602_5

    Article  Google Scholar 

  • Clements, D. H., Sarama, J., Baroody, A. J., Joswick, C., & Wolfe, C. B. (2019). Evaluating the efficacy of a learning trajectory for early shape composition. American Educational Research Journal, 56(6), 2509–2530. https://doi.org/10.3102/2F0002831219842788

  • Evans, M. A., Feenstra, E., Ryon, E., & McNeill, D. (2011). A multimodal approach to coding discourse: Collaboration, distributed cognition, and geometric reasoning. Computer-Supported Collaborative Learning, 6, 253–278. https://doi.org/10.1007/s11412-011-9113-0

    Article  Google Scholar 

  • Hallowell, D. A., Okamoto, Y., Romo, L. F., & La Joy, J. R. (2015). First-graders’ spatial-mathematical reasoning about plane and solid shapes and their representations. ZDM-Mathematics Education 47, 363–375. https://doi.org/10.1007/s11858-015-0664-9

    Article  Google Scholar 

  • Lee, J., Lee, J. O., & Collins, D. (2009). Enhancing children’s spatial sense using tangrams. Childhood Education, 86(2), 92–94. https://doi.org/10.1080/00094056.2010.10523120

    Article  Google Scholar 

  • Maheux, J., & Proulx, J. (2015). Doing|mathematics: Analysing data with/in an enactivist-inspired approach. ZDM-Mathematics Education, 47(2), 211–221. https://doi.org/10.1007/s11858-014-0642-7

    Article  Google Scholar 

  • Martin, L., & Schwartz, D. L. (2014). A pragmatic perspective on visual representation and creative thinking. Visual Studies, 29(1), 80–93. https://doi.org/10.1080/1472586X.2014.862997

    Article  Google Scholar 

  • National Council of Teachers of Mathematics (NCTM), Commission on Standards for School Mathematics. (1989). Curriculum and evaluation standards for school mathematics. The Council.

  • National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Retrieved November 2020, from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

  • Newcombe, N., & Shipley, T. F. (2015). Thinking about spatial thinking: New topology, new assessments. In J. S. Gero (Ed.), Studying visual and spatial reasoning (pp. 179–192). Springer. https://doi.org/10.1007/978-94-017-9297-4_10

  • Nurnberger-Haag, J. (2017). A cautionary tale: How children’s books (mis)teach shapes. Early Education and Development, 28(4), 415–440. https://doi.org/10.1080/10409289.2016.1242993

    Article  Google Scholar 

  • Parks, A. N., & Blom, D. C. (2013). Helping young children see math in play. Teaching Children Mathematics, 20(5), 310–317.

    Article  Google Scholar 

  • Riddell, C. (2016). Rocket ship exploration. Teaching Children Mathematics, 23(2), 64–66.

    Article  Google Scholar 

  • Sarama, J., & Clements, D. (2009). Early childhood mathematics education research. Routledge. https://doi.org/10.4324/9780203883785

    Article  Google Scholar 

  • Schroth, S., Tang, H., Carr-chellman, A., & AlQahtani, M. (2019). An exploratory study of Osmo Tangram and tangram manipulative in an elementary mathematics classroom. Journal of Educational Technology Development and Exchange, 11(1). Retrieved July 2, 2021, from https://aquila.usm.edu/jetde/vol11/iss1/1

  • Siew, N. M., Chong, C. L., & Abdullah, M. R. (2013). Facilitating students’ geometric thinking through van Hiele’s phase-based learning using tangram. Journal of Social Sciences, 9(3), 101–111. https://doi.org/10.3844/jssp.2013.101.111

    Article  Google Scholar 

  • Tchoshanov, M. (2011). Building students’ mathematical proficiency: Connecting mathematical ideas using the tangram. Learning and Teaching Mathematics, 10, 16–23.

    Google Scholar 

  • Uttal, D. H., Meadow, N. G., Tipton, E., Hand, L. L., Alden, A. R., Warren, C., & Newcombe, N. S. (2013). The malleability of spatial skills: A meta-analysis of training studies. Psychological Bulletin, 139(2), 352–402. https://psycnet.apa.org/10.1037/a0028446

  • van den Heuvel-Panhuizen, M., & Buys, K. (Eds.) (2008). Young children learn measurement and geometry: A learning-teaching trajectory with intermediate attainment targets for the lower grades in primary school (Dutch Design in Mathematics Education, Vol. 2). Sense Publishers.

  • Van Hiele, P. M. (1986). Structure and insight. Academic Press.

    Google Scholar 

  • Van Hiele, P. M. (1999). Developing geometric thinking through activities that begin with play. Teaching Children Mathematics, 5(6), 310–317.

    Article  Google Scholar 

Download references

Funding

This research was supported by a National Science Foundation DRL ITEST grant #1759254.

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Authors and Affiliations

  1. Department of Curriculum and Instruction, Purdue University, 100 N University Street, West Lafayette, IN, USA

    Laura Bofferding

  2. Rice University School Mathematics Project, Rice University, 6100 Main Street, Houston, TX, USA

    Mahtob Aqazade

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  1. Laura Bofferding
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  2. Mahtob Aqazade
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Laura Bofferding designed the study and analyzed and wrote the manuscript; Mahtob Aqazade analyzed and wrote the manuscript.

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Correspondence to Laura Bofferding.

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This research was approved by Purdue University’s Institutional Review Board.

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The authors declare no competing interests.

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Bofferding, L., Aqazade, M. “Where does the square go?”: reinterpreting shapes when solving a tangram puzzle. Educ Stud Math 112, 25–47 (2023). https://doi.org/10.1007/s10649-022-10166-0

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