Geometric Transformations and Symmetry in Primary and Secondary Education: A Review of Themes, Media, and Theoretical Frames from 1990 to Present
DOI:
https://doi.org/10.46328/ijemst.5315Keywords:
Symmetry, Geometric Transformations, Systematic Literature ReviewAbstract
There is need for increased focus on geometric transformations and symmetry in primary and secondary education, as well as increased research to support learning and teaching of geometric transformations. To identify directions for future research and teaching, we set out to map the research that has already been conducted and to identify key areas of focus and opportunity going forward. Toward these goals, this systematic review examines 62 peer-reviewed articles on teaching and learning about 2D geometric transformations and symmetry since 1990. To guide our review, we use the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) protocol. The review explores the research in terms of: (a) how students learn about transformations, (b) how teaching about transformations has been conceptualized, (c) how media have been leveraged to support learning about transformations, and (d) which theoretical frames have been leveraged and how have those frames shifted over time. Discussion and conclusions consider key areas of growth for the field going forward to better support teachers and students learning about symmetry and transformations.
References
Arnon, I., Cottrill, J., Dubinsky, E., Oktac, A., Roa Fuentes, S., & Trigeros, M. W. (2014). APOS theory: A framework for research and curriculum development in Mathematics Education. Springer. http://doi.org/10.1007/978-1-4614-7966-6
Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practises in Cabri environments. Zentralblatt für Didaktik der Mathematik, 34(3), 66-72.
Atiyah, M. (1982). What Is Geometry? The 1982 Presidential Address. The Mathematical Gazette, 66(437), 179–184. https://doi.org/10.2307/3616542
Atiyah, M. (2003). What is geometry. The changing shape of geometry. Celebrating a century of geometry and geometry teaching, 24-29.
*Avcu, S., & Çetinkaya, B. (2019). An instructional unit for prospective teachers’ conceptualization of geometric transformations as functions. International Journal of Mathematical Education in Science and Technology, 52(5), 669-698. https://doi.org/10.1080/0020739X.2019.1699966
Balacheff, N. (2013). cK¢, A model to reason on learners’ conceptions. In M. Martinez, & A. Castro, Superfine (Eds.), Proceedings of the 35th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. University of Illinois.
*Bansilal, S., & Naidoo, J. (2012). Learners engaging with transformation geometry. South African Journal of Education, 32(1), 26–39. https://doi.org/10.15700/saje.v32n1a452
Bartolini Bussi, M., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective. In L. English, M. Bartolini Bussi, G. Jones, R. Lesh, B. Sriraman, & D. Tirosh (Eds.), Handbook of international research in mathematics education. Routledge.
*Bartolini Bussi, M., & Maschietto, M. (2008). Machines as tools in teacher education. In A. N. Other, & B. N. Other (Eds.), The International handbook of mathematics teacher education: Volume 2 (pp. 183–208). Brill/Sense. https://doi.org/10.1163/9789087905460_010
Barton, B. (1996). Making sense of ethnomathematics: Ethnomathematics is making sense. Educational Studies in Mathematics, 31(1), 201–233.
Battista, M. (2011). Conceptualizations and issues related to learning progressions, learning trajectories, and levels of sophistication. The Mathematics Enthusiast, 8(3), 507-570.
Brady, C. & Lehrer, R. (2021). Sweeping area across physical and virtual environments. Digital Experiences in Mathematics Education, 7, 66–98. https://doi.org/10.1007/s40751-020-00076-2
*Breive, S. (2022). Abstraction and embodiment: Exploring the process of grasping a general. Educational Studies in Mathematics, 110, 313–329. https://doi.org/10.1007/s10649-021-10137-x
*Brijlall, D. (2017). Using questionnaires in the learning of congruency of triangles to incite formal and informal reasoning. International Journal of Educational Sciences. 17(1–3), 224 – 237. https://doi.org/10.1080/09751122.2017.1305732
Brown, J. C. (2017). A metasynthesis of the complementarity of culturally responsive and inquiry‐based science education in K‐12 settings: Implications for advancing equitable science teaching and learning. Journal of Research in Science Teaching, 54(9), 1143-1173. https://doi.org/10.1002/tea.21401
Bruce, C. D., Davis, B., Sinclair, N., McGarvey, L., Hallowell, D., Drefs, M., Francis, K., Hawes, Z., Moss, J., Mulligan, J., Okamoto, Y., Whiteley, W., & Woolcott, G. (2017). Understanding gaps in research networks: Using “spatial reasoning” as a window into the importance of networked educational research. Educational Studies in Mathematics, 95, 143-161. https://doi.org/10.1007/s10649-016-9743-2
Clements, D., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6(2), 81–89. https://doi.org/10.1207/s15327833mtl0602_1
Crosswhite, F. J., Dossey, J. A., & Frye, S. M. (1989). NCTM standards for school Mathematics: visions for implementation. Journal for Research in Mathematics Education, 20(5), 513–522. https://doi.org/10.2307/749425
*Cuturi, L. F., Cooney, S., Cappagli, G., Newell, F. N., & Gori, M. (2023). Primary Schoolers' Response to a Multisensory Serious Game on Cartesian Plane Coordinates in Immersive Virtual Reality.
Cyberpsychology, Behavior, and Social Networking, 26(8), 648-656
Davis, B., & Francis, K. (2023). Discourses on learning in education. Retrieved July 15, 2023 from https://learningdiscourses.com/
Davis, B., Sumara, D., & Luce-Kapler, R. (2015). Engaging minds: Cultures of education and practices of teaching. Taylor & Francis. https://doi.org/10.4324/9781315695891
*DeJarnette, A., González, G., Deal, J. T., & Rosado Lausell, S. L. (2016). Students’ conceptions of reflection: Opportunities for making connections with perpendicular bisector. The Journal of Mathematical Behavior, 43, 35–52. http://dx.doi.org/10.1016/j.jmathb.2016.05.001
*Dello Iacono U. & Ferrara Dentice E. (2022). Mathematical walks in search of symmetries: From visualization to conceptualization. International Journal of Mathematical Education in Science and Technology, 53 (7), 1875–1893. http://doi.org/10.1080/0020739X.2020.1850897
diSessa, A. (2000). Changing minds: Computers, learning, and literacy. MIT Press. https://doi.org/10.7551/mitpress/1786.001.0001
Driscoll, M. (2007). Fostering geometric thinking: A guide for teachers, grades 5-10. Portsmouth, NH: Heinemann.
Duval, R. (2017). Understanding the mathematical way of thinking – The registers of semiotic representations. Springer International Publishing. https://doi.org/10.1007/978-3-319-56910-9
*Edwards, L. (1991). Children’s learning in a computer microworld for transformation geometry. Journal for Research in Mathematics Education, 22(2), 122–137. https://doi.org/10.2307/749589
*Edwards, L. (1997). Exploring the territory before proof: Students’ generalizations in a computer microworld for transformation geometry. International Journal of Computers for Mathematical Learning, 2, 187–215. https://doi.org/10.1023/A:1009711521492
*Edwards, L. (2003). The nature of mathematics as viewed from cognitive science. Europena Research in Mathematics Education III, (pp. 1–10).
*Edwards, L. (2009). Transformation geometry from an embodied perpective. In W. M. Roth (Ed.), Mathematical Representation At The Interface of Body and Culture (pp. 27–44). Information Age Publishing.
*Edwards, L., & Zazkis, R. (1993). Transformation geometry: Naive ideas and formal embodiments. Journal of Computers in Mathematics and Science Teaching, 12(2), 121–145.
Eglash, R. (1999). African fractals: Modern computing and indigenous design . Rutgers University Press.
Eglash, R., Bennet, A., O’Donnell, C., Jennings, S., & Cintorino, M. (2006). Culturally situated design tools: Ethnocomputing from field site to classroom. Anthropology and Education, 347–362. https://doi.org/10.1525/aa.2006.108.2.347
Eko, Y. S. (2017). The existence of ethnomathematics in buna woven fabric and its relation to school mathematics. In International conference on mathematics and science education. Universitas Pendidikan Indonesia.
*Faggiano, E., Montone, A., & Mariotti, M. A. (2018). Synergy between manipulative and digital artefacts: A teaching experiment on axial symmetry at primary school. International Journal of Mathematical Education in Science and Technology, 49(8), 1165–1180. https://doi.org/10.1080/0020739X.2018.1449908
*Fan, L., Qi, C., Liu, X., Wang, Y., & Lin, M. (2017). Does a transformation approach improve students’ ability in constructing auxiliary lines for solving geometric problems?: An intervention-based study with two Chinese classrooms. Educational Studies in Mathematics, 96(2), 229–248. https://doi.org/10.1007/s10649-017-9772-5
*Febrian, F., & Astuti, P. (2018). The RME principles on geometry learning with focus of transformation reasoning through exploration on Malay woven motif. Journal of Turkish Science Education, 15, 31–41.
*Ferrarello, D., Mammana, M. F., & Pennisi, M. (2014). Teaching/learning geometric transformations in high-school with DGS. International Journal for Technology in Mathematics Education, 21(1), 11–17.
*Fife, J. H., James, K., & Bauer, M. (2019). A learning progression for geometric transformations (Research report No. RR-19-01). Educational Testing Service. https://doi.org/10.1002/ets2.12236
Freudenthal, H. (1971). Geometry between the devil and the deep sea. Educational Studies in Mathematics, 3(3), 413–435. https://doi.org/10.1007/BF00302305
*Gadanidis, G., Clements, E., & Yiu, C. (2018). Group theory, computational thinking, and young mathematicians. Mathematical Thinking and Learning, 20(1), 32–53. https://doi.org/10.1080/10986065.2018.1403542
*García-Lázaro, D., & Martín-Nieto, R. (2023). Mathematical and digital competence of future teachers using GeoGebra application. Alteridad [online] 18 (1), 85-98. https://doi.org/10.17163/alt.v18n1.2023.07
Goldenberg, E. P., Cuoco, A. A., & Mark, J. (1998). A role for geometry in ageneral education. In R. Lehrer & D. Chain, (Eds.). Designing learning environments for developing understanding of geometry and space (pp. 3-44). Mahwah, NJ: Routledge.
Gravemeijer, K., & Stephan, M. (2002). Emergent models as an instructional design heuristic. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing,modeling and tool use in mathematics education. Mathematics Education Library (Vol. 30, pp. 145–169). Springer. https://doi.org/10.1007/978-94-017-3194-2_10
*Gülkılık, H., Uğurlub, H. H., & Yürük, N. (2015). Examining students’ mathematical understanding of geometric transformations using the Pirie-Kieren model. Educational Sciences: Theory & Practice, 15(6), 1531–1548. https://doi.org/10.12738/estp.2015.6.0056
Gutiérrez, A., & Boero. P. (Eds.). (2006). Handbook of research on the psychology of mathematics education: Past, present, and future. Sense Publishers.
*Gutiérrez, A., Jaime, A., & Gutiérrez, P. (2021). Networked analysis of a teaching unit for primary school symmetries in the form of an e-book. Mathematics, 9(8), 832. https://doi.org/10.3390/math9080832
Gutiérrez, Á., Leder, G., & Boero, P. (Eds.). (2016). The second handbook of research on the psychology of mathematics education. Sense Publishers. https://doi.org/10.1007/978-94-6300-561-6
*Guven, B. (2012). Using dynamic geometry software to improve eight grade students’ understanding of transformation geometry. Australasian Journal of Educational Technology, 28(2), 364–382. https://doi.org/10.14742/ajet.878
*Hāwera, N., & Taylor, M. (2014). Researcher–teacher collaboration in Māori-medium education: Aspects of learning for a teacher and researchers in Aotearoa New Zealand when teaching mathematics. AlterNative: An International Journal of Indigenous Peoples, 10(2), 151–164. https://doi.org/10.1177/117718011401000205
*Hegg, M., Papadopoulos, D., Katz, B., & Fukawa-Connelly, T. (2018). Preservice teacher proficiency with transformations-based congruence proofs after a college proof-based geometry class. Journal of Mathematical Behavior, 51, 56–70. https://doi.org/10.1016/j.jmathb.2018.07.002
*Hernández-Zavaleta, J. E., Brady, C., Becker, S., & Clark, D. B. (2023). Vygotskian hybridizing of motion and mapping: Learning about geometric transformations in block-based programming environments. Mathematical Thinking and Learning, 1-35. https://doi.org/10.1080/10986065.2023.2191074
*Hershkovitz, A., Noster, N., Siller, H. S., & Tabach, M. (2024). Learning analytics in mathematics education: the case of feedback use in a digital classification task on reflective symmetry. ZDM–Mathematics Education, 1-13
Hickmott, D., Prieto-Rodriguez, E., & Holmes, K. (2018). A scoping review of studies on computational thinking in K–12 mathematics classrooms. Digital Experiences in Mathematics Education, 4(1), 48–69. https://doi.org/10.1007/s40751-017-0038-8
*Hollebrands, K. (2003). High school students’ understandings of geometric transformations in the context of a technological environment. Journal of Mathematical Behavior, 22(1), 55–72. https://doi.org/10.1016/S0732-3123(03)00004-X
*Hollebrands, K. (2007). The role of a dynamic software program for geometry in the strategies high school mathematics students employ. Journal for Research in Mathematics Education, 38(2), 164–192. https://www.jstor.org/stable/30034955
*Hoyles, C., & Healy, L. (1997). Unfolding meanings for reflective symmetry. International Journal of Computers for Mathematical Learning, 2(1), 27–59. https://doi.org/10.1023/A:1009721414546
Hoyles, C., & Noss, R. (2020, June 19). Mapping a way forward for computing and mathematics: Reflections on the UCL ScratchMaths Project. In C. Bateau, & G. Gadanidis (Eds.), Online seminar series on programming in mathematics education. Mathematics Knowledge Network. http://mkn-rcm.ca/online-seminar-series-on-programming-in-mathematics-education/
*Jacobson, C., & Lehrer, R. (2000). Teacher appropriation and student learning of geometry through design. Journal for Research in Mathematics Education, 31(1), 71–88. https://doi.org/10.2307/749820
*Johnson-Gentile, K., Clements, D. H., & Battista, M. (1994). Effects of computer and noncomputer environments on students’ conceptualizations of geometric motions. Journal of Educational Computing Research, 11(2), 127–140. https://doi.org/10.2190/49EE-8PXL-YY8C-A923
Kafai, Y., & Resnick, M. (1996). Constructionism in practice: Designing, thinking, and learning in a digital world. Lawrence Erlbaum Associates.
*Kalinec-Craig, C., Prasad, P. V. & Luna, C. (2019) Geometric transformations and Talavera tiles: A culturally responsive approach to teacher professional development and mathematics teaching. Journal of Mathematics and the Arts, 13(1-2), 72–90, https://doi.org/10.1080/17513472.2018.1504491
Kaput Center for Research & Innovation in STEM Education. (2009). SimCalc projects. http://kaputcenter.org/projects/present-work/simcalc-projects/
Koile, K., & Rubin, A. (2015). Ink-12 teaching and learning using interactive ink inscriptions in K-12. ink-12. https://ink-12.terc.edu/index.php
Kuzniak, A. (2018). Thinking about the teaching of geometry through the lens of the theory of geometric working spaces. In P. Herbst, U. Cheah, P. Richard, & K. Jones (Eds.), International perspectives on the teaching and learning of geometry in secondary schools. ICME-13 Monographs (pp. 5–21). Springer. https://doi.org/10.1007/978-3-319-77476-3_2
*Lai, Y., Lischka, A. E., Strayer, J. F., & Adamoah, K. (2023). Characterizing prospective secondary teachers’ foundation and contingency knowledge for definitions of transformations. The Journal of Mathematical Behavior, 70, 101030
Law, C.-K. (1991). A genetic decomposition of geometric transformations (Unpublished doctoral dissertation). Purdue University.
Lehrer, R., Jacobson, C., Thoyre, G., Kemeny, V., Strom, D., Horvath, J.,
Gance, S., & Koehler, M. (1998). Developing understanding of geometry and space in primary grades. In R. Lehrer, & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 138-169). Lawrence Erlbaum Associates.
*Leikin, R., Berman, A., & Zaslavsky, O. (2000). Learning through teaching: The case of symmetry. Mathematics Education Research Journal, 12, 18–36. https://doi.org/10.1007/BF03217072
Li, Y., Schoenfeld, A., diSessa, A., Graesser, A., Benson L., English, L., & Duschl, R. (2020). On computational thinking and STEM education. Journal for STEM Education Research, 3, 147–166. https://doi.org/10.1007/s41979-020-00044-w
*Mainali, B. R., & Heck, A. (2017). Comparison of traditional instruction on reflection and rotation in a Nepalese high school with an ICT-rich, student-centered, investigative approach. International Journal of Science and Math Education, 15(3), 487–507. https://doi.org/10.1007/s10763-015-9701-y
Mariotti, M.A. & Montone, A. (2021). The potential synergy of digital and manipulative artefacts. Digital Experiences in Mathematics Education, 6, 109–122. https://doi.org/10.1007/s40751-020-00064-6
*Mbusi, N., & Luneta, K. (2023). Implementation of an intervention program to enhance student teachers’ active learning in transformation geometry. SAGE open, 13(2), 21582440231179440.
*McBroom, E., Sorto, A., Jiang, Z., White, A., & Dickey, E. (2016). Dynamic approach to teaching geometry: A study of teachers’ TPACK development. In M. Niess, S. Driskell, & K. Hollebrands (Eds.), Handbook of research on transforming mathematics teacher education in the digital age (pp. 519–550). IGI Global. https://doi.org/10.4018/978-1-5225-0120-6.ch020
*Mhlolo, K. M., & Schäfer, M. (2014). Potential gaps during the transition from the embodied through symbolic to formal worlds of reflective symmetry. African Journal of Research in Mathematics, Science and Technology Education, 18(2), 125–138. https://doi.org/10.1080/10288457.2014.925269
Miles, M. B., Huberman, M. A., & Saldaña, J. (2020). Qualitative data analysis: A methods sourcebook (4th ed.). SAGE.
Mishra, P., & Koehler, M. (2006). Technological pedagogical content knowledge: A framework for integrating technology in teacher knowledge. Teachers College Record, 108(6), 1017–1054. https://doi.org/10.1111/j.1467-9620.2006.00684.x
Moher, D., Shamseer, L., Clarke, M., Ghersi, D., Liberati, A., Petticrew, M., Shekelle, P., & Stewart, L. A., & members of the PRISMA Group. (2015). Preferred reporting items for systematic review and meta-analysis protocols (PRISMA-P) 2015 statement. Systematic Reviews, 4(1), 1–9. https://doi.org/10.1186/2046
Moyer, J. C. (1978). The relationship between the mathematical structure of Euclidean transformations and the spontaneously developed cognitive structures of young children. Journal for Research in Mathematics Education, 9(2), 83–92. https://doi.org/10.2307/748871
Mullis, I. V., Martin, M. O., & von Davier, M. (2021). TIMSS 2023 Assessment Frameworks. International Association for the Evaluation of Educational Achievement.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. The National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics. (2014). Principles to actions:Ensuring mathematical success for all. The National Council of Teachers of Mathematics.
Nemirovsky, R. & Sinclair, N. (2020). On the intertwined contributions of physical and digital tools for the teaching and learning of mathematics. Digital Experiences in Mathematics Education, 6, 107–108. https://doi.org/10.1007/s40751-020-00075-3
*Ng, O. L., Leung, A., & Ye, H. (2023). Exploring computational thinking as a boundary object between mathematics and computer programming for STEM teaching and learning. ZDM–Mathematics Education, 55(7), 1315-1329
*Ng, O.‐L., & Sinclair, N. (2015). Young children reasoning about symmetry in a dynamic geometry environment. ZDM Mathematics Education, 47(3), 421–434. https://doi.org/10.1007/s11858-014-0660-5
OECD. (2018). PISA 2021 mathematics framework (Second draft). Retrieved June 14, 2024, https://www.oecd.org/pisa/sitedocument/PISA-2021-mathematics-framework.pdf
Page, M.J., McKenzie, J.E., Bossuyt, P.M., Boutron, I., Hoffmann, T.C., Mulrow, C.D., et al. (2021). The PRISMA 2020 statement: An updated guideline for reporting systematic reviews. PLoS Medicine, 18(3): e1003583. https://doi.org/10.1371/journal.pmed.1003583
*Panorkou, N., & Maloney, A. (2015). Elementary students’ construction of geometric transformation reasoning in a dynamic animation environment. Constructivist Foundations, 10(3), 338–347.
Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. Basic Books, Inc.
Papert, S. (1987). Information technology and education: Computer criticism vs. technocentric thinking. Educational Researcher, 16(1), 22–30. https://doi.org/10.3102/0013189X016001022
*Pepin, B., Gueudet, G. & Trouche, L. (2013). Investigating textbooks as crucial interfaces between culture, policy and teacher curricular practice: two contrasted case studies in France and Norway. ZDM Mathematics Education 45, 685–698. https://doi.org/10.1007/s11858-013-0526-2
Pierson, A., Clark, D., & Kelly, G. (2019). Learning progressions and science practices: Tensions in prioritizing content, epistemic practices, and social dimensions of learning. Science & Education, 28, 833–841. https://doi.org/10.1007/s11191-019-00070-0
*Pocalana, G., Robutti, O., & Ciartano, E. (2024). Resources and Praxeologies Involved in Teachers’ Design of an Interdisciplinary STEAM Activity. Education Sciences, 14(3), 333
*Portnoy, N., Grundmeier, T. A., & Grahama, K. J. (2006). Students’ understanding of mathematical objects in the context of transformational geometry: Implications for constructing and understanding proofs. Journal of Mathematical Behavior, 25(3), 196–207. https://doi.org/10.1016/j.jmathb.2006.09.002
Prahmana, R.C.I., & D’Ambrosio, U. (2020). Learning geometry and values from patterns: Ethnomathematics on the batik patterns of Yogyakarta, Indonesia. Journal on Mathematics Education, 11(3), 439–456. http://doi.org/10.22342/jme.11.3.12949.439-456
*Rawani, D., Putri, R. I. I., & Susanti, E. (2023). RME-based local instructional theory for translation and reflection using of South Sumatra dance context. Journal on Mathematics Education, 14(3), 545-562
Rowan, L., Bourke, T., L’Estrange, L., Ryan, M., Walker, S., & Churchward, P. (2021). How does initial teacher education research frame the challenge of preparing future teachers for student diversity in schools? A systematic review of literature. Review of Educational Research, 91(1), pp.112–158. https://doi.org/10.3102/0034654320979171
*Sahara, S., Dolk, M., Hendriyanto, A., Kusmayadi, T. A., & Fitriana, L. (2024). Transformation geometry in eleventh grade using digital manipulative batik activities. Journal on Mathematics Education, 15(1), 55-78.
*Savaş, G., & Köse, N. Y. (2023). Pre-service mathematics teachers’ abstraction of rotational symmetry. Journal of Pedagogical Research, 7(3), 263-286
Schwartz, J. L., Yerushalmy, M., & Wilson, B. (Eds.). (1993). The geometric supposer: What is it a case of? Lawrence Erlbaum Associates, Inc.
*Sedig, K. (2008). From play to thoughtful learning: A design strategy to engage children with mathematical representations. Journal of Computers in Mathematics and Science Teaching, 27(1), 65–101.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36. https://doi.org/10.1007/BF00302715
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14. https://doi.org/10.3102/0013189X015002004
Sinclair, N., & Moss, J. (2012). The more it changes, the more it becomes the same: The development of the routine of shape identification in dynamic geometry environment. International Journal of Educational Research, 51–52, 28–44. https://doi.org/10.1016/j.ijer.2011.12.009
Sinclair, N., Bartolini Bussi, M., de Villiers, M., Jones, K., Kortenkamp, U., Leung, A., & Owens, K. (2016). Recent research on geometry education: An ICME‐13 survey team report. ZDM Mathematics Education, 48(5), 691–719. https://doi.org/10.1007/s11858-016-0796-6
Sinclair, N., Bartolini Bussi, M. G., de Villiers, M., Jones, K., Kortenkamp, U., Leung, A., & Owens, K. (2017). Geometry education, including the use of new technologies: A survey of recent research. In Proceedings of the 13th International Congress on Mathematical Education: ICME-13 (pp. 277-287). Springer International Publishing.
*Son, J. W., & Sinclair, N. (2010). How preservice teachers interpret and respond to student geometric errors. School Science and Mathematics, 110(1), 31–46. https://doi.org/10.1111/j.1949-8594.2009.00005.x
Sterenberg, G. (2013). Considering indigenous knowledges and mathematics curriculum. Canadian Journal of Science, Mathematics and Technology Education, 13(1), 18–32. https://doi.org/10.1080/14926156.2013.758325
Stewart, I. (2013). Symmetry. A very short introduction. Oxford University Press.
Stewart, I. (2008). Why beauty is truth: a history of symmetry. Basic Books
Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151–169. https://doi.org/10.1007/BF00305619
*Takeuchi, H. & Shinno, Y. (2020). Comparing the lower secondary textbooks of Japan and England: A praxeological analysis of symmetry and transformations in geometry. International Journal of Science and Mathematics Education, 18, 791–810. https://doi.org/10.1007/s10763-019-09982-3
*Tatar, E., Akkaya, A., & Berrin Kağizmanli, T. (2014). Using dynamic software in mathematics: The case of reflection symmetry. International Journal of Mathematical Education in Science and Technology, 45(7), 980–995. https://doi.org/10.1080/0020739X.2014.902129
*Thangamani, U., & Kwan Eu, L. (2019). Students’ achievement in symmetry of two dimensional shapes using geometer’s sketchpad. Malasyan Online Jounal of Educational Sciences, 7(1), 14–22.
Usiskin, Z. P. (1972). The effects of teaching Euclidean geometry via transformations on student achievement and attitudes in tenth-grade geometry. Journal for Research in Mathematics Education, 249–259. https://doi.org/10.2307/748492
*Uygun, T. (2020). An inquiry-based design research for teaching geometric transformations by developing mathematical practices in dynamic geometry environment. Mathematics Education Research Journal. https://doi.org/10.1007/s13394-020-00314-1
van Hiele, P. M. (1957). De problematiek van het inzicht.
Gedemonstreerd aan het inzicht van schoolkinderen in meetkunde-leerstof. [Unpublished doctoral dissertation, University of Utrecht.]
*Villarroel, J. D., Merino, M., & Antón, A. (2023). Young children’s spontaneous use of isometric and non-isometric symmetries: A study regarding their unprompted depictions of plant life. Journal of Biological Education, 57(5), 971-985.
Vygotsky, L. S. (1986). Thought and language, revised edition (A. Kozulin, Ed.). MIT Press.
Walsh, D., & Downe, S. (2005). Meta-synthesis method for qualitative research: A literature review. Journal of Advanced Nursing, 50(2), 204–211.
Weintrop, D., Beheshti, E., Horn, M., Orton, K., Jona, K., Trouille, L., &
Wilensky, U. (2016). Defining computational thinking for mathematics and science classrooms. Journal of Science Education and Technology, 25(1), 127–147. https://doi.org/10.1007/s10956-015-9581-5
Wing, J. M. (2006). Computational thinking. Communications of the ACM, 49(3), 33-35.
*Xistouri, X., Pitta-Pantazi, D., & Gagatsis, A. (2014). Primary school students’ structure and levels of abilities in transformational geometry. Revista Latinoamericana de Investigación en Matemática Educativa, 14(4-1), 149–164. https://dx.doi.org/10.12802/relime.13.1747
*Xu, L., Mulligan, J., Speldewinde, C., Prain, V., Tytler, R., Kirk, M., & Healy, R. (2023). Investigating angles and symmetries in light reflection: An integrated approach. Australian Primary Mathematics Classroom, 28(2), 38-40
*Yanik, H. B. (2011). Prospective middle school mathematics teachers’ preconceptions of geometric translations. Educational Studies Mathematics, 78, 231–260. https://doi.org/10.1007/s10649-011-9324-3
*Yanik, H. B. (2014). Middle-school students’ concept images of geometric translations. The Journal of Mathematical Behavior, 36, 33–50. https://doi.org/10.1016/j.jmathb.2014.08.001
*Yanik, H. B., & Flores, A. (2009). Understanding rigid geometric transformations: Jeff’s learning path for translation. The Journal of Mathematical Behavior, 28, 41–57. https://doi.org/10.1016/j.jmathb.2009.04.003
*Yao, X., & Manouchehri, A. (2019). Middle school students’ generalizations about properties of geometric transformations in a dynamic geometry environment. Journal of Mathematical Behavior, 55, 100703. https://doi.org/10.1016/j.jmathb.2019.04.002
Yerushalmy, M., & Houde, R. A. (1986). The geometric supposer: Promoting thinking and learning. The Mathematics Teacher, 79(6), 418-422.
*Zazkis, R., & Leron, U. (1991). Capturing congruence with a turtle. Educational Studies in Mathematics, 22(3), 285–295. https://doi.org/10.1007/BF00368342
Downloads
Published
Issue
Section
License
Copyright (c) 2026 International Journal of Education in Mathematics, Science and Technology
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Articles may be used for research, teaching, and private study purposes. Authors alone are responsible for the contents of their articles. The journal owns the copyright of the articles. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of the research material.
The author(s) of a manuscript agree that if the manuscript is accepted for publication in the journal, the published article will be copyrighted using a Creative Commons “Attribution 4.0 International” license. This license allows others to freely copy, distribute, and display the copyrighted work, and derivative works based upon it, under certain specified conditions.
Authors are responsible for obtaining written permission to include any images or artwork for which they do not hold copyright in their articles, or to adapt any such images or artwork for inclusion in their articles. The copyright holder must be made explicitly aware that the image(s) or artwork will be made freely available online as part of the article under a Creative Commons “Attribution 4.0 International” license.
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
