Teacher Practices and Student Understandings in 2D Drawings of 3D Shapes

Abstract

Geometry holds a pivotal position within the realm of mathematics, yet many students struggle to grasp its concepts, posing challenges for their teachers. In the Republic of Türkiye, the Ministry of National Education has recently recommended the adoption of constructivist approaches to teach geometry. To gauge the efficacy of this instructional approach in actual classrooms, we investigated the practices of Turkish middle school mathematics teachers and their students' experiences in constructing two-dimensional drawings of three-dimensional geometrical shapes during regular lessons. The study involved four teachers from two schools and 199 students aged 12 to 14 years. Multiple data sources, including lesson observations, student worksheets and interviews with teachers and students, were utilized to gain insights into student understandings and whether the pedagogical practices they encountered were associated with this. Based on these findings, we argue that teachers may require additional support for their classrooms to transition into student-centred environments to permit students to actively engage in enriching mathematical exercises, exercise agency, and have opportunities to utilize provided manipulatives and digital tools.

Keywords

• Constructivist approaches
• Geometry education
• Middle school mathematics
• Student experiences
• Three-dimensional geometrical shapes
• Two-dimensional drawings

References

1. Aberdeen, T. (2013). Review essay: Yin, RK (2009). Case study research: Design and methods. Thousand Oaks, CA: Sage. The Canadian Journal of Action Research, 14(1), 69-71. https://journals.nipissingu.ca/index.php/cjar/article/view/73
2. Alghadari, F., Herman, T., & Prabawanto, S. (2020). Factors affecting senior high school students to solve three-dimensional geometry problems. International Electronic Journal of Mathematics Education, 15(3), em0590. https://doi.org/10.29333/iejme/8234
3. Bakó, M. (2003). Differing projecting methods in teaching spatial geometry. In K. Krainer, F. Goffre & P. Berger (Eds.), Proceedings of the Third European Research in Mathematics Education, (pp. 1–9). FMD. http://www.erme.tu-dortmund.de/~erme/CERME3/Groups/TG7/TG7_Bako_cerme3.pdf
4. Balgalmış, E., Çakıroğlu, E., & Shafer, K. (2014). An investigation of a pre-service elementary mathematics teacher's TPACK within the context of teaching practices. In M. Searson & M. Ochoa (Ed.), Proceedings of Society for Information Technology & Teacher Education International Conference 2014 (pp. 2210–2217). AACE. https://www.learntechlib.org/p/131114/
5. Battista, M. T. (2007). The development of geometric and spatial thinking. In F. K. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (2nd ed., pp. 843–908). New York: Information Age.
6. Bayrakdar-Çiftçi, Z., Akgün, L., & Deniz, D. (2013). Dokuzuncu sınıf matematik öğretim programı ile ilgili uygulamada karşılaşılan sorunlara yönelik öğretmen görüşleri ve cözüm önerileri [Teachers’ opinions and solution suggestions regarding encountered issues on the ninth grade mathematics curriculum]. Anadolu Üniversitesi Eğitim Bilimleri Enstitüsü Dergisi, 3(1), 2–21. https://dergipark.org.tr/en/pub/ajesi/issue/1528/18743
7. Bayraktaroglu, C. E. (2011). Constructivism approach in education and a critical perspective (Master's thesis). Maltepe University, Institute of Social Sciences.
8. BERA. (2011). Revised Ethical Guidelines for Educational Research. British Educational Research Association.

9. Bhagat, K. K., & Chang, C. Y. (2015). Incorporating GeoGebra into geometry learning-a lesson from India. Eurasia Journal of Mathematics, Science and Technology Education, 11(1), 77-86. https://www.ejmste.com/download/incorporating-geogebra-into-geometry-learning-a-lesson-from-india-4356.pdf
10. Bishop, A. J. (2008). Visualizing and mathematics in a pre-technological culture. In P. Clarkson & N. Presmeg (Eds.), Critical Issues in Mathematics Education: Major Contributions of Alan Bishop (pp. 109–120). Springer.
11. Brown, C. A., Kouba, V. L., Carpenter, T. P., Lindquist, M. M., Silver, E. A., & Swafford, J. O. (1988). Secondary school results for the fourth NAEP mathematics assessment: Algebra, geometry, mathematical methods, and attitudes. The Mathematics Teacher, 81(5), 337–397. https://doi.org/10.5951/MT.81.5.0337
12. Brooks, J. G., & Brooks, M. G. (1999). In search of understanding: The case for constructivist classrooms. Association for Supervision and Curriculum Development.
13. Cengizhan, S., & Koç, T. (2017). The effect of team-game-tournament technique on academic achievement and long-term retention of students in teaching mathematics and students' opinions. International Journal of Eurasia Social Sciences, 8(27), 588-608. http://www.ijoess.com/Makaleler/588182892_10.%20588-608sibel%20cengizhan.pdf
14. Christou, C., Jones, K., Mousoulides, N., & Pittalis, M. (2006). Developing the 3DMath dynamic geometry software: Theoretical perspectives on design. International Journal for Technology in Mathematics Education, 13, 168–174. https://eprints.soton.ac.uk/42114/
15. Clements, D. H. (2003). Teaching and learning geometry. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.), A research companion to principles and standards for school mathematics, (pp. 151-178). National Council of Teachers of Mathematics.
16. Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning, (pp. 420-464). Macmillan Publishing.
17. Common Core State Standards Initiative. (2022). Common core state standards for mathematics. https://learning.ccsso.org/wp-content/uploads/2022/11/Math_Standards1.pdf
18. Cooper, M., & Sweller, J. (1989). School students' representations of solids. Journal for Research in Mathematics Education, 20(2), 202–212. https://doi.org/10.5951/jresematheduc.20.2.0202
19. Creswell, J. W. (2007). Qualitative inquiry and research design: Choosing among five traditions (2nd ed.). Sage Publications, Inc.
20. Cross, D. I. (2009). Alignment, cohesions, and change: Examining mathematics teachers' belief structures and their influence on instructional practices. Journal of Mathematics Teacher Education, 12(5), 325–346. https://doi.org/10.1007/s10857-009-9120-5
21. Department for Education. (2021). National curriculum in England: Mathematics programmes of study. https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study
22. Department for Education. (2019). National curriculum assessments at key stage 2 in England, 2019 (interim). https://www.gov.uk/government/publications/national-curriculum-assessments-key-stage-2-2019-interim/national-curriculum-assessments-at-key-stage-2-in-england-2019-interim
23. Doorman, L. M., Van den Heuvel-Panhuizen, M., & Goddijn, A. (2020). The emergence of meaningful geometry. In M. Van den Heuvel-Panhuizen (Ed.), National Reflections on the Netherlands Didactics of Mathematics (pp. 281–302). (ICME-13 Monographs). Springer Nature. https://doi.org/10.1007/978-3-030-33824-4_15
24. Doruk, B. K. (2014). The educational approaches of Turkish pre-service elementary mathematics teachers in their first teaching practices: Traditional or constructivist? Australian Journal of Teacher Education, 39(10), 113–134. https://doi.org/10.3316/aeipt.204388
25. Evans, S., & Swan, M. (2014). Developing students' strategies for problem solving in mathematics: The role of pre-designed "sample student work". Educational Designer, 2(7), 1–31. http://www.educationaldesigner.org/ed/volume2/issue7/article25/index.htm
26. Fosnot, C. T., & Perry, R. S. (1996). Constructivism: A psychological theory of learning. In C. T. Fosnot & R. S. Perry (Eds.), Constructivism: Theory, perspectives, and practice (pp. 8–33). Teachers College Press.
27. Fujita, T., Kondo, Y., Kumakura, H., & Kunimune, S. (2017). Students' geometric thinking with cube representations: Assessment framework and empirical evidence. The Journal of Mathematical Behavior, 46(1), 96–111. https://doi.org/10.1016/j.jmathb.2017.03.003
28. Gutierrez, A. (1996). Visualization in 3-dimensional geometry: In search of a framework. In L. Puig, & A. Gutierrez (Eds.), Proceedings of the 20th International Conference for the Psychology of Mathematics Education, (Vol.1, pp. 3–19).
29. Gür, S. B., & Çelik, Z. (2009). Türkiye’de Milli Eğitim sistemi: Yapısal sorunlar ve öneriler [National Education system in Turkey: Structural problems and suggestions.]. SETA.
30. Haj-Yahya, A. (2022). Using theoretical and empirical background information to affect noticing of geometrical thinking. Educational Studies in Mathematics, 111, 493–513. https://doi.org/10.1007/s10649-022-10176-y
31. Hankeln, C. (2020). Mathematical modeling in Germany and France: A comparison of students' modeling processes. Educational Studies in Mathematics, 103(2), 209–229. https://doi.org/10.1007/s10649-019-09931-5
32. Hegedus, S.J., & Otálora, Y. (2022). Mathematical strategies and emergence of socially mediated metacognition within a multi-touch dynamic geometry environment. Educational Studies in Mathematics, 112, 289–307. https://doi.org/10.1007/s10649-022-10170-4
33. Huang, L., Doleck, T., Chen, B., Huang, X., Tan, C., Lajoie, S. P., & Wang, M. (2023). Multimodal learning analytics for assessing teachers’ self-regulated learning in planning technology-integrated lessons in a computer-based environment. Education and Information Technologies, 28, 15823–15843. https://doi.org/10.1007/s10639-023-11804-7
34. Kali, Y., & Orion, N. (1996). Spatial abilities of high-school students in the perception of geologic structures. Journal of Research in Science Teaching, 33(4), 369–391. https://doi.org/10.1002/(SICI)1098-2736(199604)33:4%3C369::AID-TEA2%3E3.0.CO;2-Q
35. Karaagac, M. K., & Threlfall, J. (2004). The tension between teacher beliefs and teacher practice: The impact of work setting. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, 3(1), 137–144. https://eric.ed.gov/?id=ED489550
36. Kaya, D., Keşan, C., İzgiol, D., & Erkuş, Y. (2016). Yedinci sınıf öğrencilerinin cebirsel muhakeme becerilerine yönelik başarı düzeyi [Achievement level for algebraic reasoning skills of seventh grade students]. Turkish Journal of Computer and Mathematics Education, 7(1), 142–163. https://doi.org/10.16949/turcomat.95945
37. Kirschner, P., Sweller, J., & Clark, R. E. (2006). Why unguided learning does not work: An analysis of the failure of discovery learning, problem-based learning, experiential learning and inquiry-based learning. Educational Psychologist, 41(2), 75-86.
38. Kohen, Z., & Orenstein, D. (2021). Mathematical modeling of tech-related real-world problems for secondary school-level mathematics, Educational Studies in Mathematics, 107(1), 71–91. https://doi.org/10.1007/s10649-020-10020-1
39. Konan, N., & Yilmaz, S. (2017). Teachers' views on teacher performance assessment: A mixed-method research. Proceedings of the 26th International Educational Sciences Conference, 137-160. https://dergipark.org.tr/en/pub/milliegitim/issue/39923/473428
40. Kooloos, C., Oolbekkink-Marchand, H., van Boven, S., Kaenders, R., & Reckman, G. (2022). Making sense of student mathematical thinking: The role of teacher mathematical thinking. Educational Studies in Mathematics, 110, 503–524 (2022). https://doi.org/10.1007/s10649-021-10124-2
41. Kuzniak, A., & Rauscher, J. C. (2011). How do teachers' approaches on geometrical work relate to geometry students learning difficulties? Educational Studies in Mathematics, 77(1), 129–147. https://doi.org/10.1007/s10649-011-9304-7
42. Marshall, C., & Rossman, G. B. (2011). Managing, analyzing, and interpreting data. In Designing Qualitative Research (pp. 205–227). Sage.
43. Ministry of National Education. (2013). Middle school mathematics program (Grades 5-8). MoNE.
44. Ministry of National Education. (2018). Primary and middle school mathematics program update (Grades 1-8). MoNE.
45. Ministry of National Education. (2023). Technology-supported Learning Scenarios Project in Education. MoNE. https://yegitek.meb.gov.tr/www/teknoloji-odakli-ogrenme-senaryolari-calistayi-etkimde-gerceklesti/icerik/3508
46. Ministry of National Education. (2024). Ortaokul matematik dersi öğretim programı (5, 6, 7 ve 8. sınıflar). [Middle school mathematics teaching programme (5th, 6th, 7th and 8th grades)]. https://cdn.eba.gov.tr/icerik/GorusOneri/2024ProgramlarOnayli/2024programmat5678Onayli.pdf
47. National Centre for Education Statistics. (2011). NAEP - Mathematics Content Areas. https://nces.ed.gov/nationsreportcard/mathematics/contentareas2005.aspx
48. National Centre for Education Statistics. (2018). NAEP Mathematics: National Student Group Scores and Score Gaps. https://www.nationsreportcard.gov/math_2017/nation/gaps?grade=8
49. OECD. (2019). Mathematics performance (PISA) (indicator). https://doi.org/10.1787/04711c74-en
50. Parzysz, B. (1988). Knowing vs seeing: Problems of the plane representation of space geometry figures. Educational Studies in Mathematics, 19(1), 79–92. https://doi.org/10.1007/BF00428386
51. Pittalis, M., & Christou, C. (2013). Coding and decoding representations of 3D shapes. The Journal of Mathematical Behavior, 32(1), 673-689. https://doi.org/10.1016/j.jmathb.2013.08.004
52. Saralar, İ., Işıksal, M., & Akyuz, D. (2018). The evaluation of a pre-service mathematics teacher’s TPACK: Case of 3D objects. International Journal for Technology in Mathematics Education, 25(2), 3-22. https://doi.org/10.1564/tme_v25.2.01
53. Saralar-Aras, İ. (2022). An exploration of middle school mathematics teachers’ beliefs and goals regarding a dynamic mathematics tool: GeoGebra. Journal of Research in Science Mathematics and Technology Education, 5(SI), 41-63. https://doi.org/10.31756/jrsmte.113SI
54. Simon, M. A., & Schifter, D. (1991). Towards a constructivist perspective: An intervention study of mathematics teacher development. Educational Studies in Mathematics, 22(4), 309-331. https://doi.org/10.1007/BF00369293
55. Standards and Testing Agency [STA]. (2019a). Key stage 2 tests: 2019 mathematics test materials. https://www.gov.uk/government/publications/key-stage-2-tests-2019-mathematics-test-materials
56. Standards and Testing Agency [STA]. (2019b). Mathematics paper 2: Reasoning. In 2019 National Curriculum Tests: Key stage 2.
57. Taylor, G. W., & Ussher, J. M. (2001). Making sense of S&M: A discourse analytic account. Sexualities, 4(3), 293–314. https://doi.org/10.1177/136346001004003002
58. Tekalmaz, G. (2019). Revize edilen ortaöğretim matematik öğretim programı hakkında öğretmen görüşleri [Teacher reviews about reformed high school mathematics curriculum]. Kocaeli Üniversitesi Eğitim Dergisi, 2(1), 35-47. https://doi.org/10.33400/kuje.548562
59. Tuncel, T., & Kazu, İ. Y. (2019). Ortaöğretim matematik öğretim programlarının ölçme ve değerlendirme boyutunda öğretmen görüşleri açısından incelenmesi [Examination of secondary education mathematics curriculum in terms of teacher's opinions in the dimension of measurement and evaluation]. Fırat Üniversitesi Sosyal Bilimler Dergisi, 29(2), 163-179. https://dergipark.org.tr/en/download/article-file/854639
60. Tzafilkou, K., Perifanou, M., & Economides, A.A. (2023). Assessing teachers’ digital competence in primary and secondary education: Applying a new instrument to integrate pedagogical and professional elements for digital education. Education and Information Technologies, 28, 16017–16040. https://doi.org/10.1007/s10639-023-11848-9
61. Vandenberg, S. G., & Kuse, A. R. (1978). Mental rotations, a group test of three-dimensional spatial visualization. Percept Mot Skills, 47(2), 599–604. https://doi.org/10.2466/pms.1978.47.2.599
62. Widder, M., Berman, A., & Koichu, B. (2019). An a priori measure of visual difficulty of 2-d sketched depicting 3-d objects. Journal for Research in Mathematics Education, 50(5), 489–528. https://doi.org/10.5951/jresematheduc.50.5.0489
63. Widder, M., & Gorsky, P. (2013). How students solve problems in spatial geometry while using a software application for visualizing 3D geometric objects. Journal of Computers in Mathematics and Science Teaching, 32(1), 89-120. https://eric.ed.gov/?id=EJ1006172
64. Yin, R. K. (2009). Case study research: Design and methods. Thousand Oaks, CA: Sage.
65. Yin, R. K., Clarke, C., Cotner, B., & Lee, R. (2012). Case study methods. In Handbook of complementary methods in education research (pp. 111-122). Routledge.
66. Zengin, Y. (2023). Effectiveness of a professional development course based on information and communication technologies on mathematics teachers' skills in designing technology-enhanced task. Education and Information Technolologies, 28, 16201–16231. https://doi.org/10.1007/s10639-023-11728-2