Geometry learning requires geometric experiences (Battista & Clements, 1988; Jones & Mooney, 2003). Providing rich perceptual, tactile, and kinesthetic experiences that involve both mental and physical actions greatly supports the learning of mathematical concepts (Hall & Nemirovsky, 2012; Radford, 2009). As part of geometry learning, students are expected to name, construct, draw, classify, determine the properties of, determine the length, area, and volume of, and establish spatial relationships between 3-dimensional shapes at the elementary and middle school levels (NCTM, 2000: MoNE, 2024). In the curricula of different countries, 3D shapes are limited to the learning of right cylinder, right prism, right pyramid, right cone, and sphere. In the teaching of 3D shapes, the following approaches are emphasized for the exploration of shapes: examining concrete materials (Battista & Clements, 1988), investigating unfoldings and objects formed by unfoldings (Cohen, 2023; Ng & Ferrara, 2020), circular sweeping (with the movement of a line along a curve) (Bertoline & Wiebe, 2005; Tsamir et. all, 2015), linear sweeping (with elevation along a vector) (Okumuş, 2020), using a rotational model (by rotating a 2D shape around an axis) (Okumuş, 2020), examining a daily life example (González, 2015).

Which approach or approaches are chosen is of great importance in structuring teaching. This decision has a key role both in the choice of tools/technologies to be used and in the construction of meaningful knowledge. These approaches and the learning environments in which they are employed can be considered as semiotic resources that include action, product, or process (Presmeg, et. all, 2016). For this reason, Yeh and Nason (2004) emphasize that a semiotic perspective should be considered in the construction of mathematical meaning and meaningful knowledge in the teaching of 3D geometry. Based on Peirce's semiotics triad Yeh and Nason (2004) conceptualized an “interpretant” system in which “representamen” is the “object” in the teaching of 3D geometry and created a framework in which the external material world is the object, communication is the representamen and internal spatial ability is the interpretant.

In this framework, the external material world represents all geometric objects and properties (e.g., angle, height, and length), including natural objects and idealized objects (e.g., triangle and cube shapes). Intrinsic spatial ability is the potential and capacity to perceive and know external geometric objects (Yeh & Nason, 2004). Carroll (1993) defines interpretation as the imagining, perceiving, manipulating, reorganizing, and re-acquiring of visual images of objects or forms. Communication refers to language, including spoken and written language, mathematical representations, pictures, diagrams, and kinesthetic body movements.

Approaches to be used in teaching 3-dimensional shapes should be chosen by taking these three components into account. Pre-service teachers should gain a perspective that includes these elements during their education. For this reason, the present study aimed to examine the approaches preferred by pre-service teachers in teaching cylinders, prisms, cones, pyramids, and spheres, which are the reflections of 3D geometry in curricula, in the context of the semiotic triad. Considering all these, the research question and sub-questions can be expressed as follows:

• How are pre-service middle school mathematics teachers' preferred approaches in teaching 3D objects in terms of semiotic triad of 3d geometry?
  • Do pre-service teachers' approach preferences vary on the basis of objects?
  • Which points do pre-service teachers' approach preferences emphasize in terms of the elements of the semiotic triad of 3d geometry?
  • Do the elements of the semiotic triad of 3d geometry have similarities or differences in the 3D shapes considered for pre-service teachers?

In the present study, a case study was conducted with 40 middle school pre-service mathematics teachers. Merriam (1998) conceives a qualitative case study as “an intensive, holistic description and analysis of a bounded phenomenon such as a program, an institution, a person, a process, or a social unit” (p. xiii). In this study, pre-service teachers' (PST) preferences for teaching 3D shapes are considered as a phenomenon and analyzing this phenomenon in the context of the semiotic triad of 3d geometry is considered as a case. Of the PSTS participating in the study, 30 were female and 10 were male, and the PSTs were selected by purposive sampling method in order to serve the purpose of the study. Since it was thought that their geometry knowledge and geometry teaching knowledge would affect their preferences in the selection of PSTS, those who completed the courses “Fundamentals of Mathematics 2”, which includes the definition, experience, and formation of 3D geometric shapes, and “Geometry and Measurement Teaching”, which includes knowledge, methods and techniques, and experiences for teaching 3D geometry to middle school students, were included in the study. In the interviews with PSTs, they were asked to indicate and explain the approach they would prefer to use in teaching cylinder, prism, cone, pyramid, and sphere in the curriculum. The interviews lasted 30-45 minutes, and voice recordings and researcher notes were used to collect the data. Qualitative data including PSTs' preferences and reasons were analyzed in terms of the external material world, internal spatial ability, and communication elements of the semiotic triad (Yeh & Nason, 2004). It was tried to determine how PSTs' preferences differed and resembled according to these three elements. For this purpose, PSTs’ answers were conducted through content analysis, and in this analysis, an analysis approach based on the literature was adopted to explain the data as stated by Wolcott (1994). In the analysis, the external material world, internal spatial ability, and communication elements in the model specified by, Yeh and Nason (2004) were determined as themes, and the codes within these themes were created based on the semiotic elements in the PSts' answers.

Data analysis is ongoing in the study, and early analysis shows that the approaches of examining concrete materials and examining unfoldings and the objects formed by unfolding are preferred in the exploration of 3D geometric shapes. However, it is seen that these approach preferences differ on the basis of shape class. Although the analyses continue in the context of external material world, internal spatial ability, and communication, which are the elements of the semiotic triad of 3D geometry, it is predicted that PSTs who emphasize the external material world element intensively due to the nature of the approaches will emphasize the approaches of examining concrete materials and examining daily life examples. This possible situation can be explained both by the fact that PSTs are directed to use concrete materials in geometry and measurement teaching and that the approaches in which spatial visualization will be used may remain abstract for students due to the lack of geometric experiences at the level of students to be taught. On the other hand, the preference of PSTs who emphasize intrinsic spatial ability is expected to be the approaches of investigating unfoldings and objects formed by unfoldings related to spatial orientation from the sub-areas of spatial ability, using a rotational model involving mental rotation from the sub-areas of spatial ability, and circular sweeping and linear sweeping in which spatial visualization is actively used in the exploration of shapes. Finally, the communication element is expected to be emphasized more in the circular sweeping and linear sweeping approaches in terms of gradual and changing emphasis on surfaces, heights, and bases.

Battista, M. T., & Clements, D. H. (1988). A case for a Logo-based elementary school geometry curriculum. The Arithmetic Teacher, 36(3), 11-17. Bertoline, G. R., & Wiebe, E. N. (2005). Fundamentals of Graphics Communication (McGraw-Hill Graphics). McGraw-Hill Science/Engineering/Math. Carroll, J. B. (1993). Human cognitive abilities: A survey of factor-analytic studies (No. 1). Cambridge University Press. Cohen, N. (2003). Curved Solid Nets. In N. Peterman, J. Dougherty, & J. Zillox (Eds.), Proceedings of the 27th International Conference of Psychology in Mathematics Education, Vol. 2, (pp. 229-236). University of Hawaii. González, N. A. A. (2015). How to include augmented reality in descriptive geometry teaching. Procedia Computer Science, 75, 250-256. Hall, R., & Nemirovsky, R. (2012). Modalities of body engagement in mathematical activity and learning [special issue]. Journal of the Learning Sciences, 21(2), 207–215. Jones, K., & Mooney, C. (2003). Making space for geometry in primary mathematics. Enhancing primary mathematics teaching, 3-15. Merriam, S. B. (1998). Qualitative research and case study applications in education. Jossey-Bass. Ministry of National Education [MoNE]. (2024). Middle school mathematics course curriculum (5st-8th Grades). MEB. National Council for Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. VA: Author. Ng, O., & Ferrara, F. (2020). Towards a materialist vision of ‘learning as making’: The case of 3D Printing Pens in school mathematics. International Journal of Science and Mathematics Education, 18, 925–944. https:// doi. org/ 10. 1007/ s10763- 019- 10000-9 Okumuş, S. (2020). Teaching geometric shapes: Cone and pyramid. In E. Ertekin & M. Ünlü (Eds.) Teaching Geometry and Measurement: Definitions, Concepts, and Activities. (pp.423-451). Pegem. Presmeg, N., Radford, L., Roth, W. M., & Kadunz, G. (2016). Semiotics in mathematics education. Springer Nature. Radford, L. (2009). Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings. Educational Studies in Mathematics, 70, 111–126. Tsamir, P., Tirosh, D., Levenson, E., Barkai, R., & Tabach, M. (2015). Early-years teachers’ concept images and concept definitions: triangles, circles, and cylinders. ZDM, 47, 497-509. Wolcott, H. F. (1994). Transforming qualitative data: Description, analysis, and interpretation. Sage. Yeh, A., & Nason, R. (2004). Toward a semiotic framework for using technology in mathematics education: The case of learning 3D geometry. In Full Proceedings-International Conference on Computers in Education 2004. (pp. 1191-1199). Common Ground Publishing Pty Ltd.