# A Case Study on Specialised Content Knowledge Development with Dynamic Geometry Software: The Analysis of Influential Factors and Technology Beliefs of Three Pre-Service Middle Grades Mathematics Teachers

## Keywords:

Specialized content knowledge . preservice teachers . middle grades . student work . geometry## Abstract

This study characterises the development of Specialised Content Knowledge (SCK) with dynamic geometry software (DGS) throughout a semester. The research employed a single-case study with embedded units of three pre-service middle grades mathematics teachers. Qualitative data were collected and factors affecting these three teachers' SCK development were compared and contrasted through their experiences during semi-structured clinical interviews. The cross-case comparison indicated aspects such as opportunity to justify ideas in geometry, the level of common content knowledge, and beliefs about technology as influential factors for these pre-service mathematics teachers' SCK development with DGS

## References

Advisory Committee on Mathematics Education. (2006). Mathematics in Further Education colleges. Retrieved from http://www.acme-uk.org/media/1454/acme_mathematics%20in%20further%20education%20colleges.pdf

Arcavi, A., & Hadas, N. (2000). Computer mediated learning: An example of an approach. International Journal of Computers for Mathematical learning, 5(1), 25-45.

Bair, S. L., & Rich, B. S. (2011). Characterizing the Development of Specialized Mathematical Content Knowledge for Teaching in Algebraic Reasoning and Number Theory. Mathematical Thinking and Learning, 13(4), 292-321.

Baki, A., Kosa, T., & Guven, B. (2011). A comparative study of the effects of using dynamic geometry software and physical manipulatives on the spatial visualisation skills of pre‐service mathematics teachers. British Journal of Educational Technology, 42(2), 291-310.

Ball, D. L., Hill, H. C., & Bass, H. (2005). ‘Knowing mathematics for teaching. Who knows mathematics well enough to teach third grade, and how can we decide?’ American Educator, 29(3), 14–46.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407.

Battista, M. T., & Clements, D. H. (1992). Students’ cognitive construction of squares and rectangles in Logo Geometry. In W. Geeslin & K.Graham (Eds.), Proceedings of 16th PME International Conference, 1, 57-64.

Belfort, E., & Guimarães, L. C. (2004). Teacher’s practices and dynamic geometry. In M. J. Hoines & A. B. Fuglestad (Eds.), Proceedings of the 28th PME International Conference, 2, 503-510.

Chen, R. J. (2011). Preservice Mathematics Teachers' Ambiguous Views of Technology. School Science and Mathematics, 111(2), 56-67.

Coffland, D., & Strickland, A. (2004). Factors related to teacher use of technology in secondary geometry instruction. Journal of Computers in Mathematics and Science Teaching, 23(4), 347-365.

Comiti, C. and Μοreira Baltar, P. (1997). Learning process for the concept of area of planar regions in 12-13 year-olds. Area integration rules for grades 4, 6, and 8 pupils. Proceedings of the 21st of PME Conference, 3, 264-271.

Conference Board of the Mathematical Sciences. (2001). The Mathematical Education of Teachers. Providence RI and Washington DC: American Mathematical Society and Mathematical Association of America.

Conference Board of the Mathematical Sciences. (2012). The Mathematical Education of Teachers II. Providence RI and Washington DC: American Mathematical Society and Mathematical Association of America.

Corbin, J. & Strauss, A. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory. Thousand Oaks, CA: Sage.

DeWalt, K. M., & DeWalt, B. R. (2002). Participant observation. Walnut Creek, CA: AltaMira Press.

Edwards, L. (1990). The role of microworlds in the construction of conceptual entities. In Proceedings of the XIV Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 235-242).

Ernest, P. (1989). The impact of beliefs on the teaching of mathematics. In C. Keitel, P. Damerow, A. Bishop, & P. Gerdes (Eds.), Mathematics, education, and society (pp. 99-101). Paris: UNESCO.

Forsythe, S. (2007). Learning Geometry through Dynamic Geometry Software. Mathematics Teaching Incorporating Micromath, 202, 31-35.

Gerretson, H. (2004). Pre-service elementary teachers' understanding of geometric similarity: the effect of dynamic geometry software. Focus on Learning Problems in Mathematics, 26(3), 12.

González, G., & Herbst, P. G. (2009). Students’ conceptions of congruency through the use of dynamic geometry software. International Journal of Computers for Mathematical Learning, 14(2), 153-182.

Guven, B., & Kosa, T. (2008). The Effect of Dynamic Geometry Software on Student Mathematics Teachers' Spatial Visualization Skills. Turkish Online Journal of Educational Technology, 7(4), 100–107.

Hazzan, O., & Goldenberg, P. (1997). An expression of the idea of successive refinement in dynamic geometry environments. In E. Penkonen (Ed.), Proceedings of the 21st PME International Conference, 3, 49-56.

Healy, L. (2000). Identifying and explaining geometrical relationship: Interactions with robust and soft Cabri constructions. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th PME International Conference, 1, 103-117.

Healy, L., & Hoyles, C. (2002). Software tools for geometrical problem solving: Potentials and pitfalls. International Journal of Computers for Mathematical Learning, 6(3), 235-256.

Herbst, P., & Kosko, K. (2012).Mathematical knowledge for teaching high school geometry. In L. R. Van Zoest, J-J. Lo, & J. L. Kratky (Eds.), Proceedings of the 34th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 438-444). Kalamazoo, MI: Western Michigan University.

Hill, H. C., Schilling, S. G., & Ball, D. L. (2004). Developing measures of teachers’ mathematics knowledge for teaching. The Elementary School Journal, 105(1), 11-30.

Hill, H. C., Sleep, L., Lewis, J. M., & Ball, D. L. (2007). Assessing teachers’ mathematical knowledge: What knowledge matters and what evidence counts. Second handbook of research on mathematics teaching and learning, 1, 111-155.

Hollebrands, K. F. (2007). The role of a dynamic software program for geometry in the strategies high school mathematics students employ. Journal for Research in Mathematics Education, 164-192.

Jahn, A. P. (2000). New tools, new attitudes to knowledge: The case of geometric loci and transformations in Dynamic Geometry Environment. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th PME International Conference, 1,126-136.

Jones, K. (2000). Providing a foundation for deductive reasoning: Students' interpretations when using dynamic geometry software and their evolving mathematical explanations. Educational studies in mathematics, 44(1), 55-85.

Jackiw, N. (1995). The Geometer's SketchpadTM [Computer software]. Emeryville, CA: Key Curriculum Press.

Kagan, D. M. (1992). Implication of research on teacher belief. Educational psychologist, 27(1), 65-90.

Kynigos, C., & Argyris, M. (2004). Teacher beliefs and practices formed during an innovation with computer‐based exploratory mathematics in the classroom. Teachers and teaching, 10(3), 247-273.

Laborde, C. (2000). Dynamic geometry environments as a source of rich learning contexts for the complex activity of proving. Educational Studies in Mathematics, 44(1-2), 151-161.

Laborde, C., C. Kynigos, K. Hollebrands, R. Strasser (2006). Teaching and learning geometry with technology. In A. Gutiérrez, P. Boero (Eds.), Handbook of Research on the Psychology of Mathematics Education: Past, Present and Future, Sense Publishers, pp. 275–304.

Lemos, N. (2007). An introduction to the theory of knowledge. Cambridge University Press.

Marrades, R., & Gutiérrez, Á. (2000). Proofs produced by secondary school students learning geometry in a dynamic computer environment. Educational studies in mathematics, 44(1-2), 87-125.

Moyer, P. S., Bolyard, J. J., & Spikell, M. A. (2002). What are virtual manipulatives? Teaching children mathematics, 8(6), 372-377.

National Mathematics Advisory Panel. (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. U.S. Department of Education: Washington, D.C.

Noss, R., Hoyles, C., Healy, L, & Hoelzl, R. (1994). Constructing meanings for constructing: An exploratory study with Cabri-geometry. In J. P. Ponte & J. F. Matos (Eds.), Proceedings of the 18th PME International Conference, 3, 360-367.

Pajares, F. (1992). Teachers’ beliefs and educational research: Cleaning up a messy construct. Review of Educational Research, 62(3), 307-332.

Petrou, M., & Goulding, M. (2011). Conceptualising teachers’ mathematical knowledge in teaching. In T. Rowland & K. Ruthven (Eds.), Mathematical knowledge in teaching (pp. 9–25). Dordrecht: Springer.

Sinclair, N., & Robutti, O. (2013). Technology and the role of proof: The case of dynamic geometry. Third International Handbook of Mathematics Education, New York: Springer.

Skemp, R. (1978). Relational understanding and instrumental understanding. Arithmetic Teacher, 9-15.

Stanley, G. (2008). National numeracy review report. Council of Australian Governments. Retrieved from http://www.coag.gov.au/sites/default/files/national_numeracy_review.pdf

Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455-488.

Trgalova, J., Soury-Lavergne, S., & Jahn, A. P. (2011). Quality assessment process for dynamic geometry resources in Intergeo project. ZDM, 43(3), 337-351.

Yin, R. K. (2008). Case study research: Design and methods (Vol. 5). Sage Publications, Incorporated.