HOW THE VAN HIELE THEORY AND THE PIRIE-KIEREN THEORY CAN BE USED TO ASSESS PT’s UNDERSTANDING OF CONCEPT OF REFLECTION?

Yıl 2022, Sayı: 90, 363 - 376, 28.03.2022

Murat Akarsu

Öz

This paper seeks to investigate how the van Hiele Theory and the Pirie-Kieren Theory
can be used to assess pre-service teachers’ understanding of the concept of geometric
reflection. These analyses include motivations for ultimately utilizing the van Hiele
Theory and Pirie-Kieren Theory to examine how pre-service teachers can develop
a mapping view of geometric reflection from a motion view of geometric reflection.
Additionally, I contrast previous cases which utilized the van Hiele Theory and Pirie-
Kieren Theory separately, noting that there is yet to be work done in which the Pirie-
Kieren Theory is utilized in conjunction with dynamic geometry software. While this study
is not inherently connected to these existing studies, the utilization of frameworks did
play a role in our decisions for deciding on a particular framework, namely the van Hiele
Theory. I acknowledge that both the van Viele and Pirie-Kieren frameworks offer insights
into pre-service teachers’ thinking about geometric reflection (particularly when paired
with a dynamic geometry software); however, due to certain characteristics of the van
Hiele Theory (namely providing a clear progression in-depth of knowledge), I primarily
suggest using the van Hiele Theory in teaching geometric reflection. My findings show
that the emphasis on a clear path of progression and requisite knowledge is a critical
factor in this change of perspective, as well as the importance of well-designed tasks that
illuminate characteristics for a mapping view of geometric reflection.

Anahtar Kelimeler

van Hiele theory, Pirie-Kieren Theory, Geometric reflection, Dynamic geometry software, Motion view, Mapping view.

Kaynakça

• Adolphus, T., 2011. Problems of teaching and learning of geometry in secondary schools in Rivers State, Nigeria. International Journal of Emerging Science and Engineering (IJESE), 1(2), 143-152.
• Akarsu, M. (2022). Understanding of geometric reflection: John’s learning path for geometric reflection. Journal of Theoretical Educational Science, 15(1), 64-89.
• Arzarello, F., Olivero, F., Paola, D. & Robutti, O. (2002). A cognitive analysis of dragging practices in Cabri environments. ZDM – The International Journal on Mathematics Education, 34(3), 66–72.
• Battista, M. T. & Clements, D. H. (1996). Students’ understanding of three-dimensional rectangular arrays of cubes. Journal for Research in Mathematics Education, 27, 258–292.
• Battista,M.T.(2004).Applying cognition-based assessment to elementary schoolstudents' development of understanding of area and volume measurement. Mathematical Thinking and Learning, 6(2), 185-204.
• Breen, J. J. (1999). Achievement of van Hiele level two in geometry thinking by eight grade students through the use of geometry computer-based guided instruction. (Unpublished Doctoral Dissertation), Vermillion: The University of South Dakota School of Education.
• Cavey, L. O. & Berenson, S. B. (2005). Learning to teach high school mathematics: Patterns of growth in understanding right triangle trigonometry during lesson plan study. Journal of Mathematical Behavior, 24, 171–190.
• Clements, D. H. & Battista, M. T. (1990). The effects of Logo on children's conceptualizations of angle and polygons. Journal for research in mathematics education, 21(5), 356-371.
• Clements, D. H., Sarama, J., Yelland, N. J. & Glass, B. (2008). Learning and teaching geometry with computers in the elementary and middle school. Research on technology and the teaching and learning of mathematics, 1, 109-154.
• Dubinsky, E. (1991). Constructive aspects of reflective abstraction in advanced mathematics. In Epistemological foundations of mathematical experience (pp. 160-202). Springer, New York, NY.
• 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.
• Gülkılık, H., Uğurlu, H. H. & Yürük, N. (2015). Examining Students' Mathematical Understanding of Geometric Transformations Using the Pirie-Kieren Model. Educational Sciences: Theory & Practice, 15(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.
• Hiebert, J. & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65-97). New York, NY: Macmillan.
• Hollebrands, K. F. (2003). High school students’ understandings of geometric transformations in the context of a technological environment. The Journal of Mathematical Behavior, 22(1), 55-72.
• 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, 38(2),164-192.
• İbili, E. (2019). The use of dynamic geometry software from a pedagogical perspective: current status and future prospects. Journal of Computer and Education Research, 7(14), 337-355.
• Krainer, K. (1993). Powerful tasks: A contribution to a high level of acting and reflecting in mathematics instruction. Educational Studies in Mathematics, 24(1), 65-93.
• Kutluca, T. (2013). The effect of geometry instruction with dynamic geometry software; GeoGebra on Van Hiele geometry understanding levels of students. Educational Research and Reviews, 8(17), 1509-1518.
• Luneta, K., 2014. 'Foundation phase teachers’ (limited) knowledge of geometry', South African Journal of Childhood Education 4(3), 71-86.
• Martin, L. C. (2000). Folding back and growing mathematical understanding. Paper presented at the 24th Annual Canadian Mathematics Education Study Group, Montreal, Canada.
• Martin, L. C. (2008). Folding back and the dynamical growth of mathematical understanding: Elaborating the Pirie-Kieren Theory. Journal of Mathematical Behavior, 27, 64-85.
• National Council of Teachers of Mathematics [NCTM]. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
• National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
• Pirie, S. E. B., & Kieren, T. E. (1989). A recursive theory of mathematical understanding. For the Learning of Mathematics, 9(3), 7-11.
• Pirie, S. E. B. & Kieren, T. E. (1992). Watching Sandy’s understanding grow. Journal of Mathematical Behavior, 11, 243-257.
• Pirie, S., & Kieren, T. (1994). Growth in mathematical understanding: How can we characterise it and how can we represent it? Educational Studies in Mathematics, 26(2), 165-190.
• Pirie, S., & Martin, L. (2000). The role of collecting in the growth of mathematical understanding. Mathematics Education Research Journal, 12(2), 127-146.
• Sierpinska, A. (1990). Some remarks on understanding in mathematics. For the Learning of Mathematics, 10(3), 24-36, 41.
• Skemp,R.(1976).Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20-26.
• Steffe, L. P. & Kieren, T. (1994). Radical constructivism and mathematics education. Journal for Research in Mathematics Education, 25, 711–733.
• Strutchens, M. E. & Blume, G. W. (1997). What do students know about geometry? In P. A. Kenney & E. A. Silver (Eds.), Results from the sixth mathematics assessment of the National Assessment of Educational Progress (pp. 165-193). Reston, VA: National Council of Teachers of Mathematics.
• Tikoo, M. (1998). Integrating Geometry in a Meaningful Way (A Point of View). International Journal of Mathematical Education In Science And Technology, 29(5), 663-75
• Van Hiele, P. M. (1986). Structure and insight: A theory of mathematics education. Orlando, FL: Academic Press.
• Von Glasersfeld, E. (1987). Learning as a constructivist activity. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 3-17). Hillsdale, NJ: Lawrence Erlbaum.
• Warner, L.B.(2008). How do students’behaviorsrelate to the growth oftheirmathematical ideas? Journal of Mathematical Behavior, 27, 206–227.
• Yanik, H. B. (2006). Prospective elementary teachers' growth in knowledge and understanding of rigid geometric transformations. (Doctoral Dissertation), Ann Arbor: The Arizona State University.
• Yanik, H. B. (2013). Learning geometric translationsin a dynamic geometry environment. Education and Science, 38(168), 272–287.
• Yanik, H. B. & Flores, A. (2009). Understanding rigid geometric transformations: Je’s learning path for translation. The Journal of Mathematical Behavior, 28(1), 41–57.