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

Year 2022, Issue: 90, 363 - 376, 28.03.2022

Abstract

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.

References

  • 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.
Year 2022, Issue: 90, 363 - 376, 28.03.2022

Abstract

References

  • 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.
There are 39 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Murat Akarsu This is me

Publication Date March 28, 2022
Published in Issue Year 2022 Issue: 90

Cite

APA Akarsu, M. (2022). HOW THE VAN HIELE THEORY AND THE PIRIE-KIEREN THEORY CAN BE USED TO ASSESS PT’s UNDERSTANDING OF CONCEPT OF REFLECTION?. EKEV Akademi Dergisi(90), 363-376.