Research in Mathematical Education (한국수학교육학회지시리즈D:수학교육연구)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지

Teaching the Derivation of Area Formulas for Polygonal Regions through Dissection-Motion-Operations (DMO): A Visual Reasoning Approach

  • Received : 2010.07.29
  • Accepted : 2010.09.20
  • Published : 2010.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

Utilizing a structure of operations known as Dissection-Motion-Operations (DMO), a set of mathematics propositions or area-formulas in school mathematics will be introduced through shape-to-shape transforms. The underlying theme for DMO is problem-solving through visual reasoning and proving manipulatively or electronically vs. rote learning and memorization. Visual reasoning is the focus here where two operations that constitute DMO are utilized. One operation is known as Dissection (or Decomposition) operation that operates on a given region in 2D or 3D and dissects it into a number of subregions. The second operation is known as Motion (or Composition) operation applied on the resultant sub-regions to form a distinct area (or volume)-equivalent region. In 2D for example, DMO can transform a given polygon into a variety of new and distinct polygons each of which is area-equivalent to the original polygon (cf [Rahim, M. H. & Sawada, D. (1986). Revitalizing school geometry through Dissection-Motion Operations. Sch. Sci. Math. 86(3), 235-246] and [Rahim, M. H. & Sawada, D. (1990). The duality of qualitative and quantitative knowing in school geometry, International Journal of Mathematical Education in Science and Technology 21(2), 303-308]).

Keywords

References

  1. Eves, H. (1972). A Survey of geometry: Revised Edition. Boston: Allyn and Bacon, Inc.
  2. Maturana, H. R. & Varela, F. J. (1998). The tree of knowledge: The biological roots of human understanding (Rev. ed.). Boston, Massachusetts: Shambhala.
  3. National Council of Teachers of Mathematics. (2000): Principles and standards for school mathematics. Reston, VA: NCTM. ME 1999f.03937 for discussion draft (1998)
  4. National Council of Teachers of Mathematics (NCTM)(2006). Curriculum focal points for Prekindergarten through Grade 8 mathematics: A Quest for Coherence. Reston, VA: NCTM ED495271
  5. National Council of Teachers of Mathematics (NCTM) (2008). Focus in high school mathematics: Reasoning and sense making. Reston. VA: NCTM ERIC ED507511
  6. Rahim, M. H. (1986). Laboratory investigation in geometry: A piece-wise congruence approach. Int. J. Math. Educ. Sci. Technol. 17(4),425-447. ME 1986i.02532 https://doi.org/10.1080/0020739860170405
  7. Rahim, M. H. (2003). A new proof of the Pythagorean Theorem using a compass and unmarked straight edge. Int. J. Math. Educ. Sci. Technol. 34(1), 144-150. ME 2003b.01477 https://doi.org/10.1080/0020739031000158227
  8. Rahim, M. H. & Sawada, D. (1986). Revitalizing school geometry through Dissection Motion Operations. Sch. Sci. Math. 86(3),235-246. ME 1986x.00472 ERICEJ334993 https://doi.org/10.1111/j.1949-8594.1986.tb11613.x
  9. Rahim, M. H. & Sawada, D. (1989). Inventing Tangrams through Dissection--Motion Geometry. School Science and Mathematics Journal 89(2), 113-129. ERIC EJ391157 https://doi.org/10.1111/j.1949-8594.1989.tb11899.x
  10. Rahim, M. H.; Sawada, D. & Strasser, J. (1996). Exploring shape transforms through cut and cover: The boy with the ruler. Mathematics Teaching 154, 23-29.
  11. Rahim, M. H. & Sawada, D. (1990). The duality of qualitative and quantitative knowing in school geometry. Int. J. Math. Educ. Sci. Techno!. 21(2),303-308. ME 19901.02060 https://doi.org/10.1080/0020739900210218
  12. Van Hiele, P. M. (1986). Structures and insight: A theory of mathematics education. Orlando, FL: Academic Press. ME 1988b.03491