Dr. Jackie Sack NCTM Annual Meeting March 23, 2007 B u ild I t D raw It I n t er p r e t It

2-D:The van Hiele Model of Geometric Thought Levels Language Phases  Visual  Descriptive  Relational  Deductive  Rigor  Information  Guided Orientation  Explicitation  Free Orientation  Integration

3-D Frameworks  Yakimanskaya  van Niekerk

What skills are needed?  Turn, shrink and deform 2-D and 3-D objects.  Analyze and draw perspective views, count component parts and describe attributes that may not be visible but can be inferred.  Physically and mentally change the position, orientation, and size of objects in systematic ways as understandings about congruence, similarity and transformations develop. (NCTM, 2000)

Does it make sense to begin with 2-D figures?  Rectilinearity or straightness?  Flatness?  Parallelism?  Right angles?  Symmetry?  Circles?  Similarity?

3-D Models

Conventional-Graphic Models

Conventional-Graphic Models: Functional Diagrams

Conventional-Graphic Models: Assembly Diagrams

Conventional-Graphic Models: Structural Diagrams

Intervention Program

Soma Pieces

Three visual modes  Full-scale or scaled-down models of objects  Conventional-graphic models  Semiotic models

Framework for 3-Dimensional Visualization 3-DIMENSIONAL MODEL CONVENTIONAL GRAPHIC REPRESENTATION OF THE 3-D MODEL VERBAL DESCRIPTION OF THE 3-D MODEL (oral or written) REBUILD IT TALK ABOUT IT REPRESENT IT ABSTRACTLY DRAW OR RECOGNIZE IT IN A PICTURE topfrontside SEMIOTIC OR ABSTRACT REPRESENTATION OF THE 3-D MODEL This slide is not to be reproduced in any form without the express permission of Jackie Sack.

3-Dimensional Model Stimulus Which piece? Can you rebuild it using loose cubes?

3-Dimensional Model Stimulus Rebuild it using loose cubes. Draw it. Explain how to build it. Can you make this figure using two Soma pieces?

Draw it…

2-D Conventional Graphic Model Show how these two Soma pieces can be combined to create this figure. Rebuild it using loose cubes. Draw it. Explain how to build it.

2-D Conventional Graphic Model Show how these two Soma pieces can be combined to create this figure.

2-D Conventional Graphic Model Show how these three Soma pieces can be combined to create this figure. + +

2-D Conventional Graphic Model Which two Soma pieces were combined to create this figure?

2-D Conventional Graphic Model Which two Soma pieces were combined to create this figure?

Describe it verbally Use Soma pieces 1, 2, 3, 4 and 5. 5 and 4 go on the lower front. Stand 3 behind 5, three cubes tall; and 2 next to 3 with its short leg on the ground pointing toward the front, next to 4. 1 goes on top of 2 and

Represent the figure abstractly

Upper level Lower level

Represent the figure abstractly + +

 How many and which Soma pieces do you need to build this figure?  Build the figure.

Beyond cubes…

Describe the figure’s net C

Describe the 3-D figure Y

U

2-D Implications: Reflections

2-D Implications: Rotations

Transformations: 2-D Geometry

180 o Transformations: 2-D Geometry

.9cos15 o.9cos105 o sin15 o.9sin105 o Transformations: Pre-Calculus – Calculus

Transformations: Back to Geometry

References Crowley, Mary L. “The van Hiele Model of the Development of Geometric Thought.” In Learning and Teaching Geometry, K – 12, 1987 Yearbook of the National Council of Teachers of Mathematics (NCTM), edited by Mary M. Lindquist, pp Reston, VA: NCTM, Freudenthal, Hans. Didactical Phenomenology of Mathematical Structures. Dordrecht, Holland: Reidel, Fuys, David, Geddes, Dorothy and Tischler, Rosamond. The van Hiele Model of Thinking in Geometry among Adolescents, Journal of Research in Mathematics Education, Monograph Number 3, Reston, VA: NCTM, NCTM. Principles and Standards for School Mathematics. Reston, VA: NCTM, van Hiele, Pierre. M. Structure and Insight: A Theory of Mathematics Education. Orlando, FL: Academic Press, van Niekerk, (Retha) H. M. “From Spatial Orientation to Spatial Insight: A Geometry Curriculum for the Primary School.” Pythagoras, 36 (1995a): van Niekerk, Retha. “4 Kubers in Africa.” Paper presented at the Panama Najaarsconferentie Modellen, Meten en Meetkunde: Paradigmas's van Adaptief Onderwijs, The Netherlands, 1995b. van Niekerk, Retha. “4 Kubers in Africa.” Pythagoras, 40, (1996): van Niekerk, (Retha) H. M.. “A Subject Didactical Analysis of the Development of the Spatial Knowledge of Young Children through a Problem-Centered Approach to Mathematics Teaching and Learning.” Ph.D. diss., Potchefstroom University for CHE, South Africa, Yakimanskaya, I. S. The Development of Spatial Thinking in School Children. Edited and translated by Patricia S. Wilson and Edward J. Davis. Vol. 5 of Soviet Studies in Mathematics Education, Reston, VA: NCTM, 1991.