Dr. Jackie Sack

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• Dr. Jackie Sack Irma Vazquez
• NCTM Regional Meeting
• Houston, November 30, 2007

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3-D Frameworks

• Yakimanskaya
• van Niekerk

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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)

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Does it make sense to begin with 2-D figures?

• Rectilinearity or straightness?
• Flatness?
• Parallelism?
• Right angles?
• Symmetry?
• Circles?
• Similarity?

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3-D Models
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Conventional-Graphic Models
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Conventional-Graphic Models Functional Diagrams
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Conventional-Graphic Models Assembly Diagrams
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Conventional-Graphic ModelsStructural Diagrams
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Intervention Program
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Soma Pieces
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Three visual modes

• Full-scale or scaled-down models of objects
• Conventional-graphic models
• Semiotic models

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

• Which piece?
• Can you rebuild it using loose cubes?

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3-Dimensional Model Stimulus

• Can you make this figure using two Soma pieces?

• Rebuild it using loose cubes.
• Draw it.
• Explain how to build it.

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Draw it
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Draw it
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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.
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2-D Conventional Graphic Model

• Show how these two Soma pieces can be combined to
  create this figure.

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2-D Conventional Graphic Model

• Show how these three Soma pieces can be combined
  to create this figure.

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2-D Conventional Graphic Model

• Which two Soma pieces were combined to create
  this figure?

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2-D Conventional Graphic Model

• Which two Soma pieces were combined to create
  this figure?

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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 4.
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Geocadabra (Ton Lecluse)

• Dynamic
• Computer
• Interface

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Represent the figure abstractly
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Represent the figure abstractly
Lower level
Upper level
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Represent the figure abstractly


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Represent the figure abstractly

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

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Beyond cubes
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Describe the figures net
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Describe the 3-D figure
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Describe the 3-D figure
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Geocadabra (Ton Lecluse)
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Geocadabra (Ton Lecluse)
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2-D ImplicationsReflections
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2-D ImplicationsRotations
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Transformations2-D Geometry
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Transformations2-D Geometry
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TransformationsPre-Calculus Calculus
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TransformationsBack to Geometry
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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. 1-16.
  Reston, VA NCTM, 1987.
• 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, 1988.
• NCTM. Principles and Standards for School
  Mathematics. Reston, VA NCTM, 2000.
• van Hiele, Pierre. M. Structure and Insight A
  Theory of Mathematics Education. Orlando, FL
  Academic Press, 1986.
• van Niekerk, (Retha) H. M. From Spatial
  Orientation to Spatial Insight A Geometry
  Curriculum for the Primary School. Pythagoras,
  36 (1995a) 7-12.
• 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) 28-33.
• 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, 1997.
• 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.

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More information

• Jackie Sack
• jsack_at_rice.edu
• Irma Vazquez
• ivazquez_at_houstonisd.org
• Geocadabra demo link
• http//home.casema.nl/alecluse/setupeng.exe
• Email the author to get the authorization for
  2008
• alecluse_at_casema.nl
• Geocadabra user group access through
• The Rice University School Mathematics Project
• http//rusmp.rice.edu

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