Transcript 3-D: The Foundation for Developing Geometric Thinking

Bu i l d I t
D r aw I t
Interpret
I t
Dr. Jackie Sack & Irma Vazquez
NCTM Regional Meeting
Houston, November 30, 2007
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
1
2
5
4
3
6
7
Three visual modes



Full-scale or scaled-down models of
objects
Conventional-graphic models
Semiotic models
Top V iew
Front View
Side V iew
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
1
2
top
front
1
side
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
Can you make
this figure using
two Soma
pieces?
Rebuild it using loose cubes.
Draw it.
Explain how to build it.
Draw it
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
1
5
2
3
Which two Soma
pieces were
combined to create
this figure?
6
7
4
2-D Conventional Graphic
Model
1
5
2
3
Which two Soma
pieces were
combined to create
this figure?
6
7
4
Describe it verbally
2
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.
1
3
4
5
2
1
4
5
3
Geocadabra (Ton Lecluse)
Dynamic
Computer
Interface
Represent the figure
abstractly
Represent the figure
abstractly
6
6
6
6
Lower
level
5
5
5
5
Upper
level
Represent the figure
abstractly
1
1
1
+
1
1
1
2
1
1
2
1
2
2
1
+
1
2
1
Represent the figure
abstractly


How many and
which Soma pieces
do you need to
build this figure?
Build the figure.
1
1
2
2
1
1
1
1
2
Beyond cubes…
Describe the figure’s net
C
Describe the 3-D figure
Y
Describe the 3-D figure
U
Geocadabra (Ton Lecluse)
Geocadabra (Ton Lecluse)
2-D Implications:
Reflections
2-D Implications:
Rotations
Transformations:
2-D Geometry
Transformations:
2-D Geometry
180o
median
Transformations:
Pre-Calculus – Calculus
8
6
4
2
-10
-5
5
.9cos15-2o
.9cos105o
-1.25
.9sin15o
.9sin105o
1.1
0
-4
0
1
Transformations:
Back to Geometry
6
4
2
-10
5
-5
-2
-4
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.
More information
Jackie Sack
[email protected]
Irma Vazquez
[email protected]
Geocadabra demo link
http://home.casema.nl/alecluse/setupeng.exe
Email the author to get the authorization for 2008
[email protected]
Geocadabra user group access through
The Rice University School Mathematics Project
http://rusmp.rice.edu