Dr. Jackie Sack

NCTM Annual Meeting

March 23, 2007

B u i l d I tD r aw I t

I n t e r p r e t I t

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

1 2 3 4

5 76

Three visual modes

Full-scale or scaled-down models of objects

Conventional-graphic models Semiotic models

Side ViewFront ViewTop View

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

1

12

top front side

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?

1 23 4

5

7

6

2-D Conventional Graphic Model

Which two Soma pieces were combined to create this figure?

1 23 4

5

7

6

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.

2

4

1

3

5

54 3

21

Represent the figure abstractly

Represent the figure abstractly

Upper level

Lower level

5

55

6

56

66

Represent the figure abstractly

1

1

11

1

2

1

1 2

+

+

1

1

11

1

2 2

2

Represent the figure abstractly

How many and which Soma pieces do you need to build this figure?

Build the figure.

2

2

2

1

1

1

1

11

Beyond cubes…

Describe the figure’s net

C

Describe the 3-D figure

Y

Describe the 3-D figure

U

2-D Implications:Reflections

2-D Implications:Rotations

Transformations:2-D Geometry

median

180o

Transformations:2-D Geometry

8

6

4

2

-2

-4

-10 -5 5

.9cos15o .9cos105o -1.25

.9sin15o .9sin105o 1.1

0 0 1

Transformations:Pre-Calculus – Calculus

6

4

2

-2

-4

-10 -5 5

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. 1-16. Reston, VA: NCTM, 1987.

Freudenthal, Hans. Didactical Phenomenology of Mathematical Structures. Dordrecht, Holland: Reidel, 1983.

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.