1. Contributions of Spatial Skills to Geometry Achievement:
Yvonne Kao & John Anderson
Carnegie Mellon University

2. …the first stages of removing obstacles in the way of pupils learning geometry are activities designed to improve the pupils’ understanding of their spatial world. If every teacher of geometry took steps to prepare the pupils ‘spatially’ before mathematizing the ideas, he or she would find the later work so much easier to teach on the base of those spatial experiences. And the pupils would, of course, realize that they already had some spatial foundations with which to secure their understanding of geometrical ideas (Bishop, 1986, p. 144).

3. Background
Spatial Visualization “The ability to mentally manipulate, rotate, twist, or invert a pictorially presented stimulus object” (McGee, 1979)
Correlations between spatial visualization and mathematics achievement generally fall between 0.3 and 0.6 (Battista, 1990)
Spatial visualization predicts geometry achievement (Battista, 1990; Battista, Wheatley, & Talsma, 1992; Casey, Nuttall, Pezaris, & Benbow, 1995; Connor & Serbin, 1985)

4. Background Spatial factors that also predict geometry achievement:
Spatial orientation (Connor & Serbin, 1985)
…the ability to perceive and recognize multiple perspectives or representations of an object (Tartre, 1990)
Flexibility of closure (Hoz, 1981)
The ability to “break one gestalt and form another” (Lohman, 1988)

5. But…
Problems:
Space-math correlations may not be greater than correlations between math and (Friedman, 1995):
verbal ability
reading comprehension
abstract reasoning
sports information
Studies have generally failed to establish a unique contribution of spatial skills to mathematics achievement that is distinct from general intelligence (Chipman, 2005)
The factor-analytic approach is not very useful for teachers (Bishop, 1980)

6. Goal
To better define the relationship between spatial skills and geometry achievement so that we can identify targets for instruction.
To what extent do spatial skills contribute to geometry achievement, above and beyond other cognitive factors?
Do spatial skills matter more for specific geometry subdomains?

7. Design Exploratory study
Cognitive battery from Kit of Factor-Referenced Tests (Ekstrom, French, & Harman, 1976)
Collected data on students’ standardized test scores and course grades

8. Cognitive Battery General Reasoning
Necessary Arithmetic Operations Test

9. Cognitive Battery
Verbal
Vocabulary
Word fluency

10. Cognitive Battery Spatial Visualization Orientation
Flexibility of closure

11. Participants 138 public high school students: Geometry Course N
Females
Males
Mean Age (SD)
% Caucasian
Traditional 2 teachers 84 46 38
14.41 (1.76)
96.20
Intermediate 1 teacher 41 20 21
15.18 (0.87)
97.30
TE (Cognitive Tutor) 1 teacher 13 4 9
16.08 (0.86)
92.31

12. Participants: PSSA

13. Results: Cognitive Battery
p < .0005 *
p = .077
p = .109
p = .020 *
p = .029 *
p < .0005 *

14. Correlations w/PSSA Math
Standardized math test scores were significantly correlated with:
General Reasoning r(132) = 0.67, p < .0005
Verbal Index r(132) = 0.43, p < .0005
Spatial Index r(132) = 0.39, p < .0005
PSSA Reading r(132) = 0.67, p < .0005
 = , t(129) = 6.71, p < .0005
59%
 = .383, t(129) = 6.82, p < .0005

15. Correlations w/PSSA Math
Standardized math test scores were significantly correlated with:
General Reasoning r(132) = 0.67, p < .0005
Verbal Index r(132) = 0.43, p < .0005
Spatial Index r(132) = 0.39, p < .0005
PSSA Reading r(132) = 0.67, p < .0005
 = , t(129) = 9.08, p < .0005
50%
 = , t(129) = 3.64, p < .0005

16. Intermediate Geometry
Unit
Mean Score (SD)
Language of Plane Geometry
74% (21%)
Angles
75% (19%)
Polygons and Polyhedrons
62% (18%)
Introduction to Transformations
76% (18%)
Triangles and Inequalities
69% (24%)
Congruent Triangles
73% (19%)
Parallel Lines
79% (10%)
Properties of Quadrilaterals
70% (14%)
Perimeter and Area
75% (17%)
Similarity
75% (9%)
Overall
74% (11%)

17. Intermediate Geometry
Overall test performance in Intermediate Geometry significantly correlated with:
General Reasoning
Verbal Index r(39) = 0.36, p = .010
Spatial Index r(39) = 0.26, p = .004
 = .051, t(37) = 2.85, p = .007
17%

18. Intermediate Geometry
Overall test performance in Intermediate Geometry significantly correlated with:
General Reasoning
Verbal Index r(39) = 0.36, p = .010
Spatial Index r(39) = 0.26, p = .004
Flexibility of Closure
 = .042, t(37) = 2.83, p = .005
19%

19. Intermediate Geometry
Individual unit analysis
Unit
Mean Score (SD)
Language of Plane Geometry
74% (21%)
Angles
75% (19%)
Polygons and Polyhedrons
62% (18%)
Introduction to Transformations
76% (18%)
Triangles and Inequalities
69% (24%)
Congruent Triangles
73% (19%)
Parallel Lines
79% (10%)
Properties of Quadrilaterals
70% (14%)
Perimeter and Area
75% (17%)
Similarity
75% (9%)
Overall
74% (11%)

20. Intermediate Geometry
Individual unit analysis
General Reasoning and Verbal Index did not significantly correlate with any individual units
Spatial Index and Congruent Triangles: r(51) = 0.41, p = .001
Flexibility of Closure:  = .054, t(37) = 2.16, p = %

21. Conclusions
Spatial skills do contribute to geometry achievement above and beyond other cognitive factors
…especially Flexibility of Closure
…especially for Congruent Triangles
To be continued…

22. Thank You! Joan Son and the participating teachers Cynthia Peng
This research was supported in part by the Institute of Education Sciences, U.S. Department of Education, through Grant R305B to Carnegie Mellon University and in part by NSF ROLE grant REC to Anderson.