Contributions of Spatial Skills to Geometry

Achievement:Yvonne Kao & John AndersonCarnegie Mellon University

…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).

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)

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)

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)

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?

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

Cognitive Battery• General Reasoning

• Necessary Arithmetic Operations Test

Cognitive Battery• Verbal

• Vocabulary

• Word fluency

Cognitive Battery• Spatial

• Visualization

• Orientation

• Flexibility of closure

Participants• 138 public high school students:

Geometry Course

N Females Males Mean Age (SD)

% Caucasian

Traditional2 teachers

84 46 38 14.41 (1.76) 96.20

Intermediate1 teacher

41 20 21 15.18 (0.87) 97.30

TE (Cognitive Tutor)1 teacher

13 4 9 16.08 (0.86) 92.31

Participants: PSSA

Course

TEIntermediate

Traditional

Per

cen

t o

f S

tud

ents

100.0%

80.0%

60.0%

40.0%

20.0%

0.0%

Below BasicBasicProficientAdvanced

Level

PSSA Math

Course

TEIntermediate

Traditional

Per

cen

t o

f S

tud

ents

100.0%

80.0%

60.0%

40.0%

20.0%

0.0%

Below BasicBasicProficientAdvanced

Level

PSSA Reading

Results: Cognitive Battery

p = .077p < .0005

*

p = .020

*

*

p = .029

*

p < .0005

*

p = .109

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 = .383, t(129) = 6.82, p < .0005

= 88.354, t(129) = 6.71, p < .0005

59%

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

= 60.073, t(129) = 3.64, p < .0005

= 119.659, t(129) = 9.08, p < .0005

50%

Intermediate GeometryUnit 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%)

Intermediate GeometryOverall 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%

Intermediate GeometryOverall 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%

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%)

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 = .03711%

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…

Thank You!Joan Son and the participating teachersCynthia Peng

This research was supported in part by the Institute of Education Sciences, U.S. Department of Education, through Grant R305B040063 to Carnegie Mellon University and in part by NSF ROLE grant REC-0087396 to Anderson.