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

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Contributions of spatial skills to geometry achievement

Contributions of Spatial Skills to Geometry Achievement:

Yvonne Kao & John Anderson

Carnegie Mellon University


Contributions of spatial skills to geometry achievement

…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
Background 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).

  • 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)


Background1
Background 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).

  • 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)


Contributions of spatial skills to geometry achievement
But… 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).

  • 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)


Contributions of spatial skills to geometry achievement
Goal 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).

  • 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
Design 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).

  • 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
Cognitive Battery 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).

  • General Reasoning

    • Necessary Arithmetic Operations Test


Cognitive battery1
Cognitive Battery 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).

  • Verbal

    • Vocabulary

    • Word fluency


Cognitive battery2
Cognitive Battery 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).

  • Spatial

    • Visualization

    • Orientation

    • Flexibility of closure


Participants
Participants 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).

  • 138 public high school students:


Participants pssa
Participants: PSSA 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).


Results cognitive battery

* 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).

p < .0005

p = .077

p = .109

*

*

*

*

p = .020

p = .029

p < .0005

Results: Cognitive Battery


Correlations w pssa math
Correlations w/PSSA Math 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).

Standardized math test scores were significantly correlated with:

  • General Reasoningr(132) = 0.67, p < .0005

  • Verbal Indexr(132) = 0.43, p < .0005

  • Spatial Indexr(132) = 0.39, p < .0005

  • PSSA Readingr(132) = 0.67, p < .0005

 = 88.354, t(129) = 6.71, p < .0005

59%

 = .383, t(129) = 6.82, p < .0005


Correlations w pssa math1
Correlations w/PSSA Math 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).

Standardized math test scores were significantly correlated with:

  • General Reasoningr(132) = 0.67, p < .0005

  • Verbal Indexr(132) = 0.43, p < .0005

  • Spatial Indexr(132) = 0.39, p < .0005

  • PSSA Readingr(132) = 0.67, p < .0005

 = 119.659, t(129) = 9.08, p < .0005

50%

 = 60.073, t(129) = 3.64, p < .0005


Intermediate geometry
Intermediate Geometry 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).


Intermediate geometry1
Intermediate Geometry 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).

Overall test performance in Intermediate Geometry significantly correlated with:

  • General Reasoning

  • Verbal Indexr(39) = 0.36, p = .010

  • Spatial Indexr(39) = 0.26, p = .004

 = .051, t(37) = 2.85, p = .007

17%


Intermediate geometry2
Intermediate Geometry 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).

Overall test performance in Intermediate Geometry significantly correlated with:

  • General Reasoning

  • Verbal Indexr(39) = 0.36, p = .010

  • Spatial Indexr(39) = 0.26, p = .004

    • Flexibility of Closure

 = .042, t(37) = 2.83, p = .005

19%


Intermediate geometry3
Intermediate Geometry 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).

  • Individual unit analysis


Intermediate geometry4
Intermediate Geometry 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).

  • 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
Conclusions 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).

  • 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
Thank You! 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).

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 R305B040063 to Carnegie Mellon University and in part by NSF ROLE grant REC-0087396 to Anderson.