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The Role of Visualizat ion in Geomet ric Problem S
- lving
The Role of Visualizat ion in Geomet ric Problem S olving Lisa M. - - PowerPoint PPT Presentation
The Role of Visualizat ion in Geomet ric Problem S olving Lisa M. - - PowerPoint PPT Presentation
The Role of Visualizat ion in Geomet ric Problem S olving Lisa M. Weckbacher, Ph.D. California S tate University, Northridge Yukari Okamoto, Ph.D. University of California, S anta Barbara Int roduct ion Visualization in mathematics
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Purpose and S ignificance
- f t he S
t udy
To more fully describe how visualization
functions as a problem solving tool in geometric problem solving
To extend the developing understanding that
individuals who are prone to visual-type thinking tend to be successful problem solvers in geometry
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Theoret ical Framework
Visualization is: (1) not the same as spatial ability or spatial visualization. (2) a cognitive ability used to represent types of mental images. (3) a multifaceted construct that consists of distinct imagery components to represent different obj ect- or spatial- type images (Kozhevnikov, Hegarty, & Mayer,
1999, 2002).
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(4) Verbalizer-Visualizer Dimension
(Richardson, 1977; Mayer & Massa, 2003) Represents individual differences in the ability to process words versus pictures when solving a cognitive task
- Verbalizers
Visualizers (Language S ymbols) (Visual Information)
- Obj ect-Types and S
patial-Types
- One question that remains:
Are there differences amongst spatial types in regards to geometric problem solving?
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Research Quest ions
1. What is the relation between visualization and figural geometric problem solving? 2. Among visualizers, are there obj ect types and spatial types who differ in mathematics achievement and geometric problem solving?
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Met hod
Participants
114 high school students (10th-12th
grades)
58 males, 56 females Mean age = 16.98 years PS
AT math sub-scores showed a normal distribution of mathematics achievement
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Met hod (Cont inued)
Five categories of measures
Group administration by grade level
1.
Mathematics achievement:
- Algebra II and Geometry Grades (not PS
AT)
2.
Visualization:
- S
patial imagery: Mental Rotations and Paper Folding
- Obj ect imagery: S
nowy Pictures Test
3.
Cognitive S tyle:
- Verbalizer-Visualizer Questionnaire (VVQ)
4.
S elf-Assessment in Math and Verbal Activities (S AS )
5.
Figural Geometric Problem S
- lving (FGM)
- 3D and 2D problems drawn from the NAEP
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Result s
Preliminary gender analyses
Males and females did not significantly
differ on most measures other than Mental Rotations, and geometry grades each in favor of males with the exception of S AS
- V
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(1) What is the relation between visualization and figural geometric problem solving?
Mental Rotations and Paper Folding significantly
correlated with the FGM (r = .28, p < .01 and r = .26, p < .01)
S
nowy Pictures and the FGM did not significantly correlate with one another (r = .11, p = .26)
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(2) Among visualizers, are there obj ect types and spatial types who differ in mathematics achievement and geometric problem solving?
VVQ scores revealed most participants to be high
visual
Use of composite spatial visualization divided
participants into low-, average- and high-spatial groups
Low- composite scores represented obj ect types and
high-composite scores represented spatial types
S
cores on S nowy Pictures were used to determine if the two groups represented distinct preferences for each type of imagery
Data did not support a subsample of obj ect-type
visualizers
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S pat ial-Type Visualizers
The high-spatial or spatial-type visualizers significantly
- utperformed the low- and average-spatial groups on the FGM;
the highest grades in geometry also favored these spatial-types
The three spatial groups did not significantly differ in algebra
grades
Amongst spatial types, performance differences emerged on
the 3D and 2D subscales of the FGM with fewer high scores on 3D items
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Low Spatial Average Spatial High Spatial
(n = 25) (n = 45) (n= 23) (Spatial Types) __________________________________________________________________________ Algebra II M (SD) 88.72 (6.43) 87.58 (9.00) 90.91 (5.70) Geometry M (SD) 85.84 (8.74) 87.02 (9.76) 93.22 (4.10)** FGM M (SD) 13.76 (3.03 15.42 (2.75) 16.61 (2.78)** FGM 3D M (SD) 6.72 (1.60) 7.53 (1.46) 7.78 (1.51)* FGM 2D M (SD) 7.04 (2.01) 7.89 (1.80) 8.83 (1.53)** *p < .05. **p < .01.
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Limit at ions of t he S t udy
The use of one obj ect imagery measure The use of grades as a sole index for
mathematics achievement
S
ampling bias
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General Discussion
The importance of spatial imagery as a
distinct visual process in geometric problem solving
S
trength in spatial visualization ability seemed to provide an advantage in geometric problem solving
The proper use of visualization may help
students to become better problem solvers in geometry
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Educat ional Implicat ions
Developing spatial ability at the elementary level
could help improve performance in geometry by the high school years
Classroom practices to develop spatial ability
Fall 2008 In-S
ervice for Elementary S chool Teachers
Developing S
pat ial Abilit ies Through Geomet ric Act ivit ies
Quick Draw: Developing S
patial S ense (Grayson Wheatley)
“ What did you see and how did you draw it?
”
“ What shapes do you see?
”
Teacher Questionnaire “ To date, briefly describe your experiences with
nurturing the development of spatial ability in classroom practice.”
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“ We do some work with 3D shapes… flips and turns.” “ A chapter in a math book, tangrams, pattern blocks… ” “ … I have not had much experience in developing spatial ability within the classroom.” “ S hape-making with cards… ” “ I have done estimating in j ars, legos, etc… ”
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