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 � Geometric problem solving (K-12) � Largely neglected despite a considerable need � Tends to be the weakest content area for US students (NAEP and TIMS S ) � Three-dimensional geometry in particular
Purpose and S ignificance of 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
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).
(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?
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?
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
Met hod (Cont inued) Five categories of measures Group administration by grade level Mathematics achievement: 1. • Algebra II and Geometry Grades (not PS AT) Visualization: 2. S patial imagery: Mental Rotations and Paper • Folding • Obj ect imagery: S nowy Pictures Test Cognitive S tyle: 3. Verbalizer-Visualizer Questionnaire (VVQ) • S elf-Assessment in Math and Verbal Activities 4. (S AS ) Figural Geometric Problem S olving (FGM) 5. 3D and 2D problems drawn from the NAEP •
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
(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)
(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
S pat ial-Type Visualizers � The high-spatial or spatial-type visualizers significantly outperformed 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
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.
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
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
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.”
“ 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… ”
Fut ure Work Research Question: � Why are students less apt to do well in solid geometry? Fall 2009 In-S ervice: � 3D or solid geometry in relation to spatial ability � Teacher’ s knowledge of solid geometry � Role of solid geometry in the elementary classroom Thank you! (References available upon request)
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