טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentWidder Mirela
SubjectHiding in Plane View: Visual Perception of Two-Dimensional
Representations in Teaching Spatial Geometry
DepartmentDepartment of Education in Science and Technology
Supervisors Professor Emeritus Abraham Berman
Professor Boris Koichu
Full Thesis textFull thesis text - English Version


Abstract

   The need to visualize 3-D configurations from their 2-D representations represents a main impediment for both, teaching and learning spatial geometry. This study grew out of the conviction that a priori knowledge about the inherent visual difficulty of     2-D sketches may enhance spatial geometry instruction by enabling a priori problem ranking, and dynamic monitoring of the visual difficulty when students solve 3-D geometry problems in a computer-supported learning environment. The study focused on cubes with auxiliary constructions and was conducted in three complementary stages.

   The goal of Stage 1 was to theoretically develop, conceptualize and quantify an a priori measure of visual difficulty for cubes with auxiliary constructions, based on two visual attributes embedded in 2-D sketches: (1) potentially misleading geometrical information (PMI), and (2) potentially helpful geometrical information (PHI). The cube’s orientation was also considered, as un-normatively oriented cubes can be more visually challenging. After defining the various geometric attributes of PHI and PMI in 110 normatively oriented cube-related sketches, an unequivocal counting method was developed and validated by 20 experienced high-school mathematics teachers. The ratio #PHI/#PMI was suggested as an a priori measure of visual difficulty.

  The a priori measure of visual difficulty was empirically validated at Stage II. The validation was obtained by comparing the theoretically anticipated visual difficulty of 12 normatively oriented cube-related sketches (ranked according to their ratios #PHI/#PMI), to the actual visual difficulty, as measured by the percentile of correct/desired comprehension and the study-time allocation of 174 high-school students. The same procedure was repeated for 12 corresponding un-normatively oriented sketches, obtained by a vertical flip of the original sketches. Participants were also tested for their spatial ability using a standardized test. Findings related to both correct/desired comprehension and study-time allocation suggest that a greater cognitive complexity is involved in comprehending un-normatively oriented sketches.

   Having empirically established #PHI/#PMI as an a priori measure of visual difficulty, our goal at Stage III was to find out how spatial geometry learning processes with Dynamic Geometry Software (DGS) can be characterized by the dynamic changes in visual difficulty on the computer screen. Twenty one students, seven of high, seven of medium and seven of low spatial ability level, were engaged in solving spatial geometry problems during individual work-sessions, immediately followed by semi-structured interviews. A qualitative constant comparative method was used in order to relate changes applied to #PHI/#PMI during work-sessions with other strategical, intentional and attitudinal findings. Findings show that the learners operate with DGS in order to reduce visual difficulty, in a non-linear process, which is affected by epistemological, motivational and affectional aspects, as well as by individual spatial skills.

   Focusing on visual difficulty, this study contributes a valuable theoretical contribution, as it tightens the links between spatial geometry instruction and cognitive psychology, allowing researchers and instructors a better understanding of the nature of visual impediments and their perceptual impact. Research findings have important practical implications for instruction planning, and can bring a well-desired constructive change to spatial geometry education.

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