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1 vii Table of Contents 1 Introduction Overview Combining Manipulatives and Software HyperGami JavaGami Results Reader's Guide Tools for Spatial Thinking Some Definitions of Spatial Ability Spatial Ability and Science/Mathematics Achievement Is Spatial Thinking Trainable? The Case for Manipulatives Beyond Manipulatives HyperGami Description Work with Elementary, Middle, & High-School Students Design Lessons Learned Themes Discovered JavaGami Description Implementation of JavaGami Addressing Design Issues in HyperGami Use of JavaGami Assessment of Children's Spatial Learning Overview Net and Solid Card Matching Verbal Shape Description Polyhedron Drawings Folding Net Drawings Surface Development Test Cube Rotations Chapter Summary A Case Study in JavaGami Overview Session #1: Getting Started Session #2: Capping and Stretching Session #3: A Goldfish Gift Session #4: Slicing Session #5: Mystery Shapes Discussion

2 viii 7 Stepping Back Related Work Contributions Future Plans What is REAL? References Appendix A HyperGami Orihedra Appendix B Work by Elementary and Middle-School Students in HyperGami Appendix C HyperGami Work by High School Students Appendix D Work by Elementary and Middle-School Students in JavaGami Appendix E Sample Assessment Procedures

3 ix List of Tables Table 5-1 Table 5-2 Elementary and middle-school students working with HyperGami and JavaGami, High school students working with HyperGami, Table 5-3 Assessment summary descriptions. 52 Table 5-4 Number of incorrect solid matchings (out of 5 pretests and 4 post-tests) for folding nets. 55 Table 5-5 Polyhedra used in the shape-description task. 60 Table 5-6 Table 5-7 Table 5-8 Sample categorizations of children's polyhedron descriptions. Samples of pre- and post- shape descriptions by children who worked only with paper shapes. Pre- and post- JavaGami shape descriptions by a seventh-grade girl Table 5-9 Seventh grade boy's pre- and post- descriptions Table 5-10 Polyhedra used in the shape-drawing task. 70 Table 5-11 Polyhedra given to student groups. Students were asked to draw what these shapes would look like unfolded. 75 Table 5-12 (a) Categories of folding nets. 77 Table 5-12 (b) Categories of folding nets, continued. 78 Table 5-13 Orthogonally-drawn cuboctahedron nets Table 5-14 Table 5-15 Key to the symbols used in the folding net summary tables. Pre- and post- net classifications for elementaryand middle-school JavaGami students Table 5-16 Net classifications for elementary- and middleschool HyperGami students. 84

4 x Table 5-17 Table 5-18 Pre- and post- net classifications for paper shapes students -- students worked with either six or eight shapes, depending on their general mood by the end of the sixth shape on the pre-test. ETS Surface Development Scores for JavaGami and Paper Shapes students Table 6-1 (a) Annotated transcript of the mystery shapes session. 106 Table 6-1 (b) Annotated transcript, part Table 6-1 (c) Annotated transcript, part

5 xi List of Figures Figure 1-1 Figure 1-2 Operations on an octahedron. The octahedron in (a) is stretched along the z-axis in (b); has a pyramid added to one of its faces in (c); is sliced into parts in (d); and is truncated at a single vertex in (e). Mathematical paper sculpture created in HyperGami: (a) Venus Flytrapohedron, and (b) Turtlehedron. 3 5 Figure 1-3 Overview of the JavaGami interface. 6 Figure 2-1 Figure 2-2 Figure 2-3 Figure 3-1 Figure 3-2 Figure 3-3 Samples of work by (a) Johannes Kepler, (b) M.C. Escher, and (c) Scott Kim. A sample mental rotations task. (Shepard & Metzler, 1971). Samples of mental paper-folding tasks. (Shepard & Feng, 1972). A view of the HyperGami screen in the course of a typical scenario. Applying a linear map to a polyhedral object by direct manipulation. The student (a) selects a menu choice and the system (b) displays the new solid and (c) the corresponding folding net. Creating a capped dodecahedron object from a dodecahedron. The student first (a) enters a Scheme expression in the transcript window and clicks on the face to cap. The new solid object is shown in (b) and the corresponding folding net is shown in (c) Figure 3-4 The cube (a) is capped (b); stretched along the x- axis (c); and sliced (d). Corresponding folding nets generated by the software are shown under the solid objects. 21 Figure 3-5 Figure 3-6 Folding nets for (a) an origami snail, and (b) an origami turtle decorated by a pair of girls, ages 8 and 13. Student work from Fall origami snails, turtles, frogs, and a giraffe; a rotating mathematical toy hexaflexagon in the center; a modular origami piece on the left; a dodecahedron; and a great stellated dodecahedron

6 xii Figure 3-7 Penguinhedra developed by the author illustrate the idea of polyhedra as building blocks for paper sculpture. 25 Figure 3-8 (a) A polyhedral castle designed by a 8th grade girl; (b) a polyhedral sculpture by a 6th grade boy; and (c) a dinosaur standing on a log by a 7th grade boy. 26 Figure 3-9 Figure 3-10 Figure 3-11 Figure 3-12 Figure 3-13 Figure 3-14 A caterpillarhedron which was a collaborative effort between six children and their parents. (a) A capped cuboctahedron sculpture by two 9thand 10th- grade girls (printed in black-and-white and colored with markers) (b) a paper rocket by a 9th grade boy (c) a polyhedral sculpture by a 9th grade boy based on a design by mathematician Alan Holden. The interface for changing the coordinate scale in HyperGami. (a) "Mrs. Studer rocks": A folding net for a dodecahedron for an eighth-grade boy's favorite teacher; (b) "I love dad": a folding net for a gift for a fifth-grade girl's father; (c) "Arielle": a folding net for a gift for the same fifth-grader's sister. Paper gifts received at a conference of origami artists. (a) A "friendship icosahedron" net with the initials of two girls. (b) A net for one of the pyramids of a twenty-point great stellated dodecahedron, also with the girls' initials Figure 4-1 Overview of the JavaGami system. 37 Figure 4-2 A cube net with a Martian motif designed in JavaGami by a 9 year-old girl. 38 Figure 4-3 Clicking to rotate the polyhedron about the x-axis. 39 Figure 4-4 A wireframe version of a capped cube. The wireframe object can be rotated by dragging the mouse over the window. 39

7 xiii Figure 4-5 Figure 4-6 Figure 4-7 Figure 4-8 Figure 4-9 Figure 4-10 Applying successive functions to a cube in JavaGami. The cube in (a) is capped; the capped cube (b) is truncated at a vertex; the resulting solid (c) is then stretched along the z-axis; and the resulting shape (d) is sliced into two parts: (e) and (f). Instead of menu toggles (left) to access sets of shapes, the current version of JavaGami has a "Shape Sets" picker bar (upper right). The set of shapes is loaded into the "Shapes" window at the bottom. The indicator panel at the bottom displays the current color and tool selected. A dodecahedron in the wireframe window illustrating the orientation of opposite pentagons relative to one another. A modular sculpture composed of cuboctahedra and antiprisms designed in JavaGami by a 5thgrade boy. (a) A giant-sized pencil designed by a 10 year-old boy; (b) An ice cream cone designed by a 9 yearold girl Figure 5-1 Folding nets used in net-solid matching task. 54 Figure 5-2 Figure 5-3 Figure 5-4 Figure 5-5 Figure 5-6 Figure 5-7 Polyhedra to add to the net-solid matching task. (a) snub cube and square antiprism; (b) regular pentagonal bipyramid; (c) stretched tetrahedron and stretched triangular antiprism; (d) rhombic dodecahedron; (e) dual of the cuboctahedron. Mitchelmore's classification of drawings of regular solid figures. From Mitchelmore (1978), p Drawings of an octahedron by (a) NAT (9th grade boy) and (b) AMD (10th grade girl). Drawings of a rhombic dodecahedron by (a) ALD (12th grade girl); (b) RAP (9th grade boy); and (c) RAW (9th grade boy). Drawings of (a) a tetrahedron by ELV (10th grade boy); (b) an edge-capped cube by BRH (9th grade girl); and the same edge-capped cube by (c) CAC (10th grade girl). Folding nets for the octahedron drawn by (a) MEB, grade 12 and (b)tsr, grade

8 xiv Figure 5-8 Figure 5-9 Figure 5-10 Figure 5-11 Figure 5-12 Rhombic dodecahedron nets drawn by (a) LAP, grade 9; and (b) HSW, grade 4. Pre- and post test nets drawn by a fourth-grade boy working with HyperGami. An eleventh-grade girl's (a) pre- and (b) post-test nets of a capped cube Pre- and post- nets of a cuboctahedron drawn by a ninth-grade boy. Sample question from the ETS Surface Development test Figure 5-13 Pre- and post- test scores from 37 high school 87 students on the ETS Surface Development test. Figure 5-14 Sample question from the ETS Cube Rotations test. 89 Figure 5-15 Figure 6-1 Pre- and post- test scores from 25 high school students on the ETS Cube Rotations test. An overview of the version of the software Jesse worked with in his first session Figure 6-2 Jesse's first net decorated in JavaGami. 94 Figure 6-3 Jesse's finished dodecahedron. 95 Figure 6-4 Jesse's folding net for the octahedron. 96 Figure 6-5 A new version of the software which included function buttons and separate windows for selecting shapes. 97 Figure 6-6 A folding net for the truncated cube. 98 Figure 6-7 (a) A picture of the triply-capped truncated cube, and (b) the corresponding folding net generated by the software. 98 Figure 6-8 Jesse's truncated tetrahedron with a cap. 100 Figure 6-9 The folding net for Jesse's fish body. 101 Figure 6-10 The folding net for Jesse's fish head. He took some time positioning the eyes in the correct place. 102 Figure 6-11 The folding net for the fish tail. 102 Figure 6-12 The sliced dodecahedron net. 103

9 xv Figure 6-13 Figure 6-14 Figure 6-15 (a) Jesse's sliced dodecahedron. (b) A different view of the same shape. Mystery shapes: Jesse was given paper models of these shapes and asked to recreate the shapes in JavaGami. Jesse's strategies for creating the stretched cuboctahedron Figure 6-16 (a). The cuboctahedron as it appears in the Shapes palette; (b) the cuboctahedron in the wireframe window. 110 Figure 7-1 Figure 7-2 Figure 7-3 (a) One of the shape rendering interfaces in form. Z; (b) Solid objects and their folding nets in form. Z. (a) The shape-modeling interface in Touch-3D, and (b) an unfolded model. (a). The modeling interface for tabs+, and (b) the folding net interface Figure 7-4 Plato's World in Shape Up! 115 Figure 7-5 Screen interface of The Factory 116 Figure 7-6 Figure 7-7 Scheherezade. A crank-shaft automaton designed by the author. (a) The Barecats by Paul Spooner and Matt Smith. (b) The Mill Girl and Toff by Paul Spooner

polyhedral starting shapes; a ThreeD window showing a three-dimensional

polyhedral starting shapes; a ThreeD window showing a three-dimensional 18 3. HyperGami 3.1 Description HyperGami (Eisenberg and Nishioka, 1994, 1997) is a programmable design environment for paper sculpture implemented in Scheme to run on the Macintosh platform. The interface