Walter Whiteley's Home Page

Coordinator, Graduate Diploma in Mathematics Education

On Sabbatical 2009-2010

Courses for 2008-2009

I was course director for Math 4100: Topics in Mathematics Education. It meets on Tuesday Evening 4-7. There was a moodle site for the course.
You may find the materials at http://wiki.math.yorku.ca/index.php/CMESG relevant. Theses are materials we collected for a working group at the June 2007 meeting of the Canadian Mathematics Education Study Group
You may also find some of the materials at http://wiki.math.yorku.ca/index.php/Spatial_Reasoning_Links relevant. Theses are materials from a summer project with two of the students from Math 4100.

Courses for 2007-2008

Here are links to two of my recent talks on Mathematics Education.
A public lecture at the Royal Canadian Institute on Mathematics with Eye and Hand. This is on-line as a flash document or as a webcast
A lecture at the First Canadian Computer Algebra and Dynamic Geometry Systems in Mathematics Education (CCADGME) Math with Eye and Hand: GSP, Manipulatives, and Post-Secondary Math

Courses for 2006-2007

In the fall-winter I taught Math 3050 6.0 Introduction to Geometries. For additional communication, we used pages on the Departmental Wiki: http://wiki.math.yorku.ca/index.php/MATH_3050
and a phorum on the AML server: http://forum.aml.yorku.ca/list.php?f=52.
The TA for the course was Bernd Schulze.
The outline for the course is at Math 3050 06.

The course pages from a previous version are at Math 3050.06 .

Pedagogical Projects

• I have organized a working group, under the Canadian Mathematics Education Forum. The working group is focused on early childhood geometry, with an interest in spreading a curriculum based on symmetry from preK to 14 (university). The wiki sites with more information are at: http://wiki.math.yorku.ca/index.php/CMEF_Geometry_Curriculum and at: http://wiki.math.yorku.ca/index.php/CMEF_Early_Geometry

• In the fall of 2008, I gave a public lecture at the Royal Canadian Institute on Mathematics with Eye and Hand. This is on-line as a flash document or as a webcast

• With Shelley Hunter and Donna Kotsolopoulas, I was a co-leader of a working group on Geometry, Space and Teachnology at the June 2007 meeting of the Canadian Mathematics Education Study Group. We collected materials, references, and links, at: http://wiki.math.yorku.ca/index.php/CMESG
  A document related to this is the OISE 1967 study Geometry K-13

• Work on how we learn to use visual information - and how it can be taught. A current paper "To see like a mathematician" is available for download (see below). I participated in a working group supported by SIGGRAPH creating a White Paper on Visual Learning in Science and Engineering. You can find our report here.

  Further discussion also occured at the recent International Congress on Mathematics Education 10, in Copenhagen, where I co-chaired a Topic Study Group 16 on Visualisation in the teaching and learning of mathematics TSG 16

  My own paper, distributed at ICME10, was about Claims and Questions towards a Research Program in Visualization. Download a PDF file of this paper.

  I am working with Margaret Sinclair, Faculty of Education, on an SSHRC funded project developing activities for pre-service and in-service elementary teachers developing visual and spatial reasoning. The hypothesis is that: these are important for younger children; the children can do a lot more of this that the curriculum currently provides; teachers are a key resource in supporting this critical development for school children.

  I have a series of PowerPoint presentations (with companion GSP sketches, web pages, etc.) on this and related topics. I plan to post some of them but will share them, informally, with interested folk who contact me. I have local support, including a small SSHRC grant, for this ongoing work.

  Earlier presentations were partially supported by a SCOTL grant and an SSHRC conference travel grant to attend the International Congress of Mathematical Education in Tokyo, where I gave an invited one hour regular lecture. Copies of some related papers are below, in PDF format.

• You can find an interview with me, about geometry, at the Math Mania web site. Feel free to correspond with me around any questions this raises.

• Using dynamic geometry programs for teaching (SCOTL grant 1993-95)

• with Pat Rogers, Encouraging small group work and its effect on student satisfaction (SCOTL grant 1996-97). One outcome of the project are some short pamphlets available an to support study groups which are available on the Departmental Web Pages.

  I also served as the York University representative on the Council of Ontario Universities (COU) Mathematics Curriculum Review Task Force. We provided input to the ministry (and the minister) and are working to coordinate the responses, including university admissions requirements for the new curriculum. Feel free to contact me for information.
  

• Member of the Board of Directors of the Ontario Association of Mathematics Educators (OAME)

• York MathStat Representative to the Fields Institute Mathematics Education Forum. This gathering of mathematicians and mathematics educators from all levels of education around Ontario is looking at issues such as the changing curriculum in Ontario, and support for math teachers using the new curriculum. The Mathematics Education Forum meetings are open, and I encourage anyone interested in working on these issues to contact me for further information.

Work for the York University Faculty Association

• YUFA Chief Steward.
• YUFA nominated trustee of the York Pension Fund.
• Trustee YUFA Trust Fund
• Trustee YUFA Retirees Benefits Trust

Areas of Current Research

Discrete Applied Geometry

This May, Bernd Schulze completed his Ph.D. Thesis on Combinatorial and Geometric Rigidity with Symmetry Constraints. It is available for download

Recent work, available only in preprint form, includes:

• Generic rigidity of frameworks with Symmetry: preprint with Robert Connelly, Patrick Fowler, Simon Guest, Bernd Schulze.
  Further results on rigidity of symmetric frameworks - work of Bernd Schulze, are in his thesis linked above. .
• Swapping blocks and holes in spherical frameworks with Elissa Ross and Wendy Finbow Singh. An investigation based on the duality of stresses and motions for spherical polyhedra
• The analysis of 'Assur Graphs' from mechanical engineering of linkages, with Brigitte Servatius and Offer Shair.
• Work on global rigidity of powers of graphs in dimensions 2 and 3, with Matthew Cheung.
• Computing the segmental motions and variable loop flexibility of Immunoglobulins with Sean Law (Biology) and Dean Gill Wu.
• Comparing the bar and body and bar and joint molecular models, with Tiong-Seng Tay (National University of Singapore).
• Independent direction and parallel drawing constraints in 3-D, with Natalie Stoianov, and now with Elissa Ross;
• Isostatic Almost Spherical Frameworks vis Disc Decomposition (preprint), with Wendy Finbow-Singh: a study of the rigidity and inductive constructions for 'corals': triangulated spheres with added edges, holes, etc. in specific patterns, with Wendy Finbow Singh;
• translation of first-order rigidity and rigidity averaging results among metrics derived from projective geometry (Euclidean, spherical, hyperbolic, Minkowski, ... ) with Franco Saliola;
• inductive techniques for plane rigidity using tree covereings; rigidity of generalized grids, with Laura Chavez and Lily Moshe;
• connections between the first-order rigidity of planar graphs and pseudo triangulations. with a long list of co-authors (see below)
• rigidity of generalized grids in the plane, with Joseph Aiken and Sandra Gregov.

Protein Flexibility

I am also a senior researcher on a team four year National Institute of Health grant through Arizona State University and Michigan State University on New algorithms for protein flexibility. My collaborators there are Professor Michael Thorpe and Professor Leslie Kuhn . Some of the materials from the work are now available for downloading or for use from the web, at flexweb.asu.edu

Recently, we organized a workshop at BIRS (Banff) on this topic. From the workshop, there is a Web Comptes Rendus site with a number of papers which are freely available for downloading by the public. The site also includes some 'problems' (and solutions) from the discussions around the workshop.

I am working with a Masters Student in Computer Science, two Ph.D students in mathematics, and several undergraduate students in mathematics and computer science on problems in bioinformatics, molecular flexibility and rigidity, etc..

We are now informally gathering researchers at York who are interested in bioinformatics, including protein structure and classification. If you share this interest, get in touch with me. There is a graduate course in bioinformatics COSC 5290 which is taught in the Computer Science Graduate Program. More information is available from the web pages of Mariana Kant, the course director.

Localization in a multi-agent network

For several years I have been collaborating with a group centered in computer science at Yale. Two of the senior researchers are Professor Stephen Morse and Professor Yang (Richard) Yang . Working in particular with a recent Ph.D. student (now a post-doc at Columbia) Tolga Eren, we have a number of conference papers published, and some journal articles being submitted.

Pseudo Triangulations and Rigidity of planar graphs.

A specific project, with Ileana Streinu, applied the analysis of convexifying simple polygons in the plane to one-vertex origami: when can a folded paper with one vertex (a folded cone) be flattened, or made into a convex cone, without any new folds?

Research Background

Recent Preprints

1. with Franco Saliola, Contraining Plane Configurations in CAD: circles lines and angles; preprint, accepted for publication. Avialable in PDF format from Franco Saliola's site
2. with Laura Chavez and Lily Moshe, Basis and Circuits for 2-rigidity by Tree Coverings, 2002
3. Constraining Plane Configurations in CAD: geometry of lengths and directions; March 1996.
4. with H. Crapo, 3-stresses in 3-space and projected 4-polytopes: reciprocals liftings and parallel configurations, preprint (1993).
5. with J. Owen (D-Cubed Ltd, Cambridge England), Constraining Plane Geometric Configurations in CAD: directions and distances, 1994.

Recent Publications

1. To see like a mathematician, to be presented at the Visual Representation and Interpretation conference in Liverpool England in September 2002. download
2. Dynamic Geometry and the Practice of Geometry, for distribution at ICME9 Tokyo, July 2000.
   download a PDF file of this paper.
3. The Decline and Rise of Geometry, to appear in the Proceedings of the 1999 CMESG Conference
   download a PDF file of this paper.
4. with Brigitte Servatius, Constraining Plane Configurations in CAD: combinatorics of lengths and directions; SIAM J. Discrete Math. 12 (1999), 136-153.
5. Modeling Molecular Structures, in P. Duxbury and M. Thorpe (eds.), Rigidity Theory and Applications, Birkhauser Boston.
6. An analogy in geometric homology: rigidity and cofactors on geometric graphs; Accepted for the Proceedings of the RotaFest, 1997.
7. Matroids from Discrete Geometry, in Matroid Theory, J. Bonin, J. Oxley and B. Servatius (eds.), AMS Contemporary Mathematics, (1997), 171-313.
8. Rigidity and Scene Analysis; in the Handbook of Discrete and Computational Geometry, J. Goodman and J. O'Rourke (ed.), CRC Press, 1997, 893-916.
9. with Robert Connelly, Second-order rigidity and pre-stress stability for tensegrity frameworks; SIAM J. on Discrete Methods 9 (1996), 453-491.
10. Representing geometric objects, Learning and Geometry: Computational Approaches, D. Kueker and C. Smith (eds.), Birkhauser 1996, 143-178.
11. Analytic vs synthetic geometry for computers, Learning and Geometry: Computational Approaches, D. Kueker and C. Smith (eds.), Birkhauser 1996, 121-141.
12. with Tiong-Seng Tay and Neil White, Skeletal rigidity of simplicial complexes: I; European J. of Combinatorics 16 (1995), 381-403.
13. with Tiong-Seng Tay and Neil White, Skeletal rigidity of simplicial complexes: II; European J. of Combinatorics 16 (1995), 503-523.
14. with H. Crapo, Spaces of stresses, projections, and parallel drawings for spherical polyhedra, Beitraege zur Algebra und Geometrie / Contributions to Algebra and Geometry 35 (1994), 259-281.
15. How to design or describe a polyhedron, J. of Intelligent and Robotic Systems 11 (1994), 135-160.
16. with Peter Alfeld and Larry Schumaker, The generic dimension of the space of C1 splines of degree d>8 on tetrahedral decompositions, SIAM J. Numerical Analysis (1993), 889-920.
17. with H. Crapo, Plane stresses and projected polyhedra I: the basic pattern, Structural Topology 20 (1993), 55-68.