# References

### From MathWiki

### General Reports on Geometry

All of these reports seem to share some key themes:

- the value of geometry and spatial reasoning in general eduction of citizens;

- the value of geometry and spatial reasoning in key applications of mathematics, primarily in engineering and the physical sciences;

- the value of geometry and spatial reasoning as a connection among parts of mathematics and key stage for learning 'mathematical processes' or 'mathematical habits of mind'.

- Royal Society and Joint Mathematical Council (England) 2001: Teaching Geometry 11-19

Available from http://www.royalsoc.ac.uk/document.asp?tip=0&id=1420 For you convenience, I have uploaded the recommendations, along with a few comments of areas that trouble me in this thoughtful report. Also note the appendices on spatial activities.

- Geometry K-13 OISE Report 1967 (large file). Download Now (
*http://www.math.yorku.ca/~whiteley/geometry.pdf*). Posted with permission of the Ontario Institute for Studies in Education of the University of Toronto.

This is a thoughtful, coherent report, covering the whole range of the curriculum, written by a key group including the geometer Donald Coxeter, and the Mathematician / Mathematics Educator W.W. Sawyer. It was, unfortunately, completely ignored in later revisions of the Ontario Mathematics Curriculum. In these revisions, the main approach to geometry has been to reduce pieces, without a major review of the goals of geometry curriculum, the connections of geometry to other parts of the mathematics (and science) curriculum, or the possible introduction of new material to match the new tools (e.g. dynamic geometry software GSP which is licensed for all teachers and all students in Ontario), or new findings about students' capacities. The dominant force in Ontario has been to reduce the schooling by one year (from K-13 to K-12), and the compress the curriculum while preserving Calculus as the high-end final course. Under this pressure, the lack of geometry knowledge and vision among curriculum writers means geometry content and spatial abilities have been reduced with each revision over the last few decades. See the next two items for reflections on these processes.

- W. Whiteley (1999) The Decline and Rise of Geometry, in the Proceedings of the 1999 CMESG Conference Download now (
*http://www.math.yorku.ca/~whiteley/cmesg.pdf*).

This is my overview of some of the means by which geometry has declined (starting with University Mathematics programs at the graduate, then the undergraduate programs), as well as the pedagogical need and the external needs in applied areas why (and how) it will revive.

- Salisbury Report on Senior Level Mathematics (in Ontario) (2006).

This report addressed some choices for the Grade 12 Curriculum, under sever constraints. I have extracted the commentary on Geometry (including why even more chunks were set aside, and why 'vectors' survived). This extract is here

- Hans Freudenthal: Geometry Between the Devil and the Deep Sea. Educational Studies in Mathematics, 3, pp. 413-435. Available through JSTOR: http://www.jstor.org/ (requires login).

This is a wonderful, hopeful overview by one of the masters of the field of mathematics education. In this context, the 'devil' is the focus on geometric proofs and axiomatics, and the 'deep sea' is where we should be diving to build on the richness of geometry and the capacity of students to learn in diverse ways over the plurality of geometries.

- Conference Board of the Mathematical Sciences. (2001). The mathematical education of teachers. Providence RI and Washington DC: American Mathematical Society and Mathematical Association of America. Available at: http://www.cbmsweb.org/MET_Document/ Relevant to the role of geometry in the education of future teachers.

- Kuhane Commission (France] Rapport d'étape sur la géométrie et son enseignement http://ups.prepas.org/maths/kahane/geometrie.pdf

A recent French report as part of a larger review of the mathematics curriculum.

- Paul Goldenberg, Albert Cuoco, June Mark: A role for Geometry in General Education. In Designing Learning Environments for Developing Understanding of Geometry and Space, Lawrence Erlbaum, 1998, 3-44.

This places geometry and visual reasoning, at the heart of a 'Habits of Mind' approach to mathematics curriculum. The entire volume is rich in material and reflections.

- Zalman Usiskin: The Implications of "Geometry for All" http://www.ncsmonline.org/NCSMPublications/1997journals.html#oct97mel

### Articles

Eisenberg, Ann: Pages on Craft Technology Group, (the use of craft materials, including paper material). For a chapter of her thesis on Spatial Reasoning see: http://www.cs.colorado.edu/~eisenbea/thesis/final/Chapter2.pdf

Gal, H., & Linchevski, L. (2005). Changes in teachers' ways of coping with problematic learning situations in geometry instruction. from 15th ICMI Study Conference: The Professional Education and Development of Teachers of Mathematics, http://stwww.weizmann.ac.il/G-math/ICMI/log_in.html, Águas de Lindóia, Brazil.

Hanna, G. and Jahnke, H.N. (1999). Using Arguments from Physics to Promote Understanding of Mathematical Proofs. Proceedings of PME23. http://fcis.oise.utoronto.ca/~ghanna/PME99.pdf

Hoyles, C., Foxman, D., & Küchemann, D. (2002). A Comparative Study of Geometry Curricula. London, QCA.

Jones, K., & Fujita, T. (2001). Developing a new pedagogy for geometry. In J. Winter (Ed.), Proceedings of the British Society for Research into Learning Mathematics 21(3) November 2001 (pp. 6). University of Southampton, UK. http://eprints.soton.ac.uk/14686/

T Fujita, K Jones, Y Shinya Geometrical intuition and the learning and teaching of geometry, 2004 http://eprints.soton.ac.uk/14687/01/Fujita_Jones_Yamamoto_ICME10_TSG10_2004.pdf

Jones, Keith, Fujita, Taro and Ding, Liping (2005) Developing geometrical reasoning in the classroom: learning from expert teachers from China and Japan. In, 4th Biennial Conference of the European Society for Research in Mathematics Education (CERME4), Sant Feliu de Guíxols, Spain, 17-21 February 2005. , 10pp. http://eprints.soton.ac.uk/18818/

McLeay, Heather: Imagery, Spatial Ability and Problem Solving From the ATM, mounted by permission.

Olkun, Sinan (2003): Making Connections: Improving Spatial Abilities with Engineering Drawing Activities http://www.cimt.plymouth.ac.uk/journal/sinanolkun.pdf

Shulman, L. S. (1987). Knowledge and teaching: Foundation of the new reform. Harvard Educational Review, 57(1), 1-22.

from the American Chemical Society Journal Crystal Growth and Design (Froebel was also a crystallographer). Froebel was the inventor of kindergarten in the 1840's, worked with 3-D geometric (and aesthetic) activities prior to introducting 2-D activities.

SIGGRAPH White Paper on Visual Learning in Science and Engineering: http://education.siggraph.org/conferences/other/visual-learning/ This is an eclectic report with a vast bibliography from people ranging from visual arts educators and psychologists to a mathematician. A follow up article Visual learning for science and engineering McGrath, M.B.; Brown, J.R. pp. 56 - 63 (2005) in: IEEE Computer Graphics and Application Volume: 25, Issue:5, 2005 A related article, on the use of CAD type / sketching programs for teaching visualization and spatial reasoning, in the same issue, is: Improving visualization skills in engineering education Contero, M.; Naya, F.; Company, P.; Saorin, J.L.; Conesa, J. pp. 24 - 31

Teaching Symmetry In the Elementary Curriculum. Christy Knuchel www.montanamath.org/TMME/TMMEv1n1a1.pdf

Here are links to a couple of articles by Walter Whiteley on visualization and mathematics: Teaching to See Like a Mathematicians: http://www.math.yorku.ca/~whiteley/Teaching_to_see.pdf From the conference on Visual Representation and Interpretation.

The Decline and Rise of Geometry in 20th Century North America: http://www.math.yorku.ca/~whiteley/cmesg.pdf This is a summary of the plenary talk at the CMESG conference.

Dynamic Geometry and the Practice of Geometry, for distribution at ICME9 Tokyo, July 2000; http://www.math.yorku.ca/~whiteley/Dynamic.pdf

Claims and Questions towards a Research Program in Visualization (ICME 10 paper): http://www.math.yorku.ca/~whiteley/Visualization.pdf

### Links

Go to the page of links

Page on Big Ideas in Geometry: procedures, concepts, patterns

Return to the working group main page