Last Update: 21 September 2009
Updated: Math Ed Article, Dynamic Geometry Sketches, Math e-Newsletter, Quote & Cartoon (at bottom)
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Bi-Monthly/Quarterly Math e-Newsletter (Sept 09)
Updated!!!Java Sketchpad, Cinderella & GeoGebra Sketches - interactive geometry on the web
View a video clip illustrating the hexagon result from my article in The Montana Maths Enthusiast, July 2007, further down, by clicking on (then clicking play button) hexagon centroids
Have a look at some of my geometry students' comments by clicking feedback
New!!! Read latest published review of my Rethinking Proof book in Mathematics in School, May '07 by clicking Rethinking Review
For other reviews of Rethinking Proof go to Other Rethinking Reviews
Telecommunications Tower on Bluff, Durban showing an approximate cone cut by plane giving an ellipse
NEW!!! Downloadable PDF papers/books for sale
Free Articles/papers/talks
Please note that most of the article/items below are in pdf format and require Adobe Acrobat Reader which can be downloaded for free from Adobe Acrobat Reader
The articles and materials below may be used for non-profit educational purposes only, provided that their source/origin is properly referenced at all times.
Kindly please inform me of any links or downloads that are NOT working properly.
Mathematics Education
NEW!!! Border Patterns, Tessellations & Problem Solving with Transformations. Invited paper AMESA Congress, 2007, 1.1 MB, PDF
Research Evidence on Hierarchical Thinking, Teaching Strategies and the Van Hiele Theory: Some critical comments. RUMEUS Report, pp. 36, 1987, 1.5 MB, PDF
ICMI-19 Discussion Document: Proof and Proving in Mathematics Education. pp. 12, 2007/8, 136 KB, PDF
Problemsolving and Proving via GeneralizationLearning & Teaching Mathematics, no. 5. pp. 19-25, 2008, 900 KB, PDF
Mathematical Applications, Modeling and TechnologyInvited Plenary at Annual Congress of AMESA, July 2007, AMESA Proceedings, 1.2 MB, PDF
Some pitfalls of dynamic geometryDiscusses some pitfalls of using dynamic geometry in teaching geometry. Feb 2007, Teaching & Learning Mathematics, 276 KB, PDF
Recycling cyclic polygons dynamicallyThis article includes zipped Sketchpad sketches and deals with testing by construction, and proving the converse of a well-known cyclic quad theorem, and then considering some generalizations. May 2006, Mathematics in School, 64 KB.
Rol en Functie van het Bewys in de Dynamische Meetkunde (A somewhat modified and translated version into Dutch of my "Role & function of proof in Dynamic Geometry", pdf, 300 KB, EUCLIDES, Feb 2006, 81(4), pp. 184-188)
Mathematical Modeling and Proof (pdf, 24 KB, AMESA '04 paper)
Using dynamic geometry to expand mathematics teachers' understanding of proof, 2004, IJMEST 35 (5), pp. 703-724, 176 KB
Clough's Conjecture - A Sketchpad Investigation (Workshop on a variation of Viviani's theorem presented at AMESA 2004, Potchefstroom & NCTM 2004, Philadelphia.
Rethinking Proof with Sketchpad - in Hebrew (pdf, 2003 in ALEH, 30, May, 19-26. English versions are in ARCHIVE articles below)
A further Pythagorean Variation on a Fibonacci theme (MiS, 2002, pdf)
All cubic polynomials are point symmetric (TLM, 2004, pdf)
The Role and Function of a Hierarchical Classification of Quadrilaterals (FLM, 1994, pdf)
The role of experimentation in mathematics (AMESA 2003 paper)
A Sketchpad Investigation of "Arithmetic Sequence" Parabola (AMESA 2003 Workshop)
Developing understanding for different roles of proof in dynamic geometry (ProfMat 2002, pdf)
Para uma Compreensao dos Differentes Paeis La Demonstracao em Geometria Dinamica (ProfMat 2002, pdf)
The Role and Function of Proof with Sketchpad (1999, pdf)
Constructive Evaluation of Definitions in a Sketchpad Context (Amesa 2002 paper, pdf file)
Mathematics
NEW!!! A Further Generalization of Turnbull's Theorem ( pdf, 464 KB, IJMEST, 1993. Generalizes the familiar "opposite angle supplementary" theorem for cyclic quadrilaterals to other cyclic polygons.)
Solving a Locus Problem via Generalization (Contains link to a Sketchpad sketch, pdf, 20 KB, Reflections, Spring 2008, pp. 20-21. Applies circle geometry and similarity to solving a locus problem.)
A Question of Balance: An Application of Centroids (Contains link to a Sketchpad sketch, pdf, 416 KB, Mathematical Gazette, Nov 2007, pp. 525-528, and March 2008. Applies similarity and centroids to proving a result about similar quadrilaterals on the sides of a quadrilateral.)
An example of the discovery function of proof(Contains link to a Sketchpad sketch, pdf, 32 KB, Mathematics in School, Sept 2007, pp. 9-11. Generalizes a 9th grader's discovery for a triangle to a convex quadrilateral.)
A hexagon result & its generalization via proof(Contains link to a Sketchpad sketch, pdf, 88 KB, The Montana Mathematics Enthusiast, June 2007, Vol. 4, No. 2. Illustrates the discovery function of proof by generalizing a hexagon result.)
Further Reflection on a SA Mathematics Olympiad Problem(Contains link to a Sketchpad sketch, pdf, 40 KB, Learning & Teaching Mathematics, March 2007, pp. 25-27. Discusses underlying hidden theorem related to a special type of quadrilateral, and how this in turn can be viewed as a special case of Dandelin's Theorem.)
More on hexagons with opposite sides parallel (Contains link to a Sketchpad sketch, pdf, 120 KB, Mathematical Gazette, Nov 2006)
A generalization of the Spieker circle and Nagel line (Contains link to Sketchpad sketches, pdf, 100 KB, Pythagoras, July 2006)
A generalization of the Nine-point circle and Euler line (Contains links to Sketchpad sketches, a further generalization of the Euler line and a synthetic proof, pdf, 40 KB, Pythagoras, Dec 2005)
The nine-point conic: a rediscovery and proof by computer (Describes a generalization of nine-point circle and associated Euler line & proof by computer, pdf, 216 KB, IJMEST, Jan 2006)
Feynman's Triangle: Some feedback and more (pdf, 16 KB, Math Gazette, March 2005)
A dual to a BMO problem (pdf, 12 KB, March 02 Math Gazette)
Ezit's Rule: An Explanation & Proof (From LTM, Feb 2005, PDF, 20 KB)
An Extended Classification of Quadrilaterals (An excerpt from my Some Adventures In Euclidean Geometry booklet, 8 KB, pdf)
Crocodiles and Polygons: A generalization of Viviani's theorem to equi-angled polygons. An article from March 2005 issue of Mathematics in School, pdf.
Stars: A Second Look (Pdf, A 1999 article in Math in School as a short response to another article on the interior angle sum of star polygons.)
An interesting property of a quadrilateral circumscribed around a circle. (An excerpt from my book Some Adventures in Euclidean Geometry, 1996, pdf.)
'n Bewys deur matematies induksie van 'n bekende bewering - in Afrikaans (jpg, 1988, . Spectrum 26(3), August, p.38. Will be rewritten in English at some stage & posted here).
A Fibonacci Generalization and its Dual (2000, IJMEST, pdf)
Generalizations of maltitudes (IJMEST, 1999, pdf) (includes a generalization of Euler line to cyclic polygons)
The Affine Equivalence of Cubic Polynomials (2003, pdf)
How does the wind affect road-running achievement? (1991, pdf)
From Miquel Triangles to Miquel Distances (2002, pdf)
Dual Generalizations of Van Aubel's Theorem (pdf)
A Generalization of the Fermat-Torricelli Point (pdf)
A Further Generalization of the Fermat-Torricelli Point (pdf)
More articles
Please click on the links below for more of my articles & publications.
ARCHIVE OF MATHEMATICS EDUCATION (Articles uploaded earlier; e.g. Role of Proof - also in Spanish & Portuguese; Teaching Definitions; Future Of Secondary School Geometry - also in Spanish; Role of Technology in Math, etc.)
ARCHIVE OF MATHEMATICS (Articles uploaded earlier; e.g. on theorems of Van Aubel, Kosnita, Haruki, side-angle duality, applications in rugby, tennis; etc.)
SKETCHPAD COMPETITION SOLUTIONS
UPDATED!!! READER INVESTIGATIONS from KZN AMESA MATHEMATICS JOURNAL
Sketchpad Sketches
Experience the POWER and BEAUTY of moving geometric figures; i.e. DYNAMIC GEOMETRY!
Download FREE Sketchpad 3 sketches for investigations or demonstrations by clicking on SKETCHPAD 3 SKETCHES
Download FREE Sketchpad 4 sketches for investigations or demonstrations by clicking on SKETCHPAD 4 SKETCHES
Dynamic Geometry sketches for my paper THE ROLE OF PROOF IN INVESTIGATIVE, COMPUTER-BASED GEOMETRY: SOME PERSONAL REFLECTIONS in the MAA Book "Geometry Turned On: Dynamic Software in learning, Teaching, and Research", 1997, edited by James King & Doris Schattschneider (as well as other sketches & files from the book) can be found at geometry turned on
Conjecture From RETHINKING PROOF proved! (pdf file)
Free 12 pp booklet BUILDING RELATIONSHIPS: UNCOVERING MATH WITH YOUR FAMILY is available from Texas Instruments and can be downloaded from family math
The Association for Mathematics Education of South Africa (AMESA)
GEOMETRY STEP BY STEP FROM THE LAND OF THE INCAS
FIBONACCI A site on FIBONACCI numbers and the GOLDEN SECTION is a useful resource for encouraging students into math.
Find out about latest developments in mathematical research. Written for the layperson. What's New in Mathematics
Comprehensive List of Links to Math Sites
Quote of the Month
Contact me at: profmd@mweb.co.za
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