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Downloadable Articles, Papers & Materials

Michael de Villiers/Dynamic Math Learning

(Currently back at University of KwaZulu-Natal, South Africa)

Last Update: 18 July 2011

Updated: Math Article, Dynamic Geometry Sketches, Math e-Newsletter, Quote & Cartoon (at bottom)

Click Sketchpad Demo to download a FREE DEMO copy of SKETCHPAD (required for viewing free downloadable sketches further down)

To order a full version of Sketchpad in Southern Africa go to SKETCHPAD

Click FATHOM DEMO to download a FREE DEMO of Fathom, the premier dynamic statistics package.

Click TINKERPLOTS to download a FREE DEMO of TinkerPlots, an exciting & simple data handling & statistics package for young children.

Bi-Monthly/Quarterly Math e-Newsletter (July 2011)

Updated!!!Java Sketchpad, Cinderella & GeoGebra Sketches - interactive geometry on the web

sphere.jpg: no description

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

ellipsetower.jpg: Telecommunications Tower on Bluff, Durban showing an approx cone cut by plane giving an ellipse

Telecommunications Tower on Bluff, Durban showing an approximate cone cut by plane giving an ellipse

Downloadable PDF papers/books for sale

Most cited papers and books according to Google Scholar (7 April 2011):

1) The Role and Function of Proof in Mathematics (1990) - cited by 141.

2) Rethinking Proof with Sketchpad (1999/2003) - cited by 119.

3) The role and function of a hierarchical classification of quadrilaterals (1994) - cited by 58.

4) To teach definitions in geometry or teach to define? (1998) - cited by 40.

5) The Role of Proof in Investigative, Computer-based Geometry: Some Personal Reflections (1997) - cited by 38.

6) The Future of Secondary School Geometry (1996) - cited by 27.

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!!! Worksheet: Order of operations Worksheet and Teacher Notes to develop understanding of the need for 'order of operations' in order to avoid 'ambiguity', Pythagoras, 1992, 1.9 MB, PDF

NEW!!! 2010 Van Hiele plenary paper now in Portuguese & Croatian

A Comparative Study of two Van Hiele Testing Instruments Poster paper presented by a then student of mine, Eddie Smith, at PME-13, Paris 1989

Some Reflections on the Van Hiele theory. Invited paper and video made at the Program Post-Graduate Studies in Mathematics Education at the Pontifical Catholic University of Sao Paulo, Brazil in April 2010, as well as invited plenary at the Croatian Mathematical Conference in Zagreb, Croatia in June 2010

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)


NEW!!! Equi-angled cyclic and equilateral circumscribed polygons ( pdf, 565 KB, Mathematical Gazette. Generalizes rectangles and rhombi.)

From the Fermat points to the De Villiers points of a triangle ( pdf, 1.2 MB, AMESA July 2009 paper. Describes the discovery and proof of the so-called De Villiers points of a triangle, tracing its origin from the Fermat points of a triangle.)

Vanaf die Fermat punte na die De Villiers punte van 'n driehoek ( pdf, 344 KB, SA Tydskrif vir Natuurwetenskap & Tegnologie, Sept 2010. Beskryf die ontdekking en bewys van die sogenaamde De Villiers punte van 'n driehoek, en trek die oorsprong terug na die Fermat punte van 'n driehoek.)

Some hexagon area ratios: Problem solving by related example ( pdf, 324 KB, Mathematics in School, 2010. Describes the discovery and proof of an interesting area ratio result for a parallelo-hexagon.)

Generalizing the Nagel line to Circumscribed Polygons by Analogy & Constructive Defining ( pdf, 548 KB, Pythagoras, 2008. First discusses the genetic approach & the importance of the history of math, reasoning by analogy, and the role of constructive defining before presenting the result - with downloadable GSP sketch.)

LTM Cover Problem ( pdf, 248 KB, Learning & Teaching Mathematics Journal, 2009. Gives a proof of the result, and two additional problems - with links to interactive sketches.)

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 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

maa.jpg: GSP pic on p.4

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

My Favorite Links:

The Association for Mathematics Education of South Africa (AMESA)




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 @ (use without spaces)

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