Recommended Background Reading
The following two introductory chapters from my Rethinking Proof book (free to download) outlines the rationale for the book as well as the Van Hiele learning theory, which provided important guidance for the design, structuring & sequencing of the activities. However, these activities are not intended as a complete, self-contained geometry curriculum - these are more like snapshot suggestions for mathematics teachers to use and/or adapt as they see fit.
The Role and Function of Proof with Dynamic Geometry
The van Hiele Theory — Defining and Proving Within a Dynamic Geometry Context
Activities
The dynamic geometry activities below are from my book Rethinking Proof (free to download). As mentioned above these activities are intended as possible exemplars to introduce and illustrate different functions of proof to learners/students - see De Villiers (1990, 1999). To save valuable classroom time, many of the sketches are supplied ready-made in order to focus on the main tasks at hand, namely observing, experimenting, conjecturing & proving. However, some sketches require users to measure certain quantities, or to make some constructions or calculations, but Tools are provided in the WebSketchpad sketches below.
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Active links are underlined and shown in blue.
Proof as Explanation
Distances in an Equilateral Triangle (Viviani's theorem & further generalizations)
Water Supply I: Four Towns (Introducing perpendicular bisectors)
Water Supply II: Three Towns (Concurrency of perpendicular bisectors)
Triangle Angle Sum (Using tessellation)
Quadrilateral Angle Sum (Using tessellation)
Crossed Quadrilateral Angle Sum (Lakatosian learning experience; defining internal angles)
Exploring Isosceles Trapezium Properties (Incl. Logical Explanation using Symmetry)
Cyclic Quadrilateral Alternate Angles Sum (Incl. generalization to cyclic 2n-gons)
The Center of Gravity of a Triangle (Concurrency of medians; Ceva's theorem)
Proof as Discovery
Kite Midpoints (Incl. generalizations to Varignon's theorem & Orthodiagonal quadrilateral)
Logical Discovery (Varignon Parallelogram Perimeter)
Isosceles Trapezoid Midpoints (Incl. generalization to Equidiagonal quadrilateral)
Logical Discovery: Circumscribed (Tangential) Quadrilateral (Pitot's theorem; incl. further generalization)
(Note: The Fermat-Torricelli point activity further on (under Challenge) is also designed to illustrate the 'discovery function' of proof.)
Proof as Verification
Areas Ratios
Varignon Parallelogram Area (Incl. generalization to area crossed quad)
Logical Paradox
Cyclic Quadrilateral Alternate Angles Sum Converse
Concurrency Conjecture
Triangle Altitudes (Concurrency)
Light Ray in a Triangle (Fagnano's Minimal Path)
Parallel Lines (Thomsen's Hexagon)
Proof as Challenge
Parallelogram Angle Bisectors (Incl. generalization to any non-tangential quadrilateral)
Parallelogram Squares (Incl. generalization to Van Aubel's theorem)
The Fermat-Torricelli Point (Incl. generalization from right △ to arbitrary △)
Airport Problem (More about Fermat-Torricelli)
Napoleon's Theorem
Miquel's Theorem
Proof as Systematization
Reasoning Backward: Triangle Midpoints (Proving earlier assumed lemma)
Reasoning Backward: Parallel Lines (Proving earlier assumed lemma)
Systematizing Rhombus Properties (Descriptive Defining of a Rhombus)
Systematizing Isosceles Trapezoid Properties (Descriptive Defining of an Isosceles Trapezoid)
References
De Villiers, M. (1990). The Role and Function of Proof in Mathematics. Pythagoras, 24, pp. 17-24.
De Villiers, M. (1991). Pupils' Needs for Conviction and Explanation within the Context of Geometry. Pythagoras, 26 (July), pp. 18-27.
De Villiers, M. (1992). Children's Acceptance of Theorems in Geometry. Poster presented at PME 16, 6-11 August 1992, University of New Hampshire, USA.
De Villiers, M. (1998). To teach definitions in geometry or teach to define?. In A. Olivier & K. Newstead (Eds), Proceedings of the Twenty-Second International Conference for the Psychology of Mathematics Education (PME): Vol. 2. (pp. 248-255). University of Stellenbosch: Stellenbosch,
12-17 July 1998.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
De Villiers, M. (2017). Revisiting the Van Hiele theory. Invited paper at the EIEM Conference 2017 at the University of Lisbon, Portugal, organized by the Portuguese Society for Research in Mathematics Education.
Shongwe, B. (2021). Learners’ beliefs about the functions of proof: building an argument for validity. Educational Studies in Mathematics, 107:503–523; DOI: https://doi.org/10.1007/s10649-021-10047-y
Shongwe, B. (2026). A student’s espoused beliefs about functions of proof and their behaviour in proving. International Journal of Mathematical Education in Science and Technology, Volume 57, Issue 6, pp. 1127-1149. DOI: https://doi.org/10.1080/0020739X.2025.2483507
Associated Sketchpad Sketches for Rethinking Proof
Rethinking Proof Sketches (zipped file)
Published Reviews of Rethinking Proof
Published Reviews of Rethinking Proof
Some Related Links & Videos
Invited Nacome Plenary 2021: Does Dynamic Computer Verification imply the End of Proof? (YouTube video)
Rethinking Proofs & Proving (AI produced YouTube video)
Some Van Hiele theory video clips (From Plenary Talks & Presentations in Brazil & Croatia)
Introducing, Classifying, Exploring, Constructing & Defining Quadrilaterals
Some External Links
Van Hiele model (Wikipedia)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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Michael de Villiers, created with WebSketchpad, 14 June 2025; updated 21 June 2025; 4/11/19/27 July 2025; 3 August 2025; 14 Sept 2025; 12 Jan 2026; 12 Feb 2026; 16 March 2026; 13 April 2026; 18 June 2026.
