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December 2019 MATH e-NEWSLETTER

This more or less twice a year Math e-Newsletter is sent out mainly in Southern Africa, but has information that may be of interest to others from different parts of the world. Please e-mail me at profmd @ mweb. co. za (without spaces) if you'd like to be on the mailing list. (Also please say if you specifically want it sent to you as an e-mail or whether you prefer downloading/reading it from my homepage and just receiving a notification).

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Dear Colleagues & Friends

Wishing everyone a happy Festive end of the year. Please note I am sending this Math e-Newsletter from my g-mail account(s), which I don't check regularly.


My Downloadable Articles Homepage has been updated with the following new items:
1) Tiling with a Trilateral Trapezium and Penrose Tiles Learning & Teaching Mathematics, March 2019, no. 25.
2) Why does it Work? A Mathematical Explanation and Further Generalization of a Card Trick A card trick is mathematically explained using elementary school algebra and then generalized. Learning & Teaching Mathematics, Dec 2018, no. 24.
3) An Interesting Collinearity Learning & Teaching Mathematics, July 2019. No. 26.
4) mathematics/science cartoon and mathematics quote (at bottom of page)

My Dynamic Mathematics Sketches site has been updated with the following new sketches (as well as some updates to older sketches):
1) Area Parallelogram Partition Theorem: Another Example of the Discovery Function of Proof

My Student Explorations site has been updated with the following new sketches (as well as some updates to older sketches):
1) Diagonal Division Ratios in a Quadrilateral
2) A Diagonal Property of a Rhombus constructed from a Rectangle

WebSketchpad: My latest Dynamic Geometry Sketches all use WebSketchpad, which are more easily accessible across a range of platforms from cellphones to tablets as well as laptops and PC's without the need to constantly update Java, or to change your security settings.


Exciting news is that Sketchpad can now be downloaded for free from Free Sketchpad. You can then use any valid license code on 1, 2, 3, or 1,000 machines and Sketchpad will be fully functional on all those machines. When you are prompted, and you've forgotten or don't have a license code, use the License Name and Authorization Code provided below to unlock your license for Sketchpad. Use all characters and spaces as shown, including the 6-character code that appears at the end of the License Name.

License name: Pythagoras Single User License 1863 KJUJAM
Authorization code: BDFBF968


All parents, teachers and students please note that the closing date for the South African Mathematics Olympiad (SAMO) entries is 31 January 2020. Download the entry form here. Currently about 100 000 high school learners participate, but we'd like to see even more learners involved and stimulated.


1) Long time colleague and friend, Dirk Laurie, from Stellenbosch University, whom I've known and worked with for many years passed away in August 2019. He was very involved with the SA Math Olympiad, and will be sorely missed by many. Read a short tribute to him at: "Passing of a remarkable South African mathematician" or this scanned report (in Afrikaans) in the 18 August 2019 Rapport newspaper at: "'Briljantste' mens skryf oor 'Onvolkome Kennis'".
2) Rosamund Sutherland from the School of Education at the University of Bristol (UK) passed away in February 2019. She was a pioneer in demonstrating how students' learning in school can surpass expectations with the support of expert teachers and digital technology. Read a tribute to her in the Guardian at: "Obituary".


a) Read this classic, influential UNESCO 1992 book "The Influence of computers and informatics on mathematics and its teaching". Editors, Cornu, B & Ralston, A.
b) Visit the free "Worldwide Center of Mathematics YouTube" video channel for many great videos on Problems of the Week, Algebra, Calculus, Differential Equations, Statistics, Combinatorics, History of Math, etc.
c) The UD JRME Editorial Team has been writing a series of editorials. Here is a list of the recent four:
i) "March 2019 - Posing Significant Research Questions"
ii) "May 2019 - Theoretical Framing as Justifying"
iii) "July 2019 - Choosing and Justifying Robust Methods for Educational Research"
iv) "November 2019 - So What? Justifying Conclusions and Interpretations of Data".
d) An excellent new book by Petar Mladinic & Nikol Radovic is available for purchase at: "The Geometry of Nature (Mathematics)".
e) A lovely collection o stimulating mathematical puzzles presented in a comic book form by Petar Mladinic & Ninoslav Kunc is available for purchase at: "Wonderful World of Math (Collection of mathematical puzzles)".
f) NCTM will be launching a new practitioner journal in January 2020, the "Mathematics Teacher: Learning and Teaching PreK-12", which will reflect current practices of mathematics education, as well as maintain a knowledge base of practice and policy in looking at the future of the field. Content is aimed at mathematics educators teaching preschool to 12th grade with peer-reviewed and invited articles.
g) A book for researchers on the history of mathematics by Bernard Beauzamy on Archimedes is available for purchase at: "Archimedes' Modern Works".


a) Eighth International Conference to Review Research in Science, Technology and Mathematics Education, epiSTEME 8, January 3 - 6, 2020, Mumbai, India. URL:

b) The 28th Conference of the Southern African Association for Research in Mathematics, Science and Technology Education, SAARMSTE, Nelson Mandela University, Port Elizabeth, 14-17 January 2020. URL:

c) Joint Mathematics Meetings Exhibition of Mathematical Art, Denver, Colorado, January 15-18, 2020. URL:

d) NCTM Centennial Annual Meeting and Exposition, April 1-4, 2020, "Celebrating 100 Years: Looking Back and Moving Forward", Chicago. URL:

e) Joint Annual Conference Mathematical Association (MA)/NANAMIC, 15 - 18 April 2020, Wyboston Lakes Resort, Wyboston, Bedfordshire. URL:

f) The 2020 Annual Congress of AMESA, Tshwane University of Technology, Pretoria West Campus 21-26 June 2020 with the theme "Developing equitable mathematical teaching practices that empower teachers and learners in the 4IR era". Website: 

g) 14th Annual International Conference on Mathematics: Teaching, Theory & Applications, 29-30 June & 1-2 July 2020, Athens, Greece (Academic Responsible: Dr. Codruta Simona Stoica, Professor and Vice-Rector, Aurel Vlaicu University of Arad, Romania).

h) International Association for Statistical Education (IASE) 2020 Roundtable Conference,July 6-10, 2020, Nanjing, China. URL:

i) 2020 Classroom Teaching Research for All Students (CTRAS) Conference, July 9-11, 2020, School of Mathematics & Science, Shanghai Normal University. URL: CTRAS.

j) The 14th International Congress on Mathematical Education (ICME-14), Shanghai, China, July 12-19, 2020. URL:

k) The International Study Group on the Relations Between the History and Pedagogy of Mathematics (HPM Group), University of Macau (China), July 21-25, 2020. URL:

l) Tenth ERME Topic Conference: Mathematics Education in the Digital Age (MEDA), Linz (Austria), September, 16-18, 2020. URL:


"A Mathematician who is not also something of a poet will never be a complete mathematician" -Karl Weierstrass.

"The mathematical meaning of proof carries three senses. The first is verification or justification, concerned with the truth of a proposition; the second is illumination, in that a good proof is expected to convey an insight into why the proposition is true; this does not affect the validity of a proof but its presence in a proof is aesthetically pleasing. The third sense of proof is the most characteristically mathematical, that of systematisation, i.e. the organisation of results into a deductive system of axioms, major concepts and theorems, and minor results derived from these." - Alan Bell (1976:24-25), A study of Pupils' Proof-Explanations in Mathematical Situations. Educational Studies in Mathematics.


Variation Law - If you change lines (or traffic lanes), the one you were in will always move faster than the one you are in now (works every time).

Law of the Result - When you try to prove to someone that a machine won't work, it will.

Law of Probability - The probability of being seen is directly proportional to the stupidity of your act.

Proof by accident: "Hey, what have we here?!"

Proof by symbolic mumbo-jumbo: " (B ÔøΩ P ... ) , $(C ÔøΩ W ) ..."

Proof by illegibility: "... (scribble, scribble) ... QED!"


Mathematically yours


Prof Dr Michael de Villiers
(Dynamic Mathematics Learning)
South Africa
Dynamic Geometry Sketches:
Dynamic Mathematics Learning Store:


Visit the SA Mathematics Olympiad at