The standard proof, the elegant proof, and the proof without words of tasks in geometry, and their dynamic investigation

References

• National Council of Teachers of Mathematics. Principles and standards for school mathematics. Reston (VA): National Council of Teachers of Mathematics; 2000. p. 4.
   Google Scholar
• Hardy GH. A mathematician`s apology. Cambridge (UK): Cambridge University Press; 1940. p. 40.
   Google Scholar
• Gadanidis G, Hoodland C. The aesthetics in mathematics as story. Can J Sci Math Technol Educ. 2003;3(4):487–498.
   Google Scholar
• Rota GC. The phenomenology of mathematical beauty. Syntbase. 1997;111(2):171–182.
   Web of Science ®Google Scholar
• National Council of Teachers of Mathematics. Curriculum and evaluation standards for school mathematics. Reston (VA): National Council of Teachers of Mathematics; 1989. p. 79.
   Google Scholar
• Sinclair N, Crespo S. What makes a good problem? An aesthetics lens. In: Novotna J, Moraova H, Kratka M, Stehlikova N, editors. Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education; Prague: PME; 2006, Vol. 5, p. 129–136.
   Google Scholar
• Sinclair N. The aesthetic as a liberating force in mathematics education. ZDM. 2009;41(1):45–60.
   Google Scholar
• Dreyfus T, Eisenberg T. On different facets of mathematical thinking. In: Sternberg RJ, Been-Zeen T, editors. The nature of mathematical thinking. Mahwah (NJ): Lawrence Erlbaum; 1996. p. 253–284.
   Google Scholar
• Stylianides GJ. Reasoning and proving in school mathematics textbooks. Math Think Learn. 2009;11(4):255–288.
   Web of Science ®Google Scholar
• Lowrie T, Logan T, Scriven B. Perspectives on geometry and measurement in the national curriculum. In: Atweh B, Goos M, Jorgensen R, Siemon D, editors. Engaging the Australian curriculum mathematics: perspectives from the field. Australia: MERGA Inc; 2012. p. 71–88. Available from: http://www.merga.net.au/node/223.
   Google Scholar
• Leikin R, Lev M. Multiple solution tasks as a magnifying glass for observation of mathematical creativity. In: Wo JH, Lew HC, Park KS, Seo DY, editors. Proceedings of the 31st International Conference for the Psychology of Mathematics Education; Seoul: The Korea Society of Educational Studies in Mathematics; 2007, Vol. 3. p. 161–168.
   Google Scholar
• Stupel M, Ben-Chaim D. Plane geometry and trigonometry – related fields: do they work hand in hand? Far East J Math Educ. 2013;11(1):43–74.
   Google Scholar
• Stupel M, Ben-Chaim D. One problem, multiple solutions: how multiple proofs can connect several areas of mathematics. Far East J Math Educ. 2013;11(2):129–161.
   Google Scholar
• Rav Y. Why do we prove theorems? Philosophic Math. 1999;7(7):5–41.
   Google Scholar
• Aharoni R. Mathematics poetry and beauty. Tel-Aviv: KM Press; 2007. p. 7. Hebrew.
   Google Scholar
• Hanna G. Proof, explanation and exploration: an overview. Educ Stud Math. ( Special issue on Proof in Dynamic Geometry Environments). 2000;44(1–2):5–23.
   Google Scholar
• Bell A. A study of pupils, proof-explanations in mathematical situations. Educ Stud Math. 1976;7:23–40.
   Web of Science ®Google Scholar
• de Villiers M. The role and function of proof in mathematics. Pythagoras. 1990;24:17–24.
   Google Scholar
• de Villiers M. Rethinking proof with the Geometer's Sketchpad. Emeryville (CA): Key Curriculum Press; 1999.
   Google Scholar
• Hanna G, Jahnke HN. Proof and proving. In: Bishop A, Clements K, Keitel C, Kilpatrick J, Laborde C, editors. International handbook of mathematics education. Dordrecht: Kluwer Academic Publishers; 1996. p. 877–908.
   Google Scholar
• Sinclair N. Notes on the aesthetics dimension of mathematics education. Burnaby; 2008. Available from: www.unige.ch/math/EnsMath/Rome2008/WG5/Papers/SINCL.pdf
   Google Scholar
• Brown JR. Philosophy of mathematics: an introduction to the world of proofs and pictures. New York, (NY): Routledge; 1999.
   Google Scholar
• Nelsen RB. Proof without words I. Exercises in visual thinking, Vol. 1. Washington (DC): The Mathematical Association of America; 1993.
   Google Scholar
• Nelsen RB. Proofs without words II. More exercises in visual thinking (classroom resource materials). Vol. 2. Washington (DC): The Mathematical Association of America; 2001.
   Google Scholar
• Eves H. Great moments in mathematics before 1650. Washington (DC): The Mathematical Association of America; 1983.
   Google Scholar
• Vinner S. The avoidance of visual consideration in calculus students. Focus Learn Problem Math. 1989;11(2):149–156.
   Google Scholar
• Dreyfus T. On the status of visual reasoning in mathematics and mathematics education. In: Furinghetti F, editor. Proceedings of the Fifteenth International Conference on the Psychology of Mathematics Education; Assisi: PME;1991. p. 33–48.
   Google Scholar
• Rao AR. Proof without words. At Right Angles. 2012;1(1):14–16.
   Google Scholar
• Segal R, Stupel M. An investigative task incorporating computerized technology with conserved property and generalization. Res J Math Technol. 2015;4(2):1–16.
   Google Scholar