zbMATH — the first resource for mathematics

Dynamic visual models: ancient ideas and new technologies. (English) Zbl 1443.00012
Bailey, David H. (ed.) et al., From analysis to visualization. A celebration of the life and legacy of Jonathan M. Borwein, Callaghan, Australia, September 25–29, 2017. Cham: Springer. Springer Proc. Math. Stat. 313, 189-201 (2020).
Summary: We provide dynamic visual models of the following facts established by ancient mathematicians:
\(\sum_{i=1}^n(2i-1)=n^2\), \(n\in\mathbb{N}\),
\(\sum_{i=1}^ni=\frac{n(n+1)}{2}\), \(n\in \mathbb{N}\),
\(\sum_{i=1}^ni^2=\frac{n(n+1)(2n+1)}{6}\), \(n\in\mathbb{N}\),
\(\sum_{i=1}^ni^3=\left(\frac{n(n+1)}{2}\right)^2\), \(n\in \mathbb{N}\).
We contrast the clarity of the models by outlining formal mathematical proofs based on those timeless ideas. We also reflect about the place that proofs play in the calculus classroom.
For the entire collection see [Zbl 1442.00024].
00A35 Methodology of mathematics
97E50 Reasoning and proving in the mathematics classroom
Full Text: DOI
[1] Alsina, C., Neslon, R.B.: An invitation to proofs without words. Eur. J. Pure Appl. Math. 3(1), 118-127 (2010) · Zbl 1209.00020
[2] Bailey, D.H., Borwein, J.M., Kapoor, V., Weisstein, E.: Ten problems in experimental mathematics. Am. Math. Mon. 113, 409-481 (2006) · Zbl 1153.65301
[3] Beery, J.: Sums of powers of positive integers. In: Loci (2009). https://doi.org/10.4169/loci003284
[4] Borwein, J.M.: The experimental mathematician: the pleasure of discovery and the role of proof. Int. J. Comput. Math. Learn. 10(2), 75-108 (2005)
[5] Borwein, J.M.: The life of modern homo habilis mathematicus: experimental computation and visual theorems. In: Monaghan, J., Trouche, L., Borwein, J. M.: Tools and Mathematics: Instruments for Learning. Mathematics Education Library, vol. 110. Springer, Berlin (2016)
[6] Borwein, J. M., Bailey, D.H.: Mathematics by Experiment: Plausible Reasoning in the 21st Century, 2nd edn. A. K. Peters, Natick (2008) · Zbl 1163.00002
[7] Borwein, J., Jungić, V.: Organic mathematics: then and now. Not. Am. Math. Soc. 59(3), 416-419 (2012) · Zbl 1273.01065
[8] Bostock, M.: Visualizing algorithms, posted on https://bost.ocks.org/mike/algorithms/. Accessed 9 Nov 2017
[9] Cox, J.A.: Proofs of the sum of squares formula, posted on http://www.fredonia.edu/faculty/math/JonathanCox/math/SumOfSquares/SumOfSquares.html. Accessed 9 Nov 2017
[10] Descartes, R.: A Discourse on Method. The Project Gutenberg EBook \(\#59\), posted on http://www.gutenberg.org/files/59/59-h/59-h.htm. Accessed 12 Nov 2017
[11] Graham, R., Rothschild, B., Spencer, J.: Ramsey Theory. John Wiley and Sons, New York (1990)
[12] Gravina, M. A.: Dynamical visual proof: what does it mean?. In: Santos, M., Shimizu, Y. (eds.) Proceedings of the 11th International Congress on Mathematical Education, Monterrey, Mexico (2008)
[13] Guy, R.: The strong law of small numbers. Am. Math. Mon. 95(8), 697-712 (1988) · Zbl 0658.10001
[14] Hanna, G.: A critical examination of three factors in decline of proof. Interchange 31(1), 21-33 (2000)
[15] Heath, T.L.: A History of Greek Mathematics, vol. 1. The Clarendon Press, Oxford (1921) · JFM 48.0046.01
[16] Heath, T.L.: A History of Greek Mathematics, vol. 2. The Clarendon Press, Oxford (1921) · JFM 48.0046.01
[17] O’Connor, J. J., Robertson, E. F.: Francesco Maurolico, posted on http://www-history.mcs.st-and.ac.uk/Biographies/Maurolico.html. Accessed 12 Nov 2017
[18] Edwards, C., Penney, J.: Calculus, Early Transcendentals. 7th edn. Pearson, London (2012)
[19] Knott, R.: Fibonacci puzzles, posted on http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibpuzzles2.html. Accessed on 12 Nov 2017
[20] Math 100 — Learning objectives, posted on https://www.math.ubc.ca/ yhkim/yhkim-home/teaching/Math100-2017/pdfs/maths100_180_objectives.pdf. Accessed 22 Feb 2018
[21] Monaghan, J., Trouche, L., Borwein, J. M.: Tools and Mathematics. Mathematics Education Library, vol. 110. Springer, Cham (2016)
[22] Nelsen, R.: Proofs Without Words: Exercises in Visual Thinking. Mathematical Association of America, Washington (1993) · Zbl 1160.00303
[23] PISA 2015 — Canada. http://www.compareyourcountry.org/pisa/country/can?lg=en. Accessed 22 Feb 2018
[24] Stewart, J.: Calculus, Early Transcendentals. 7th edn. Brook/Cole (2012) · Zbl 1270.00017
[25] Sum of \(n\) Integers, at GeoGebra. https://geogebra.org/m/ufxG9eHn. Accessed 22 Feb 2018
[26] Sum of Odd Numbers, Wolfram Demonstration Project. http://demonstrations.wolfram.com/SumOfOddNumbers/. Accessed 22 Feb 2018
[27] Sum of Squares, on GeoGebra. https://geogebra.org/m/JBnrZdn7. Accessed 22 Feb 2018
[28] Sum of the Cubes of the First \(n\) Natural Numbers, on GeoGebra. https://geogebra.org/m/Z8tq2Usw. Accessed 22 Feb 2018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.