Abstract
We describe our adventures in creating a new first-year course in Experimental Mathematics that uses active learning. We used a state-of-the-art facility, called The Western Active Learning Space, and got the students to “drive the spaceship” (at least a little bit). This paper describes some of our techniques for pedagogy, some of the vignettes of experimental mathematics that we used, and some of the outcomes. EYSC was a student in the simultaneously-taught senior sister course “Open Problems in Experimental Mathematics” the first time it was taught and an unofficial co-instructor the second time. Jon Borwein attended the Project Presentation Day (the second time) and gave thoughtful feedback to each student. This paper is dedicated to his memory.
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Notes
- 1.
Here, \(x_0 = 1\),
, so on and so forth. - 2.
Avogadro’s number is \(6\cdot 10^{23}\), about.
- 3.
For more complicated functions one shouldn’t simplify for numerical stability reasons. But for \(x^2 - a\), it’s okay.
, so on and so forth.References
Artacho, F.J.A., Bailey, D.H., Borwein, J.M., Borwein, P.B.: Walking on real numbers. Math. Intell. 35(1), 42–60 (2013)
Borwein, J.M., Borwein, P.B.: Strange series and high precision fraud. Am. Math. Mon. 99(7), 622–640 (1992)
Borwein, J., Devlin, K.: The computer as crucible: an introduction to experimental mathematics. Aust. Math. Soc. 208 (2009)
Borwein, J., van der Poorten, A., Shallit, J., Zudilin, W.: Neverending Fractions: An Introduction to Continued Fractions, vol. 23. Cambridge University Press, Cambridge (2014)
Corless, R.M.: Error backward. Contemporary Mathematics, vol. 172, pp. 31–31 (1994)
Corless, R.M.: Computer-mediated thinking. In: Proceedings of Technology in Mathematics Education (2004). http://www.apmaths.uwo.ca/~rcorless/frames/PAPERS/EDUC/CMTpaper.pdf
Corless, R.M., Fillion, N.: A graduate introduction to numerical methods. AMC 10, 12 (2013)
Corless, R.M., Jankowski, J.E.: Variations on a theme of Euler. SIAM Rev. 58(4), 775–792 (2016)
Corless, R.M., Jeffrey, D.J.: Scientific computing: one part of the revolution. J. Symb. Comput. 23(5), 485–495 (1997)
Corless, R.M., Essex, C., Sullivan, P.J.: First year engineering mathematics using supercalculators, 2nd edn. SciTex, The University of Western Ontario, London (1995)
Griffiths, D.F., Sanz-Serna, J.M.: On the scope of the method of modified equations. SIAM J. Sci. Stat. Comput. 7(3), 994–1008 (1986)
Hamming, R.W.: Methods of mathematics applied to calculus, probability, and statistics. Courier Corporation, Chelmsford (2012)
Handelsman, J., Ebert-May, D., Beichner, R., Bruns, P., Chang, A., DeHaan, R., Gentile, J., Lauffer, S., Stewart, J., Tilghman, S.M., et al.: Scientific teaching. Science 304(5670), 521–522 (2004)
Hardy, G.: Divergent series. 1949. Theorem 65, 122 (1949)
Karamichalis, R., Kari, L.: MoDMaps3D: an interactive webtool for the quantification and 3D visualization of interrelationships in a dataset of DNA sequences. Bioinformatics (2017)
Karamichalis, R., Kari, L., Konstantinidis, S., Kopecki, S.: An investigation into inter-and intragenomic variations of graphic genomic signatures. BMC Bioinform. 16(1), 246 (2015)
Kari, L., Hill, K.A., Sayem, A.S., Karamichalis, R., Bryans, N., Davis, K., Dattani, N.S.: Mapping the space of genomic signatures. PloS One 10(5), e0119815 (2015)
Labelle, G.: Sur l’inversion et l’itération continue des séries formelles. Eur. J. Comb. 1(2), 113–138 (1980)
Olds, C.D.: Continued Fractions, vol. 18. Random House, New York (1963)
Schröder, E.: On Infinitely Many Algorithms for Solving Equations (1993)
Strang, G.: Too much calculus (2001). http://www-math.mit.edu/~gs/papers/essay.pdf
Von Neumann, J., Goldstine, H.H.: Numerical inverting of matrices of high order. Bull. Am. Math. Soc. 53(11), 1021–1099 (1947)
Acknowledgements
We thank David Jeffrey for his early encouragement, Steven Thornton and Torin Viger for their help with the course material, and the membership of USAM: Julia Jankowski, Andy Wilmot, and Anna Zhu. Gavan Watson, Stephanie Oliver, and Wendy Crocker were very helpful with active learning and the use of WALS. We thank Lila Kari for introducing us to the chaos game representation. We also thank the Rotman Institute of Philosophy and the Fields Institute for Research in the Mathematical Sciences for their sponsorship of the Computational Discovery conference (www.acmes.org), at which some of these ideas presented here were refined. We also thank the Center for Teaching and Learning for a Fellowship in Teaching Innovation Award 2018–2019.
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Chan, E.Y.S., Corless, R.M. (2020). A Random Walk Through Experimental Mathematics. In: Bailey, D., et al. From Analysis to Visualization. JBCC 2017. Springer Proceedings in Mathematics & Statistics, vol 313. Springer, Cham. https://doi.org/10.1007/978-3-030-36568-4_14
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