Abstract
Over the past 30 years or so, the authors have been teaching various programming for mathematics courses at our respective universities, as well as incorporating computer algebra and numerical computation into traditional mathematics courses. These activities are, in some important ways, natural precursors to the use of Artificial Intelligence in Mathematics Education. This chapter reflects on some of our course designs and experiences and is therefore a mix of theory and practice. Underlying both is a clear recognition of the value of computer programming for mathematics education. We use this theory and practice to suggest good techniques for and raise questions about the use of AI in Mathematics Education.
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Notes
- 1.
For example, (Slagle, 1963), which took a “Good Old-Fashioned Artificial Intelligence (GOFAI)” approach, and concluded “The solution of a symbolic integration problem by a commercially available computer is far cheaper and faster than by man”. Of course, this was from the era when people still believed in GOFAI. We are grappling with different problems today, using much more powerful tools. Yet some important things can be learned by looking at the effects of the simpler and older tools. The riposte to Slagle (1963) was the development of Computer Algebra (Davenport, 2018) as a separate discipline.
- 2.
Much interest in computation and proof for pure mathematics was generated by the very successful polymath project. Because computation has always been perceived as instrumentally important, a corresponding but much larger scale project on the Applied Mathematics side might be the Intergovernmental Panel on Climate Change.
- 3.
We had intended to give the reference (Rosati et al., 1992) for this; however, that journal seems to have disappeared and we can find no trace of it on the Web, which is a kind of testimony to ephemerality. Some of the lessons of that article were specific to the calculator, which was too advanced for its era and would be disallowed in schools today. We shall not much discuss the current discouragingly restricted state of the use of calculators in schools hereafter.
- 4.
Students were enrolled in one of three tutorial hours, but often went to all three hours.
- 5.
Often, these lists rely on fairly baseless claims or are derived from the number of Internet searches for a language; here, we base this claim on proportion of code on GitHub as measured by https://madnight.github.io/githut/ visited on 2021-02-09.
- 6.
Active learning is defined, for instance, in a well-known teaching and learning website at Queen’s University, Kingston, Ontario: “Active learning is an approach to instruction that involves actively engaging students with the course material through discussions, problem solving, case studies, role plays and other methods. Active learning approaches place a greater degree of responsibility on the learner than passive approaches such as lectures, but instructor guidance is still crucial in the active learning classroom. Active learning activities may range in length from a couple of minutes to whole class sessions or may take place over multiple class sessions.”.
- 7.
The elementary functions of the calculus are not “elementary” in the sense of being simple, but instead they are “elementary” in a similar sense to the elementary particles of physics.
- 8.
Aphra Behn 1640–1689 has one of the most interesting, if only dubiously accurate, biographies that we have read.
- 9.
Ambrose Bierce 1844–1914(?) was an American satirist, critic, and journalist, perhaps most famous for his collection of definitions published as “The Devil’s Dictionary”.
- 10.
https://stackexchange.com/ Stack Exchange is a network of websites for communities where contributors ask and answer questions and then vote on responses. Stack Overflow is for general programming but specialist communities exist, e.g., for Maths, AI, and Data Science. While generally good, the quality does vary, and is poor for computer security (Fischer et al., 2017).
- 11.
https://www.chegg.com/ Chegg is a homework help website.
- 12.
- 13.
Given the economic constraints of the large class model, we mean. Even then, there may be alternatives, such as so-called “mastery grading” (Armacost & Pet-Armacost, 2003). We look forward to trying that out. Exam stress is often counterproductive, and the current university assessment structures do encourage and reward successful cheating. We would like a way out of this, especially now in COVID times.
- 14.
One British citizen in 25,000 is a graduate of XX10190.
- 15.
Although it’s true that, sometimes, simply reading a question aloud can be surprisingly useful, of course tone matters, here. Reading the question aloud as if it were a reminder to the instructor can be less painful for the student.
- 16.
Strubell et al. (2019) report that training a big model with neural architecture search can generate as much CO\(_2\) as five cars during their lifetime, including fuel.
- 17.
Except as an important stepping stone to the real truth—see the entry “Lies to Children” in Wikipedia. Sometimes a simplistic story is the right first step.
- 18.
Aristotle may have done us a disservice by looking down on crafts and craftspeople; the term Software Carpentry is not likely to induce respect for the discipline in academia, for instance. We lament this prejudice.
- 19.
See also Bradford et al. (2009), which shows that computational tools can affect the basic meaning of equality: pedagogical equality is not the same as mathematical equality. It is perfectly possible for two expressions to be mathematically equal, but only one expression to be the desired student response.
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Acknowledgements
RMC thanks the Isaac Newton Institute for Mathematical Sciences and the staff of both the University Library and the Betty and Gordon Moore Library at Cambridge for support and hospitality during the programme Complex Analysis: Tools, techniques, and applications, by EPSRC Grant # EP/R014604/1 when some of the work on this project was undertaken. RMC likewise thanks the University of Bath for an invitation to visit Bath, at which this project was started. EYSC and RMC also thank Western University for a grant to work on the project Computational Discovery on Jupyter, some of whose results are discussed here.
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Betteridge, J., Chan, E.Y., Corless, R.M., Davenport, J., Grant, J. (2022). Teaching Programming for Mathematical Scientists. In: Richard, P.R., Vélez, M.P., Van Vaerenbergh, S. (eds) Mathematics Education in the Age of Artificial Intelligence. Mathematics Education in the Digital Era, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-030-86909-0_12
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