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Bill Blyth, Teaching Experimental Mathematics: Digital Discovery Using Maple Teaching Experimental Mathematics: Digital Discovery using Maple Bill Blyth 1; 2 [email protected] 1 Australian Scientific & Engineering Solutions (ASES) 2 School of Mathematical and Geospatial Sciences RMIT University Australia Abstract One of the \Recommendations (by Klaus Peters) on Jon Borwein's LinkedIn page starts with: \Jonathan Borwein is a world-renowned research mathematician and one of the leaders in the emerging field of experimental mathematics to which he has made many original contributions. He has written several books that have helped to define the field. Examples include finding closed forms of integrals by numerically computing high precision approximations (accurate to hundreds of decimal digits) which is used to identify the answer. Knowing the correct answer facilitates the proof of its correctness. Here this experimental mathematics approach is taken with a couple of traditional “find the maximum . " problems suitable for senior school or first year undergraduate students (who could even be pre-calculus students). We use the Computer Algebra Sys- tem, Maple, for experimentation (and visualization) to compute high accuracy numerical approximations to the answer. Maples identify( ) command is used to identify the exact answer. Knowing the answer, we prove that it is correct - without using calculus. We use small group collaborative learning in the computer laboratory, so we parameterize the problem and recommend the use of Computer Aided Assessment (such as provided by the package MapleTA). Students engage with the visualization, are active learners with deep learning of the concept of maximum and gain a gentle introduction to the important new field of exper- imental mathematics. 1 Introduction Since the mid 1990s, we have often stated our view that all mathematics graduates from uni- versities should be expert users of one of the two professional level Computer Algebra Systems, CAS: Maple being one of these. A recent study [9] of UK university students reported \that 59% of students find their lectures boring half the time and 30% find most or all of their lec- tures boring." A range of different teaching methods were investigated: the most boring were lab sessions; computer sessions ranked as the second most boring. \Computer sessions too have the potential to be stimulating or tedious; the findings of this study suggest that too many fall into the later category. This could be due to the manner in which sessions are conducted (e.g. are the computer tasks relevant and interesting?), . etc." These findings are not surprising given that most courses are teacher centric and use a tra- ditional teaching approach (lectures) with \tutorials" and assignments which are mostly done without use of technology. When technology such as CAS is used (at both school and under- graduate levels), it is often a minor supplement where the usual \by hand" exercises completed by using CAS following the \by hand" approach. We suggest that innovative teaching materi- als which exploit the strengths of the CAS for tightly integrated e-Learning and e-Assessment, such in the Maple component of our first year course (and in higher year courses taught entirely using Maple), and a pedagogy of collaborative learning in small groups as active learners in a computer lab need to replace (at least in part) boring lectures and boring computer lab sessions. In this paper we consider a standard calculus problem: find which cylinder inscribed in a unit sphere has the maximum volume. We provide visualization and show how to obtain an accurate decimal approximation to the radius that gives the maximum. This leads to an identification of the possible exact answer. By foregoing calculus, we give a gentle introduction to experimental mathematics: we provide several algebraic proofs that introduce important ideas in mathematics, such as approximation in the small. We also support our intuitive notion that the maximum of a graph of a function \looks like a parabola" near the maximum by algebraically calculating the quadratic approximation of the function near its maximum. This paper is organized as follows. Section 2 introduces experimental mathematics. Sec- tion 3 has a brief overview of first semester, first year Maple sessions which provide a suitable preparation to undertake the Cylinder in a Sphere problem described here. Section 4 pro- vides a description of the Cylinder in a Sphere problem and visualization (including multiple representation and animations). Section 5 zooms in on the plot of volume versus radius of the cylinder to approximate the r that maximizes the volume; and a possible exact answer is identified. Several proofs without using calculus are discussed: in Section 6, volume squared is maximized; in Section 7, volume is maximized. Several more proofs are indicated in Section 8. Section 9 introduces a parameterized problem (of a Cylinder in a Spheroid) and discusses e-Marking and Computer Aided Assessment. This is followed by a brief Conclusion. 2 Experimental Mathematics Jon Borwein is a pre-eminent research mathematican who is one of the leaders in developing and popularizing experimental mathematics. We suggest that every aspiring or practising mathematician should read some of his work: we particularly recommend the book [6] and the short article [7]. In the later, Borwein shares Polya's view on intuition: Intuition comes to us much earlier and with much less outside influence than formal arguments . Therefore, I think that in teaching high school age youngsters we should emphasize intuitive insight more than, and long before, deductive reasoning. George Polya (1887{1985) [10], p.128 Borwein continued with [Polya did go] on to reaffirm, nonetheless, that proof should certainly be taught in school. [ . ] My musings focus on the changing nature of mathematical knowledge and in consequence ask questions such as How do we come to believe and trust pieces of mathematics?, Why do we wish to prove things? and How do we teach what and why to students?. [7] In our experience with our own mathematics undergraduate and PhD students is that they have little appreciation of intuition in mathematics: my discussions about the importance of intuition (and the development of it) come as a surprise to them. Borwein [7] gives an informal summary of what is meant by experimental mathematics: • Gaining insight and intuition; • Discovering new relationships; • Visualising math principles; • Testing and especially falsifying conjectures; • Exploring a possible result to see if it merits formal proof; • Suggesting approaches for formal proof; • Computing replacing lengthy hand derivations; • Confirming analytically derived results. Another eminent mathematician, Doran Zeilberger [11] states that The purpose of mathematical research should be the increase of mathematical knowledge, broadly defined. We should not be tied up with the antiquated notions of alleged \rigor". A new philosophy of and attitude toward mathematics is devel- oping, called \experimental math" (though it is derided by most of my colleagues; I often hear the phrase, \Its only experimental math"). Experimental math should trickle down to all levels of education, from professional math meetings, via grad school, all the way to kindergarten. Should that happen, Wigners \unreasonable effectiveness of math in science" would be all the more effective! Lets start right now! A modest beginning would be to have every math major undergrad take a course in experimental mathematics. We agree, but note that many senior school mathematics courses use CAS so we suggest \the beginning" should sensibly occur in the secondary school. 3 Preliminaries For about a decade, we designed and ran Maple topics for our first year university \Calculus" course comprised of four traditional lectures and one Maple lab session each week. The weekly lab sessions had no lectures: students work collaboratively in small groups. For the first semester, the topics were secondary school level mathematics. Students start by taking two sessions to work through an Introduction to Maple which includes a simple assignment that is printed and submitted as a hardcopy, but all other work is completely electronic. An Animation worksheet is followed by a major Animation Assignment: students pick a project (from a list) that is on a school level topic, see [1]. After these activities, students are expert at preparing plots and animations, so these are recurring themes. The next activity is Spot the Curve. The Maple file Spot the Curve is accompanied by another Maple binary file which contains all the procedures for randomly generating the curves, their translated curves and automatic marking procedures: this binary file is read into the student Spot the Curve file so that all of the required procedures are invisible but active in the student file. Each student group is presented with several plots of a curve and a translated copy of that curve: the task is to identify what the translation is (by using plots and animations). Students use the provided automatic marking procedure to immediately mark their result. Feedback shows that students really enjoy this activity: for details, see [2]. Another style of auto marked problem is the FishPond problem, a straightforward applica- tion of using the trapezoidal rule to approximate area. It has multiple steps with numerical answers: the marking procedure is used by students to immediately mark their work: for de- tails, see [2]. The FishPond problem is followed by working through a Maple file of a structured Polya approach to “finding the maximum area", followed by an assignment on finding the max- imum area of the Norman Window problem. For a discussion of the Norman Window problem, multiple representations, e-Marking and Computer Assisted Assessment, see [3]. With this (or similar) background, students are prepared to investigate various solution methods (proofs) of a particular related problem, see [5]. These activities provide students with an appropriate background to undertake the maxi- mum problem(s) described next. 4 The Cylinder of Maximum Volume inscribed in a unit Sphere This problem is a classic one that appears in many textbooks.
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