Jonathan Borwein, FRSC
Canada Research Chair, Dalhousie
Professor Laureate
School of Mathematical & Physical Sciences
University of Newcastle
Callaghan NSW 2308, Australia

October 30, 2008

Dear Helge, Ann and Vaishali: I am currently teaching a second year one-semester course called Mathematical Software here in Newcastle to about 30 students. (A weekly two hour lecture and a two hour tutorial for 13 weeks.))

See

There is no text for such a course that I know of and the current syllabus while well intended has no intellectual heart. I describe some more about the current course in the Appendix below

I’m presently preparing the material for a revised course.I intend to write a companion book called something like

Introduction to Modern Mathematical Computing

My intention is to have a draft by the end of February 2009. I have a Research assistant hired for the four summer months (November through February).I envisage producingno more than 250 pages aimed at a second year European, UK, Australian-Canadian audience or second to third year US audience.

I would like to target it early enough in the curriculum so it leaves students a significant proportion of their program in which to exploit their new found skills. I shall assume that the reader has a first year calculus background and a little discrete math and is probably taking or has taken a first linear algebra course.

Iplan to organize the course and book around topics in the following order.

  • Discrete Mathematics and Elementary Number Theory
  • Linear Algebra and Applications
  • Calculus, Analysis and Applications
  • Geometry, Dynamics and Topology

This will lead to the following six major chapters.

  1. Introduction and Maple Preliminaries
  2. Discrete Mathematics and Elementary Number Theory
  3. Linear Algebra and Applications
  4. Calculus, Analysis and Applications
  5. Geometry, Dynamics and Topology
  6. Further topics

The book will thus have a coherent curriculum---and one that I do not think has real competitors. The ordering of the topics allows for an increasing sophistication of mathematical concepts and computational ideas (how to use the math not why the algorithms work) to be developed together.

Each section will finish with a selection of Tutorial like problems. Each chapter will conclude with a section of Excursions of a more ambitious nature and set up so as to be useful for directed study or term projects.Where possible the routine examples will actually foreshadow more interesting phenomena but these matters will only be discussed in the Excursion section.

I will primarily teach and write forMaple and, especially in Chapters five and six, Cinderella (sold by Springer). I shall, however, make reference to numerical computation and use of MATLAB accessed through Maple (which now has a MATLAB add-on), toother visualization and web tools, etc.

With a coauthor, I could handle Maple and Mathematica but probably only by making the book much larger than I wish. It would also be possible to have parallel Maple version and Mathematica editions.

My colleague Jan de Gier at Melbourne is teaching an Experimental Mathematics course Mathematica.

Moreover:

  • The book would allow for a variety of topics to be touched upon: topics that never get discussed for most undergraduates.
  • It could be taught be taught in many contexts (with a change in emphasis depending on the instructor).
  • The first module could be built so as tobe used independently to introduce Maple in first year in a Calculus tutorial mode.
  • The material in each module can be arranged so the only real prerequisites are a calculus course and linear algebra (here that is taken in the same semester).
  • Auxiliary web material is of course both possible and likely.

So I’d like to discuss whether you see this as a possible fit with SUMAT. I’d certainly prefer to begin writing with a publisher (and style files) settled.

"I can calculate the motions of the heavenly bodies," he said, "but not the madness of people." Isaac Newton quoted by Christopher Reed, "The Damn'd SouthSea”, Harvard Magazine, May-June 1999.

Appendix

Background on the Current Course

  1. The curriculum has “traditionally” been 50% an Introduction to Maple and 50% how to use Latex. There are two term tests but 60% of the marks coming from a project in which students use their new skills in mathematical type setting to perform a Maple based exploration project. Due to prior teachers of the course, the projects have been heavily in numerical ODE’s and the mid-terms much too hard.
  1. It turns out only about 20% are generic math honours students and the rest are in CS, Engineering, or in more than 50% of the cases aim to be teachers. I’ve had each lecture run out of a Maple worksheet which we then annotate and put on the course website for tutorial use.
  1. The ‘latex’ instruction in Maple produces quite adequate output that can be dropped into MathType (Word or PowerPoint) or TexPoint (PowerPoint)and the Maple worksheets are quite good themselves. Moreover, Maple now can read Mathematica notebooks and at least on an expression level translate most Mathematica into Maple.
  1. This year I got Frank Garvan’s permission to use his nice but now dated Maple V primer (only 90 pages) and took Mike Doob’s advice on a free Latex manual. I also had MathType (which now accepts Latex input) and TexPoint installed in the Tutorial Laboratory along with my own company’s products.
  1. I have come to the conclusion that teaching conventional latex skills makes no sense for students who will not go on to graduate work. Moreover, the process of going from Maple to latex to MathType or TexPoint to Word is really easy now:

For instance, I start in Maple with

J:=Int(x^4*(1-x)^4/(1+x^2),x=0..1); (1)

EQ:=J=value(J);

latex(EQ);

\int _{0}^{1}\!{\frac {{x}^{4} \left( 1-x \right) ^{4}}{1+{x}^{2}}}{dx }={\frac {22}{7}}-\pi (2)

Then I cut and paste into MathType and out comes:

Likewise in TexPoint(see the next picture).

As a bonus we can handle and format pretty nice images easily. Moreover, it is easy to show the rudiments of latex along the way. I point out the general form of the syntax we obtain by comparing the Maple in (1) to the latex in (2) and we see what tweaking a subscript or changing an entry in \frac does.

Or how the donut was drawn?

  1. Not teaching latex (except inter alia) allows for a much richer course. For 90% of users, teachers and producers of math I can focus on showing what is possible with Mapleor out on the Web and not on the dull details of Tex.
  1. While there is a lot of experimentation under the hood when I teach the course, this also provides a method to reach a bigger and different audience. I had 40 faculty turn up for a seminar I gave here entitled “My experiences with math software” last September. Only three had seen Sloane’s Encyclopedia and the numbers were similar for almost everything in my portal