"Interactive mathematics texts from Computer Algebra Systems:
use in teaching engineering students"

Published in Vol2 of Proceedings of the Australian Computers in Education Conference, Perth, July 9-13 1995 : ISBN 1 875534 10 5.

Added AFTER PUBLICATION Since this paper was published, PWS have moved their efforts from disk publication of Maple/Mathematica materials to Web publication. End of addition.

Abstract

The problem being addressed now is how to provide, in these Worksheet/Notebook formats, materials which are worth being called "texts", rather than "little essays". One method is to hold the series of "little essays" together by arranging that they accompany an existing conventional printed text. The PWS Notebook series of disks covers, for both CAS, various texts put out by the publishing firm PWS. PWS also distribute CAS materials from their Web site. Our own efforts related to preparing materials for possible inclusion in the PWS Notebook distribution to accompany Zill and Cullen "Advanced Engineering Mathematics" (PWS) will be displayed.

Introduction

We believe that the Worksheet/Notebook formats, combined with student editions of CAS running on relatively inexpensive, e.g. home, computers, will result in a massive increase in the use of CAS. CAS are quite old, so it is a valid to indicate why there is still scope for the big increase CAS have been mathematicians' research tools from the 70s. However, in the early period, they were not user-friendly. From the early 80s, the more adventurous introduced CAS into the upper levels of teaching - Honours and later third-year undergraduate. From the late 80s, CAS began to be used in large-class teaching, e.g. to second-year Engineering Mathematics classes. On-line helps in the packages were crucial to this. The large classes also made it essential to produce some printed materials to support the use of CAS. Cooperative efforts - making materials available across the net - enabled the load of preparing such material to be shared. By about 1992, some of the printed materials aimed to bridge between existing text books and the CAS, a development which definitely assists integration of the students' CAS experience with their other experiences of mathematics. Another major change is now in progress, brought about by the Worksheet/Notebook formats, henceforth abbreviated to 'notebook'. Rather than providing extensive printed materials for the CAS, the "disk/electronic-supplement with the text", "interactive text" is delivering the information in a more immediately usable form. The PWS Notebook series of disks began in 1994, PWS distribution via the Web began in 1995. This article explores issues related to these new "interactive text" CAS possiblities.

With or without "interactive texts", we see the computer work is a supplement to, and not a substitute for, more conventional methods of teaching - text, lectures, assignments of exercises from the text, tutors marking these and tutorial discussion.

The two CAS which are now most widely used in engineering and in upper levels of undergraduate Engineering Mathematics teaching are Maple and Mathematica. Both were released in the late 80s, e.g. for Mac in 1987 and 1988 respectively. The PC versions were later than Unix and Mac versions, because Maple and Mathematica both needed more than the 640K common in early DOS machines. As new versions appear, the two 'Ma-' packages leap frog in their capabilities and features. Both have good on-line help. Both provide very good support for graphics and animation.

There are at least three factors which will cause rapid development of CAS "interactive texts".

1. Notebooks now look good enough so that a 10 or 20 page mathematical document can be written in them in a way which is pleasant to read.
2. About half the students have access to home computers with 4Meg RAM - enough to run the Maple Student Version. This fraction will increase. Access to machines at home encourages students to use the CAS when they have a problem to tackle - from an engineering course or wherever -, not just when their mathematics lecturer asks them to do a particular mathematics assignment.
3. Several publishers of mathematics texts understand that the provision of CAS materials supporting their texts will enhance their market share.

There remains a lot of uncertainty. Will formats more akin to Matlab's Notebook structure, which build on existing word-processors, using DDE or OLE or OpenDoc become more popular? In March 95, Maple released MathOffice which combines, via DDE, MSWord and Maple. Will it be a CD- ROM which accompanies the mathematics book, rather than a disk with the book? Will it be distribution via the Web? Irrespective of the answers to these questions, CAS notebooks are tools worth exploiting. Given the varous uncertainties though, it would seem prudent to "keep notebooks simple" so that they can be maintained.

Background

The leading CA-in-teaching sites in Australia are now UNSW and RMIT, both of which use Maple. The leading Mathematica-in-mathematics-teaching sites are Tasmania and Otago. There can be more to CAS than just presenting mathematics: UNSW and Tasmania use CAS for marking assignments.

There are Australian user groups associated with the packages. The Maple e- mail based one is
maple-news@solution.maths.unsw.edu.au

GK, and Rod James, from UNSW, have pre-Worksheet linear algebra course materials on the Maple Share Library. These are at the level of 2nd year Engineering Mathematics courses. Like the "interactive text" items described below, these materials supplement established texts.

General educational issues

There is now an extensive literature on educational issues associated with CAS: see references at end of this paper. Some conference proceedings have bibliographic articles listing hundreds of references. There are hundreds of texts for students associated with using particular CASs. Frequently authors write two books, one for each of the two Ma- packages, for example, E. Johnson's books published by Brooks/Cole. (Sometimes groups of authors write similar books for three or more CA packages, including DERIVE, Theorist, etc. as well.) While there are a few gems amongst these CAS-specific books, we suggest that "the disk, etc. with the book", working with established mass-selling texts is likely to become relatively more important in the very near future. The vendors' WWW sites are convenient for finding out about commercially published materials, as well as many other things.

There is still much to learn about writing notebooks. For example, here is an item concerning "interactive texts" from Scheftic (1990). When writing on-line materials you should direct the reader by using a variety of layout details in place of the denser format you would use in a printed document. Even a highly motivated person with good concentration will find it easier to follow than full paragraphs of text on a display screen. Compare this paragraph with the following.

		
		When presenting text on line:
		  - avoid paragraphs,
 		  - guide the reader's attention,
 		  - set off key ideas.
		Compare this format with the para. shown above.

Methods of CAS use, outside "interactive texts", which remain valid Although we believe that the "interactive mathematics texts" in production will result in a large increase in CAS use, it will join time-honoured, and working, methods of using CA. Here is a check-list of these.

• A modest number of CAS-capable computers needs to be provided in a campus lab. These are useful for initial induction into packages even when student home computers are widely used.
• Some printed guidelines to using the packages, emphasising how to get on- line help, and which notebooks to read first, will continue to be needed. These will become shorter as better notebooks appear.
• Computer lab Practice Classes are useful to answer students' questions.
• With video over-head projection and large-font capabilities of the CAS, CAS can be demonstrated in lectures. In spite of the improved look of the CAS, we find that 20 minutes to be the upper limit for maintaining students' attention. CAS are better when used actively by students than passively watched.
• Setting some exam questions involving giving CA input and output enables the lecturer to answer "yes" if students ask "will CA be on the exam".
• Once a CAS is introduced, every assignment should have a CAS problem. Regular use is helpful. We recommend that CA code be given in the first use of any new CA method but after this, for obviously related problems, the students should be expected to apply their skills, i.e. to write simple pieces of CA code. Sometimes, for motivational reasons, it is worth attempting to get the students to do larger calculations a little closer to engineering than might have been done before.
• Engineering lecturers sometimes require mathematical techniques before the mathematics course has reached that technique. With CAS, it is possible to tell the students the command to get the answer, along with a promise that the mathematics course will later explain "how it works". (At UWA in 1994, the "Machine Dynamics" lecturer needed to use Laplace Transforms before the Mathematics course reached them, and we expect that the same will occur in 1995.)
• A number of East Australian universities have Maple computer laboratory work, sometimes for assessment via one-page, easy-to-mark, short-answer problems detachable from the rest of the lab sheet.
• Small-group student project work, for assessment, using the CA packages, has also been tried. Though this is popular with some groups of students, it possibly requires too much staff time in marking for use with large classes.

When the "disk/electronic-supplement for the book" is in place, it is expected that the amount of printed materials the students need will be reduced. Library copies of longer printed materials, rather than student copies should suffice. The PWS Notebook materials contain questions, just as a text does: these could be used in the assignment questions set.

Engineering mathematics and its teaching

1. Engineers have sums to do, and a CAS can be a convenient "interactive reference source".
2. Engineers are most motivated to use CAS when they are applied to problems which they see as relevant to their engineering interests, or which arise in their engineering courses.
3. It appears most efficient to introduce a CAS when the students are sufficiently advanced in their courses to be moderately sophisticated in use of computers and in mathematics. Some programming skills - familiarity with if and for statements, etc. is useful. The students mathematics needs to be sufficiently advanced that they are able to appreciate, understand and make real use of the CAS capabilities.
4. A combination of a CAS and a traditional text seems to work better, for most students and teachers, than more radical efforts based entirely on Mathematica NoteBooks or Maple Worksheets.

PWS Notebook disks/electronic-supplements:
adding "interactivity" to PWS textbooks

Starting from another point, there are huge numbers of Maple Worksheets and Mathematica Notebooks available. This leads to the question of how best to organise interactive text materials for students. There is no complete answer, but a partial solution, mentioned as (4) above, is to produce materials close to widely selling text-books. Some publishers, of which PWS is an example, are very supportive. PWS publish a series of inexpensive disks, "PWS Notebook disks". These are written to accompany existing texts they publish. So far these disks have been produced mostly for first year Calculus texts with large volumes of sales, including the books by Swokowski, and Dick and Paton. Neither of these texts are used in Perth, but they are used at various East Australian universities. There are other PWS Notebook disks associated with engineering texts not just engineering mathematics texts. PWS also runs a Web site.

The PWS Notebook disks' organization mirrors that of the texts, with chapter and section headings. Each section includes at least one fully worked example with problem solving hints, followed by ready-to-work exercises. A style guideline is available to PWS Notebook disk authors. An important restriction, needed to stop the tendency to add material, is to stay very close to the text. For first year texts this is rigorously enforced. In second-year engineering- mathematics texts, many results are merely presented, but not proved. Where the proofs are very short with a CAS it may be possible to include them in the electronic supplement. (The order of the truncation errors in the Runge-Kutta formula of S15.6 of Zill and Cullen is an example.) The requirement for notebook disks to stay close to the text means that there remains plenty of scope for individual lecturers using the text and its accompanying supplements to develop there own CAS applications which they may use as a focus of additional materials which reflect their special interests.

Maintenance of CAS materials, like the PWS electronic-supplements, as both text books and software move through different editions and versions requires workers. It makes sense, at least for the PWS texts which do not have first-year level volume sales, for at least some of these workers to be drawn from CAS- competent lecturers actually using the text in their classes.

For next year or so GK will be producing notebook materials to accompany Zill and Cullen "Advanced Engineering Mathematics" (PWS : 1992), a text in use at UWA and various East Australian universities. It is expected that PWS will be producing Notebook electronic supplements to accompany this text, there being a team of US CAS users to do this. Assuming GK's contributions are of sufficiently high quality, it is expected that they will be included in the PWS Notebook supplements. The other authors of this paper expect to check some of GK's efforts. The authors hope to convince the publishers that they should use the same team to produce similar supplements to accompany O'Neil "Advanced Engineering Mathematics" (in 1994 with Wadsworth - a sister company to PWS, and, now, 1995 in PWS), which is used at more Australian universities than is ZC. GK is specialising in the linear algebra and vector calculus parts (Part II of ZC), and the numerical methods parts (Part V of ZC). Many useful ideas can be salvaged from GK's Maple written earlier, separate from ZC, though much work remains to be done to achieve the uniformity of style required for the PWS supplement. GK's materials are available by anonymous ftp on 130.95.16.1 in subdirectories of /pub/keady. (Added Dec 95. Now also on the Web: click here.) (If a Notebook disk is produced, PWS will have rights to world sales for the disks accompanying the 2nd edition of ZC. The first version of GK's materials, for the first edition of ZC, will be available across the net, as this facilitates exchange between members of the team producing the PWS Notebook materials.)

References

E. Kreyszig "Advanced Engineering Mathematics" (Wiley, 7th ed.: 1993) and "Maple manual". (Wiley: 1994).

L.C. Leinbach, J.R. Hundhausen, A.M. Ostebee, L.J. Senechal, D.B. Small (eds) The Laboratory Approach to Teaching Calculus. Mathematics Association of America Notes series, volume 20. (MAA: 1991).

C. Scheftic Interactive mathematics texts: ideas for developers. In T. Lee (ed.) Mathematical computation with Maple V. (Birkhauser: 1993).

T. Scott, M.B. Monagan, R. Pavelle and D. Redfern "Maple in Education - Part II", Maple Technical Newsletter, Northern Spring 1994, pp69-78.

D.G. Zill and M.R. Cullen "Advanced Engineering Mathematics" (PWS:1992).