Educational institutions are complex organisations. Each one has its own set of rules, relations, dependencies and established practice. Teaching staff frequently report that the system at their place of work is inflexible and hinders their development. For mathematics staff who may be interested in incorporating CAS in their teaching, a variety of logistical obstacles can impair their progress. These include the availability of IT facilities, provision of staff training, time to prepare new lessons, availability of support materials and the means of sharing expertise among faculty staff.

In a survey of teachers in the eastern counties of England [Bowers & Schuller, 1995], 85% of teachers at schools where CAS was available but who did not use it themselves in their teaching cited logistical reasons for this, with only 15% indicating some kind of ideological opposition. The main reason for non-use was lack of awareness and/or confidence (44%), highlighting the need for improved provision of in-service training and support for staff. Then came lack of time to prepare new lessons and work schemes (22%) and difficulty in booking adequate computer lab space (19%). It is clear that creative and pro-active ways of dealing with the organisational issues of time, training, resources and support can remove perceived obstacles and encourage staff to become more involved with computer algebra.

In the following paragraphs we outline some strategies for facilitating CAS instruction by addressing the underlying logistics problems. Many of the sugggestions are based on examples of good practice observed in a variety of institutions in various countries. The precise details naturally depend on the nature of the particular school or college concerned. The emphasis here is on workable strategies which are not expensive to implement. The main requirement is for committed and enthusiastic leadership, with the desire to help pave the way for colleagues who are initially more reluctant.

STAGE-MANAGE THE LEARNING ENVIRONMENT

If teachers are to be encouraged to incorporate CAS (or any other new development) in their teaching, they must feel that they are doing so within a supportive environment. Experienced colleagues can facilitate this by "stage-managing" the classrooms, computer labs, workshops and corridors where the mathematics staff and students work. Posters extolling the virtues of the relevant computer algebra software are readily available from publishers, and can be liberally displayed. Neat examples of CAS applications can be posted on notice boards, and annotated to show just how powerful the software is. Examples of students’ own work can be displayed. Banks of catalogued worksheets and handouts should be easily accessible - even left lying around - in classrooms, staff work areas and libraries.

One example of observed good practice was in a school where mathematics noticeboards are updated weekly to display examples of DERIVE applications for the work due to be covered that week by particular groups. The teaching staff took at least as much interest in these as the students!

The fundamental objective here is that no teacher can claim to be unaware of the CAS used in the institution, how it can be meaningfully deployed in mathematics teaching, or where to find support materials. Stage-managing the learning environment in this way can encourage both staff and students to take a greater interest in the value of computer algebra.

Naturally, a particularly dedicated teacher must take the lead in organising the display materials. It is always disheartening if your carefully crafted handouts or beautiful posters are ignored, tossed to one side or defaced. Unfortunately, pioneers never have an easy life . . .

VARY IN-SERVICE TEACHER TRAINING

Certainly, a programme of staff training and updating is a key way to impart awareness and confidence to staff. In new areas of technology such as computer algebra, where so many developments have taken place within only a few years, the busy classroom teacher will find it difficult to remain curent. Indeed, in his survey of 129 teachers in eastern England, all of whom were currently teaching mathematics at A-level (pre-university courses for 16-19 year olds), Bowers found that 49% had no first-hand experience of CAS, and 66% worked at institutions where CAS was not available [Bowers & Schuller, 1995].

Sending staff out to externally run short courses can be expensive. However, such courses usually have the benefit of credibility and authority in the eyes of the participants, and provide a neutral and undisturbed environment for work. On the other hand, in-house training events organised by the colleagues with most experience and willingness to share it cost very little to arrange, but may not be taken seriously by the participants and are often liable to disruption.

One compromise may be for neighbouring schools or colleges to work together. Staff from school A travel to and are given a course by the "experts" of school B, and vice-versa. This maintains the advantages of an externally run course, without incurring the expense of a commercial provider.

However the programme of in-service training is organised, it is considered important for it not to be seen in isolation, and not to be planned before the participants are in a position to make full use of the outcomes. ACDCA conference participants Bowers, Marshall and Watkins from the UK all noted independently how enthusiasm for DERIVE imparted to participants on one-day short courses soon becomes diluted and ultimately forgotten if the participants’ institutions do not yet have a CAS of their own for these staff to reinforce their newly learned skills. Sending staff on "taster" courses before an institutional commitment to CAS exists has little value.

This is also an appropriate place to re-iterate four golden rules of teacher in-service training in computer algebra:

a) Don’t underestimate the lack of basic IT/keyboard skills of many competent and experienced mathematics teachers.

b) Don’t try to cover all the features of the CAS - explain, and let the delegates try, the essential ones, and wait for the question: "But can DERIVE do this . . .". Now you’ve got them hooked!

c) Don’t just show them examples of good classroom material - get them to try producing a worksheet themselves. Teachers also take pride in achievement.

d) Don’t alienate novices. Rather than demonstrating all the new areas of mathematics that CAS opens up, concentrate primarily on how it has benefits for familiar topics.

RESCHEDULE THE START OF COURSES

The main thrust of this chapter is to encourage teachers to use CAS in their mathematics teaching. It may be perceived as unfair if students are denied the opportunity of using CAS in their studies because their teachers lack confidence themselves in the new technology. This may be the case particularly in higher education, where a significant amount of self-directed independent study is expected of the mathematics students. In such a situation, the initiative should be taken centrally in order to help facilitate equal access for all.

On possible model is to allow an induction period at the beginning of the course/year, lasting one or two weeks. Regular mathematics classes would be postponed, replaced by "Introduction to DERIVE" workshops in the computer lab, run by the CAS experts in the department. Although aimed primarily at the students, staff who were less confident with CAS would also be encouraged to attend.

Having experienced the facilities of the CAS before the actual mathematics course starts, the students may well refer to it in class, and their teacher might become more inclined to take an interest. Even if this is not immediately the case, the students of the reluctant staff will be at less of a disadvantage having learned the basics of the CAS in advance and knowing where to find it for their own use.

If such a model is implemented, some discussion with other disciplines is necessary to appropriately schedule content. For example, Physics classes may await the introduction of the derivative in mathematics before beginning Newtonian mechanics. A two week delay in content may require some adjustment of the physics course syllabus.

PHASE IN COURSE CONTENT CHANGES

If teachers are to be encouraged to use CAS in the classroom, it must be made to appear as an educational opportunity rather than a threat to established, successful methods. . For this reason, we recommend no modification to mathematics course content when the CAS is first introduced, and to continue to use the same textbooks.

Although computer algebra systems such as DERIVE will fundamentally affect the teaching and learning of mathematics in the long term, teachers who are initially sceptical or lack confidence with the new technology are more likely to be carried through an evolutionary process than a revolutionary one. Any sudden change changes to courses or syllabi which are driven by the technology available might satisfy that minority of teachers who are enthusiastic CAS practitioners, but would most likely alienate their less experienced colleagues. In addition, we find it is pedagogically unsound to allow technology to master the curriculum. Rather, it is appropriate curriculum objectives that may be served by technological advances in educational methods.

In England, the pre-university examinations (A-levels) for 18 year old school leavers are set and marked by external examining boards which have been hesitant in acknowledging CAS as a permissible mathematical tool. While frustrating for some, this has had the advantage that there has been time to encourage and persuade the CAS "novices" in the department of the value of computer algebra independent of external pressures.

At university level, more and more textbooks on calculus or linear algebra already contain problems annotated with a symbol such as "PC", meaning that the use of a computer is recommended. Traditionally such problems are generally ignored, since they go beyond the scope of what the students may expect in the final examination. A teacher just starting to experiment with computer algebra thus has a bank of examples from an already familiar setting, and will be able to appreciate how useful a tool the CAS is. Furthermore, the traditional textbooks frequently contain topics which are difficult to explain on the chalkboard - relying for example on repeated calculations or accurate graphical representation - and these areas tend to be touched on only briefly. A teacher who is aware of the need for an improved presentation medium within existing course materials should welcome the benefits of computer algebra systems more readily than if a whole new unfamiliar CAS-based syllabus was suddenly imposed.

UPGRADE THE STATUS OF COURSE DEVELOPMENT

A common complaint from teachers is that they do not have enough time to develop new schemes of work or lesson plans which include technological innovations such as computer algebra. Even if the use of CAS in the classroom is phased in gradually while continuing to use existing traditional teaching materials, there will come a point when new curriculum resources and assessment schemes become desirable in order to exploit the educational advantages of computer algebra more fully.

Course development of this nature, which entails a fundamental re-appraisal of what mathematics should be learned, as well as the way mathematics is taught and learned, should be an issue for the whole faculty rather than left to individual teachers to implement on their own, especially if some of the teachers still lack confidence or expertise in computer algebra. Without a co-ordinated approach, there is the danger of "re-inventing the wheel", as well as the likelihood of different styles and notations in the CAS materials produced. Course development for the next century, incorporating computer algebra and other technological innovations, is arguably the greatest challenge facing mathematics departments at the moment, and the significance of the undertaking should be recognised by managment. By attaching not just status but resources to this project, support can be directed to those staff members who might otherwise lack the confidence to take part.

Additional prestige can be awarded to the use of computer algebra systems in mathematics education by encouraging original research into their use and benefits. This is a valid area for scholarly activity, and several publications now exist (for example, The International DERIVE Journal) which provide an outlet for such research. This being a relatively young field, teachers at "grass roots level" are able to add to their list of publications by reporting scientifically on their own work and classroom experience. This may convince otherwise reluctant teachers that involvement with CAS is valuable and worthwhile.

EXPLOIT THE MATHEMATICS WORKSHOP

By Mathematics Workshop we mean an open-access mathematics study centre which is equipped with a range of learning resources (textbooks, worksheets, study guides, tutors) and a suite of computers running various mathematical software. Such workshops are common in the UK. They can be reserved by teachers for activity-led group sessions to supplement more traditional classroom lessons, and are otherwise open to all students on a drop-in basis for additional private study. It is the mathematics workshop - or a similar computer laboratory - where students will have access to computer algebra software. It should be utilised in such a way that teachers, too, are given every encouragement to make use of the available computer algebra system.

Difficulty in booking adequate computer lab space is often cited as an obstacle by teachers. It might be possible to overcome this by re-appraising the utilisation of the mathematics workshop or the computer labs. These are often scheduled on a "first come, first served" basis, or occupied by default over the whole year by groups who only have occasional need for the facilities. The not uncommon sight of students sitting alongside a computer terminal but writing down pen-on-paper notes copied from the board indicates less than optimal use of highly desirable space.

The logistical problem of getting the right students to the right machines at the right times may be overcome by prohibiting block bookings, so that classes only occupy the space when they explicitly need it. Furthermore, if the workshop manager prioritises bookings for a transitional period from those teachers who intend to use the computer algebra software, there will be less reason for staff not to start to incorporate the CAS in their teaching.

An alternative approach to workshop utilisation runs contrary to the one above. Here, the mathematics classes of those teachers who are less confident with the CAS are timetabled exclusively in the workshop or computer lab. The availability of machines is thus no longer an obstacle, and together with strategies to minimise other potential difficulties (as outlined elsewhere in this chapter) this would provide positive logistical support for colleagues who might otherwise hesitate to start using computer algebra in their teaching.

Since the size, location, resources and organisation of the mathematics workshops and computer laboratories vary so much between institutions, it is difficult to generalise on strategies for optimal utilisation. The fundamental point, however, is that creativity is essential in order that rooming and machines are manipulated to fit the needs of the students and teachers, and not the other way round. The key element is an effective management strategy, with open communication to enable the laboratory administrator to schedule the lab efficiently.

ENCOURAGE AND SCHEDULE TEAM TEACHING

Teachers with little experience with computer algebra systems may feel hesitant or uncertain about how to deploy it effectively in the classroom. Scheduled team-teaching - sharing a class with a colleague with more experience - can allow students to benefit from using CAS in their lessons while at the same time imparting ideas to the colleague with less experience.

Team-teaching can have two versions. Double-staffing a class for all or part of the week is expensive. A case would have to be made to justify this cost as part of a planned transition period during which CAS novices are essentially mentored by their more experienced colleagues. If school/college facilities allow, two parallel classes could be timetabled together, say for computer lab sessions, which has no implications for additional staffing costs.

An alternative method of team-teaching is for two (or more) staff to divide between them the number of weekly mathematics hours for a particular class. This also forces the teachers to work together in planning and delivering the course. The CAS specialist then has a reason for encouraging and supporting the CAS novice in this aspect of their shared work. Students would also bring their CAS awareness and enthusiasm from the classes with the existing practitioner into the classes with the less experienced colleague, motivating the latter to adapt his/her teaching strategy.

In the UK, Watkins has noted that inexperienced teachers benefitted from short periods when external DERIVE consultants came into their classes and worked together in the lesson planning. Initially the consultant took the lead in planning and delivering DERIVE-based lessons, but by the end of the period the teacher felt more and more confident in putting together his/her own ideas, relying less and less on support from the consultant. Using a consultant in this way has cost implications, but external development funds may be available to support this.

TARGET RESOURCES, CASCADE EXPERIENCE

There is a school of thought that says any problem can be solved by throwing money at it. In terms of computer algebra, this would imply equipping each school with adequate hardware and software so that no teacher could claim to be unaware of what a CAS is. However, Nocker notes that the Austrian experience (where all secondary schools were indeed provided with resources for DERIVE) was not particularly efficient in terms of actual in-class use by the teaching staff. Not all schools valued an imposed gift.

In the English county of Suffolk, a government grant scheme were used to provide two selected secondary schools (students aged 11 to 18 years) with a site licence for DERIVE, notebook computers for the mathematics staff and compulsory initial training. In return for this infusion of resources, the staff at the two schools who had bid successfully for this funding were "contracted" to present a report to teachers at other local schools and cascade their experience and new-found enthusiasm for computer algebra.

This method of targeting resources based on competitive bids and with a commitment attached was intended to add weight and seriousness to the introduction of the new technology. The schools concerned did indeed make good use of the opportunity and became converted to the use of CAS in their mathematics classes.

Unfortunately, it was found that the other local schools declined to take part in the subsequent dissemination events. The presumed reason was that they had no particular interest in hearing about other peoples’ work in a project in which they had no direct involvement. Would they have reacted differently if they had been the ones targeted with the resources?

THE PURIA MODEL

The strategies outlined above for overcoming the logistical barriers which might discourage teachers from using a computer algebra system in the classroom are aimed at smoothing the way forward. The route towards a new style of mathematics teaching is evolutionary rather than revolutionary. The underlying principle is to give the teacher the freedom to acquire adequate experience and confidence.

The novel PURIA acronym is proposed here as a model for the way many teachers respond to the opportunities of computer algebra systems. When first introduced to a CAS such as DERIVE, they play around with it and try out its facilities. Once the initial novelty has worn off, they then realise they can use it meaningfully for their own work - for example, to generate questions and answers for test papers, to check solutions to problems or to produce screen dumps for overhead transparencies. In time, they find themselves recommending it to their students, albeit essentially as a checking tool and in a fairly piecemeal fashion at this stage. Only when they have actually observed students using the software to good effect do they feel confident in incorporating it more directly in their lessons, for example through DERIVE-based handouts or investigations. Finally they feel they should assess their students’ use of the CAS, at which point it becomes firmly established in the teaching and learning process.

PURIA reflects a fairly laissez-faire development of the teacher’s response which, unforced, may take some time to unfold. It should, we maintain, underlie any structured or phased approach to attracting teachers to become more involved with computer algebra systems, for example through in-service training or peer support and encouragement.

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We have outlined models for overcoming logistics obstacles to the employment of computer algebra in mathematics education. Models must be tailored to fit the needs and budget constraints of the particular situation, which depends on school level, county and national setting. By creating a CAS-enthused environment, teachers are encouraged to experiment with computer algebra.

Teacher training is essential, and a variety of models exist to make this valued by colleagues yet affordable for the institution. Providing hardware, software, classroom materials and appropriate learning environment is necessary for teachers reluctant to experiment in a less than ideal setting. Finally, a system of mentoring provides novice teachers with access to experts and an opportunity for feedback.

The primary variable which can facilitate or block these changes is time. We emphasise a phased-in approach to CAS use. From a practical viewpoint, current budgets may not suffice to implement all the enabling situations described here. We advocate that content focus and levels of computer algebra implementation keep pace with evolving attitudes of teachers. Certainly, we cannot wait for ever for attitudes to change, but with a mentor carefully nudging and mechanisms in place which facilitate CAS use, the teacher can develop an appreciation for computer algebra systems alont the lines of the PURIA model.