It is generally accepted that computer algebra systems (CAS) will have an increasing influence on the way mathematics is taught, learnt and assessed. Many articles and publications exist which deal with innovative ways of using CAS in mathematics education, and a number of studies have been carried out into the ways students react to and benefit from the incorporation of CAS in their classes. However, very little has been written about the role of the teacher in all this. There appears to be an implicit assumption in much of the literature that computer algebra is a "good thing", and that teachers cannot wait to start using it with their students. However, those involved with implementing policies for computer algebra at grass-roots level in schools and colleges often give anecdotal evidence of a reluctance of classroom teachers to take part, and a scepticism towards claims of the advantages of CAS in the learning process.

It goes without saying that the classroom teacher is the key to the successful introduction of new methods and new technologies. Of course, it is possible for the student to come across these independently - in the case of CAS, the recent development of a new generation of hand-held calculators means that a home computer is no longer a pre-requisite - but it is the endorsement of the teacher which imparts seriousness and respectability. With the increasing speed of technological development, it is crucial that teachers keep themselves informed so that they are in a position to make valid judgements and adapt their teaching accordingly. If they fail to do so, they are doing a disservice to their students and, ultimately, the next generation of mathematicians.

In an attempt to quantify current levels of involvement among teachers, in particular with reference to their experience of CAS and their attitude towards it, two surveys were recently carried out (independently) in England and in Austria. Although the methodology and content of these surveys differed, the results bear comparison, in particular with respect to the conclusions drawn about the continued need for staff development and training.

A summary of the main results of the surveys is given below.

This survey was carried out in the English counties of Essex, Norfolk and Suffolk during the spring of 1995. The aim was to gain an insight into how widespread the use of computer algebra is among mathematics teachers at GCE A/AS-level. For those who do use CAS, how do they use it and what do they think of it? For those who do not use CAS, what are the reasons for this? The intention was to highlight possible areas for future in-service teacher training, as well as providing a contribution towards establishing a wider picture of teachers' attitudes and experience with computer algebra.

It was decided to concentrate on GCE A-level and AS-level mathematics classes. These are pre-university courses in England for 16-19 year old students. The mathematical "core" content (which includes algebra, calculus, functions and their graphs) is such that CAS can provide a valuable tool for the investigation and solution of standard problem areas. The course syllabuses at A/AS-level are prescribed centrally, so that the teachers are working under similar constraints. Since A/AS-level does not (yet) specify CAS as an essential classroom tool, teachers are free to decide whether or not, and to what extent, to use it. It was hoped that that this scenario would provide a "barometer" for how teachers behave with regard to CAS when it is left to their own judgement. It should be pointed out that under current regulations computer algebra systems are not permitted in the final A/AS-level examinations.

A questionnaire was sent to all teachers in Essex, Norfolk and Suffolk who were teaching A/AS-level mathematics in the 1994/5 academic year. This was to provide an easily identifiable target, to facilitate the delivery of the questionnaires. A set of questionnaires was addressed to the head of mathematics of each school or college where A/AS-levels were taught, with the request to issue one questionnaire to each teacher with a current A/AS-level mathematics group and discard the remainder. A further advantage of only approaching teachers actually teaching A/AS-level mathematics that year was that it would target those for whom computer algebra is a relevant issue to address. This would not necessarily have been the case for those who "usually" taught A/AS-level mathematics, but who had not done so for the past couple of years, such has been the pace of recent developments.

After initially establishing a few personal and professional details from the respondent, the questionnaire divided into four parts, according to how the respondent identified his/her experience of CAS from the four options:

Follow-up questions to establish further detail of experience and attitudes were then asked for each of these four subgroups.

The questions were generally multiple-choice in nature, with the respondent choosing the most appropriate response from a list. The response options provided were based on the experience of the originator in running in-service training courses in DERIVE for local teachers for a number of years. Space was also provided on the questionnaire for additional responses where appropriate. Questionnaires were returned anonymously.

A total of 129 completed questionnaires were returned. It is difficult to express the response rate in percentage terms, since smaller schools may only have one member of staff teaching A/AS-level, whereas larger schools and colleges would have several. Sets of questionnaires were sent to all institutions in the three counties where A/AS-level mathematics was taught, almost 140 institutions in total.

Of the teachers who returned the questionnaire, 46% (59/129) worked at LEA (state-sector) secondary schools with sixth forms, 36% (46/129) at independent or grant-maintained schools with sixth forms, and 18% (24/129) at tertiary colleges or colleges of further and higher education.

The response to the key question of the teachers’ experience of CAS was as follows:

Thus almost half of all teachers in the survey had no first-hand experience of computer algebra systems. Indeed, several noted on the questionnaire that this survey was the first they had ever heard about CAS! Two-thirds of the teachers worked at institutions which did not have CAS available for staff or students, even though all the institutions surveyed delivered mathematics at A/AS-level or beyond. Of those teachers who taught A/AS-level mathematics at an institution which had CAS available for staff and/or students, the majority did not actually use it in their teaching. Overall, fewer than one in seven teachers in the survey used computer algebra systems on occasion in their teaching.

Of the 63 respondents who had only heard about CAS (option (a) above), DERIVE was clearly the most well known software (75%). The most common source of awareness was magazine/journal articles, ahead of conversations with colleagues, LEA advisers and attendance at conferences. When asked to select a statement which most closely reflected their attitude towards CAS based on what they already knew, the vast majority (81%) indicated "I don’t really know enough about them to comment". In a similar vein, "a general lack of awareness/interest among staff" was the main reason (56%) teachers in this category gave for their institution not having a CAS available for staff or student use. Overall there was no significant difference in the proportion of teachers without first-hand experience of CAS in each of the three types of institution surveyed. It was also not significantly associated with age group. However, almost all (8/10) of the part-time teachers who responded to the questionnaire belonged in this category.

Among the 22 respondents who had tried out some computer algebra systems but whose institution had not (yet) acquired one for staff or student use (option (b) above), DERIVE was easily the most familiar (82%). The majority (15/22) had gained experience of CAS on an organised in-service training course, but this involvement does not seem to have been taken further, since only 4 of the 22 had their own copy of the software at home to use further. Based on what they had already experienced of computer algebra, 50% (11/22) selected the attitude statement "They offer an opportunity to enrich the experience of students". Only 14% (3/22) felt that they would "lead inevitably to a drop in standards", and 9% (2/22) considered them "not relevant to mathematics teaching at A/AS-level". The remaining 27% (6/22) felt that they still "did not really know enough about them to comment". "Inadequate resources" was the main reason cited by 77% (17/22) of teachers in this category for their institution not having a CAS. Only 1 of the 22 teachers was aware of plans at their institution to purchase a CAS in the near future.

There were 27 questionnaires returned from teachers whose institution had a computer algebra system available, but who did not use it in their A/AS-level teaching (option (c) above). In 74% (20/27) of these cases, DERIVE was the software available. The general picture was that teachers in this category were not necessarily ideologically opposed to CAS - only 22% (6/27) chose "negative" attitude statements concerning falling standards or irrelevance, whereas 45% (12/27) believed CAS could "enrich the experience of students", and 33% (9/27) admitted that they "didn’t know enough to comment". When asked to select the main reason from a list of five why they did not deploy CAS in their A/AS-level teaching even though it was available, 44% (12/27) said that they knew too little about it and lacked confidence, 22% (6/27) claimed there was not enough time to prepare new lessons and work schemes, 19% (5/27) said there were not enough computers or it was too difficult to book them, 11% (3/27) did not consider CAS to offer anything of particular value to A/AS-level mathematics students, and 4% (1/27) cited falling standards for being opposed to letting students do algebra on computer. Finally, in response to a question on whether teachers in this category used graphics calculators with their A/AS-level students, 96% (26/27) used them at least occasionally, and 59% (16/27) used them "often and regularly".

The fourth category of respondents, namely those whose institutions had computer algebra software available and who used it in their A/AS-level teaching (option (d) above), comprised 17 teachers. For 14 of these 17 teachers (82%), DERIVE was the software used. When asked how they deployed the CAS, the response was as follows (more than one choice allowed):

14 of the 17 teachers (82%) who used CAS with their students advocated introducing its facilities immediately, with the rest preferring to wait until after a consolidation of basic numerical and algebraic skills by hand. When asked to select the main reason for incorporating CAS in their teaching, the response was:

The 17 teachers who had experience of using CAS with their students were asked to assess the validity of a range of arguments commonly encountered against introducing CAS into A/AS-level mathematics classes. In their view:

In addition to questions about their own attitude towards computer algebra, the teachers who had CAS at their institution and who used it in their teaching were also asked about how their students reacted to it. 15 of the 17 teachers (88%) in this category worked at institutions where the CAS was available for students in private study as well as class time, but only 5 of these 15 teachers (33%) claimed that the majority of their students actually used it in private study time. Overall, most teachers (10/17) considered that their students found CAS "interesting and useful"; none reported that the majority of their students disliked using CAS; the rest were "indifferent - it is just another thing they were expected to do". In most cases (11/17) no correlation between students’ ability and enthusiasm for computer algebra was identified; some (5/17) thought that the "better" students were more enthusiastic; none thought that the "better" students were less enthusiastic; an one respondent noted that the most confident students were those who also had a PC at home. With regard to gender, 5 respondents considered that male students were more enthusiastic users of CAS, 1 considered female students more enthusiastic, but the majority (11/17) reported no observed association between gender and attitude among students.

One of the most worrying aspects to emerge from this survey is that almost half of teachers currently delivering A/AS-level mathematics courses have no first-hand experience of computer algebra systems, and that two-thirds of such teachers work in institutions which do not have CAS available for staff or students to use. Moreover, if one assumes that those teachers who do use CAS would be keen to share their views and complete the questionnaire whereas those who know little about CAS would be more inclined to ignore the questionnaire, the true figures may well be higher. The imminent arrival of pocket computers running a version of DERIVE, and the fact that at least one A/AS-level examining board will soon be piloting an examination paper that allows the use of CAS, means that the students of teachers without experience of CAS may be disadvantaged. There is clearly scope for further in-service training and awareness raising across all types of educational institutions which prepare students for A/AS-level mathematics.

In those institutions which did not have CAS, the reasons were perceived by the teachers to be fairly evenly split between "inadequate resources" (47%) and "general lack of awareness/interest" (44%). Only one respondent claimed that it was "departmental policy" not to use CAS at A/AS-level. This would appear to reflect a teaching profession that is hard-pressed in terms of time and money with respect to technological advances rather than ideological opposition.

For those institutions which did have CAS available to staff and/or students, the fact that the majority (61%) of teachers who responded to the questionnaire did not actually use it in their teaching also highlights scope for further professional updating and support. This is reinforced by the fact that 44% of these non-users claimed that it was due to not knowing enough/not feeling confident enough about the potential of computer algebra. The next highest reason given, namely inadequate time to prepare new lessons and work schemes to incorporate CAS (22%) reinforces the need for continued publication of resource material for such teachers to use. Reluctance to use CAS could not be attributed to technophobia, since all but one respondent in this category used graphics calculators with students at least occasionally. Overall, teachers in tertiary/FHE colleges were significantly less likely to use the CAS available at their institution for A/AS-level than school teachers, possibly because the colleges may have purchased the CAS primarily for use in other courses.

Teachers at LEA maintained schools with sixth forms were significantly more likely to use CAS in their A/AS-level teaching than those at independent schools or tertiary/FHE colleges. This may be due to the fact that the work at LEA schools is overviewed by a county advisory team, which has the responsibility of facilitating the introduction of new curricular developments. However, "LEA advisers" were cited as a source of information on the existence of CAS by only 36% of those teachers at LEA maintained schools who did not yet have first-hand experience of computer algebra.

Only 17 out of 129 (less than 1 in 7) completed questionnaires came from teachers who actually used CAS with their A/AS-level mathematics classes. The overall response from these teachers (as detailed in the previous section) was positive. It is worth highlighting that the vast majority considered it most appropriate to introduce students to the computer algebra system straight away. The only one of the commonly articulated arguments against using CAS that was considered valid by the majority of the teachers who actually used it was the difficulty of finding/producing good teaching materials. It is speculated here that this might be one reason for most of these teachers finding that the majority of their students do not routinely use the CAS in their private study time.

The key points to emerge from this survey, then, are:

(i) there is still a disturbing lack of awareness and experience of CAS among mathematics teachers at A/AS-level;

(ii) the perceived lack of suitable support material is felt a hindrance by teachers who both do and do not use CAS in their classes;

(iii) DERIVE is by far the most common CAS in England for A/AS-level mathematics.

The Austrian project "Modern Mathematics Engineering using Software-Assisted Approaches" deals with the use of the latest technologies in teaching mathematics in secondary technical colleges, with students aged 15 to 19 years. This project is concerned not just with computer algebra systems (CAS), but also spreadsheets and programmable graphics calculators. Among the aims of the project are the investigation of the actual take-up of the new media, the analysis of the reactions of teachers and students, and the generalisation of experiences at the secondary technical colleges to the delivery of mathematics at other school levels. In early summer 1994 a questionnaire was used to gain feedback on these points from teachers.

A questionnaire was posted to all 350 subscribers of the AMMU publications. These publications, consisting of discussion papers and supplementary teaching materials, are produced by the Austrian Working Group on Modern Instruction in Mathematics, and are made freely available to all who join the mailing list. Since subscription to AMMU is purely voluntary, it was assumed that the teachers surveyed would be particularly interested in new technology. The results of the questionnaire would thus provide a representative view of those currently active in this field. Approximately three quarters of the recipients of the questionnaire worked at secondary technical colleges, the rest at a variety of other institutions.

The questionnaire consisted mainly of a set of 53 statements which had to be rated on a five-point Likert scale (for example, strongly agree - generally agree - undecided - generally disagree - strongly disagree). There were also questions to establish some personal details of the respondent, and some open-ended questions. The questionnaires were returned anonymously.

A total of 123 completed questionnaires were returned, a response rate of 35%. 98 of the 123 respondents (80%) were from secondary technical colleges, the type of school of particular interest to the project. It should be re-iterated that receiving and subsequently completing and returning the questionnaire was taken to be an indicator of interest and involvement with new technology in mathematics teaching, and that the responses are assumed representative of this subgroup of teachers.

The majority of the respondents (56%) came from teachers in the age range 36 to 45 years. It was not surprising that the age group above 45 years was relatively small. The approaching end of the professional career as well as a "classical" education (without a pocket calculator) often go hand in hand with conservative views in the field of mathematics, making a low readiness for innovations seem natural. However, it came as a complete surprise that the lowest percentage of respondents (19%) came from the age group below 36 years, since one might expect that it is usually young people who enthusiastically press forward with innovations. Possible reasons for this perceived anomaly include:

· the absolute percentage of practising teachers in the 36-45 age range is greater than that for below 36. However, it is not as extreme as a ratio of 3:1.

· at the beginning of a young teacher’s career, he/she is fully occupied with learning the fundamentals of the job, and with personal and family matters, leaving little time for innovative development work.

However, neither of these arguments is fully satisfactory. Follow-up interviews with young graduates have shown that teacher training courses for mathematics at Austrian universities generally do not use the new media didactically. Such involvement is essentially voluntary, and computer algebra in particular is scarcely mentioned. This deficiency must be eliminated as a matter of urgency.

The questionnaire asked teachers to appraise their attitude to using new technology in the classroom, both retrospectively before they had started to use it, and currently given their present experience. The responses were on a similar five-point scale ranging form "disapproving" to "convinced":

The positive change in attitudes is striking. Only 12 respondents revised their attitude negatively, whereas 52 were more convinced now than before. Teachers were clearly more convinced of the value of electronic calculation tools after actually using them in class with students.

An interesting response was received when the teachers were asked to comment on their perception of the motivation of students when new technology was used in the classroom. They were asked to state their level of agreement with the statement "When using modern calculation tools, students are motivated to put more effort into mathematical problems". In a parallel questionnaire (not reported fully here), the students themselves were asked the same question. The results were:

There is an obvious discrepancy here. One may assume that reality rather lies on the side of the students’ statements. As motivation is not objectively measurable, we may assume that a high commitment and a certain amount of expectation blur the teachers’ point of view. The potential danger in relying on the reports of "enthusiasts" when evaluating educational developments is hereby noted in passing.

The teachers were asked to assess the importance of various calculating tools for the teaching of mathematics:

As expected, the simple scientific calculator is generally regarded as essential. It is noteworthy that fewer than 15% of teachers currently see the calculating tools mentioned here as overall unimportant, including the more sophisticated ones. In particular, computer algebra systems have almost caught up with programmable graphics calculators as being perceived as an important tool. This is a clear statement on the part of involved teachers of the methodological and didactic opportunities provided by CAS.

The questionnaire asked teachers for the advantages and disadvantages of using the new calculating tools in their classes. More than one choice was allowed. The most frequently cited advantages were:

The most frequently cited disadvantages were:

It is remarkable that the feature of "mathematical understanding" ranks very high, both as an advantage and a disadvantage! Attention is drawn without comment to the relatively low ranking given to the time saved by students and the possibility of experimental mathematics. Loss of calculation skill was, as expected, high on the list of perceived disadvantages, but at the same time only mentioned by 1 in 5 respondents.

Finally, the implications of the new technologies for the teacher outside the classroom was considered. The majority (59%) tended to agree that assessment had become more difficult since the latest calculating tools were introduced. Besides the organisational problems, for example how the technology can be made available in examinations, difficulties are also perceived in terms of content. There is a shift from rote skills, which the calculating tools can perform, towards concepts and understanding. This in turn means a new style of examination papers and, most of all, new criteria for marking.