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Mathematics Education Research Journal

1996, Vol. 8, No.1, 38-57

Using Computers in the Mathematics Classroom: The Role of the Teacher Michael Thomas & Jackie Tyrrell

John Bullock

The University ofAuckland

Massey High School

This paper examines the issues involved in assisting teachers in their implementation of computers in the mathematics classroom. It documents them in the light of a year-long New Zealand study involving teachers using co!TIputers in their mathematics teaching. Our results suggest that putting a computer in the mathematics classroom is unlikely to result in changes in learning or teaching unless the personal philosophy of classroom practice held by each teacher undergoes a major transformation. Further, the teachers' transition to a positive view of computer use needs to develop from their experiences and this requires time. The paper describes some of the factors influencing this change, including different classroom management styles and a new view of the role of the computer, and explains these as the product of a change of mindset on the part of the teachers.

Introduction When computers first became available to teachers in schools there was much optimism about the benefits which were likely to accrue from their use in mathematics teaching. Later, a UK Mathematical Association review (Ball, Higgo, Oldknow, Straker & Wood, 1987), which looked at the potential of the computer, stated: Styles of teaching which involve the use of microcomputers can aid greatly the acquisition of '" important high-level skills which are exactly those which will be required in the future. (p. 8)

However the reality has often not been the same as that perceived, or hoped for, by many educationists. As Freudenthal (1981) predicted, the transition to the computer paradigm has proved to be one of the major problems of mathematics education throughout the eighties and into the nineties. The Cockcroft report (1982) claimed that, for the computer era, one of the fundamental concerns of mathematics educators needs to be "the w~ys in which calculators and microcomputers can be used to assist and improve the teaching of mathematics in the classroom" (p. 374). In spite of the widely recognised potential of the computer in mathematics teaching, the UK Mathematical Association review (Ball, Higgo, Oldknow, Straker & Wood, 1987) noted that, at that time in the UK, only a minority of both primary and secondary teachers of mathematics are using computers to support these styles of learning. A recent survey suggests that nearly three-quarters of secondary mathematics teachers use computers either very rarely or never in their work ... the position in primary schools is little if any better. (p. 8)

There is a clear gap between rhetoric and reality. . Why then is it that so many mathematics teachers do not make use of

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technology which is widely acknowledged to have great potential for mathematics learning? One reason for this apparent discrepancy may be the failure of researchers to involve teachers in the ownership of research, and to work in conjunction with them to develop good practice. Hoyles (1992) indicated that research has only recently acknowledged the complexity of the role of the teacher in the implementation of technology, and the crucial role of teacher beliefs and assumptions. Reflecting on her work with teachers in the Microworlds project, Hoyles described how the computer was: a window on our teachers' beliefs and a mirror on our own. 'We set out with a strong commitment to conduct a course focused on mathematics education and not. on technology but it was not until the project was up-der way that what we meant by mathematics education 'became dearer to us and our students-particularly with respect to the importance we accorded to group work, social interaction, problem solving and learner contro1. (p. 39)

Understanding this complexity must occur before technology can be used effectively in the mathematics classroom.

Aim The year-long research· study described in this paper looked at the issues arising from the introduction of technology in mathematics from the teachers' perspective. Initially our aim was to describe what the teachers considered successful implementations from their perspective. It soon became clear, however, that due to the difficulties they experienced, our aim needed to encompass an analysis of teachers' attitudes and' aspirations and how these were related to the manner and extent of their use of the computer in the mathematics classroom.

Background Studies One of the most common attitudes described in the context of initiating teachers' classroom use of computers in mathematics is the teachers' apprehension at using the computer. For example, Smith (1984) described the real anxiety which many novice teacher-users face when confronted with using technology in their lessons for the first time. Looking at the problems facing such teachers, Smith commented on the important role of support and the way in which others interact with such teachers: Too many pseudo-casual, off-hand, rapidly executed demonstrations ... might prove something to the demonstrator/ counsellor whose help has been sought, but they do absolutely nothing for the novice ... except, perhaps, reinforce beliefs about personal inadequacy. (p. 29)

More recently Dunn and Ridgway (1991, 1994) found among pre-service primary teachers that lack of confidence was a major reason for not using computers in class, with 40% of one sample saying they were unsure or anxious' about using computers. Several studies which have looked at teachers using computers in the mathematics classroom have provided some insight into the criteria for successful

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implementation. One such study was the Queensland Sunrise Centre Project in Australia which began in 1990 and was a major initiative charting the direction in which technology use in education could proceed. Each of the students involved in the project had a personal computer for use at school and at home, and each of the teachers was given the personal use of a powerful laptop computer. Finger and Grimmett (1993, p. 86) discussed 12 major issues which need to be addressed when managing and supporting technology initiatives in schools. They noted that it is vital to recognise the importance of people, and to build ownership of the rationale of the initiative with key participants. They also pointed out the importance of appreciating that technology has implications for classroom organisation and management, and stressed the need for teachers to have both training and professional development in the use of the new technology. Logan and Sachs (1987) found that, in teacher development programs which produced successful learning, participants believed in the importance of the task on which they were working. Finger and Grimmett (1993) listed strategies which they believed were likely to assist teachers in the successful implementation of technology in schools. These strategies included the involvement of teachers in all aspects of the project, including the'negotiation, development and writing of policy documents, and the holding of regular meetings of all personnel with responsibility for the innovation. Finger and Grimmett also noted that it is imperative that teachers be provided with the opportunity to vary their

professional behaviour in such a way that they can facilitate the adoption of changes in educational practice afforded through the use of technology for learning. (p. 88)

Acceptance of the ownership of the project by those involved was also identified by Treagust and Rennie (1993) as one of the factors necessary for success. The Primary Laptop Project at Warwick University in the UK (Ainley & Pratt, 1995), investigated the effect of high levels of personal access to computers in mathematics on Year 7 (age 12) and Year 5 (age 10) students. Commenting on possible reasons for the project's success they observed that: in order to achieve these results, much more than access to personal technology was needed. It seemed to us that the support given to teachers at the planning stage helped them in a fundamental way to appreciate how activities could be set up which would have the potential for this learning to take place ... we have also seen dramatic improvements in the teachers' confidence in both these areas [how to intervene and the potential for technology to facilitate learningJ, and feel that their sharing of expertise and reflections in their journals have contributed significantly to this. (p . 447)

The importance of the support provided for teachers cannot be overestimated. The Melbourne Technology Enriched Algebra Project reported by Asp, Dowsey and Stacey (1992), which involved 14 and 15 year old students in a typical Melbourne secondary school using technology in their algebra learning, noted that the teachers were able to handle the lessons well, and encountered few of the chaotic moments which many teachers had anticipated. The two aspects described above, namely the need for teacher support in order to raise confidence levels and the significance of the participating teachers'

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attitudes to the project, appear as a common thread in much research. Teachers particularly need to feel involved in the ownership of the research and have a positive attitude towards it if it is to be successfuL In the New Zealand project reported here, these were major initiatives right from the start. We hoped to build up the teachers' beliefs in the importance and value of the research, and to provide encouragement through regular joint meetings, discussions and classroom visits.

Methodology The research began in May 1993 and formed part of. the Technology in Mathematics Education (TIME) project funded by the New Zealand Ministry of Education. The project involved twelve teachers from six schools and three research support staff. All the teachers volunteered to take part in the project. The teachers were. to engage in cycles of action research in their classrooms in areas which they chose but involving the computer, with guidance from the support staff. The research design involved the support staff in dual roles-supporting the teachers' classroom-based research and investigating the factors influencing the success of the teachers' work. All of the teachers agreed to use the computer in their daily mathematics teaching. They were asked to keep daily logs of their work in the project, recording what they did and their thoughts and feelings with respect to its success or otherwise. The cluster support team arranged regular meetings and workshops to give encouragement and to share ideas. At some meetings the teachers were interviewed (both individually and in groups) about their work, and they also completed questionnaires. In addition, support team members visited the teachers' classrooms (often weekly) to support and encourage, and were often actively involved in using the computers with groups of students. The research took place in the second and third school terms of 1993 and the first term of 1994. Each of the six schools was provided with two computers which remained in the schools at the end of the project. The principal software supplied was a high-quality spreadsheet, and was chosen with the aim of encouraging teachers (and thus students) to use the computer as a multi-purpose tooL The teachers selected one of their normal teaching classes to take part in the project. The aim was to have the computer available for most of the lessons with this class, and the teachers agreed to document these lessons. The classes which the teachers decided to involve are shown in Table I, where students in J2 (Junior 2) are approximately 6 years of age and students in Fl (Form 1) are 12 years old. Table 1

The Classes Chosen by the Teachers in Each of the Years Teacher

1

2

3

4

5

6

7

8

9

10

11

12

1993 1994

F3 F4,

F3 F3

F2 F2

F2 F2

F4 F4

F3 F3

J2/3 J2/3

F1/2

F3 F3

F4 F3

F1/2 F2

Fl/2

Fl/2

NIL

For the last term of the project, most teachers had the choice of continuing with the same class of students or beginning with a fresh group. However, none chose to

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continue with the same group of students. This appeared to be a reflection of the learning process that the teachers felt they had been through. On the whole, they wanted to begin afresh and use their growing confidence with students who had not yet been involved in the project. Table 1 shows that there is a spread of classes covering every form from J2 to F4 (age 15 years), with some senior students peripherally involved. All ability groups were covered: many of the classes were mixed ability, but there were also streamed groups, described by their teachers as low ability, below average, average,and top ability classes. The schools included students from a wide range of backgrounds, including 54% European, 21% Maori, 16% Polynesian and 8% Asian. The ratio of female students to male students was 49%:51%, and the socio-economic level varied from severely low to upper-middle as described by the New Zealand Ministry of Education Elley scale. Hence the study involved students with a wide range of ages and abilities, and from a wide range of socio-economic, ethnic and cultural backgrounds, and is novel in this respect. The teachers in the project comprised six males and six females and although all were experienced teachers, their length of service varied from 5 to 35 years.

The Arrival of the Computer-A Familiarisation Stage

Teacher Backgrounds Once the teachers had taken delivery of the computers and the software, they entered a stage of familiarising themselves with the new technology. It is important to note that most of these teachers have many years' experience with computers, and some had used them in their classrooms. However~ none of the teachers had used computers extensively in their mathematics teaching. The number of years of computing experience for the teachers averaged 9.5 years. This usually included some use in higher education and training, leading to some experience with wordprocessing or databases. Full details of the teachers' stories and the data used for analysis in the study are available in the final report (Bullock, Barton, Buzeika, Ellis, Regan, Thomas & Tyrrell, 1994). The following excerpts from their writing and from interview transcripts give a typical picture of this experience. I first became involved in computers across the curriculum in primary schools about 10 years ago ... my skills are basic: one [University] paper emphasising Logo, and what has been needed for initiating and continuing programmes for computers in schools. [TI2] I first used a computer ... in 1982 ... At my previous school I used a markbook program and a report program, and as Junior Dean I was involved in inputting and accessing data from the school student database ... my only experience of using computers with students was taking two classes into the computer room to use MousePlotter .. , I bought a laptop for word processing, and use Word Perfect quite a lot now. When I took the IBM home in the holidays, I started with the mouse tutorial -that's how much I needed to know. [T9]

In addition, five of the teachers had some experience of programming computers, with at least one having extensive experience in this area:

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In my first year I taught F6 computing. I used computers extensively when studying for my Bachelor of Engineering (Engineering Science). As well as using a wide variety of applications (forecasting, stochastic methods of operations research, etc.) I programmed in Pascal, Fortran, Simscript and Basic, and studied aspects of electronics and hardware. I used CADCAM with the DSIR on a project to do with the design of nevy technology. Since starting teaching, I've been on a couple of "Technology in the Classroom" courses. [TI]

These details from their backgrounds allows us to see clearly that the teachers were not novice computer users. However, surprisingly, in spite of their wide range of computer experiences, eight of the twelve teachers in the project had not previously used a computer in their mathematics teaching and only two had ever used a spreadsheet iri. their classroom teaching. 'In view of the aim of the study, this seems to be a significant point. Computer experience alone is not necessarily sufficient to enable teachers to have the confidence and ability to use the computer to nurture student learning of mathematics.

Initial Difficulties During the first two terms of the project, the teachers' main concern was how to adapt what happened in their classrooms to take account of the technology. Initially, they looked for ways of fitting computer use into existing classroom procedures, and were generally appalled at the degree of organisation, the preparation time, and the seemingly impossible amount of attention demanded by the computer users. The upheaval resulting from presence of the computer had several components: • • • • •

Finding a place (in more than one sense) in the classroom for the technologYi Coping with the hardware and operating systemi Considering the need to extend one's classroom management techniquesi Developing a feel for the potential of open-ended softwarei and . Recognising some of the implications of technology for school mathematics.

The ways in which the teachers experienced and dealt with this initial upheaval are described in the following discussion.

Finding a Place for the Computer For most of the project teachers the sudden physical presence of a computer in their classroom represented a major challenge to established notions of teaching practice and classroom organisation. The physical introduction of one or two computers into an already crowded classroom is unsettling. Although the teachers put a great deal of thought and effort into rearranging their rooms, there were no simple solutions. It began to seem inevitable that changes made to accommodate the computer or computers would in fact result in changes to the whole fabric of teaching and learning which could take place in that room. For example, when four desks were removed from circulation each time the computers were in the room, one teacher had to rethink

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his usual strategy of starting the lesson with a written quiz. Instead, the class went straight into group work. Towards the end of1993, several teachers echoed this teacher's strategies. For example, another teacher commented: The computers take up quite a bit of room and joining desks into groups helps to accommodate them. [I am] using group work more often, and definitely using groups as a classroom management strategy. [TlO]

Thus the physical presence alone of technology in the classroom forced teachers to rethink their approaches to the structure of their lessons.

Sharing Computers The project provided two computers to each school, with two teachers participating. Each of the pre-secondary teachers kept their computer in their room (often with another which was already there), the secondary schools moved the pair of computers between the teachers' rooms, because they felt that one computer was not enough to be worthwhile. Developing routines for shared use took considerable planning, and the whole notion of sharing computers continued to be a situation which most teachers would have avoided if possible. The computers could not be based in my teaching room, which was used for detention. This meant ... I had to plan ahead for transporting the computers to and from my room. This sometimes created a problem at the beginning of a period. [TIO] Having to move [the computers] from room to room is a hassle. It is ideal to have one or two computers in the room all the time. [T2] .

Rooms were often widely separated, wheels sometimes fell off trolleys and special equipment such as carry trays were expensive. .

Managing a Mathematics Classroom with Computers Once a teacher had dealt with the logistics of getting the computers into the mathematics classroom, another issue demanded urgent resolution: how to manage the classroom activities once the computer was present. When asked whether classroom management had been an issue, an experienced. teacher commented "It's stupid not to make it an issue. The whole year will fall apart if you don't keep your discipline" [T6]. . Again, there were no quick and easy solutions. As with finding a home for the computer, it began to seem inevitable that developing successful management techniques for the new classroom situation involved far-teaching changes~ and the teachers found this a difficult and uncomfortable process. In the first term of the project, one researcher described the "anarchic" effect of the arrival of the computer. Finger and Grimmett (1993), describing the Queensland Sunrise Centre Project, concluded that: .

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Computers are inherently subversive ... it was not uncommon for the teacher to have a technological problem or process explained to her/him by an eleven or twelve year old student ... As students interact with a personal intellectual technological tool, they repeatedly set their own tasks, develop curricula ... to s1!ccessfully complete their task, evaluate their progress towards their goal and restructure the~r work program. This process is at the heart of the subversiveness of technology. (p. 91)

Even when students were not working as purposefully as implied in this statement, the teachers no longer had the feeling of being in total control of the learning which was taking place.

A Crisis of Confidence . At the same time as they were struggling to fit the computer into their teaching, most of the teachers were also tackling their own lack of confidence with computers and they were reluctant to use the computer (in or out of class) without support. Some were als0 aware that the depth and breadth of their own mathematical understanding was not adequate to cope with some situations arising from the spreadsheet. All questioned their ability to produce worthwhile or relevant materiaL Not uncommon feelings were: I was thinking what the hell am I doing. [T6] ... amazed at lackof self-confidence. [T2]

Several of the teachers who had the spreadsheet as their only powerful software had one main question in the first two terms: When are we going to get some more software? [T2, TlO, T12]

This may reflect the difficulty of dealing with open-ended software-of not being able to use the computer in its more familiar role of reward or baby-sitter. One reason for the teachers' general lack of confidence may be that they often did not see their previous computer involvement as relevant to the mathematics teaching in the project.

What Has Happened to the Mathematics? Hoyles (1992) observed that "Innovation ... because it inevitably perturbs the dynamics of a classroom, makes more apparent the mathematical beliefs and understandings of teachers and students" (p. 37). Our experience was that the teachers were very much aware that their primary role was to teach mathematics, and their dominant apprehension appeared to be an educational one-the prospect of using the computer to teach mathematics was even more worrying than the actual use of the computer. The starting point for most of the teachers was to try to reproduce electronically the kind of work they were doing already. However, from this viewpoint, the computer appeared unsuitable for many topics. As one teacher [TID] commented, "[My main worry was] how was I going to teach certain concepts on [the computer."

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At cluster group meetings, the teachers enjoyed the experience of working in groups using the spreadsheet on rich, open-ended problems. However, few immediately perceived these experiences as appropriate for classroom use. Three features of these problems challenged their assumptions about mathematics learning. They found that mathematical thinking was required to set up the spreadsheet; repetitive calculations could be done electronically; and the results of calculations required interpretation-how can I use these numbers to answer the original question? Some teachers, keen to implement the recently released national curriculum, boldly tried these problems in the classroom, and were then prepared to use similar problems if these were generated by the support team. Other'teachers felt that problems of the type described above were inappropriate for their classes. It soon became clear that the period of familiarisation would be a long process.

A Sense of Disorientation Initial problems such as these were very disorientating for some of the teachers. This sense of dislocation which teachers feel was recognised by Kaput (1992): Computers are too difficult for the average teacher to use in the typical classroom ... The high price of entry in terms of effort, especially in the context of the historic curricular pressure towards easily measured computational skill, is likely to continue. (p. 517)

It is becoming increasingly clear (Thompson, 1985; Fraser et al., 1988; Ainley & Pratt, 1995; Finger & Grimmett, 1993) that successful implementation of computers

involves transforming the individual teacher's classroom in ways which enable the technology to be integrated in learning and teaching rather than existing as an addon. Ball, Higgo, Oldknow, Straker & Wood (1987) commented that there is " no substitute for having computers available for use within the classrooms in which mathematics is taught" (p. 48). Although the teachers found it difficult, initially, to integrate the computer into their classroom procedures, its constant availability proved a key factor in their eventual progress to acceptance and comfortable use.

A Change of Viewpoint During the first two terms of the project, the teachers struggled to adapt what happened in their classrooms totake account of the co.mputer. Their initial strategy of trying to fit computer use into existing classroom procedures led to frustration. This was a period of uncomfortable uncertainty and intense questioning. However, early in the third term of the project, some teachers experienced a dramatic change in their thinking and their approach to the use of the computer. It seemed to be sudden, as if a switch had been thrown. [Earlier in the project] I was thinking, "What the hell am I doing?" It was like going through a jungle. But all of a sudden, something clicked. I was able to get above everything and look down and get an overview. I could go on just about forever now, I think. [T6]

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I was struck by my sudden shift from totally hopeless to completely convinced through the course of one good activity-one day. [T9] I suddenly realised, about 40 minutes ago, that it's not that "I've got a computer, what maths shall I do," but that "I've got some maths to do, how can the computer be used to do it?" [T5]

We identified five characteristics of this changed perspective: • a changed perception of the classroom; • a change in the role of the teacher to guide and co-learner; • an appreciation of students' new freedom to explore the mathematics as well . as learn t e c h n i q u e s ; . • the development oia shift of mathematical focus; the ability to "see" ways in which the computer could be used to enhance mathematics learning; and • a new and broader perspective on the context in which teaching takes place. These factors appeared to be inter-related and perhaps symbiotic.

A Changed Perception of the Classroom Finding new ways of managing the classroom was often the first major breakthrough; teachers who changed their viewpoint exhibited a changed classroom perception. Their classroom was more democratic, and more open to different learning styles and discussion. In a strange way the impact is of a more democratic· classroom, certainly more chaotic and a little bit scary at times. My students are enthused by it (back at lunchtime etc.). [T9]

There was considerable variety in the management strategies adopted. For example, one teacher found that having the computers in an adjoining area had some positive effects. Initially I thought it would be better to have the machines in the classroom. But the room adjoining mine had been the computer room, and because it had alarms, that Was where the computers were kept. The room had an alcove with a flexi-fold door, so even when there was another class in there I found it was all right to send four students off at a time ... This [has resulted in] a cooperation that was not in evidence in the whole-class situation. [T6]

The teachers found that group work and discussion fitted quite naturally with computer use. [I'm] not a group person myself, but putting students in front of a screen, [it] seems to be natural for them to discuss things. [TIO] [I now] allow students more time [to] experiment and test different ideas '" to discuss and share their ideas and findings. [IS]

The Teacher as Guide and Co-Learner A second characteristic of the changing view on the part of the teachers was the idea that they too could, and indeed needed to, become learners in the classroom.

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They no longer found it possible or desirable to be the source of all knowledge for their students. Many of them discovered, to their surprise, that there was no sudden loss of respect or teacher control when they embraced the idea of learning alongside their students. The level of noise increased, but it was "interested" noise. The students were more on task than when they were given "busy" work; the change appeared to have positive effects on student learning. It's new for me to be "discovering" at the same pace as the kids.... The kids like to "do it" for themselves and not necessarily be shown. [T9] I have found that the student discussions are valuable with the benefit for me being that I have learned a number of things about [the spreadsheet]. There has been a definite increase in the level of mathematics communication between students. [TIO] The kids are as good as I am at using the computer. I avoid telling them what to do. .. , The kids take over in a very nice sort of way, and show one another. [T6] The kids help each other and free the teacher. [T2]

The pressure on the teacher to perform was reduced. It appeared that part of becoming at ease with the computer in the mathematics classroom was getting used to saying, "I don't know, but I'll find out," or perhaps "Let's see if we can find out." But then I say "I don't know" to my kids all the time. [T8]

It seemed inevitable that when computers were being used as an investigative tool

or as an enhancement to the method of solution, the role of the teacher changed from provider to questioner and guide. The teacher needs to become a guide to the source of knowledge rather than the source of knowledge itself. [TS]

The students became happier with the teach~r leading a guided discovery mode of learning.

An Appreciation of Students' Freedom to Explore The computer, due to its processing power, enables students to be exposed to a far greater range of problems and solutions than is possible with pen, paper and even calculator. This has implications for the kinds of learning that take place. Students could now explore possibl~ solutions without vast expenditure of effort in calculating or graphing data. Thus in a single lesson they tried a variety of approaches to solving a problem. Unsatisfactory' results provided learning experiences, rather than being discarded as "wrong answers." Also students could quickly generate results for a large number of examples and then use this information to make decisions or generalisations. [The time saved] is better spent on drawing conclusions. [TIl] Examples of the computers speeding things up abound: F3E graphing in Operation Death Mission; F3W sorting data in Cemetery Maths; problem solving in F6 Alternative Maths. [T9] I like to pose things ... such as, "What will that do?" ... and it's up to them to find out. I believe in kids finding out rather than me telling them. [T7]

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The computer aiso allowed teachers another freedom which was valuable promoting understanding:

ill

[It seems that we] can ask the student to be critical of what's written on the screen in a way that we can't criticise what they have in their books. [T9]

The development of critical analysis based on mathematical results became a real possibility. I think the potential that maths offers is for them to become conscious critical thinkers in the widest sense. There are opportunities for this with [the spreadsheet]. [T9]

The Development ofa Shift ofMathematical Focus The teachers' change perspectives produced a dramatic increase in confidence in their ability to plan activities involving the computer. Tasks became more openended, taking on the characteristics of the preliminary tasks described earlier. For example, TI (teaching an unstreamed class) prepared an early spreadsheet which required students to track down errors in formulas. Later in the project, the same students were finding "Flats/Houses to Let" datain the newspaper and entering it in a spreadsheet with appropriately headed columns. A simple activity li~e' this provides clearly defined stopping points to give a sense of achievement to all groups of students, but the open-ended nature of the potential analysis of the data provided a stimulus to discussion and further exploration. One teacher, experienced in computing and accustomed to working in the computer laboratory, reflected on the year-long project and the various strategies that he had tried: I can look at the exercises now and see how I can apply the spreadsheet to them, it's great. I never thought I'd be able to do it to start with. [TS]

Teachers continued to develop this skill in creating activities over the final term of the project: In preparation for today's lessons I find little need to attempt a question with the [spreadsheet] before the class does. My knowledge has grown and so has the efficiency. In fact I have sometimes created a problem and let the students attempt to solve it and often watch and learn from their methods .. , The uses and ideas came much more freely as computer/spreadsheet experience developed. [TID] ... changed thinking and opened up different methods and types of problems. I can look at an exercise now and think, oh yes I could do it this way or that, use a spreadsheet here etc. I didn't think I would ever be able to do it last year. [T8]

The teachers had experienced a shift in emphasis from the computer and what it can do, to the mathematics and how new areas can be explored. The computer was seen as a tool for the learning and understanding of new and interesting mathematics. We have referred to this as a shift of mathematical focus. This shift has allowed them to see the computer as something which can be used at any time in a lesson, not requiring them to "stop doing mathematics and start doing some computer work."

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A Broader Perspective on Context Our research leads us to believe that this change of viewpoint is not a minor shift which can be easily accommodated. The successful integration of computers into the mathematics classroom implies that major changes are needed in the teachers' views of the context of their teaching, and that these changes have to come from experience. Teachers began to recognise the need for a global perspective on their computer use in mathematics: A total rethink on lesson and unit plans has been required. [T10] In future, the computer will be integrated into my normal preparation time-but more importantly, written into the [school's mathematics] scheme. [TS] I think I've now reached the 21st century, I can't see how we could go on teaching mathematics.in a humdrum, "textbooky" sort of a way without bringing [in] the hands-on, both practical and electronic. It's a new generation of kid. They're motivated towards this sort of stuff. [T6]

The teachers' early concerns centred around issues of storing and marking student work. However, at the conclusion of the project their focus of concern had shifted to the mathematics.

Facilitating the Change of Viewpoint The teachers in the study recognised that it is not easy to take a step into the unknown in one's teaching. Many teachers work within their comfort zone, and this can create a barrier against the introduction of technology: There is a great deal of effort required for teachers to move outside their "comfort zone," a huge supporting beam of the teacher barrier to the implementation of technology. [TS]

The research reported in this paper indicates that if teachers are to make the shift of viewpoint necessary for a full and successful utilisation of the computer in mathematics teaching, two factors are of particular importance: computers must be ever-present and available in the classroom, and the support of experienced staff must be on hand as required.

The Omnipresent Computer A striking feature of successful computer use in the classroom was the way in which both students and teachers came to view the computer as an integral part of the classroom. The computer needs to be just another tool to aid the learning process. When I adopted this attitude, preparing work suitable for using the computer became much easier. [TS] If you start the year with it they accept it as normal. It's no big deal to them; they just get on with it. [T6]

Both teacher and students became as comfortable with the computer as they were with any other mathematical tool. However, the teachers recognised that their

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students had to go through the same processes of change that they themselves had gone through. Many students associate the computer with games and it takes time for them to accept that mathematical thinking is required to use software such as a spreadsheet. It seems likely that any change in student viewpoint will take much longer if students only experience intermittent exposure to computers through highly planned trips to the computer room for a particular unit of work.

Effective Support When we reflected on the support provided for teachers, ~nd their comments . on this support, we were struck by the diversi,ty of teachers' requirements and approaches. We believe that effective support recognises this variety and sees it as a strength. Creating a climate in which teachers can share their different problems, strategies, and solutions is an important aspect of teacher support. Regular visits to teachers in their classrooms were appreciated by them for several reasons. Regular visits: • were seen as a way of helping teachers to give the project priority over the numerous other demands on their time; • became an opportunity for teachers to sound out and clarify their current ideas on the impl~cations of the project as a whole; • provided an opportunity for specific ideas for using general-purpose software in current or imminent topics to be discussed and refined; • meant that the support team member could suggest ways of dealing with problems that had arisen since the last visit; and • provided an opportunity to suggest useful strategies. For example, teachers often found that another person in the room was extremely valuable when the computer was in use.

Allowing Time for Change The familiarisation period for teachers in the study was relatively long. We feel that this may, in part, be related to the mindset of the teachers: the process of reevaluating one's preconceptions about "suitable" use for a computer in the mathematics classroom took considerable time. All the teachers were deeply engaged in this process. The two teachers who outwardly made least progress in integrating the computer in their mathematics teaching provided insights into the familiarisation process. During the year of the project, they examined in detail their preconceptions about computers and the needs of the children they taught (aged 11-13, from very low socio-economic backgrounds, with limited experience of computers outside school). The teachers began tentatively to question these preconceptions. They argued strongly.that working with the computer did little to meet the primary needs of their students for constant reassuring contact with the teacher. Two of the teachers shared their experiences with colleagues and one remarked, reflecting on her own experience: You don't expect anything in the first hour, or the first week. You might have three sessions with them, and in a year's time they'll say, "I'm feeling more comfortable with this now." [T9]

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Similar conclusions were reached by Hoyles (1992) when she described her work in the Logo Maths Project, and commented that: "the social norms and expectations of the student group could not be ignOl:ed in any analysis of group interactions" (p. 38). , There is a time factor involved in gaining confidence, and we have found no short-cut to the adoption of the new perspectives required for effective technology use. This is consistent with the findings of research in teacher development. The Teachers Raising Achievement in Mathematics project (Britt, Irwin, Ellis & Ritchie, 1993) found that it took much of the two years of the project for teachers to move towards a constructivist viewpoint in their thinking about math~matics teaching and learning.

The Effect of Teaching Styles It could easily-be assumed that good practice for integrating the computer into mathematics lessons might require all teachers to adopt similar teaching styles. However, our experience indicates that teachers who use the computer effectively in their mathematics classroom do not all teach in the same way-they display a wide variety of teaching styles. The various approaches to effective integration of the computer reflect the different approaches taken to the preparation of lessons involving the computer.

The Significance of Preparation For any software which can be used to encourage mathematical understanding, there is a minimum level of teacher preparation which is essential. For example, the teacher needs some understanding of the concepts on which the software is based, and some notion of the kinds of situations for which it can be used. With microworlds or other open-ended, software, it takes considerable time and experience to develop an understanding of the power of the underlying concepts, and of the kinds of situations that can provide a springboard for student learning.

Approaches to Lesson Preparation The teachers tended to fall into two categories: those who explore,' and those who plan. A preference for one approach emerged early in the project, and tended to persist even when the teacher had adopted a changed viewpoint. The "explorers" are generally teachers who are thoroughly conversant with both the computer and the mathematics they are teaching. They expect their students to share their confidence, and are likely to model an enthusiastic, openended exploratory behaviour. Their belief is that, given the right conditions, students have the inclination and ability to find their way around the software, and learn for themselves what they need to do for a particular task. Because these teachers have built up a thorough working knowledge of the computer (and perhaps the software) by trial and error rooted in their wide experience, they believe the students will do the same. They believe that computer skills are incidental; picked up in context as needed. One such teacher knew when she first met an early model that microcomputers were for her. She recognises that she enjoys using the computer and so she does not demand efficiency:

Using Computers in the Mathematics Classroom

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I've spent an awful lot of time producing an awful lot on the computer, when I know that if I sat down with a felt pen and a piece of cardboard it would be done much quicker. It's not as much fun, though! [T11]

Teachers who plan appear to do so for a variety of reasons. They may feel confident about themathematics but unsure about their computer knowledge. Some feel that to.o much uncertainty is inappropriate for their students. Their class may include a wide range of abilities, attitudes and social adjustment, and the teacher may want to be available to all students rather than spending time sorting out computer hassles. The teacher may wish to have time available during lessons to observe students, or to work with small groups on social skills: such as working collaboratively. One teacher of 11 to 13-year-old, children gave a vivid picture of her preparation for using new software. Her approach is to prepare carefully specified objectives, prepare resources very thoroughly, prepare instructions on how to get started-then leave the children to it. My planning's always thorough: What I want to achieve, and how I'm going to achieve it. [Doing new things without careful preparation:] that's when I get the speed wobbles. [T8]

At the end of the year, she reflected in her journal on the style of teaching she had adopted, and its consequences: During the lesson, the teacher was readily accessible, because the tasks themselves encouraged the students to be self-teaching rather than teacher-dependent. The teacher, in fact, often just moved around the room observing pupil behaviour and interactions rather than being directly involved in "teaching" ... The students were given the tools and experiences and tasks to learn for themselves the skill that the teacher had set. [T8]

During the year the teachers moved between planning and exploring, trying a variety of ways of making effective use of the computers. Although the teachers initially found that planning for lessons involving the computer was time-consuming, and required an enthusiasm and dedication which they sometimes felt they lacked, by the end of the project they were no longer spending disproportionate amounts of time on preparation. They looked forward to spending even less: [Currently preparations for] computer aCtivities take much more thinking and preparation. [I think that] in future it will take less time. [T2]

A Teacher-Change Questionnaire Once it became obvious that the teachers had needed to make an attitudinal change in order to be successful in their implementation of technology, it was decided to give them a questionnaire to attempt to measure their individual attitudinal shifts. They were given a list of terms (see the first column of Table 2) based on the questionnaire developed by Claxton and Carr (1991, pp. 7-9). The categories described by these terms are intended to represent a continuum. This continuum was illustrated by a passage describing a fictional teacher going through the process of change, with each of the terms clarified in the context of her

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experiences at that stage. The teachers spent some time reflecting on their year and then filled in the questionnaire during the last two writing sessions. After reading and discussing the passage, the teachers marked on the list what they considered to be their start and finish positions (represented by sand f respectively) with regard to each of the four categories: 1. Their style of assessing student learning in relation to the integration of

technology within the assessment item. 2. Their style of teaching as related to integrating computers into their work. 3. Their view of mathematics and the assessment of computers within the programme. 4. Their view of mathematics and its effect on their teaching practice. The statistical analysis which follows could be questioned on the basis of the sampling procedure and the need to define the population to which inferences can be drawn, However, the extent of the differences is such that there can be little doubt that participation in the project had educationally significant effects on all involved. Tables 2 and 3 show a relatively consist starting position across the teachers surveyed. At the start of the research, nine of the teachers placed themselves as no stronger than agreement" with the principle of using computers in their teaching and none of them placed themselves beyond this with regard to using the computer to assess students. Two of the teachers even rated their view as opposed to the use of computers in their teaching of mathematics. By the end of the study these two teachers had made the biggest shifts in viewpoint, representing themselves as at the recuperation and permeation levels. We conclude that the teachers felt far from committed to the use of technology in the classroom at the beginning of the year in spite of their previous experience with computers. Looking at the teachers' views of computers in their teaching practice we see that, apart from the teacher who placed himself at point 17 on the 21 point scale, no teacher thought that their attitude had shifted by fewer than 5 points, with a mean shift of 10.0 points (p