Epistomological Pluralism (when? - at ACER)

Can Epistemological Pluralism make Mathematics Education more Inclusive?

Liddy Nevile
Australian Council for Educational Research

There is growing concern about the exclusive nature of higher achievement in mathematics in schools at a time when there is an increasing use of computers in schools. In this paper, a course for undergraduate primary teachers is explained and the theory that the friendliness of the computational environment can make it a useful remedial mathematics medium is examined. What has been termed 'epistemological pluralism' by Turkic' and Papert (1990) is advocated as an alternative approach to the more traditional one which aims to help students reach an intellectual maturity which is based on the graduation from concrete to abstract processes for the construction of knowledge.

There is a continuing problem with what might be called inclusive mathematics education in schools. Many students are alienated from mathematics by barriers which lie outside the domain of mathematics itself, things such as cultural cringe (as it is known in Australia) can make students who profess positive attitudes to mathematics part of a cultural 'elite' which soon becomes a rejected minority in schools.

While not suggesting that an isolated incident or comment can make a difference; we are all conscious of times when students have been brought to reassess their 'mathematical shyness' and with new-found confidence work to achieve new heights of personal success. The problem confronted in the course reported in this paper was how to provide experiences for a large number of students which would have this effect.

The theory underlying the course reported was that if a group of 'math shy' undergraduate primary teachers could be given positive reinforcement of their own abilities and a more mathetic perspective (see Higginson, 1972) from which to approach mathematical activity by working at mathematics in a suitably constructed computer environment, they would learn more mathematics as undergraduates and be better equipped to help fight the mathematical poverty of the schools in which they would be working. The course did not rely simply on providing 'nice' experiences. Logo programming was used to provide a formal, concrete environment in which to do mathematics in order to foster or rekindle interest in mathematics and understanding of mathematical processes in a curriculum which depended on a surrounding culture of 'epistemological pluralism' (Turkle & Papert, 1990). This did not mean that students could avoid the formality of mathematics but it did mean that they could work legitimately as bricoleurs, piecing together and polishing their work as apprentice craftspeople.

The aim was to provide the students with an alternative path to achievement of mathematical products, not to lower the standard of the mathematics in any way. Papert and Turkle argue in a paper about similar problems within the domain of computer science (Turkle & Papert, 1990) that this demands the rejection of the notion of bricolage as an immature thinking process and the acceptance of it as an alternative process. They argue that computer virtuosos frequently employ concrete processes to achieve their high standards and that the preference for the use of one style of knowledge construction in preference to the other should no longer be considered inferior.

Analyzing the data collected from 60 students during a course which promoted the legitimacy of mathematical bricolage provided an opportunity to evaluate the conjecture reported recently by Turkle and Papert (1990 p. 153):

The development of a new computer culture would require more than environments where there is permission to work with highly personal approaches. It would require a new social construction of the computer, with a new set of intellectual and emotional values more like those applied to harpsicords than hammers.

They, and we, have chosen to adopt the definition of bricolage used by Levi-Strauss (Turkle & Papert, p 136),that:

Bricoleurs construct theories by arranging and rearranging, by negotiating and renegotiating with a set of well-known materials.

Initially, the 60 students in our course responded to a survey of attitudes to mathematics and computers. Then there were weekly discussions about the nature of mathematics and mathematical problems were discussed for their diversity of subject-matter (from areas such as information theory, chaos, number systems and cultural differences); the students spent one and a half hours working in Logo microworlds per week, often returning to the same problem-domains for three or four consecutive weeks; the students wrote weekly journals in which they reflected upon their experiences in the course and their previous experiences; some students were video­taped during working sessions and all were audio-taped during a final discussion of the nature of mathematics, and each student submitted a final essay relating their current understanding of the nature of mathematics and its relevance to them with their previous understanding, as reported in their journals throughout the semester. In addition, the students weekly submitted activity sheets on which they proposed activities for young maths students.

The following is a summary of the information gathered about the students' attitudes before the course started. There were 35 questions in the survey and of the total sample of 60, 13 students were boys.

ATTITUDE TO COMPUTERS

Computers would be interesting at home; I'd like to learn to use one, it would be easy, I'd use it more than other members of my family, I enjoy computing and mathematics would be more fun with computers and it is not relevant that they were not necessary in the past, not true that they are not good for the world, not fun, might get too powerful, or would be hard for me to use.

More than 60% agreed , 30% were not sure and 8% disagreed.

ATTITUDE TO MATHEMATICS

I do extra work, it's easy, it's important that I understand, and I feel good about it and it is not the case that I feel tense, don't do well, have to remember things, get upset, can't understand and only do mathematics because I have to.

Almost 40% agreed, 20% were uncommitted and nearly 30% disagreed.

SOCIAL ATTITUDE TO COMPUTERS

Smart, science-type people like computers and they are not very sociable or athletic.

Only 3 students agreed with these statements but 25% were not prepared to say they were not true.

In the course at Harvard as reported by Turkle and Papert, the students identified computers with mathematics and those who were mathematics shy were similarly computer shy. Many of these students, according to the conjecture, while not denied physical access to computing, were loath to traverse the mental barrier they perceived to lie between them and computing, the knowledge construction techniques of the canonical elite.

In a large survey of Canadian students (Collis, 1988), it was found that the majority of students associated computing with a masculine, mathematical model and for many lower-achieving girls, in particular, studying computing was a daunting prospect. The aspects of computing which were said to contribute to this negative attitude were school policies and practices; social expectations, and personal factors. Collis et al stated that in Canada, where at the time students did not use computers for instructional purposes or within a broad range of curriculum areas as tools, the students do not develop an image of it as something which will be of interest/value to them and (Collis, 1988, p. 122):

Through this omission, adolescent females as well as males are deprived of access to many valuable aspects of constructive computer usage.

The Canadians concluded that of the major changes necessary, more scope for collaborative work was one and gender-typing another. In order to correct the imbalance, the report suggested that it would be necessary to change teacher education and school practice to widen the range of experiences and use the computer as a tool in the curriculum; to organize the computer use better to make it more equitable, and to counsel the girls who don't like computers. Many of these changes have been tried in schools and it is their continuing failure to achieve all the results desired that brings our attention to the ones advocated in this paper. In our work with computers we have come to believe that the problem goes deeper and that the focus must be on what sort of tool the particular user is likely to be able to use best, on whether beyond what Turkle has called 'hard mastery' (Turkle, 1984), which for some is perfect, there lie other types of usage which are equally successful in the hands of hitherto non-participating users.

In continuing work relating to the introduction of computers into education, we have developed a series of metaphors for the representation of computers as tools and we have found they operate with varying success. We prefer to think about how students are to be encouraged to use the computer than in within which particular disciplinary area.

In the Melbourne course, the situation was very different from that reported by Collis et al and Turkle and Papert. By 1988, Victorian schools had been persuaded by and large to use computers across the curriculum, particularly in subjects such as history, geography and language, as well as for vocational training for those students who were to remain at school only until they could gain employment. The survey shows that the students in the course did not agree with the Harvard students about computers. The few who had managed to stay within the mathematics streams at their schools had rarely used computers and knew that if they did it would be to work on programs and problems which they tended to associate with mathematics and science. They did not think programming was a trivial exercise. The majority of our students, more likely to be lower-achieving mathematically and so more involved in word-processing classes, had a different attitude. Any programming they had done would most probably have been elementary Logo programming and most of their experiences would have been designed to convince them that mastery of the computer is easy. It is not surprising that these students considered programming to be easy and likely to cause them no particular problems despite their difficulties with mathematics. This meant that for our students, using the computer and Logo was not likely to pose a problem which they would associate with previous failure in the same way as more traditional mathematical activities in a traditional medium would have done. That was our contention and the premise upon which the course was designed.

In our course, the students worked in Logo in microworlds which focussed on mathematical facts related to numbers and geometry. The activities they undertook were aimed to have them participating in, and so concretely representing, mathematical processes such as hypothesizing and proving their conjectures, often doing this by particularizing and generalizing, and collaborating with other students to negotiate problem posing and solving methods. Mathematics was promoted throughout the course as dynamic knowledge: it was said to entail both the processes and products of mathematical activity. The students' set-work involved the investigation of problems which they developed from worksheets such as the one shown:

how many sides do stars have?                     what is a lemma?

                               what theorems are there?

is there a general rule?           are stars polygons?

          are there some special circumstances?

what connects :angle and :sides?                           what :angle does it need?

The unusual aspects of the activities for these students included the following: the sustained nature of the work; the open-endedness of the work; the freedom of choice with respect to the processes of working and the accompanying lack of direction, which led to a diversity being developed within the group to which common standards essential for such processes as proof could be applied meaningfully; the collaborative nature of mathematics within this culture; the concrete nature of the mathematical processes; the friendliness of the Logo environment which is so instantly forgiving of corrected mistakes compared to the white-page medium where errors are for ever; and the graphical nature of the work which contributed to the students' appropriation of problems as given and often resulted in personally meaningful decoration being added to work in progress.

In the words of the students:

In the Sunrise School we were not given a bunch of mathematic facts or sums as I had been so used to in primary and secondary school which had led me to believing that's all mathematics was. That's the reason that I was so surprised to learn of the huge amount I took part in each day. We were encouraged to find things out on own, to take risks and to experiment with the little information that we were given. I found this to be an effective way of teaching mathematics as we were not given a question-answer situation where there was only one right answer. Each person discovered different methods and worked at their own pace. No one knew exactly what the teacher expected them to come up with at the end of each lesson so we were made to do the thinking ourselves and to work out our own problems that we created out of interest.

The result was that new ways of learning were soon on the agenda:

At school I always learnt geometry by the process of problem solving and an endless amount of writing. However, I still never fully understood it. Learning how is difficult, but learning why ... seemed impossible. The Sunrise school seemed to change my ideas of geometry by converting problem solving activities to an activity where I was able to actually see the formation of angles according to different shapes. Turtle geometry allowed me to change numbers in a fixed program and allowed me to actually see up on a screen the value that they numbers had. By using a simple program such as to polygon sides, repeat sides forward twenty left 360 divided by sides end and placing numbers where appropriate I was able to work out different shapes of different sizes and therefore different angles. As I was able to see the shapes forming and relating the shapes to the numbers I used, geometry became understandable.

The students were not all immediately post-schooling and one of the mature students showed all the typical signs of anxiety of a mother returning to the work-force: teacher training was a long way from dress-making. She wrote in her journal after a couple of weeks:

Logo presents as an easy step by step method of programming. When I was first introduced to Logo, I could not have been more intimidated by the prospect of having to learn this strange new skill. However, after a few weeks of practice it became clear that this computer language has many parallels with simple logical task solving. For instance, in using Logo turtle graphics I began by running a polygon generating formula of which I had no direct understanding. This lack of understanding left me quite frustrated at first and I was afraid of not ever being able to understand. It was not until a closer examination of the formal steps described in the formula itself that I gained the necessary insight to create a specific formula of my own. Logo, like common household and work related tasks, was seen to involve fundamental concepts of practical, logical, and step by step thinking. ...

This course has revealed not only the broad, wide-ranging applications of mathematics generally, but also the relationship between virtually every task and the mathematical thinking I once thought was the sole province of mathematicians, accountants, engineers and the like. As a consequence of this revelation I have realised the importance of gaining a deeper understanding of the mental steps involved in virtually every practical task. This experience has enhanced by appreciation of the subtleties of the teaching process.

For the students it was important to provide a culture which would scaffold them as they came to terms with their own understanding:

In her second week, one student wrote:

Great satisfaction!! These are two words to describe my feelings. Although I've used computers for school work before, I never realised the full potential of their functions!! They are created with the use of a few signals to the computer a series of shapes of different sizes and colours on the computer display. It's wonderful to experience a certain amount of control over what is displayed on the screen. My satisfaction, is only a small portion of the total feeling in the room, however. Everyone had created images hence today I learnt that knowledge is the key to control.

Later she added:

Suddenly my perfect uncomplicated image of maths is changed from black and white to an array of psychedelic splashes of colour. Suggestions of dots being potentially rich ... stunned me, that is until I looked at my eight year old cousin's colouring book, aha, I cried, dot to dot activities. Yes I realise their potential, number recognition, number sequences had started to make amazing sense to me and I had begun to critically investigate my current concepts of maths but also my childhood experiences in maths.

Many of the students reported their interactions with other students and throughout the semester they became critically aware of the value of good collaboration:

I yearn to be able to understand everything I can about mathematics. I desperately want to be able to teach the children well and with confidence. So far I have become more aware of the way my mind works and using computers in the course the way we have been has helped me to learn how to just sit and figure problems out. I now have more of a desire to want to sit down and figure out a problem just for my own satisfaction. My mind has become more open and more open to thinking up ways of solving these problems. I am now able to put more pressure on myself to find out something because I am more confident that I can finally reach an answer. The activities on the computer has helped proved this to me. As time went on I was able to offer more suggestions as to why things were so and eventually both of us (my partner Anna) benefited from each other's gaining and growing knowledge.

But the aim of the course was to give the students opportunities to work on their own personal meaning of mathematics. The introspectivity fostered by the course and recorded by the students was most revealing:

This course I've attempted to talk about maths and the problems I have had. At the beginning most of my discussions were attempting to get a better understanding about the course. I don't think that I have ever talked about maths as much as I did at the start of the semester. Towards the end I began to talk about maths and how it effects me, the type of things I do and how maths relates to my everyday living. I have also noticed the difference in my attitude towards numbers. I didn't place great emphasis on the importance of numbers and how much children need to understand them, but doing and learning different activities made me see that numbers can be enjoyable. Children should be able to experiment and this could lead to talking about maths using everyday language.

The aim of the course was to use computers to bring students back into the mathematical culture to enable them to work productively in the future in a field from which they had been alienated, in many cases officially for lack of achievement and in others, frequently by bathers built of negative associations and alien practices. The evidence showed that there were many students in the group who preferred to work closely with objects in what increasingly is being described as a typically feminine way. We prefer to associate this style with craftspeople and virtuoso artists. We suggest that since the introduction of computers into schools, providing a concrete medium in which bricoleurs can work at mathematics, there is no further excuse for losing from mathematics those students who prefer to work in this way.

References:

Collis, B J., 1988. "Adolescent Females and Computers: Real and Perceived Barriers" in (eds.) J.S. Gaskell and A.T. McLaren, Women and Education A Canadian Perspective, pp 117 - 131. Alberta: Detselig Enterprises Ltd.

Higginson, W., 1972. Towards Mathesis, unpublished PhD thesis. Alberta: University of Alberta.

Turkle, S., 1984. The Second Self: Computers and the Human Spirit. New York: Simon & Schuster.

Turkle, S. and Papert, S., 1990. Epistemological Pluralism: Styles and Voices within the Computer Culture in

Signs: Journal of Women in Culture and Society 1990, vol.16, no.l. Chicago: University of Chicago.