The Role of Culture and Mathematics in a Creative Design Activity in Papua New Guinea
Kay Owens
University of Western Sydney Macarthur
A study of Architecture students undertaking their second design project (a paper sculpture without glue) involved analysing the interviews of 14 students. It was found that their cultural backgrounds influenced their attitudes to themselves as designers, and their past experiences influenced the ways in which they used mathematics. Other feelings, imagery, and their environment (other students and their changing sculptures) also influenced them. Their responsiveness during the project was an important aspect of their creativity. Mathematics was seen as part of their cultural as well as school background.
Curricula and Culture
Different approaches have been taken to develop mathematics curricula. One approach has revolved around the organisation of the content in a logical sequence and another around the psychological development of the student so that topics are reconsidered over the years in a spiralling design. A third approach has been the use of rich activities from which, for example, mathematics can be socially constructed (see, for example, Howson, Keitel, and Kilpatrick, 1981; Yackel & Cobb, 1995). The effect of a creative design activity on students’ understanding of architecture and mathematics has been considered in this study.
The report also considers cultural and social issues related to learning. Cultural background influences ways of thinking that will impact on learning in the classroom (Boero et al., 1995; Christie, 1994) in the sense of situated cognition. Lave (1988), and Carraher (1988) found that people in all classes and walks of life are capable of performing quite complex mathematical operations provided that the context in which the mathematics is presented links with the learners’ personal worlds. So, for example, children used complex arithmetic skills when selling in the market (Carraher, 1988). Millroy (1992) studied the mathematical ideas and thinking of carpenters and suggested that tacit knowledge manifests itself through activities although it may not be spoken. The carpenters used what school-mathematics would call symmetry ideas in making designs on boxes but they were created and centred on the boxes in culturally influenced ways. The current study used qualitative approaches to investigate the possible influences of culture on design and mathematical understandings.
The impact of culture on architecture is easily noted in the use of decorative designs in societies such as Papua New Guinea. However, this may appear to be superficial compared to building design or construction. Ward and Wong (1996) have argued that architecture provides for basic needs and design services but should not be limited to matters of visual aesthetics, separate from issues of power and social transformation. In their development of a pedagogy of the design process, they encouraged participants to “demythologize their [the teachers’ and students’] own beliefs about architecture, education and racial prejudice and about the relationship between them.”
Eisenman (1988) argued that second languages like architecture (and mathematics) were avoiding cultural roots and class, a view that Ward and Wong (1996) illustrate is not possible. The Whare Wananga project (Ward & Wong, 1996) avoided demonstrable Maori carvings but considered precise placement of building elements such as the use of the atrium as a meeting place. Another project that considered cultural roots was the design for a museum for the Mashantucket Pequot Nation for which the guiding principles were couched initially in ideas “of cultural diversity and place identity as generating principles in architecture” (Atkin & Krinsky, 1996, p.237). Interestingly, these views from architecture are salient to the social and cultural aspects of mathematics education and education encountered in Papua New Guinea.
The Study
Each first year Architecture student at the Papua New Guinea University of Technology made a sculpture out of paper. The students produced quite varied and attractive designs. We, the lecturers and I, wanted to know how students were thinking in order for them to develop such unique and interesting sculptures. We expected that the answers would inform our mathematics and architecture education.
In order to develop the sculpture, students were faced with a problem. In the light of the literature on problem solving (see, for example, Nisbett, Putt, & Taplin, 1996) the focus of enquiry for the current study was on students’ motivation and cognitive processing. My personal comprehensive view of problem solving emphasised responsiveness during problem solving (Owens, 1996). My theoretical discussion was based on a grounded theory study of primary students engaged in practical spatial activities. Feelings, knowledge, imagery, attention, and problem-solving approaches influence students’ responsiveness. Responsiveness in turn affects the students’ actions (in the present study, on their sculptures) and prompts others to comment which in turn influence the student’s cognitive processing. This was my background but I did not set out deliberately to apply this model in the current study. I was looking for motivational influences, cognitive processes and whether mathematics played any part in their thinking. A further question concerned the influence of traditional culture on the students’ work and how they perceived traditional mathematics, especially within the context of building.
Procedure
Students’ sculptures. Students had previously made a sculpture using glue and any available materials that they wished to use (e.g. “junk”, cardboard, sticks). For their second sculpture, students were able to use cardboard (3 mm thick) and paper that came in three colours. They were given certain restrictions such as the maximum size of the sculpture and told not to use glue. They were encouraged to work directly with three dimensional space. The students were motivated by this activity and spent many hours in building their sculptures.
Interviewees. A third of the first year architecture class of 34 students were interviewed by the author after completing their sculptures. Two of the 14 students were female. Eleven were from coastal provinces of Papua New Guinea, two from highland provinces, and one was from a neighbouring Melanesian country, the Solomon Islands. Two students lived with their parents on oil palm plantations away from their home province, six lived in cities, and a few lived in rural towns. Several boarded at high school and most boarded at senior high school. Comments were made about the art classes and carvings at one of the senior high schools¾three attended this school. It is significant that most parents and one guardian had incomes and half had post-school training.
Analysis. The other lecturers were impressed at the honesty, naturalness, and reflectivity of the students’ comments. Codes were chosen to cover the different kinds of responses and these were grouped under themes. By considering the coded transcripts, a summary of results was developed to illustrate how students were motivated and worked and how they viewed mathematical and cultural influences. (Transcripts are used verbatim without clarification on English grammar.)
Results
Intrinsic Motivation and Responsiveness
Positive feelings. Half the students spontaneously noted that their sculptures were pleasing them and they were particularly intrigued that they were producing such a beautiful or different or imaginative sculpture. For example, David started with the idea of a trophy (Figure 1 is his final sculpture) but when I asked him what he liked about his design he said:
It is beautiful and it did not symbolise [anything]. It didn’t exist and doesn’t represent anything in this place, and I came up with this thing!
Willie (Figure 2) said:
The most [pleasing] thing about the sculpture which really intrigued me was about the curves and the way I made the curves.
Figure 1. David used pattern, stability, repetition, and measurement.
Figure 2. Willie’s holistic sculpture illustrates his use of curves and repetition.
Figure 3. TKeps developed a functional idea.
Imagery. Students noted spontaneously their imagination or their use of imagery. TKeps (Figure 3) said:
Most of it was from my imagination and creativity so it gives me idea of which things to fit into each part.
When describing their imagery, 50% referred to some physical object that might have started them off but 20% spontaneously noted that they moved away from being bound by the initial physical idea as they responded to their work and began to imagine ideas or solve the problem of joining parts together. Four students (30%) specifically referred to imagery that did not relate to anything physical; one student said he did not imagine anything but based his sculpture on his feelings at the time. Three students (20%) had initially wanted to make moving objects.
Beliefs about Themselves as Architects
Students saw themselves as creative designers of buildings. Two students specifically commented that they liked buildings in cities and that they were intrigued by them and studied them. Others commented on their personal study of designs at the National Art School or in traditional carvings.
Traditional backgrounds were mentioned spontaneously as important in 60% of cases but only 30% directly noted that they incorporated traditional ideas into their sculpture. Their traditional background seemed to be part of a belief that their cultural experiences were valuable for creativity and were a part of their identity as a designer. Interestingly, this did not seem to be as strong with the two female students. Cultural ideas led to the use of curves, spirals or symmetry. For example, Fred (Figure 4) noted:
When I started to create the sculpture, it came out of my imagination. What I learnt back at home I added onto the materials and it came to what it looks like now. ... It was totally flat [cardboard] but when I started to put bits and pieces in, I think it looks like a house and I make like a house and I started to think about making house back at home ... It is the spiral thing that looks like it swirls around and the woven part that makes it attractive. ... like ancestors in traditional types they used to make spiral bits they just bend hard woods to make small juts and to fish...
Ian (Figure 5) perceived village life and stories as influential.
Ian: Village life, talking to elders, they were telling traditional stories and I tried to incorporate some of the traditional designs into it as well. Partly some of my sculptures they tell people how I feel. If I am angry the sculpture looks very scary and taunted and dull. It depends on the mood I am in.
I: Can you explain how stories are in there [the sculpture]?
Ian: Not really. Patterns on main mast and shapes on curved areas are found in my tradition. In it there are lots of zigzags.
Figure 4. Fred incorporates traditional decoration and curves.
Figure 5. Ian used three dimensional shapes in his compact sculpture.
Figure 6. Taurus used the sea devil and counterpoint balance.
Only in four cases was a deliberate attempt made to transfer a two-dimensional traditional drawing or mask as a major part of the sculpture. In addition to the last student, one student, Taurus from the Solomon Islands, based his sculpture (Figure 6) on a sea devil traditional design and story, and another incorporated a mask face. Others used weaving techniques for effect. Often the idea of bilas (decoration) seemed to encourage consideration of the use of colour or extra features. Three used intricate traditional designs for decoration. Taurus says:
The part I like the most is the rising face as it expresses features of what the sea devil is like with its two big eyes, the erecting tongue devouring and the two birds in opposite, which he used to navigate the unknown seas. I like it because eyes are bent, circular, and it is out of single paper. ... It is symmetry. The other half must be equal afterwards.
Cultural decoration had significance culturally and architecturally, and this depth is beginning to be vocalised.
Architectural Ideas
Use of mathematics. Students readily applied mathematics to draw circles or to measure sizes and equal lengths. Some began with shapes like triangles, circles and spheres and tried to build on these. Other aspects of mathematics that were mentioned were repetition and pattern, perspective, the use of measurement to improve accuracy, the use of mathematics to develop different, more difficult designs, and the importance of three- and two-dimensional images.
Difference and similarity. For many, creating designs meant creating something original and different from what already existed in the environment or in another’s work. Some wanted difference on different sides of their sculpture while others wanted some similarity in, say, opposite sides. Others strived not to have symmetry but yet to have balance in the artistic sense. This is noticeable in comments made by Fing (Figure 7).
When I visited the National Art School I saw paintings and they made me interested, lines going here and there and gives me a depth of feeling I really like this and I see my own work and I can put it here and there and it looks good.... When I needed strips I measured the same length with ruler but ... I looked at it and cut them to make them look better and not too symmetrical and all the same [see parts at the top of Figure 7]. ... It was time consuming to build back structure with circles (see the repeated design, Figure 7).
Fing’s interest in difference is manifest in his thoughts and actions. Fing used mathematical ideas in several ways and linked architectural and mathematical ideas in an abstract way:
I was just thinking and feeling structures are just like building numbers that go on and on and structures go on and on, more like infinity or repetition or what goes on and comes back; big, small and just like that.
Repetition and symmetry. The view of Fing’s sculpture (Figure 7) illustrates how some students used repetition for effect. Repetition can be seen in Willie’s sculpture (Figure 2), and David’s sculpture (Figure 1) as he said:
[The curves] are about 50 cm long and I measured it from one join to another and cut out to fit papers around, I tried it out and I saw a pattern, then I drew it and I would join it up with slits in right place.
Others used symmetry effectively as seen in the sea devil design (Figure 6).
Figure 7. Fing developed ideas from modern buildings, repetition and asymmetry.
Figure 8. Bell’s sculpture shows traditional influences.
Proportion and wholeness. Nearly all students made reference to these ideas. One used spheres and circles to achieve this; others used curves. As Ian (Figure 5) said:
I don’t like to make symmetrical things because I have the idea that nothing is perfect so when I created my sculpture I didn’t want it to look symmetrical. ...The more complex I designed sculpture, it challenges the mathematical side of me so when I design it with awkward shapes, it challenges the mathematical side of me how it fits together. e.g. circular bit that protrudes, I didn’t want it to protrude too much so I had to think about its size. If it was too big it would stand out and dominate when someone looked at it and if it was too small it would not be noticed. (The comment about nothing being perfect is a cultural attitude that frequently influences the making of objects.)
Bell’s comment shows how ideas changed to keep an holistic perspective in the sculpture:
At last minute, I removed some of it, especially unnecessary and didn’t fit picture when look from all angles, and I asked opinions from friends who gave some ideas, and what was missing.
Firm foundations and balance. Half the students noted the importance of either a firm foundation or some means of making the sculpture sturdy so it would not topple over. This problem frequently led to interesting creative changes to the sculpture. Willie (Figure 2) said:
I thought I had better start with sketch of anything, I was sketching anything that came into mind of statue and got sort of things around it and made different kinds of sketches. I had a sketch with base and these curves and I represented curves with coloured paper and have intersect them into middle and come down to base again and had two objects (diamonds) on the sketch but I didn’t include them on the sculpture but pushed them right down. ... One problem was I had difficulty in balancing curves and they were wobbling and would not stand up straight so I put cardboards in and it was firm and didn’t move ... it was trying to come out so I had to every now and then hold curves together and the two cardboards were supporting whole thing. ... The two cardboards were like this - ... one reached to the top and other to edge of curved paper.
Several students measured carefully to mark the centre of a part. Ian used stays to hold his mast firm, and summed up his ideas on architecture:
I think [foundation] is important in architecture. If it is right then rest will stand up. [An important idea in architecture] is like actual design of building and to look peculiar and to experiment with shapes and ideas.
Taurus used circular weights on a stick to balance the high part (Figure 6). He said:
Mainly the fitting in of joints was the bit that needed cautious work. But with patience and advice from my colleagues and the successive trials that I made, I finally made it. It was constructed in a simple manner. The solid foundation were then followed by the bits and pieces that made up the rest. For example the scalloped papers. Extra weights were forced into it on various areas requiring it so as to balance the entire figure.
Functionality. The students’ works were considered as sculptures rather than models of buildings but TKeps (Figure 3) particularly talked about the functions of different parts of his music studio.
I began with cardboard as a strong foundation. After setting up cardboard, I started cutting paper and bend the shapes to make the building. ... That part and four corner was apartment where musical instruments can be played and stored and top part is entertainment centre. ... It is a building not on ground floor so I want it to be a few metres high above ground level so there is a bit of cardboard as base for uplifting building. The floor level can become some sort of foot track for others coming in.
The integrity of his sculpture came from his consideration of it as a music studio so all aspects, position and shape of parts, nature of decorations, were related to this function.
Views on Mathematics
Uses in architecture. As mentioned under the section on architecture, students saw that mathematics could be used in architecture and ideas such as symmetry, repetition, balance, and relative proportion were seen as part of this mathematical perspective on architecture. In most cases, students considered mathematics as measurement and calculation. Several students noted that they or traditional workers or architects had used measurement to get symmetrical sides, to make equal slots when joining, and to help with balance. This was thought to be a practical and important use of mathematics for architecture. In one case triangular and circular shapes had to be developed from the measurements so that they would sit together. Shapes were also considered as part of mathematics. Ian began by “playing around with common shapes” and Clive discussed the nature of a cone. The importance of estimation and accuracy was noted in terms of the purpose for which mathematics was being used. For example, in building a bush materials house in the village, estimation was most appropriate although equality of lengths as marked by the length of a stick or a piece of rope was seen as necessary. For architects, measurement and calculation improved accuracy. Ratio was seen in terms of scale but was linked to the idea of parts of a holistic structure without certain parts dominating; this was also referred to as being in proportion. One student specifically noted perspective and another the importance of plans. Two students commented that it was mathematical to be able to judge how big a structure might be when one considered the plan of a building and its height.
Language adequacy with mathematics. Students frequently expressed themselves inadequately when they first tried to explain how they used mathematics. Clive only linked in his school mathematics when he tried to form a cone with a piece of paper. Other students referred to triangles when they meant three-dimensional shapes but others used the terms prism and pyramid although one used the term prism when he in fact had a frustrum.
Traditional mathematics. All students could suggest some type of traditional activity that involved systematic thinking. Order of events was one idea expressed by students. For example, it was needed in preparing gardens, in the kitchen, in fishing, in house-building (order in which parts are done), and in using the stars.
Most students referred to getting houses straight by use of ropes and sticks or by planting sticks in a line. They commented that it was a skill to get the lines straight. One specifically commented that a good builder would have a plan in his mind but the on-looker would only know what it was when they saw the finished building. They noted that good builders could vary the plans and would know how many uprights, or sago-leaf bundles would be needed for the particular house. One commented that there were differences in the ways of building houses. Sometimes the outside plan was decided and then divided up while others considered the rooms to be needed before finalising the plan. Diagonal struts were not in all houses.
Views on Problem Solving
About half the students commented on the problem of getting started. Most overcame this problem by drawing or cutting papers or watching others. Once started they usually kept going. Time was also a problem noted by about a third of the students. Some noted that it took time to fix aspects of their sculpture or to make the many parts.
Sometimes change in ideas and imagery came about because they had physically to make the object without glue so this problem led to invention. For example, Fing used holes and rolled paper to hold parts together and then he used this idea for effect (see Figure 7). Often they set their own problem¾how to make the sculpture more pleasing. One approach was to think of an idea, physically cut it out and try it checking its effectiveness by holding up pieces of paper to see how it would look. Others drew sketches to start themselves thinking.
Affect. As mentioned earlier, their pleasure at seeing their own sculptures was evident and motivating. Some also noted that they put their feelings into their design. Ian said:
I designed it on how I was feeling at the time. I didn’t want to build something big, I felt small inside, I was pressured and school was mounting on me.
Evaluating their ideas (metacognition). Students commented on the difficulties of stability and joining parts together and on their thinking in order to solve the problems. Their reflection and development of solution ideas was an important aspect of their designing (see comment on pattern by David).
Concrete materials. Interestingly, the use of physical objects rather than drawings flaired their imagination. However, as with all concrete materials that are used for learning, students really needed to be faced with a problem in order for there to be connections between past experiences and imagery, new imagery, methods of connection, holistic ideas, and application of ideas.
Use of past experiences. Three students spontaneously noted that they used ideas from primary school experiences in making structures, or in building with small objects or blocks, or in playing. One student noted that she lost a lot of her creativity at school as she had not played and built things when she was small.
Problem solving involved the students’ feelings, knowledge, imagery, past experiences, metacognition, and use of concrete materials. They responded idiosyncratically to solve the problem and learn from the activity.
Working with Others
Most students spontaneously made a comment about the influence of others. Although many of them noted that they tried not to be influenced and to make something original, they saw an idea like a woven mat and decided to use it, or a friend had talked about an idea which was then used such as Japanese ikabana flowers. One noted that his friends thought his sculpture was different things and that pleased him as it was in the eye of the beholder that a sculpture should be interpreted. Others had specific help to overcome a problem like how to make joins. A couple of students noted that, although they were given comments to start them or help with a problem, what they did was still their own idea.
In three cases, students noted that others had started and that they were having difficulties getting started and this was pressuring them. They then just took papers and cut them and started. In this way, their responsiveness, pushed by strong inadequate feelings, helped them positively to begin to create.
One said it was important for students to work in groups both in mathematics and when creating structures because students could learn best that way and give each other ideas.
Discussion
Responsiveness During Rich Activity
The interviews provided an opportunity for students to reflect on their thinking, and they indicated that the sculpture making was beneficial but that it would have been of further value if they reflected and reported in the classroom on important values, ways of thinking, and architectural concepts. Further, it could have been a good starting point for the development of mathematical ideas and interest. For example, if students had shared an idea about modifying shapes or using shapes for balance, then the various properties of shapes and how changes affect shapes could generate a practical creative understanding of shape in architecture. Properties may remain invariant (e.g. angles opposite the two equal sides of an isosceles triangle are equal) while other changes lead to variation (e.g. the length of the equal sides can vary and the apex angle can be acute or obtuse.
The responsiveness model (summarised in Figure 9) was developed from young students solving two-dimensional spatial problems (Owens, 1996; Owens & Clements, 1998) but it was found to be applicable to tertiary students (Owens, Perry, Conroy, Geoghegan, & Howe, 1998) and now to students working in three-dimensional space problems.
Responsiveness
Person ...
Imposes concepts and imagery on materials Manipulates materials Applies heuristics Records, displays, describes Notices aspects of materials / people Expresses feelings Communicates with the teacher / student
|
Context |
Cognitive Processing |
|
Influence
Context ...
Influences perceptions especially
seeing and hearing
Affects feelings
Affects the opportunity to manipulate
Disrupts / prompts thinking
Encourages / discourages communication
Figure 9. Aspects of problem solving (Owens, 1996).
By being responsive, reacting to what they saw, reacting to their imagination, or the comments of others and their feelings about their work, students were being responsive. Without responsiveness, the problems would not have been overcome but more importantly their imagination and creativity would not have flowed. Responsiveness is a compound variable; its components are dependent on a balance of cognitive and affective processing. Responsiveness is the movement forward, the risk-taking of problem-solving. Often multiple thoughts have to be held for consideration and action over several seconds or minutes until the context reacts to the development.
Cultural Influence
While culture was an underlying influence on students, it was clear that the students were using only minor aspects of tacit knowledge (Millroy, 1992) with both architectural and mathematical concepts and processes. It seems that explicit discussion of this cultural knowledge would assist students to develop a recognition of the social, political, class, and colonial aspects of their school, architecture, and mathematics education (Ward & Wong, 1996). Such discussions would assist students to develop their cultural heritage in their architectural design, to recognise the interactions between some of these less apparent aspects of architectural and mathematical education, and to create culturally rich design.
References
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Acknowledgement. This research was carried out in conjunction with the students and staff of the Papua New Guinea University of Technology.