Mathematics and Crafts in Andalusia: An Anthropological-Didactic Study

María Luisa Oliveras Contreras
Granada University

Introduction

In this paper I shall attempt to set out the work plan and the methodology from an anthropological study in which I explored the uses of concepts, properties, theorems, etc. of different aspects of Mathematics in performing typical crafts in Andalusia, Spain. I used an ethnomathematical approach for the theoretical foundations, and an interpretative-qualitative methodology.

From the point of view of positivist research, it is very difficult to try to explain the plan of work, the methodology, and the results of any ethnographical research on the Didactics of Mathematics. Positivist research presupposes a structured design in which the hypotheses and techniques are determined a priori. This is in contrast to the flexibility which is appropriate for an interpretative and qualitative methodology which proceeds with very open and interactive plans, and emerges as the first phase of the field work is being carried out.

I shall attempt to set out the general questions which motivated me to start this work, as well as the characteristics of the field work and the scenarios which form it. The origin of this work was my desire to discover the degree of the social use of concepts, properties, relationships, theorems, etc. corresponding to intuitive Geometry in the performance of the tasks which constitute the production process of certain hand-made products which have been in Andalusia in a traditional way for many centuries. This curiosity was inspired by my personal involvement with Didactics of Mathematics from my work on methodology, as well as from my interest in the social and cultural manifestations in the environment in which I live.

Objectives and Scenarios for the Research

It is clear that the "teaching of Mathematics" should give way to "mathematical education or enculturation", because it does not merely involve isolated cognitive variables, but the intricate reality of a human being. A person's education is completed within a society, in the different institutions which form that society: school, family, etc. These subgroups which make up a person's environment may have a defined curriculum or may have aims and messages which emanate from the cultural heritage. Many of the conflicts within teaching arise from the conflict between planned studies and real life experiences.

Also, in Mathematics, it may be stated that there are specific creations in each cultural environment, within the generalization that belongs to scientific truths, in spite of the fact that the process of "removing facts from their original context" (abstraction), makes us forget the social and cultural bases which precede abstract ideas. When we delve into the processes which surround personal construction of mathematical knowledge, we have no options but to go back to the cultural roots of the social groups of which that person is the result. To a large extent we must accept an approach to these didactical questions that is much more anthropological than psychological.

Therefore, I am attempting to shed some light onto the unclear relationship between school mathematics education and popular mathematics culture in Andalusia. In order to accomplish that, the general objectives of this work are:

A. To discover the mathematics contained within the context of certain cultural products, which have been kept alive up to our days, handed down orally and by experience. Accomplishment of this first objective should contribute to the knowledge of the:

a-1 most hidden facts of our popular culture, and

a-2 social requirements for a mathematical preparation suitable for certain fields of work.

B. Explore the various didactic implications of this knowledge through an analysis of:

b-1 the relationship between the school curriculum and popular knowledge and awareness of the divergence between culture and education related to Mathematics,

b-2 the didactic relationship underlying the learning process,

b-3 the sources of information and the utility of education, using case studies of the professional histories of the informants, and

b-4 the learning involved and its application to the preparation of teachers and resultant models of teacher preparation.

The field to be studied is made up of a selection of "scenarios" or craft work which I considered to be representative of our culture and which will allow us, a priori, to select those processes in which mathematics is applied. Table 1 contains a list of the crafts that were considered.

Table 1

Crafts that Were Studied

1. Marquetry (inlaid work)

2. Musical instruments, particularly guitars

3. Landscaping, particularly with stone laying

4. Granada pottery

5. Carpentry, particularly hand-made and designer furniture (bargueños, jamudas)

6. Dressmaking, preparation of patterns

7. Sculpting and carving in marble and wood

8. Hand-woven carpets

9. Stained glass and lamps

10. Crochet work, lacemaking

11. Bronze and glass craftsmanship, specifically in Granada lamps

12. Iron and copper forging, railings and decorations

13. Hand-made Alpujarra textiles, design and production

14. Goldsmithing and jewelry making

15. Embroidery using tule and gold

16. Graphics arts, sign painting, illustrations, design

17. Other products of construction work: plastering, from carpentry

This list, although it is not totally exhaustive, is a good representation of the work and cultural traditions which were at their cultural zenith during our parents' generation and are much less evident today.

By way of an hypothesis, with respect to objective A, it was expected that we would observe geometrical, topological and measurement therein. With respect to objective B, it was expected that there would be little influence of the school curriculum on their mathematical knowledge prior to specific training or apprenticeship.

Theoretical Foundations and Work Plan

I had a firm conviction that the interpretative methodological approach involved a paradigm that was suitable for many studies in the Didactics of Mathematics and that it cast doubt on the relevance of the positivist approach for explaining current problems. Consequently, this research has emerged from a "context" or natural situation, i.e. the performance of everyday tasks by craftsman when they make their products. The researcher and her collaborators collected the data with the intention of perceiving meaning and widespread relationships in the observed phenomena. This approach, according to Lincoln and Guba (1985), has the advantages of adaptability, immediate processing, holistic capacity, and possibilities for clarifying responses and detecting those which have unusual or idiosyncratic characteristics.

I used direct qualitative techniques for collecting data: participatory and non-participatory observation, interviews, and abbreviated professional case studies. I began by working out an outline of the aspects to be borne in mind during the observation and for predicting the contents to be included, worked out on the basis of the theoretical knowledge of the scenarios to be observed. The semi-structured interviews of the informants, who were the most expert or experienced craftsmen, contained the request that they should tell their own professional story, emphasizing the initial training period and the stages of greatest difficulty. The final interviews, since there were initially informal contacts, were audio or video recorded, so that they could be later analyzed. These audiovisual techniques may be considered to be included under the heading of direct techniques, since they were obtained directly by the researcher. The work plan covered two phases, which were developed over four academic years (1989-1993), according to a qualitative time series design. The sample was divided into two smaller samples, the first in those professions numbered 1 to 10 in Table 1, and the second involved those numbered 11 to 17, and 1, 3 and 8 for more profound study.

The inductive analysis of the data, as a first step in processing, was by means of a plan suggested by Miles and Huberman (1984).

The next, highly time consuming step, was that of reducing these in accordance with the most important objectives of the study. The third step was the display of the abridged or simplified data.

In order to break down the data I chose to select and simplify them by making the initial reduction by centering my attention on one topic: mathematics. Therefore, all other information was ignored. The relationships between the different sequences of the whole process, as well as the processing of the data are cyclical and not linear as in positivist frameworks, which causes a partial advance in aspects which at the same time shall reinforce those that follow.

There may be many ways of displaying the information: matrices, double entry tables, diagrams,

etc. I chose double entry tables as most appropriate in this case.

A Presentation of Some Results

For reasons of space, I cannot go into detailed descriptions of each of the crafts that was analyzed. I shall present a table (Table 2) with the relationships between the mathematical contents discovered and the craft from the first phase of the study, using for this the numbers from Table 1. The mathematical contents discovered are labeled as follows:

A. Shapes: interior, exterior and borders

B. Angles and movements, symmetry, translations, turns. Axes, planes, centers of symmetry, guide vectors.

C. Tessellations of the plane and space

D. Similarities and dilations. Thales' Theorem

E. Measurement and units. Optimization of amounts under given conditions. Relationships between length, surface area and volume.

F. Two-dimensional portrayal of three-dimensional space. Making flat designs in space. Maps and graphs. Surface areas of turns.

G. Theorem of Pythagoras. Applications.

H. Specific and incorporated symbols, graphic languages.

Table 2

Relationship between the Crafts and the Mathematical Content

	A	B	C	D	E	F	G	H
1	x	x	x		x			x
2	x	x			x	x
3	x	x	x	x	x		x
4	x	x				x
5	x	x			x		x
6	x	x		x	x	x
7	x	x			x	x
8	x	x		x	x
9	x	x	x
10	x	x	x					x

It may be observed that craft No. 3, landscaping and stone laying, is the one which has the highest number of relationships with the mathematical contents. No. 6, dressmaking, follows it with five. As may be observed, the geometrical shapes and symmetry are concepts applied in all the crafts, whereas only in crafts 1 and 10 is there any specific symbolization. However, the most interesting point is hidden when the summarized data is presented. For example, the different uses of symmetry and tessellation which include an incredible scope, ranging from the designs and patterns for stonework and sculptures to controlling the weaving process for carpets, which is carried out by means of copying the work of another more experienced person, who follows the design. The inlays for the inlaid work, using millimetric measurements, are astounding given their great accuracy, whereas in stonework the mastering of large areas is required.

The use of symbolization by means of areas with different designs and the interpretation of plans, as well as the creations of designs, used in landscaping, carpentry and brickwork, are crafts with a symbolization and interpretation which I consider to be a form of Ethnomathematics. About 90% of the craftsmen were not aware of their use of geometrical elements or of calculations and measurements which are related to Mathematics. They think that they do not need any training in this field and believe that they only need practical training with a master craftsman. The teaching relationship involved is based on authority and uses as its principal teaching techniques simple observation and the copying of actions.

References

Lincoln, V. and Guba, E. (1985). Naturalistic Inquiry. Beverly Hills: Sage.

Miles, M. and Huberman, A. (1984). Drawing valid meaning from qualitative data: Toward a shared craft. Educational Researcher, 13, pp. 20-30.