Putting Kolams: Mathematical Thinking in a Women's Art

Putting Kolams: Mathematical Thinking in a Women's Art

Sunita VatukCity College of New York, CUNY, USA

[email protected]

Most visitors to Tamil Nadu are struck by the ubiquity of designs made of white quartz dust and/or rice powdernewly made each morning on the ground outside of homes. They are called kolams, and are “put” almostexclusively by Hindu, and some Christian, women. Observant mathematicians and math educators areimmediately struck by links between kolams and various branches of mathematics that include (but are notlimited to) school mathematics or the practical mathematics of everyday life; kolams suggest introductory topicsin discrete math, number theory, abstract algebra, sequences, fractals, and computer science alongside those inalgebra and arithmetic. Early work connecting kolams with research in mathematics and computer science wasconducted by Siromoney, Siromoney, and Krithivasan (1974). In addition, the learning of kolams can be aidedby mathematical knowledge and techniques, suggesting that proficiency in kolams might be accompanied by anaffinity for mathematics.

In contrast with this outsider's view, there is no doubt that for most Tamils, including the vast majority of thewomen who put them, kolams are not considered to be mathematical objects. This interview-based studyexamines the relationship between the thinking of the women who put kolams and mathematical thinking. Due tospace constraints, a discussion of the cultural context of kolams is not included in this paper. The reader curiousabout kolams outside of this paper's narrow focus can find more information elsewhere, for example in Layard(1937), Nagarajan (1993), Dohmen (2004), Mall (2007), and Laine (2009).

Motivation and Research Question

The question of what it means to think mathematically has been revisited many times by many thinkers in manyfields. This paper is a very early report on an ongoing empirical study conducted in the spirit of grounded theoryand, as such, cannot hope to answer that question in general.

The larger project does hope to contribute to an understanding of whether any part of mathematical thinking (asunderstood by mathematicians and math educators) has resonance for women involved in a mathematically richculturally-embedded practice squarely outside of an academic setting, and whether the way these women thinkabout what they do (as opposed to the objects they make) has resonance for mathematicians or math educators.Such an understanding may have broader implications, perhaps for mathematicians who wish to be effectiveinstructors to non-mathematicians and who might struggle with what it means to think mathematically outside ofthe highly technical arena of research. Paulus Gerdes wrote about his encounter with the Angolan sand drawingscalled sona: “[e]ver since my first personal contact with the sona, I 'felt' – trained in Europe as a researchmathematician – that I was dealing with mathematical ideas.” (Gerdes, 2001). Mathematicians who have lookedat kolams have a similar reaction. However, for the most part, the voices of the women who make kolams havenot been a significant part of the mathematical discussions around them.

This paper is based on the analysis of initial open-ended interviews of over 40 kolam experts and a small numberof teachers and mathematicians familiar with kolams. It examines a much simpler question: To what degree do

women who are experts in putting kolams express ideas, demonstrate strategies, and participate in discussions

that are recognisably mathematical, as defined either by math educators or mathematicians?

Tamils who do not themselves put kolams will express admiration for a beautifully put kolam along with theopinion that the main skills involved are hand-eye coordination and memorisation. The initial data indicate thatwhile none of the women spontaneously made conjectures, generalised or proved theorems in ways that look likethe final products of research mathematics, kolams occupy a rich space between the purely functional and thepurely intellectual, and involve many features of mathematical thought. For many women there is a culturalimperative to put a kolam outside her home everyday, but it is a personal choice and a significant commitment tolearn a large number of complex kolams. This is especially true for working women who have little leisure time.

82 Proceedings of epiSTEME 5, India

Context

In the ethnomathematics and math education literatures, philosophical and practical questions have arisen bothabout the view of mathematics as a European creation and about the relationship between research mathematicsas it is conducted in universities, the mathematics taught to undergraduates in technical majors, schoolmathematics as it is taught in K-12, and the mathematics that exists outside of academic settings. Gerdes (2006)and Eglash (1999) highlight the mathematical knowledge that can be gained through exploring traditional designand the process of making patterns by hand. Researchers and educators such as Selin and D’Ambrosio (2000),Ascher (2004), Frankenstein and Powell (1997), and Joseph (2010) influenced views of the history ofmathematics and led to the wider use of materials from non-European cultures in math classrooms. While someeducators continue to defend the teaching of school mathematics in terms of providing valuable tools foreveryday life, others such as Lave (1988), Nunes, Carraher, and Schliemann (1993) and Dowling (1998) havehighlighted the disconnect between the math of everyday life and school mathematics. Some, such as Evans andTsatsaroni (2000) and Lockhart (2009), have also made the case that promoting math as a utilitarian subject hurtsthe discipline of math. And many in the mathematics education community, such as Driscoll (2007) have beenpromoting teaching mathematics as “habits of mind” rather than emphasising specific topics.

Methodology

A series of open-ended interviews with kolam experts is underway. Thus far, 49 women have been interviewed.Eight are high-caste (Brahman), approximately two-thirds grew up in villages, 7 still live in villages, and theyhave between zero years of formal schooling to post-graduate degrees. Most are either housewives or domesticworkers, a few are teachers, college students, office workers, and professors. They were all identified as expertsthrough kolam contests, neighbours, family members, employers, or others in the study. The interviews aresimilar to clinical interviews developed as part of research in math education (Ginsburg, 1981), starting with asimple request to teach the researcher some everyday kolams that the interviewee likes to put, followed up withquestions such as “Are any of these kolams alike?” “Can you make another one like this one that is smaller orbigger?” or “Can you finish that kolam in another way?” The women were interviewed for between one andthree hours depending on their availability, level of interest and knowledge. The most skilled women could draw(or put in powder) on the order of 50 different kolams from memory in a single hour. Follow-up interviews withthe most skilled participants are ongoing.

Where possible, the interviews were conducted either with a video camera, or using a pen that links an audio fileto pen-strokes. This allows for the sequence in which different parts of the kolam are drawn, and the facility andspeed with which they are drawn, to be part of the data. The interviews were transcribed in the original Tamiland English and then translated into English. Notes about which kolam was being drawn at which points in theinterview were added to the transcripts, along with any observations about how it was drawn.

Data Analysis

The initial interviews are being coded to create more focused research questions. A set of structured follow-upinterviews based on this analysis are being piloted. The initial analysis is inspired in part by Lobato (2003),making a distinction between an actor-oriented analysis and an observer-oriented analysis.

The first approach is an attempt to understand as clearly as possible how the creators think. In this analysis, thefocus is on language the women use, the manner in which they put the kolams, their discussions about kolamswith relatives and neighbours, and so on. This is not without pitfalls: the mere fact of a researcher paying suchclose attention to something they are not used to thinking of as important has an impact. Some of theinterviewees say it is the first time they have thought about one or more of the questions.

The second approach is to analyse the kolams from the perspective of a university-trained mathematician. Theresearcher has been learning to draw kolams from notebooks, videos and photographs, keeping track of when sheis or is not consciously using mathematics to learn and analyse them. Interviews with math teachers and othermathematicians will eventually inform this part of the analysis.

Preliminary Observations

During the initial round of data collection and analysis some potentially interesting lines of inquiry havesurfaced. Four of these are discussed below in detail; and some others are briefly noted in the final section. Forthe sake of brevity, the issues discussed here are limited to a type of kolam called kambi kolam. An array of dots(pulli) is placed on the ground, and then one or more curves (kambi) are put around the dots according to fairly

Putting Kolams: Mathematical Thinking in a Women's Art 83

strict, but unarticulated, rules. All but a handful of women claimed that these kolams provided the greatestmental challenge, and mathematicians familiar with kolams find the richest possibilities in them.

Structure: The Importance of the Pulli

Those who are not practitioners of kolams, whether from Tamil Nadu or from elsewhere, often miss theimportance of the array of pulli, focusing instead on the beauty and complexity of the designs formed by thekambi. However, for the women who make them, the pulli, and the structure they provide, are paramount. Thisimportance reveals itself in several ways.

One participant explained “if the dots aren’t right, the kolam won’t come,” and this sentiment was repeated inone form or another by almost everyone interviewed. In fact, often when a mistake is made while putting acomplex kolam, the woman will say “there is a mistake in the dots,” whether or not that is the case. Only once(in over 50 such incidents analysed so far) did a woman go on to correct the dots and finish the kolam; usually akolam was abandoned if the woman perceived the dots to be incorrect.

In another context, when asked whether there are other kolams similar to those a woman has put in theresearcher's notebook, the reply was often “yes, there is another one, but I don’t remember the dots.” This wasoften followed by a refusal to try to recreate the kolam.

The dot arrays also loom large in the responses of these experts when asked about which kolams are alike. It isthe first (and often only) reason given for saying that two kolams are not alike. The converse does not hold: it isnot enough for two kolams to have the same dot array to be considered alike. In such a case, often she will say“the pulli are the same, but the models are different.” (The English word “model” is commonly used, and noteasy to define.)

For many mathematicians, the array of dots can be viewed as a problem for which a particular kolam is asolution. For example: “Is it possible to put a single kambi kolam with 90-degree rotational symmetry startingwith a 5 by 5 array of dots?” Though none of the women expressed the challenge in such an explicit way, anunexpected outcome from when the researcher gave ST some pages with pre-printed dot arrays is suggestive.ST did not attend school and is not accustomed to pencil and paper. She had trouble drawing straight dots in thenotebook which was adding to her difficulties in drawing the kambi. Once given the pre-printed sheets, sheexpressed delight, and started to put kolams she had never put before. She said that the interview had “broughtsomething out of her” and expressed sudden enthusiasm for a follow-up interview despite her exhausting workschedule as a live-in maid. After that day, giving out pre-printed dot arrays towards the end became part of someopen-ended interviews, and is part of the structured interview being piloted.

Classification: Tilings and Sequences

During the interviews, the researcher often asked whether any kolams on given pages were the same, or whetherthe interviewee could put another kolam of the same type as one already in the notebook. The questions come upnaturally in the context of how to make a kolam bigger for the Tamil month of Margazhi, when it is important toput large kolams each day. These simple questions gave insight into how women classify kolams.

There are two distinct approaches that came up regularly. The first is to repeat a basic unit in a kind of tiling. Forthe women who expressed the most enthusiasm for that approach, the structure of the tiling was clearly separatedfrom the unit used. For example, SK put nine different kolams starting with a 5 by 5 square array of dots, andsaid that any one of them could be expanded to fill as big a space as desired, with the corner dots shared bytouching tiles. She showed the researcher one such example, using what she (and the researcher) considered tobe the simplest basic unit: “This one is easiest, so madam can learn it.”

Another common practice of enlarging kolams creates single larger units rather than tiling the same small unit.One of the simplest such examples is shown below. The figures show three kolams that were usually labeled 4-dot, 8-dot, and 12-dot versions of the same “model”. (The numbers refer to the number of dots in the longestrow.) During a gathering with another mathematician, a group of math teachers and a group of cleaners at aschool, one of the cleaners put the 24-dot version in powder on the ground. After she was finished, the researcherasked her if there was a smaller kolam of the same type. She put the 4-dot version, and claimed it was thesmallest. Both mathematicians were surprised by this answer, having decided beforehand that the 8-dot version“ought to be” the smallest example of that type. The mathematicians were assuming that the smallest kolamshould carry necessary information


Figure 1. 4-dot version, 8-dot version and 12-dot version

about how to “go onto the next size.” At first glance the 4-dot version suggested the next in the sequence shouldbe a larger square, rather than having the stepping stone array of the other kolams in the sequence. Uponreflection, the researcher decided that perhaps for the woman who put the 4-dot version, the smallest kolam wasnot necessarily the generator of the larger versions; rather she appeared to be following an algorithm for goingdown from 24 to 20 and so on down to the 4-dot version.

Since the women do not care about creating a rigorous classification scheme, as a mathematician might, theirclassifications are not consistent across types of kolams. There are examples of sequences of kolams where thesmallest exemplar differs from the rest and examples of sequences in which they do not. In some “models” thesmallest may even have a different symmetry group from the rest of the sequence, even though the symmetrygroup is the same for all larger elements of the sequence.

Mathematical Dispositions

Exploring the question of when larger kolams are the same as smaller kolams led to an extended interaction witha group women in a village. The 9-dot version of the kolam below is fairly common; the 15-dot version muchless so, and to date no woman in the study has drawn one larger than 15. NT's attempt inspired the researcher tomake what she thought ought to be the next three in the sequence, with 21, 27, and 33 dots. To do this, she used akind of “dual” that shows the structure of the kolam and acts as a guide, like that traced in the right hand drawingon the previous page. After KA tried to draw a 21-dot version that had 17 closed curves instead of 1, and threeother women in the village all tried, unsuccessfully, to draw a 21-dot version with only 1 curve, the researchershowed her attempts and asked them if they were “correct.” The women discussed the different versions atlength and ultimately decided that the researcher's version was correct. The discussion was taken very seriouslyby all three women, and the decision, once reached, was accepted by all.

Although this was an exceptional sequence of events, several features of the discussion were seen in otherinterviews. In particular, the idea that some kolams are correct and others are not, while not universal, iscommon. And the persistence with which they approach often difficult tasks is also common, with women ofteninsisting on fixing difficult kolams, despite sometimes impatient husbands and children waiting for dinner to becooked. Many of the women said that they preferred one-curve kolams to the (often easier) multi-curve kolams,and preferred to put the one-curve kolams all at once rather than in pieces, because of the feeling of satisfactiongained from succeeding in a challenging task.


Figure 2. 15-dot kolam and 15-dot kolam with tracing of “dual”

In addition, during the process of building larger kolams out of smaller units some women used a strategy to jointogether modules that this researcher and several other mathematicians independently “discovered” as a way tocreate variations and prove some simple theorems.

Doing vs. Saying

One of the methodological issues that needs to be grappled with in this study is how to balance the evidencefrom what women do with what women say. Understandably, most are much more comfortable drawing kolamsthan they are talking about them, and this discomfort is evident in the relatively sparse language they use. Forexample, when a woman is trying to remember a complex kolam that she has not put since Margazhi, there isstrong evidence that she is aware of the symmetry group. Of over 1000 kolams put in the researcher’s notebook,and over 1500 photographed in situ, only a handful show what the researcher identified as mistakes insymmetries, and almost all of those were made during improvisations of complex kolams. Usually, if a half-remembered kolam has four-fold rotational symmetry, a woman will typically start by drawing a motif at onecorner until she is not sure how to continue. Then she will repeat that at the other corners before working outhow to connect the kambi in the middle. If there is mirror symmetry as well as rotational symmetry, she breaksthe process down even more: drawing one half of the motif at a corner until she runs of out steam, then drawingthe mirror image before moving on as before. Many women acknowledge this as a strategy for recreatingkolams, even as they express a preference for putting them as a single curve. ST explained it thus when askedwhy she was rotating the notebook and putting a kolam in pieces: “I used to be able to put this kolam in one line,but I haven’t put it for many years.”

However, none of the women distinguish linguistically between rotations and reflections, using general termssuch as “it's the same on both sides” for either situation. An analysis that focuses on language alone may misssomething: suggesting a potential confusion or lack of knowledge that their actions refute. The structuredinterviews should shed light on the specific question about symmetry groups, but the larger methodological issueremains.

A particularly dramatic example of this issue arose in a follow-up interview with VS, a skilled and versatilepractitioner. Several other women appeared to favour a particular structure. One of the simplest such examples isgiven here in two forms, a finished kolam on the left, decomposed into three separate curves on the right. Notethat the lightly-drawn curve is identical to the a 90 degree rotation of the dark curve. The dashed curve is neededto finish the kolam, enclosing the remaining dots.

Of the 46 kolams VS drew in her initial interview, only 4 were structured in this way. The researcher asked herhow one such was formed, and she pointed out, with no hesitation whatsoever, that there were exactly threecurves and that “this curve” (the repeated curve) was on “both sides”. When the researcher pointed to anotherfew with that same structure, she again pointed to the rotated curve and said it was on “both sides.” At this point,the researcher asked her if she could draw another kolam “like these” and her response was to draw one that hadthe same dot pattern as the last one discussed, rather than one structured in this fashion. The researcher againpointed to the way the other four were structured, and asked if she thought they were alike in some way. VSagreed they were, but laughed at the idea of discussing it further, saying “we just put the kolams!” She went onto draw a completely unrelated kolam she particularly wanted to show. However, in the hour following this shortconversation, almost a quarter of the kolams VS put had this structure. She had not previously demonstrated a


particular interest in these types of kolams. Therefore, it is tempting to say that what she did (putting thosekolams) is a stronger statement than what she said (that she didn’t think about them in this way).

Figure 3.

Continuing Research

The initial interviews also shed light on the differences between how experts and novices classify kolams,awareness the women had about the unarticulated rules of kambi kolams, additional strategies for rememberingkolams, algebraic thinking involved creating sequences, problem-solving strategies used in creating new kolams,and the disconnect that appears to exist between mathematical thinking displayed in kolam-making and thatlearned or taught in school. These, along with some of the more speculative ideas discussed in this paper are thefocus of ongoing data collection, through structured interviews, observations of groups, and conversations withexperts about mathematicians' understanding of kolams. Interviews with math teachers, mathematicians andmath students, whether kolam experts or not, will add to an understanding of what it means to look at kolamsthrough mathematical eyes.

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Putting Kolams: Mathematical Thinking in a Women's Art

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