Subject:

Drawing
(Myths and legends)

Drawing (Social aspects)

Drawing (Social aspects)

Authors:

Thirumurthy, Vidya

Simic-Muller, Ksenija

Simic-Muller, Ksenija

Pub Date:

09/01/2012

Publication:

Name: Childhood Education Publisher: Association for Childhood Education International Audience: Academic; Professional Format: Magazine/Journal Subject: Education; Family and marriage Copyright: COPYRIGHT 2012 Association for Childhood Education International ISSN: 0009-4056

Issue:

Date: Sept-Oct, 2012 Source Volume: 88 Source Issue: 5

Topic:

Event Code: 290 Public affairs

Geographic:

Geographic Scope: India Geographic Code: 9INDI India

Accession Number:

305838937

Full Text:

[ILLUSTRATION OMITTED]

It is 5:00 a.m. in the morning in Thanjavur, India. Sita wakes up to the call of a rooster. As she gets up from her bed, her 2-year-old son and 3 1/2-year-old daughter follow her. After quickly brushing her teeth and washing her face, Sita reaches for a small dish containing white chalk powder with a gritty feel. With her children in tow, she carries the dish and a pail of water to the threshold of the house. Setting the dish and pail of water to the side, she sweeps to even the muddy ground at the entry to the house. Then, she lifts up the pail and showers the ground with water to purify it. Once the dust settles, Sita scoops out some of the white powder and places it in her left palm. She pinches a little bit of the white powder between her right thumb and index finger. As she rubs her fingers, the white powder flows evenly and steadily from between her fingers to the ground in dots (pulli). Sita frequently pinches the white powder from the dish. Her movements are swift and it appears as though her fingers kiss the ground. Soon, she has nine equidistant dots placed in every row and column (9 x 9) and the infrastructure for her kolam is ready. With the framework of dots complete, she uses a steady hand to draw a curvilinear line (kambi) with the same powder, moving her fingers forward and never going over the same line again, weaving in and around the dots to make sure every dot is encased. Against the dark brown, wet ground, the kolam in white stands out. The completed drawing took Sita about 10 minutes. As she worked, the children participated peripherally, occasionally grabbing the white powder and attempting to mimic their mother.

Each day begins with the purification of the threshold, followed by creation of the kolam drawing. These geometric drawings adorn the ground at the entrance of even the humblest of homes in south India, creating an aesthetic local social space (Laine, 2009). As hundreds of traditional kolam designs exist, the same kolam seldom is repeated in a month. The conscious and routine female activity of creating the kolam takes place early each morning without fail. Through peripheral participation (Lave & Wengar, 1991) as well as direct engagement in this pursuit, young children are nurtured into their cultural way of life.

WHAT IS KOLAM?

In the art form of kolam, dots called pulli are arranged in rhombic, square, triangular, or free shapes, and a single, uninterrupted linear or curvilinear line, called the kambi, intertwines the dots (Yanagisawa & Nagata, 2007). While there are no written or verbally stated rules, Yanagisawa and Nagata have deduced some of the basic rules of pulli and kambi kolam from examining the designs: "(1) Loop drawing-lines, and never trace a line through the same route. (2) The drawing is completed when all points are enclosed by a drawing-line. (3) Straight lines are drawn along the dual grid inclined at an angle of 45 degrees. (4) Arcs are drawn surrounding the points. (5) Smooth drawing" (p. 32). They do point out that some exceptions to these rules are made, although rarely. We would also like to add, based on our observations, a sixth rule: there must be symmetry in the drawings. Girls somehow understand these implicit regulations and operate within the parameters.

In this article, we discuss the significance of this sociocultural activity and provide an analysis of the inherent academic concepts, skills, and dispositions children may gain from observing and participating in the activity. Then, we explore the concept of (ethno)mathematics in kolam and the potential opportunities to learn mathematical concepts and skills. We will briefly discuss the importance of everyday activities to children's learning and provide a theoretical framework and the potential skills--visual, spatial, and algebraic--and academic concepts nested in the kolam activity.

SOCIOCULTURAL CONTEXT AND SIGNIFICANCE OF KOLAM

For generations, women and girls of the household have made these drawings in front of their homes. Although such practices can be found in different parts of India, called by different names (kolam, muggu, or rangoli), it is only in the south that this activity is prevalent, occurring every morning.

Different beliefs about the importance of kolam exist. Some south Indians are welcoming the goddess of wealth, Lakshmi, into their homes for her blessing (Kilambi, 1985). Others believe that performing the kolam before or at sunrise welcomes the Sun god, in hopes that he will shine his blessings on their home. The ground is thus purified with water in preparation for this spiritual drawing. During the course of the day, the kolam is obliterated, and so is redone the next morning.

It [kolam] is usually the first household task that girls learn to perform, and after marriage it is part of women's daily duties.... The kolam is considered to complete the image of the house, both in the sense of a beautiful material image at the entrance, and a notion of a prosperous home where things are in order. (Laine, 2009, p. 59)

These drawings also announce to the world the well-being of the family. For wedding and festive occasions, kolams are elaborate and a red border is added. In contrast, its absence indicates a death in the family.

Kolam exposes children to "historically accumulated and culturally developed bodies of knowledge and skills essential for household or individual functioning and well-beings (Gonzalez, Moll, & Amanti, 2005, p. 133). Mothers, aunts, and neighbors slowly nurture young girls in this activity. At first watching from the sidelines during the early years, girls begin participating when their fine motor skills are developed and coordinated enough to draw these magical images. Over time, mothers completely transition responsibility for this chore to their young daughters. In families with more than one daughter, the girls work together to create the kolam. The novices use chalk pieces or pencils to practice making the drawings until they become proficient. The powder can be easily wiped off to correct mistakes; repeated mistakes and wipes, however, can spoil the image. Therefore, girls capture, encode, and decode the image in their memory with much clarity before reproducing it on the ground.

City dwellers in high-rise buildings may appoint a maid to execute a common kolam for the entire building. Individuals make simpler ones at their own thresholds. On festival days and during the months of December and January, the drawings at the threshold become much more elaborate. As a commemorative celebration of the Sun god in Chennai a major city in the south, in the month of January, a kolam contest is organized by the state on the temple street. The street is blocked off and several large squares are drawn adjacent to each other in which the contestants draw their kolam. A contest for young girls is held simultaneously.

RELATIONSHIP TO MATHEMATICS

Kolam can be called an "ethnomathematical" activity. The field of ethnomathematics examines the mathematical accomplishments of different (typically, nonwestern) cultural groups. Marcia Ascher, an ethnomathematician, notes that while cultural practices are often mathematically rich (e.g., the kolam, the Sona storytelling tradition of the Angola and Congo regions of Africa, and the basketry traditions of the Bora in the Peruvian Amazon), it is unusual for such cultural practices to be accepted and studied in-depth academically, as has been the case with kolam (Ascher, 2002).

Kolam lends itself well to mathematical explorations at all levels. The mathematical aspects of kolam have prompted mathematicians and computer scientists to explore its properties. In fact, the Indian computer scientist Gift Siromoney and his colleagues have written computer programs to generate families of kolam (Siromoney, Siromoney, & Robinson, 1989). Mathematicians have studied the relationships between kolam and such higher-level mathematical concepts as fractals and knot theory. Chenulu (2007) describes a series of lessons she conducted in her 6th-grade classroom in the United States that allowed students to discover kolam's mathematical properties. The lessons included topics in graph theory, algebra, and geometry. Students considered how a kolam could be drawn so that no line is traced more than once; used patterns and reasoned algebraically to discover ways to find the total number of pullis needed in a kolam; and investigated the symmetry of kolam patterns.

Visual Imagery

Kolam is a visual art using images to capture the creator's ideas and thoughts. For the purpose of this article, we define "visual imagery" as captured images with details and intricacies, and "visual memory" as the ability to recall something after the object is removed from sight ("Visual Memory Skills," 2007). The "imagery is particularly useful where the need for processing is high" (Lowrie & Kay, 2001, p. 248). Visual imagery is a skill needed when completing tasks that require geometric reasoning, even if the tasks themselves are not geometric in nature. Visual imagery is central to the transformation of concepts and ideas (Marshall, 2007). When an image is stored in duplicate, superior recall is possible (Winnick & Brody, 1984). A kolam task requires one to store and recall images. Before reproducing the kolam on the ground, children first call up an image in their visual imagery, and then register the image in their visual memory in terms of pulli and kambi, parsed constituent units, and symmetry. The dual coding theory (Paivio, 1986) explains how the visual image is received, to be coded by a dual coding system (verbal and visual), which makes recall easy. Thus, one could argue that kolam possibly aids in the development of the visual and verbal imagery systems.

In elementary mathematics, creating a visual image of a problem greatly enhances children's problem-solving skills. Consider this story problem: "A snail is climbing out of an 8-foot deep hole. Every day the snail climbs 3 feet, and every night, it slides 2 feet. How long will it take the snail to climb out of the hole?" Without visualizing the problem, a child may give the incorrect answer of eight days. Drawing a diagram, however, will reveal the answer to be six days, as the child will see that the snail will be able to leave the hole on the sixth day before sliding down 2 feet again.

Drawing diagrams is an integral part of kolam, present in all aspects of re-creating pulli arrangements and kambi placement. Therefore, drawing diagrams to solve a mathematical problem like the one above should be a familiar notion to a kolam practitioner.

Blueprint for Thinking

The kolam ritual is a complex activity that engages children physically and cognitively, challenging them to visualize/recall the kolam and figure out ways to connect the dots, and allowing them to practice these skills in an everyday activity (Gauvain, 2005). Rogoff (1982) argues that "in everyday situations, thought is in the service of action" (p. 7); in the case of kolam, such thinking occurs when children develop the infrastructure of kolam dots and how to develop the pattern. Even though the activity may appear mechanical, such activities, one could argue, can help children develop a blueprint for thinking. For example, the visual imagery involved in developing the kolam can assist them in developing spatial thinking.

In addition, membership in the practicing community and a joint focus of adult and child mediate interaction among the members of a culture and their physical world, which, in turn, affects the psychological functions of the participating individuals, contributing to the individual's proficiency (Wertsch, 1991). Thus, thinking is viewed as a shared activity.

Kolam provides innumerable opportunities for children to think mathematically with support from adults and skilled peers. In particular as they graduate from one level to the next in their apprenticeship community of female relatives and peers, children may learn to employ their visual imagery and memory (without which they cannot recall what they observed previously), as well as their spatial and algebraic reasoning, with increasing sophistication (e.g., when children exercise visual imagery to develop a complex framework of dots and consider the various possible options for connecting them). Children can use these skills to refine their mathematical reasoning.

Spatial Reasoning

Manipulating the visual imagery and seeing the "between-ness" in two things requires spatial skills, those that are constructed mentally or internally (Mackenzie, 1980). Spatial skills are those required to understand relationships between objects, and to manipulate them (Tartre, 1990). Spatial reasoning is considered a predictor of success in science, technology, engineering, and mathematics (STEM), and research shows that students with high spatial skills are more likely to be interested in mathematics and science, and to pursue advanced degrees in these fields (Newcombe, 2010). Spatial skills can be developed through frequent exposure to certain tasks. As reproducing the kolam from memory involves manipulation and rotation of the pulli and kambi and the ability to see geometric patterns and symmetry, kolam creation is especially important for girls in India, who learn complex spatial relationships through apprenticeship from their female relatives and community members. Visual imagery and spatial reasoning are inseparable in the kolam activity.

[FIGURE 1 OMITTED]

An analysis of kolam designs confirms that the complex designs are often built from constituent basic units, joined after a transformation (Ascher, 2004). Interviews with kolam practitioners show that they remember patterns by directions, knowing which way to move or rotate to get to the next step (Siromoney et ah, 1989). Rotation and translation in the basic kolam design is demonstrated in Figure 1 (Ascher, 2004, p. 172).

The basic design (b) is rotated 90 degrees clockwise around the pulli, enclosed by its "tail loop," and translated (i.e., shifted) two distances between pullis in the direction of the tail to obtain its next position (c). The process is repeated three times. Women and girls creating the designs will not memorize each line in the finished design, but only the basic unit, the rotation patterns, and the actions needed--in this case, rotation and translation (and adding a border at the end). An unskilled observer sees only the encasement of dots and the number of dots, whereas the practitioner immediately decomposes, and distinguishes, the rotation and the inherent patterns.

The creation of kolam designs requires algebraic as well as spatial reasoning. Many kolam designs are recursive families: a pattern can be generated in arbitrary sizes, by either extending the basic unit in a uniform way, or merging multiple copies of a smaller design to form a more complex one. The family of designs known as Anklets of Krishna (Asher, 2004, p. 173), shown in Figure 2, is an example of a recursive family of kolam: design (b) consists of four copies of design (a), design (c) consists of four copies of design (b), and so on. ]he process can continue indefinitely, as the basic unit (a) can be translated repeatedly to obtain the more complex designs (b) and (c). The practitioner only needs to remember the basic design, and how to connect copies of the previous design to form the next one. Through this process, she is engaging in algebraic reasoning.

The publication Principles and Standards of School Mathematics (National Council of Teachers of Mathematics [NCTM], 2000) states that algebra should be taught in the mathematics curriculum as early as kindergarten. The document also argues that the primary goal in algebra instruction should be to enable all students "to understand patterns, relations, and functions" (NCTM, 2000), rather than mere manipulation of symbols. As evidenced in the previous example, the kolam activity introduces algebra concepts in a meaningful way, even to very young children. Consider, for comparison, this problem from the NCTM Standards, suggested for use with 3rd- to 5th-grade students, with the purpose of developing algebraic sense (NCTM, 2000):

For example, a teacher might ask students to describe patterns they see in the "growing squares" display [see Figure 3] and express the patterns in mathematical sentences. Students should be encouraged to explain these patterns verbally and to make predictions about what will happen if the sequence is continued.

The solution to this mathematical problem and the creation of the Anklets of Krishna design are based on one of the foundational processes of algebra, if not all of mathematics--that of recognizing and generalizing patterns. The kolam artist's creative process undoubtedly and serendipitously develops her algebraic reasoning.

CONCLUSION

The ethnomathematical activity of kolam is an everyday ritual performed by thousands of women and girls in South India. Practitioners may not be conscious of its mathematical intensity, as they focus primarily on the art. Over the last two decades, however, researchers have begun showing interest in studying this art form for mathematical interests. Yet it is seldom explored for its mathematical thought by either educators or by its practitioners. Consequently, creating or analyzing kolam designs is not common in mathematics classrooms, in India or elsewhere. Therefore, our recommendation to mathematics teachers, especially those of Indian children (including children of Indian descent, as well as children from countries where African Sona is practiced), is to incorporate kolam activities into their curriculum. Teachers of Indian children hold the power of unleashing a new level of understanding of mathematical concepts in their students by connecting classroom mathematics to the funds of knowledge they bring from participating in their cultural group (Gonzalez et al., 2005).

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Being introduced to kolam is beneficial for non-Indian children as well. Children often feel disconnected from the mathematics they are learning in school and question its relevance to their daily lives. Chenulu (2007), a teacher of Indian origin, has successfully demonstrated that it is possible for teachers to introduce children to mathematical practices of different cultures through ethnomathematics and expose the deliberate connection between art and mathematics. "Them are numerous ways teachers can use kolam-inspired activities to make mathematics more relevant and engaging, yet rigorous. Chenulu worked in collaboration with the art teacher, who introduced kolam in art class prior to the mathematics unit, and thus made a meaningful connection between two curricular strands.

As the field of ethnomathematics has succeeded in showing, mathematical accomplishments of non-Europeans abound, and teachers of mathematics should share these accomplishments with their students in an effort to broaden understanding of contributions by various cultural groups. Finally, children's everyday sociocultural activities should have entrenched academic concepts, skills, and dispositions. As education around the world becomes more accessible, specifically to girls and to children who are the first in their family to attend school the need for researchers and teachers to uncover the treasure children bring from home has never been greater. It is important for educators to anchor new learnings to what children bring from home so learning becomes an enjoyable and meaningful process.

References

Ascher, M. (2002). The kolam tradition. American Scientist, 90(1), 56-63.

Ascher. M. (2004). Mathematics elsewhere: An exploration of ideas across cultures. Princeton, N J: Princeton University Press.

Chenulu, S. (2007). Teaching mathematics through the art of kolam. Mathematics Teaching in the Middle School, 12(8), 422-428.

Gauvain, M. (2005). Sociocultural contexts of learning. In A. E. Maynard & M. I. Martini (Eds.). Learning in cultural context (pp. 9-40). New York. NY: Kluwer Academic/Plenum Publishers.

Gonzalez, N., Moll. L., & Amanti, C. (2005). Funds of knowledge: Theorizing practices in households and classrooms. Mahwah, NJ; Lawrence Erlbaum.

Kilambi, J. K. (1985). Toward an understanding of the Muggu: Threshold drawings in Hyderabad. Anthropology and Aesthetics, 10, 71-102.

Laine, A. (2009). Complementarity between art and anthropology experiences among kolam makers in South India. Suomen Antropologi: Journal of the Finnish Anthropological Society, 34(2), 58-69.

Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge, England: Cambridge University Press.

Lowrie, T., & Kay, R. (2001). Relationship between visual and nonvisual solution methods and difficulty in elementary mathematics. The Journal of Educational Research, 94(4), 248-255.

Mackenzie, J. (1980). Reasoning and logic. Synthese, 79(1), 99-117.

Marshall. J. (2007). Image as insight: Visual images in practice-based research. Studies in Art Education: A Journal of Issues and Research, 49(1). 23-41.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

Newcombe. N. S. (2010). Picture this: Increasing math and science learning by improving spatial thinking. American Educator, 34(2), 29-35.

Paivio, A. (1986). Dual coding and episocid memory: Subjective and objective sources of memory trace components. In F. Klix & H. Hafgendorf (Eds.), Human memory and cognitive capabilities: Mechanisms and performances (pp. 225-236). Amsterdam. The Netherlands: Elsevier Science.

Rogoff. B. (1982). Integrating context and cognitive development. In M. E. Lamb & A. L. Brown (Eds.), Advances in Developmental Psychology, Volume 2 (pp. 125-170). Hillsdale, NJ: Lawrence Erlbaum.

Siromoney, G., Siromoney. R., & Robinson, T. (1989). Kambi kolam and cycle grammars. In R. Narasimhan & G. Siromoney (Eds.), A perspective in theoretical computer science: Commemorative volume for Girl Siromoney (pp. 267-300). Chennai, India: World Scientific.

Tartre, L.A. (1990). Spatial orientation skill and mathematical problem solving. Journal for Research in Mathematics Education. 21(3), 216-229.

Visual memory skills. CDA training module. (2007). Charlotte, NC: HighReach Learning.

Wertsch, J. (1991). Voices in the mind. Cambridge, England: Harvard University Press.

Winnick. W. A., & Brody, N. (1984). Auditory and visual imagery in free recall. The Journal of Psychology 118, 17-29.

Yanagisawa, K., & Nagata. S. (2007). Fundamental study on design system of Kolam pattern. Forma, 22, 31-46.

Vidya Thirumurthy is Associate Professor, Instructional Development and Leadership, Pacific Lutheran University, Tacoma, Washington, and a 2011-2012 Fulbright-Nehru Research Scholar in India. Ksenija Simic-Muller is Assistant Professor, Department of Mathematics, Pacific Lutheran University, Tacoma, Washington.

It is 5:00 a.m. in the morning in Thanjavur, India. Sita wakes up to the call of a rooster. As she gets up from her bed, her 2-year-old son and 3 1/2-year-old daughter follow her. After quickly brushing her teeth and washing her face, Sita reaches for a small dish containing white chalk powder with a gritty feel. With her children in tow, she carries the dish and a pail of water to the threshold of the house. Setting the dish and pail of water to the side, she sweeps to even the muddy ground at the entry to the house. Then, she lifts up the pail and showers the ground with water to purify it. Once the dust settles, Sita scoops out some of the white powder and places it in her left palm. She pinches a little bit of the white powder between her right thumb and index finger. As she rubs her fingers, the white powder flows evenly and steadily from between her fingers to the ground in dots (pulli). Sita frequently pinches the white powder from the dish. Her movements are swift and it appears as though her fingers kiss the ground. Soon, she has nine equidistant dots placed in every row and column (9 x 9) and the infrastructure for her kolam is ready. With the framework of dots complete, she uses a steady hand to draw a curvilinear line (kambi) with the same powder, moving her fingers forward and never going over the same line again, weaving in and around the dots to make sure every dot is encased. Against the dark brown, wet ground, the kolam in white stands out. The completed drawing took Sita about 10 minutes. As she worked, the children participated peripherally, occasionally grabbing the white powder and attempting to mimic their mother.

Each day begins with the purification of the threshold, followed by creation of the kolam drawing. These geometric drawings adorn the ground at the entrance of even the humblest of homes in south India, creating an aesthetic local social space (Laine, 2009). As hundreds of traditional kolam designs exist, the same kolam seldom is repeated in a month. The conscious and routine female activity of creating the kolam takes place early each morning without fail. Through peripheral participation (Lave & Wengar, 1991) as well as direct engagement in this pursuit, young children are nurtured into their cultural way of life.

WHAT IS KOLAM?

In the art form of kolam, dots called pulli are arranged in rhombic, square, triangular, or free shapes, and a single, uninterrupted linear or curvilinear line, called the kambi, intertwines the dots (Yanagisawa & Nagata, 2007). While there are no written or verbally stated rules, Yanagisawa and Nagata have deduced some of the basic rules of pulli and kambi kolam from examining the designs: "(1) Loop drawing-lines, and never trace a line through the same route. (2) The drawing is completed when all points are enclosed by a drawing-line. (3) Straight lines are drawn along the dual grid inclined at an angle of 45 degrees. (4) Arcs are drawn surrounding the points. (5) Smooth drawing" (p. 32). They do point out that some exceptions to these rules are made, although rarely. We would also like to add, based on our observations, a sixth rule: there must be symmetry in the drawings. Girls somehow understand these implicit regulations and operate within the parameters.

In this article, we discuss the significance of this sociocultural activity and provide an analysis of the inherent academic concepts, skills, and dispositions children may gain from observing and participating in the activity. Then, we explore the concept of (ethno)mathematics in kolam and the potential opportunities to learn mathematical concepts and skills. We will briefly discuss the importance of everyday activities to children's learning and provide a theoretical framework and the potential skills--visual, spatial, and algebraic--and academic concepts nested in the kolam activity.

SOCIOCULTURAL CONTEXT AND SIGNIFICANCE OF KOLAM

For generations, women and girls of the household have made these drawings in front of their homes. Although such practices can be found in different parts of India, called by different names (kolam, muggu, or rangoli), it is only in the south that this activity is prevalent, occurring every morning.

Different beliefs about the importance of kolam exist. Some south Indians are welcoming the goddess of wealth, Lakshmi, into their homes for her blessing (Kilambi, 1985). Others believe that performing the kolam before or at sunrise welcomes the Sun god, in hopes that he will shine his blessings on their home. The ground is thus purified with water in preparation for this spiritual drawing. During the course of the day, the kolam is obliterated, and so is redone the next morning.

It [kolam] is usually the first household task that girls learn to perform, and after marriage it is part of women's daily duties.... The kolam is considered to complete the image of the house, both in the sense of a beautiful material image at the entrance, and a notion of a prosperous home where things are in order. (Laine, 2009, p. 59)

These drawings also announce to the world the well-being of the family. For wedding and festive occasions, kolams are elaborate and a red border is added. In contrast, its absence indicates a death in the family.

Kolam exposes children to "historically accumulated and culturally developed bodies of knowledge and skills essential for household or individual functioning and well-beings (Gonzalez, Moll, & Amanti, 2005, p. 133). Mothers, aunts, and neighbors slowly nurture young girls in this activity. At first watching from the sidelines during the early years, girls begin participating when their fine motor skills are developed and coordinated enough to draw these magical images. Over time, mothers completely transition responsibility for this chore to their young daughters. In families with more than one daughter, the girls work together to create the kolam. The novices use chalk pieces or pencils to practice making the drawings until they become proficient. The powder can be easily wiped off to correct mistakes; repeated mistakes and wipes, however, can spoil the image. Therefore, girls capture, encode, and decode the image in their memory with much clarity before reproducing it on the ground.

City dwellers in high-rise buildings may appoint a maid to execute a common kolam for the entire building. Individuals make simpler ones at their own thresholds. On festival days and during the months of December and January, the drawings at the threshold become much more elaborate. As a commemorative celebration of the Sun god in Chennai a major city in the south, in the month of January, a kolam contest is organized by the state on the temple street. The street is blocked off and several large squares are drawn adjacent to each other in which the contestants draw their kolam. A contest for young girls is held simultaneously.

RELATIONSHIP TO MATHEMATICS

Kolam can be called an "ethnomathematical" activity. The field of ethnomathematics examines the mathematical accomplishments of different (typically, nonwestern) cultural groups. Marcia Ascher, an ethnomathematician, notes that while cultural practices are often mathematically rich (e.g., the kolam, the Sona storytelling tradition of the Angola and Congo regions of Africa, and the basketry traditions of the Bora in the Peruvian Amazon), it is unusual for such cultural practices to be accepted and studied in-depth academically, as has been the case with kolam (Ascher, 2002).

Kolam lends itself well to mathematical explorations at all levels. The mathematical aspects of kolam have prompted mathematicians and computer scientists to explore its properties. In fact, the Indian computer scientist Gift Siromoney and his colleagues have written computer programs to generate families of kolam (Siromoney, Siromoney, & Robinson, 1989). Mathematicians have studied the relationships between kolam and such higher-level mathematical concepts as fractals and knot theory. Chenulu (2007) describes a series of lessons she conducted in her 6th-grade classroom in the United States that allowed students to discover kolam's mathematical properties. The lessons included topics in graph theory, algebra, and geometry. Students considered how a kolam could be drawn so that no line is traced more than once; used patterns and reasoned algebraically to discover ways to find the total number of pullis needed in a kolam; and investigated the symmetry of kolam patterns.

Visual Imagery

Kolam is a visual art using images to capture the creator's ideas and thoughts. For the purpose of this article, we define "visual imagery" as captured images with details and intricacies, and "visual memory" as the ability to recall something after the object is removed from sight ("Visual Memory Skills," 2007). The "imagery is particularly useful where the need for processing is high" (Lowrie & Kay, 2001, p. 248). Visual imagery is a skill needed when completing tasks that require geometric reasoning, even if the tasks themselves are not geometric in nature. Visual imagery is central to the transformation of concepts and ideas (Marshall, 2007). When an image is stored in duplicate, superior recall is possible (Winnick & Brody, 1984). A kolam task requires one to store and recall images. Before reproducing the kolam on the ground, children first call up an image in their visual imagery, and then register the image in their visual memory in terms of pulli and kambi, parsed constituent units, and symmetry. The dual coding theory (Paivio, 1986) explains how the visual image is received, to be coded by a dual coding system (verbal and visual), which makes recall easy. Thus, one could argue that kolam possibly aids in the development of the visual and verbal imagery systems.

In elementary mathematics, creating a visual image of a problem greatly enhances children's problem-solving skills. Consider this story problem: "A snail is climbing out of an 8-foot deep hole. Every day the snail climbs 3 feet, and every night, it slides 2 feet. How long will it take the snail to climb out of the hole?" Without visualizing the problem, a child may give the incorrect answer of eight days. Drawing a diagram, however, will reveal the answer to be six days, as the child will see that the snail will be able to leave the hole on the sixth day before sliding down 2 feet again.

Drawing diagrams is an integral part of kolam, present in all aspects of re-creating pulli arrangements and kambi placement. Therefore, drawing diagrams to solve a mathematical problem like the one above should be a familiar notion to a kolam practitioner.

Blueprint for Thinking

The kolam ritual is a complex activity that engages children physically and cognitively, challenging them to visualize/recall the kolam and figure out ways to connect the dots, and allowing them to practice these skills in an everyday activity (Gauvain, 2005). Rogoff (1982) argues that "in everyday situations, thought is in the service of action" (p. 7); in the case of kolam, such thinking occurs when children develop the infrastructure of kolam dots and how to develop the pattern. Even though the activity may appear mechanical, such activities, one could argue, can help children develop a blueprint for thinking. For example, the visual imagery involved in developing the kolam can assist them in developing spatial thinking.

In addition, membership in the practicing community and a joint focus of adult and child mediate interaction among the members of a culture and their physical world, which, in turn, affects the psychological functions of the participating individuals, contributing to the individual's proficiency (Wertsch, 1991). Thus, thinking is viewed as a shared activity.

Kolam provides innumerable opportunities for children to think mathematically with support from adults and skilled peers. In particular as they graduate from one level to the next in their apprenticeship community of female relatives and peers, children may learn to employ their visual imagery and memory (without which they cannot recall what they observed previously), as well as their spatial and algebraic reasoning, with increasing sophistication (e.g., when children exercise visual imagery to develop a complex framework of dots and consider the various possible options for connecting them). Children can use these skills to refine their mathematical reasoning.

Spatial Reasoning

Manipulating the visual imagery and seeing the "between-ness" in two things requires spatial skills, those that are constructed mentally or internally (Mackenzie, 1980). Spatial skills are those required to understand relationships between objects, and to manipulate them (Tartre, 1990). Spatial reasoning is considered a predictor of success in science, technology, engineering, and mathematics (STEM), and research shows that students with high spatial skills are more likely to be interested in mathematics and science, and to pursue advanced degrees in these fields (Newcombe, 2010). Spatial skills can be developed through frequent exposure to certain tasks. As reproducing the kolam from memory involves manipulation and rotation of the pulli and kambi and the ability to see geometric patterns and symmetry, kolam creation is especially important for girls in India, who learn complex spatial relationships through apprenticeship from their female relatives and community members. Visual imagery and spatial reasoning are inseparable in the kolam activity.

[FIGURE 1 OMITTED]

An analysis of kolam designs confirms that the complex designs are often built from constituent basic units, joined after a transformation (Ascher, 2004). Interviews with kolam practitioners show that they remember patterns by directions, knowing which way to move or rotate to get to the next step (Siromoney et ah, 1989). Rotation and translation in the basic kolam design is demonstrated in Figure 1 (Ascher, 2004, p. 172).

The basic design (b) is rotated 90 degrees clockwise around the pulli, enclosed by its "tail loop," and translated (i.e., shifted) two distances between pullis in the direction of the tail to obtain its next position (c). The process is repeated three times. Women and girls creating the designs will not memorize each line in the finished design, but only the basic unit, the rotation patterns, and the actions needed--in this case, rotation and translation (and adding a border at the end). An unskilled observer sees only the encasement of dots and the number of dots, whereas the practitioner immediately decomposes, and distinguishes, the rotation and the inherent patterns.

The creation of kolam designs requires algebraic as well as spatial reasoning. Many kolam designs are recursive families: a pattern can be generated in arbitrary sizes, by either extending the basic unit in a uniform way, or merging multiple copies of a smaller design to form a more complex one. The family of designs known as Anklets of Krishna (Asher, 2004, p. 173), shown in Figure 2, is an example of a recursive family of kolam: design (b) consists of four copies of design (a), design (c) consists of four copies of design (b), and so on. ]he process can continue indefinitely, as the basic unit (a) can be translated repeatedly to obtain the more complex designs (b) and (c). The practitioner only needs to remember the basic design, and how to connect copies of the previous design to form the next one. Through this process, she is engaging in algebraic reasoning.

The publication Principles and Standards of School Mathematics (National Council of Teachers of Mathematics [NCTM], 2000) states that algebra should be taught in the mathematics curriculum as early as kindergarten. The document also argues that the primary goal in algebra instruction should be to enable all students "to understand patterns, relations, and functions" (NCTM, 2000), rather than mere manipulation of symbols. As evidenced in the previous example, the kolam activity introduces algebra concepts in a meaningful way, even to very young children. Consider, for comparison, this problem from the NCTM Standards, suggested for use with 3rd- to 5th-grade students, with the purpose of developing algebraic sense (NCTM, 2000):

For example, a teacher might ask students to describe patterns they see in the "growing squares" display [see Figure 3] and express the patterns in mathematical sentences. Students should be encouraged to explain these patterns verbally and to make predictions about what will happen if the sequence is continued.

The solution to this mathematical problem and the creation of the Anklets of Krishna design are based on one of the foundational processes of algebra, if not all of mathematics--that of recognizing and generalizing patterns. The kolam artist's creative process undoubtedly and serendipitously develops her algebraic reasoning.

CONCLUSION

The ethnomathematical activity of kolam is an everyday ritual performed by thousands of women and girls in South India. Practitioners may not be conscious of its mathematical intensity, as they focus primarily on the art. Over the last two decades, however, researchers have begun showing interest in studying this art form for mathematical interests. Yet it is seldom explored for its mathematical thought by either educators or by its practitioners. Consequently, creating or analyzing kolam designs is not common in mathematics classrooms, in India or elsewhere. Therefore, our recommendation to mathematics teachers, especially those of Indian children (including children of Indian descent, as well as children from countries where African Sona is practiced), is to incorporate kolam activities into their curriculum. Teachers of Indian children hold the power of unleashing a new level of understanding of mathematical concepts in their students by connecting classroom mathematics to the funds of knowledge they bring from participating in their cultural group (Gonzalez et al., 2005).

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Being introduced to kolam is beneficial for non-Indian children as well. Children often feel disconnected from the mathematics they are learning in school and question its relevance to their daily lives. Chenulu (2007), a teacher of Indian origin, has successfully demonstrated that it is possible for teachers to introduce children to mathematical practices of different cultures through ethnomathematics and expose the deliberate connection between art and mathematics. "Them are numerous ways teachers can use kolam-inspired activities to make mathematics more relevant and engaging, yet rigorous. Chenulu worked in collaboration with the art teacher, who introduced kolam in art class prior to the mathematics unit, and thus made a meaningful connection between two curricular strands.

As the field of ethnomathematics has succeeded in showing, mathematical accomplishments of non-Europeans abound, and teachers of mathematics should share these accomplishments with their students in an effort to broaden understanding of contributions by various cultural groups. Finally, children's everyday sociocultural activities should have entrenched academic concepts, skills, and dispositions. As education around the world becomes more accessible, specifically to girls and to children who are the first in their family to attend school the need for researchers and teachers to uncover the treasure children bring from home has never been greater. It is important for educators to anchor new learnings to what children bring from home so learning becomes an enjoyable and meaningful process.

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Vidya Thirumurthy is Associate Professor, Instructional Development and Leadership, Pacific Lutheran University, Tacoma, Washington, and a 2011-2012 Fulbright-Nehru Research Scholar in India. Ksenija Simic-Muller is Assistant Professor, Department of Mathematics, Pacific Lutheran University, Tacoma, Washington.

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