ObjectiveTo provide an account of the mathematical thinking involved in playing the uril, and of the pragmatics of construction, negotiation, and validation of theorems-in-action by its players.

SignificanceThe study is significant not only for the understanding the object of study, but for a greater dialogue between ethnomathematics and anthropology.

Studying mathematics from an anthropological perspectiveTheorists in mathematics education have recently acknowledged anthropology as a useful tool for understanding the development of mathematical knowledge.Ethnomathematics as a research program was conceptualized as lying on the border between mathematics, education, history, and anthropology. However, we find the collaborative and interdisciplinary nature initially envisioned for the program is still lacking..A major problem has occurred when specialists in one field have “borrowed” theories from another field. The result has been the application of outmoded ideas to the new field (see Connors, 1990). Unfortunately, those outdated ideas were in great part accepted by the specialists in the new discipline, having shaped many of the current studies in the area. On the other hand: often not being engaged with mathematics in their own culture, anthropologists seldom ask questions with mathematics in mind or may have a restricted view of what is of importance as mathematical thought.

An account of the construction of mathematical knowledge in uril, a Capeverdean game: Theorems-in-action

IntroductionThis project focuses on uril, a traditional game played in Cape Verde. Uril is a type of mancala, the generic name by which the many versions of this board game is known. Mancalas are played primarily in African countries, but also in other regions of the world.

We acknowledge two aspects of mathematical activity:• the process of problem posing, problem solving, and theory building – an activity that often occurs individually.• the process of validation within the group, activity eminently social. (Muniz, 1999)

Analytical framework•Vergnaud (1998): It is in situ that we form our concepts and theories, and although we organize knowledge into theoretical statements concerning a class of objects and situations, these are preceded in human activity by "concepts-in-action" and "theorems-in- action".•Simmel (1971): Society must be considered a reality in a double sense. On the one hand are the individuals, the bearers of the processes of association, who are united by these processes into the higher unity which is the society. On the other hand, there are the interests which, living in the individuals, motivate such a union.As a form of sociability, play gets the symbolic significance that distinguishes it from pure pastime.

Games of no chance•Two players move alternately;•No chance devices are used to affect outcomes;•Players have complete information about other player’s “hand”;•Players have complete information and control in relations to their own “hand” – no “draws”, for example, to affect it. (Erickson, 1996)

Conclusions•Participants in the study subdivided game into three stages: 1) Send off , 2) development, dialogue or negotiation, 3) end of game.•Accumulation of seeds in one pit is only one among the many strategies used to set up a situation in the “dialogue” phase. This observation leads us to discharacterize Retschitzki’s (1990) subdivision of the game into: beginning, accumulation, and end of game.•End of game provides many opportunities for the practice of theorems-in-action. Many theorems are shared and known by good players. Knowledge of these theorems is a decisive factor into solving the situation put forth in the dialogue phase of the game. This contradicts the depiction in the literature of end of game as less interesting. From the point of view of creating theorems-in-action it is the most fertile phase.•Players learn create strategies that become crystallized as theorems-in-action, or knowledge that is validated by the group and that no longer needs to be investigated. For example, see description of the 3-4 strategy above. Other theorems-in-action are used for the so-called 3-open-4 and 3-5 configurations.•We have accounts of individual problem-posing and problem-solving, and identified players that we may seek in a posterior study for case-studies.•Concepts of concreteness or abstractness seem dubious to characterize mathematical knowledge.•Playing, as a form of sociability, combines the individual and social aspects of mathematical activity.

The rules•The game starts with four seeds in each pit.•Players take turns in “sowing”: emptying the pit of choice and distributing its contents, one seed per pit, counterclockwise.•If one plays a pit that has enough stones to go completely around the board (12 or more), the original house is skipped and left empty.•If the last stone is dropped into a house on the opponent's side, resulting in that house having with 2 or 3 stones, the player captures all the stones in that house.•A capture includes consecutive previous houses which also contain 2 or 3 stones.•If a player does not have any seeds left, the opponent must make a move which will give him/her seeds. If no such move can be made, the game is over and each player gets the remaining seeds on his/her side of the board.  

Works CitedConnors, J. (1990). When mathematics meets anthropology: The need for interdisciplinary dialogue. Educational Studies in Mathematics 21, (5), 461 – 469.Erickson, J. (1996). Sowing games. In R. J. Nowakowsky (Ed.), Games of no chance , 29 (pp. 285-297). New York: Cambridge University Press.Muniz, C. A. (1999). La construction de la connaissance mathématique, en jouant. Doctoral dissertation. Paris, L'Université Paris 13. Retschitzki, J. (1990). Stratégies des joueurs d'awélé. Paris : L'Harmattan.Simmel, G. (1971). On Individuality and Social Forms. Donald Levine (org.). Chicago: The University of Chicago Press. Vergnaud, G. (1998). Toward a cognitive theory of practice. In A. Sierpinska & J. Killpatrick (Eds.), Mathematics education as a research domain : a search for identity. Dordrecht: Kluwer Academic Publisher.

Ana Lúcia Braz DiasDepartment of MathematicsCentral Michigan University

Juliana Braz DiasDepartment of Anthropology

Universidade de Brasília

Uril is a game of no chance or pure strategy, as opposed to games of chance.

The game can be conceptualized as a mediator for the development of mathematical knowledge:•it is are a fertile source of problems – individuals work at posing new situations for which they have no results, and work diligently at solving them.•it provides a space where individuals feel confident to create, test, validate, and discuss his or her own schema of action.•it favors frequent navigation between the individual and social aspects of mathematical activity.

MethodologyIn conformity with the anthropological perspective adopted, the study is ethnographical in nature, anchored fundamentally in the immersion in the local (Capeverdean) life and culture through participant observation. We selected one of the sites where the game was played in the city of Mindelo, São Vicente island, for focus in this phase of the study. We frequented the site daily, afternoons and evenings. We observed others playing, talked to players, and played the game, having been taught rules and strategies by local players.The language spoken during data collection were Portuguese and Capeverdean Creole.Fieldnotes and a field diary were written daily, immediately after the day’s participant observations.

Uril is played on a 2 x 6 board (two rows of 6 pits) and 48 uril seeds.

AcknowledgementsThis research was partially funded by a Faculty Research & Creative Endeavors (FRCE) grant from Central Michigan University.

DiscussionOnce players reach certain configurations the game does not need to continue because they know what the outcome is going to be. For example, if players reach a configuration, which we will call 3-4, in which one player has three seeds and the other has four, as shown in the top left grid in the figure to the right (Figure 8), they know that the player with four seeds will end up capturing either all seeds or four seeds, depending on is to play next.Figure 8 shows three columns of grids representing game configurations. Each 2x6 grid represents a board, and the numbers in the cells correspond to the number of seeds in a pit at a certain moment. In each 2x6 grid, a cell is shaded. The shading indicates what pit the player is going to “sow” next.For example, from the figure’s initial position the player on top plays the yellow cell, sowing its content counterclockwise, and the new configuration is . Next, the player on the bottom is going to play the pit indicated by the blue shading.The players we observed assume the other player is going to do the best move available.The figure is to be read in three columns, from top to bottom and left to right.

1 1 1 1 2

1 1 1 1 1 1 1 3 1 1 1

1 1 1 1 1 2

1 1 1 1 1 1 1 2 1

1 1 1 1 1

1 1 1 1 1 1 1 1 1 2 1

1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 2

1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 2

1 1 1 1 1 1

1 1 1 1 1 1 1 1 1

1 1 1 1 1 1

1 1 1 1 1 1 1 1 1

1 1 1 1 1 1

1 1 2 1 1 1 1 1

2 1 1 1 1

1 1 2 1 1 1 1 1

2 1 1 1 1

1 1 2 1 1 1 1 1

2 1 1 1 1

1 1 2 1 1 1 1 1

2 1 1 1 1

1 1 2 1 1 1 1 1

3 1 1 1

1 1 2 1 1 1 1 1

3 1 1 1

2 2 1 1 1 1 1

1 1 1

1 1 1 2 2 1 1 1 1 1

1 1 1 1

1 1 1 3 1 1 1 1 1

1 1 1

1 1 1 3 1 1 1 1 1 1

1 1 1

1 1 1 3 1 1 1

Dr. Ana Lúcia Braz DiasAssociate ProfessorCentral Michigan University, Department of Mathematics214 Pearce HallMount Pleasant, MI, 48859 – U.S.A.E-mail: dias1al@cmich.edu

Dr. Juliana Braz DiasProfessor AdjuntoUniversidade de Brasília, Departamento de Antropologia. Campus Universitário Darcy Ribeiro - ICC Centro - Sobreloja B1-34770910-900 - Brasilia, DF – BrasilE-mail: jbrazdias@hotmail.com

Authors’ contact information:

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Figure 1: Map with location of Cape Verde.

Figure 2: Mindelo, São Vicente.

Figure 3: One of the researchers receives instruction on game strategies.

Figure 4: Mindelo street, near research site.

Figure 5: Uril as played in Mindelo, Cape Verde.

Figure 6: The playing of uril, a social activity.

Figure 7: Players alternate turns, but expect a certain movement by opponent and play nearly simultaneously.

Figure 8: Coding of one of the “theorems-in-action” validated by the players.