RESUMO
Resumen La presente investigación tuvo como objetivo identificar y describir los niveles de desarrollo de la noción de "mitad" en un grupo de niños de dos comunidades indígenas pertenecientes al pueblo shipibo-konibo de la región Ucayali, en la Amazonía del Perú. Se trabajó con 14 estudiantes de edades entre 7 y 13 años, seis de ellos pertenecientes a la comunidad de Bethel y ocho a la comunidad de Bena Jema. Todos fueron evaluados utilizando el método clínico-crítico de Jean Piaget. Los estudiantes de la comunidad de Bethel fueron evaluados con una tarea centrada en cantidades discretas y los de la comunidad de Bena Jema, con la misma tarea y con otra centrada en cantidades continuas. Los desempeños de los participantes evidenciaron cuatro niveles de desarrollo en cada una de las tareas. Sus respuestas fueron consistentes con otras investigaciones que evaluaron la misma noción en contextos occidentales. Los resultados apoyan la universalidad en la construcción de la noción de "mitad", pero evidencian un retraso en la adquisición de los niveles por parte de los niños evaluados, si se toma como referencia los currículos nacionales. Se discuten los hallazgos resaltando la universalidad de las estructuras lógico-matemáticas y la necesidad de repensar el momento y la forma en que la noción de "mitad" aparece en el currículo.
Abstract This research aims to identify and describe developmental levels of the notion of "half" in a group of children from two Shipibo-Konibo native communities of the Ucayali region, in the Peruvian Amazon rainforest. Fourteen students aged between 7 and 16 participated, six from Bethel community and eight from Bena Jema community. The community of Bethel is located approximately 6 hours by river from the city of Pucallpa, while Bena Jema is located within Pucallpa's boundaries, in the Yarinacocha district. All participants were assessed using Jean Piaget's clinical-critical method, with two tasks. In the first task, focusing on discrete quantities: different sets of cards with the picture of a fish were presented sequentially, some with an even number of units and some with an odd number. Participants were asked to choose and present back half of each set. In the second task, focusing on continuous quantities: participants were asked to choose and present half of a single raw spaghetti, which they needed to break with their hands. If the resulting pieces were unequal, they were asked to break them again and redistribute the results. Students from Bena Jema community were assessed with both tasks, while students from Bethel community were assessed only with the discrete quantities task. The results showed four developmental levels in the discrete quantities task. In the first level, participants took any one of two parts of the set to be "half", without checking if they were equal or whether putting them back together reconstituted the original whole. In the second level, participants took any one of two equal parts to be "half", but were inconsistent and accepted the possibility of the parts being unequal. In the third level, participants only took any one of two equal parts to be "half" but had difficulties splitting odd-numbered sets, including sets with fewer elements than the even-numbered ones they had previously split correctly. In the fourth level, the task was solved successfully with both even-numbered and odd-numbered sets. For the continuous quantities task, three levels were found. In the first level, any one of two raw spaghetti pieces was taken to be "half", without checking their evenness. In the second level, participants took any two equal pieces of the raw spaghetti to be "half", but without checking whether putting them back together reconstituted the original whole, with no extra parts remaining. In the third level, participants took any one of two equal pieces to be "half", checking whether putting them back together reconstituted the original whole. These results are consistent with reports from previous research assessing the notion of "half" in Western contexts. Results support universality in the development of the notion of "half", but show a delay in the participants' level of acquisition, taking Peru's national curriculum as reference. This study is a contribution to the understanding of the development of the notion of "half" in indigenous children living in Amazonian native communities, and shows the relevance of the Piagetian clinical-critical interview in these sociocultural contexts. The implications of these findings are discussed in relation to the universality of logical-mathematical knowledge, as well as the need to rethink the timing and manner in which the notion of "half" appears in the curriculum.
RESUMO
Se presentan los resultados de una evaluación de las operaciones matemáticas de adición y sustracción, en un grupo de niños de enseñanza primaria (5 niñas y 2 niños) de 7, 9 y 11 años, de una comunidad nativa Shipibo-Konibo de la región Ucayali, en la Amazonía del Perú. Se realizó con dos métodos distintos, la Entrevista Clínico-crítica de Piaget, que incluyó problemas culturalmente contextualizados y material representativo y manipulable como figuras de animales y bolitas para armar collares, y una prueba tradicional de lápiz y papel. Ambas pruebas evaluaron el mismo tipo de operaciones, extraídas de lo que propone el diseño curricular nacional para esos grados. Los resultados muestran que los niños tienen dificultades tanto en la adición como en la sustracción y no logran resolver con éxito todas las tareas. Sin embargo, se pueden observar mejores resultados con la evaluación cualitativa por medio de la entrevista clínico-crítica, en contraste con la evaluación tradicional de lápiz y papel que evidencia resultados mucho más pobres. La primera otorga además mayor información sobre el proceso operativo de los niños y muestra que los niveles de desarrollo de sus competencias matemáticas van de la mano con lo que plantea la teoría piagetiana. Los resultados se discuten señalando la universalidad del conocimiento lógico-matemático y su pertinencia para comprender los procesos de aprendizaje en contextos de diversidad cultural y analizando críticamente el modo en que la evaluación constructivista ofrece mayor información y recursos para los docentes de educación intercultural bilingüe de comunidades amazónicas del Perú.
Elementary-school children from a Shipibo-Konibo indigenous community in the Ucayali region, in the Amazonian rainforest of Perú, were evaluated regarding their abilities to solve addition and subtraction problems. These operations were assessed by two means: through Jean Piaget's clinical-critical method (using culturally contextualized problems and concrete materials such as pictures of Amazonian fish, pictures of arrows, or beads and thread to make necklaces), and by a traditional pencil-and-paper test. Both the clinical interview and the pencil-and-paper test evaluated the same type of operations, which were taken from the national curricular program for these school grades. The Shipibo-Konibo people are an Amazonian indigenous group that speaks a native language in the Panoan family; since most members of this group are fluent in Spanish, however, no translator was needed and the assessments were conducted in Spanish. The Shipibo-Konibo people are principally settled along the Ucayali River in the Amazon rainforest in Perú, although currently many of them have relocated to other areas of the country, including Lima the capital city, in search of better work or education opportunities. After the Asháninka and the Awajún, the Shipibo-Konibo is the third largest Amazonian indigenous group in Perú. Informed consent was obtained following the guidelines of Frisancho, Delgado, and Lam (2015), which are based on previous experience working with Amazonian indigenous communities in the Ucayali region of Peru. As research has shown that individuals from cultural diverse backgrounds may have different expectations for the research process, and may perceive it in a different way than people from industrialized nations (Lakes, Vaughan, Jones, Burke, Baker, & Swanson 2012), informed consent included both individual consent and a communitarian meeting. It also included the donation of gifts (tools, groceries, and other useful items) for the community, and a debriefing meeting with the community's school teachers. The assessment was conducted by two researchers in a school classroom. It took around 30 minutes with the older children and 45 with the younger ones. In all cases the clinical-critical interview was applied first (addition and then subtraction), and finally the pencil-and-paper test. Although a native speaker of shipibo was present during the evaluation, his services were not needed as children were fluent in Spanish. Results show that children have difficulties in the development of both addition and subtraction. They make counting mistakes and have trouble understanding the logic of subtraction (taking a number from another, larger one). Some do not conceptualize subtraction as the opposite of addition and, in consequence, cannot foresee that joining two numbers that were previously separated will result in the same original quantity. Children aged seven were unable to solve any of the operations in the pencil-and-paper test. However, better results were obtained through qualitative, Piagetian assessments, in contrast to the quantitative, pencil-and-paper assessments. During the interviews, children were able to show their cognitive processes and ways of thinking while solving the problems, and with scaffolding, the use of concrete materials such as pictures or beads, and strategic help from the evaluators (Parrat, 2016a, 2016b), many of them were also able to develop a better comprehension of the problem and self-correct their initial answers. Developmental levels for addition and subtraction consistent with Piagetian theory were identified. These levels show a progression from the impossibility of grasping logical addition or subtraction, to the capabilities of psychological reversibility and logical composition of inverse and direct operations. The results are discussed using Piaget's theory and the problem of particulars and universals in cognitive development, and analyzing critically how this kind of assessment can help elementary school teachers respond to the needs of intercultural bilingual education in Peru's Amazonian indigenous communities.
