Number sense has been considered as one of the most important mathematical notions to be addressed in school mathematics in the 21st century. In this paper, we identify how students of a public middle school, located in a rural area in Mexico, showed several aspects of number sense by performing tasks involving arithmetic operations in a shopping context. Students invented mental and written calculation strategies that provide evidence of understanding regarding numbers and operations. Students used fantasy money as a representation system to decompose quantities in non-standard ways, or referred to strategies employed by unschooled people. Compensation strategies that involve modifying the quantities before applying the written algorithm for subtraction, in order to perform a subtraction without regrouping, also appeared. That is, it was identified a transfer from mental calculation strategies to a paper and pencil scenario for facilitating calculations during problem solving, which is an indicator of student' creativity and number sense. Additionally, the proposed tasks provided evidence that contextualized tasks could promote the understanding of decimal numbers.
ResumenEn este artículo reportamos los resultados de una investigación exploratoria, cuyo interés fue identificar cómo las creencias que sostienen profesores en servicio sobre el Teorema de Pitágoras, son indicadores de un reduccionismo didáctico relativo a este resultado matemático. Se aplicaron cuestionarios a cinco profesores de matemáticas que laboran en un bachillerato público y una entrevista semi-estructurada a un profesor quien imparte cursos de física y matemáticas en una carrera de ingeniería en una universidad pública. Los datos recolectados se analizaron con base en seis categorías de creencias. Un resultado relevante del trabajo es que los profesores conciben al Teorema de Pitágoras como un hecho aislado y no como un conocimiento integrado a una
Mathematical tasks are crucial elements for teachers to orient, foster and assess students' processes to comprehend and develop mathematical knowledge. During the process of working and solving a task, searching for or discussing multiple solution paths becomes a powerful strategy for students to engage in mathematical thinking. A simple task that involves the construction of an equilateral triangle is used to present and discuss multiple solution approaches that rely on a variety of concepts and ways of reasoning. To this end, the use of a Dynamic Geometry System (GeoGebra) became instrumental in constructing and exploring dynamic models of the task. These model explorations provided a means to generate novel mathematical results.By genuine inquiry, we mean the process of raising and evaluating questions grounded in experience, proposing and developing alternative explanations, marshalling evidence from various sources, representing and presenting that information to a larger community, and debating the persuasive power of that information with respect to various claims. [10,p.11] Zaslavsky [11] offers an approach to transform standard mathematical tasks into openended ones by changing initial conditions of the statement. For example, the standard task
Resumen: Las estrategias de cálculo mental juegan un rol importante en el desarrollo del sentido numérico. Al respecto, buscamos determinar cómo alumnos de una telesecundaria, ubicada en una comunidad rural del estado de Hidalgo, México, pueden desarrollar estrategias de cálculo mental al realizar tareas que involucran sumas y restas en contextos de compraventa. Identificamos ocho estrategias diferentes, algunas de las cuales emplearon el dinero de fantasía como sistema de representación, que apoyó la descomposición de las cantidades en forma novedosa. Los estudiantes hicieron referencia a estrategias que utilizan personas sin escolaridad al recibir el cambio de compras en el mercado; también aparecieron estrategias como quitar uno al minuendo en una resta con reagrupación, para obtener una sin reagrupación. Esto es, se observó una transferencia de las estrategias de cálculo mental, cuando los participantes modificaron el algoritmo estándar para la resta, con el objetivo Fecha de recepción: 6 de diciembre de 2017. Fecha de aceptación: 17 de septiembre de 2018.
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