Las ideas fundamentales de probabilidad en el razonamiento de estudiantes de bachillerato

Abstract

El objetivo de este trabajo es explorar las inferencias que los estudiantes de bachillerato formulan a partir de su conocimiento de las interpretaciones frecuencial y clásica de probabilidad. Se describen y analizan los razonamientos de 30 estudiantes del 12º grado, quienes cursaban la materia de Probabilidad y Estadística II. La recolección de datos fue mediante tres versiones de un cuestionario en los que se pide hacer predicciones y estimar probabilidades. El análisis de las respuestas revela la tendencia de los estudiantes al cálculo de probabilidades, mayormente apoyados en razonamientos informales inadecuados en los que intervienen las ideas de variabilidad, aleatoriedad e independencia. Con base en este resultado, se sugiere que el primer objetivo en la enseñanza de la probabilidad debe ser el desarrollo de un razonamiento informal sobre estas ideas. Además, se presenta una propuesta sobre el razonamiento informal que puede potenciar el razonamiento probabilístico de los estudiantes.

Prospective Mathematics Teachers Understanding of Classical and Frequentist Probability

“…These prospective teachers were given a questionnaire consisting of two open-ended tasks, as shown in Figure 1. In Task 1, adapted from Sánchez and Valdez [25], the participants were asked to estimate the number of white and black balls in the urn, using the result of 1000 extractions. The probability of drawing a ball of each colour in the urns will be given by w/10 and b/10, where w is the number of white balls, b the number of black balls, with w + b = 10.…”

“…Therefore, the correct answer is that urn B is preferable, which, according to the estimation of its composition in Task 1, has a higher number of favourable cases. As a distractor, the subjects were given the results of 10 new draws from each urn, In Task 1, adapted from Sánchez and Valdez [25], the participants were asked to estimate the number of white and black balls in the urn, using the result of 1000 extractions. The probability of drawing a ball of each colour in the urns will be given by w/10 and b/10, where w is the number of white balls, b the number of black balls, with w + b = 10.…”

“…About 6% of the prospective teachers provided a range of values for the composition of black and white balls in the urns. Although these prospective teachers used the idea of variability in estimation, which is fundamental to relating the classical and frequentist approaches to probability [25], they misinterpreted this variability, since, as we have determined, a composition of the urns different to what is expected would be highly improbable. These participants did not correctly consider sampling variability, although they related the classical and frequentist views, and therefore they manifested an informal quantitative reasoning (level 3), not reaching the higher level of understanding of variability in the classification of Sánchez and Valdez [32].…”

“…Prospective teachers must have a good understanding of the characteristics and differences of these two approaches and their relationship to each other. Such an understanding requires knowledge of the LLN [21,24], as well as of the fundamental stochastic ideas of randomness, variability, and independence [25]. This type of relational understanding is the one that we aim to assess in this paper for a sample of prospective Spanish secondary and high school mathematics teachers.…”

“…Sánchez and Valdez [25] studied the inferences made by a group of 30 high school students in Mexico, using their knowledge of the classical and frequentist conceptions of probability. The students were divided into three groups of 10 students, each of which was given a different questionnaire, with variations of the same task that combined probability comparison and sampling.…”