Se presenta una síntesis de la investigación sobre el razonamiento probabilístico de los niños hasta los 7 años. Tras una breve mención a los trabajos de Piaget e Inhelder y Fischbein, se revisan estos y otros estudios asociados. Analizamos los estudios sobre la comprensión del azar en los niños, su lenguaje probabilístico, la identificación y enumeración de posibilidades, estrategias de comparación de probabilidades y razonamiento combinatorio. El maestro y las familias pueden utilizar esta síntesis para apoyar la educación temprana del razonamiento probabilístico, que beneficiará al niño en su desarrollo, posterior comprensión de la probabilidad y toma de decisiones en situaciones de incertidumbre.
The purpose of this research was to analyse high school students’ probabilistic reasoning when interpreting news from the media. To fulfil this goal, 76 high school students were given six questions related to a report on traffic accidents taken from the media news. The questions involved some conditional probabilities and the critical reading of the information presented. Although the computation of the complementary event probability was easy, there were more troubles in dealing with other conditional probability questions. Some students relied too much on the authority principle to interpret the data and did not reach a level of critical reading of the information. Finally, few students were able to identify the missing information needed to apply Bayes’ theorem.
Ultrafiltration membranes have several advantages over conventional drinking-water treatment. However, this technology presents major limitations, such as irreversible fouling and low removal of natural organic matter. Fouling depends heavily on the raw-water quality as well as on the operating conditions of the process, including flux, permeate recovery, pre-treatment, chemical cleaning, and backwashing. Starting with the premise that the optimisation of operating variables can improve membrane performance, different experiments were conducted in a pilot plant located in Granada (Spain). Several combinations of permeate and backwashing flow rates, backwashing frequencies, and aeration flow rates were tested for low-quality water coming from Genil River with the following results: the effluent quality did not depend on the combination of operating conditions chosen; and the membrane was effective for the removal of microorganisms, turbidity and suspended solids but the yields for the removal of dissolved organic carbon were extremely low. In addition, the threshold transmembrane pressure (-0.7 bar) was reached within a few hours and it was difficult to recover due to the low efficiency of the chemical cleanings. Moreover, greater transmembrane pressure due to fouling also increased the energy consumption, and it was not possible to lower it without compromising the permeate recovery. Finally, the intensification of aeration contributed positively to lengthening the operation times but again raised energy consumption. In light of these findings, the feasibility of ultrafiltration as a single treatment is questioned for low-quality influents.
El intervalo de confianza es un procedimiento básico en inferencia estadística y, por ello, su estudio se incluye en las Matemáticas Aplicadas a las Ciencias Sociales II para alumnos de dicha modalidad de Bachillerato. Además, en las pruebas de acceso a la universidad se propone con frecuencia un problema relacionado con este contenido. Con el objetivo de evaluar la comprensión del tema, en este trabajo se analizan las respuestas dadas por 58 estudiantes de segundo curso del citado Bachillerato a un cuestionario que consta de seis ítems de opción múltiple y un problema abierto tomado de anteriores pruebas de acceso la universidad. Los resultados muestran una pobre comprensión del tema, con pocas respuestas correctas en los ítems de opción múltiple, que evalúan la comprensión conceptual, y con sólo un 40% de estudiantes que logran una resolución completa y correcta del problema.
In this paper, we develop a personal synthesis of the most outstanding research on the teaching and learning of probability in the past years. We conducted a systematic search to examine publications on this topic in mathematics education, statistics education, education, and psychology journals. This exploration was complemented by additional studies published in conference proceedings or books. We classified these papers to highlight the main recent research tendencies in the field, according to the theme studied and considering the research objectives. Epistemological analyses suggest that informal inference based on simulation diminishes the topic abstraction but reduces probability to its frequentist view. Topics receiving particular attention include children’s probabilistic knowledge, the effect of visualizations on solving conditional probability problems, teachers’ education and probability modelling. In the final section, we recommend relevant points in which more investigation is needed to complete our knowledge about teaching and learning. In particular, we miss research on teachers’ mathematical knowledge of many probability concepts and on their didactic knowledge.
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