Influencia del razonamiento proporcional en la comparación de probabilidades y resolución de tareas probabilísticas

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Currently, mathematics curricula attach greater importance to probability teaching in primary and secondary education due to the need for probabilistic reasoning in decision-making dealing with situations of uncertainty and for the future study of statistical inference. Given this situation, the research literature in mathematics education notes that intuition in probability, and more specifically the ability to compare probabilities, progresses in stages. Moreover, these stages may be related to those corresponding to the development of proportional reasoning, although they are not equivalent. However, there is little research on the level of probabilistic and proportional reasoning among students in Spain, and none in Costa Rica once they have received training in probability. This doctoral thesis, carried out as a compendium of publications, describes two studies on the evaluation of probabilistic reasoning with students in primary and secondary education, analysing its relationship with proportional reasoning. As the main theoretical foundation, we rely on the Ontosemiotic Approach to mathematical knowledge and instruction (EOS), the Spanish and Costa Rican curricula on probability and proportionality contents, and the main precedents on the assessment of probabilistic reasoning in children and adolescents, with special attention to the comparison of probabilities and the assessment of proportional reasoning levels. The first empirical study was of an exploratory nature and was conducted only with Costa Rican students in the sixth grade of primary education who received instruction in probability. Based on a quantitative and qualitative analysis of the students’ answers to a questionnaire adapted from the research literature, the strategies employed, and the semiotic conflicts detected are described. Approximately half of the sample presents an adequate performance in the comparison of probabilities, typical of the developmental stage of the participants, according to the research literature; however, greater difficulty is shown in the construction of sample spaces when asked to identify certain and impossible events; likewise, difficulties were evidenced in a task on fair play, where the winnings of two players had to be equated according to their probability of winning. Few differences were found between the results obtained and those of previous research, with students of the same age who had not been taught the subject. In the second study, a larger sample of students from Spain and Costa Rica, from different school years, was used to analyse the relationship between the level of proportional and probabilistic reasoning in students in the last year of primary education (6th year of General Basic Education) and those in secondary education (ages 13 to 16). For this purpose, a questionnaire is constructed using a valid and reliable method that allows us simultaneously to evaluate both types of students’ reasoning. In the probabilistic part, the comparison of probabilities is assessed using two different contexts, ballot boxes and roulette wheels, with different levels of proportional reasoning used, which are also considered in the ratio comparison tasks. In addition, items on fair play, sample space construction, and the detection of possible biases in the comparison of probabilities in roulette wheels are included. The results of the different proportional reasoning tasks and their relationship with the rest of the skills assessed are analysed. For some of the variables, the results are compared with those of Spanish students of equivalent ages, whereas for the rest, we work only with Costa Rican students. In general, it is observed that the age at which a certain level of reasoning is reached is higher than that assumed by Noelting (1980a, 1980b); however, a low percentage of subjective beliefs is also found in the resolution of probabilistic tasks posed in the questionnaire. Likewise, statistically significant correlations were found (although of small intensity) between the level of proportional reasoning in which a student was placed and his or her results in the level of probabilistic reasoning shown in the different tasks proposed. We consider that both the design of the evaluation questionnaire and the results obtained in our research and described in this report provide new knowledge regarding proportional and probabilistic reasoning and their relationship, which will help teachers, teacher trainers, and researchers in the design and evaluation of training plans and new lines of research on the subject.