Análisis epistemológico y evaluación de la comprensión del concepto de variable aleatoria en estudiantes de secundaria chilenos

Résumé

This study delves into the comprehension of random variables, binomial and normal distributions within the context of Chilean schools. The primary focus is analyzing how these subjects are addressed in the Chilean school curriculum and textbooks. Additionally, it aims to evaluate the understanding of these topics among students who have completed their education in Chilean schools, given that these subjects are typically covered in the latter grades of secondary education (between grades 10 and 12). The theoretical framework for this research relies mainly on the Ontosemiotic Approach to Mathematical Knowledge and Instruction. The doctoral thesis is structured into six studies, encompassing background research, examining textbooks and Chilean curricular regulations, developing and validating an assessment tool, and evaluating students' comprehension in the Chilean educational context. Study 1 involved conducting a comprehensive literature review on the instruction and learning of random variables and binomial and normal distributions. This review specifically focused on the most pertinent research related to i) the teaching of these subjects, addressing both epistemological and didactic aspects in both school and university settings, and ii) the learning of these topics, examining cognitive aspects within the school and university context. The findings from the literature review indicate that studies conducted from an epistemological standpoint have only superficially identified elements of the random variable (such as problems, procedures or representations) crucial for effective teaching at both school and university levels. Additionally, studies that offer guidance on the key elements of teaching the binomial distribution have yet to be identified. In studies conducted from a cognitive perspective, similar investigations were found addressing the understanding of random variables and their probability distributions in school or university education. However, notable gaps exist in the literature, warranting further exploration. One such gap pertains to the need to investigate the level of understanding of random variables and probability models among students upon completion of their school education. The second and third studies examine how random variables, as well as binomial and normal distributions, are addressed in the current Chilean curricular regulations and textbooks intended for students in grades 10 to 12 (ages 15 to 18) during the timeframe of this research. Specifically, Study 2 identified the problem situations related to the subjects of interest promoted by the school curriculum and the textbooks under investigation. The findings reveal that certain problem situations are endorsed in the school curriculum but are either absent or minimally covered in the surveyed textbooks. Examples include the calculation of probabilities associated with binomial and normal distributions using technological tools and distinguishing between random variables and variables with functional dependence. Additionally, observations indicated instances where textbooks introduce situations or problems not suggested in the Chilean curriculum. These include differentiation between discrete and continuous random variables, calculating the value of unknowns to establish a probability function, determining probabilities in a standard normal distribution, graphically representing the normal distribution, and calculating values corresponding to a given probability within the context of the normal distribution. Similarly, in Study 3, the examination focused on recognizing the language, concepts, propositions, procedures, and arguments associated with the topics under consideration, as suggested by Chilean curricular guidelines and school textbooks. The results reveal the following: (i) Concerning the language related to the random variable and normal distribution, it was evident that there is congruence between the curricular guidelines and the surveyed books. However, discrepancies were observed between the analyzed documents in the context of the binomial distribution. (ii) In terms of concepts linked to the random variable, coherence was evident between the school curriculum and the textbooks. Conversely, inconsistencies were observed between the documents analyzed in the context of probability models. (iii) Regarding propositions linked to the binomial distribution, there was harmony between the curricular guidelines and the textbooks. However, certain incongruities were identified between the surveyed documents in the context of the random variable and normal distribution. (iv) Concerning procedures related to the random variable and probability models, there was disharmony between the school curriculum and the analyzed books. (v) In the arguments associated with the random variable, binomial distribution, and normal distribution, incongruities were found between the surveyed documents. In the fourth and fifth studies, an assessment instrument was developed and validated to evaluate the comprehension of random variables and binomial and normal distributions within the Chilean school context. In Study 4, a guide for problem situations about interest was initially crafted based on the analysis of Chilean curricular guidelines (GPS-RVPD). Subsequently, an initial set of items representing the problem situations outlined in the guide was chosen from prior research or previously examined textbooks. Following this, the content validity of these items was evaluated by a panel of experts, whose assessment led to the final selection of items. The results indicate that the GPS-RVPD encompasses various problems, giving rise to the remaining primary mathematical components. The tool is deemed suitable for identifying items that represent its problem situations through expert judgment, as the final set of items achieved a favorable validity and agreement coefficient of 0,87. Study 5 examined the instrument's construct validity and reliability, involving administering the initial version of the questionnaire to a targeted sample of 80 Chilean school graduates. The instrument's construct validity was assessed through both exploratory factor analysis (EFA) and confirmatory factor analysis (CFA), while reliability was gauged using Cronbach's alpha coefficient. The EFA findings revealed that the questionnaire's empirical structure consists of six factors, explaining 58% of the total variance. Each factor is associated with a specific problem domain, comprising three to four items related to the topics under consideration: random variable, probability function and distribution function of a discrete random variable, values of central position or dispersion associated with a random variable, binomial, and normal distributions. In the CFA, most of the factor-item coefficients demonstrated adequate values, and the overall structure of the instrument exhibited acceptable measures of fit. Furthermore, a Cronbach's alpha coefficient of 0.824 indicated that the questionnaire is reliable, demonstrating consistency between the items and the construct. Consequently, the final version of the questionnaire (consisting of 19 items) provides evidence of being a valid and reliable tool for investigating the understanding of random variables and probability distributions among Chilean school graduates. In Study 6, the assessment of the understanding of random variables and binomial and normal distributions was conducted among 101 Chilean school graduates using the previously validated questionnaire. The findings indicate that the participants addressed most of the problem situations evaluated in the questionnaire, albeit with a low success rate. Furthermore, a limited percentage of students in the sample demonstrated the ability to utilize the various mathematical components essential for solving the assessed tasks. Consequently, Chilean school leavers must demonstrate a greater understanding of random variables. In terms of comprehending probability models, the participants resolved all the problem situations assessed in the questionnaire, though at a relatively low proportion. Additionally, only a few participants successfully employed the diverse primary mathematical components of solving problem situations related to the binomial distribution. Similarly, a few students in the sample applied some primary objects linked to the resolution of tasks involving the normal distribution. Hence, participants displayed a limited understanding of both binomial and normal distribution. Finally, the primary contributions of this dissertation encompass (i) insights into the tasks related to random variables, the binomial model, and the normal model, as well as the various primary mathematical components involved in their solution, as promoted in Chilean school textbooks; (ii) the development of the tool, "Guía de Situaciones-Problemas sobre Variable Aleatoria y sus Aplicaciones en Distribuciones de Probabilidad según el Currículo Escolar Chileno" (Guide of Situations-Problems on Random Variable and its Applications in Probability Distributions according to the Chilean School Curriculum); (iii) the creation of the assessment instrument, a questionnaire to evaluate the understanding of random variables, and binomial and normal distributions in school leavers; and (v) the presentation of results from the comprehensive evaluation of the understanding of random variables and probability models among Chilean school leavers.