The Content Knowledge of Preservice Elementary Teachers about the Possibility: An Essential Factor in the Effective Teaching of Probability Concepts

Introduction: The probability has entered the content of mathematics textbooks of Iranian schools. Simultaneously with the addition of this area of mathematics to the textbooks, it is necessary for the teachers who are responsible for teaching probability concepts to receive the appropriate training for the optimal teaching of these topics. This is because, successful teaching of probability requires adequate preparation of teachers (Batanero & Álvarez-Arroyo, 2024).
    The results of some international studies indicate that teacher training curricula, especially elementary teachers, do not lead to their mastery of probability concepts, as they should (Kurt & Coşkuntuncel, 2020; Ruz et al., 2021; Batanero, 2022; Brückler & Milin Šipuš, 2023). Earlier findings suggest that there are different approaches for dealing with these difficulties, the most common of which is to prioritize the development of preservice teachers of early- or primary-age students content knowledge (Denton, 2023). The acquisition of the main part of content knowledge and knowledge of content education for teachers should happen during the teacher training period. The present study only focuses on preservice elementary teachers, as the first educators who contextualize the concepts of probability in students, to determine at what level they know the concepts of probability and to what extent they are able to correctly solve the problems that include these concepts.
Research questions: At what level of the Structure of Observed Learning Outcome (SOLO) model is the understanding of preservice elementary teachers about the probability concepts?
Method: This study is descriptive-survey type. The research population includes all preservice elementary teachers in the Farhangian University campuses of Markazi Province who have successfully completed the Basic Mathematics course. The research sample is 137 preservice elementary teachers from two campuses of Farhangian University in Markazi Province, who were selected by randomized clustering method. The researcher-made tool included four tasks, three of which included more than one question; in fact, students answered nine questions. These tasks were designed based on the participants' understanding of five commonly used probability concepts, namely sample space, probability of event, comparison of probabilities, fairness and conditional probability.  
SOLO model was used to analyze the answers provided by preservice teachers to the test questions. In this way, based on the levels of this model, a framework was adapted by the first researcher and after its review and approval by experts in the field of mathematics education, statistics and curriculum planning, it was used as a tool for categorizing student teachers' answers.
Results: The general results of the data analysis indicate that preservice teachers did not have an acceptable performance in any of the five probability concepts investigated in this study and the relative frequency of complete answers, which is placed at the relational level, in each probability concept, is below 40%. In understanding the concept of sample space, about 31% of the sample group members managed to provide complete answers and were placed at the relational level. In addition, only about 37% of the participants were able to provide an answer at the relational level in response to the questions related to probability calculation. Student teachers' understanding of the fairness of a game was not better, and in this sense, only 27% of all members of the sample group gave a complete answer. In the probability comparison, the lowest complete response rate compared to other probability concepts was observed. Because only about 25% of the participants had provided complete answers. Finally, about 34% of the elementary student teachers had provided an answer at the relational level to the conditional probability questions. The results of this study are in line with Kaedi (2017) and Alonso-Castaño and colleagues (2021), who described the content knowledge of student teachers about probability as unfavorable.
Discussion and Conclusions: In explaining the results obtained from this study, at least two basic factors can be mentioned. The first factor is the educational background of preservice teachers in high school. The second essential factor is the quality and content of basic mathematics instruction at Farhangian University.
Based on the results obtained, the authors suggest that a compensatory math lesson should be defined for preservice teachers who studied humanities in high school, so that these students have more opportunity to learn and master basic math concepts, including probability. In addition, it is necessary for Farhangian University, with sufficient planning and investment, to use experts with experience in the field of mathematics education and teacher training in order to compile a comprehensive resource for the basic mathematics course, which according to the relevant curriculum must include the concepts of probability. moreover, the teachers of this course should pay special attention to the mathematical content of the mentioned source, including probability.