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Research in mathematics education can be understood as a system of activities addressing the basic and applied problems related to teaching and learning of mathematics. Such a system includes the activities of foundation, planning, implementation, evaluation of mathematics instruction, and teacher professional development, which are supported by different theories. This diversity of theories raises interest in their comparison, coordination, and possible integration. The paper aims to present a case of application of the Cultural Historical Activity Theory (CHAT), in its 3rd and 4th generation versions, to analyze the emergence of the Onto-semiotic Approach to mathematical knowledge and instruction as a theoretical framework that addresses the study of the five partial activities mentioned above. This use of the CHAT can be useful in studies on theory articulation by focusing not only on the subjects, the object, and the instruments but also on the community context, the ecological-normative environment in which these activities take place, and the dilemmas or contradictions between theories.

Educating students to apply probability literacy and reasoning in out-of-school contexts requires the previous preparation of teachers, which should include the assessment of their mathematical and didactic knowledge. Consequently, we investigated three research questions concerning the probabilistic reasoning and literacy of 66 prospective secondary school teachers when solving probability questions related to a report taken from the media news, their ability to identify the fundamental stochastic ideas needed to solve the task, and their capacity to predict their students' potential difficulties. The participants needed to compute the probability of a complementary event, several conditional probabilities, and perform critical reading of the information in the report. Despite the participants' good mathematical knowledge, a high percentage showed reasoning biases, such as the fallacy of the transposed conditional, confusion between conditioning and causation, and few identified the information missing to solve a question. There was a poor capacity to determine the stochastic ideas needed to solve the questions, beyond probability, sample space, conditional probability, and sampling. The participants quoted a few potential errors of their students, mainly conceptual mistakes with scarce recognition of interpretation errors. These results reveal the need to reinforce teachers' probabilistic literacy, reasoning, and related didactic knowledge.

This paper analyzes the relationship between proportional reasoning and understanding fair games in Costa Rican students. We conducted a quantitative and qualitative analysis of the answers to six items on comparing ratios of increasing difficulty level and another item on prize estimation in a fair game. We describe the strategies employed and the semiotic conflicts detected in 292 Costa Rican students from Grades 6 to 10 (11-16-year-olds), comparing the findings with those established in previous research. The results show an increase in the level of proportional reasoning with the grade, although the age at which the higher levels are reached is lower than that assumed by Noelting. The percentage of students applying correct strategies in the fair game problem also increases with grade, and a relationship between the understanding of fair game and the level of proportional reasoning is observed.

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.

We present an exploratory study of Costa Rican and Spanish students' (11-16-year-olds) competence to compare probabilities in urns and compare ratios in mixture problems. A sample of 704 students in Grades 6 through to Grade 10, 292 from Costa Rica and 412 from Spain, were given one of two forms of a questionnaire with three probability comparison and three ratio comparison problems each. The full questionnaire consisting of both forms covers six different proportional reasoning levels for each type of problem. We analysed the percentages of correct responses to the items and the strategies used by students in each grade and in each country. The results suggest students had the highest difficulty in comparing probabilities for items at the same proportional reasoning level. The results also point to the probabilistic biases in students' responses, although these biases were less frequent than in previous research.