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Although student covariation reasoning has been explored in depth to improve understanding of the correspondence between variables, research has focused on studying existing reasoning about variables in Cartesian representations. The working method had a qualitative approach, with a descriptive exploratory scope, the spontaneous representations that the participants evidenced under the level of covariational reasoning of the variables present in three contextualized situations were explored, posed to a population of third-grade middle school students. The students argued with concrete and abstract drawings the general behavior of the variables. At low levels of reasoning they used pictorial representations, at higher levels they used graphical diagrams and tables. The exploratory study shows a relationship between the type of spontaneous representations and the student's level of covariational reasoning, as well as the rigor of the description of the problem.

The present educational intervention study focused on addressing some areas of improvement in the teaching and learning of trigonometric ratios in high school students. A diagnostic evaluation was performed that revealed an instrumental understanding of trigonometric concepts by the students. Subsequently, a didactic sequence was implemented that incorporated practical applications of trigonometric ratios in the context of displacement vectors. The results of the study indicated significant improvements in students' understanding after the intervention, marking a shift toward relational understanding. This proposal is presented as an effective strategy and suggests promising ways to improve pedagogy in this field.

This research reports the implementation of an evaluation instrument of the slope concept in high school students. The design of this study was based on four dimensions: Skills, Properties, Uses and Representations (SPUR model; Thompson & Kaur, 2011) and on three conceptualizations: constant ratio, behavior indicator and trigonometric conception. This work adopts a qualitative approach to analyze the students' productions and a quantitative approach when obtaining the percentages of the student's responses. The general objective of the work is to evaluate the effect produced by a group of tasks designed with the SPUR model on the slope concept in high school students. The results show that students have traditional conceptualizations of the slope as a constant ratio and trigonometric conception. However, these conceptualizations emphasize more procedural aspects than conceptual ones. This finding could partially explain why students can solve certain tasks of procedural nature but not conceptual tasks that usually require a multifaceted view of the slope.

This report delineates the outcomes of an intervention conducted with in-service high school educators, focusing on elucidating three distinct scenarios within geometric and arithmetic domains: the infinitely large, infinitely numerous, and infinitesimally close. Grounded in the theoretical framework of conceptual change, it is posited that when an individual exhibits entrenched conceptions, it signifies a misclassification of the pertinent concept, necessitating a categorical shift to effectuate a transformation in their cognitive schema, particularly concerning the notion of infinity. Thus, the principal objective of this investigation was to ameliorate the entrenched conceptions held by educators pertaining to infinity through a workshop-based intervention. Preceding the workshop, educators predominantly exhibited conceptions aligned with natural and potential infinities. However, after the workshop, a discernible transition was observed, with educators engendering an actual conception of infinity or an omega-epsilon position, exemplified by their acceptance of equivalences such as 0.999…=1 and the parity in the cardinality of sets comprising natural numbers, even numbers, and perfect squares. Nonetheless, notwithstanding this progress, confident educators evinced resistance to embracing the concept of actual infinity, particularly in instances such as the hypothetical scenario depicted in Hilbert's Grand Hotel. Consequently, drawing upon the framework of conceptual change theory, it can be postulated that a complete categorical shift was not universally realized among educators due to their reluctance to revise entrenched beliefs concerning natural or potential infinity.

The present research focuses on analyzing how fourth-grade elementary school students (ages 9 to 10) solve and interpret the result of non-routine problems, precisely division measurement and division-partition with remainder. The methodology is qualitative, with a descriptive and interpretative approach. The information was collected using a questionnaire consisting of three problems (two of quotitive division and one of partitive division) and a clinical interview. The results showed the importance of using the division, multiplication, and addition algorithms to give a realistic answer to the problems. In the same way, it was possible to demonstrate the graphic strategy combined with counting to give a realistic answer to the problem. However, students were found to use division correctly but without an interpretation of the remainder or quotient. Likewise, they struggled to choose the correct procedure to solve the problem. These data suggest including realistic problems in mathematics classrooms to make sense of mathematical concepts in real life or the student's context. Likewise, this study provides implications on the mathematical problems that the teacher proposes in the classroom, where not only the division algorithm should be taught mechanically, nor focus on posing routine problems that lead the student to use a single heuristic resolution strategy. Essentially, the teacher is required to include real-world problems, where the student can awaken creativity to represent in different ways the understanding of a problem and, therefore, different strategies to solve it. In addition, that the student has the ability to check the result of the problem, with the conditions, situations or circumstances imposed by reality or everyday life. [This is a reprint of an article originally published in "Mathematics Teaching Research Journal" v14 n5 p159-180 Win 2022.]

The present research focuses on analyzing how fourth-grade elementary school students (ages 9 to 10) solve and interpret the result of non-routine problems, precisely division measurement and division-partition with remainder. The methodology is qualitative, with a descriptive and interpretative approach. The information was collected using a questionnaire consisting of three problems (two of quotitive division and one of partitive division) and a clinical interview. The results showed the importance of using the division, multiplication, and addition algorithms to give a realistic answer to the problems. In the same way, it was possible to demonstrate the graphic strategy combined with counting to give a realistic answer to the problem. However, students were found to use division correctly but without an interpretation of the remainder or quotient. Likewise, they struggled to choose the correct procedure to solve the problem. These data suggest including realistic problems in mathematics classrooms to make sense of mathematical concepts in real life or the student's context. Likewise, this study provides implications on the mathematical problems that the teacher proposes in the classroom, where not only the division algorithm should be taught mechanically, nor focus on posing routine problems that lead the student to use a single heuristic resolution strategy. Essentially, the teacher is required to include real-world problems, where the student can awaken creativity to represent in different ways the understanding of a problem and, therefore, different strategies to solve it. In addition, that the student has the ability to check the result of the problem, with the conditions, situations or circumstances imposed by reality or everyday life.