This paper focuses on the importance of covariational reasoning within the processes of mathematics teaching and learning. Despite the internationally recognized relevance of covariation, research shows that only a small percentage of students and teachers is able to adopt covariational reasoning and the majority of mathematics curricula do not contain explicit references to covariational skills. In particular, when covariational reasoning manifests as conceptual knowledge, it becomes challenging to assess, and the need for innovative methods of assessment emerges; there is a need for suitable assessment to highlight the characteristics of covariation and capture the various features that characterize conceptual understanding. Comparative judgement (CJ) is an innovative assessment method based on collective expert judgements of students’ work rather than requiring scoring rubrics. Due to its holistic approach, CJ is particularly suitable for assessing complex mathematical competencies, and, as we shall see in this study, it proved to be appropriate for the covariation’s assessment. In details, our study aims to investigate the perception and relevance attributed by mathematics teachers to covariation as a theoretical construct and the way CJ can help in the assessment of covariational reasoning skills underlying a less structured modelling task.

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Meaning-making in teaching-learning mathematical processes is a relevant issue analysed through different philosophical and educational frames. In particular, the use of digital tools in mathematics education affects the meaning-making processes. This paper discusses meaning-making from a   phenomenological standpoint, in which interpretative activities are relevant. This approach requires a careful analysis of the semiotic resources’ evolution, including those related to the used digital tools. The paper aims to introduce an analytical tool, the Timeline. This tool is an elaboration on previous analysis tools, like the interaction flowchart and the semiotic bundle. Such a tool allows the analysis of relationship among interactions, semiotic resources, and meaning-making. In this paper, the Timeline is used to analyze two episodes from two different learning experiments where GeoGebra and augmented reality are used. High school students from Italy and Israel participated in this study. Video recording has been used to document the entire learning experiments. The analysis provides evidence that the Timeline enables investigating the relationship between students-teacher-artifacts interactions and meaning-making. Moreover, results may give teachers ideas for using digital tools to foster students’ meaning-making.

The mathematical modelling of a real-life phenomenon is an elaborated activity, and it often requires complex forms of covariational reasoning, such as second-order covariation. This study aims to characterize how students use several forms of covariational reasoning when modelling a real-life phenomenon. To achieve this research goal, it is proposed the analysis of a teaching experiment conducted in an 11th-grade classroom and focused on the mathematical modelling of the relationship between three quantities, temperature, absolute humidity, and relative humidity, which is mathematically represented in the psychrometric chart. The qualitative analysis was focused on covariational reasoning and the students’ processes of mathematical modelling of the real-life phenomenon under investigation. Findings from five representative episodes showed an interlacing of several forms of covariational reasoning, the emergence of qualitative, quantitative, and global characterizations of covariational reasoning, and three different roles of covariation throughout the various steps of the modelling activities. From an educational point of view, the modelling activities described here offer practical insights for the design of activities aimed at promoting the modelling of real-life phenomena through a covariational approach.