In this paper, the aim is to make a theoretical contribution by focusing on the origin, characteristics, and potential values of the concepts of instrumental and relational understanding. Five characteristics are identified to make it easier to operationalise the concepts and use them as an analytical framework. There is also a focus on how the…

Transitioning from material to theoretical geometry is often considered a significant challenge in compulsory education. Over the last twenty years, research rooted in the Theory of Didactical Situations (TDS) has explored how material geometry can support the teaching and learning of theoretical geometry, promoting continuity in teaching. These…

Since statistical inference is a probabilistic generalization about a population analyzed on the basis of a sample, inferential reasoning demands producing reasons ("statistical" and "contextual") to substantiate and validate generalizations. To convey an understanding of students' inferential reasoning, we present a…

Grasping mathematical objects as related to processes is often considered critical for mathematics understanding. Yet, the ontology of mathematical objects remains under debate. In this paper, we theoretically oppose internalist approaches that claim mental entities as the endpoints of process-object transitions and externalist approaches that…

Within the commognitive perspective, ritual and explorative routines are used in a very particular way to distinguish students' routines according to whether they are driven by social reward or by generating a substantiated narrative. Explorative routines in this theorisation may refer not to inquiry-based activity but to the result of a student's…

This paper combines recent developments in theories of knowledge (complex dynamic systems), technologies (embodied interactions), and research tools (multimodal data collection and analysis) to offer new insights into how conceptual mathematical understanding can emerge. A complex dynamic system view models mathematics learning in terms of a…

This study assessed the effectiveness of a spatialized mathematics intervention on the mathematical performance and spatial visualization skills of students from an underprivileged Indonesian community. Grade 8 students (N = 407) were assigned to one of twelve experimental classes, receiving the spatial mathematics intervention, or one of seven…

Thirty-seven third graders and thirty-two first graders engaged in solving a tangram puzzle in the shape of a fox. They had five minutes to solve the puzzle, and after this time, they received guidance on the particular piece they had difficulties with. Through the lenses of navigating flexible abstraction, reinterpretation, combinations, and…

We are interested in understanding how university students learn to use programming as a tool for "authentic" mathematical investigations (i.e., similar to how some mathematicians use programming in their research work). The theoretical perspective of the instrumental approach offers a way of interpreting this learning in terms of…

The aim of this research is to study how students recognise the concept of gradient of a function of two variables in its different representations and how they make conversions between these representations. Three groups of university students with different profiles took part in this research, and Duval's theory of semiotic representations was…

Representations are key to mathematical activities and meaning-making processes as they are part of modeling, connecting, communicating, and understanding mathematical ideas and concepts. The current study sought to examine a group of novice algebra learners' interactions with different representations from an onto-semiotic approach. A case study…

Argumentative problem solving in mathematics classrooms is a crucial practice that supports important student learning goals via collaborative deliberation and consensus building, but also places substantial cognitive and affective demands on both students and teacher. In this in-depth qualitative study, we considered how students' emotions during…

We investigate how students make sense of irrational exponents. The data comprise 32 interviews with university students, which revolved around a task designed to examine students' sensemaking processes involved in predicting and subsequently interpreting the shape of the graph of f(x)=x[superscript square root of 2]. The task design and data…

Noticing the strengths in students' mathematical thinking is a critical skill that teachers need to develop, but it can be challenging due to the prevalence of deficit-based thinking in mathematics education. To address this challenge, a teacher education course was designed to encourage prospective teachers to engage in critical reflection on…

This paper substantiates the notion of inclusive materialism by illustrating the role of signs and micro-perception in the learning process of multiplication. Inclusive materialism, offered by de Freitas and Sinclair, considers mathematical learning a process of becoming a learning assemblage, which embodies various materials in a Deleuzian sense.…