This paper aims to reveal the potential of inferentialism, an emerging philosophy in mathematics education, to extend contemporary constructivist research regarding conceptual development. Going against the traditional view of a notion as the name of a corresponding concept and a constructivist way of naming second-order models, we provide a new…

Mathematical knowledge and skills are critical in the development of students. In this article, we begin to unpack the concept of mathematics disabilities and how they are presented across classrooms. We explain commonly used terms such as, specific learning disability and dyscalculia, and emphasize how students qualify for math support. We…

The author has been influenced throughout his time in mathematics education by the work of Caleb Gattegno. Gattegno made extensive use of the word awareness whereas much educational literature from a psychological perspective talks about memory (for example, Justicia-Galiano, MartÌn-Puga, Linares & Pelegrina, 2017). This has, amongst other…

The purpose of this paper is to propose an effective learning environment for the initial stages of mathematical learning. Basic numerical skills and the objects and actions that trigger those skills are conceptualized as a mathematics-learning environment. We discuss numerical learning mechanism and the basic skills and environments we use to…

The purpose of this article is to describe and illustrate learning trajectories (LTs) and introduce a new, comprehensive set of resources that support teaching meaningfully and joyfully. An LT for a topic has three parts: (1) a mathematical goal; (2) a developmental progression of levels of thinking; and (3) teaching practices (environments,…

The author believes that in mathematics education, with ourselves AND with students or learners, people need to embrace failure as a beautiful thing -- an incredibly wonderful opportunity to learn more about the math or the tool or the people. The five characteristics of provocative objects can help to understand how to design such lovely failure,…

In this methodological paper, I present an enactivist approach to study how a student becomes coherently aware within a mathematics classroom, observing on the basis of my own observations of the learning of the students, acknowledging myself as part of the context, engaging with features of what I am seeing, demonstrating how through the…

This article is about utilizing the process of building and exploiting models as a framework for inquiry in mathematics education research. We introduce and discuss a model of such a process and compare this model with key methodological features of design research in mathematics education. We argue that despite several communalities, educational…

The notions of meaning and sense, which are closely related to understanding, play an essential role in educational processes in general and, therefore, in the teaching and learning of mathematics. However, the terms 'meaning' and 'sense' are unevenly used in the theory of language, semiotics, philosophy and psychology. After describing the main…

Learning progressions have become increasingly prevalent in mathematics education as they offer a fine-grain map of possible learning pathways a child may take within a particular domain. However, there is an opportunity to build upon this research in ways that consider learning from multiple perspectives. Many current forms of learning…

Learning trajectories (LTs) can inform teaching by contributing a variety of pedagogical and content-related insights and strategies. They support sequencing topic introduction and development. Because ideas evolve gradually and often rely on a careful introduction of new representations, operations, cases of numbers, structures, and definitions,…

Throughout my teaching career, I have believed it is important to develop positive relationships with my students and help my students develop positive relationships with each other. In my personal and professional experiences, I have observed the positive differences strong relationships have on how my students engage in class and how comfortable…

TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) is a multi-year, multi-institution collaborative project funded by the US National Science Foundation aimed at developing and testing student projects based on primary historical sources related to the original or subsequent development of core topics…

A subject of growing interest in mathematics education is the affective domain and its effects on the teaching and learning processes, giving rise to different models of its components and conditioning factors. In this paper, we apply the ontological and semiotic categories from the Onto-Semiotic Approach (OSA) to research in mathematics…

This paper illustrates that focusing on processes of interaction is crucial to a closer understanding of mathematical learning processes in mixed-ability groups. In doing so, the paper's focus is on expounding a theoretical-methodological framework of an interactionist perspective in mathematics education. This framework interlinks sociological…