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…

Earlier approaches to sense-making in mathematics have looked at the ways students comprehend a mathematical concept. Recent research suggests that some students make sense not only of mathematical objects that have a being, but also of objects that have yet to become. In such cases, learning mathematics is not just an act of comprehending a given…

In this article, I argue that the common practice across many school mathematics curricula of using a variety of different representations of number may diminish the coherence of mathematics for students. Instead, I advocate prioritising a single representation of number (the number line) and applying this repeatedly across diverse content areas.…

In this paper we analyze common solutions that students often produce to isomorphic tasks involving proportional situations. We highlight some key distinctions across the tasks and between the different equations students write within each task to help elaborate the different interpretations of equivalence at play: numerical, transformational, and…

"Where Mathematics Comes From" (Lakoff & Núñez 2000) proposed that mathematical concepts such as arithmetic and counting are constructed cognitively from embodied metaphors of actions on physical objects, and four actions, or 'grounding metaphors' in particular: collecting, stepping, constructing and measuring. This article argues…

Pairing the mid-century work of Ada Dietz (1882-ca. 1970) with two compelling contemporary projects from Sonya Clark (1967- ), this article considers the ways in which normative mathematical ideas are remade through their engagement with weaving practice. Highlighting recent efforts to further ethnomathematics' original decolonial intentions, I…

Within a study of student reasoning in abstract algebra, we encountered the claim "division and multiplication are the same operation." What might prompt a student to make this claim? What kind of influence might believing it have on their mathematical development? We explored the philosophical roots of "sameness" claims to…

Critical to constructing and interpreting graphs is an individual's understanding of the underlying coordinate systems, yet coordinate systems are often overlooked or taken-for-granted in both mathematics education research and curricula. In this paper, we foreground coordinate systems and present a distinction between two uses of coordinate…

Literature has already shown that gestures play a relevant role in classroom interactions between students and teacher. We integrate the perspective of Theory of Semiotic Mediation with the notion of Semiotic Bundle to illustrate how gestures can be used as "pivot signs" in semiotic chains. This means that gestures can be performed by…

The aim of this article is to advise readers that natural numbers may be introduced as ordinal numbers or cardinal numbers and that there is an ongoing discussion about which come first. In addition, through several examples, the authors demonstrate that in the process of answering the question "How many?" one may, if convenient, use…

This paper seeks to integrate a scholarship of "ritual" with a scholarship of "improvisation" and relate the intersection thereof with mathematics teaching and learning and with mathematics education research. Drawing on their experiences as classroom teachers, as practicing improvisers, and as current education researchers,…

Certain aspects of theoretical frameworks in mathematics education can remain unexplained in the literature, hence unnoticed by the readers. It is thus interesting to participate in a dialogue where they can be discussed and compared in terms of the aims, objects studied, results and their relation to teaching. This study is focused on how APOS…

How do students cope with and make sense of polysemy in mathematics? Zazkis (1998) tackled these questions in the case of 'divisor' and 'quotient'. When requested to determine the quotient in the division of 12 by 5, some of her pre-service teachers operated in the domain of integers and argued for 2, while others adhered to rational numbers and…

The literature concerning students' understanding of the equal sign has focused narrowly within the context of formal mathematics. Researchers have put forth a hierarchy of conceptions of the equal sign with only the top level regarded as correct. Meanwhile, we find that the equal sign is used widely and in a variety of ways in advertising and…

This story is a playful retelling of ideas related to infinity. Presented as a historical fiction, the story reflects the thinking of research participants who addressed the ping pong ball conundrum, and where indicated, the individuals who contributed to modern formal understandings of infinity. This story offers a way of engaging with questions,…