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…

Across two experiments, we investigated how verbal labels impact the way young children attend to proportional information, well before the introduction of formal fraction education. Five- to seven-year-old children were introduced to equivalent non-symbolic proportions labeled in one of three ways: (a) a single, categorical label for multiple…

Many of the applied mathematics courses we teach touch upon a range of rich and important ethical issues -- issues that, for a number of reasons, are rarely openly discussed in class. In this work, we describe a sequence of activities co-designed and co-taught by philosophy and mathematics faculty in the hopes of bridging the divide between the…

We investigated the effect of conceptual transparency in the physical structure of manipulatives on place-value understanding in typically developing children and those at risk for mathematics learning disabilities. Second graders were randomly assigned to one of three manipulatives conditions: (a) attachable beads that did not make the…

A new classification for semantic structures of one-step word problems is proposed in this paper. The classification is based on illustrations of word problem situations in Common Core State Standards (CCSSM, 2010) and related historical studies (e.g. Weaver, 1973, 1979, 1982), as well as conceptual elaborations of embodied and grounded nature in…

This study utilized a partially nested randomized control design to investigate the impact of Fusion, a first-grade math intervention. Blocking on classrooms, students were randomly assigned to one of the three conditions: a Fusion two-student group, a Fusion five student group, or a no-treatment control group. Two primary research questions were…

Research Findings: The aim of this study was to investigate U.S. preschool teachers' math pedagogical content knowledge in number comparison. Seventy-four in- and pre-service teachers completed a set of scenario-based questions from a researcher-designed questionnaire that is composed of learning scenarios. Through a mixed-method approach, the…

Many tertiary institutions with mathematics programmes offer introduction to proof courses to ease mathematics students' transition from primarily calculation-based courses like Calculus and differential equations to proof-centred courses like real analysis and number theory. However, unlike most tertiary mathematics courses, whose mathematical…

In this article, we review research literature concerning number sense from several related fields. Whereas other authors have pointed to difficulty defining number sense or to some degree of inconsistency in the literature, we argue instead that this is a case of polysemy: three constructs given the same name. Our purpose is to clarify the…

Three rational number notations--fractions, decimals, and percentages--have existed in their modern forms for over 300 years, suggesting that each notation serves a distinct function. However, it is unclear what these functions are and how people choose which notation to use in a given situation. In the present article, we propose quantification…

How do children construct a concept of natural numbers? Past research addressing this question has mainly focused on understanding how children come to acquire the cardinality principle. However, at that point children already understand the first number words and have a rudimentary natural number concept in place. The question therefore remains;…

Recent work has probed the developmental mechanisms that promote fair sharing. This work investigated 2.5- to 5.5-year-olds' (N = 316; 52% female; 79% White; data collected 2016-2018) sharing behavior in relation to three cognitive correlates: number knowledge, working memory, and cognitive control. In contrast to working memory and cognitive…

In this paper, we provide an overall perspective on the teaching and learning of discrete mathematics. Our aim is to highlight what research has been conducted in this area and to connect it to existing research ideas for future work. We begin by characterizing discrete mathematics and its role in the school curriculum, highlighting themes,…

The study aims to investigate students' conceptualisation of the quantum numbers concept through their verbal descriptions and pictorial representations of the corresponding atomic structures. The study was conducted with the participation of 192 12th-grade students from six secondary schools in Northern Greece, via a paper-and-pencil test,…

Young children's symbolic magnitude understanding, or knowledge of how written numerals and number words can be ordered and compared, is thought to play an important role in their mathematical development. There is consistent evidence that symbolic magnitude skills predict mathematical achievement in later childhood and adulthood. Yet less is…