This article describes how the authors developed the activity Speaqual to enable their students to "speak" about the "equal" sign and foster communication of their reasoning. The article explains how to play Speaqual and how the students were thoroughly engaged in a friendly competition, centralizing algebraic concepts and…

Semiotics is the study of symbols and the creation of their meaning through social interactions. In this paper, we explore more deeply the socially situated nature of semiotics by extending current semiotics theories and frameworks from the perspective of emergent modeling. The first expansion is to replace the interpretant (meaning associated…

Fostering students' bidirectional conception of the equal sign (viewing the equal sign as indicating an equivalence of two sides rather than a 'show result' symbol) is challenging, and students' misconception of the equal sign is persistent. Some studies mention that in China, the pedagogical approach to introducing the equal sign supports…

Incorporating visual representations, rather than strictly relying on symbolic representations, is a research-based strategy for supporting fraction learning. However, students must also make transitions between visual and symbolic fraction representations to apply the conceptual understanding they gained from visual representations to symbolic…

This study investigates how the approximate number system (ANS) and young children's symbolic skills jointly develop and interact. Specifically, the study aims at disentangling the directionality of the association between ANS acuity and a wide range of symbolic skills that reflect 4- to 5-year-olds' symbolic quantitative knowledge (enumeration…

We conducted a descriptive exploratory study in which we analyzed 313 sixth to eighth grade students' answers to a word problem, accompanied by diagrams, involving generalization in an algebraic functional context. In this research, we jointly addressed two objectives: (a) to determine the strategies deployed by students to generalize and (b) to…

Developing structure sense is an important part of learning algebra. We investigated learners' structure sense of algebraic expressions involving brackets. This led us to propose the constructs "surface structure" sense and "systemic structure" sense. Using a random sample of 58 Grade 10 learners scoring above 40% in a test, we…

This study sought to use indigenous ("Adinkra") artefacts to present "Concreteness Fading" in multiplication of one-digit and one-digit numbers. The researcher used simple random sampling technique to select 51 participants from 300 student teachers. Two sets of tests were used to collect the data. The results were equally…

Learning fractions is essential for academic and daily life success. A critical first step in acquiring fractions is learning to transcode them (e.g., writing ½ when hearing "one half"). However, little is known about how students master fraction transcoding. We addressed this gap by assessing fraction writing in two groups of Brazilian…

This paper presents data from the first of three iterations of teaching experiments conducted with secondary teachers. The purpose of the experiments was to investigate how teachers' combinatorial reasoning could support their development of algebraic structure, specifically structural relationships between the roots and coefficients of…

The ability to choose the larger between two numbers reflects a mature understanding of the magnitude associated with numerical symbols. The present study explores how the knowledge of the number sequence and memory capacity (verbal and visuospatial) relate to number comparison skills while controlling for cardinal knowledge. Preschool children's…

Although the roles of symbolic numerical magnitude processing (SNMP) and working memory (WM) in mathematics performance are well acknowledged, studies examining their joint effects are few. Here, we investigated the profiles of SNMP (1- and 2-digit comparison) and WM (verbal, visual and central executive) among Norwegian first graders (N = 256),…

Crystalline concept is the main concept used as the reference by students in algebraic verification. This concept divided the way of solving algebraic verification into two types: symbolic and embodied compression. This research aimed to explore the students' mathematical thinking process in solving algebraic verification based on the Crystalline…

Mathematical notations not only contribute to but also confound the development of the understanding of mathematical concepts (Leinhardt, Zaslavsky & Stein,1990). Inaccurate use of calculus notations is likely to have a significant influence on student conceptual understanding and student assessment scores in the Calculus 1 course. This…

This study examined the effects of training involving spatial versus nonspatial representations of numerical magnitude for promoting arithmetic fluency. The key goal was to advance theoretical understanding of the relation between spatial and math learning, while simultaneously laying the groundwork for the development of future educational…