This study examines the solutions of 34 kindergarten children as they create equal groups from n bottle caps, where n was equal to 8, 9, 22, and 23. For each n, children were asked to find as many different solutions as possible. The number of solutions they found, i.e., children's fluency, as well as the strategies used to create equal groups,…

We are all fascinated by loops and their formation in space. When a line cuts itself, it forms an intersection point and creates a space. This is an experimental study done by analyzing several loops, forming a concrete formulation by visualizing the patterns observed, and then proving the formulations proposed using the known standard…

Mathematical coherence is a goal within the Common Core State Standards for Mathematics. One aspect of this coherence is how student mathematical thinking is developed across concepts. Unfortunately, mathematics is often taught as isolated ideas across grades. The multiplicative field is an area of study that needs to be examined as a space to…

Studies have found that some teacher professional development programs that are based on Cognitively Guided Instruction (CGI) can increase student mathematics achievement. The mechanism through which those effects are realized has been theorized, but more empirical study is needed. In service of this need, we designed a novel measure of…

Understanding of the equal sign is associated with early algebraic competence in the elementary grades and equation-solving success in middle school. Thus, it is important to find ways to build foundational understanding of the equal sign as a relational symbol. Past work promoted a conception of the equal sign as meaning "the same as".…

Around the globe, young students are expected to learn about time, yet how is it that they themselves make sense of this topic? From a sociocultural perspective, sense-making about time emerges in relation to properties of available tools and representations, such as analog clocks or digital notation. Such interactions with the symbols and…

When teaching the multiplicative domain in New Zealand primary schools, teachers tend to spend a greater proportion of time on the meaning and processes of multiplication, to the detriment of a specific focus on understanding the concept of division. When division is taught, it tends to be by reversing the context and turning the division…

"Productive Struggle" refers to a strategy that gives students an opportunity to increase their background mathematical knowledge. Productive Struggle helps students connect key concepts, determine how and where an error occurs, and supports students in figuring out how to use their own thinking and reasoning skills to correct an error.…

Background: Problem-solving as a vehicle to develop independent thinking skills is mostly underestimated and is often either overlooked or not given adequate attention within the existing South African mathematics curriculum. Consequently, numerous learners often display limited skills or lack skills to adequately crack Mathematics problems by…

The aim of this study is to reveal the meanings that fourth grade students in elementary school attach to the four operation symbols (+, -, x, :) that are abstract, in different representations. The pattern of the study is a case study of qualitative research methods. In the study, the evaluation tool developed by the researchers was applied to…

The purpose of this study is to compare the differences of four essential aspects (i.e., representational forms, contextual features, cognitive demand levels, and response types) embedded in mathematical problems within the topics of numbers and operations in mathematics textbooks used in Finland, Indonesia, Malaysia, Singapore, and Taiwan. This…

This article reports on an exploration of how second-graders can learn mathematics through programming. We started from the theory that a suitably designed programming language can serve children as a language for expressing and experimenting with mathematical ideas and processes in order to do mathematics and thereby, with appropriate tasks and…

Numeracy is one of the essential competencies that the objectives of teaching math to primary students should be towards. However, many research findings show that the problem of "innumeracy" frequently exists at primary schools. That means children still do not feel at home in the world of numbers and operations. Therefore, the paper…

Conceptual challenge is often considered a necessary ingredient for promoting deep learning in an inquiry-based environment. However, challenge alone does not support conceptual development. In this paper, we draw on complexity theory as a theoretical lens to explore how a primary teacher facilitated students' conceptual change through repeated…

Problem solving has been the focus of the Common Core State Standards for Mathematical Practice. Helping students acquire critical-thinking and problem-solving skills has become the primary goal of mathematics education across all grade levels. However, research has found that many students struggle with word problems because of poor text…