Children's knowledge of the ordinal relations among number symbols is related to their mathematical learning. Ordinal knowledge has been measured using judgment (i.e., decide whether a sequence of three digits is in order) and ordering tasks (i.e., order three digits from smallest to largest). However, the question remains whether performance on…

Investigators often rely on the proportion of correct responses in an assessment when describing the impact of early mathematics interventions on child outcomes. Here, we propose a shift in focus to the relative sophistication of problem-solving strategies and offer methodological guidance to researchers interested in working with strategies. We…

The purpose of this study was to investigate what factors third-grade students took into consideration when posing problems for their peers and how these factors affected the mathematical complexities of the problems. Free and semi-structured problem-posing tasks were given to 27 third-grade students, and the problems they created for their peers…

The present paper describes beautiful conservation relations between areas formed by different geometrical shapes and area relations formed by algebraic functions. The conservation properties were investigated by students at an academic college of education using a computerized technological tool and were subsequently accompanied by justified…

One of the fundamental topics taught in the microeconomics class is minimizing economic costs. It includes understanding the concept of derivatives and applying them. However, most of the first-year undergraduate students find calculus difficult to understand, which also results in poor knowledge of optimization. We use the method based on the…

When, how, and why students use conceptual knowledge during math problem solving is not well understood. We propose that when solving routine problems, students are more likely to recruit conceptual knowledge if their procedural knowledge is weak than if it is strong, and that in this context, metacognitive processes, specifically feelings of…

Mathematical knowledge is constructed hierarchically from basic understanding of quantities and the symbols that denote them. Discrimination of numerical quantity in both symbolic and non-symbolic formats has been linked to mathematical problem-solving abilities. However, little is known of the extent to which overlap in quantity representations…

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…

Understanding how environments influence learning requires attending not only to what is present but also to what is absent. In the context of mathematics learning, this means attending not only to problems that children encounter frequently in textbooks but also to ones that appear rarely. We present research in this article showing that students…

Understanding how environments influence learning requires attending not only to what is present but also to what is absent. In the context of mathematics learning, this means attending not only to problems that children encounter frequently in textbooks but also to ones that appear rarely. We present research in this article showing that students…

The development of mathematical reasoning is a key proficiency for mathematics within the Australian Curriculum. However, reasoning can be difficult for teachers to assess, particularly with pen and paper tests. In this study, interview tasks were designed across three curriculum areas at three different levels to assess student reasoning through…

In this article, the authors share frameworks for problem types and students' reasoning about integers. The authors found that all ways of reasoning (WoRs) were used across grade levels and that specific problem types tended to evoke particular WoRs. Specifically, students were more likely to use analogy-based reasoning on all-negatives problems…

Functional thinking is an established route into algebra. However, the learning mechanisms that support the transition from arithmetic to functional thinking remain unclear. In the current study we explored children's pre-instructional intuitive reactions to functional thinking content, relying on a conceptual change perspective and using mixed…

In Asia, some children are taught a calculation technique known as the 'mental abacus'. Previous research indicated that mental abacus experts can perform extraordinary feats of mental arithmetic, but it disagrees as to whether the technique improves working memory. The present study extended and clarified these findings by contrasting performance…

This article concerns the purpose, function, and mechanisms of students' rhythmic behaviors as they solve embodied-interaction problems, specifically problems that require assimilating quantitative information structures embedded into the environment. Analyzing multimodal data of one student tackling a bimanual interaction design for proportion,…