Children struggle with exact, symbolic ratio reasoning, but prior research demonstrates children show surprising intuition when making approximate, nonsymbolic ratio judgments. In the current experiment, eighty-five 6- to 8-year-old children made approximate ratio judgments with dot arrays and numerals. Children were adept at approximate ratio…

Fractions are a challenging mathematics topic for many elementary and middle school students, and even for adults. However, a growing body of developmental research suggests that young children can reason about visually presented proportions, well before fraction instruction, providing insight into how fractions might be introduced to improve…

Understanding fraction magnitudes is foundational for later math achievement. To represent a fraction "x/y," children are often taught to use "partitioning": break the whole into "y" parts, and shade in "x" parts. Past research has shown that partitioning on number lines supports children's fraction…

Understanding fraction magnitudes is foundational for later math achievement. To represent a fraction x/y, children are often taught to use "partitioning": Break the whole into y parts and shade in x parts. Past research has shown that partitioning on number lines supports children's fraction magnitude knowledge more than partitioning on…

This article illustrates and differentiates the unique role cognitive interviews and think-aloud interviews play in developing and validating assessments. Specifically, we describe the use of (a) cognitive interviews to gather empirical evidence to support claims about the intended construct being measured and (b) think-aloud interviews to gather…

This paper gives an example of how to address the challenge of designing a learning progression that describes student thinking, with the necessary specificity to align instructional opportunities and assessment tools. We describe the Conceptual Underpinnings of Evolution project and the iterative process of developing a novel learning progression…

By bringing everyday phenomena into the classroom, teachers can more readily engage students in authentic scientific inquiry. When working with young children, the best phenomena are those that students can directly experience and investigate. Meaningful phenomena can be identified by watching children at play, listening to the conversations they…

In the present study, the relationship between the mathematics grade and the three basic cognitive abilities (inhibition, working memory, and reasoning) was analyzed regarding possible alterations during elementary school. In a sample of N = 244 children, the mathematics grade was best predicted by working memory performance in the second grade…

This paper reports on exploratory research into visual imagery in the learning of early mathematics. It aims to present a theoretical position in relation to perceptual perspectives of visual imagery. Students' visual imagery is explored in one example of a teacher's use of a protocol to encourage students to replicate dot patterns. This example…

How people see the world, even how they research it, is influenced by beliefs. Some beliefs are conscious and the result of research, or at least amenable to research. Others are largely invisible. They may feel like "common knowledge" (though myth, not knowledge), unrecognized premises that are part of the surrounding culture. As we…

Process-oriented dynamic testing aims to investigate the processes children use to solve cognitive tasks, and evaluate changes in these processes as a result of training. For the current study, a dynamic complex figure task was constructed, using the graduated prompts approach, to investigate the processes involved in solving a complex figure task…

Background: Outcomes of static tests provide an indication of what children have learned in the past, up to the moment of testing, and can therefore underestimate the cognitive abilities of atypically developing children, such as children with language difficulties. In contrast, dynamic tests aim to examine children's potential for learning. The…

This study evaluated a new measure for analyzing the process of children's problem solving in a series completion task. This measure focused on a process that we entitled the "Grouping of Answer Pieces" (GAP) that was employed to provide information on problem representation and restructuring. The task was conducted using an electronic…

Recent studies reveal that children can solve proportional reasoning problems presented with continuous amounts that enable intuitive strategies by around 6 years of age but have difficulties with problems presented with discrete units that tend to elicit explicit count-and-match strategies until at least 10 years of age. The current study tests…

In this study we document how a mathematical idea, shared by a student, changes the development of the class discussion. Our work characterizes the flexible and open dynamics in mathematics instruction that a teacher adopts to give her students control of their own mathematical ideas. Our work argues that this flexibility is necessary for teachers…