The present study examined patterns of number-related utterances and actions directed to 9-month-old infants by their parents. An ethnically and economically diverse sample of 86 families participated in structured play sessions conducted separately with the mothers and fathers of each infant. Data were coded for eight math talk categories, and…

The present study tests two predictions stemming from the hypothesis that a source of difficulty with rational numbers is interference from whole number magnitude knowledge. First, inhibitory control should be an independent predictor of fraction understanding, even after controlling for working memory. Second, if the source of interference is…

Competence in fractions is important in achieving advanced mathematics such as algebra and calculus. To foster students' understanding of fractions, intervention studies have found number lines to be an effective representational tool. Yet, it is unclear whether or not number lines are equally effective regardless of the ways in which they are…

This article reports on students' problem-solving approaches across three representations--number lines, coordinate planes, and function graphs--the axes of which conventional mathematics treats in terms of consistent geometric and numeric coordinations. I consider these representations to be a part of a "hierarchical representational…

This report summarizes the findings across two case studies conducted as part of WestEd's evaluation of the California Statewide Early Math Initiative (CAEMI). The case studies aimed to understand the planning and implementation of professional learning and coaching in local communities. They also examined changes in educators' math knowledge,…

Before children learn the specific meanings of numerals like six, do they know that they represent precise quantities? Previous studies have reported conflicting evidence and have found that children expect numerals to label precise quantities in some tasks but not in others (Condry & Spelke, 2008; Sarnecka & Gelman, 2004). In this…

We interviewed 40 students in Grade 7 to investigate their integer reasoning. In one task, children were asked to write and interpret equations related to a story problem about borrowing money from a friend. Their responses reflect different perspectives concerning the relationship between this real-world situation and various numerical…

This paper presents the results of a pre-test, instruction, post-test study that investigated students' integer mental models and how their models changed based on instruction. Sixty-one first graders' responses to questions about the values and order of negative numbers were categorized according to a series of mental models. The models reveal…

We identify and document 3 cognitive obstacles, 3 cognitive affordances, and 1 type of integer understanding that can function as either an obstacle or affordance for learners while they extend their numeric domains from whole numbers to include negative integers. In particular, we highlight 2 key subsets of integer reasoning: understanding or…

Tasks that ask students to label rational number points on a number line are common not only in curricula in the upper elementary school grades but also on state assessments. Such tasks target foundational rational number concepts: A fraction (or a decimal) is more than a shaded part of an area, a part of a pizza, or a representation using…

We interviewed 40 students each in grades 7 and 11 to investigate their integer-related reasoning. In one task, the students were asked to write and interpret equations related to a story problem about borrowing money from a friend. All the students solved the story problem correctly. However, they reasoned about the problem in different ways.…

A single-case multiple baseline design was used to evaluate the effects of supplemental mathematics instruction with Numicon on the number sense skills of kindergarteners with varying degrees of math difficulty. Nine students participated in the study. Intervention consisted of 20 minutes of supplemental mathematics instruction using the…

How can children's natural perceptuo-motor skills be harnessed for teaching and learning mathematical structure? We address this question in the case of the integers. Existing research suggests that adult mental representations of integers recruit perceptuo-motor functionalities involving symmetry. Building on these findings, we designed a…

The Educational Policy Improvement Center (EPIC) conducted an investigation of the Intersegmental Committee for the Academic Senates (ICAS) Statements of Competencies for Mathematics and Academic Literacy. The purpose of this work is to understand how the ICAS competencies relate to college and career readiness, as represented by the augmented…

The California Department of Education is in the process of revising the Career and Technical Education (CTE) Model Curriculum Standards. The Educational Policy Improvement Center (EPIC) conducted an investigation of the draft version of the Health Sciences and Medical Technology Standards (Health Science). The purpose of the study is to…