This study investigated discursive positioning moves that facilitated Latino/a English learners' (ELs) opportunities to take on agentive problem-solving roles in group mathematical discussion. A focus on mechanisms that support students' agentive participation is consistent with our view that recurrent experiences participating and being positioned in particular ways contribute to identity development. Findings suggest several ways that discursive positioning facilitated ELs' agentive participation, including via: (a) explicit statements that validated ELs' reasoning, (b) invitations to share, justify, or clarify thinking that positioned ELs as competent problem solvers, and (c) inviting peers to respond to an EL's idea in ways that positioned the idea as important and/or mathematically justified.

Although equal sharing problems appear to support the development of fractions as multiplicative structures, very little work has examined how children's informal solutions reflect this possibility. The primary goal of this study was to analyze children's coordination of two quantities (number of people sharing and number of things being shared) in their solutions to equal sharing problems and to see to what extent this coordination was multiplicative. A secondary goal was to document children's solutions for equal sharing problems in which the quantities had a common factor (other than 1). Data consisted of problem-solving interviews with students in 1st, 3rd, 4th, and 5th grades (n = 112). We found two major categories of strategies: (a) Parts Quantities strategies and (b) Ratio Quantities strategies. Parts strategies involved children's partitions of continuous units expressed in terms of the number of pieces that would be created. Ratio strategies involved children's creation of associated sets of discrete quantities. Within these strategies, children drew upon a range of relationships among fractions, ratio, multiplication, and division to mentally or physically manipulate quantities of sharers and things to produce exhaustive and equal partitions of the items. Additionally, we observed that problems that included number combinations with common factors elicited a wider range of whole-number knowledge and operations in children's strategies and therefore appeared to support richer interconnections than problems with relatively prime or more basic number combinations.

In this study, the author represented what matters to bilingual students in their everyday lives-namely bilingualism and everyday experiences-in school-based mathematical problems. Solving problems in pairs, students demonstrated different patterns of organizing and coordinating talk across problem contexts and across languages. Because these patterns bear consequences for how bilinguals experience mathematics learning, the author takes these patterns as the basis to argue that mathematics education for bilingual students should capitalize on bilingualism and experiences as cognitive resources.

No abstract

This research focuses on eliciting bilingual students' problemsolving reasoning by providing mathematical tasks designed to involve the students in the action described in the problem by making connections to contexts familiar to them. Results showed these tasks allowed children to articulate their mathematical reasoning through gestures and speech. Two questions guided this investigation: (a) How do second-grade bilingual students communicate their mathematical reasoning? and (b) What is the role of the mathematical tasks in eliciting this mathematical reasoning? Analysis of seven second-grade Spanish-English bilingual students solving addition and subtraction problems showed that students simultaneously used words and gestures to communicate their mathematical reasoning to others and to regulate their own cognitive activity. In general, the students demonstrated that their developing bilingual proficiency did not constitute an impediment for guiding mathematical tasks to resolution; rather, students imparted mathematical meaning to each task by mutually supporting their verbal and nonverbal behaviors. The paper discusses how researchers and teachers can benefit from parallel attention to bilingual students' verbal and nonverbal communication. Both processes invite reflection on what it means to know and learn mathematics bilingually in early grades.