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Capturing the breadth and variety of children's understanding is critical if studies of children's mathematical thinking are to inform policy and practice in early childhood education. This article presents an investigation of young children's counting. Detailed coding and analyses of assessment interviews with 476 preschoolers revealed understandings that would be overlooked by solely assessing the accuracy of their responses. In particular, many children demonstrated understandings of counting principles on a challenging task that were not captured by other, simpler tasks. We conclude that common approaches to capturing young children's mathematical understanding are likely underestimating their capabilities. This study contributes to researchers' understanding of what making sense of counting looks and sounds like for preschool age children (3-5 years), the development and relations among counting principles (one-to-one, cardinal, and patterns of the number sequence), and the affordances of challenging, open-ended tasks. We close by considering the implications of recognizing and building from what children know and can do for researchers, practitioners, and policymakers.

This survey-based study reported how the intensity of the beliefs of novice in-service teachers about reformbased mathematics instruction changed as they crossed the threshold into their first year of teaching. It also investigated the relationship between the influence of school-based mentors, or "transitional faculty," including mentor teachers, instructional coaches, and administrators, along with curriculum materials, and the changes in belief intensity. Results obtained indicated belief changes were highly influenced by transitional faculty and curriculum materials. This study highlights the importance of creating systems to improve mentor selection and training, so the work of university personnel is not undone during the novice's first year of teaching. [ABSTRACT FROM AUTHOR]

Understanding algebraic properties is a key component of building mathematical thinking across grades K-8, yet less is known about the development of students' use of mathematical properties within their strategies. This article presents results from four conversations with 24 5th-grade students over a school year, that focus on examining the development of students grouping within strategies for multiplication and division problems. Findings add to previous research on student strategies within multiplication and division by detailing some of the nuances in students' use of grouping. Additionally, a focus on student strategies reveals students' progression in more explicit use of grouping underlies the development of more planful use of the distributive and associative properties of multiplication. • A progression of grouping begins within count/add strategies and continues into invented algorithms. • Students' use of grouping within strategies gives evidence of their development of the use of algebraic properties. • More explicit and planful use of grouping connects to more planful use of the distributive and associative properties. • Student grouping and algebraic property use connect student strategies across whole number and fractional work. [ABSTRACT FROM AUTHOR]