The teacher displayed counting cards that included both dots and numerals in order from one to five, as she counted them with her students. She then turned the cards facedown, keeping them in order, and began an identify-a-hidden-card activity with the class. This class was engaged in the third of three card activities that develop number sense…

Use young children's quick attention to numerosity to evaluate their grasp of number while they engage in game play.

A single fact-focused number-string discussion is unlikely to result in all students adopting more flexible and efficient fact strategies; however, these authors have found that repeated engagement in such discussions over time does promote student reasoning about relationships, which in turn leads to increased fact fluency. By cultivating a…

The standards from the Operations and Algebraic Thinking (OA) domain in the Common Core State Standards for Mathematics (CCSSM) build on one another across the grades (CCSSI 2010). Understanding each grade level's OA standards is necessary within the broader context of the K-grade 5 OA standards. In this article, the authors share their…

The Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) outlines ambitious goals for fraction learning, starting in third grade, that include the use of the number line model. Understanding and constructing fractions on a number line are particularly complex tasks. The current work of the authors centers on ways to successfully…

This article reports on a study in which researchers asked children to "make up as story and a picture about marbles for this number sentence: 3 x 5 = 15." Students in this study came from pre - dominantly low- to average-income families living in three distinct geographical areas within the United States. A similar division task was…

This article highlights four routines that use visual images and open questions to invite students to discuss, justify, make arguments, and question their own thinking as well as their classmates' thinking. The authors share specific classroom examples from each routine that highlight the meaningful discourse they can inspire. Teacher prompts and…

This article examines the importance of developing the notion of place value as a rate of ten. In exploring how to nurture this concept, the author looks at the role of the language of value, the problem types of multistep multiplication and addition along with measurement division, each with ten as an organizing unit, as well as strategically…

Skilled instructors observe consistent errors that young children may make in particular situations and seek ways to help children reform their thinking through conceptual understanding. Such teacher observations as those presented at the beginning of this article, combined with similar observations by teachers in other preschool and kindergarten…

This article introduces the formative assessment probe--a powerful tool for collecting focused, actionable information about student thinking and potential misconceptions--along with a process for targeting instruction in response to probe results. Drawing on research about common student mathematical misconceptions as well as the former work of…

As students progress from working with whole numbers to working with integers, they must wrestle with the big ideas of number values and order. Using objects to show positive quantities is easy, but no physical negative quantities exist. Therefore, when talking about integers, the author refers to number values instead of number quantities. The…

When asked to determine the number of tens in twenty-five, most second graders who have had instruction on place value can quickly provide the correct answer of two. However, when asked to show how the numeral 2 is represented in a set of twenty-five objects, many children struggle to draw a connection between the digit 2 and twenty objects in the…

Young children spend a much greater amount of time on practicing multiplication facts compared to understanding the concept of multiplication. When students have long-term, foundational concepts rather than a series of fragmented algorithms or facts, they are more likely to understand and generalize the mathematics. Using generalized models that…

Mrs. Fisher's class was learning about arithmetic functions by pretending to operate real-world "function machines" (Reeves 2006). Functions are a unifying mathematics topic, and a great deal of emphasis is placed on understanding them in prekindergarten through grade 12 (Kilpatrick and Izsák 2008). In its Algebra Content Standard, the…

In 2000, the "National Council of Teachers of Mathematics" recommended that Algebra Standards, "instructional programs from prekindergarten through grade 12 should enable all students to use mathematical models to represent and understand quantitative relationships." In this article, the authors suggest the "Balance"…