Kyle, a kindergartner from a low-income family, is shown a set of three black dots on a white mat. His teacher then hides the dots with a small box lid and lays down an additional set of two dots. She pushes the two dots under the cover, one at a time. Kyle must now choose the number of dots "hiding" under the box from a set of four…

Representations that create informative visual displays are powerful tools for communicating mathematical concepts. The National Council of Teachers of Mathematics encourages the use of manipulatives (NCTM 2000). Manipulative materials are often used to present initial representations of basic numerical principles to young children, and it is…

Over the course of her teaching career, this author learned to create environments in which both the teacher and learners embrace understanding. She introduced new concepts with a general question or word problem and encouraged students to find solutions with a strategy of their choice. By using this instructional method, she allowed her students…

Mathematics professor Julie Cwikla decided that she needed to investigate young children's understandings and see what precurricular partitioning notions young minds bring to the fraction table. Cwikla realized that only a handful of studies have examined how preschool-age and early elementary school-age students solve fraction problems…

In this article, associate professor Chrystal Dean describes how teachers can challenge their upper elementary students' understanding of area beyond a memorized formula. Herein she describes an activity that will show students the "why" behind using A = l × w to solve rectangular area problems. The activity will help deepen…

In this article, fourth-grade teacher Tabetha Finchum describes how a program called Arizona Master Teacher of Mathematics (AZ-MTM), a Noyce grant funded through the National Science Foundation, helped boost her confidence and broaden her understanding of the philosophies, curricula, and instructional decisions being implemented by other teachers.…

Overgeneralizing commonly accepted strategies, using imprecise vocabulary, and relying on tips and tricks that do not promote conceptual mathematical understanding can lead to misunderstanding later in students' math careers. In this article, the authors present thirteen pervasive mathematics rules that "expire." With the…

Designed for a broad audience, including educators, camp directors, afterschool coordinators, and preservice teachers, this investigation aims to help individuals experience mathematics in unconventional and exciting ways by engaging them in the physical activity of building geometric shapes using ropes. Through this engagement, the author…

The Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) define what children should understand and be able to do in K-grade 12. This document also includes a description of key mathematical processes and proficiencies, the Standards for Mathematical Practice (SMPs), which provide an important overview for the kind of robust thinking…

The idea of elementary school students working together on mathematical tasks is not new, but recent attention to creating purposeful discourse in mathematics classrooms prompts teachers to revisit discourse-promoting strategies for mathematics lessons. The Common Core's Standards for Mathematical Practice (CCSSI 2010) encourage teachers to…

Geometry is the branch of mathematics that addresses spatial sense and geometric reasoning. Students begin to understand geometry through direct interaction with their physical world. Because it is the study of the physical attributes of the environment, geometry has relevance for every student; the world becomes a big classroom. As students see,…

The study of functions has traditionally received the most attention at the secondary level, both in curricula and in standards documents--for example, the Common Core State Standards for Mathematics (CCSSI 2010) and "Principles and Standards for School Mathematics" (National Council of Teachers of Mathematics [NCTM] 2000). However, the…

Thoughtful implementation of the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) presents an opportunity for increased emphasis on the development of mathematical understanding among students. Granted, ascertaining the mathematical understanding of an individual student is highly complex work and often exceedingly difficult.…

Technology permeates every aspect of daily life, from the sensors that control the traffic signals to the cameras that allow real-time video chats with family around the world. At times, technology may make life easier, faster, and more productive. However, does technology do the same in schools and classrooms? Will the benefits of technology…

The purpose of this article is to provide strategies for recognizing meaningful mathematics in common play contexts in early childhood classrooms and to offer suggestions for how teachers might intervene in these moments to help children attend to the mathematical ideas embedded in their play. In particular, the author's focus on the concepts…