Educators often speak of teaching as "listening," yet they seldom encounter it in practice. What converts pedagogical space into authentic educational experience is a climate of respect that is born of serious, just, humble, and generous relationships in which both the authority of the teacher and the freedom of the students are ethically…

Representations play a key role in mathematical thinking: They offer "a medium" to express mathematical knowledge or organize mathematical information and to discern mathematical relationships (e.g., relative household expenditures on a pie chart) using text, symbols, or graphics. They also furnish "tools" for mathematical processes (e.g., use of…

By providing sustained, embedded professional development over the course of a school year--as well as involving teachers in creating high-quality assessment tasks, implementing the assessments with students, and tying assessment directly to instruction--the authors increased the value that their teaching community places on mathematical…

Children are naturally curious and want to make sense of their world. To implement mathematical tasks that nurture children's desire to reason, it is valuable for teachers to have experienced for themselves comparable tasks and learning environments (Ball and Bass 2000). In this article, the author describes three strategies to facilitate…

"Teaching mathematics from a sense-making perspective entails more than solving non-routine problems" (NCTM 2007, p. 85). When teachers want students to see mathematics as something that makes sense, they must furnish opportunities for them to solve problems in their own way and use structures that nurture their reasoning and sense making. The…

"Classroom number talks," five- to fifteen-minute conversations around purposefully crafted computation problems, are a productive tool that can be incorporated into classroom instruction to combine the essential processes and habits of mind of doing math. During number talks, students are asked to communicate their thinking when presenting and…

The crucial need for a mathematically literate society, coupled with a sustained focus on mathematics standards, continues its center-stage presence. At the same time, "Principles and Standards" states that it is imperative to offer "all students high-quality programs that include significant mathematics presented in a manner that respects both…

Traditionally, the first mathematical task for primary grade students to master is addition. Mastering addition facts is truly a positive experience. Then one turns to subtraction. After mastering addition facts, many students think that subtraction facts are a whole new set of facts to learn that have nothing to do with addition facts. They do…

In their overview for the prekindergarten-grade 2 Standards, the National Council for Teachers of Mathematics (NCTM) documents the value of early mathematical environments. During these early years, young children are building beliefs about what mathematics is and learning about themselves as early mathematicians. What young children learn about…

Regular education teachers may have received inadequate preparation to work with the variety of student disabilities encountered in the classroom, or they may have received limited training regarding the full range of learning disabilities and their effects on classroom performance. Along with problems in the area of math, students may also have…

Traditionally, learning basic facts has focused on rote memorization of isolated facts, typically through the use of flash cards, repeated drilling, and timed testing. However, as many experienced teachers have seen, "drill alone does not develop mastery of single-digit combinations." In contrast, a fluency approach to learning basic addition…

Exploring number systems of other cultures can be an enjoyable learning experience that enriches students' knowledge of numbers and number systems in important ways. It helps students deepen mental computation fluency, knowledge of place value, and equivalent representations for numbers. This article describes how the author designed her…

As elementary school students attempt to understand mathematical concepts, engaging in the processes of reasoning and sense making is important. To do so, students should be expected to listen to and challenge their classmates' ideas. Disagreements provide students with the impetus to think deeply about mathematics in an effort to make sense of a…

When a child understands number relationships, he or she comprehends the meaning of numbers by developing multiple, flexible ways of representing them. The importance of developing number relationships in the early years has been highlighted because it helps children build a good foundation for developing a more sophisticated understanding of…

Developing an understanding of place value and the base-ten number system is considered a fundamental goal of the early primary grades. For years, teachers have anecdotally reported that students struggle with place-value concepts. Among the common errors cited are misreading such numbers as 26 and 62 by seeing them as identical in meaning,…