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…

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…

In this article, the authors discuss their approach to column addition and subtraction algorithms. Adapting an original idea of Paul Cobb and Erna Yackel's from "A Contextual Investigation of Three-Digit Addition and Subtraction" related to packing and unpacking candy in a candy factory, the authors provided an analogous context by designing…

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,…

Considerable evidence shows that the number line is a powerful learning tool for children in elementary school. Diezmann and Lowrie (2006) noted several cognitive advantages for users, including opportunities to demonstrate the continuity aspect of numbers as well as the provision of a useful tool for representing and solving problems. However,…

In the third century, Diophantus, the "Father of Algebra" no less, described equations of the form x + 20 = 4 as "absurd." The absurdity stemmed from the fact that the result of four is obviously less than the addend of twenty. And more than 1300 years later, Pascal argued that subtracting four from zero leaves zero because of the impossibility of…

The topic of fractions is an important concept in elementary school mathematics. The study of fractions not only enables students to perform computations but also provides a foundation for later work with rates, percents, slope, and other topics in secondary school mathematics. However, research has long reported that many students and even adults…

How a school is organized--or, more specifically, the extent to which teachers work together and treat one another as professionals--can influence elementary school students' mathematics achievement. So suggest the authors of a recent study on the impact of school organization on longitudinal student achievement (Moller, Stephanie, Roslyn A.…

In late 2009, the National Council of Teachers of Mathematics released "Focus in High School Mathematics: Reasoning and Sense Making" (NCTM 2009). This new NCTM policy publication is designed to offer guidance for high school mathematics teachers in providing focus and coherence that parallels what "Curriculum Focal Points for Prekindergarten…

For nearly three decades, during which the author taught everything from basic algebra to advanced placement calculus, the author thought of himself as a secondary school mathematics teacher. The notion of teaching elementary school math never appealed to him because of its simplicity. The author stresses that anyone could teach children to count,…

"The World Is Flat" (Friedman 2005) describes the globalization that advances in technology have imposed on the world economy in recent decades. For the U.S. to maintain its stature in the world, future citizens must be prepared to problem solve and apply their skills to new situations. These future citizens are sitting in elementary school…

Although increasing emphasis is being placed on mathematical justification in elementary school classrooms, many teachers find it challenging to engage their students in such activities. In part, this may be because the teachers themselves have not had an opportunity to learn what it means to justify solutions or prove elementary school concepts…