The sandwich problem described in this article would likely be considered more appropriate for third grade or above. However, as members of the research team for a lesson study involving teachers from kindergarten through fourth grade, the authors saw kindergarten students rise to the challenge. The authors invite readers to review two scenarios,…

When the focus of instruction is on learning to use mathematical tools--such as a ruler or a calculator--as a procedure, students tend not to incorporate their own mathematical reasoning. This unreflective use of tools hinders students' questioning of the reasonableness of their answers. This article describes how children expressed their…

Paper folding is an easy and inexpensive way to engage students artistically, culturally, and mathematically. The pinwheel base is the source for many origami playthings, from a pinwheel to a boat to a butterfly. This article explores a fourth grade activity that repurposes the pinwheel base into a rich and inexpensive manipulative for the review…

Understanding the decimal system is challenging, requiring coordination of place-value concepts with features of whole-number and fraction knowledge (Moloney and Stacey 1997). Moreover, the learner must discern if and how previously learned concepts and procedures apply. The process is complex, and misconceptions will naturally arise. In a…

In this article, the authors first present the Hands Together! task. The mathematics in this problem concerns the relationship of hour and minute durations as reflected in the oft-overlooked proportional movements of the two hands of an analog clock. The authors go on to discuss the importance of problem solving in general. They then consider…

One of the most important jobs mathematics teachers have is supporting students and helping them to develop confidence in their own capabilities as mathematical thinkers. Children who do not believe in themselves are unlikely to share their ideas, persevere through challenges, or take risks that lead to new insights. Yet many students have no…

Is anything more refreshing than going for a nice, long swim? The math scenarios presented in this article will take the reader back to hot summer days and remind the reader what a cool dip in the water feels like. Solving these problems is enjoyable and encourages the solver to think of the many ways that math is all around--even in the middle of…

Children's intuitive understandings of mathematical ideas--both correct, generalizable strategies alongside misconceptions--showcase the complexity of their thinking. However, recognizing children as complex thinkers is one thing but it is another thing altogether to leverage their ideas to plan for and carry out mathematics instruction. The…

Acknowledging the critical role of mathematics coaches in enhancing teacher capacity, positively influencing teacher beliefs, and increasing teacher investment in professional development, NCTM's "Principles to Actions: Ensuring Mathematical Success for All" (2014) calls on coaches to "take action." However, coaches seek…

Students' success with fourth-grade content standards builds on mathematical knowledge learned in third grade and creates a conceptual foundation for division standards in subsequent grades that focus on the division algorithm. The division standards in fourth and fifth grade are similar; but in fourth grade, division problem divisors are only one…

A lack of fractional understanding is a well-documented obstacle to student achievement in upper elementary and middle school math (National Center for Educational Statistics [NCES] 1999; Lamon 1999; National Research Council [NRC] 2001). Lamon (1999) notes that one major conceptual hurdle that students must overcome is the idea that fractions are…

Facilitating productive discussions in the mathematics classroom is a challenge for many teachers. Discourse--student communication of mathematical ideas with teachers and peers--provides a platform on which students share their understanding, clarify misperceptions, and evaluate ideas. If students are unable to access and participate in…

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…

Two perspectives of the term "Venn diagram" reflect the typical differences in the uses of Venn diagrams in the subject areas of mathematics and language arts. These differences are subtle; nevertheless, they can potentially be confusing. In language arts, the circles in a Venn diagram typically represent things that can be compared and…

Based on a study with six elementary school teachers and 115 students in grades 3, 4, and 5, this article discusses ways that teachers might support an understanding of variable as students explore pattern-finding activities. The practices emerged from interviews with teachers and a yearlong observation of teaching algebraic thinking during an…