The first author's initial attempts at a whole-class mathematics discussion were woefully dysfunctional. Ten second graders showing their answers to a missing addend question about the number of children left riding a school bus did not result in the mathematical epiphany that the textbook promised. She did not give up on using mathematical…

In a classroom teaching experiment, we explored fifth graders' ability to identify relationships in and between two patterns--what we may refer to as functional relationships. Conventional functions curricula are dominated by defining relationships between two patterns via a rule, describing how to find y or f(x) given a particular value for x…

After teaching the concept of division with remainders for many years, Vanessa Battreal, Vanessa Brewster, and Juli Dixon noticed that interpreting remainders in contextualized problems is particularly challenging for elementary school students. To effectively interpret remainders, students must attend to problem context and consider carefully…

In many classrooms, students solve problems posed by others--teachers, textbooks, and test materials. These problems typically describe a contrived situation followed by a question about an unknown that students are expected to resolve. Unsurprisingly, many students avoid reading these problems for meaning and instead engage in a suspension of…

A second-grade class visits Saint John's Food Pantry each year and spends the morning learning about the needs of the community. Then students pitch in and help the staff for a few hours. When the children return to school, they create signs advertising a canned food drive, and they place collection boxes at various locations around the school.…

In this article, the authors discuss a special type of multiplication-and-division-of-fractions problem that elementary school teachers can use to promote children's understanding of fractional quantities and their relationships. These problems are accessible to students working at different levels of fraction understanding, and they can be solved…

Assessing student learning traditionally involves determining whether students can solve a certain percentage of problems correctly, under the assumption that this achievement indicates they have the knowledge and understanding they need to progress to new topics. This article explores what teachers can learn from looking closely at student…

Supporting students to develop an understanding of the meaning of fractions is an important goal of elementary school mathematics. This involves developing partitioning strategies, creating representations, naming fractional quantities, and using symbolic notation. This article describes how teachers can use a formative assessment problem to…

"Ensuring Mathematical Success for All" (NCTM 2010) outlines eight teaching practices for effective teaching and learning of mathematics. One of the teaching practices, building procedural fluency from conceptual understanding, states the following: Effective teaching of mathematics builds fluency with procedures on a foundation of…

This article describes a process called "Backing Up" which is a way to preassess student understanding of a topic and gauge student readiness to move forward in the learning process. This process of backing up begins with using responses to a word problem to identify categories of students' understandings in relation to the expectations…

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…

Recognizing the challenges that students face when they may lack background and experiential knowledge with regard to coins is important so that teachers can address this during instruction. Exposure to coins is one of the basic necessities to learning about the coins, but that is just the initial part of the learning sequence. Unless students are…

Mathematical modeling is an important and accessible process for elementary school students because it allows them to use mathematics to engage with the world and consider if and when to use it to help them reason about a situation. It fosters productive struggle and twenty-first-century skills that will aid them throughout their lifetime.

This investigation presents students with the challenge of determining the total number of eggs in a photograph of a stack of egg trays. The perspective of the photograph does not allow students to count all the individual eggs in the top tray but does allow them to see the number of eggs in one row and one column of the top tray. During this…

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…