Teachers often use number talks at the beginning of class to hear the ways that students understand mathematical topics so that they can base future instruction on student thinking. Students who engage in number talks become accustomed to thinking about mathematics in multiple ways, sharing their thoughts and critiquing the thoughts of others. The…

A framework using contexts, language, and representations can support middle school students' justifications. This article introduces classroom tasks and student work to provide a vision of framework implementation in classroom settings. Although the students in this article are future elementary and middle school teachers, the framework tasks and…

Two studies (Trautwein 2007 and Dettmers et al. 2010) show a positive correlation between high-quality homework and mathematics achievement. Students who completed their homework assignments scored better on assessments. However, these studies also showed no relationship between time spent on homework and resulting student achievement. This helped…

In April 2018, the National Council of Teachers of Mathematics (NCTM) released "Catalyzing Change in High School Mathematics: Initiating Critical Conversations," a long-needed and well-articulated position for transforming the core purposes, structures, instructional practices, essential mathematics concepts, and organization of high…

In this article, the authors engage middle-grades students in a series of tasks to develop the mathematical idea of equal versus equivalent, culminating in a gerrymandering task with social, political, economic, and educational implications. The primary mathematical goals of this exploration were to involve students in solving real-world problems…

Modeling is an effective tool to help students access mathematical concepts. Finding a math teacher who has not drawn a fraction bar or pie chart on the board would be difficult, as would finding students who have not been asked to draw models and represent numbers in different ways. In this article, the authors will discuss: (1) the properties of…

In their ongoing professional development project, the authors had an opportunity to work with a group of teachers who were particularly insightful. These teachers shared written reflections about their thinking, prior experiences, and changing self-awareness of their mathematics practice. In one session, teachers were provided the…

The vision of mathematics learning advocated by the National Council of Teachers of Mathematics (NCTM) for twenty-five years (1989, 1991, 2000, 2014) positions students as active learners, constructing their knowledge of mathematics through exploration, discussion, and reflection. World-class standards put into place by states and provinces over…

The authors explore how one teacher uses play and enacts the Effective Teaching Practices as a way to increase student engagement and autonomy and to meet learning goals when teaching mathematics with interactive simulations.

Percentages have proven to be a challenging concept in school mathematics. At the surface, a percentage is merely a rational number, representing a ratio between a number and 100. At the conceptual core, however, a percentage is sensitive to the context, making sense with respect to a network of related quantities. In the effort of the authors to…

Middle-grades teachers and students can have different perspectives on the value of discussing students' mathematical mistakes, despite various classroom evidence that such discussions can help foster strong conceptual understanding. Some teachers consider student mistakes to be an opportunity to correct errors in individual student thinking.…

Two mathematics teachers in a small rural school decided to create a task that would engage seventh graders. The goal of the real-world activity was to help students develop geometric and spatial reasoning and to support their understanding of volume of rectangular prisms. The impetus for the task came from the teachers' desire to engage students…

Inquiry-based instruction is a student-centered approach to teaching that focuses on active learning (Barron and Darling-Hammond 2008) in which students engage with "tasks that promote reasoning and problem solving" (NCTM 2014). Specifically, such tasks encourage a variety of solution strategies and stimulate use of the NCTM Process…

Middle school students need opportunities to craft informal mathematical arguments beyond justification of steps in an algorithm. Pentominoes provide an excellent vehicle for such activity. In this article, the authors describe an exploration with pentominoes that engaged a group of ninth-grade students in proportional reasoning, dilation, and…

The Puppy Love problem asked fifth and sixth grade students to use their prior knowledge of measures of central tendency to determine a data set when given the mean, mode, median, and range of the set. The problem discussed in this article is a task with a higher level of cognitive demand because it requires that students (1) explore and…