This article explores three processes involved in attending to evidence of students' thinking, one of the Mathematics Teaching Practices found in "Principles to Actions: Ensuring Mathematical Success for All." These processes, explored during a classroom activity on proportional relationships, are discussed in this article, another…

Ever try to captivate students with real-world applications only to discover that they live in a very different world? Assigning an Extreme Word Problem activity to students alleviates this dilemma by placing the authorship of a problem squarely into the hands of learners. Rather than trying to be hip and finding word problems that resonate with…

This article explores facilitating meaningful mathematics discourse, one of the research-based practices described in "Principles to Actions: Ensuring Mathematical Success for All." Two tools that can support teachers in strengthening their classroom discourse are discussed in this, another installment in the series.

Take a moment to think about the students class. What are their mathematical strengths? How can a teacher know a mathematical strength when they see it? Can teachers identify at least one mathematical strength for every student? More important, how are teachers using students' strengths in their mathematics classrooms? Teaching is a complex…

Engaging in argumentation can lead to an increase in the development of critical-thinking skills, not to mention academic achievement, emotional maturity, and academic and communication skills. If argumentation is such an effective format for developing rigorous academic ideas, how might it be used in discussion-oriented classrooms to engage…

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…

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…

Watching a debate can help people understand multiple points of view. At the conclusion of a debate, a consensus might occur after a deep analysis and discussion of the issues. Most students will participate in a class debate at some point in their schooling, but chances are that debate will not take place in math class. What could there be to…

A learning trajectory describes the progression of student thinking and strategies over time in terms of sophistication of both conceptual understanding and procedural fluency. Currently, learning trajectories exist in the research literature for many mathematical domains, including counting, addition and subtraction, multiplicative thinking,…

This article discusses how teaching via problem solving helps enact the Mathematics Teaching Practices and supports students' learning and development of the Standards for Mathematical Practice. This approach involves selecting and implementing mathematical tasks that serve as vehicles for meeting the learning goals for the lesson. For the lesson…

Empirical arguments rely on examples without necessarily addressing all cases. Students should be skeptical of empirical evidence and should seek more secure arguments for generalizations, such as those that explain why a generalization is true for all cases. Generalizing on the basis of patterns in data is an important mathematical practice;…

Students engage in proportional reasoning when they use covariance and multiple comparisons. Without rich connections to proportional reasoning, students may develop inadequate understandings of linear relationships and the equations that model them. Teachers can improve students' understanding of linear relationships by focusing on realistic…

Journals provide a history of each student's thinking over time and allow this history to be easy to review. Journaling in mathematics has been found to be a valuable tool both for students and for teachers. Students benefit from journaling because it advances their mathematical understanding and ability to communicate in mathematics; teachers…

The Common Core's Standards for Mathematical Practice encourage teachers to develop their students' ability to reason abstractly and quantitatively by helping students make sense of quantities and their relationships within problem situations. The seventh-grade content standards include objectives pertaining to developing linear equations in…

Middle school students need relevant, meaningful contexts to apply emerging mathematical ideas. In this project, through the context of an architecture investigation, seventh-grade students engaged in mathematics involving area, surface area, volume, ratios and proportional thinking, number sense, and technology integration. Students, working in…