In "Principles to Actions: Ensuring Mathematical Success for All," NCTM (2014) defines productive struggle as students delving "more deeply into understanding the mathematical structure of problems and relationships among mathematical ideas, instead of simply seeking correct solutions" (p. 48). Hiebert and Grouws (2007, p. 387)…

As current mathematics standards, such as the Common Core, are being implemented throughout the United States, it has become evident that teachers need support to enact the tenets of those standards. To help in this endeavor, this article was published as a guideline to emphasize to mathematics education stakeholders that "effective teaching…

Researchers have extensively documented, and math teachers know from experience, that algebra is a "gatekeeper" to more advanced mathematical topics. Students must have a strong understanding of fundamental algebraic concepts to be successful in later mathematics courses. Unfortunately, algebraic misconceptions that students may form or…

Students can use geometric representations of numbers as a way to explore algebraic ideas. With the help of these representations, students can think about the relations among the numbers, express them using their own words, and represent them with letters. The activities discussed here can stimulate students to try to find various ways of solving…

Geometric pattern tasks can be a useful tool for developing students' capacities to reason algebraically and in helping establish classroom norms and practices at the beginning of the school year. In this article, the authors consider the mathematical and social purposes that pattern tasks can serve by examining one middle school's…

Identifies and addresses fundamental ideas that build the foundations of algebraic thinking and reasoning. Focuses on developing appropriate instructional approaches to these concepts. Explains what is meant by "composite unit" and how this concept framed the development of an instructional sequence to help students make sense of and find meaning…

One problem that students have with mathematics is that they often view the topic as a series of unrelated ideas. Sometimes they are aware that they have to know one concept to move to the next, but what is done in geometry is not necessarily related to anything in algebra. This failure to recognize mathematical connections limits students'…

Shows how sequenced learning activities can help students build conceptual bridges from arithmetic to algebra. Focuses on understanding equivalent algebraic expressions, an area in which U.S. students scored poorly on the Third International Mathematics and Science Study (TIMSS). (YDS)

Reflects on one teacher's strategies for facilitating discourse in his classroom--creating an environment in which student ideas are the basis of discussion yet ensuring productive mathematics. (YDS)

Describes different solution strategies used by students for a problem on the cost of talking on the phone. (YDS)

Presents an activity, "Lisa's Lemonade Stand," that actively engages students in algebraic thinking as they analyze change by investigating relationships between variables and gain experience describing and representing these relationships graphically. (YDS)

Presents a measurement problem that can be explored by students of varying ability. Discusses aspects that make it accessible in a heterogeneous class. (YDS)

Describes a series of lessons in which 6th grade students explore notions of rates of change and their effect on the shapes of graphs. Addresses aspects of the algebra content standard for the middle grades. (YDS)

Demonstrates the transformations that are possible to construct a variety of magic squares, including modifications to challenge students from elementary grades through algebra. Presents an example of using magic squares with students who have special needs. (YDS)

Describes a lesson plan in which students apply their mathematical intuition to determine win-loss possibilities of teams in a sports league and develop a formula for calculating the magic number for clinching first place. (YDS)