The Building Painted Cubes Task is a groupworthy algebraic task. Students build cubes using linking unit cubes, search for algebraic patterns, and report findings on posters. This task can create spaces for students to see themselves as doers of mathematics.
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Construct It! Building Painted Cubes Task: Serena’s Case
Elizabeth Suazo-Flores and Lisa Roetker
Building Algebraic Procedures From Concepts: Like Terms
Leah M. Frazee and Adam R. Scharfenberger
This task sequence for adding and subtracting like terms—grounded in the concepts of equivalence and algebra as generalized arithmetic—helps students see connections between concepts and procedures in algebra.
Problems to Ponder
Chris Harrow and Justin Johns
Problems to Ponder provides 28 varying, classroom-ready mathematics problems that collectively span PK–12, arranged in the order of the grade level. Answers to the problems are available online. Individuals are encouraged to submit a problem or a collection of problems directly to mtlt@nctm.org. If published, the authors of problems will be acknowledged.
Reconsidering Mathematical Authority
Michael D. Hicks, Jessica Pierson Bishop, Christina Koehne, and Mai Bui
Who has mathematical authority in your classroom, and what does authority look like? Find out different ways you can help students gain authority.
Two Technology Frameworks: Their Use in Lesson Design
Elizabeth B. Harkey, Angela T. Barlow, and Victoria Groves-Scott
The authors used two technology frameworks to design three seventh-grade mathematics lessons, focusing on opportunities to address collaboration, authenticity, or personalization.
Ways to Help Students Become Powerful Mathematical Thinkers
Alan H. Schoenfeld
Ear to the Ground features voices from several corners of the mathematics education world.
Discuss It! Collaborating on the Tortoise and Hare Task
K. Ann Renninger, Maria Consuelo De Dios, Annie Fetter, Maeve R. Hogan, Moe Htet Kyaw, Ana G. Michels, Marina Nakayama, Richard Tchen, Stephen A. Weimar, Helena Werneck de Souza Dias, and Feven Yared
The authors share an online collaborative problem-solving activity that integrates support for students’ developing conceptual understanding, focused engagement, and positive feelings of agency and identity.
Characterizing Secondary Teachers’ Structural Reasoning
Stacy Musgrave, Cameron Byerley, Neil Hatfield, Surani Joshua, and Hyunkyoung Yoon
The Common Core State Standards for Mathematical Practice asks students to look for and make use of structure. Hence, mathematics teacher educators need to prepare teachers to support students’ structural reasoning. In this article, we present tasks and rubrics designed and validated to characterize teachers’ structural reasoning for the purposes of professional development. Initially, tasks were designed and improved using interviews and small pilot studies. Next, we gave written structure tasks to over 600 teachers in two countries and developed and validated rubrics to categorize responses. Our work contributes to the preparation and support of mathematics teachers as they develop their own structural reasoning and their ability to help students develop structural reasoning.
The Do Nothing Machine
Keith Dreiling
The Trammel of Archimedes traces an ellipse as the machine’s lever is rotated. Specific measurements of the machine are used to compare the machine’s actions on GeoGebra with the graph of the ellipse and an ellipse formed by the string method.
Filling Vases and Making Tanks
Jana Dean
Two classic hands-on tasks address conceptual understanding of functions. The tasks center student discourse and rough draft mathematics as students grapple with the relationship between input and output.
