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.
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Characterizing Secondary Teachers’ Structural Reasoning
Stacy Musgrave, Cameron Byerley, Neil Hatfield, Surani Joshua, and Hyunkyoung Yoon
Developing Meaning for Mathematical Expressions
John K. Lannin, Christopher Austin, and David C. Geary
Explore two ways that algebra students interpret mathematical expressions. Learn instructional tasks to help students develop meaning.
GPS: Making Meaningful Use of Structure in PK–12
Katherine Ariemma Marin and Natasha E. Gerstenschlager
Growing Problem Solvers provides four original, related, classroom-ready mathematical tasks, one for each grade band. Together, these tasks illustrate the trajectory of learners’ growth as problem solvers across their years of school mathematics.
Contemplate Then Calculate
Amy Lucenta and Grace Kelemanik
Teaching students to apply structural thinking instead of automatically following procedures and algorithms can result in efficient, elegant strategies and fewer errors.
A Quadratic to a Quadratic? This Is New!
Michael S. Meagher, Michael Todd Edwards, and S. Asli Özgün-Koca
Using technology to explore a rich task, students must reconcile discrepancies between graphical and analytic solutions. Technological reasons for the discrepancies are discussed.
Weaving the Rainbow (Odd I)
Lucy Rycroft-Smith
This piece is a rumination on flow, pattern, and edges/transitions, focusing on polynomials of odd degree and overlaying/underlaying the flow of the graphical structure with a rainbow to suggest the central importance of queer visibility in mathematics.
The Opportunities of No-Solution Problems
Nicholas J. Gilbertson
When students encounter unusual situations or exceptions to rules, they can become frustrated and can question their understanding of particular topics. In this article, I share some practical tips.
