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## Developing Preservice Teachers’ Understanding of Area Through a Units Intervention

Preservice elementary teachers (PSTs) often enter their teacher preparation programs with procedural and underdeveloped understandings of area measurement and its applications. This is problematic given that area and the area model are used throughout K–Grade 12 to develop flexibility in students’ mathematical understanding and to provide them with a visual interpretation of numerical ideas. This study describes an intervention aimed at bolstering PSTs’ understanding of area and area units with respect to measurement and number and operations. Following the intervention, results indicate that PSTs had both an improved ability to solve area tiling tasks as well as increased flexibility in the strategies they implemented. The results indicate that PSTs, similar to elementary students, develop a conceptual understanding of area from the use of tangible tools and are able to leverage visualizations to make sense of multiplicative structure across different strategies.

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## Visualizing the Arithmetic Mean

The two provided activities are geared for students in middle school to facilitate and deepen their understanding of the arithmetic mean. Through these activities, students analyze visual representations and use a special type of statistical thinking called transnumerative thinking.

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## Teacher Goals and Dilemmas in the Use of Mathematical Representations

The purpose of this article is to describe the reported goals of teachers participating in professional development related to their efforts to consider multiple representations in their teaching. Through analysis of monthly written reflections and group discussion, we describe 3 teacher dilemmas that emerged related to their efforts to consider multiple representations in their teaching: (a) equitable practices when particular representations result in differential success, (b) a teacher's need to balance exposure and choice, and (c) a potential dilemma related to conflicts between competing goals. We provide suggestions of how mathematics educators might use these findings to support future professional development efforts.

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## Systematic Interventions for Teaching Ratios

Consider this research-based intervention approach to ratio using concrete, semiconcrete, and abstract representations to help students who struggle with this concept.

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## Radical Thoughts on Simplifying Square Roots

A geometric approach using exact square manipulatives can promote an understanding of the algorithm to dismantle radical expressions.

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## Teaching Structure in Algebra

Exercises about glue and trees and addresses can inoculate students against their notorious tendency to reduce incorrectly when simplifying expressions.

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## Exploring Function Transformations Using the Common Core

Tying your teaching approach to the Common Core Standard for Geometry and Congruence will help students understand why functions behave as they do.

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## Norms and Mathematical Proficiency

How do classroom behavioral expectations support the development of students' mathematical reasoning? A sixth-grade teacher and his students developed this example while discussing a ratio comparison problem.

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## The healthy vending machine

A good problem can capture students' curiosity and can serve many functions in the elementary school classroom: to introduce specific concepts the teacher can build on once students recognize the need for additional mathematics or to help students see where to apply already-learned concepts. We encourage teachers to use the monthly problem in their own classrooms and report solutions, strategies, reflections, and misconceptions to the journal audience.

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## Is That Square Really a Circle?

Considering circles in taxicab geometry helps students with Euclidean concepts.