Many students have a dominant part-whole conception of fractions. We examine why this is problematic and explore strategies to move students beyond this limitation.

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## Beware of “Gaps” in Students’ Fraction Conceptions

### Patrick L. Sullivan, Joann E. Barnett, and Kurt Killion

## 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.

## Exploring Prospective Teachers’ Development of the Cartesian Form of Complex Numbers

### Gülseren Karagöz Akar, Merve Saraç, and Mervenur Belin

In this study, we investigated prospective secondary mathematics teachers’ development of a meaning for the Cartesian form of complex numbers by examining the roots of quadratic equations through quantitative reasoning. Data included transcripts of the two sessions of classroom teaching experiments prospective teachers participated in, written artifacts from these teaching sessions, and their answers to pre-and-post written assessment questions. Results point toward prospective teachers’ improved meanings regarding the definition of complex numbers and the algebraic and geometrical meanings of the Cartesian form of complex numbers. Implications for mathematics teacher education include providing specific tasks and strategies for strengthening the knowledge of prospective and in-service teachers.

## Experience First, Formalize Later

### Sarah Stecher, Luke Wilcox, and Lindsey Gallas

The EFFL model empowers students to build strong conceptual understanding of mathematics through carefully designed, equity-minded activities that disrupt the traditional lecture-based classroom.

## GPS: Iterating and Partitioning

### Daniel K. Siebert and Monica G. McCleod

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.

## Learning Trajectories for Vertical Coherence

### Jere Confrey, Meetal Shah, and Alan Maloney

Three learning trajectories and their connections show how to promote vertical coherence in PK–12 mathematics education.

## Using TileFarm to Support Emerging Multiplication

### Matt B. Roscoe

Symmetric dot patterns are a particularly powerful object for investigation, providing opportunities for foundational learning across PK–5. We found that second-grade students naturally used repeated addends to count symmetric dot patterns created using the new software TileFarm.

## Using Artwork to Explore Proportional Reasoning

### Sarah B. Bush, Karen S. Karp, Jennifer Nadler, and Katie Gibbons

By examining ratios in paintings and using a free educational app, students can size up artists' use of proportional reasoning in their creations.

## Chivalry, at Least “Math Chivalry,” Is Not Dead!

A cartoon exploring a problem about order of operations is coupled with a full-page activity sheet.

## Questioning the Order of Operations

### Kami M. Dupree

Abandon mnemonics and make stronger connections between the operations and properties of arithmetic.