### Christian Rüede, Sog Yee Mok, and Fritz C. Staub

This article shows that enabling teachers to integrate comparing solution strategies into their teaching fosters student flexibility in algebra. We designed two professional development (PD) programs that either focused exclusively on comparing solution strategies, or additionally introduced the accountable talk approach to guiding productive classroom discussions. The effects of both PD programs were investigated in an experimental field study (*N* = 39 teachers, 739 students). In both experimental groups, student posttest gains in strategy flexibility and procedural knowledge were greater than in the control group. The accountable talk group also increased conceptual knowledge. Significant effects in strategy flexibility were still observed 2.5 months later. We discuss recommendations for PD programs to foster flexibility in algebra using comparing.

### Patricio Herbst, Daniel Chazan, Percival G. Matthews, Erin K. Lichtenstein, and Sandra Crespo

### Douglas H. Clements, Julie Sarama, Carolyn Layzer, and Fatih Unlu

A follow-up of a cluster-randomized trial evaluated the long-term impacts of a scale-up model composed of 10 research-based guidelines grounded in learning trajectories. Two treatment groups received the intervention during the prekindergarten year, and one of these groups received follow-through support in kindergarten and first grade. Business-as-usual curricula were used in all other cases, including all years for the control group. Early effects on mathematics achievement decreased through fourth grade but reemerged at fifth grade. These results support both a latent trait hypothesis, whereby stable characteristics of students explain differences in achievement, and a latent foundation hypothesis, whereby early mathematical knowledge and skills provide a foundation for competence in mathematics in later years, especially those that involve challenging mathematics.

### Wayne Nirode and Brian Boyd

### Robert Schoen, Ian Whitacre, and Zachary Champagne

Changes in U.S. textbooks indicate that U.S. first-grade students in the Common Core era were exposed to a wider variety of word problem types than students in previous generations were. We compared the performance of U.S. first graders in the Common Core era with that of previous generations in solving 11 types of additive word problems to investigate a decades-long debate—whether certain types of word problems are inherently more difficult than others or whether relative difficulty is influenced by exposure. We found that overall patterns of relative difficulty persist; however, U.S. first graders in the Common Core era outperformed their historical counterparts when solving the types of problems that rarely appeared in textbooks used in the 1980s.

### Eric Milou and Steve Leinwand

The standard high school math curriculum is *not* meeting the needs of the majority of high school students and that serious consideration of rigorous alternatives is a solution whose time has come.