### Karen Hollebrands, Heather West, Emily Elrod, and Valerie Faulkner

### Natasha Gerstenschlager, Angela T. Barlow, Alyson Lischka, Lucy Watson, Jeremy Strayer, D. Christopher Stephens, Kristin S. Hartland, and James C. Willingham

### Mark A. Creager, Rachel B. Snider, and Christopher W. Parrish

Cognitively demanding tasks provide important opportunities for students to develop an understanding of mathematics; however, they are challenging to launch and implement. The authors designed a secondary methods unit on launching tasks. Participants in the study were enrolled in five different methods courses. Using a noticing framework, findings suggest that by engaging in the unit, preservice teachers developed a greater understanding of the four aspects of an effective task launch. When viewing video examples, preservice teachers were able to talk about the four aspects of a task launch with increased specificity. Additionally, they began to identify ways of developing common language without reducing cognitive demand. We discuss implications of this work and offer suggestions for future teacher education research.

### Megan Staples and Jillian Cavanna

To support teachers in implementing ambitious reform efforts, professional developers and teacher educators need to know more about teachers’ thinking about argumentation. Specifically, there is a need to understand more about teachers’ views and evaluations of *students’* mathematical arguments as they play out *in practice*. In this article, we share a tool developed to elicit teachers’ pre- and postevaluations of students’ mathematical arguments on a problem-solving task. We discuss the design of the tool and provide evidence of its utility. Our findings indicate that the tool can be used to (a) identify changes in teachers’ evaluations of student mathematical arguments over time and (b) inform the design of professional learning experiences.

### Charmaine Mangram and Kathy Liu Sun

The pervasiveness of digital technology creates an imperative for mathematics teacher educators to prepare preservice teachers (PSTs) to select technology to support students’ mathematical development. We report on research conducted on an assignment created for and implemented in secondary mathematics methods courses requiring PSTs to select and evaluate digital mathematics tools. We found that PSTs primarily focused on pedagogical fidelity (ease of use), did not consider mathematical fidelity (accuracy), and at times superficially attended to cognitive fidelity (how well the tool reflects students’ mathematical thinking processes) operationalized as the CCSS for Mathematical Practice and Five Strands of Mathematical Proficiency. We discuss implications for implementing the assignment and suggestions for addressing PSTs’ challenges with identifying the mathematical practices and five strands.

### Sarah Theule Lubienski, Colleen M. Ganley, Martha B. Makowski, Emily K. Miller, and Jennifer D. Timmer

Despite progress toward gender equity, troubling disparities in mathematical problem-solving performance and related outcomes persist. To investigate why, we build on recurrent findings in previous studies to introduce a new construct, “bold problem solving,” which involves approaching mathematics problems in inventive ways. We introduce a self-report survey of bold problem-solving orientation and find that it mediates gender differences in problem-solving performance for both high-achieving middle school students (*n* = 79) and a more diverse sample of high school students (*n* = 222). Confidence mediates the relation between gender and bold problem-solving orientation, with mixed results for mental rotation skills and teacher-pleasing tendencies as mediators. Overall, the new bold problem-solving construct appears promising for advancing our understanding of gender differences in mathematics.

### Kelly Curtis, Katrina Lindo, and Amanda Jansen

When a ninth-grade teacher used discourse moves aligned with responding to students’ thinking and explicitly promoting productive dispositions, her students reported having positive experiences.