The first standard presented in the *Professional Standards for Teaching Mathematics* (NCTM 1991) highlights the importance of choosing and using *worthwhile mathematical tasks*. Teachers are curriculum architects charged with ensuring the quality of the mathematical tasks in which their students engage.

### Barbara J. Reys and Vena M. Long

### Margaret Schwan Smith and Mary Kay Stein

What features of a mathematics classroom really make a difference in how students come to view mathematics and what they ultimately learn? Is it whether students are working in small groups? Is it whether students are using manipulalives? Is it the nature of the mathematical tasks that are given to students? Research conducted in the QUASAR project, a five-year study of mathematics education reform in urban middle schools (Silver and Stein 1996). offers some insight into these questions. From 1990 through 1995, data were collected about many aspects of reform teaching, including the use of small groups; the tool that were available for student use, for example, manipulatives and calculators; and the nature of the mathematics tasks. A major finding of this research to date, as described in the article by Stein and Smith in the January 1998 issue of *Mathematics Teaching in the Middle School*, is that the highest learning gains on a mathematicsperformance assessment were related to the extent to which tasks were et up and implemented in ways that engaged students in high levels of cognitive thinking and reasoning (Stein and Lane 1996). This finding supports the position that the nature of the tasks to which students are exposed detennines what students learn (NCTM 1991), and it also leads to many questions that should be considered by middle school teachers.

### Milan F. Sherman, Charity Cayton and Kayla Chandler

This article describes an intervention with preservice mathematics teachers intended to address the use of Interactive Geometry Software (IGS) for mathematics instruction. A unit of instruction was developed to support teachers in developing mathematical tasks that use IGS to support students' high-level thinking (Smith & Stein, 1998). Preservice teachers used the IGS Framework (Sherman & Cayton, 2015) to evaluate 3 tasks, to revise a task, and ultimately to design a task using the framework. Results indicate that a majority of preservice teachers in this study were successful in creating a high-level task where IGS was instrumental to the thinking demands, and that the IGS Framework supported them in doing so. The article concludes with suggestions for use by fellow mathematics teacher educators.

### Allison W. McCulloch, Nina G. Bailey, Kristen Fye, and Gayle Scott

-based instructor of the TPC course partnered with two high school teachers; together, they had a goal of creating a space in which the PSTs would draw upon both academic and practitioner knowledge as they learned to design technology-based mathematical tasks for

### Heather R. Gallivan

Utilizing knowledge of students' culture, home, and community lives can support students' mathematics learning (e.g., Civil, 2007). It is important for prospective teachers (PTs) to develop strategies for doing so in their future classrooms. This article reports on a framework and intervention designed to support PTs in learning to use information gathered about a student's culture, interests, and home and community life to revise a high-level mathematics task to be more culturally relevant for that student. PTs demonstrated both success and different culties during the phases of the intervention, but were able to use what they learned to revise a high-level mathematics task for a student. The article also discusses the potential of the intervention and the implications for other mathematics teacher educators when enacting this intervention.

### Tiffany G. Jacobs, Marvin E. Smith, Susan Swars Auslander, Stephanie Z. Smith and Kayla D. Myers

Teacher preparation programs face increasing demands to demonstrate the competencies of prospective teachers in their programs, while maintaining a focus on developing high-leverage instructional practices in their methods coursework. This study used a mixed methods approach to examine how the implementation of a simulated edTPA elementary mathematics task influenced prospective teachers' experiences in an elementary mathematics methods course. The course curriculum featured cognitively guided instruction (CGI) as the exemplar for understanding and implementing problembased, cognitively oriented pedagogy. Our findings indicate that elements of both CGI and the simulated edTPA mathematics task worked synergistically to enhance opportunities for CGI-type lesson enactment, support productive changes in beliefs, and contribute to the prospective elementary teachers' (PTs) preparation.

### Marjorie Henningsen and Mary Kay Stein

In order to develop students' capacities to “do mathematics,” classrooms must become environments in which students are able to engage actively in rich, worthwhile mathematical activity. This paper focuses on examining and illustrating how classroom-based factors can shape students' engagement with mathematical tasks that were set up to encourage high-level mathematical thinking and reasoning. The findings suggest that when students' engagement is successfully maintained at a high level, a large number of support factors are present. A decline in the level of students' engagement happens in different ways and for a variety of reasons. Four qualitative portraits provide concrete illustrations of the ways in which students' engagement in high-level cognitive processes was found to continue or decline during classroom work on tasks.

### Glendon W. Blume, Judith S. Zawojewski, Edward A. Silver and Patricia Ann Kenney

Worthwhile mathematical tasks engage the problem solver in sound and significant mathematics, elicit a variety of solution methods, and require mathematical reasoning. Such problems also prompt responses that are rich enough to reveal mathematical understandings. Just as good classroom practice engages students in worthwhile mathematical tasks, sound professional development does the same with teachers. Providing teachers with opportunities to engage in worthwhile mathematical tasks and to analyze the mathematical ideas underlying those tasks promotes the vision of the *Professional Standards for Teaching Mathematics* (NCTM 1991).

### Courtney R. Nagle and Jodie L. Styers

Perspective from the top of a roller coaster engages students in reasoning about slope and functions.

### Diana V. Lambdin and Kathleen Lynch

How Would Your Middle-Grades students approach this algebraic thinking task?