With the publication of the National Council of Teachers of Mathematics' Curriculum Standards document in 1989, nurturing students' *mathematical thinking* secure a prominent place in the discourse surrounding school curriculum and instructional redesign. Although the standards document did not provide a definition for mathematical thinking, the authors highlighted processes that could support its development, including problem solving, communicating ideas, building and justifying arguments, and reasoning formally and informally about potential mathematical relationships. Less articulated were ways that mathematical thinking may be supported toward the development of proving and prooflike reasoning among students (Maher and Martino 1996).

### Azita Manouchehri, Pingping Zhang and Jenna Tague

### 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.

### Stephanie Casey and Joel Amidon

title rather than building on the students’ mathematical thinking to reach the goal of the lesson. Erica, on the other hand, spent the debriefing discussing the instructor’s classroom management moves and the behavior of students in the class . Dr

### Madhuvanti Anantharajan

researchers emphasize the importance of attending to young children’s mathematical thinking ( Carpenter et al., 1996 ; Carpenter et al., 2017 ; Clements & Sarama, 2009 ; Ginsburg, 1983 ). Attending to children’s understanding of counting involves attending

### Jane F. Schielack, Dinah Chancellor and Kimberly M. Childs

Suggestions for an elementary school activity and questions to promote five types of mathematical thinking: modeling, logical analysis, inference, optimization, and abstraction.

Flexible mathematical thinking—the ability to generate and connect various representations of concepts—is useful in understanding mathematical structure and variation in problem solving. Of the many important reasoning habits listed in NCTM's *Focus in High School Mathematics: Reasoning and Sense Making* (2009, pp. 9–10), four habits complement flexible mathematical thinking.

### Leone Burton

This paper argues that mathematical thinking is not thinking about the subject matter of mathematics but a style of thinking that is a function of panicular operations, processes, and dynamics recognizably mathematical. It further suggests that because mathematical thinking becomes confused with thinking about mathematics, there has been little success in separating process from content in the classroom presentation of the subject. A descriptive model of mathematical thinking is presented and then used to provide a practical response to the questions, Can mathematical thinking be taught? In what ways? The reacher is encouraged to recognize both what constitutes mathematical thinking, whether in the mathematics class or some other, and what conditions are necessary to foster it.

### Terry Woods, Gaye Williams and Betsy McNeal

The relationship between normative patterns of social interaction and children's mathematical thinking was investigated in 5 classes (4 reform and 1 conventional) of 7- to 8-year-olds. In earlier studies, lessons from these classes had been analyzed for the nature of interaction broadly defined; the results indicated the existence of 4 types of classroom cultures (conventional textbook, conventional problem solving, strategy reporting, and inquiry/argument). In the current study, 42 lessons from this data resource were analyzed for children's mathematical thinking as verbalized in class discussions and for interaction patterns. These analyses were then combined to explore the relationship between interaction types and expressed mathematical thinking. The results suggest that increased complexity in children's expressed mathematical thinking was closely related to the types of interaction patterns that differentiated class discussions among the 4 classroom cultures.

### Dianne S. Goldsby and Barbara Cozza

NCTM's Principles and Standards for School Mathematics emphasizes the need for all students to organize and consolidate their mathematical thinking through communication and to communicate their mathematical thinking coherently to others (NCTM 2000). Writing helps students focus on their own understandings of mathematics: “Students gain insights into their thinking when they present their methods for solving problems, when they justify their reasoning to a classmate or teacher, or when they formulate a question about something that is puzzling them” (NCTM 2000, pp. 60–61).

### Victoria R. Jacobs, Lisa L. C. Lamb and Randolph A. Philipp

The construct *professional noticing of children's mathematical thinking* is introduced as a way to begin to unpack the in-the-moment decision making that is foundational to the complex view of teaching endorsed in national reform documents. We define this expertise as a set of interrelated skills including (a) attending to children's strategies, (b) interpreting children's understandings, and (c) deciding how to respond on the basis of children's understandings. This construct was assessed in a cross-sectional study of 131 prospective and practicing teachers, differing in the amount of experience they had with children's mathematical thinking. The findings help to characterize what this expertise entails; provide snapshots of those with varied levels of expertise; and document that, given time, this expertise can be learned.