This article addresses the nature and extent of reasoning and proof in the written (i.e., intended) curriculum of 20 contemporary high school mathematics textbooks. Both the narrative and exercise sets in lessons dealing with the topics of exponents, logarithms, and polynomials were examined. The extent of proof-related reasoning varied by topic and textbook. Overall, about 50% of the identified properties in the 3 topic areas were justified, with about 30% of the addressed properties justified with a general argument and about 20% justified with an argument about a specific case. However, less than 6% of the exercises in the homework sets involved proof-related reasoning, with developing an argument and investigating a conjecture as the most frequently occurring types of proof-related reasoning.

### Denisse R. Thompson, Sharon L. Senk and Gwendolyn J. Johnson

### Jennifer Noll and J. Michael Shaughnessy

Sampling tasks and sampling distributions provide a fertile realm for investigating students' conceptions of variability. A project-designed teaching episode on samples and sampling distributions was team-taught in 6 research classrooms (2 middle school and 4 high school) by the investigators and regular classroom mathematics teachers. Data sources included survey data collected in 6 research classes and 4 comparison classes both before and after the teaching episode, and semistructured task-based interviews conducted with students from the research classes. Student responses and reasoning on sampling tasks led to the development of a conceptual lattice that characterizes types of student reasoning about sampling distributions. The lattice may serve as a useful conceptual tool for researchers and as a potential instructional tool for teachers of statistics. Results suggest that teachers need to focus explicitly on multiple aspects of distributions, especially variability, to enhance students' reasoning about sampling distributions.

### Kimberly Cervello Rogers and Michael D. Steele

Graduate teaching assistants serve as instructors of record for numerous undergraduate courses every semester, including serving as teachers for mathematics content courses for elementary preservice teachers. In this study, we examine 6 teaching assistants' teaching practices in the context of a geometry content course for preservice teachers by focusing on their enactment of reasoning-and-proving tasks. Results indicate that teaching assistants engaged preservice teachers in a variety of reasoning-and-proving activities. For 42 of 82 tasks observed, preservice teachers' engagement in reasoningand-proving processes decreased relative to the potential for reasoning and proving in mathematical tasks. This investigation into teaching assistants' teaching practices identifies factors associated with their enactment of reasoning-and-proving tasks (e.g., generating student participation). This research has implications for professional development to support college mathematics instructors' teaching.

### Laurie H. Rubel

This article describes a subset of results from a larger study (Rubel, 2002) that explored middle school and high school students' probabilistic reasoning abilities across a variety of probabilistic contexts and constructs. Students in grades 5, 7, 9, and 11 at an urban, private school for boys (*n* = 173) completed a Probability Inventory, comprising adapted tasks from the research literature, which required students to provide answers as well as justifications of their responses. Supplemental clinical interviews were conducted with 33 students to provide further detail about their reasoning. This article focuses specifically on the probabilistic constructs of compound events and independence in the context of coin tossing. Analyses ofjustifications of correct and incorrect answers are provided, offering insight into students' strategies, reasoning, and underlying cognitive models. A belief framework is supported by the results of this study. Potential implications for research and instruction are also discussed.

### Helen M. Doerr and Lyn D. English

A modeling approach to the teaching and learning of mathematics shifts the focus of the learning activity from finding a solution to a particular problem to creating a system of relationships that is generalizable and reusable. In this article, we discuss the nature of a sequence of tasks that can be used to elicit the development of such systems by middle school students. We report the results of our research with these tasks at two levels. First, we present a detailed analysis of the mathematical reasoning development of one small group of students across the sequence of tasks. Second, we provide a macrolevel analysis of the diversity of thinking patterns identified on two of the problem tasks where we incorporate data from multiple groups of students. Student reasoning about the relationships between and among quantities and their application in related situations is discussed. The results suggest that students were able to create generalizable and reusable systems or models for selecting, ranking, and weighting data. Furthermore, the extent of variations in the approaches that students took suggests that there are multiple paths for the development of ideas about ranking data for decision making.

### Marilyn Carlson, Sally Jacobs, Edward Coe, Sean Larsen and Eric Hsu

The article develops the notion of covariational reasoning and proposes a framework for describing the mental actions involved in applying covariational reasoning when interpreting and representing dynamic function events. It also reports on an investigation of high-performing 2nd-semester calculus students' ability to reason about covarying quantities in dynamic situations. The study revealed that these students were able to construct images of a function's dependent variable changing in tandem with the imagined change of the independent variable, and in some situations, were able to construct images of rate of change for contiguous intervals of a function's domain. However, students appeared to have difficulty forming images of continuously changing rate and could not accurately represent or interpret inflection points or increasing and decreasing rate for dynamic function situations. These findings suggest that curriculum and instruction should place increased emphasis on moving students from a coordinated image of two variables changing in tandem to a coordinated image of the instantaneous rate of change with continuous changes in the independent variable for dynamic function situations.

### Henri Picciotto

I enjoyed George Roy Allen, and Thacker (2020) article, “Linking Factors and Multiples to Algebraic Reasoning.” The authors described how an engaging problem, combined with savvy teacher moves, can help students engage in mathematical thinking

### Sean P. Yee, George J. Roy and LuAnn Graul

As middle school students explore number patterns, they use the evidence in the pattern to generate a conclusion and then justify that conclusion with logic. However, a challenge exists when middle school students use conditional reasoning to come

### George J. Roy, Jessica S. Allen and Kelly Thacker

). Importantly, the context of the task has to be relatable to a student and as a result provide both the incentive as well as a rationale for students to engage in mathematical reasoning and sense making. Second, the task must provide an opportunity for a wide

## Redeem Reasoning

### readers speak out

### Kuo-Liang Chang

This opinion piece discusses how simplicity, ease, and efficiency—in the guise of shortcuts, tips, and packed procedures—kill mathematical reasoning.