### Yael Luz and Michal Yerushalmy

We report on an innovative design of algorithmic analysis that supports automatic online assessment of students’ exploration of geometry propositions in a dynamic geometry environment. We hypothesized that difficulties with and misuse of terms or logic in conjectures are rooted in the early exploration stages of inquiry. We developed a generic activity format for if–then propositions and implemented the activity on a platform that collects and analyzes students’ work. Finally, we searched for ways to use variation theory to analyze ninth-grade students’ recorded work. We scored and classified data and found correlation between patterns in exploration stages and the conjectures students generated. We demonstrate how automatic identification of mistakes in the early stages is later reflected in the quality of conjectures.

### Patricio Herbst

My academic family lost our patriarch, Jeremy Kilpatrick, on September 17, 2022. As I write this in October, reflecting on his legacy to our field feels timely. After a brief biographical sketch, I explore how Jeremy's influence, particularly in his role as Editor in Chief of JRME, shaped our field—and my own work as editor.

### Yang Jiang and Gabrielle A. Cayton-Hodges

This exploratory study investigated the behaviors and content of onscreen calculator usage by a nationally representative sample of eighth-grade students who responded to items from the 2017 National Assessment of Educational Progress mathematics assessment. Meaningful features were generated from the process data to infer whether students spontaneously used calculators for mathematical problem solving, how frequently and when they used them, and the nature of the operations performed on calculators. Sequential pattern mining was applied on sequences of calculator keystrokes to obtain patterns of operations that were representative of students’ problem-solving strategies or processes. Results indicated that higher scoring students not only were more likely to use calculators, but also used them in a more goal-driven manner than lower scoring students.

### Merav Weingarden and Einat Heyd-Metzuyanim

One of the challenges of understanding the complexity of so-called reform mathematics instruction lies in the observational tools used to capture it. This article introduces a unique tool, drawing from commognitive theory, for describing classroom discussions. The Realization Tree Assessment tool provides an image of a classroom discussion, depicting the realizations of the mathematical object manifested during the discussion and the narratives that articulate the links between these realizations. We applied the tool to 34 classroom discussions about a growing-pattern algebraic task and, through cluster analysis, found three types of whole-class discussion. Associations with classroom-level variables (track, but not grade level or teacher seniority) were also found. Implications with respect to applications and usefulness of the tool are discussed.

### Dana L. Grosser-Clarkson and Joanna S. Hung

This Perspectives on Practice manuscript focuses on an innovation associated with “Engaging Teachers in the Powerful Combination of Mathematical Modeling and Social Justice: The Flint Water Task” from Volume 7, Issue 2 of *MTE*. The Flint Water Task has shown great promise in achieving the dual goals of exploring mathematical modeling while building awareness of social justice issues. This *Perspectives on Practice* article focuses on two adaptations of the task—gallery walks and What I Know, What I Wonder, What I Learned (KWL) charts—that we have found to enhance these learning opportunities. We found that the inclusion of a gallery walk supported our students in the development of their mathematical modeling skills by enhancing both the mathematical analyses of the models and the unpacking of assumptions. The KWL chart helps students document their increase in knowledge of the social justice issues surrounding the water crisis. Using the mathematical modeling cycle to explore social justice issues allows instructors to bring humanity into the mathematics classroom.

### Kuo-Liang Chang and Ellen Lehet

Defining a quadratic function through the slopes of its secant/tangent lines leads to the fundamental theorem of calculus (FTC) and an alternative way of understanding integration.

### Gwyneth Hughes, Michele B. Carney, Joe Champion, and Lindsey Yundt

Two broad categories of instructional practices, (a) explicitly attending to concepts and (b) fostering students’ opportunities to struggle, have been consistently linked to improving students’ mathematical learning and achievement. In this article, we describe an effort to build these practices into a framework that is useful for a diverse set of professional development (PD) offerings. We describe three examples of how the framework is used to support teacher learning and classroom instructional practice: a state-mandated course, lesson studies, and a large-scale teacher–researcher alliance. Initial findings suggest that consistently emphasizing this framework provides both content and structural guidance during PD development and gives coherence and focus to teachers’ PD experiences.

### David B. Custer and Ksenija Simic-Muller

We reflect on recent presentations at the NCTM annual conference and articles in MTLT that address statistics, data modeling, and data science. We observe that such presentations and articles are increasingly common, and encourage readers to use them in their teaching and write about their own adventures with data.