Samuel Otten

• PhD
• Professor at University of Missouri

Professional development (PD) for mathematics teachers often emphasizes transformative instructional change. However, a more modest, incremental approach may offer a higher likelihood of success in ways that complement transformational efforts. This Editorial discusses the potential advantages of incremental PD where teachers make small but meaning...

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This article considers the rise of generative Artificial Intelligence (GenAI) in the context of secondary mathematics education, focusing on its responses to cognitively demanding tasks and the pedagogical implications of these interactions. Using tools such as ChatGPT (OpenAI) and Gemini (Google), we investigate how GenAI engages in complex mathem...

Mathematics educators have written a great deal about cognitively-demanding tasks but this study of 141 lessons across 47 different algebra classes found cognitively-demanding tasks to be rare in practice. Of 2,378 coded tasks, 93% were low cognitive demand, predominantly procedures without connections to meaning. Only 6% were high cognitive demand...

For thousands of years, there was a gradual increase of carbon in Earth’s atmosphere. Students can model the historical data using linear functions. Then they can learn about climate change in the modern era by, depending on their grade level, using a combination of exponential and periodic functions to explore carbon’s multi-faceted variation, dis...

Growing Problem Solvers provides four original, related, classroom-ready mathematical tasks, one for each grade band. Together, these tasks illustrate the trajectory of learners’ growth as problem solvers across their years of school mathematics.

In response to the COVID-19 pandemic, schools transitioned to Emergency Remote Teaching (ERT). In May 2020, as part of an existing study of flipped Algebra instruction, we interviewed eleven Missouri teachers to understand how their instruction changed as they moved to ERT. Drawing on practical rationality, we found the pandemic led to a breach of...

In high school geometry, proving theorems and applying them to geometry problems is an expectation for students. An essential part of most geometry proofs is the diagram because it not only helps encapsulate the claim being proved but can also be a tool in reasoning or communicating an argument. This interview-based study investigated how high scho...

Mathematics professional development (PD) has had many small victories but has not brought about a widespread change in what constitutes typical mathematics instruction. This theoretical essay argues that many PD projects have been based on an assumption that the aims of the PD should be ambitious, but ambitious PD requires that a large set of crit...

Reasoning-and-proving is a crucial part of students’ mathematical experiences in secondary school. There is scholarly debate, however, on the extent to which proving at the secondary level needs to be formal and whether all students should be held to disciplinary standards of rigor. In this study, we investigated the notion of “proof for all” from...

Teachers are implementing flipped instruction in an increasing number of mathematics classes but the research base is not yet well developed on this topic. Many studies of flipped instruction in mathematics have involved a small number of classes utilizing flipped instruction being compared to classes with non-flipped instruction, but this study de...

The planful use of boardspace can help move the structure and regularity to the visual realm and make it more readily perceivable by students.

Reasoning-and-proving is viewed by many scholars to be a crucial part of students' mathematical experiences in secondary school. There is scholarly debate, however, about the necessity of formal proving. In this study, we investigated the notion of "proof for all" from the perspective of secondary mathematics teachers and we analyzed, using the fra...

In this study, we investigated the notion of “proof for all” from the perspective of secondary mathematics teachers. Using the framework of practical rationality we analyzed the justifications teachers gave for whether or not all students should learn proof. We conducted interviews with twenty-one secondary teachers from a socioeconomically-diverse...

We discuss how mathematics teachers can share authority with students during proving through a manageable and productive way of breaking proving into three phases — proof initiation, proof construction, and proof validation — where authority can be shared gradually and strategically.

Encourage student collaboration in problem solving by altering the who, when , and what of the homework videos you use in flipped lessons.

Teachers who are flipping instruction face the challenging task of selecting or creating high-quality videos for their students. This article presents a framework for evaluating videos and describes the benefits of including interactive features and considering options beyond lecture videos.

In this paper we discuss different ways teachers can integrate science and mathematics into their curriculum. In particular, we focus on science and mathematics integration via the disciplinary practices.

These proceedings collect the plenaries, research reports, brief research reports, poster summaries, and working group summaries that were presented as part of PME-NA 41 in St Louis, MO, USA.

The discipline of mathematics values precision and teachers are accountable for promoting and supporting their students in attending to precision (ATP), which in the USA is an explicit standard for mathematical practice included in the Common Core State Standards for Mathematics. This study used thematic discourse analysis to examine how eight midd...

Teaching methods courses are a crucial factor in the preparation of secondary mathematics teachers. Throughout the United States, secondary methods courses have diverse curricula, including variation in the topics covered in these courses. To assess this variation, the authors identified 41 topics potentially valued by secondary methods instructors...

The authors discussed 2 paths that the mathematics education community should consider with regard to citation-based metrics of journal quality: either working within the system to enhance positioning or resisting or modifying the system itself.

The rise of digital resources has had profound effects on mathematics curricula and there has been a concurrent increase in teachers flipping their instruction—that is, assigning instructional videos or multimedia for students to watch as homework and completing problem or exercise sets in class rather than vice versa. These changes have created a...

Students’ experiences with proving in schools often lead them to see proof as a static product rather than a negotiated process that can help students justify and make sense of mathematical ideas. We investigated how authority manifested in whole-class proving episodes within Ms. Finley’s high school geometry classroom. We designed a coding scheme...

Educational research communities bear responsibility for establishing a substantial body of evidence to support claims that drive the field. For example, one commonly accepted claim is that there is a relationship between the cognitive demand of mathematical task enactments and students’ learning. One study that is often cited in association with t...

Flipped instruction in school mathematics has been occurring more frequently. This study investigated two teachers’ motivations for, conceptions of, and experiences with flipped mathematics instruction. We found that the teachers were motivated to flip based on colleagues’ recommendations and potential benefits for students. The teachers discussed...

Flipped instruction is becoming more common in the United States, particularly in mathematics classes. One of the defining characteristics of this increasingly popular instructional format is the homework teachers assign. In contrast to traditional mathematics classes in which homework consists of problem sets, homework in flipped classes often tak...

Studies have found Educational Studies in Mathematics (ESM) and the Journal for Research in Mathematics Education (JRME) to be more highly regarded by mathematics education scholars than the other journals. However, these survey studies provide only a snapshot in time, and only from a subset of the researchers in the field. Scholarly databases, suc...

One defining characteristic of flipped instruction is the homework teachers assign, which typically consists of an instructional video rather than problem sets (Bergmann & Sams, 2012). We present a framework for flipped homework that categorizes types of homework and draws on existing literature to discern quality for each type (see Figure 1). This...

Although an increasing number of teachers report flipping a lesson or their entire class, little is known about what leads teachers to this decision or how they implement flipped instruction. Understanding these factors is important because it can allow teacher educators and researchers to investigate ways to support teachers in enacting practices...

Students make strategic choices—and justify them—to solve a system of two linear equations.

In this study, we triangulate the rankings in Table 1 with other measures of journal quality offered through database algorithms. In particular, we address the following questions: – What are the rankings of journals in mathematics education? – To what extent is there overlap and agreement among the survey and database rankings? Method • Collected...

As new computation technologies become available, algebra teachers can choose to ban them, limit their use, or use them as an opportunity to reevaluate learning goals.

Attending to precision (ATP) is essential in mathematics. This study examined ATP instances through the lens of univocal (functioning to convey information) and dialogic (functioning to generate new meaning) discourse. Analysis of data from five secondary mathematics classrooms focused on whole-class instances of ATP with coding based on the univoc...

Mathematics teacher education has been criticized, both internally and externally, for failing to identify shared practices and goals within teacher preparation programs. Work has begun to address this criticism at the elementary level but less exists at the secondary level. This paper reports on a national survey with responses from 116 secondary...

In this article, we present examples from Mr. Forrest’s advanced algebra class of students discussing definitions of two previously learned concepts. In the first example, Mr. Forrest presses students to express a precise definition of polynomial. In the second example, he and the students engage in deeper reasoning as they explore the implications...

In this article, we present examples from Mr. Mingley’s 7th grade classroom where attention to precision allowed for the emergence of other standards for mathematical practices, such as looking for patterns that generalize across examples and reasoning about those patterns during a lesson on the traceability of networks.

Univocal discourse, characterized by its function of conveying information from one person to another, is common in mathematics classrooms but dialogic teaching aims at students coming to participate in dialogic discourse, that is, discourse functioning to generate new meaning within a community. Many mathematical practices are directed at the deve...

Presentation to the faculty and doctoral students at Middle Tennessee State University of our JRME article which was under review at the time.

Although many reasons exist to support the Common Core State Standards for Mathematics (CCSSI 2010), different groups have found many reasons to oppose them. One particular criticism circulating on social media is an attack on problems embodying the Standards for Mathematical Practice (SMPs), which are mathematical ways of thinking, or habits of mi...

Instructional leadership is integral to improving mathematics teaching in secondary schools. However, administrators often lack sufficient content knowledge in mathematics to be effective in this role. This study examined the impact of professional development focused on developing leadership content knowledge in algebra. Data included written asse...

Reconsider typical discourse strategies when discussing homework and move toward a system that promotes the Standards for Mathematical Practice.

This study examines a sequence of four middle school algebra tasks through their enactment in three teachers’ classrooms. The analysis centers on the cognitive demand—the kinds of thinking processes entailed in solving the task—and the participatory demand—the kinds of verbal contributions expected of students—of the task as written in the instruct...

Attending to precision is an important aspect of mathematical thinking, whether it be an awareness of the precision of measurements and calculations or a concern for the precision of mathematical communication. In the U.S., attending to precision has been explicitly included as a Standard for Mathematical Practice within the Common Core State Stand...

International calls have been made for reasoning-and-proving to permeate school mathematics. It is important that efforts to heed this call are grounded in an understanding of the opportunities to reason-and-prove that already exist, especially in secondary-level geometry where reasoning-and-proving opportunities are prevalent but not thoroughly st...

Strategies for engaging students in constructing arguments and critiquing reasoning in advanced algebra can establish a productive classroom culture.

Confusion can arise from the subtle difference between proving a general and a particular statement, especially when general statements are presented by textbooks in ways that make them appear particular in nature. The authors discuss the implications for teaching proof in light of the current opportunities in high school geometry textbooks.

Explicit reasoning-and-proving opportunities in the United States are often relegated to a single secondary geometry course. This study analyzed the reasoning-and-proving opportunities in six U.S. geometry textbooks, giving particular attention to the chapter that introduced proof. Analysis focused on the types of reasoning-and-proving activities e...

This brief report analyzes mathematical task enactments in 9 middle school algebra classrooms using the complementary lenses of cognitive demand and participatory demand. Whereas cognitive demand is well known in the field as a conceptualization of the thought processes entailed in solving a mathematical task, participatory demand is a new construc...

An optimization problem from a calculus class can be made accessible to algebra and prealgebra students. Are you smarter than a Welsh corgi?

Background: Homework is a key component of students' school mathematics experiences, especially at the secondary level. Past studies have shown that because a substantial portion of class time is spent going over homework assignments, homework does not remain an at-home activity. Yet, little is known about what takes place during the classroom acti...

This article offers a particular analytic method from systemic functional linguistics, thematic analysis, which reveals the mathematical meaning potentials construed in discourse. Analyses of two middle school classroom excerpts focusing on area—one that derives triangle area formulas from the rectangle area formula and another that connects parall...

As calls are made for reasoning-and-proving to permeate school mathematics, several textbook analyses have been conducted to identify reasoning-and-proving opportunities outside of high-school geometry. This study looked within geometry, examining six geometry textbooks and characterizing not only the justifications given and the reasoning-and-prov...

Our answers to students' questions about the relevance of what we teach might paint mathematics into an undesirable corner.

The outer billiard dynamical system models the motion of a particle around a compact domain, such as a planet orbiting a star. When considering outer billiards in hyperbolic space, an interesting problem is to determine precisely the conditions in which an orbiting particle breaks orbit and escapes to infinity. Past work has classified triangular a...

This article introduces a mathematical entity, the varimeter, that can be used to engage future mathematics teachers in reflection about the structure of measurement systems, thus aiding in the development of insight and understanding which may be beneficial to the future teachers when they work with their own students on measurement. Furthermore,...

This article presents a model of the hyperbolic plane—the Missing Disk model—and discusses ways in which it may be used in a non-Euclidean geometry class to engage students in authentic mathematical activities.

This article examines mathematics teacher collegiality by focusing on both the ways in which teachers interacted as critical colleagues in a long-term professional development project and the evolving role of the teacher–educator–researcher as the facilitator of this project. The professional development collaboration comprised two phases: one focu...

This study builds on the mathematical tasks framework developed during the QUASAR project by considering the ways in which enacted mathematical tasks are concluded. Though the framework originated from a cognitive tradition, this study takes a sociocultural perspective and reinterprets task phases through the lens of activity structure. Based on ob...

As students move onward and upward through collegiate mathematics they are often impressed by the power of advanced techniques (aren't they?), techniques allowing problems that were previously difficult or near-impossible to be solved with relative ease. For example, once students have learned the residue theorem in complex analysis they are able t...

A vignette from an early algebra class reveals a rich opportunity for generating proof before geometry.

This study applied thematic discourse analysis (Lemke, 1990) to a section of a middle school lesson focused on the relationship between the area of parallelograms and rectangles. This analysis provides a way to show the structure of the semantic relations between mathematical terms, shedding light on points of convergence and divergence between par...

The study of student learning of and difficulties with undergraduate calculus has been a major part of mathematics education research for over a decade (Ferrini-Mundy & Graham, 1994). This study used the Lesson Study experience of a group of mathematics Ph.D. students to investigate the potential of lesson study as a research tool for mathematics e...

In smooth manifold theory, the notion of a tangent space makes it possi-ble for differentiation to take place on an abstract manifold. In this paper, the notion of a distribution will be presented which makes it possible for in-tegration to take place on an abstract manifold. The first section introduces terminology and builds intuition via an anal...

It is a common misconception (at least among students and laypeople) that mathematical discovery in elementary geometry is complete. However, by mixing together cyclic polygons and cyclic product relations, a new result has been uncovered. The theorem and its proof are easily accessible for the average undergraduate, and generalizations are suggest...