Trigonometry is an important subject in the high school mathematics curriculum. As one of the secondary mathematics topics that are taught early and that link algebraic, geometric, and graphical reasoning, trigonometry can serve as an important precursor to calculus as well as collegelevel courses relating to Newtonian physics, architecture, surveying, and engineering. Unfortunately, many high school students are not accustomed to these types of reasoning (Blackett and Tall 1991), and learning about trigonometric functions is initially fraught with difficulty. Trigonometry presents many first-time challenges for students: It requires students to relate diagrams of triangles to numerical relationships and manipulate the symbols involved in such relationships. Further, trigonometric functions are typically among the first functions that students cannot evaluate directly by performing arithmetic operations.

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### Keith Weber

The purpose of this article is to investigate the mathematical practice of proof validation—that is, the act of determining whether an argument constitutes a valid proof. The results of a study with 8 mathematicians are reported. The mathematicians were observed as they read purported mathematical proofs and made judgments about their validity; they were then asked reflective interview questions about their validation processes and their views on proving. The results suggest that mathematicians use several different modes of reasoning in proof validation, including formal reasoning and the construction of rigorous proofs, informal deductive reasoning, and examplebased reasoning. Conceptual knowledge plays an important role in the validation of proofs. The practice of validating a proof depends upon whether a student or mathematician wrote the proof and in what mathematical domain the proof was situated. Pedagogical and epistemological consequences of these results are discussed.

### Manya Raman and Keith Weber

According to the NCTM *Standards* (2000), conjecturing and proving should be central activities throughout students' mathematical education. However, the question of how we can help students generate proofs, especially the formal proofs expected at the high school level, is still a difficult one. In this article, we argue that one cause of students' troubles with proof is that they are not accustomed to making the important, but difficult, connections between their intuitive sensemaking and the formal proofs they are supposed to produce. Thus as teachers, we should provide opportunities for students to make these connections.

### Jennifer A. Czocher and Keith Weber

To design and improve instruction in mathematical proof, mathematics educators require an adequate definition of proof that is faithful to mathematical practice and relevant to pedagogical situations. In both mathematics education and the philosophy of mathematics, mathematical proof is typically defined as a type of justification that satisfies a collection of necessary and sufficient conditions. We argue that defining the proof category in this way renders the definition incapable of accurately capturing how category membership is determined. We propose an alternative account—proof as a cluster category—and demonstrate its potential for addressing many of the intractable challenges inherent in previous accounts. We will also show that adopting the cluster account has utility for how proof is researched and taught.

### Keith Weber and Kathryn Rhoads

Understanding what mathematics teachers know, what they need to know about mathematics, and how that knowledge is learned are important goals in mathematics education. Research on mathematics teacher knowledge can be divided into two categories: (a) what knowledge mathematics teachers have or need to have to teach effectively (e.g., Hill, Rowan, & Ball, 2005; Kahan, Cooper, & Bethea, 2003), and (b) how teachers' mathematical knowledge for teaching can be developed (e.g., Bell, Wilson, Higgins, & McCoach, 2010; Proulx, 2008). This book describes research of the second type. To date, research in this area has focused primarily on how mathematical knowledge develops in university or researcher-led teacher preparation or professional development programs. This book is novel in that it concerns how and what teachers learn through the process of teaching itself. In his contribution to this book, Ron Tzur (chapter 3) lays out three reasons why this research is essential. First, he argues, teacher preparation programs simply do not contain enough time for teachers to learn all they need to know, so teachers' learning through teaching is essential. Second, for teachers to develop knowledge of how students think about mathematics and how students receive mathematical lessons, teachers must have classroom experience. Third, the experiences that teachers encounter when teaching have the potential to give rise to meaningful changes in their beliefs and practice.

### Juan Pablo Mejiía-Ramos and Keith Weber

We report on a study in which we observed 73 mathematics majors completing 7 proof construction tasks in calculus. We use these data to explore the frequency and effectiveness with which mathematics majors use diagrams when constructing proofs. The key findings from this study are (a) nearly all participants introduced diagrams on multiple tasks, (b) few participants displayed either a strong propensity or a strong reluctance to use diagrams, and (c) little correlation existed between participants' propensity to use diagrams and their mathematical achievement (either on the proof construction tasks or in their advanced mathematics courses). At the end of the report, we discuss implications for pedagogy and future research.

### Keith Weber and Juan Pablo Mejía-Ramos

In a recent article, Inglis and Alcock (2012) contended that their data challenge the claim that when mathematicians validate proofs, they initially skim a proof to grasp its main idea before reading individual parts of the proof more carefully. This result is based on the fact that when mathematicians read proofs in their study, on average their initial reading of a proof took half as long as their total time spent reading that proof. Authors Keith Weber and Juan Pablo Mejía-Ramos present an analysis of Inglis and Alcock's data that suggests that mathematicians frequently used an initial skimming strategy when engaging in proof validation tasks.

### David Roach, David Gibson, and Keith Weber

You might want to try this brief experiment in one of your mathematics courses.

### Keith Weber, Juan Pablo Mejía-Ramos, and Tyler Volpe

Many mathematics educators believe a goal of instruction is for students to obtain conviction and certainty in mathematical statements using the same types of evidence that mathematicians do. However, few empirical studies have examined how mathematicians use proofs to obtain conviction and certainty. We report on a study in which 16 advanced mathematics doctoral students were given a task-based interview in which they were presented with various sources of evidence in support of a specific mathematical claim and were asked how convinced they were that the claim was true after reviewing this evidence. In particular, we explore why our participants retained doubts about our claim after reading its proof and how they used empirical evidence to reduce those doubts.

### Kristen Lew, Timothy Patrick Fukawa-Connelly, Juan Pablo Mejía-Ramos, and Keith Weber

We describe a case study in which we investigate the effectiveness of a lecture in advanced mathematics. We first videorecorded a lecture delivered by an experienced professor who had a reputation for being an outstanding instructor. Using video recall, we then interviewed the professor to determine the ideas that he intended to convey and how he tried to convey these ideas in this lecture. We also interviewed 6 students to see what they understood from this lecture. The students did not comprehend the ideas that the professor cited as central to his lecture. Based on our analyses, we propose 2 factors to account for why students did not understand these ideas.