Keith Weber

Students’ understanding of trigonometric functions

In this article students’ understanding of trigonometric functions in the context of two college trigonometry courses is investigated. The first course was taught by a professor unaffiliated with the study in a lecture-based course, while the second was taught using an experimental instruction paradigm based on Gray and Tall’s (1994) notion of procept and current process-object theories of learning. Via interviews and a paper-and-pencil test, I examined students’ understanding of trigonometric functions for both classes. The results indicate that the students who were taught in the lecture-based course developed a very limited understanding of these functions. Students who received the experimental instruction developed a deep understanding of trigonometric functions.

How Mathematicians Obtain Conviction: Implications for Mathematics Instruction and Research on Epistemic Cognition

The received view of mathematical practice is that mathematicians gain certainty in mathematical assertions by deductive evidence rather than empirical or authoritarian evidence. This assumption has influenced mathematics instruction where students are expected to justify assertions with deductive arguments rather than by checking the assertion with specific examples or appealing to authorities. In this article, we argue that the received view about mathematical practice is too simplistic; some mathematicians sometimes gain high levels of conviction with empirical or authoritarian evidence and sometimes do not gain full conviction from the proofs that they read. We discuss what implications this might have, both for mathematics instruction and theories of epistemic cognition.

Mathematicians' perspectives on their pedagogical practice with respect to proof

In this article, nine mathematicians were interviewed about their why and how they presented proofs in their advanced mathematics courses. Key findings include that: (1) the participants in this study presented proofs not to convince students that theorems were true but for reasons such as conveying understanding and illustrating methods, (2) participants believed students did not appreciate the complex processes involved in reading a proof but often did not teach these processes to students, (3) the participants used superficial methods to assess students’ understanding of a proof and (4) some participants questioned whether proof was the best way to convey mathematics to all of their students.

Mathematicians' Perspectives on Features of a Good Pedagogical Proof.

In this article, we report two studies investigating what mathematicians value in a pedagogical proof. Study 1 is a qualitative study of how eight mathematicians revised two proofs that would be presented in a course for mathematics majors. These mathematicians thought that introductory and concluding sentences should be included in the proofs, main ideas should be formatted to emphasize their importance, and extraneous or redundant information should be removed to avoid distracting or confusing the reader. Study 2 is a quantitative study assessing the extent to which a larger group of mathematicians (N = 110) agreed or disagreed with the eight mathematicians interviewed in Study 1. This quantitative study confirmed the findings of Study 1 by demonstrating a high degree of agreement among mathematicians regarding how they would revise proofs for pedagogical purposes.

On Mathematicians' Different Standards When Evaluating Elementary Proofs

In this article, we report a study in which 109 research-active mathematicians were asked to judge the validity of a purported proof in undergraduate calculus. Significant results from our study were as follows: (a) there was substantial disagreement among mathematicians regarding whether the argument was a valid proof, (b) applied mathematicians were more likely than pure mathematicians to judge the argument valid, (c) participants who judged the argument invalid were more confident in their judgments than those who judged it valid, and (d) participants who judged the argument valid usually did not change their judgment when presented with a reason raised by other mathematicians for why the proof should be judged invalid. These findings suggest that, contrary to some claims in the literature, there is not a single standard of validity among contemporary mathematicians.

Challenges in Mathematical Cognition: A Collaboratively-Derived Research Agenda

This paper reports on a collaborative exercise designed to generate a coherent agenda for research on mathematical cognition. Following an established method, the exercise brought together 16 mathematical cognition researchers from across the fields of mathematics education, psychology and neuroscience. These participants engaged in a process in which they generated an initial list of research questions with the potential to significantly advance understanding of mathematical cognition, winnowed this list to a smaller set of priority questions, and refined the eventual questions to meet criteria related to clarity, specificity and practicability. The resulting list comprises 26 questions divided into six broad topic areas: elucidating the nature of mathematical thinking, mapping predictors and processes of competence development, charting developmental trajectories and their interactions, fostering conceptual understanding and procedural skill, designing effective interventions, and developing valid and reliable measures. In presenting these questions in this paper, we intend to support greater coherence in both investigation and reporting, to build a stronger base of information for consideration by policymakers, and to encourage researchers to take a consilient approach to addressing important challenges in mathematical cognition.

Expanding Participation in Problem Solving in a Diverse Middle School Mathematics Classroom.

In this paper, we discuss our experiences with an after-school program in which we engaged middle-school students with low socioeconomic status from an urban community in mathematical problem solving. We document that these students participated in many aspects of problem solving, including the posing of problems, constructing justifications, developing and implementing problem-solving heuristics and strategies, and understanding and evaluating the solutions of others. We then delineate what aspects of our environment encouraged the students to take part in these activities, particularly emphasising the proactive role of the teacher, the tasks the students completed, and the social norms of our after-school sessions. Finally, we discuss the relationship between our study and the literature on equity research in mathematics education.

Mathematics majors’ beliefs about proof reading

We argue that mathematics majors learn little from the proofs they read in their advanced mathematics courses because these students and their teachers have different perceptions about students’ responsibilities when reading a mathematical proof. We used observations from a qualitative study where 28 undergraduates were observed evaluating mathematical arguments to hypothesize that mathematics majors hold four specific unproductive beliefs about proof reading. We then conducted a survey about these beliefs with 175 mathematics majors and 83 mathematicians. We found that mathematics majors were more likely to believe that when reading a good proof, they are not expected to construct justifications and diagrams, they can understand most proofs they read within 15 minutes, and understanding a proof is tantamount to being able to justify each step in the proof.

The role of affect in learning Real Analysis: a case study

This paper presents a description of the interaction between one students emotional states and mathematical learning during an introductory Real Analysis course. This case study illustrates the important influence affect can have on a students success or failure in learning Real Analysis. In particular, it shows how frustration and anxiety led the observed student to place more emphasis on rote learning strategies and avoid engaging in the course material. The case study also describes how the students small gains in her understanding provided her with feelings of pleasure and encouragement, which in turn motivated her to seek out opportunities to study the material further.

On the different ways that mathematicians use diagrams in proof construction

The processes by which individuals can construct proofs based on visual arguments are poorly understood. We investigated this issue by presenting eight mathematicians with a task that invited the construction of a diagram, and examined how they used this diagram to produce a formal proof. The main findings were that participants varied in the extent of their diagram usage, it was not trivial for participants to translate an intuitive argument into a formal proof, and participants’ reasons for using diagrams included noticing mathematical properties, verifying logical deductions, representing ideas or assertions, and suggesting proof approaches.