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In this paper, we provide an in-depth account of traditional IBL instruction. Understanding the nature and effects of this form of instruction is of growing importance due to the strength and breadth of the IBL movement and its connections to other forms of inquiry in undergraduate mathematics. In this case study of one real analysis course taught by a very experienced instructor and leader in the IBL community, we identify the primary goals this professor set forth for his students and the instructional strategies he employed to help them attain these goals. We observe that many of these practices seek to manage the double bind of proving in the classroom (Herbst Journal for Research in Mathematics Education, 33(3), 176-203, 2002). Accordingly, we relate and categorize student storylines in the course according to their goal orientation and their level of buy-in to the IBL learning environment. We operationalize orientation and buy-in according to the double bind of the didactic contract by observing whether students focused more on the professor’s obligation to provide resources for students to successfully complete the work of the course (and earn good grades) or on his obligation to provide an authentic experience in mathematical proving. Students’ with different goal orientation and buy-in profiles took advantage of different aspects the professor’s instructional strategies to engage in the course in ways that were productive for them.

Responding to students' questions is a critical part of teaching mathematics (Cavey, Whitenack, and Lovin 2006; NCTM 1991; Schifter 1996). A particular response may either stifle a student's inquiry or, ideally, stimulate his or her interest in mathematics. Although formulating responses that have the potential to engage students in developing new mathematical insights is challenging, we believe that this skill can be developed. We have found it helpful to view this challenge as a matter of recognizing the mathematics of a student's question—that is, the mathematics the student understands as well as the mathematics he or she may be ready to learn

The careful use of notation and language in the statement of both the definitions and problems of calculus can begin the process of making students mathematically literate while allowing them to enjoy working on challenging problems and applications without the aid of numerous examples. Engaging the students to participate in the precise use of the language inculcates a philosophy of careful use of language that benefits all students regardless of major. Not only will this benefit those few students who decide to major in mathematics, but the precision of language will especially benefit both majors and non-majors outside of their mathematical studies. This pedagogical piece exhibits anecdotal evidence that such an approach provides motivation to students, better preparation for “proofs” classes, and a desire to pursue higher mathematics. Examples of calculus problems presented in the traditional fashion alongside subtly more precise presentations illustrate the process.

We describe a two-semester numerical methods course that serves as a research experience for undergraduates without requiring external funding or modification of current curriculum. The first semester introduces traditional material and builds a proper set of tools that the students use in the second semester to approach a more research oriented problem. Our vehicle is an engineering problem associated with the hydro-dynamics of keel design which is used to introduce students to constrained optimization via a variation of the traditional isoperimetric problem of finding the curve with fixed end-points, fixed perimeter, and maximum area.

Sobolev descent has long been established as an efficient method for numerically solving boundary value problems, ordinary differential equations and partial differential equations in a small number of iterations. We demonstrate that for any linear ordinary differential equation with initial value conditions sufficient to assure a unique solution, there exists a Hilbert space in which gradient descent will converge to the solution in one iteration. We provide two elementary examples, one initial value problem and one boundary value problem, demonstrating the effectiveness of the theory in numerical settings. As there are ample efficient numerical methods for solving such problems, the significance of the paper is in the approach and the question it raises. Namely, do such spaces exist for wider classes of differential equations?

“That student is taught the best who is told the least.” R. L. Moore This chapter addresses the generation of materials to be used to support Moore Method teaching. Development and usage of materials by the prospective user is treated from various perspectives, including guidelines for writing your own problem sequences, adapting textbook treatments, adapting pre-existing Moore Method course notes, modifying materials to fit the realities of a particular class, addressing learning-theory issues, and obtaining support when using another's notes. These perspectives illustrate principles that may guide the construction of problem sets, resolve the question of authoring notes or adapting pre-existing notes, and provide sources for course notes that have been written for use in Moore Method classes other than your own. The essays will likely offer the greatest benefit if read in the order presented, especially for the reader entertaining thoughts about using the Moore Method for the first time. May's essay is both highly practical and succinct in its outlook. Mahavier deals, in some detail, with making the decision whether to author notes or use a pre-existing problem sequence. He then segues into guidelines for penning materials. Coppin focuses exclusively on authoring one's own notes, addressing issues from learning theory and illustrating how course notes can address these theories directly. Parker writes from the viewpoint of generating course notes, then adapting them consequent to classroom dynamics.

The Moore Method: A Pathway to Learner-Centered Instruction offers a practical overview of the method as practiced by the four co-authors, serving as both a 'how to' manual for implementing the method and an answer to the question, 'what is the Moore method?'. Moore is well known as creator of The Moore Method (no textbooks, no lectures, no conferring) in which there is a current and growing revival of interest and modified application under inquiry-based learning projects. Beginning with Moore's Method as practiced by Moore himself, the authors proceed to present their own broader definitions of the method before addressing specific details and mechanics of their individual implementations. Each chapter consists of four essays, one by each author, introduced with the commonality of the authors' writings. Topics include the culture the authors strive to establish in the classroom, their grading methods, the development of materials and typical days in the classroom. Appendices include sample tests, sample notes, and diaries of individual courses. With more than 130 references supporting the themes of the book the work provides ample additional reading supporting the transition to learner-centered methods of instruction.

“All the instruments have been tried save one, the only one precisely that can succeed: well-regulated freedom.” —Jean-Jacques Rousseau In this chapter, we discuss the results of educational research that has been conducted evaluating the impact of the Moore Method and describe the results of educational research on similar teaching approaches, draw parallels to the Moore Method, and make suggestions for further assessment of the effectiveness of the Moore Method for interested readers. Introduction Whenever an innovation emerges in teaching, there are important questions to be asked. Does the innovation actually work? That is, does it truly improve students' learning, or does it just give the appearance of doing so? Is it feasible that the innovation can be implemented by other instructors in other educational settings, or does it rely upon the talent of a few gifted instructors in a particular context? Is the innovation appropriate for all learners, or just for a particular subgroup? Though the proponents of such innovations are often quite enthusiastic in their efforts to describe and promote the innovation, they are often only able to provide anecdotal evidence for its success. While important and frequently convincing, such evidence is not sufficient to answer the questions posed above. Rather, careful, systematic research should be done to establish the effectiveness of the innovation.