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In this article, I will discuss a collective research project that I designed for my History of Mathematics course. My students, who are by and large pre-service teachers, explored online, digital versions of 18th-century British almanacs that contained question-and-answer sections for mathematics. In a multi-stage research process, they explored the primary sources, investigated selected mathematical problems from the almanacs, conducted prosopographies of the contributors to the almanacs, and finally surveyed secondary sources. They then presented their results in pairs to their classmates. Through these presentations, as well as the prosopographies and notes they shared on our online classroom management system, the working pairs of students wrote research papers that placed their findings within the context of the research of the whole class. This collective research project allowed my students to begin focused and purposeful research within the first week of classes. Moreover, this project encouraged them to become highly engaged with their primary sources and enabled them to work as a team. This collective research project framework could be applied to a variety of groups of primary sources from the history of mathematics (such as textbooks and journals), now widely available online, to a variety of mathematics courses. [ABSTRACT FROM PUBLISHER]

In this paper, the authors discuss their experiences with ungrading at a small public university in the U.S. and a large public university in Germany. The courses described are Calculus 1 and a content course for pre-service secondary teachers of mathematics. The professors teach differently: one with lecture, the other with inquiry-based learning, and choose to use elements of ungrading in different ways. In one case, the professor lets students self-evaluate their homework, while in the other the professor uses a mix of specifications grading and ungrading for the final grade. We outline and compare our assessment approaches and discuss student performance and feedback. We also present some patterns relating to gender: in both cases, we observed that women students chose lower self-evaluations than men students. We end this paper with a general reflection on our teaching experience and our plans to use ungrading in the future. [ABSTRACT FROM AUTHOR]

Lesson study, a popular method of professional development in which instructors collaborate to critique and improve a lesson, can be used for curriculum development. This paper describes an implementation of lesson study that went through two iterations across different, related, contexts: an undergraduate mathematics content course for secondary preservice teachers and a professional development session with practicing middle and secondary teachers. We describe the ways in which the lesson study process helped us develop curriculum for advanced undergraduate courses for future teachers and provide our insights into the benefits and challenges of this approach. [ABSTRACT FROM AUTHOR]

We provide an introduction to the special issue on Teaching Inquiry, through its motivation and themes. We focus here on Part I: Illuminating Inquiry. [ABSTRACT FROM PUBLISHER]

This paper describes a course designed to introduce students to mathematical thinking and a variety of lower level mathematics topics using baseball while satisfying the goals of quantitative reasoning. We give suggestions for sources, topics, techniques, and examples so any mathematics teacher can design such a course to fit their needs. The course assumes no previous college level mathematics and includes statistics, linear functions, modeling, and simulation. [ABSTRACT FROM AUTHOR]

This paper aims to illustrate a design cycle of inquiry-based mathematics activities. We highlight a series of questions that we use when creating inquiry-based materials, testing and evaluating those materials, and revising the materials following this evaluation. These questions highlight the many decisions necessary to find just the right tasks for our students. Throughout the paper the use of multiple representations (graphical, numerical, symbolic, and narrative) and the distinction between facts, skills, methods, and conceptual understanding is explained and illustrated with examples. Additionally, we present evidence of student learning through excerpts from student journals and exam analysis. [ABSTRACT FROM PUBLISHER]

This paper introduces lessons designed to incorporate applications to teaching high school mathematics in an undergraduate Discrete Mathematics course. Because many prospective high school teachers do not take courses that are specifically designed for teachers, providing materials aimed at teacher preparation that can be easily integrated into courses that serve a general mathematics major is one strategy for addressing mathematics teacher preparation. We developed lessons using four guiding features: choosing appropriate content; making school mathematics connections; incorporating active learning; and providing robust lesson notes. Our interview-based findings document how the lessons were used by instructors and analyze the mathematical ideas and understandings that arose from the use of the lessons at two different sites. [ABSTRACT FROM AUTHOR]

Much has been written about the benefits to be derived by incorporating the history of mathematics into its teaching. Of the various ways to do this, the use of original sources is among the most thrilling for the insights and the challenges it offers for students and instructors alike. In this paper, we describe examples of one particular approach to address these challenges that preserves the thrill of the read. [ABSTRACT FROM PUBLISHER]

Preparing students placing into developmental mathematics for success in undergraduate mathematics and preparing future teachers for the increasing demands of K–12 school settings are both persistent, yet seemingly divergent, problems facing higher education. In this paper, we shed light on a model that attempts to address aspects of each problem and share preliminary evidence of success in a multi-year application of this model. The paper highlights the benefits of implementing the model, particularly to support collaboration between faculty in mathematics and teacher education. [ABSTRACT FROM AUTHOR]

Mathematics education faculty working in mathematics departments are commonly involved in teaching undergraduate mathematics courses, teaching and supervising postgraduate students of mathematics education, and developing mathematics teachers (pre-service and/or in-service). This paper proposes a new elective course that can be taken by either undergraduate mathematics majors or by graduate teaching assistants pursuing degrees in mathematics and teaching undergraduates themselves. Twenty-eight topics are suggested, and the rationale for each is briefly described. Offering such a course provides different opportunities for students; for example, familiarizing them with effective ways to teach and learn mathematics, which has the potential to improve their effectiveness as tutors within the department. A similar course has been successfully taught to undergraduate students majoring in mathematics at an Iranian university for nearly two decades. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to describe the use of a problem involving superfactorials (a specific product of factorials) that provides an in-depth and comprehensive mathematical experience, encompassing skills that mathematicians view as tantamount to exploration. The problem is easily accessible and fosters creativity and perseverance, thereby allowing participants to practice standard steps in developing a mathematical argument: from the problem, to simplifications, then conjecture, and finally proof. We present a brief solution to this problem and then discuss how the experience in trying to solve this problem may evolve during a professional development session for teachers. By providing teachers with extended time to investigate a single problem, we aim to provide professional development opportunities beyond the standard mathematical content or instructional practices. This problem also makes a welcome addition to any undergraduate mathematics curriculum and could be used in a mathematics for liberal arts course, capstone course, or any course with a focus on developing problem-solving skills and the process of doing mathematics. [ABSTRACT FROM AUTHOR]

An introduction is presented in which the editor discusses articles in the issue on topics including Inquiry-Based Modeling of Population Dynamics with Logistic Differential and Difference Equations; how to have students discover techniques for understanding the solutions in differential equations.

Teaching teachers to participate in mathematical inquiry has the potential to both transform belief systems about mathematics and to transform teachers from consumers of mathematics to producers of mathematics. The focus of this paper is to describe the use of a problem, based on a non-traditional binary operation, to encourage and teach pre-service and in-service teachers inquiry and problem-posing. We present a brief mathematical solution for the problem and we then discuss problem-posing activities and discussions of connections to the Standards of Mathematical Practice from the Common Core State Standards. We discuss several key aspects of the problem that make it well-suited for the purposes of teaching inquiry and problem-posing. [ABSTRACT FROM PUBLISHER]

This paper brings together general principles for teaching inquiry across disciplines with those generally employed in mathematics classrooms. It describes some broad principles of teaching inquiry to first-year writing students and shows how these can be employed in the context of a course on game theory in popular culture. Although undergraduate mathematics may seem miles away from undergraduate writing, especially in first-year courses, we show how common educational practices in inquiry-based mathematics courses are easily adapted to teaching a writing course and how these pedagogical methods can be used to encourage students to develop habits of inquiry. [ABSTRACT FROM PUBLISHER]

Mathematics teachers need to teach students more than just a laundry list of mathematical skills. Many of us also want to train our students to use mathematical habits of mind. However, traditional grading schemes usually do not allow us to reward intellectual courage and modesty. In this paper, I report on my implementation of specifications grading in a mathematics content course for preservice elementary teachers. This scheme supports my students in revising their mathematical thinking, taking mathematical risks, and effectively communicating their ideas through writing. [ABSTRACT FROM AUTHOR]

Secondary school mathematics teachers often need to answer the “Why do we do that?” question in such a way that avoids confusion and evokes student interest. Understanding the properties of number systems can provide an avenue to better grasp algebraic structures, which in turn builds students’ conceptual knowledge of secondary mathematics. This paper investigates the number systems from K-12 mathematics, discusses the properties held in each, and diagrammatically establishes connections between the number systems and algebraic structures. [ABSTRACT FROM PUBLISHER]

Initially designed to be an interdisciplinary experiment that would change attitudes about mathematics, the semester-long collaboration between a writing instructor and a mathematics instructor yielded unexpected long-term results. The collaboration served as an immersion in methods and techniques used by writing instructors. Description of jointly supervised assignments demonstrates how this experience enhanced a mathematician's ability to use formal writing assignments in other mathematics classes. [ABSTRACT FROM PUBLISHER]

Educators often argue that mathematics should be taught so that the students in the course are actually "doing mathematics." Is there a consensus among mathematicians and mathematics educators as to the meaning of "doing mathematics?" In an effort to answer this question, we administered a survey to hundreds of university-level mathematics and mathematics education faculty members. Participants ranked the importance of various mathematical activities and also responded to several open-ended questions. Responses to the open-ended questions were analyzed qualitatively to identify patterns. In this paper, we discuss the patterns we observed in analyzing the survey data, with a particular focus on the similarities and differences between mathematicians and mathematics educators. [ABSTRACT FROM AUTHOR]

Over the last 10 years faculty members in the Department of Mathematical Sciences at the United States Air Force Academy have incorporated individual oral exams into mathematics courses. We have experimented with various approaches, shared results and ideas with other department members, and refined our techniques. We have found that this alternative assessment gives the instructor considerable insight into students’ understanding, and, in general, students value the opportunity to demonstrate their abilities via this kind of assessment. In this paper, we provide suggestions on successfully administering oral exams and offer evidence supporting their use in mathematics courses. [ABSTRACT FROM PUBLISHER]

Writing and communication are essential skills for success in the workplace or in graduate school, yet writing and communication are often the last thing that instructors think about incorporating into a mathematics course. A mathematical modeling course provides a natural environment for writing assignments. This article is an analysis of the type of writing assigned in my mathematical modeling course. Where the assignments are unusual, they are thoroughly described and I share my experiences on what worked well and what was not as successful. Where the assignments are more traditional, I offer an analysis of problems and pitfalls in teaching writing. Formal project reports for the class teach students how to write scientific articles. Unusual reflective assignments help students integrate their learning and apply it in new contexts. [ABSTRACT FROM PUBLISHER]

In the context of a Capstone course for prospective secondary teachers (PSTs) of mathematics, this paper examines how a journal assignment was implemented and how it can help us understand PSTs’ views of mathematics and mathematics teaching. First, the course design and specifics of the assignment are described. Then, details about implementation of the assignment are included. Finally, affordances of incorporating this type of assignment are discussed with examples of PSTs’ responses. The identified benefits include PSTs: (a) connecting new mathematical ideas to their prior knowledge; (b) identifying what they do not know; and (c) articulating the importance of reasoning and proving in mathematics teaching and learning. These benefits support the notion that using a journal assignment in mathematics content courses designed for PSTs can help them develop a profound understanding of mathematics content and practices. [ABSTRACT FROM PUBLISHER]

We describe the use of browsing and searching (in libraries, online, inside sources, at meetings, in abstracts, etc.) as a way to stimulate the teacher of undergraduate mathematics, specifically in differential equations. The approach works in all other areas of mathematics. Browsing can help build new and refreshing teaching materials based on how mathematics is used and explored in places other than mathematics. These “other” places are where almost all of our students will be going after they study with us and we should: (i) know about their journey and arrival points; and (ii) understand the disciplinary approaches for those areas which sent these students to us in the first place for their mathematics studies. We describe a personal browsing experience that spanned almost 40 years and proved to be very worthwhile in finding applications of differential equations to modeling Lysergic Acid Diethylamide in the human body. [ABSTRACT FROM AUTHOR]

A single case study approach was used to provide an in-depth examination of the special events that take place in the experiences of a graduate mathematics teaching assistant (MTA) during adaptation to a variety of physical, emotional, and psychological issues. Through intervention by a faculty member, professional counselor, and medical doctor, the MTA eventually achieved positive coping and adaptation away from suicidal ideations. In the context of attribution theory, we will look at the particular stressors and the how their concomitant effect led the MTA to consider suicide. The case study documents the interactions between the MTA and the faculty member as well as how the MTAs attributions impacted their tone and tenor. Drawing from this account and others encountered by other MTAs, the second half of the paper presents a case story to help MTAs consider how to navigate situations that contribute to stress. [ABSTRACT FROM AUTHOR]

This article describes key elements for mathematics teachers, through the faculty development process, to increase student engagement and motivation in learning mathematics. It addresses common barriers facing student motivation and achievement in the field of mathematics and provides potential remedies from a faculty development viewpoint to alleviate these barriers. The paper offers practical examples from the authors' own faculty development programs and experiences in teaching mathematics to both foster and grow the intellectual curiosity of students and offers suggestions for successful faculty development programs. [ABSTRACT FROM AUTHOR]

This paper describes writing assignments from a modern algebra course which were used to foster the development of proof-writing skills. Each assignment consists of proofs and mathematical work supposedly written by fictional characters, where the proofs are designed to illustrate common misconceptions, difficult points, and typical writing flaws. Students critique the fictional work and write short essays on their analysis. The response from students has been very favorable, especially among future teachers. [ABSTRACT FROM AUTHOR]

This mixed-methods study examined prospective elementary teachers' (N = 32) mathematical knowledge and mathematics self-efficacy while learning about the Singapore Modeling Method in a university mathematics course. Pre- and post- quantitative data were collected via two content knowledge assessments and one belief survey, and qualitative data were gathered through individual interviews. The quantitative data analysis showed that content knowledge did not change significantly, whereas mathematics self-efficacy did change significantly. Three themes from the interviews evidenced growth in mathematical knowledge: justifying reasoning in problem solving with visual clarity, variety of valid strategies, and changes in mathematics self-efficacy and connections with content knowledge. [ABSTRACT FROM AUTHOR]

Graphs about Us (GaU) is an activity for building classroom community within mathematics classrooms. Through the creation and analysis of graphical representations of data about the students' lives in and out of school, this activity aims to allow each student to feel seen and validated in the class, create shared knowledge of students' experiences, and to identify and challenge dominant narratives that may be unproductive for learning mathematics. By situating this activity within a mathematical context, GaU promotes students' learning about each other and about mathematics. This article describes three phases of the GaU activity, connects the activity with these goals for building classroom community, and discusses challenges that have arisen when using GaU with students. [ABSTRACT FROM AUTHOR]

We describe the design and implementation of lessons in undergraduate abstract algebra that integrate applications to teaching high school mathematics. Each lesson consists of a pre-activity, class activity, homework set, assessment questions and detailed instructor notes. The included applications intend to bridge advanced mathematics concepts to secondary school mathematics content for prospective secondary mathematics teachers, as well as provide instructors with a useful resource. Each lesson includes four guiding features: addressing meaningful content; making connections to mathematics teaching; emphasizing active learning; and providing comprehensive notes for instructors. Interviews with instructors and their students document how the instructors adapted the lessons to their classrooms and how the lessons influenced undergraduates' mathematical understanding. [ABSTRACT FROM AUTHOR]

The authors describe ways in which a history of mathematics course can help prospective teachers of mathematics develop knowledge they will need for teaching. The types of knowledge include content knowledge of mathematics, knowledge of mathematics for teaching, and pedagogical knowledge. [ABSTRACT FROM AUTHOR]

This article describes reflections of two mathematicians and a mathematics teacher educator who collaborated on the development and implementation of courses (probability and statistics connections, number concept connections, and middle school mathematics methods) for middle school mathematics preservice teachers. The instructors of the courses, two in mathematics and one in mathematics education, worked together to more explicitly link course materials, assignments, and the pedagogical approaches. Collectively, the courses were designed to address the five components of preservice teachers' mathematical knowledge for teaching (PT-MKT), and to model effective teaching practices. Using their collective experiences co-planning and implementing these course adjustments were made in the subsequent year. The instructors were pleased by their implementation and student outcomes in all three courses. We describe how each component of the PT-MKT framework was approached in these courses and discuss challenges experienced by the instructors, who were part of a larger effort to develop and implement a middle school teacher preparation program. The information shared is based on data collected as part of a program evaluation effort, and is bolstered by the instructors' recollection of events. Overall, the instructors enhanced the curricula and their instructional practices and found that the attention placed on developing PT-MKT support the mathematical development of middle school mathematics preservice teachers. [ABSTRACT FROM AUTHOR]

It is widely accepted in the mathematics education community that pedagogies oriented toward inquiry are aligned with a constructivist theory of learning, and that these pedagogies effectively support students’ learning of mathematics. In order to promote such an orientation, we first separate the idea of inquiry from its conception as a collection of methods. Then, by grounding those methods in a generally accepted theory of learning, we construct an inquiry-oriented pedagogy from a constructivist perspective. We then discuss the implications of this pedagogy for the design of mathematical tasks that democratize student access to inquiry. This work has implications for educators who wish to enact an inquiry-oriented pedagogy in their classroom in order to support their students’ problem solving and problem posing. [ABSTRACT FROM PUBLISHER]

In this article we present three group activities designed for math students: a balloon-twisting workshop, a group proof of the irrationality ofπ, and a game of Math Bingo. These activities have been particularly successful in building enthusiasm for mathematics and camaraderie among math faculty and students at Kenyon College. [ABSTRACT FROM PUBLISHER]

Mathematics teachers often give cumulative final exams, but little research has been done on the effects that cumulative exams given throughout the semester have on student grades and content knowledge. This study went beyond a cumulative final exam, and investigated the benefits of cumulative versus non-cumulative semester exams on students’ final exam scores. Comparing control and experimental sections of an introductory mathematics course showed a positive effect of cumulative exams on final exam scores. Additionally, low-scoring students in the experimental sections performed better on the final exam compared with the average of their semester exams than low-scoring students in the control section. This interaction was not statistically significant, however. [ABSTRACT FROM PUBLISHER]

Teachers often promote care in doing calculations, but for most students a single mistake rarely has major consequences. This article presents several real-life events in which relatively minor mathematical errors led to situations that ranged from public embarrassment to the loss of millions of dollars’ worth of equipment. The stories here are appropriate to share in middle school through undergraduate classrooms. [ABSTRACT FROM PUBLISHER]

Real Analysis is a required class for most undergraduate mathematscs majors, but it is also one of the most difficult classes they will take. In this article, the author compares two approaches to teaching in the two analysis classes she has taught. The first one was taught in a traditional lecture-homework-exam format, while the second was taught using active learning tasks, such as before-class reading assignments and in-class discussions of some homework problems. The students in the second class were more actively engaged with the material, and their performance on graded homework and exams was better. This article also discusses the author's transition from conducting a lecture-based class to teaching with a more active format. [ABSTRACT FROM PUBLISHER]

We propose the use of finite topological spaces as examples in a point-set topology class especially suited to help students transition into abstract mathematics. We describe how carefully chosen examples involving finite spaces may be used to reinforce concepts, highlight pathologies, and develop students' non-Euclidean intuition. We end with a project in which finite spaces are featured. [ABSTRACT FROM PUBLISHER]

The study documents the extent to which high school teachers reflect on their need to revise and extend their mathematical and practicing knowledge. In this context, teachers worked on a set of tasks as a part of an inquiring community that promoted the use of different computational tools in problem solving approaches. Results indicated that the teachers recognized that the use of the Cabri-Geometry software to construct dynamic representations of the problems became useful, not only to make sense of the problems statement, but also to identify and explore a set of mathematical relations. In addition, the use of other tools like hand-held calculators and spreadsheets offered them the opportunity to examine, contrast, and extend visual and graphic results to algebraic approaches. [ABSTRACT FROM PUBLISHER]

The Mathematics Education Debate is an assignment designed for and implemented in an undergraduate mathematics methods course for prospective secondary school mathematics teachers. For the assignment, students read and analyze current research and policy reports related to mathematics education, prepare and present their positions, offer rebuttals, and evaluate the presentations of their classmates. This article describes the assignment, discusses the results of its implementation, and offers suggestions for adapting the assignment for other courses. [ABSTRACT FROM AUTHOR]

This article examines the transition process of 14 international teaching assistants (TAs), who teach mathematics courses in U.S. college contexts. In particular, this study documents several major aspects of changes in their teaching, including their curriculum choice and instructional practices over time. This study reveals how the institutional contexts and the TAs' subjective knowledge shaped their teaching as well as prospects for professional development. Findings reveal that their instructional adaptation process reflects their psychological conversion process, which entailed ironic consequences in their final choice of curriculum and instructional practices. Implications for their professional development are discussed from constructivist teaching perspectives. [ABSTRACT FROM AUTHOR]

External performance assessments for teachers of mathematics have traditionally been accomplished through reviews of student comments as well as through formal evaluations by superiors. Additionally, evaluating lecture-based teaching (as opposed to more active learning environments) focuses primarily on classroom performance, curriculum planning, materials development, and assessment of student performance. In this article we identify four additional qualities that are desirable when one is teaching mathematics in a problem-solving curriculum. We then offer various modes of feedback that teachers can use to help achieve a comprehensive assessment of their own teaching effectiveness. [ABSTRACT FROM AUTHOR]

This article describes key elements for faculty development programs to prepare mathematics teachers for technology-rich environments. We offer practical examples from our experiences in teaching mathematics with technology and in teaching others to incorporate technology-based pedagogies. We address challenges faced by faculty using technology, offer suggestions for successful faculty development programs, and provide a list of resources for technology-based teaching. [ABSTRACT FROM AUTHOR]

One of the most publicly celebrated figures in mathematics teaching during the last century was Jaime Escalante (dramatized in the film Stand and Deliver). The Escalante story aptly fulfills three prevailing myths about effective teachers: the Teacher is self-made, the Teacher is the expert, and everything depends on the Teacher. These myths are so predominant in our culture that we can also see their representation in recent legislative actions such as No Child Left Behind and the American Competitiveness Initiative. This demonstrates the strength that the myth has in reflecting, influencing, and legalizing our perceptions of good mathematics teaching. [ABSTRACT FROM AUTHOR]

Given the increase in the number of fictional women mathematicians and scientists on television and in the movies, educators who wish to incorporate pop culture into their classrooms need tools with which to evaluate these portrayals. In this article we summarize studies related to the impact of Hollywood representations on girls and then we provide a case study example of the character Fred Burkle from the television series Angel. In addition we provide a theoretical foundation and popular culture role model checklist that can be used to analyze other representations. [ABSTRACT FROM AUTHOR]