We discuss a general formula for the area of the surface that is generated by
a graph $[t_0, t_1] \to \mathbb R^2$ sending $t \mapsto \bigl(x(t), y(t)
\bigr)$ revolved around a general line $L: A x + B y = C$. As a corollary, we
obtain a formula for the area of the surface formed by revolving $y = f(x)$
around the line $y = m x + k$.
: In a multivariable calculus course, students must master a large number of concepts in order to successfully learn the material. This paper will discuss one way of addressing this di#culty through the use of ConcepTests, that is, multiple choice questions given in the lecture that test understanding as opposed to calculation. In particular, we will look at various types of ConcepTests and the material they can cover. Keywords: multivariable calculus, ConcepTests, small group work 1
We advocate teaching introductory discrete mathematics by first teaching equational propositional and predicate logic and then using it as tool (and not simply viewing it as an object of study) throughout the study of later topics — e.g., set theory, induction, relations, theory of integers, combinatorics, solving recurrence relations, and modern algebra. The in-depth (6–7 weeks) treatment of logic emphasizes rigorous proofs, strategies for developing proofs, and much practice, so that students develop a skill in formal manipulation. Care is taken to explain all concepts clearly and rigorously. Later topics are dealt with by viewing them as theories — extensions of the predicate calculus. The course should motivate the use of logic as a pervasive tool. It must enlighten, and not stifle and oppress. Our experience shows that this is possible. Anecdotal evidence shows that students come away with less fear of mathematics, proof, and notation, more confidence in their mathematical abilities, an appreciation for rigor, and the urge to try out their new skills in other courses.
This article begins the development of a taxonomy of mathematical prose, describing the precise function and meaning of specific types of mathematical exposition. It further discusses the merits and demerits of a style of mathematical writing that labels each passage according to its function as described in the taxonomy.
This study explored the problem posing abilities and attitudes towards mathematics of students in a university pre-calculus class and a university mathematical proof class. A measure of attitude towards mathematics and corresponding two sample t-test revealed a significant (p = .001) difference in the attitude levels of students in the two classes. A measure of problem posing ability explored students problem posing abilities in both numeric and non- numeric contexts. Two sample t-tests showed no significant difference in the problem posing ability (numeric or non-numeric) of students in the two classes. A matched pairs t-test showed a significant difference in numeric posing versus non-numeric posing ability in both classes. Lastly there was no correlation between students attitudes towards mathematics and problem posing abilities.
In 2002, The United States Military Academy at West Point celebrates its bicentennial. For most of those two hundred years the academy has taught its courses by what is known as the Thayer method of instruction. In short, the philosophy is that cadets are responsible for their own learning. They study the material prior to attending class. The learning is then reinforced in class through a combination of group learning and active learning exercises done primarily at the blackboards.I will give a short overview of how instruction at West Point has evolved in two hundred years, along with a discussion of our current teaching methods and how they can be integrated into any science classroom.
We provide in-depth information and analysis on three activities for use in History of Mathematics courses taught either in a traditional semester format for undergraduates or in a summer professional development course for middle school teachers. These activities require students to be active participants in their own learning. They also complement the more traditional mathematics content of such courses. The first focuses on accuracy of historical information; the second on biographical aspects of major contributors to mathematics; and the third on the accuracy of historical infomation, problem posing and questioning, and effective research. For each activity, we discuss the goals, implementation, outcomes, observations, and possible modifications.
We describe the roles and duties of a director in developing an introductory actuarial program. Degree plan design, specialized exam courses, internship classes, coordination of efforts with Economics and Finance Departments, opportunities for creating a minor in actuarial mathematics, actuarial clubs, career advice, and interaction with actuarial societies are discussed.
Some challenges to increasing actuarial science program size through recruiting broadly among potential students are identified. Possible solutions depend on the structures and culture of the school. Up to three student cohorts may result from partition of potential students by the levels of academic progress before program entry: students entering college for the first time, undergraduate transfers, and baccalaureate degree holders. Serving full-time students from these diverse academic levels—and also part-time students who may be from any of the cohorts—involves adaptations such as added recruiting methods, modified curriculum sequencing, extra one-on-one advising, and adjusted professional development activities. As a starting point for faculty dialogue, this paper presents some data and gives references and discussion points.
Students earning a degree in mathematics often seek information on how to apply their mathematical knowledge. One option is to follow a curriculum with an actuarial emphasis designed to prepare students as an applied mathematician in the actuarial field. By developing only two new courses and utilizing existing courses for Validation by Educational Experiences credit, our university developed an actuarial track to accompany a major in mathematics. An actuarial path attracts additional students to the major in mathematics and is a program of study sustainable by smaller departments.
The Casualty Actuarial Society (CAS) believes that the most effective way to advance the actuarial profession is to work in partnership with universities. The CAS stands ready to assist universities in creating or enhancing courses and curricula associated with property/casualty actuarial science. CAS resources for university actuarial science programs and courses are described.
This paper focuses on the development of an actuarial science track for the mathematics major at Salisbury University (SU). A timeline from the initial investigation into such a program through the proposal and approval processes is shared for those who might be interested in developing a new actuarial program. It is wise to start small and take advantage of as many existing courses as possible. Two new courses were developed to fill the void at SU. These courses as well as seminars for preparing students for SOA/CAS exams are described. The extent to which the program prepares students for taking the actuarial exams and/or finding employment is also discussed.
In this introduction to the special issue devoted to actuarial education, we provide an overview of the actuarial profession and what is involved in developing a program to train future actuaries. In addition we point to articles in this issue that include information on the two major actuarial organizations, along with detailed examples of various types of actuarial programs at many levels from very small introductory programs to advanced programs. There are articles on how to start up and run a program, how to handle advising and recruiting problems, and recommendations about courses to prepare students for the actuarial examinations.
Many institutions wish to offer a path for students pursuing actuarial careers but lack the student demand to offer new courses or hire additional faculty. Fortunately, a program training students to enter the profession can often be constructed using existing courses and well-informed advising.
In 2007, the B.S. in Applied Mathematics program consisting of five concentrations, including Actuarial Science, began at West Chester University of Pennsylvania, and we graduated our first class (of one) that December. We describe our program, some ideas to consider when planning your own program, and share some of the successes of our program which have led to our department having close to 40 Actuarial Science majors in six short years! We will focus on our courses and exam preparation, our internship program, how we built connections with businesses and organizations, and our plans for the future.
Building an Actuarial Science program designated as advanced requires dedicated faculty, support from the administration, and a core group of strong students. Washburn University may serve as a model for those wishing to start or enhance such a program at their institution. We face three main ongoing challenges: first, the hiring and retention of high-caliber statisticians; second, attraction and retention of top-quality students; and third, concerns over small upper-division classes. We examine the relevant history of the program at Washburn, the actuarial science curriculum, necessary computing experience, the Society of Actuaries/Casualty Actuarial Society exam series, work experience including internships, and some useful actuarial science sites on the World Wide Web.
Bryant University was originally a school for business majors and offered only a few mathematics courses. After becoming accredited by the New England Association of Colleges and Universities in the 1960s, the college was required to upgrade its offerings in the area of mathematics. In the 1970s, the department offerings were increased to include computer electives, advanced topics in mathematics, and graduate-level quantitative business courses. As the department grew, majors were established in the department, including a major in Actuarial Mathematics. This major has grown from its first graduating class in 1986 of five students to a current total student enrollment today of 172. This paper discusses the evolution of the major and the success of the program. We provide the curriculum and highlight changes, which were necessary to conform to the ever-changing requirements for becoming an actuarial Fellow. We also discuss unique aspects of our program such as the SAS Certification in Data Mining that has greatly increased the marketability of our students.
The purpose of this article is to describe the actuarial science program at our university and the development of a course to enhance students’ problem solving skills while preparing them for Exam P/1 of the Society of Actuaries (SOA) and the Casualty Actuary Society (CAS). The Exam P/1 prep course, formally titled Mathematical Foundations of Actuarial Science, has been offered twice by our institution. Since its offering, nine students have sat for Exam P/1 and seven students passed the exam (three students passed on their first attempt). This course has created a new culture emphasizing studying for actuarial examinations and has established camaraderie among actuarial science students at our institution. The Spring 2010 and Spring 2012 versions of the course will be described and compared and future changes to the course will also be presented.
We describe specific curricular decisions employed at Butler University that have resulted in student achievement in the actuarial science major. The paper includes a discussion of how these decisions might be applied in the context of a new actuarial program.
The article provides details of the process of starting an actuarial science major at a small, liberal arts college. Some critique of the major is included, as well as some challenges that may be faced by others wanting to start such a major at their institution.
The Society of Actuaries (SOA) is the world’s largest actuarial organization. This article describes the SOA with particular attention paid to its education and qualification processes and resources available for university and college programs.
In many post-secondary, introductory mathematics courses failure and withdrawal rates are reaching as high as 50% and average GPA is steadily decreasing. This is a problem that has been witnessed across the globe. With widespread reforms taking place in K-12 mathematics education, many innovative teaching strategies have been created, implemented, and analyzed in an attempt to increase the conceptual knowledge of mathematics students at all levels. However, most of these strategies are not appropriate for large, lecture-style classrooms as are usually seen in large universities. This paper outlines some teaching and assessment strategies that have been modified to accommodate the needs and restrictions of such classrooms.
Albrecht Dürer is well known as a Renaissance artist, but his 1525 work, the Underweysung der Messung, or Manual on Measurement, also contains a significant amount of mathematics. This article outlines four projects inspired by parts of this work, with additional material coming from Staigmüller’s 1891 “Dürer als Mathematiker.” Before-and-after questionnaires were given to the students who worked the projects; their responses are analyzed.
The Mat-Rix-Toe project utilizes a matrix-based game to deepen students’ understanding of linear algebra concepts and strengthen students’ ability to express themselves mathematically. The project was administered in three classes using slightly different approaches, each of which included some editing component to encourage the improvement of the students’ mathematical thinking and writing. Differences in the implementation of the project illustrate the benefits and drawbacks of various methods of editing in the mathematics classroom and highlight recommendations for improvements in future implementations of the project.
Concept-focused quiz questions required College Algebra students to write about their understanding. The questions can be viewed in three broad categories: a focus on sense-making, a focus on describing a mathematical object such as a graph or an equation, and a focus on understanding vocabulary. Student responses from 10 classes were analyzed. Students’ writing provided insight into misunderstandings they had about linear and quadratic equations and systems of linear equations and difficulties using correct mathematical vocabulary. Instructors can use this type of information, which may be undetected with other forms of assessment, to make instructional decisions that meet students’ needs.
The inverted classroom is a course design model in which students’ initial contact with new information takes place outside of class meetings, and students spend class time on high-level sense-making activities. The inverted classroom model is so called because it inverts or “flips” the usual classroom design where typically class time is spent on information transfer (usually through lecturing) while most higher-order tasks are done outside the classroom through homework. The inverted classroom model is particularly well-suited for linear algebra, which mixes relatively straightforward mechanical calculation skills with deep and broad conceptual knowledge. In this paper, we discuss how the inverted classroom design can be applied to linear algebra in three modes: as a one-time class design to teach a single topic, as a way to design a recurring series of workshops, and as a way of designing an entire linear algebra course.
The present study aimed to investigate in what way and to what extent Taiwanese college students’ epistemological views of mathematics had evolved during a history-based liberal arts mathematics course titled: “When Liu Hui Meets Archimedes—Development of Eastern and Western Mathematics.” The course was designed to help college students comprehend the cultural features of mathematics by highlighting the similarities and differences in the formation of mathematical knowledge in the Eastern and Western worlds. Through comparing and contrasting collected data, we found students demonstrated different epistemological views of mathematics in three aspects: first, they were able to recognize the diverse features of mathematics; second, they better realized how mathematics had interacted with societies; and third, they tended to comprehend the different mathematical cultures that exist in the Eastern and Western worlds. However, students were unable to appreciate intuition-based inductive approaches and were less likely to acknowledge how creativity and imagination were involved in mathematical thinking.
We have used wiki technology to support large-scale, collaborative writing projects in which the students build reference texts (called WikiTextbooks). The goal of this paper is to prepare readers to adapt this idea for their own courses. We give examples of the implementation of WikiTextbooks in a variety of courses, including lecture and discovery-based courses. We discuss the kinds of challenges that WikiTextbooks address and focus on critical design decisions. Finally, we conclude with a suggested template wiki project that is approachable for new users and appropriate for many course structures.
Liberal arts mathematics courses can provide non-majors the opportunity to connect mathematical topics with areas of personal interest. This article describes two end-of-unit writing assignments (on voting and graph theory) that have been structured so that each student is able to synthesize course material in a unique way, while ensuring a consistent level of depth and efficient assessment for the instructor. Students are free to choose the context for their project, but they must incorporate a “skeleton” of data that is common to the entire class.
History of math courses are commonly offered in mathematics departments. Such courses naturally lend themselves to writing assignments, and a growing body of research supports writing as a means to learn mathematics. This article details two such assignments, providing an overview of the course in which they are situated, and a student-led feedback and editing cycle is described. Feedback from student surveys and instructor observations indicates that these assignments were well-received and enhanced student learning. These assignments and feedback protocol lend themselves well to other mathematics courses with a substantial writing assignment.
Providing audio files in lieu of written remarks on graded assignments is arguably a more effective means of feedback, allowing students to better process and understand the critique and improve their future work. With emerging technologies and software, this audio feedback alternative to the traditional paradigm of providing written comments seems a natural progression to improve student learning. Its utility is especially significant for feedback on writing in mathematics, when it becomes easier for instructor corrections to not only point to deficiencies, but also offer more in depth explanations on the root causes of error in student formulations or mathematical modeling. This paper details the benefits of implementing an audio feedback technique for writing in mathematics, and provides results of a case study performed during four semesters of a collegiate freshman mathematics course. Given instructor initiative and startup time to acclimate to this new way of assessment, the benefits of the audio feedback alternative towards student learning appear substantial.
This paper outlines a method for teaching topics in undergraduate mathematics or computer science via historical curricular modules. The contents of one module, “Networks and Spanning Trees,” are discussed from the original work of Arthur Cayley, Heinz Prüfer, and Otakar Borůvka that motivates the enumeration and application of trees in graph theory. Cayley correctly identifies a pattern for the number of (labeled) trees on n fixed vertices. Prüfer’s paper provides a rigorous verification of this pattern, whereas Borůvka’s paper offers one of the first algorithms for finding a minimal spanning tree over the domain of labeled trees. These latter two papers in juxtaposition offer a pleasing confluence of concepts and applications, written verbally before the modern terminology of graph theory had been formulated.
Ithaca College, in New York, has developed and tested a projects-based first-year calculus course over the last 3 years which uses the graphs of functions and physical phenomena to illustrate and motivate the major concepts of calculus and to introduce students to mathematical modeling. The course curriculum is designed to: (1) emphasize on the unity of calculus; (2) focus on the effective teaching of the central concepts of calculus; (3) increase geometric understanding; (4) teach students to be good problem solvers; and (5) improve attitudes toward mathematics. The course centers on large, often open-ended, problems upon which students work both in and outside of class in groups, and individually, spending 2 to 3 weeks on each problem. Most of these projects are presented in such a way as to require a top-down analysis, in which the top level forces attention to a main idea, while the computations are required at the lowest levels. This approach enables students to recognize calculations as the "nuts and bolts" of a larger problem-solving process. Students' active participation, and clear written presentations of results are required. The course design is best represented by a spiral, which emphasizes the unity of calculus, while allowing for the continual review of the discipline's key skills and concepts, such as graphing, distance and velocity, multiple representations of functions, modeling, and top-down methodology. Three sample course problems are provided. (MAB)
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.
This article presents two approaches to using original sources in a capstone writing project for a History of Mathematics course. One approach involves searching local libraries and is best suited to schools in metropolitan areas. A second approach involves online resources available anywhere. Both projects were used in a course intended for mathematics majors with an education concentration. The specific details of both projects will be discussed, including the motivation and setting, grading scheme, and revision process.
This paper provides suggestions for preparing students to take the actuarial examination on financial mathematics, SOA/CAS Exam FM/2. It is based on current practices employed at Slippery Rock University, a small public liberal arts university. Detailed descriptions of our Theory of Interest course and subsequent Exam FM/2 prep course are provided along with suggestions for course scheduling and strategies for improving performance on the exam.
This article discusses the creation of a student project about linear difference equations using primary sources. Early 18th-century developments in the area are outlined, focusing on efforts by Abraham De Moivre (1667–1754) and Daniel Bernoulli (1700–1782). It is explained how primary sources from these authors can be used to cover material appearing in most discrete mathematics courses while revealing how the mathematical ideas evolved. It is argued that this project works well in a discrete mathematics course that also makes the transition to higher mathematics, developing mathematical reading and writing skills in a natural setting.
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.
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.
We developed, piloted, and evaluated a successful model for integrating other disciplinary perspectives while learning a specific discipline. Faculty who taught first-year biology and mathematics courses produced examples of how these disciplines complement each other. These faculty presented students with the connecting threads that integrate mathematics and biology within their own courses. Intervention group students (those exposed to the integration) were block scheduled in biology and mathematics sections taught by participating faculty to evaluate the outcomes. Compared to the control group students, who had not been exposed to integration, the intervention group students perceived that they understood the application and relatedness of mathematics and biology, and they appreciated this integration in their courses. Similar integrations could be achieved elsewhere using the model we present. (Contains 1 table.)
Solving indeterminate algebraic equations in integers is a classic topic in the mathematics curricula across grades. At the undergraduate level, the study of solutions of non-linear equations of this kind can be motivated by the use of technology. This article shows how the unity of geometric contextualization and spreadsheet-based amplification of this topic can provide a discovery experience for prospective secondary teachers and information technology students. Such experience can be extended to include a transition from a computationally driven conjecturing to a formal proof based on a number of simple yet useful techniques.
Described are student/consultant reports, turned in as homework, that reflected their abilities to distinguish useful from irrelevant information, solve the underlying calculus extremum problem, and reconcile the individual mathematical solutions with the ethical considerations surrounding the moral dilemma of the hypothetical consulting client. (JJK)
This article provides a framework for creating and using writing assignments based on four types of writing: personal, expository, critical, and creative. This framework includes specific areas of student growth affected by these writing styles. Illustrative sample assignments are given throughout for each type of writing and various combinations thereof. Also discussed are the assessment of mathematical writing and suggestions for beginning users of writing in mathematics courses.
We consider a variety of writing assignments used for two general education mathematics courses. The student writing ranges from brief informal pieces to more formal assignments that address the full scope of a course, with an emphasis on encouraging students to have a richer experience of mathematical people and ideas and of their own learning. The assignments incorporate multiple media, including novels and films, and are spread throughout a course so that students have frequent and regular exposure to writing.
This editorial introduces the Special Issue on Writing in the Mathematics Curriculum. This issue begins with an article detailing reasons and ways to incorporate writing into math courses. The issue continues with an exploration of writing in a variety of forms: project-oriented articles, uses of technology in the writing and editing process, and course-specific approaches. The articles include concrete ways in which to introduce writing into courses in the mathematics curriculum, and also give practical guidelines for evaluating student writing.
To address the challenge of developing students' mathematical intuition in our nonlinear optimization course, we crafted a novel methodology that integrates guided classroom discussions, graphical explorations using a custom procedure written to support unique MAPLE graphics, and an instructor led outside-the-classroom exercise that provides a common referential experience against which students can communicate their understanding of nonlinear search techniques that form the core of our course content. Rather than attempting to fabricate our perception of what should be a shared experience for our students, we created an actual terrain walk exercise to fill this need. Our students use this experience as a reference point for explaining their insights, critiquing suggested improvements or modifications in basic local search algorithms, and for illuminating the costs and benefits of adopting various search strategies in light of underlying problem characteristics.