Collegiate Mathematics Research Articles

When less is more: visualizing basic inequalities

Inequalities permeate mathematics, from the Elements of Euclid to operations research and financial mathematics. Yet too often, especially in secondary and collegiate mathematics, the emphasis is on things equal to one another rather than unequal. While equalities and identities are without doubt important, they don’t possess the richness and variety that one finds with inequalities. The objective of this book is to illustrate how the use of visualization can be a powerful tool for better understanding some basic mathematical inequalities. Drawing pictures is a well-known method for problem solving, and the authors will convince you that the same is true when working with inequalities. They show how to produce figures in a systematic way for the illustration of inequalities and open new avenues to creative ways of thinking and teaching. In addition, a geometric argument cannot only show two things unequal, but also help the observer see just how unequal they are. The concentration on geometric inequalities is partially motivated by the hope that secondary and collegiate teachers might use these pictures with their students. Teachers may wish to use one of the drawings when an inequality arises in the course. Alternatively, When Less Is More might serve as a guide for devoting some time to inequalities and problem solving techniques, or even as part of a course on inequalities.

Teacher lust: reconstructing the construct for mathematics instruction

Two collegiate mathematics courses for prospective elementary and middle grades teachers provide the context for the examination of Mary Boole’s construct of teacher lust. Through the use of classroom observations and instructor interviews, the author presents a refined conception of teacher lust. Two working aspects of the construct were identified: (1) enacted teacher lust; an observable action that may remove an opportunity for students to think about or engage in mathematics for themselves; and (2) experienced teacher lust; an internal impulse to act in the manner described. Empirical examples of each facet, differences between conscious and unconscious interactions with teacher lust, along with potential antecedents are discussed.

The notion of reducing abstraction in quadratic functions

One possible approach students can cope with abstract algebra concepts is reducing abstraction. This notion occurs when learners are unable to adopt mental strategies as they deal with abstraction level of a given task. To make these concepts mentally accessible for themselves, learners unconsciously reduce the level of the abstraction of the concepts (O. Hazzan and R. Zazkis, Reducing abstraction: the case of school mathematics, Educ. Stud. Math. 58 (2005), pp. 101–119). Reducing abstraction as a theoretical framework has been used for explaining students’ thought process in areas of advance mathematical topics in collegiate mathematics. By analysing two tenth-grade algebra honour students’ ways of thinking on quadratic functions, this article shows how the notion of reducing abstraction can be used for analysing students’ mental processes in secondary school mathematics and provides insights into students learning about graphs as well as reports two different strategies on translation tasks as an act of reducing the level of abstraction: one provides an effective method, while the other becomes misleading.

Undergraduate mathematics achievement in the emerging ethnic engineers programme

This paper addresses the issue of collegiate mathematics achievement of underrepresented minority students as it investigates the impact of a cooperative learning calculus programme on the first-year calculus experience of non-Asian ethnic minority engineering students. The Emerging Ethnic Engineers Programme in the College of Engineering at the University of Cincinnati is a successful, comprehensive programme that focuses on the recruitment, retention, academic success, professional development, and timely graduation of underrepresented coloured students. The objectives of the programme are accomplished through three interrelated phases: pre-college science and mathematics programmes, first-year collegiate programmes, and upper-division programmes. The underlying principles of the first-year programme include academic achievement and establishing a strong sense of community among the cohort. This report will focus on the cooperative learning calculus programme that has been successful in improving retention and academic success rates for coloured freshmen engineering students.

The Eccentricity of a Conic Section

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsAyoub B. AyoubAyoub B. Ayoub (aba2@psu.edu) is Professor of Mathematics at Abington College of the Pennsylvania State University. His interests include number theory, mathematics history and collegiate mathematics. He received the Penn State's 1990 George Atherton Award for Excellence in Teaching.

Responding to Calls for Change in High School Mathematics: Implications for Collegiate Mathematics

This article briefly discusses the recent history of mathematics reform in high school, and then reports on research evidence for one of the new NSF-funded curriculum projects, the Core-Plus Mathematics Project (CPMP). Implications for collegiate mathematics are also discussed. A summary of the results of several studies using a range of achievement measures comparing CPMP students to comparable students in more conventional high school mathematics curricula is given on page 114: “Thus, research to date indicates that CPMP students perform particularly well [and better than the comparison students] on measures of conceptual understanding, interpretation of mathematical representations and calculations, and problem solving in applied contexts. Their performance is also relatively strong in content areas like statistics and probability that are emphasized in the curriculum. On measures of algebraic manipulative skills, CPMP students usually, but not always, score as well as students in more conventional curricula.” A study of student performance on a mathematics placement test used at a major university is summarized on page 116: “On the algebra subtest, the means of the [conventional] precalculus students and the CPMP Course 4 students were virtually identical. On the intermediate algebra subtest, the mean of the precalculus group was greater than that of the Course 4 group. The only statistically significant difference in means was on the calculus readiness subtest (t = 4.93, p < 0.01). That difference favored the CPMP students.” Responding to Calls for Change in High School Mathematics: Implications for Collegiate Mathematics Harold L. Schoen and Christian R. Hirsch

Responding to Calls for Change in High School Mathematics: Implications for Collegiate Mathematics

Responding to Calls for Change in High School Mathematics: Implications for Collegiate Mathematics

Mathematical Literacy and Abstraction in the 21st Century

Ed Dubinsky received his Ph.D. in mathematics from the University of Michigan in 1962 and for the next 25 years engaged in research in theoretical mathematics. In the mid‐80′s he became interested in mathematics education and has worked exclusively in the area since then. In his research, he tries to understand how a person's mind might be working when he or she tries to understand mathematical concepts at the postsecondary level. Based on this research, he has conducted large‐scale curriculum development projects in calculus, discrete mathematics, abstract algebra and cooperative learning. He has been editor or co‐editor of UME Trends, Research in Collegiate Mathematics Education, and the Journal of Computers in Mathematics and Science Teaching. He has held faculty positions at 8 universities in 5 countries on 3 continents: Fourah Bey College (Sierra Leone), University of Ghana, Tulane University, McMaster University, Polish Academy of Sciences, Clarkson University, Purdue University, and Georgia State University. Dr. Dubinsky is presently retired and consults with several universities on education matters.

Research in Collegiate Mathematics Education II

Research in Collegiate Mathematics Education II

Locating starting points in differential equations: a realistic mathematics education approach

The paper reports on ongoing developmental research efforts to adapt the instructional design perspective of Realistic Mathematics Education (RME) to the learning and teaching of collegiate mathematics, using differential equations as a specific case. This report focuses on the RME design heuristic of guided reinvention as a means to locate a starting point for an instructional sequence for first-order differential equations and highlights the cyclical process instructional design and analysis of student learning. The instance of starting with a rate of change equation as an experientially real mathematical context is taken as a case for illustrating how university students might experience the creation of mathematical ideas. In particular, it is shown how three students came to reason conceptually about rate and in the process, develop their own informal Euler method for approximating solution functions to differential equations.

Instructional qualities of a successful mathematical problem‐solving class*

* A short version of this article will appear in a special issue of Research in Collegiate Mathematics Education. American Mathematical Society. Problem solving has been an important issue in mathematics education during the last 25 years. Research studies in this area have documented extensively what students show while working with mathematical problems (resources, beliefs, and cognitive and metacognitive strategies). However, there is little information about how mathematical problem‐solving activities should be implemented in regular classrooms. This paper shows what activities have been successful implemented by an expert during a mathematical problem‐solving course. It focuses on the identification of qualities of the problems used to promote the development of students strategies and values that reflect mathematical practice in the classroom. Specia’ attention is paid to the type of question that help students evaluate their own problem‐solving processes and the importance of allowing the students to work in small group activities. Although the course itself does not address a particular sequence of content (algebra, geometry or calculus), there is indication that it could be used as a model to be adjusted to a regular mathematical course.

Lecturing at the "Bored"

Even a superficial review of the literature on collegiate mathematics education reform reveals a common thread: the imperative nature of the college mathematics student's active participation in the learning of mathematics. Active participation in the classroom may take the form of working cooperatively with classmates in small groups, spending time in the computer lab using class-related software, presenting problems or concepts on the board to classmates, or other activities germane to the subject bestdes passively listening to a lecture presented by the instructor. There is a wide range of lectures styles, but the stle to which I refer is one characterized by little expectation of student engagement besides perhaps following the logic of the lesson. Because active learning in the mathematics classroom is advocated by several professional groups ([1], [4], [5], [6], [8]), it is certainly reasonable for mathematics instructors to reflect on their rationale for maintaining a lecture style of teaching, even part of the time. During the 1995-1996 academic year, that is exactly what I did. I was teaching a section of reformed calculus, with a class size of forty. I had planned on extensive use of cooperative groupwork, and had accordingly divided the class into ten groups of four people each. I attempted to make the groups heterogeneous (in gender and ability) by collecting quantitative measures of the students' previous mathematical performance. Several times during the first two sveeks of class, I set aside time for group work. However, due to the physical layout of the classroom (five rows of auditorium seating with twelve seats per row and an aisle up the middle) my students were having a difficult time actually communicating in their groups. Classtime initially devoted to guided discovery in the groups degenerated into work time for forty individuals, certainly not the vision I had foreseen when planning earlier that summer. To restructure this time, I tried to view my entire class as a large cooperative group, with me as the group leader. Instead of breaking the class into four-person groups as before, I would present examples of concepts we were currently studying, and pose open-ended questions to the class. For example, while developing a geometrical approach to Newton's method for finding the roots of an equation, I asked my students which solution they thought the graphical process would yield when the initial guess for the x-value was directly between two of the roots. Was there a pattern that allowed us to predict the general result? Another time wheh discussing inverse functions, we did an impromptu in-class exploration of which Itnear functions, and later, which general functions, are their own inverses. Such discussions involved much conjecturing, arglling, and scratchpaper seatwork, individually and in pairs, among the students, and were opportunities for algebraic, geometric, and numerical exploration. l Because of my class's general enthusiasm and willingness to participate, If felt we had developed a classroom atmosphere free of criticism, and open to conjecturing. The students often presented problems for each other, furthering their own mathematical communication. Although I was somewhat disappointed not to be able to watch my students develop cooperative skills, I felt that we had made the

How to Teach Mathematics: A Personal Perspective

`[A] professor will sometimes prepare for a lecture not by writing some notes or browsing through the book but by lounging in the coffee room with his colleagues and bemoaning (a) the shortcomings of the students, (b) the shortcomings of the text, and (c) that professors are overqualified to teach calculus. Fortified by this yoga, the professor will then proceed to his class and give a lecture ranging from dreary to arrogant to boring to calamitous...The good news is that it requires no more effort, no more preparation, and no more time to be a good teacher than to be a bad teacher. The proof is in this booklet.' ---from the Preface Lively and humorous, yet serious and sensible, this book is a practical guide to the teaching of mathematics. Eschewing generalities, Krantz emphasizes specifics---from how to deal with students who beg for extra points on an exam to mastering blackboard technique to how to use applications effectively. In addition, the book also deals with such sensitive subjects as cheating, bribery, and sexual harassment. Those teaching collegiate mathematics for the first time will find Krantz's advice especially helpful, and more experienced instructors will appreciate the book's elucidation of the fine points of excellent teaching. This book is intended for graduate students preparing for a career in college teaching of mathematics, mathematics instructors and professors.

Review: A New Era in Collegiate Mathematics Education Research: Good Learning of College Mathematics: Can We Recognize, Understand, Foster, and Document It?

Research in college mathematics education is neither new (Suydam, 1975) nor unusual (Becker &amp; Pence, 1994). However, the literature of this field is still very scattered, with many of the articles published in journals that mathematicians do not usually read. Although in recent years a few books have appeared that deal with college mathematics, books that collect papers devoted to research in the field are certainly needed.

REFLECTIONS ON THE USE OF TECHNOLOGY IN THE MATHEMATICS CLASSROOM

ABSTRACT The use of technology in collegiate mathematics classrooms is increasing. The most effective means of using technology, however, is an open question. During the summer of 1993, the Department of Mathematical Sciences at Montana State University offered an experimental course in which the programming language ISETL served as a tool for exploring concepts in abstract algebra. This paper describes these efforts and offers informal observations regarding the effectiveness of this approach. In addition, the paper identifies instructional strategies that (in the opinion of the author) facilitate the successful use of ISETL in abstract algebra. These strategies are offered as a preliminary model of technology use in the mathematics classroom.

Review: A New Era in Collegiate Mathematics Education Research: A Mathematician's View of RCME

For the mathematician interested in issues of learning and teaching, these are, ostensibly, the best of times. The American Mathematical Society has joined the Mathematical Association of America in making choice time slots available at national and regional meetings for discussions of educational issues. These sessions have been well attended, not only by young Ph.D.'s, but also by senior mathematicians, full professors with the protection of tenure, who have the potential to substantially modify the way in which college mathematics teaching is done.

Collegiate Mathematics Education Research: What Would That Be Like?

Collegiate Mathematics Education Research: What Would That Be Like?

Collegiate Mathematics Education Research: What Would That Be Like?

(1993). Collegiate Mathematics Education Research: What Would That Be Like? The College Mathematics Journal: Vol. 24, No. 5, pp. 431-445.

Reviews: Words to Solve by

Unlike most books reviewed in the journal for Research in Mathematics Education, Problem Solving in the Mathematics Curriculum (PSMC) does not report research. Instead, it seems designed to (a) recommend that problem solving be consistently included in collegiate mathematics instruction, (b) describe some considerations in and ways of teaching problem solving, (c) present an extensive bibliography chosen to help those initiating or teaching problem-solving courses or problem-solving sequences within courses, and (d) give the results of a survey conducted by the Committee on the Teaching of Mathematics of the Mathematical Association of America; the survey provided the impetus for PSMC. Accordingly, the book is divided into four parts. The short first part describes the evolution of PSMC and the recommendations of the Committee on the Teaching of Mathematics. The second part, a more-or-less personal essay by Alan Schoenfeld, gives suggestions for teaching problem solving. The third and most extensive part is an annotated bibliography of journals, books, and articles that might be used to develop in struction in problem solving or to find appropriate problems for such instruction. The last part presents both the survey instrument and the results of the survey.

Alternatives to Research

A Formidable challenge to excellence is posed by two trends in collegiate mathematics education: one is the admirable flow of numerous students into fields requiring mathematics, with a consequent demand for good college teachers; the other is the lamentable tendency for undergraduate mathematics faculties to be understaffed and overworked. The two trends together generate a challenge which the CUPM Panel on College Teacher Preparation now confronts: How can undergraduate faculty members achieve and maintain high quality in mathematics teaching?