Mathematical Association Of America Research Articles

Seven Characteristics of Successful Calculus Programs

In these days of tight budgets and pressure to improve retention rates for science and engineering majors, many mathematics departments want to know what works, what are the most productive means of improving the effectiveness of calculus instruction. This was the impetus behind the study of Characteristics of Successful Programs in College Calculus undertaken by the Mathematical Association of America. The study consisted of a national survey in fall 2010, followed by case study visits to seventeen institutions that were identified as “successful” because of their success in retention and the maintenance of “productive disposition,” defined in [NRC 2001] as “habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.” Our survey revealed that Calculus I, as taught in our colleges and universities, is extremely efficient at lowering student confidence, enjoyment of mathematics, and desire to continue in a field that requires further mathematics. The institutions we selected bucked this trend. This report draws on our experiences at all seventeen colleges and universities but focuses on the insights drawn from those universities that offer a PhD in mathematics, the universities that both produce the largest numbers of science and engineering majors and that often struggle with how to balance the maintenance of high-quality research with attention to undergraduate education. Case studies were conducted in the fall of 2012 at five of these universities: two large public research universities, one large private research university, one public technical university, and one private technical institute. We shall refer to these as: PrTI: Private Technical Intitute. Private university. Data from nine sections of calculus with an average enrollment of 33.

The MAA undergraduate poster session 1991–2013

We provide a historical report of the undergraduate poster session at the annual joint meetings of the American Mathematical Society and the Mathematical Association of America from its inception in 1991 until 2013. We also provide data on the number of undergraduates attending the joint meetings and the number giving talks. From a humble beginning in 1991 with 12 posters involving fewer than 20 students, no judges, and no prizes, the Mathematical Association of America (MAA) undergraduate poster session at the Joint Mathematics Meetings (JMM) has grown to 304 posters representing the work of 487 students, 234 judges, and 50 prizes in 2013. The idea for the poster session came from the newly formed MAA subcommittee of the Committee on the Undergraduate Program in Mathematics (CUPM) on research by undergraduates chaired by Lester Senechal. Because of the small number of participants in 1991, there were no follow-up poster sessions in 1992 or 1993. John Greever organized the 1994 and 1995 sessions, followed by Judy Palagallo in 1996, Mario Martelli in 1997, and Palagallo again in 1998. Participation doubled in 1999 when Martelli and Aparna Higgins were organizers and doubled again in 2000 with Martelli as organizer. From 1999 until 2005 Higgins was host of the award ceremony.

From Visualization to Computer Animation Approaches in Mathematics Learning: the Legacy throughout History of Human Endeavours for Better Understanding

Presently, there has been growing interests in using mathematics' history in teaching mathematics [Katz, V. & Tzanakis, C. (Eds.) (2011). Recent Developments on Introducing a Historical Dimension in Mathematics Education. Washington, DC: Mathematical Association of America]. Thus, this article introduces some work of scholars from ancient East Indian culture like Bhaskara (AD 1114-1185) and Arabic culture such as Ibn Qurrah (AD 9th c) that are related to Pythagoras Theorem. In addition, some Babylonian creative works related to Pythagorean triples found in a tablet known as 'Plimpton 322', and an application of the Pythagorean Theorem found in another tablet named 'Yale Tablet' are presented. Applications of computer animation of dissection Motion Operations concept in 2D and 3D using dynamic software like Geometer's-Sketchpad and Cabri-II-and-3D. Nowadays, creative minds are attracted by the recent stampede in the advances of technological applications in visual literacy; consequently, innovative environments that would help young students, gifted or not, acquiring meaningful conceptual understanding would immerge.

The History of the Undergraduate Program in Mathematics in the United States

This article describes the history of the mathematics major and, more generally, collegiate mathematics, in the United States. Interestingly, the Mathematical Association of America was organized about 100 years ago, around the same time that academic majors came into existence.

On the mean value of dedekind sum weighted by the quadratic gauss sum

Various properties of classical Dedekind sums S(h, q) have been investigated by many authors. For example, Wenpeng Zhang, On the mean values of Dedekind sums, J. Théor. Nombres Bordx, 8 (1996), 429–442, studied the asymptotic behavior of the mean value of Dedekind sums, and H.Rademacher and E.Grosswald, Dedekind Sums, The Carus Mathematical Monographs No. 16, The Mathematical Association of America, Washington, D.C., 1972, studied the related properties. In this paper, we use the algebraic method to study the computational problem of one kind of mean value involving the classical Dedekind sum and the quadratic Gauss sum, and give several exact computational formulae for it.

Senior Capstone Seminar: A Comprehensive Learning Experience

Over the last four years of the senior capstone seminar at Western Carolina University, we have redesigned the course substantially to comply with our institutional Quality Enhancement Plan for engaged student learning and to follow the guidelines proposed by the Mathematical Association of America's Committee on Undergraduate Programs in Mathematics. The capstone course has been structured around critical student objectives representing recommendations from both initiatives. The redesigned course balances between the dual roles that it is asked to play while preserving a comprehensive and holistic mathematical learning experience.

Beyond Swinging: Hinged Dissections that Twist or Fold

A geometric dissection is a cutting of a geometric figure into pieces that can be rearranged to form another figure. Some dissections can be connected with hinges so that the pieces form one figure when swung one way on the hinges, and form the other figure when swung another way. In addition to using “swing hinges”, which allow rotation in the plane, we can use “twist hinges”, which allow one piece to be flipped over relative to another piece via rotation by 180° through a third dimension. Furthermore, we can use “fold hinges”, which allow rotation along a shared edge, a motion that is akin to folding. This talk will introduce a variety of twist-hinged and fold-hinged dissections of regular polygons and stars, and other figures such as polyominoes. The emphasis will be on both appreciating and understanding these fascinating mathematical recreations. I will employ algorithmic and tessellation-based techniques, as well as symmetry and other geometric properties, to design the dissections. The goal will be to minimize the number of pieces, subject to the dissection being suitably hinged. Animations and video will be used to demonstrate the hinged dissections, in addition to actual physical models. Bio: Greg N. Frederickson is a Professor of Computer Science at Purdue University, in West Lafayette, Indiana. His primary area of research is the design and analysis of algorithms, and he has served on the editorial boards of SIAM Journal on Computing, SIAM Journal on Discrete Mathematics, Algorithmica, and IEEE Transactions on Computers. He also pursues interests in mathematical recreations, specifically geometric dissection. On this topic he has published three books and a number of articles. He has twice won the George Polya Award from the Mathematical Association of America.

Does gender bias influence awards given by societies?

AGU is a participant in a U.S. National Science Foundation (NSF)–funded project called Advancing Ways of Awarding Recognition in Disciplinary Societies (AWARDS), which seeks to examine whether gender bias affects selection of recipients of society awards. AGU is interested in learning why there is a higher proportion of female recipients of service and education awards over the past 2 decades. Combined with a lower rate of receipt of research awards, these results suggest that implicit (subconscious) bias in favor of male candidates still influences awardee selection. Six other professional societies (American Chemical Society, American Mathematical Society, American Society of Anesthesiologists, Mathematical Association of America, Society for Neuroscience, and Society for Industrial and Applied Mathematics) are participating in the project. Volunteers from each participant society attended an Association for Women in Science (AWIS)–sponsored workshop in May 2010 to examine data and review literature on best practices for fair selection of society awardees. A draft proposal for implementing these practices will be brought before the AGU Council and the Honors and Recognition Committee at their upcoming meetings.

Achieving Recognition Equity

PROFESSIONAL AWARDS continue to be important contributors to career advancement for scientists, especially those in academic institutions. In 2009, the Association for Women in Science (AWIS) received a three-year grant from the National Science Foundation for a new project entitled Advancing Ways of Awarding Recognition in Disciplinary Societies (AWARDS). AWARDS is designed to create a sustainable framework for ensuring progress toward more equitable awards and recognition for women and members of underrepresented groups in a wide range of scientific communities. AWIS recruited a team of disciplinary societies as partners in this project, including the RAISE Project (Recognition of the Achievements of Women in Science, Medicine & Engineering), the American Geophysical Union, the American Mathematical Society, the American Statistical Association, the Mathematical Association of America, the Society for Neuroscience, the Society for Industrial & Applied Mathematics, and the American Chemical Society. Thes...

A Short Proof of Combinatorial Nullstellensatz

In this note we give a short, direct proof of the combinatorial Nullstellensatz.

Is mathematics inevitable? a miscellany by Underwood Dudley (editor), The Mathematical Association of America, 2008 324 pages, hardcover

No abstract available.

A computer laboratory program in engineering mathematics to enhance mathematical conceptualisation

This article describes a laboratory supplementary program that was integrated into a traditional mining engineering mathematics unit. The practical classes consisted of computer investigations designed to help develop mathematical concepts. The program described here was mainly web based and did not directly rely on a computer algebra system for its implementation. An evaluation of the program is included. References S. Cunningham. The visualization environment for mathematics education. In Visualization in Teaching and Learning Mathematics , ed. W. Zimmermann and S. Cunningham, 67--76. USA, Mathematical Association of America, 1991. A. Franco, P. Franco, A. Garcia, F. Garcia, F. J.Gonzalez, S. Hoya, G. Rodriguez, and A. de la Villa. Learning calculus of several variables with new technologies. The International Journal of Computer Algebra in Mathematics Education , 7 (4), 295--309, 2000. B. E. Garner and L. E. Garner. Retention of concepts and skills in traditional and reformed applied calculus. Mathematics Education Research Journal , 13 (3), 165--184, 2001 S. Habre. Visualization enhanced by technology in the learning of multivariate calculus. The International Journal of Computer Algebra in Mathematics Education , 8 (2), 115--130, 2001. B. H. Hallet. Visualization and calculus reform. In Visualization in Teaching and learning Mathematics , ed. W. Zimmermann and S. Cunningham, 121--126, 1991. USA, Mathematical Association of America F. Marton and R. Saljo. Approaches to learning. In eds. F. Marton, D. Hounsell and N. Entwistle, The Experience of Learning , 36--55, 1984. Scottish Academic Press, Edinburgh. R. Moreno and R. Mayer. Verbal redundancy in multimedia learning; When reading helps listening. Journal of Educational Psychology , 94 (1), 153--163, 2002. L. D. Murphy. Computer algebra systems in calculus reform, MSTE, University of Illinois at Urbana-Champaign, 1999. http://mste.illinois.edu/users/Murphy/Papers/CalcReformPaper.html M. Pemberton. Integrating web-based maple with a first year calculus and linear algebra course. Proceedings of the 2nd International Conference on the Teaching of Mathematics , Hersonissos, Greece, July 2002. http://www.math.uoc.gr/ ictm2/Proceedings/pap316.pdf R. Pierce and K. Stacey. Observations on students' responses to learning in a cas environment. Mathematics Education Research Journal , 13 (1), 28--46, 2001. M. D. Roblyer. Integrating Educational Technology Into Teaching (4th Ed.), 2006. Pearson, New Jersey, USA. J. Stewart. Calculus (5th Ed.), 2003. Brooks/Cole, Belmont, USA. E. J. Tonkes, B. I. Loch and A. W. Stace. An innovative learning model for computation in first year mathematics. International Journal of Mathematical Education in Science and Technology , 36 (7), 751--759, 2005. L. M. Villarreal. A step in the positive direction: Integrating a computer laboratory component into developmental algebra courses. Mathematics and Computer Education , 37 (1), 72--78, 2003. S. Vinner. The pseudo-conceptual and the pseudo-analytical thought processes in mathematics learning. Educational Studies in Mathematics , 34 (2), 97--129, 1997. P. Vlachos and A.K. Kehagias. A computer algebra system and a new approach for teaching business calculus. The International Journal of Computer Algebra in Mathematics Education . 7 (2), 87--104, 2000.

Yueh-Gin Gung and Dr. Charles Y. Hu Award for 2010 to Kenneth A. Ross for Distinguished Service to Mathematics

Ken Ross has made many contributions to mathematics over a long career. From 1971 to 1980 he served the American Mathematical Society (AMS) as Associate Secretary for the Western Section, and in 1983 he was elected Secretary of the Mathematical Association of America (MAA). When the responsibilities of the MAA Secretary were split in 1989, he became the MAA’s first Associate Secretary and continued to be responsible for the MAA presence in the national joint AMS-MAA meetings. He had long been involved with meetings; during his appointment at the AMS he was in charge of AMS regional meetings as well as any national meetings held in a Western city. In these assignments for the AMS and the MAA he traveled extensively to check out possible meeting sites. At the time of the meetings Ken was everywhere at once to ensure that everything was going smoothly. Of course, as MAA Secretary he was also responsible for keeping track of all the committees, preparing materials for Executive Committee and Board of Governors meetings, and generally keeping the business of the Association on track. In these various roles he showed great talent for working well with people, quietly making sure that everything got done. And, at least on the outside, he always appeared calm and in control. In 1993 Ken was elected MAA President and successfully led the MAA during a particularly challenging time due to financial problems. Following his term as President, he went on to chair the Response Group on the NCTM Standards, a potentially

How to introduce historically the normal distribution in engineering education: a classroom experiment

Little has been explored with regard to introducing historical aspects in the undergraduate statistics classroom in engineering studies. This article focuses on the design, implementation and assessment of a specific activity concerning the introduction of the normal probability curve and related aspects from a historical dimension. Following a teaching–learning approach inspired by history, this idea was put into practice at an engineering school. For this purpose a historical module on the normal curve elaborated by Katz and Michalowicz [V.J. Katz and K.D. Michalowicz (eds.), Historical Modules for the Teaching and Learning of Mathematics, The Mathematical Association of America, Washington, 2005] was adapted to develop different aspects of the topic. Since the activity fostered cross-curricular competences, it was designed in keeping with the guidelines of the European Higher Education Area.

The Sixty-Ninth William Lowell Putnam Mathematical Competition

The results of the Sixty-Ninth William Lowell Putnam Mathematical Competition, held December 6, 2008, follow. They have been determined in accordance with the regulations governing the Competition. The contest is supported by the William Lowell Putnam Prize Fund for the Promotion of Scholarship, an endowment established by Mrs. Putnam in memory of her husband. The annual Competition is held under the auspices of the Mathematical Association of America. The first prize, $25,000, was awarded to the Department of Mathematics of Harvard University. The members of the winning team were Zachary R. Abel, Iurie Boreico, and Arnav Tripathy; each was awarded a prize of $1,000. The second prize, $20,000, was awarded to the Department of Mathematics of Princeton University. The members of the winning team were Peter Z. Diao, John V. Pardon, and Adrian I. Zahariuc; each was awarded a prize of $800. The third prize, $15,000, was awarded to the Department of Mathematics of the Massachusetts Institute of Technology. The members of the winning team were Qingchun Ren, Xuancheng Shao, and Yufei Zhao; each was awarded a prize of $600. The fourth prize, $10,000, was awarded to the Department of Mathematics of Stanford University. The members of the winning team were Young Hun Jung, Nathan K. Pflueger, and Jeffrey Wang; each was awarded a prize of $400. The fifth prize, $5,000, was awarded to the Department of Mathematics of the California Institute of Technology. The members of the winning team were Jason C. Bland, Zarathustra E. Brady, and Brian R. Lawrence; each was awarded a prize of $200. The five highest ranking individual contestants, the Putnam Fellows, in alphabetical order, were Brian R. Lawrence, California Institute of Technology; Seok Hyeong Lee, Stanford University; Arnav Tripathy, Harvard University; Bohua Zhan, Massachusetts Institute of Technology; and Yufei Zhao, Massachusetts Institute of Technology. Each received an award of $2,500. The next eleven highest ranking individual contestants, in alphabetical order, were Iurie Boreico, Harvard University; Adam C. Hesterberg, Princeton University; William A. Johnson, University of Washington; Cedric Lin, University of British Columbia; Anton S. Malyshev, Princeton University; John V. Pardon, Princeton University; Qingchun Ren, Massachusetts Institute of Technology; Oleg O. Rudenko, Florida Atlantic University; Colin P. Sandon, Massachusetts Institute of Technology; Jacob N. Steinhardt, Massachusetts Institute of Technology; and Alex Zhai, Harvard University. Each received an award of $1,000. The next nine highest ranking individual contestants, in alphabetical order, were Zachary R. Abel, Harvard University; Thomas D. Belulovich, Massachusetts Institute of Technology; Jason C. Bland, California Institute of Technology; Gabriel T. Bujokas, Massachusetts Institute of Technology; Konstantin Matveev, University of Toronto; Nathan K. Pflueger, Stanford University; Aaron H. Potechin, Princeton University; Dong Uk Rhee, University of Waterloo; and Adrian I. Zahariuc, Princeton University. Each received an award of $250.

Review ofCalculation vs. Context: Quantitative Literacy and Its Implications for Teacher Educationby Bernard L. Madison and Lynn Arthur Steen (Editors)

Madison, Bernard L. and Steen, Lynn Arthur (Eds.). Calculation vs. Context: Quantitative Literacy and Its Implications for Teacher Education. (Washington, DC: Mathematical Association of America, 2009). 197 pp. Softcover. ISBN 978-0-88385-908-7. Available free on the MAA website at http://www.maa.org/ql/calcvscontext.html The papers in Calculation vs. Context discuss the role of quantitative literacy in the K-12 curriculum and in teacher education. The papers present a varied set of perspectives and address three themes: the changing environment of education in American society; the challenges, and the necessity, of preparing teachers to teach quantitative literacy and of including quantitative literacy in the K-12 education; and cross-disciplinary approaches to quantitative literacy. While the conclusion reached by several of the authors is that the best place to teach quantitative literacy is at the college level, the book offers serious considerations of how quantitative literacy can and should inform the K-12 curriculum. The book also marks a turning point in the quantitative literacy movement as “QL explorers,” as Lynn Steen calls them, move beyond issues of definitions and content to a discussion of how to bring quantitative literacy into a broader setting.

Approximations to π Derived from Integrals with Nonnegative Integrands

Paper 18: Stephen K. Lucas, “Approximations to π derived from integrals with nonnegative integrands,” American Mathematical Monthly, vol. 116 (2009), p. 166–172. Copyright 2009 Mathematical Association of America. All Rights Reserved.

What Mathematics Should Adults Learn? Adult Mathematics Instruction as a Corollary to Two Decades of School Mathematics Reform

Twenty-five years ago, the United States Department of Education published A Nation at Risk: The Imperative for Educational Reform, sounding alarm that the United States educational foundations were in grave danger, eroded by a rising tide of mediocrity that threatens our very future as a Nation and a people (National Commission on Excellence in Education, 1983, p. 5). The National Research Council (1989, 1990, 1993) followed with a trilogy of texts specifically addressing the condition of United States mathematics education. In 1989, the National Council of Teachers of Mathematics (NCTM) released their groundbreaking Curriculum and Evaluation Standards for School Mathematics, followed by two additional publications promoting standards for the profession and for assessment (1991, 1995). The three original Standards volumes were revised and consolidated in 2000, and in 2006, NCTM released Curriculum Focal Points, a handbook concerning the grade at which specific mathematics content should be taught from pre-school through 8th grade. In 2008, the National Mathematics Advisory Panel released its report with recommendations concerning K-12 mathematics education in the United States. In the intervening years since 1989, other mathematics professional organizations have addressed the issue of standards for post-secondary institutions. The American Mathematical Association of Two-Year Colleges (AMATYC) published Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus and, recently, Beyond Crossroads: Implementing Mathematics Standards in the First Two Years of College. The Mathematical Association of America (MAA) released Undergraduate Programs in the Mathematical Sciences: CUPM Curriculum Guide 2004, making recommendations for service and mathematics major courses. This year, the MAA released the latest entry in its sequence, Calculation vs. Context: Quantitative Literacy and Its Implications for Teacher Education, which explores the disconnect between the education of prospective teachers and the practical applications of mathematics in adult life. Those practical applications, called interchangeably numeracy and quantitative literacy in the reports to be examined, have been explored within and outside the mathematics communities. With the exception of the Equipped for the Future project, adult mathematics education experts were noticeably absent from the preparation of the standards documents summarized here. A corollary is, by definition, an immediate consequence or easily drawn conclusion, a natural consequence or result (Stein, 1975, p. 300). This paper terms our work as adult educators a corollary to the reforms of the past 20 years because it is primarily subsequent and reactive to the principles proposed by the school and academic establishment. While there is substantial overlap, the recent shift by the National Math Panel towards a traditional view of mathematics education represents a divergence from many of the values we hold for adult mathematics education that challenges the corollary analogy and gives cause to suggest a new and different theorem for our students. School Mathematics The National Council of Teachers of Mathematics (NCTM) is the largest and most powerful player on the school mathematics field. In their Principles and Standards for School Mathematics, they describe a curriculum as than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades (NCTM, 2000, p. 14). This definition of curriculum was global, and, over time, it became evident that the community could benefit from a more directive NCTM statement suggesting a judicious sequence of topics within the curriculum framework. Furthermore, a mapping of topics to grade levels would provide a consistent timeline for designers, authors, and publishers of curricula. The resulting product, Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics, provides descriptions of the most significant mathematical concepts and skills at each grade level (NCTM, 2006, p. …

University of Wisconsin-Milwaukee mathematics focus courses: mathematics content for elementary and middle grades teachers

There has been much debate in recent years as to the amount and type of mathematical knowledge that teachers need to acquire. One set of recommendations is provided by the Mathematical Education of Teachers (MET) report, produced jointly by the American Mathematical Society and the Mathematical Association of America. This paper reports on efforts at the University of Wisconsin-Milwaukee to implement the MET report recommendations for pre-service elementary and middle grades teachers, in the contexts of teacher education programmes and the teacher licensing structure of the state of Wisconsin. Data is also presented on the results of teaching the resulting courses to in-service teachers.

New Einsteins need positive environment, independent spirit

The article “Why No ‘New Einstein’?” and the ensuing letters perhaps raise a similar question in fields other than physics. In the mathematical arena, one could ask “Why no new Euler?” Of course, Leonhard Euler’s name could be replaced by the names of several other great mathematicians, but an argument could be made that Euler shared with Einstein an amazing intuition that, it seems, is a trait of a select few. I believe William Dunham’s wonderful book gives insight to Euler’s intuition. 1 1. W. Dunham, Euler: The Master of Us All, Mathematical Association of America, Washington, DC (1999). It may be the common opinion among modern mathematicians that many of Euler’s methods would not stand up to current mathematical rigor. And, as an engineer, I dare not take issue with that. But it seems one reason why no new Euler has arisen is that for scientists and engineers, at least, the flame of intuition too often is extinguished in the very first university mathematics class they take. Certainly mathematical rigor has its place. But an intuitive line of thought that leads to a correct mathematical result ought not to be discouraged, beyond a possible admonition about where such thinking could lead one astray. In fact, intuitive thinking ought to be celebrated, as long as we non-mathematicians do not make any claims to rigor or demand that mathematicians strictly agree with us.A new Euler would not necessarily emerge from the non-mathematician class, although that possibility cannot be ruled out either. Paul Dirac, Richard Feynman’s hero, comes to mind immediately as one who resembled Euler in the way he did some of his mathematics. His book on quantum mechanics shows how he masterfully created a new mathematical formulation in order to do his physics. 2 2. P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford U. Press, New York (2000). The mathematicians were left the task of showing that his results could also be proven rigorously. After all, who will argue with one whose non-rigorous mathematics leads to the discovery of a new particle?REFERENCESSection:ChooseTop of pageREFERENCES <<1. W. Dunham, Euler: The Master of Us All, Mathematical Association of America, Washington, DC (1999). Google ScholarCrossref2. P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford U. Press, New York (2000). Google Scholar© 2006 American Institute of Physics.