AMATYC Review

Editor: Barbara S. Rives, Lamar State College
Production Manager: John C. Peterson

Table of Contents

• From the Editor's Keyboard

  Greetings! It hardly seems possible another semester is underway and the fall AMATYC conference in Washington, DC will be here. Time passes quickly and my five-year commitment as editor ends at the conclusion of the fall 2008 conference in Washington, DC. This column is the last "From the President's Keyboard" column written by your current editor. The five years have been interesting, challenging, enlightening, and rewarding. Each issue of The AMATYC Review involved the contributions of many people and a special "thank you" goes to each of the following:

• Production Manager: John Peterson, whose experience and expertise prepared the journal for publication by typesetting articles, designing many of the journal covers, making changes to the proofs, and completing a multitude of other items needed to finalize the journal for publication.
• AMATYC Board liaisons: Wanda Long, Irene Doo, and Jane Tanner who provided guidance, proofreading, and help whenever needed.
• The AMATYC office staff: Cheryl Cleaves, Beverly Vance, and Christine Shott who answered many questions, provided guidance, and resources.
• Authors: Without your contributions, The AMATYC Review would not have been possible. Thank for your patience during the review process.
• Reviewers and Editorial panelists: See pages ?? and ?? for the names of these contributors to The AMATYC Review. They contributed their time and expertise to reviewing articles and making recommendations to the editor.
• Feature editors: Brian Smith (software review editor), Sandra DeLozier Coleman (book review editor), and Stephen Plett (problems section editor). A special thanks to the late Robert Stong who died in April (see page ??). He served as the solutions editor for the problems section of The AMATYC Review for 20 years.
• University and college support: Abilene Christian University, Abilene, TX and Lamar State College-Orange, Orange, TX for their administrative support of the editor by providing space for the AMATYC editorial materials, released time for AMATYC work, and general encouragement and office materials.
• AMATYC Presidents: Judy Ackerman, Kathy Mowers, and Rikki Blair for their guidance, support, and vision for AMATYC.

• One of the most frustrating aspects of being editor was not being able to publish all the excellent articles that could have been published - if space had been available. These manuscripts were forwarded to the AMATYC office in Memphis in anticipation of the selection of the new editor and production manager.

  Best wishes to the new editor and production manager as they take responsibility for developing and producing the journal. A wonderful adventure awaits them. I look forward to receiving the future publications.

  

  Barbara S. Rives, Editor
  E-mail: ReviewEditor@amatyc.org

• Areas and Volumes in Pre-Calculus
  Joscelyn A. Jarrett

  Joscelyn A. Jarrett is a professor of mathematics at Gordon College in Barnesville, GA. He received his BA (Hons) in mathematics from Fourah Bay College in 1967, an MS in mathematics from the University of Toronto in 1970 and a PhD in secondary mathematics education from the University of Iowa in 1980. Joscelyn is an active member of both the National Council of Teachers of Mathematics and the American Mathematical Association of Two-Year Colleges. E-mail: j_jarrett@gdn.edu

  This article suggests the introduction of the concepts of areas bounded by plane curves and the volumes of solids of revolution in Pre-calculus. It builds on the basic knowledge that students bring to a pre-calculus class, derives a few more formulas, and gives examples of some problems on plane areas and the volumes of solids of revolution that could be solved at the pre-calculus level. More students will benefit from the exposure to these concepts, as not all pre-calculus students go on to take calculus. Furthermore, when students do get to calculus, they would have already acquired some skills in visualizing mental images or drawing sketches of solids of revolution.

• In Memoriam: Robert Stong (1936-2008)

• The Mathematics of Starry Nights
  Farshad Barman

  Farshad Barman received his PhD in electrical engineering from the University of California in Santa Barbara in 1979. He taught and worked in that eld until 1992. He received his master's degree in mathematics from Portland State University in 1995 and has been teaching mathematics at Portland Community College since then. His current interests are the mathematics of astronomy, stargazing, and baseball. E-mail: fbarman@pcc.edu

  The mathematics for finding and plotting the locations of stars and constellations are available in many books on astronomy, but the steps involve mystifying and fragmented equations, calculations, and terminology. This paper will introduce an entirely new unified and cohesive technique that is easy to understand by mathematicians, and simple enough to fit on one line, and easy to program into a graphing calculator. The result will be a 2 xn matrix of star coordinates that will model the positions of naked-eye visible stars and constellations for a given date and time and location of the observer. This technique is based on coordinate transformations in  and mapping from  to . The precession of the equinoxes will be explained and included in the calculations, and will therefore make the star plots accurate for approximately two thousand years into the future or the past. This paper provides examples for the application of linear transformations and mappings, using one of the most natural physical phenomena, and is written for readers with limited knowledge of astronomy.

• Lucky Larry #89

• The Principal Square Root of Complex Numbers, Terence Brenner

  It is stated in any algebra book that the principal square root of a positive number is . In this article, the definition of the principal square root is expanded to include complex numbers.

• Lucky Larry #90

• On the Presentation of Pre-Calculus and Calculus Topics: An Alternate View
  Aleksandr Davydov and Rachel Sturm-Beiss

  Aleksandr Davydov is an assistant professor of mathematics at the Kingsborough Community College (KCC) of the City University of New York (CUNY). He earned his MS in mathematics from Samarkand State University (Russia) and his PhD in mathematics from Ural State University (Russia). His primary area of interest is differential equations and their applications. E-mail: ADavydov@kbcc.cuny.edu Rachel Sturm-Beiss is an associate professor of mathematics in Kingsborough Community College (KCC) of the City University of New York (CUNY). She earned her PhD in pure mathematics from the Courant Institute of New York University. Her primary area of interest is statistical processes and modeling. E-mail: RSturm@kingsborough.edu

  The orders of presentation of pre-calculus and calculus topics, and the notation used, deserve careful study as they affect clarity and ultimately students' level of understanding. We introduce an alternate approach to some of the topics included in this sequence. The suggested alternative is based on years of teaching in colleges within and outside the US, and on our careful review of textbooks currently used in two-year and four-year colleges.

• How to Design Your Own  to e Converter, Harlan J. Brothers

  Harlan Brothers is Director of Technology at The Country School in Madison, CT where he teaches programming, fractal geometry, and guitar. Having worked for six years with Michael Frame and Benoit Mandelbrot at the Yale Fractal Geometry Workshops, he now lectures on the subject of fractal music. Harlan is also an inventor with five US patents. E-mail: harlan@thecountryschool.org

  A simple restatement of its limit definition formula allows one to derive trigonometric approximations for e. These novel closed-form expressions can then be used as functions that will "convert" the digits of  into those of e. Maclaurin series expansions are used to assess rates of convergence for these expressions.

• Lucky Larry #91

• Meet Me at the Crossroads: Over-Fishing to Meet the Standards
  John E. Donovan, II

  John teaches mathematics and mathematics education at the University of Maine. In addition to developing and discovering practical applications of math, he enjoys fly shing in the Penobscot River from his kayak, spending time with his wife and 4 kids, all things Mac, and long walks listening to novels on his iPod. E-mail: john.donovan@maine.edu

  To achieve the vision of mathematics set forth in Crossroads (AMATYC, 1995), students must experience mathematics as a sensemaking endeavor that informs their world. Embedding the study of mathematics into the real world is a challenge, particularly because it was not the way that many of us learned mathematics in the first place. This article is about one such example, the effects of fishing on fish populations, but the method of analysis used is widely applicable. The fishing model developed is based on intuitions about how populations change over time. Traditionally such examples are reserved for the study of calculus and differential equations, but qualitative methods of analysis make them accessible to students in precalculus. This example, and others like it, should not be considered add-ons to an already over-burdened curriculum. Rather, such problems provide launching points for students to develop deep understandings of mathematics through investigation of things that are real.

• Lucky Larry #92

• Successful Developmental Mathematics Education: Programs and Students - Part III, Irene M. Duranczyk

  Irene is an assistant professor in the Department of Postsecondary Teaching and Learning with an EdD from Grambling State University, Louisiana. She taught developmental mathematics since 1990 and was an administrator of developmental programs for over 20 years. Irene is the recipient of the 2007 National Association for Developmental Education's (NADE) Outstanding Research Conducted by a Developmental Education Practitioner Award. E-mail: duran026@umn.edu

  This is the third and final article in this series. The first article reviewed the literature for research studies on successful developmental programs and students. The second article reported on the qualitative research methods and results documented from a purposive sample of twenty successful developmental mathematics students 3-5 years after completing their developmental studies. This article presents more detail on what shaped this qualitative study, identifies specific implications for developmental mathematics educators, and makes recommendations for further research on success in developmental mathematics.

• On Moving a Couch Around a Corner
  Jawad Sadek and Russell Euler

  Russell Euler is a professor in the Department of Mathematics and Statistics at Northwest Missouri State University where he has taught since 1982. His mathematical interests include analysis, differential equations, geometry and number theory. Presently he is the Problem Editor for the Fibonacci Quarterly. Russell enjoys construction, volunteer work at his church, and learning from his three daughters. E-mail: reuler@nwmissouri.edu Jawad Sadek is a professor in the Department of Mathematics and Statistics at Northwest Missouri State University. His main mathematical interest is complex analysis. Jawad enjoys soccer and traveling around the world. E-mail: JAWADS@nwmissouri.edu

  Finding the longest rectangular couch with a given width that can be maneuvered around a corner is an old and interesting problem. It has been the subject of numerous research articles. In this note, two open questions that were raised in Moretti's article (2002) about the subject are discussed. In addition, the maximum area of a couch rounding a corner is also found. Reference Moretti, C. (2002). Moving a couch around a corner. The Coll. Math. Journal, 33(3), 196-200.

• Collinear Points Problem, Harris S. Shultz and Ray C. Shiflett

  Harris S. Shultz received the Southern California Section of the Mathematical Association of America's Award for Distinguished College or University Teaching in 1992. He has directed numerous institutes for secondary mathematics teachers, has designed online professional development programs and has been a frequent contributor to The AMATYC Review. E-mail: hshultz@fullerton.edu

  Ray C. Shiflett received his PhD at Oregon State University. He has published in operator, measure, matrix, and number theory, topology, optometry, science fiction, and mathematics education. He served as Chair of Mathematics at Wells College, Dean of the College of Science at Cal Poly Pomona, and Executive Director of the National Research Council's Mathematical Sciences Education Board. He enjoys golf, fly fishing, writing songs, and wood working. E-mail: rcshiflett@roadrunner.com

  Students were asked to find all possible values for A so that the points (1, 2), (5, A), and (A, 7) lie on a straight line. This problem suggests a generalization: Given (x, y), find all values of A so that the points (x, y), (5, A), and (A, 7) lie on a straight line. We find that this question about linear equations must be resolved using the more advanced tools of quadratic equations. The number of possible values of A can be zero, one or two, depending upon the given point (x, y). Moreover, the three cases are partitioned by an oblique parabola having its axis at an angle of 45 degrees to the Cartesian plane coordinate axes.

• Lucky Larry #93

• Sighting the International Space Station 
  Donald Teets

  Donald Teets has taught at the South Dakota School of Mines and Technology since obtaining his doctorate from Idaho State University in 1988. He received the Allendoerfer award from the Mathematical Association of America in 2000 for an article on the astronomical work of Gauss, and the Distinguished Teaching Award from the Rocky Mountain Section of the MAA in 2004. E-mail: donald.teets@sdsmt.edu

  This article shows how to use six parameters describing the International Space Station's orbit to predict when and in what part of the sky observers can look for the station as it passes over their location. The method requires only a good background in trigonometry and some familiarity with elementary vector and matrix operations. An included set of exercises leads the reader step-by-step through the computations. Specific instructions are included for implementation of the method using a spreadsheet tool such as Excel. This article gives students the rare opportunity to use classroom mathematics to solve a complicated real-world problem, and to observe the results of their solution in real time.

• A Binary Divisibility Theorem For Mersenne Numbers, Travis Thompson

  Travis Thompson received the PhD degree in mathematics from the University of Arkansas in 1977. He is currently the dean of the college of sciences at Harding University in Searcy, Arkansas. E-mail: thompson@harding.edu

  Arithmetic tests for divisibility of an integer by another integer are well known. This article states and proves conditions for divisibility in binary form.

• Lucky Larry #94

• Book Review
  Edited by Sandra DeLozier Coleman

  YEARNING FOR THE IMPOSSIBLE - The Surprising Truths of Mathematics, John Stillwell. A.K. Peters, Ltd., Wellesley, Massachusetts, 2006. Hardcover. xiii + 244 pp. ISBN 978-1-56881-254-0.

• The Problems Section
  Edited by Stephen Plett and Robert Stong

  The BA Problem Set consists of four new problems.

  Set AY Solutions

  Solutions are given to the five problems from the AY Problem Set from the Fall 2007 issue of The AMATYC Review. In addition, addenda were provided for the solvers of the AW Problem Set from the Fall 2006 issue.

Editor: Barbara S. Rives, Lamar State College
Production Manager: John C. Peterson

Table of Contents

• Lucky Larry #83

• From the Editor's Keyboard

  The fall semester is underway and hopefully, you and your colleagues have carefully read Beyond Crossroads, and have begun implementing the content of this document into your departments and classrooms. Another article focusing on Beyond Crossroads is included this issue--see the article written by Ham. Due to the importance of Beyond Crossroads, articles focusing on implementation are welcomed for review and consideration for possible publication in future issues of The AMATYC Review. Each issue for the foreseeable future will have at least one article published that shares the implementation
  successes at your campuses.

  Please submit the implementation manuscripts using the following guidelines:

• Length: 58 pages, typed in 12 point font
• Style: APA Publication Manual, 5th edition--this means the tables, figures, and references should be in APA (American Psychological Association) Style. If they are not in APA format, your materials will be returned to you to make the changes. This slows the review process even more, so please use APA format in your submission. For more information on the APA Style go to http://apastyle.apa.org/.
• Submission: Submit five hard copies of the manuscript to Barbara Rives, Editor, 204 Hardin Administration Building, Box 29140, Abilene, Texas 79699-9140. Please also include the following in the lower left corner of the package--"Attention: Beyond Crossroads implementation article."
• Send a digital copy of the manuscript as an e-mail attachment to ReviewEditor@amatyc.org. List the following in the subject line of the e-mail: [; Beyond Crossroads implementation article:
  ]. For example, if I submitted a manuscript, the subject line would read--Rives; Beyond Crossroads implementation article: Implementing the Standards for Student Achievement and Success.

• A special "thank you" goes to all the authors who have submitted manuscripts for possible publication. The review process has taken much longer than the authors (and the editor too) would like; however, there is "light at the end of the tunnel." If all goes as planned (manuscripts reviewed and returned), all authors who submitted manuscripts prior to June 1, 2007 should know the final determination of their manuscript by the time you receive this journal. Many excellent articles have been received for review and consideration. I wish more manuscripts could be published; however, this is not possible due to page limitation of each issue of The AMATYC Review. Have a wonderful fall semester. See you in Minneapolis.

  Barbara S. Rives, Editor
  E-mail: ReviewEditor@amatyc.org

• Successful Developmental Mathematics Education: Programs and Students-- Part I
  Irene M. Duranczyk

  Irene is an assistant professor in the Department of Postsecondary Teaching and Learning with an EdD from Grambling State University, Louisiana. She taught developmental mathematics since 1990 and was an administrator of developmental programs for over 20 years. Irene is the recipient of the 2007 National Association for Developmental Education's (NADE) Outstanding Research Conducted by a Developmental Education Practitioner Award. E-mail: duran026@umn.edu

  This article, the first in a three-part series, will explore the existing body of research regarding successful developmental mathematics education. The three-part series will present qualitative research conducted at a large Midwest public university. The qualitative study was conducted three to five years after students completed their developmental mathematics course work. The purpose was to collect students' points of view regarding what, if any, aspects of the developmental mathematics program contributed their success. Students do not read the literature that professional educators read and educators often do not check back with students after program completion to assess what parts of the educational experience have contributed the students' growth once they have completed their educational requirements. The second and third articles in the series will report on the research methods and results. The second article will specifically address aspects of the developmental mathematics program that students attributed to their successful experiences in life as well as their subsequent successful educational experiences. The last article in this series will provide some of the research tools used in this study and identify specific implications--what do developmental educators need to consider as they evaluate the effectiveness of their developmental mathematics programs.

• Differintegration: The One Branch of Calculus, Andrew J. Berry

  Andrew J. Berry received his BS and MS degrees in mathematics at the University of Illinois at Urbana-Champaign, and his PhD at New York University. He is associate professor of mathematics at LaGuardia Community College, City University of New York. E-mail: ajberry@nyc.rr.com

  How might one define a functional operator DIf (x), say for f (x) = 1 + x2 + sin x, such that D+1(1 + x2 + sin x) = 2x + cos x and D-1(1 + x2 + sin x) = x + x3/3 − cos x? Our task in this article is to describe such an operator using a single formula involving the limit of a sum which depends only on a single parameter specifying the order of the operation of differintegration.

• How to Compute the Partial Fraction Decomposition Without Really Trying, Richard Brazier and Eugene Boman

  Richard Brazier received his BA from Bath University in the UK and his Masters and PhD degrees in applied mathematics from University of Arizona in Tucson. His interests include his family, seismology, gardening, home remodeling and philately. E-mail: rab27@psu.edu Eugene Boman received his BA in mathematics from Reed College in 1984 and his MS and PhD in applied mathematics from the University of Connecticut in 1986 and 1993 respectively. He has been teaching at the Dubois campus of Penn State since 1996. E-mail: ecb5@psu.edu

  For various reasons there has been a recent trend in college and high school calculus courses to de-emphasize teaching the Partial Fraction Decomposition (PFD) as an integration technique. This is regrettable because the Partial Fraction Decomposition is considerably more than an integration technique. It is, in fact, a general purpose tool which crops up naturally in a wide range of applications. The techniques for computing the Partial Fraction Decomposition are numerous to say the least and tend to fall into two categories, general methods which will work for any decomposition and specialized methods which work only for special cases. Unfortunately, the general techniques are often cumbersome and tend to make relatively simple decompositions seem complex, and the specialized techniques, while often very easy to use, tend to roliferate to the point of chaos because there is a lot of variation in the kinds of decompositions that occur. We present an algorithm for computing the Partial Fraction Decomposition that is based on Heaviside's "cover-up" method-possibly the simplest of the known specialized techniques. The "cover-up" method is extended to a general technique which can be used for any decomposition. Our algorithm is simple to use and teach and is usually more efficient than other known algorithms, specialized or general.

• An Alternative Method to the Classical Partial Fraction Decomposition
  Chokri Cherif

  Chokri Cherif is an assistant professor of mathematics at the Borough of Manhattan Community College (BMCC) of the City University of New York (CUNY) and a 2006-2007 (Cohort 3) Project ACCCESS Fellow. He earned his MA in Pure Mathematics from the City College of New York and his PhD in Pure Mathematics from the Graduate Center of the City University of New York. His primary area of interest is functional analysis and its application to image processing. E-mail: ccherif@bmcc.cuny.edu

  PreCalculus students can use the Completing the Square Method to solve quadratic equations without the need to memorize the quadratic formula since this method naturally leads them to that formula. Calculus students, when studying integration, use various standard methods to compute integrals depending on the type of function to be integrated. Before integrating rational functions, students often need to know how to decompose the function by using the Partial Fraction Decomposition. In some cases, extending the Completing the Square Method beyond polynomial functions, to include rational functions, can be very helpful in avoiding lengthy computations where the potential of error is high. In this manuscript we propose an alternative method to the lengthy Partial Fraction Decomposition, used in standard calculus textbooks, to compute the indefinite integral of a family of rational functions. We will also demonstrate how the integral of the rational function, one over one plus x to the fourth power, can be thought of as a special case of the integral of the family of rational functions.

• Lucky Larry #84

• Beyond Assessment
  Jim Ham

  Jim Ham is a professor of mathematics at Delta College in University Center, Michigan, near Saginaw. He served on the Beyond Crossroads National Advisory Committee and was a section writer for Beyond Crossroads. He is also actively involved in MichMATYC and AMATYC's Placement and Assessment Committee. E-mail: jaham@delta.edu

  Stakeholders are interested in accountability in public education. College professors are doing innovative things in the classroom to help students learn mathematics and, when required, are documenting this learning. This article provides several hypothetical examples of how documented assessments of student learning at the classroom, course and programs levels, can provide evidence of accountability. A well-documented collection of assessment results and actions responding to these results can be the bridge between assessment and accountability. If we take care of the little things (documented classroom, course, and program assessments), the big thing (accountability) will take care of itself.

• A Couple of "lim (h=>0)-is-missing" Problems 
  Ko Hin Lau

  Ko Hin Lau is an assistant professor in the mathematics department at State University of New York (SUNY), College of Agriculture and Technology at Cobleskill. He obtained his PhD in mathematics from Indiana University. His academic interests include analysis, operator theory, and mathematics education. Email: laukh@cobleskill.edu

  Since most students "hate" the concept of limit, in order to make them "happier," this article suggests a couple of naive "lim (h=>0)-is-missing" problems for them to try for fun. Indeed, differential functional equations that are related to difference quotients in calculus are studied in this paper. In particular, two interesting observations are made in this article, namely, (1) it is possible to solve a differential functional equation just by some basic algebra; and (2) a certain class of smooth functions is characterized by imposing a simple condition on the value c, where c is guaranteed by the Mean Value Theorem for any smooth functions defined on any interval [a, b].

• Exploring Measurement Error with Cookies: A Real and Virtual Approach via Interactive Excel
   Scott A. Sinex, Barbara A. Gage, and Peggy J. Beck

  Scott A. Sinex is professor and chair of the physical sciences and engineering department of Prince George's Community College in Largo, MD. He received a PhD in geochemistry from the University of Maryland at College Park. He is involved with using technology to develop dynamic and interactive visualization of science and mathematical concepts for guided-inquiry instruction. E-mail: ssinex@pgcc.edu Barbara A. Gage is professor in the physical sciences and engineering department of Prince George's Community College in Largo, MD. She received her PhD in curriculum and instruction with emphasis in chemical education from the University of Maryland at College Park. When she's not in a chemistry classroom, she is designing activities for and teaching pre- and in-service teachers in Earth and space sciences. E-mail: bgage@pgcc.edu Peggy J. Beck is professor in the mathematics department of Prince George's Community College in Largo, MD. She received her MA degree in mathematics from The Pennsylvania State University. She has used the cookie module in both intermediate and college algebra, as part of the Peer-Led Team Learning approach to teaching mathematics. E-mail: pbeck@pgcc.edu

  A simple, guided-inquiry investigation using stacked sandwich cookies is employed to develop a simple linear mathematical model and to explore measurement error by incorporating errors as part of the investigation. Both random and systematic errors are presented. The model and errors are then investigated further by engaging with an interactive Excel simulation and a variety of what if scenarios. A conceptual understanding is developed by hands-on manipulation combined with further virtual experimentation. Numerous higherorder thinking and science process skills are used throughout the investigation.

• Lucky Larry #85

• A Study on Student Performance in the College Introductory Statistics Course 
  Jen-Ting Wang, Shu-Yi Tu, and Yann-Yann Shieh

  Jen-Ting Wang has a PhD in statistics from University of California at Santa Barbara. She is an associate professor in the department of mathematics, computer science, and statistics at the State University of New York College at Oneonta. Her research areas includes applied statistics, Bayesian statistics, and statistical education. E-mail: WangJ@Oneonta.Edu Shu-Yi Tu received her PhD in mathematics from University of California at Santa Barbara. She is an assistant professor of Mathematics at University of Michigan, Flint. Her research interests include applied mathematics, nonlinear wave equations, and statistics. E-mail: sytu@umflint.edu Yann-Yann Shieh is a statistician at Office of Special Education and Rehabilitative Service, US Department of Education. She has a PhD in educational psychology. Her areas of specialization are multilevel modeling. E-mail: yshieh@air.org

  Introductory Statistics is a required course for most college students in order to graduate. Research has been conducted for determinants of achievement in college mathematics courses; however, there has been little investigation for statistics courses. In this exploratory study, data concerning students' grades received in this course, the academic performance in high school and in college, as well as numbers of collegiate credits earned were collected from a public four-year liberal arts college. This study aims to identify the most significant factors of students' grades in this course. In addition, a comparison between performances of male and female students, as well as those of freshmen and non-freshmen was also examined. Class size effect was discussed as well. In addition to searching for the most important factors, the prediction model for the course grade was also established from multiple linear regressions. Findings suggest that a student with a good college and high school GPAs, as well as high SAT math score may perform well in the introductory statistics course. High school math grades were also found to be an important predictor.

• Book Reviews
  Edited by Sandra DeLozier Coleman

  THE PARROT"S THEOREM: A Novel, Denis Guedj, Translated by Frank Wynne, Thomas Dunne Books, an imprint of St. Martin's Press, New York, 2000, ISBN 0-312- 30302-5 (pbk).

  CRIMES AND MATHDEMEANORS, Leith Hathout, Illustrated by Karl H. Hofmann, A.K. Peters, Ltd., Wellesley, Massachusetts, ISBN-10: 1-56881-260-4.(back to top)

• Software Reviews
  Edited by Brian E. Smith

  An Overview of Several Popular Web-Enhanced
  Instructional Products: Part I

  In order to better understand the products that will be discussed here, one needs to be aware of the three systems that were developed for use in an e-learning environment.

  The first system, a Course Management System (CMS), was designed primarily for use in academia. This system offers its users the ability to place course materials online, create various assessment features such as tests and quizzes, communicate with students, and track student and course statistics. The most common CMS products on the market are WebCT, Blackboard, e-College, and ANGEL. Because the high price of these products can be prohibitive, free "Open Source" products such as Moodle and Saki have surfaced.

  The second system, a Learning Management System (LMS), is similar to the CMS but was designed primarily for use in corporate training. This system offers its users the ability to register students, track student participation and completion, transfer information to other systems, process course charges and tuition payment/transfers, manage skill development, and create reports. A few of the most common LMS products on the market are NetDimensions EKP, Saba, and SumTotal Systems.

  The third and newest system, a Learning Content Management System (LCMS), was designed to combine the learner and administrative capabilities of an LMS with the content creation and storage capabilities of a CMS (see Figure 1).

With the increase in popularity of CMS, the desire to add text specific ready made content available for use within CMS increased as well. Instructional designers were employed to create products that would satisfy this need. Of course, the popularity for these products grew with the increase in distance learning offerings and the need to easily manage multiple section offerings of a course. Hence, it became a major challenge to make these products more dynamic (interactive), more robust, and web-compatible. Due to the efforts and vision of the major players in education: Pearson Education (Addison-Wesley/Prentice Hall), McGraw-Hill, and the ALEKS Corporation, many of these challenges have been realized. The most common web-enhanced instructional products currently on the market (in order of their development) are ALEKS^® (ALEKS Corporation 1965), MyMathLab^® (Pearson Education 2000), Math Zone^® (McGraw-Hill 2004), Thompson NOW^® (Thompson
2005), and Eduspace (Houghton Mifflin 2006). More recently, Thompson-Brooks/Cole has introduced WebAssign^®. However, this product tends to fall under the LMS category and is, therefore, not discussed here.

As in most web-enhanced instructional products, there is both a student module and an instructor module to the product. The instructor module of the product includes all of the necessary tools for development, assessment, and implementation of a course whether it is tied to a specific text or not. In many instances, it permits cloning of a course, making management of multiple sections of a course possible. The student module of the product minimally includes instructor prepared practice quizzes/tests and course documents. However, the more sophisticated product also includes algorithmically generated interactive practice problems, quizzes, and tests, mini-lecture video clips, animations, power points, and access to an e-book.

To create a good web-enhanced instructional product, instructional designers need to consider the functionality of the product within the following theoretical context:

1. Learning Theories

1. Behaviorism
2. Constructivism
3. Cognitivism

2. Learning Styles of the Student

1. Visual/Haptic
2. Visual/Verbalizer
3. Leveling/Sharpening

1. Traditional
2. Distance Education (e-learning)
3. Computer Supported Collaborative Work
4. Computer Aided Instruction

1. Use (how and where)
2. Assessment (is it working well?)

5. Multimedia Technologies

1. Communication
2. Inquiry

    

6. Goals of Multimedia Design

1. Information Acquisition
2. Knowledge Construction

    

7. Goals of Multimedia Learning

1. Remembering: recall & retention
2. Understanding: transfer

User experience and interface design in the context of creating software represents an approach that puts the user, rather than the system, at the center of the process. This philosophy, called user-centered design, incorporates user concerns and advocacy from the beginning of the design process and dictates the needs of the user should be foremost in any design decisions [5].

With the theory of web-enhanced instructional product design in mind, an overview of each of the most popular products (in order of their development) is presented here.

ALEKS^® 2.0 Overview

ALEKS is an acronym for Assessment and LEarning in Knowledge Spaces. A bulk of the development of the ALEKS online interactive system began as a result of a multimillion dollar NSF grant. The ALEKS system was based on Knowledge Space Theory which basically asserts that a complete conceptual knowledge of a subject like Algebra can be separated into various disjoint and/or overlapping elements of knowledge within the subject area. Using a series of complex algorithms and interactive math problems, ALEKS is theoretically able to determine a student's knowledge state at any particular time within the learning process and "intelligently" lead the student into the concept that he/she is most ready to learn next. A more detailed discussion of the theoretical basis of ALEKS can be found in "Knowledge Spaces" by Jean-Paul Doignon and Jean-Claude Falmagne, (Springer, 1965). ALEKS requires the appropriate Java Runtime environment and a math plug-in to run properly. These items are automatically detected and downloaded upon registration.

Administrator Module

Administrators are required to register for their course using an instructor access code. An ALEKS instructor access code can be obtained by contacting your local sales representative. After registration and upon login, ALEKS will detect and install the required plug-ins and then present the instructor with a new message board. Instructors can read messages or go on to the Main page where they can select from the following options:

• How Do I: where instructors can obtain help for all features of ALEKS
• Course Administration: where instructors can:
  º Create a new course
  º Display the number of students in each course and its corresponding course code
  º Change the name or topic of a course
  º Change the password of a student
  º Change personal preferences (password, message options, e-mail forward, etc.)
  º Change account preferences of a student
  º Move a student from one course to another
  º Un-enroll a student from a course
  º Delete a course containing no student
• College Administration: where instructors can: º Create a new instructor account
  º Change the password of another instructor, or of a teaching assistant.
  º Change account preferences (name, messaging options, e-mail forwarding, etc.) of an instructor.
  º Move a course from one instructor to another.
  º Delete an Instructor Account
• Reporting: where an instructor can generate a status report (progress, time spent on ALEKS, etc.) in a variety of styles
• Taking Actions: where an Administrator can:
  º Schedule a new assessment
  º Cancel an assessment
  º Change the name, date, grading scale of an assessment
  º Edit the grading scale, date or name of a past or upcoming scheduled/requested assessment
  º Create a Quiz
  º Edit a Quiz
  º Delete a Quiz
  º Send a message to communicate with students or instructors.
• Advanced: this economical mode contains all of the above features and is available for the more experienced ALEKS user.

From an administrative standpoint, the Results & Progress menu gives the course administrator the ability to create a quiz for all sections of a course, e-mail all students from a specific section of a course, create a new course section, add a new instructor, review student progress for all sections of a course, and obtain reports for all sections of a course. Students can also be conveniently "draged and dropped: from one section of a course to another (see Figure 2).

Also from an administrative standpoint, the Standards & Syllabi menu gives the course administrator the ability to set standards for the sections as well as to adjust the course syllabus for each section.

Student Module

Students are required to register for their course using a purchased access code. The student would generally purchase this access code from their campus bookstore bundled with a text order from the instructor or course administrator. The student module of the ALEKS product consists of both an assessment and a learning mode. Each will be discussed separately below.

Assessment Mode

Upon registration and plug-in check and installation, each student is required to navigate through a tutorial on proper data entry and use of the ALEKS system. This tutorial takes approximately 10-20 minutes depending upon the computer skills of the student. When the tutorial has been completed the student is given an initial assessment test. The first question that a student encounters is always based upon the course content, but each question thereafter is selected by the system according to the way the student has answered a previous question. The number of questions within an assessment varies depending upon the answers to questions within the assessment. Although no feedback is provided during an assessment, when the assessment has been completed, ALEKS generates an individualized pie chart report that tells the student what knowledge elements ALEKS has deemed the student knows.

Learning Mode

Once the student has seen the ALEKS generated report, the student must then exit the report pie and enter the learning mode pie. By selecting an available element (concept) within a slice of the pie, a student is able to navigate through the course material. The student can attempt to solve the problem or can read an explanation of the problem's solution. The student is then presented with a similar problem. If the student incorrectly answers the new problem, the ALEKS system evaluates the type of error that could have occurred and then offers the student options. Students are given an assessment when ALEKS perceives that the student is ready for one, unless an assessment has been assigned by the Administrator. Students always have access to an overview of items that they can do and items that they need to learn next.

Product Functionality--Comments

Administrator Module

ALEKS has a robust administrative component. Multiple sections of a course can be created with relative ease. Although students can be easily "dragged & dropped" from one section of a course to another, their work was, at the time of this review, not able to be moved with them. It is unclear at this time whether the product revision provides this functionality. ALEKS generates a variety of useful student and class reports that give a quick overall view of the class's progress.

ALEKS has recently undergone a revision adding the following enhancements:

• Automatic Textbook Integration
• New Instructor Module
• Instructor-Created Quizzes
• New Message Center

It is difficult for instructors to follow a text since students are usually in different chapters or sections of a chapter at any given time.

Student Module

Although the student assessment module of ALEKS is typically only supposed to offer the student between 15-25 questions, some students have found themselves taking assessments that have contained over 80 questions. In the learning mode, students have found themselves sent back to elements that they had previously learned. Students have been known to be caught in infinite loops and had difficulty moving forward in the course. It is not readily apparent how to exit the initial assessment pie and enter the learning mode. Students are instructed to click on "Exit," but in doing so, are immediately logged out of the product. It is hard for students to follow a textbook since they are permitted to select from any section of the pie that ALEKS has deemed them ready to learn.

General

Norton Antivirus has presented a problem for ALEKS users! In general, the overall design and functionality of this product appears to be theoretically strong in items 3, 4, 6, and 7 but weak in items 1, 2 and 5.

MYMATHLAB Overview

Course Compass (CC) is an easy to use Course Management System (CMS) environment developed by Pearson Education using Blackboard technology. Addison-Wesley and Prentice Hall offer a wide variety of textbooks within the CC environment, with 250 of these titles enhanced by MyMathLab (MML). MML is a series of text-specific, customizable courses for Addison-Wesley and Prentice Hall textbooks in mathematics and statistics. MML is powered by CC andMathXL (MXL), Pearson Education's robust stand-alone online homework, tutorial, and assessment system (see Figure 3).

As a stand-alone system, MXL is fully functional outside of the CC/MML environments and is used primarily in the development of single courses. MXL is placed within the CC/MML environment when more control over multiple sections of a course is necessary. MML permits the delivery of online courses using the content of MXL and the online tools within CC. Moreover, instructors who wish to add their own content, documents, and videos, or want to customize the learning environment for their students can only do so in MML. Thus, MXL is the essence of the dynamic course materials for selected mathematics and statistics courses. MXL provides instructors with the following rich set of course options:

• a powerful homework and test manager
• a custom exercise builder
• comprehensive gradebook tracking
• complete online course content and customization tools
• the ability to copy or share courses and manage course groups

MXL is also a dynamic learning tool that provides students with:

• interactive tutorial exercises
• an e-book with multimedia learning aids
• individualized study plans
• tutoring service

In order to operate properly, MXL requires the MXL player which is a proprietary program developed by Pearson Education to deliver mathematics online. Although Java is used to deliver mathematics for older statistics and calculus titles, new editions of these texts will require the MXL player as well.

Administrator Module

Administrators are required to register for their course using an instructor access code. The access code is provided to instructors who adopt the MML product through their local sales representative or their course administrator. After registration and upon login, instructors are given the opportunity to take a tour of the product. As in most web-enhanced instructional products, certain plugins are necessary. These plugins can be user installed or installed by a computer administrator in the event that the instructor does not have administrator access to the computer. The administrator can create a Master Syllabus within a "coordinator" course and copy the coordinator course as many times as necessary to a "member" course. The instructor of the member course enrolls as a student and is given TA status by the course administrator.

From within a selected course, the administrator and TA have both student and instructor access, although the TA privileges are restricted. Instructor access is gained by selecting the tab labeled "Control Panel". In the control panel area, an instructor can upload or modify course documents, send e-mails, and manage the course menu. However, only the administrator has the ability to modify chapter contents and delete students from the course. The administrator can also modify MML components of the course; assign text specific algorithmically generated homework and tests, set gradebook options, etc. One should note that there are two gradebooks available from within the control panel. The first is CC dependent while the second is from within MML and keeps track of all web-enhanced assignments (see Figure 4).

Although the CC gradebook can be used for additional assignments, since it does not track student work done in MML, most instructors do not use it.

Student Module

Students need to have an access code in order to use the MML or MXL product. The course materials are generally purchased as a complete bundled package that includes the textbook and MML or MXL student access kit. Additional resources can be packaged but must be specially requested. A standalone access code can be purchased online via credit card. MML access codes remain active as long as the instructor keeps the course open. MXL student codes are good for 12 months or 24 months depending on the text (one term or two term course). Students have access to a variety of features like "Help Me Solve This", "View an Example", section lecture video, animations, and power points. An individualized study plan is generated for the student after every test to allow students to work on material that needs to be studied further.

Product Functionality--Comments

Administrator Module

MML has a clean Administrator appearance. Navigation from one stage of course/section development to another is relatively easy and cloning of a course can be done fairly quickly. Algorithmically generated assignments and tests can be copied and/or modified using the samples provided from within the product, or algorithmically generated problems can be selected from a test bank. Static or algorithmic tests can be uploaded from a Test Generator and made available for the web; however, the tests must be in multiple choice format. All interactive problems coincide with the selected text. Material that has been deleted from the course syllabus is automatically inaccessible from the MML test bank, homework, or study plan. The Administrator has the ability to simplify the course management interface.

Student Module

Students have complained that the math palette occasionally disappears, however, the new MXL player release appears to have diminished or resolved this issue. Having an MML access code remain active after it has been redeemed for as long as the instructor keeps the course open is helpful to students who have for some reason not completed the course on time. For students who have either not done well in the course or needed to drop the course, there is no need to purchase another code if they enroll in another course using the exact same text. The student interface appears to be easy to navigate and assignments
easy to access.

General

In general, the overall design and functionality of this product appears to be theoretically strong in items 1, 2, 3, 4, 5, 6, and 7.

Summary

Table 1 on the next page provides the reader with a quick overview of the instructional products that were discussed in Part I of this manuscript. A complete table of all of the instructional products discussed will be provided in Part II.

References

• ALEKS Corporation. (2006). ALEKS [Online]. Available: http://www.aleks.com/ [2006, October 05].
• de Leeuwe, Marcel, (2001). e-LearningSite [Online]. Available: http://www.e-learningsite.com/lmslcms/whatlms.htm [2006, September 05]
• Doignon, J.P., & Falmange, J.C. (1965). Knowledge Spaces. New York: Springer.
• Microsoft Corporation. (2006). MSDN [Online]. Avaliable: http://msdn.microsoft.com [2006, September 05].
• Martin-Gay, Beginning Algebra, 4th Edition, Prentice Hall, 2005.
• Pearson Education. (2000). CourseCompass/MyMathLab [Online]. Available: http://www.coursecompass.com/ [2006, September 01].

Reviewed by Annette M. Burden, Associate Professor, Mathematics and Statistics, Youngstown State University, College of Arts and Sciences, (Youngstown, OH). Burden is an associate professor of mathematics at Youngstown State University. She is beginning algebra coordinator and coordinator of the mathematics distance program. Annette also develops upper level mathematics courses for Empire State College. She is a member of numerous mathematics associations and the recipient teaching and service awards. She also serves on several multimedia advisory panels. Her e-mail address is aburden@as.ysu.edu.

Send software reviews to:

Brian E. Smith
AMATYC Review Software Editor
Department of Management Science
McGill University
1001 Sherbrooke St. West
Montreal, QC, Canada H3A 1G5

or e-mail: brian.smith@mcgill.ca

• The Problems Section
  Edited by Stephen Plett and Robert Stong

  New Problems

  The AY Problem Set consists of five new problems.

  Set AW Solutions

  Solutions are given to the four problems from the AW Problem Set that were in the
  Fall 2006 issue of The AMATYC Review.

• Mathematics For Learning With Inflammatory Notes for the Mortification of Educologists and the Vindication of "Just Plain Folks"
  Alain Schremmer

  In the Spring 2004 issue of The AMATYC Review, Schremmer introduced his idea for an open-source serialized text: Mathematics For Learning. The Preface to the text appeared in the Spring 2004 issue with a new chapter in each subsequent issue of The AMATYC Review. This issue contains Chapter 6: Repeated Multiplications and Divisions, with sections on "A Problem With English" and "Templates." 

• Lucky Larry #86

Editor: Barbara S. Rives, Lamar State College
Production Manager: John C. Peterson

Table of Contents

• Change: It Comes Straight from the Heart
  Richelle (Rikki) Blair

  Rikki Blair is President-elect of AMATYC, editor of Beyond Crossroads, and professor emeritus of mathematics at Lakeland Community College. Her professional interests are curriculum development, incorporating active student learning experiences into the classroom, and increasing professional development opportunities for faculty. She received her PhD from Kent State University in curriculum and instruction. E-mail: richelle.blair@sbcglobal.net

  An important component of transitioning from a classroom instructor to a practicing teaching professional is a commitment to continuous growth and lifelong learning. The professionalization process is dynamic, producing a state of professionalism with changes in one's values, philosophy, and classroom activities. When considering a change in behavior or practice, the power of past practices, rules, or paradigms to influence current judgments and choices cannot be underestimated. In order to embrace a particular change in behavior or practice, it must be seen, felt, touch the heart, and resonate with one's own values. Mathematics professionals who find implementing change challenging may find the strategies and step-by-step process of the Implementation Cycle of Beyond Crossroads helpful. The goal of the document is to empower the mathematics professional to embrace change and strengthen the learning and teaching of mathematics.

• Moving Beyond Crossroads: Opportunities and Paradoxes
  Lynn Arthur Steen

  Lynn Arthur Steen is a professor of mathematics and special assistant to the Provost at St. Olaf College in Northfield, Minnesota. A former president of the Mathematical Association of America, Steen is the editor or author of several books including Math and Bio 2010, Achieving Quantitative Literacy, Mathematics and Democracy (2001), Why Numbers Count, On the Shoulders of Giants, Everybody Counts, and Calculus for a New Century. Steen holds a PhD in mathematics from the Massachusetts Institute of Technology as well as several honorary degrees. E-mail: steen@stolaf.edu

  Beyond Crossroads addresses many of the challenges facing American higher education and offers members of AMATYC an ideal opportunity to respond energetically and constructively to these challenges. Notwithstanding the many efforts to improve secondary schooling, most students enter postsecondary education well behind where they should be in mathematics. If two-year colleges are to succeed in preparing students for life and work in the 21st century, AMATYC members will necessarily play an increasingly central role.

• Beyond Crossroads: Putting Standards into Action
  Gregory D. Foley

  Gregory D. Foley is senior scientist for secondary school mathematics improvement for the Austin Independent School District, Austin, Texas. Greg has taught elementary arithmetic through graduate-level mathematics, as well as upper division and graduate-level mathematics education. He has presented over 200 lectures, workshops, and institutes throughout the United States and internationally, and has directed a variety of funded projects. In 1998, Foley received the AMATYC Award for Mathematics Excellence. E-mail: gfoley@austinisd.org

  Beyond Crossroads is a call to action. Within this call, AMATYC has updated its 1995 Crossroads standards, developed a new set of guiding principles, and created a blueprint for implementing these revised principles and standards. The principles guiding Beyond Crossroads are a significant overhaul of their predecessors and are bold statements that embrace changes in practice to improve student learning. The Standards for Intellectual Development now include linking multiple representations, and the Standards for Content add measurement to the geometry standard and data analysis to the probability and statistics standard. Beyond Crossroads has a clear focus on implementation, with six of the ten chapters devoted to the new Implementation Standards. This action-oriented focus on implementation is reminiscent of NCTM's Agenda for Action. Many of the critical themes of Beyond Crossroads--problem solving, quantitative literacy, technology, accommodating diverse needs, professionalism, and public support--have their roots in the Agenda for Action. The article concludes with a critical question: Will we heed this latest call for change and act on it? We have at our disposal the materials and methods we need; now we must act on what we know. This requires hard work, tenacity, and mutual support.

• Four Steps to a Standards-Embracing Department
  Alan Jacobs, Sally Jacobs, Ted Coe, and Connie Carruthers

  Alan Jacobs, recently retired from the mathematics department of Scottsdale Community College, Scottsdale, Arizona, served as a section writer and reviewer for Beyond Crossroads. He is past mathematics department chair at Scottsdale CC and coauthor of The Maricopa Project. He received the AMATYC Teaching Excellence Award in 2005. E-mail: salnal@cox.net Sally Jacobs, recently retired from the mathematics department of Scottsdale Community College, Scottsdale, Arizona, was a contributing writer for Beyond Crossroads. She was involved with implementation of various reform initiatives at Scottsdale CC and was the faculty liaison between Maricopa faculty and mathematics education research projects at Arizona State University. E-mail: sally.jacobs@sccmail.maricopa.edu Ted Coe, evening chairperson of the mathematics department at Scottsdale Community College, Scottsdale, Arizona, was a contributing writer for Beyond Crossroads. E-mail: Ted.Coe@sccmail.maricopa.edu Connie Carruthers, professor of mathematics and daytime chairperson at Scottsdale Community College in Scottsville, Arizona, received a BA at University of California and a MS at California State University, Northridge. E-mail: connie.carruthers@sccmail.maricopa.edu

  How did it happen that both full-time and adjunct faculty at Scottsdale Community College embrace a standards-based curriculum from beginning algebra through differential equations? Simply put, it didn't just happen. Not only did it take well over a decade, but it was also the result of a sequence of initiatives, decisions, discussions, targeted faculty development, and a willingness to take risks. This article summarizes that sequence of initiatives in four steps: Invest in a manageable change, with a plan to bring the entire department along. We made this investment when we adopted a reform-calculus book in 1994. Engage the adjunct and full-time faculty in activities that build mutual respect. We began joint professional development seminars that led to learning communities. Implement your initiative to solve the problem only when you have agreement about what the problem is. The key word is "agreement." Our initiatives were most successful when we were patient enough to come to agreement. Build on past successes. After several initiatives you develop a department process. Use your unique department process on each new initiative.

• Quantitative Literacy--Beyond Crossroads Gets It Right
  William G. Steenken

  William G. Steenken retired from GE Aviation, Cincinnati, OH in 2001 after a 29 year career, but continues to consult with them on a regular basis. He holds a PhD in Mechanical Engineering and is the author of over 34 papers in the field of Inlet/Engine Compatibility and Engine Operability. He has been involved with education since 1977 through service on school, and mathematics and science advisory and coalition boards. For the last ten years, he has been deeply involved in supporting the efforts to improve K12 mathematics and science education at the policy level in Ohio. E-mail: steenken@worldnet.att.net

  In this article, Steenken conveys some thoughts about when parents and our citizenry will believe that today's students know mathematics. It will be when students have significant Quantitative Literacy (QL) skills as set forth in AMATYC's new standards, Beyond Crossroads. He supports the strong call for QL to be imbedded across all curricula and as a method for assuring that students who leave two-year programs are prepared for the world that will confront them. He further supports the need for faculty to see QL as a daily part of their lives, especially as they make the Implementation Cycle presented in Beyond Crossroads as a method for assuring continuous improvement in their professional activities. He ends with a statement that Beyond Crossroad's strong call for QL will be met with strong support from industry and the business world.

• Setting a Course for Change Based on Beyond Crossroads
  John A. Dossey

  John A. Dossey is the Distinguished University Professor of Mathematics (Emeritus) at Illinois State University. Prior to his 30-year career at Illinois State University, he taught middle and senior high school mathematics. John served as President of the National Council of Teachers of Mathematics (NCTM) and Chair of the Conference Board of the Mathematical Sciences (CBMS). He received his BS and MS degrees at Illinois State University and his PhD from the University of Illinois at Urbana-Champaign. E-mail: jdossey@math.ilstu.edu

  The article provides a vision of how Beyond Crossroads can serve as a departmental guide to inducing systemic change at a two-year college. Curricular change does not mean just changing the content taught, it also means changing the way it is taught, the way learning may be assessed, and the ways in which faculty and administrators may evaluate curricular effectiveness. Beyond Crossroads provides a starting point, the rest remains in the hands of faculty committed to providing the best program possible for their students.

• Implementing Change in College Algebra
  William E. Haver

  Bill Haver is professor of mathematics at Virginia Commonwealth University (VCU). He received his PhD in the area of infinite dimensional topology from SUNY Binghamton. He has also held appointments at the University of Tennessee (UT), the Institute for Advanced Study, and the National Science Foundation. He is chair of the Curriculum Renewal Across the First Two Years Committee of the Mathematical Association of America. He has taught college algebra at the UT, VCU, Rutgers University, Bates College, and J. Sargeant Reynolds Community College. E-mail: wehaver@vcu.edu

  In this paper, departments are urged to consider implementing the type of changes proposed in Beyond Crossroads in College Algebra. The author of this paper is chair of the Curriculum Renewal Across the First Two Years (CRAFTY) Committee of the Mathematical Association of America. The committee has members from two-year colleges, four-year colleges, and research universities. CRAFTY recently organized 11 workshops, each bringing together representatives from partner disciplines to explore the mathematical needs of students in their discipline. The recommendations from the various disciplines were remarkably consistent and lead to College Algebra Guidelines that provide a vision of what all students enrolled in College Algebra should experience. The Guidelines contain specific recommendations concerning topics in functions, equations and data analysis that need to be contained in the course. They also address appropriate pedagogical and assessment practices. These are at a more specific level than Beyond Crossroads. However, there is a very strong correlation between the College Algebra Guidelines and the Basic Principles of Beyond Crossroads. College Algebra indeed provides an important place to begin the implementation proposed in Beyond Crossroads.

• Lucky Larry #73

• Beyond Crossroads: Impressions of a Statistics Educator
  Richard L. Scheaffer

  Richard L. Scheaffer received his PhD in statistics from Florida State University, whereupon he joined the faculty of the University of Florida and remained on that faculty ever since. Now professor emeritus of statistics, he was chairman of the department for a period of twelve years. Research interests are in the areas of sampling and applied probability, especially with regard to applications of both to industrial processes. He has published numerous papers in the statistical literature and is co-author of five college-level textbooks covering aspects of introductory statistics, sampling, probability, and mathematical statistics. E-mail: rls907@bellsouth.net

  Beyond Crossroads recognizes that success in the modern world demands higher-level thinking across the mathematical sciences. Broad quantitative literacy skills are essential for the college graduates of today and tomorrow if they are to be informed citizens and productive workers. Such skills include the quantitative aspects of daily life and work that allow educated people to make intelligent decisions based on knowledge rather than being manipulated through guile or fear. Quantitative literacy is largely akin to statistical thinking, because many of the quantitative areas of life and work involve understanding data--how it is collected, what it represents, and what conclusions can be drawn from it. In that respect, the document is in concert with the American Statistical Association's Guidelines for Assessment and Instruction in Statistics Education (GAISE). All colleges should seriously consider how following the recommendations of Beyond Crossroads can impact their mathematics programs for students in their first two years.

• Developing and Implementing a Quantitative Reasoning Program at BMCC
  Klement Teixeira

  Klement Teixeira is a deputy chair of the mathematics department at Borough of Manhattan Community College, CUNY, in New York City. He earned an MA in mathematics specializing in probability and statistics from City College, CUNY, an MS in mathematics at the Courant Institute of Mathematical Sciences, New York University, and a PhD in mathematics education from the Steinhardt School of Education, New York University. E-mail: kteixeira@bmcc.cuny.edu

  The case study approach is commonly used in the fields of law, medicine, and business administration to help apply theory to practice. This approach is equally useful in the teaching and learning of mathematics since various categories of coping strategies used to alleviate math anxiety become more meaningful when they are used to assist "real" students. A number of case studies were developed to apply the coping strategies presented in Beyond Crossroads to students who have found themselves at the "crossroads" between success and failure. Each case represents a student at the author's institution who is confronted with a potentially stressful situation when attempting to study mathematics.

• Lucky Larry #74

• Using Chapter 6 of Beyond Crossroads as a Catalyst for Curriculum Change
  James W. Hall

  James W. Hall is a Parkland College Professor Emeritus. He was department chair of mathematics at Parkland College for 7 years and has written numerous textbooks in undergraduate mathematics. He is also writing team chair for Chapter 6 on Curriculum and Program Development in Beyond Crossroads. He celebrated his first hole-in-one on May 15, 2006 near his home in Sun Lakes, Arizona. E-mail: jhall@wbhsi.net

  This article is written from the perspective of a department chair who recognizes that there are often significant barriers within the department to changing the curriculum. This article makes the case that changes are needed and suggests actions that can be the catalyst for change.

• Time to Re-evaluate: Am I Implementing the Standards? 
  Nancy J. Sattler

  Nancy Sattler is an adjunct mathematics teacher at Terra Community College in Fremont, Ohio. She has been a member of AMATYC's Distance Learning Committee since its inception and was a section writer for Beyond Crossroads. E-mail: nsattler@terra.edu

  Beyond Crossroads states that mathematics faculty should (a) select technology that is accessible to students enrolled in their distance learning mathematics course, (b) advise students on the expectations of their distance learning mathematics course and orient them to the distance learning environment of their course, (c) provide students with course information outlining course objectives, concepts, ideas, and learning outcomes for their distance learning mathematics course, (d) engage in ongoing professional development to enhance their mathematics course presentation and support their teaching practice in the distance learning environment, and (e) assure that learning outcomes in mathematics distance learning sections are consistent with those of similar mathematics courses taught in the classrooms. Nancy Sattler, past chair of the Distance Learning Committee, explains how she is addressing Beyond Crossroads strategies in her online mathematics class as Terra Community College changed from quarters to semesters and the curriculum changed. She adheres to the philosophy that technology should facilitate a kind of learning that is durable, has substance, is engaging to students, and provides mathematical insights through a high level of understanding of the mathematics being taught.

• Building Consensus and Providing Guidance among Professional Societies? 
  Johnny W. Lott

  Johnny W. Lott is the director of the Center for Teaching and Learning Excellence at The University of Mississippi. He is a past president of the National Council of Teachers of Mathematics and was professor of mathematics education at The University of Montana until his recent move to Ole Miss. E-mail: jlott@olemiss.edu

  Beyond Crossroads has as a stated objective having "two-year college mathematics faculty and institutions collaborate with professional societies, government agencies, and educational institutions to build consensus and provide guidance to practitioners." Issues involving two-year faculty and university faculty members in conversations about common issues has at times been challenging. With school teachers (typically members of the National Council of Teachers of Mathematics (NCTM)) at the pre-collegiate level in the mix, the conversation becomes even more difficult. With students taking dual enrollment courses at high school and all levels, the Mathematics Association of America (MAA) writing placement examinations in conjunctions with Maplesoft for use throughout the collegiate levels, and teacher preparation being distributed across two- year colleges and four-year schools, the conversations are needed and desirable. In order for the conversations to happen, suggestions include working with the NCTM/MAA Joint Committee on Common Concerns, possibly adding a member of the Board of Directors of NCTM who is a two-year college person, adding a specific member of the Board of Governors for the MAA to represent two-year colleges and adding a representative from the pre-collegiate level and the four-year level to the American Mathematics Association for Two-Year Colleges (AMATYC) Board of Directors. None of this will be easy, but each could help move the conversations along. Beyond Crossroads is a document with a wide vision of what could happen in this arena. It will take the concentrated work of the three named professional organizations to make this happen.

• Lucky Larry #75

• Commentary: Beyond Crossroads, Joan R. Leitzel

  Joan Leitzel is President Emerita, University of New Hampshire; Professor Emeritus, The Ohio State University. Dr. Leitzel is an accomplished leader, having served as President of the University of New Hampshire, Senior Vice Chancellor for Academic Affairs at the University of Nebraska Lincoln, Director of the Division of Materials Development, Research, and Informal Science Education at NSF; and Associate Provost at Ohio State. She received her Ph.D. in mathematics at Indiana University and was a Professor of Mathematics at Ohio State for 25 years. She is a former chair of the Mathematical Sciences Education Board at the National Research Council. E-mail: joan.leitzel@unh.edu

  The Commentary salutes AMATYC for its significant contributions to standardsbased education in mathematics and discusses possible audiences for Beyond Crossroads, in addition to the primary audience of faculty members and departments in two-year institutions. Because Beyond Crossroads focuses on lower division college mathematics, it helps clarify the connections between secondary school mathematics and baccalaureate programs. Consequently, it can be a valuable tool for both high school teachers of mathematics and faculty in baccalaureate programs and can help with efforts to create a more coherent curriculum across grades 916. Beyond Crossroads is also seen to be a potential resource for those working on assessment and placement instruments at several levels, for those providing professional development to teachers of middle school and high school mathematics, and for those attempting to implement content standards in instruction. In this Commentary, Beyond Crossroads is viewed as more than a resource for two year institutions and is highlighted as potentially important to several areas of mathematics education.

• Lucky Larry #76

• Assessment: Key to Teaching and Learning
  Judy Marwick

  Judith Marwick is vice president of instruction and student services at Kankakee Community College in Illinois. Earlier, she was department chair and professor of mathematics at Prairie State College. Judy served as a writing team chair for Beyond Crossroads and was chair of the AMATYC Placement and Assessment Committee from 19992003. She holds an MS in mathematics from Purdue University and an EdD in community college leadership from the University of Illinois. E-mail: jmarwick@kcc.edu

  Assessment of student learning is key to all educational endeavors and required by governmental and accrediting bodies. Faculty initiate and implement assessment strategies to be sure that students are learning what is being taught. Classroom assessment is generally easier for faculty to embrace than course or program level assessment because classroom assessment techniques can be developed by individual instructors and implemented within a single classroom. Course and program assessment requires collaboration among all faculty involved in teaching a section of a course or within a program or sequence of courses. While it may be difficult to reach a consensus about what is most important for students to learn or how best to measure their learning, the discussion and introspection among colleagues wrestling with these issues is of great value. Faculty at community colleges have never stopped at what is easy. In fact, they move mountains and make a difference in students lives every day. Assessment should be seen as one more endeavor that, while difficult to implement, has the potential for significant results

• Lucky Larry #77

• Lucky Larry #78

• Using Assessment of Student Learning As A Catalyst for Change
  Myra Snell

• Math Anxiety Case Studies: A Beyond Crossroads Companion
  Fred Peskoff

  Fred Peskoff is chairperson of mathematics at Borough of Manhattan Community College, City University of New York. He has made numerous presentations both nationally and internationally on math anxiety and its impact on students and faculty. His work has been published by the Harvard Graduate School of Education. Peskoff won the 2003 AMATYC Teaching Excellence Award for the Northeast Region. E-mail: fpeskoff@aol.com

  The case study approach is commonly used in the fields of law, medicine, and business administration to help apply theory to practice. This approach is equally useful in the teaching and learning of mathematics since various categories of coping strategies used to alleviate math anxiety become more meaningful when they are used to assist "real" students. A number of case studies were developed to apply the coping strategies presented in Beyond Crossroads to students who have found themselves at the "crossroads" between success and failure. Each case represents a student at the author's institution who is confronted with a potentially stressful situation when attempting to study mathematics.

• Lucky Larry #79

• AMATYC's Role for Improvement of Future Learning
  Linda P. Rosen

  Linda P. Rosen is President of Education and Management Innovations, Inc. Previously, Rosen served as Senior Advisor to Secretary of Education Richard W. Riley and as the Executive Director of the National Commission on Mathematics and Science Teaching for the 21st Century (known as the Glenn Commission). She was also the Executive Director of NCTM and the Associate Executive Director of the Mathematical Sciences Education Board.

  The educational landscape has shifted significantly in the past few months with a new call for national standards and national tests as well as for accountability in higher education. Beyond Crossroads must be implemented with full understanding of these shifts and with agility to adapt to further seismic changes. A brief history puts the magnitude of recent shifts in context. Those in mathematics education often claim the title as "godparent" of the standards movement after the 1989 release of the NCTM Curriculum and Evaluation Standards. Yet, when President George H.W. Bush and the nation's governors announced America 2000 to create "world class standards" and achievement tests, politicians laid claim to "godparent" status. In the 1990s, the business community also laid claim as the "godparent" of the standards movement. Of course, parentage is unimportant as long as high quality, well-conceived standards get put into practice. And, therein lays the problem for K12 and for higher education: defining high quality standards and, more importantly, implementing them. The Commission on the Future of Higher Education recently identified three As for the renewal of higher education: access, affordability, and accountability. It is the third A--accountability--that is pertinent to the release of Beyond Crossroads. Knowing that administrators and policymakers are weary of calls for excellence without commensurate, steady progress towards that vision and know-ing that external pressures on them to "deliver" are increasing, it behooves AMATYC to take seriously the need to improve every component of mathematics education in the first two years of college.

• Lucky Larry #80

• Why is it Essential to Involve Stakeholders in Implementing Beyond Crossroads?
  Sue Parsons

  Sue Parsons is currently the Director of Teacher TRAC and Learning Community Programs and an associate professor of mathematics at Cerritos College. She served on the National Academy of Science MSEB Board 20012004. She also served as AMATYC West Region Vice President, Co-PI on an AMATYC NSF Teacher Preparation grant, and as a writing team chair for the AMATYC Beyond Crossroads Project. E-mail: parsons@cerritos.edu

  Most two-year mathematics faculty initially won't gravitate toward the chapter on stakeholder involvement in implementing Beyond Crossroads. Faculty most likely will search out the chapter on curriculum and instruction. In fact, some mathematics faculty may not consider the relevancy of other stakeholders as an important factor for improving their students learning in mathematics. The thought may exist that, "I am a two-year college mathematics professor. Why do I need to collaborate with entities outside my department? Why do I need to be involved with other stakeholders? I am well versed in my content area and have the mathematical background to teach two-year college students." Part of implementing Beyond Crossroads is the recognition that improving student learning in mathematics will not be fully realized without meaningful involvement of many stakeholders. The article addresses questions and discussion that are meant to emphasize that we, as mathematics faculty members, are stakeholders and Beyond Crossroads is a call to action for all stakeholders to work together to improve student success in mathematics courses and programs in the first two years of college.

• Lucky Larry #81

• Book Reviews
  Edited by Sandra DeLozier Coleman

  THE CALCULUS WARS: Newton, Leibniz, and the Greatest Mathematical Clash of All Time, Jason Socrates Bardi, Thunder's Mouth Press, an imprint of Avalon Publishing Group, Inc., New York, 2006, ISBN 1-56025-706-7.

  TOM STOPPARD: PLAYS 5--Arcadia, The Real Thing, Night and Day, Indian Ink, Hapgood, Tom Stoppard, Faber and Faber Limited, London, 1999, ISBN 0-571-19751-5. (back to top)

• Lucky Larry #82

• Software Reviews
  Edited by Brian E. Smith

Software Reviews

Reviewed by Patrick J. DeFazio, Onondaga Community College

Edited by Brian E. Smith

powerOne™ Graph v4.2

Producer and Distributor: Infinity Softworks, Inc.
Web addresses: www.infinitysw.com
Price: Retail Price $59.99

As a qualified educator, administrator or director, you may be eligible to purchase a single copy of powerOne™ Graph graphing-scientific calculator software or powerOne^® Finance financial calculator software for a 75% discount for your personal use.

Platforms: Palm^® OS

PowerOne™ Graph v4.2 by Infinity Softworks, Inc. is a graphing calculator software title for Palm^® handheld devices. Its functionality is robust and it is both expandable and customizable through downloads from Infinity Softwork's web site. The software includes capabilities for computation, conversion, graphing (with analysis), business, matrix, probability, statistics, regression, and more. The user can select their desired input mode (algebraic, RPN, chain, order of operations) to meet their individual needs. Entries are made through the touch screen using the stylus and pop-up keypad calculators (when needed), or through Palm's^® Graffiti writing software. Data and results can often be copied and pasted to/from the system clipboard and the software documentation indicates that a user can also export results to spreadsheet and word processing applications (which may require add-ons). Use of the Palm^® device's wireless communications is also enabled to allow the user to "beam" selected data, functions, or results to others.

This review outlines the main calculator interface, illustrates the use of a statistics template, and then demonstrates a few of the more commonly needed graphical features. Screen shots were obtained using a PC emulator. It should be noted that the resolution of the images from the emulator do not adequately indicate the resolution obtained when using the software on a Palm^® handheld. [Ed.: Because this journal is printed in black and white, the color features of the software cannot be seen.]

The main calculator interface (see Figure 1) has many features that make calculation input and access (to the many additional software features) very easy. Calculations are entered through the use of the keypad and function buttons (on the function bars). These buttons call individual functions (or function categories) and appear in two rows that the user can scroll to see additional available buttons. The buttons can be customized to include templates as well. The list of available function buttons may change with the use of alternate skins (from the website). The user can also select functions through the functions

Within the view window are some additional features. The H3 in the screen shot in Figure 4 indicates the memory location where the current calculation results are stored. When thebutton within the view window is tapped with the stylus a calculations log opens showing a recent list of calculations and results. These can be individually recalled to the view window for use in the current calculation. The D in the view window pictured in Figure 1 indicates that the calculator is in decimal mode. This can be selected to open a menu that allows the user to change the base for calculations or convert results (to binary, octal, or hexadecimal). Fraction and mixed number modes/conversions are also available here. (See Figures 24.)

The powerOne button at the top of the main calculator screen provides access to the preference settings. It also contains the copy/paste commands that access the Palm^® system clipboard. Additional features mentioned in this review may also be accessed through this button.

New data (variables, constants, matrices, tables, etc.) can be entered through the"My Data" navigation button at the top of the main calculator screen. User created macros (specific equations for recall in other calculations) can also be entered here. Many useful constants come stored in the "My Data" area including e, , the speed of light, gravity acceleration, electron mass, and others. Tables are easily created by selecting New from the "My Data" screen. The user can name the table, enter its imensions (see Figure 5), and then, using the pop-up keyboard calculator (which automatically appears when needed), input the individual entries, as shown in Figure 6. After completion and returning to the "My Data" screen the table can later be edited, duplicated, beamed, or have notes attached (Figure 7). The ability to attach notes is a nice feature that would be very useful if multiple tables are needed for an assignment or project.

Templates are a nice feature of this software. Some templates come pre-installed, others can be created by the user or obtained from Infinity Softworks Inc.'s website. Access to the templates is obtained through the "My Templates" navigation button at the top of the main calculator screen. When this button is selected with the stylus a menu of templates (sorted into categories) is opened. Business, calendar, conversion, and many statistical templates are available here. Some of the statistical choices are shown in Figure 8. The 2-Var Stats template was chosen in Figure 9 to illustrate the process.

The two columns of the Sample table are selected from drop down lists as the data source. When OK is tapped by the stylus the template runs and approximately two screens of statistics are shown in Figures 10 and 11. Tapping the button at the top of this screen provides a nice summary of all of the statistics calculated in this template, as indicated in Figure 12. Selecting Graph from this screen shows a plot of the data with an automatically fitted window. From there, Analysis can then be selected with Regression and Quadratic chosen from the subsequent menus to produce a graph of the quadratic regression function (see Figures 13 and 14). Details of the regression function can be found using the button as demonstrated in Figure 15.

The function f(x) = sin(x) + 1 will be used to demonstrate a few of the graphical features of the software (graphing, finding extrema, tangent lines, and intersections) that are in common use in a mathematics classroom. Some very nice features of this handheld device software (color, naming, categorizing, notation, and use of stylus) that differentiate it from many popular graphing calculators are also highlighted.

The "Details" tab (Figure 21) is the location for assigning the new graph a name (optional) and a category (optional), as indicated in Figure 22. These options could be very useful for an instructor wishing to categorize multiple graphs by different courses or for a student categorizing graphs by assignment. Naming a graph by its homework exercise would also prove a beneficial use of this feature. Graphs without names are listed by their function rule. Color and line styles can also be assigned here. The "Prefs" tab (Figure 21) allows the user to select window settings. One additional (and very nice) feature found on this screen is the Notes option. Selecting Notes from the New Graph screen opens a text input area allowing the user to enter annotations, comments, or questions that can be saved with the graph (see Figure 23). This would be very useful as a pedagogical tool.

Now that the graph's information has been entered, return to the "My Graphs" screen. The drop menu at the upper right-hand corner allows the user to have only the graphs in a desired category to be shown (Figure 24). The desired individual graph(s) are checked, window settings adjusted (if desired) (see Figure 25), and then Graph is selected to view the graph shown in Figure 26.

The Palm^® stylus works very well as an input/selection device for many common graphical analysis procedures. The steps required for obtaining local extrema and tangent lines, tracing, zooming, and determining points of intersection all highlight this fact. These can all be initiated from the "Graph" screen, most from the Analysis menu. Other tools are available on this menu as well, including Roots, Derivative, and Inflection.

Select Maximum from the Analysis menu and then use the stylus to drag open a box defining a region for the software to calculate the maximum function value within. The results are displayed on the graph screen (Figure 27). Additional boxed regions can then be defined with the stylus to find additional maximums.

Zooming can be controlled by selecting Zoom In (or Zoom Out) and then tapping the desired focal area of the graph with the stylus. Alternatively, the user can select Zoom Box from the Zoom menu and define the box by dragging the stylus over the desired area of the graph. Tracing (evaluating) is accomplished by selecting Trace/Eval from the Analysis menu and then using the stylus to tap the desired point on the graph, as indicated in Figure 29. Again, the stylus can also be slid along the graph to see the function values instantly change.

The intersection of two graphs can be found by having both graphs shown on the same axes, selecting Intersection from the Analysis menu, and then using the stylus to drag open a box around the desired point (see Figure 30).

This review has illustrated some of the abilities and features of this very functional software title. The main interface and subsequent menus and screens are clear and easily navigated. The Palm^® stylus is an intuitive input device and the software utilizes it well. Pop-up keyboards and calculators provide an easy way to enter data for those who frustrate easily with PDA-type handwriting recognition. The ability to add-on or customize this software adds to its strength as an educational tool. In addition, the full color and higher resolution screen outclass the common graphing calculator screens that are widely used. Any mathematics instructor or student currently owning a Palm^®device would find this software very useful.

Reviewed by Patrick DeFazio, Assistant Professor, Department of Mathematics, Onondaga Community College (Syracuse, NY). DeFazio received his BS (Mathematics) from the State University of New York (SUNY) at Fredonia and both a BA (Philosophy) and MA (Mathematics) from SUNY Brockport.

Send reviews to:

Brian E. Smith
AMATYC Review Software Editor
Department of Management Science
McGill University
1001 Sherbrooke St. West
Montreal, QC, Canada H3A 1G5

or e-mail: brian.smith@mcgill.ca

• The Problems Section, Edited by Stephen Plett and Robert Stong

New Problems

The AX Problem Set consists of four new problems.

Set AV Solutions

Solutions are given to the four problems from the AV Problem Set that were in the
spring 2006 issue of The AMATYC Review.

Editor: Barbara S. Rives, Lamar State College
Production Manager: John C. Peterson

Table of Contents

• Journey to Beyond Crossroads: A Reflection
  Susan S. Wood, Philip H. Mahler, and Sadie C. Bragg

  Susan S. Wood is Assistant Vice Chancellor for Educational Programs and Instructional Technology for the Virginia Community College System. Prior to joining the administrative offices of the 23-college Virginia system, she was professor of mathematics at J. Sargeant Reynolds CC in Richmond, Virginia, for 32 years. Susan served as AMATYC president from 19992001 and is Lead Project Director for the AMATYC Beyond Crossroads Project. Susan has a doctorate in Mathematics Education from the University of Virginia. E-mail: swood@vccs.edu Sadie C. Bragg is Senior Vice President of Academic Affairs and professor of mathematics at Borough of Manhattan CC, CUNY. Sadie served as AMATYC president from 1997-1999 and is currently a co-director of Project ACCCESS, an AMATYC professional development program. Sadie holds a doctorate in the College Teaching of Mathematics, from Teachers College, Columbia University. E-mail: sbragg@bmcc.cuny.edu Philip H. Mahler teaches at Middlesex CC, Bedford, Massachusetts. He is a past president of AMATYC and NEMATYC and a leader in the recent updating of the AMATYC standards. He participated in activities at the national level on quantitative literacy and college algebra reform. Phil has a BA in Modern Languages from Assumption College and an MAT in Mathematics from the University of Florida. E-mail: mahlerp@middlesex.mass.edu

  In 1995, the American Mathematical Association of Two-Year Colleges (AMATYC) published its standards document, Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus. AMATYC's Strategic Plan for 2000-2005 called for reviewing and revising the AMATYC Standards. In 2001, this task began under the leadership of AMATYC's President, Past President, and President-Elect. A National Advisory Committee provided guidance throughout the standards revision project. Through the dedication of hundreds of AMATYC volunteers, drafts of what is now called Beyond Crossroads: Implementing Mathematics Standards in the First Two Years of College were created, reviewed, revised, and improved. The document, with official release in November 2006, describes five new standards to implement the 1995 standards for content, pedagogy, and intellectual development. The five new implementation standards address student learning and the learning environment, assessment of student learning,curriculum and program development, instruction, and professionalism. Accompanying the standards are implementation recommendations and action items. Central themes include embracing change, an implementation cycle, and the involvement of stakeholders. This article is a reflection from the three project directors on the five years of the development of Beyond Crossroads.

• The Lost Divisibility Rules for 7 and Beyond
  A. J. Berry 

  Andrew J. Berry received his BS and MS degrees in mathematics at the University of Illinois at Urbana-Champaign, and his PhD at New York University. He is associate professor of mathematics at LaGuardia Community College, City University of New York. E-mail: ajberry@nyc.rr.com

  As a precursor to lessons on prime decomposition and reducing fractions, rules are generally presented for divisibility by 2, 3, 5, 9, and 10 and sometimes for those popular composites such as 4 and 25. In our experience students often ask: "What about the one for 7?" and we are loathe to simply state that there isnt one. We have yet to see a rule for divisibility by primes 7 or greater in any standard textbook. Maybe these are slightly more involved than the other divisibility rules, yet we find that they should be included, or at least mentioned, so as not to suggest to the student that such algorithms are only possible for a few special integers. Divisibility criteria are arithmetic methods that determine whether or not one integer divides another without having to actually carry out the division. These methods in question offer a simpler course than by performing the division itself to resolve the question of divisibility. We suggest that introducing some of these techniques into the algebra/precalculus curriculum might generate some interest in the "higher arithmetic."

• On the Applications of Axial Representation of Trigonometric Functions
  M. Vali Siadat

  M. Vali Siadat is Distinguished Professor and Chair of the Mathematics Department at Richard J. Daley College, Chicago. He received his BSEE from UC, Berkeley, and MSEE from SJSU. Subsequently, he earned his MS, PhD, and DA in mathematics, all from the University of Illinois at Chicago. Dr. Siadat is the 2005 Carnegie Foundation for the Advancement of Teaching Illinois Professor of the Year. E-mail: vsiadat@ccc.edu

  In terms of modern pedagogy, having visual interpretation of trigonometric functions is useful and quite helpful. This paper presents, pictorially, an easy approach to prove all single angle trigonometric identities on the axes. It also discusses the application of axial representation in calculus - finding the derivative of trigonometric functions.

• Double Negatives
  Timothy Mayo

  Tim Mayo teaches developmental mathematics, intermediate algebra, and calculus at Mohave Community College in Lake Havasu City, Arizona. E-mail: TMAYO@imail.mohave.edu

  "Hey, man, you know I didn't do nothing." You mean, my friend, that you did something. There's a double negative in your speech. Your meaning's the opposite of what you preach. When two negatives come together There is a fast change in the weather. Two negatives cannot remain. They'll cause each other grief and pain. You will find double "nos" in Greek, But in your tongue they stink and reek. In math they cannot live in peace. On paper please give them release. And so two negatives must part. I mean this with all my heart. In their place a plus appears. They part forever, no more tears! 1993

• People vs. Collins: Statistics as a Two-Edged Sword
  Jean McGivney-Burelle, Katherine McGivney, and Ray McGivney

  Jean McGivney-Burelle is assistant professor of mathematics at the University of Hartford. She earned her PhD in Curriculum and Instruction from the University of Connecticut in 1999. As director of the secondary education certification program, her interests are in the area of mathematics education and teacher preparation. E-mail: burelle@hartford.edu Katherine McGivney is associate professor at Shippensburg University. In 1997, she received her PhD in mathematics from Lehigh University. Her current interests are in the areas of discrete mathematics and probability. E-mail: gmcgi@ship.edu Ray McGivney is professor of mathematics at the University of Hartford. He earned his AB and MA in mathematics at Clark University and his PhD in mathematics at Lehigh University. He has served as mathematics consultant for several school systems in Connecticut and presented at numerous local, regional and national professional meetings. E-mail: mcgivney@hartford.edu

  Real-life applications of the use (and misuse) of mathematics invariably pique students' interest. This article describes a legal case in California that occurred in the 1960's in which a couple was convicted of robbery, in part, based on the expert testimony of a statistics instructor. On appeal, the judge noted several mathematical errors in this testimony and overturned the conviction. In fact, he observed that at least one of the instructor's arguments actually pointed to the innocence of the accused couple. This article gives the details of the alleged crime itself, the main points of the instructor's testimony, and the judge's corrections. It ends with an interesting mathematical footnote from the judge, the details of which surprisingly involve an application of L'Hˆospital's Rule.

• Packing Infnite Number of Cubes in a Finite Volume Box 
  Haishen Yao and Clara Wajngurt

  Haishen Yao is assistant professor of mathematics at Queensborough Community College/CUNY. He received his PhD from the University of Illinois at Chicago under the guidance of Charles Knessl. His research interests lie in applied mathematics as well as pedagogical research. E-mail: HYao@qcc.cuny.edu Clara Wajngurt is professor of mathematics at Queensborough Community College/ CUNY where she has taught since 1983. She holds a doctorate in Mathematics from City University of New York Graduate Center and she has published several papers on number theory and related topics. She is involved in mentoring new faculty and curriculum development and teaches all levels of mathematics. E-mail: CWajngurt@qcc.cuny.edu Packing an infinite number of cubes into a box of finite volume is the focus of this article. The results and diagrams suggest two ways of packing these cubes.Specifically suppose an infinite number of cubes; the side length of the first one is 1; the side length of the second one is 1/2 , and the side length of the nth one is 1/n. Let n approach infinity so that an infinite number of cubes is obtained. Note that the total volume of these cubes is finite and the purpose is to determine how to pack these infinite cubes into a finite dimensional box.

• Sketching Curves for Normal Distributions--Geometric Connections
  Michael J. Bossé

  Michael J. Boss´e is an associate professor of Mathematics Education at East Carolina University. He received his PhD from the University of Connecticut. His professional interests within the field of mathematics education include elementary and secondary mathematics education, pedagogy, epistemology, learning styles, and the use of technology in the classroom. E-mail: bossem@ecu.edu

  Within statistics instruction, students are often requested to sketch the curve representing a normal distribution with a given mean and standard deviation. Unfortunately, these sketches are often notoriously imprecise. Poor sketches are usually the result of missing mathematical knowledge. This paper considers relationships which exist among graphs of all normal distributions and then extends these ideas to the geometric understanding of the area under the curve.

• Examining Students' Conceptions Using Sum Functions by Kevin Ratliff and Joe Garofalo

  Kevin Ratliff is an associate professor of mathematics at Blue Ridge Community College in Weyers Cave, Va. He is currently pursuing an EdD in Mathematics Education at the University of Virginia. E-mail: ratliffk@brcc.edu Joe Garofalo is Co-Director of the Center for Technology and Teacher Education and coordinator of the mathematics education program area in the Curry School of Education at the University of Virginia. Joe's interests include mathematical problem solving, the use of technology in mathematics teaching, and mathematics teacher education. E-mail: jg2e@cms.mail.virginia.edu

  Students' understanding of functions is a topic that has been researched extensively. In this qualitative study, five university students of varying mathematical backgrounds were interviewed to reveal strategies and misconceptions as they struggled with graphical and analytical tasks relating to sum functions. Weaker students are seen to rely heavily on algebraic approaches to solving problems and to have a strong urge to average graphically. Selection of an appropriate scale is problematic, as is the confusion of slope and height. Understanding functions as objects emerges as beneficial for the stronger students while function as process seems preeminent for the weaker ones. Implications for teaching are presented.

• Lucky Larry #68

• What Does Conceptual Understanding Mean?
  Florence S. Gordon and Sheldon P. Gordon

  Florence S. Gordon is recently retired as professor of mathematics at New York Institute of Technology. She is a co-author of Functioning in the Real World, co-author of Contemporary Statistics: A Computer Approach and co-editor of the MAA Notes volumes, Statistics for the Twenty First Century and A Fresh Start for Collegiate Mathematics: Rethinking the Courses Below Calculus. She has published extensively in mathematics and statistics education. E-mail: fgordon@nyit.edu Sheldon Gordon is Distinguished Teaching Professor at Farmingdale State University of New York. He is a member of a number of national committees involved in undergraduate mathematics education and is leading a national initiative to refocus the courses below calculus. He is the principal author of Functioning in the Real World and a co-author of the texts developed under the Harvard Calculus Consortium. E-mail: gordonsp@farmingdale.edu

  All advocates of curriculum reform talk about an increased emphasis on conceptual understanding in mathematics. In this article, the authors use many examples to address the following issues: What does conceptual understanding mean, especially in introductory courses such as college algebra, precalculus, or calculus? How do we recognize its presence or absence in students? How do we develop a high level of conceptual understanding in students? How do we alter courses introductory courses to make conceptual understanding an important component? How do we assess whether students have actually developed their conceptual understanding? How do we recognize and reward students who display unexpected conceptual insights?

• Book Reviews
  Edited by Sandra DeLozier Coleman

  
  FROM ZERO TO INFINITY: What Makes Numbers Interesting, 50th Anniversary Edition, Constance Reid, A. K. Peters, Ltd., Wellesley, Massachusetts, 2006, ISBN 1-56881-273-6.

• Lucky Larry #69

• Lucky Larry #70

• Mathematics For Learning With Inflammatory Notes for the Mortification of Educologists and the Vindication of "Just Plain Folks,"
  Alain Schremmer

  
  In the Spring 2004 issue of The AMATYC Review, Schremmer introduced his idea for an open-source serialized text: Mathematics For Learning. The Preface to the text appeared in the Spring 2004 issue with a new chapter in each subsequent issue of The AMATYC Review. This issue contains Chapter 5: Multiplication, with sections on "Metric Headings," "Multiplication As Dilation," "Multiplication as Co-multiplication." and "Multiplication as Area of a Rectangle." 

The Problems Section
Edited by Stephen Plett and Robert Stong

New Problems

The AW Problem Set consists of four new problems.

Set AU Solutions

Solutions are given to the four problems from the AU Problem Set that were in the Fall 2005 issue of The AMATYC Review.

• Point of Distinction
  Sandra DeLozier Coleman

  A point in space begins to move
  creating endpoints-clearly two!
  A new dimension is defined
  as point evolves into a line.
  This segment, we shall call an edge,
  and on its motion now will hedge
  the growth of what we call a face,
  as likewise edge a path doth trace.
  But note, the path's particular.
  It must be perpendicular!
  So, long before the face is through,
  of matching edges there are two!
  Two others grow as we progress,
  but two are instantaneous!
  With length that equals width attained
  we change the way we move again,
  and once more, right away, it's clear,
  two matching faces just appear.
  Four more develop over time,
  but two are instantly defined!
  Extending to the hypercube,
  assuming a new way to move,
  the cube which has six matching faces,
  a path analogous now traces,
  where slightest motion yields in full
  two separate cubesidentical!
  These move apart in such a fashion,
  their pathway we can scarce imagine,
  but, by analogy, in time,
  six other cubes will be defined.
  At this point what results we call
  a cube that's four dimensional.
  There's nothing special about four.
  There could be any number more.
  We try within our space to learn
  to see them through the twists and turns
  and slices that don't show the whole,
  but rather how the form unfolds.
  But always it would seem to me
  the thing most difficult to see
  is that small speck of space and time,
  where separateness is first defined!

  June 20, 2003

• Lucky Larry #71

• Lucky Larry #72

Table of Contents

• Lucky Larry #66

• The Radical Axis: A Motion Study

  Ray McGivney and Jim McKim

  	Ray McGivney is Professor of Mathematics at the University of Hartford. He earned his AB and MA in mathematics at Clark University and his PhD in mathematics at Lehigh University. He has served as mathematics consultant for several school systems in Connecticut and presented at numerous local, regional and national professional meetings. E-mail: mcgivney@hartford.edu
  	James McKim, now at Winthrop University, holds a PhD in mathematics from the University of Iowa. He has taught mathematics and computer science for more than 30 years, the last 15 mainly to working professionals. He is the coauthor (with Ray McGivney and Ben Pollina) of two mathematics textbooks and the author of several articles in both computer science and mathematics. E-mail: mckimj@winthrop.edu Interesting problems sometimes have surprising sources. In this paper we take an innocent looking problem from a calculus book and rediscover the radical axis of classical geometry. For intersecting circles the radical axis is the line through the two points of intersection. For nonintersecting, nonconcentric circles, the radical axis still exists. We are interested in the relationship of this line to the two circles in this latter case. We take an algebraic approach to its formula so we can see this relationship as we move and scale the defining circles. This approach culminates in the discovery that if the two circles grow so that their areas increase at equal rates then the radical axis remains constant and in fact is the eventual line of intersection of the two circles.

Graphical Representation of Complex Solutions of the Quadratic Equation in the xy Plane

Todd McDonald

• Extending the Rule of 72 Through Linear Approximations (no abstract available)

• The Harmonic Series Diverges Again and Again

  Steven J. Kifowit and Terra A. Stamps

  Steve Kifowit is an associate professor of mathematics at Prairie State College. He has a BS degree in physics and applied mathematics and an MS degree in computational mathematics, both from Northern Illinois University. E-mail: skifowit@prairiestate.edu Terra Stamps is an associate professor of mathematics and the mathematics department chair at Prairie State College. She holds a bachelor's degree in mathematics from the University of Montevallo and a master's degree in pure mathematics from the University of Alabama. E-mail: tstamps@prairiestate.edu

  The harmonic series is one of the most celebrated infinite series of mathematics. A quick glance at a variety of modern calculus textbooks reveals that there are two very popular proofs of the divergence of the harmonic series. In this article, the authors survey these popular proofs along with many other proofs that are equally simple and insightful. A common thread connecting the proofs is their accessibility to first-year calculus students.

• Universal Paradox

  Sandra DeLozier Coleman

  One gigantic set made of all that there is
  Boggles the mind with paradoxes.
  For it is greater than all, but smaller than this--
  The set which consists of the subsets of it.

  June 1986

• Extending Rules for Exponents and Roots Utilizing Mathematical Connections

  Michael J. Bossé and Stephen Kcenich

  	Stephen Kcenich is currently a lecturer at the University of Maryland in College Park. His main interest is actuarial mathematics and its application to undergraduate curriculum. E-mail: stephenkcenich@yahoo.com
  This paper considers rules for multiplying exponential and radical expressions of different bases and exponents and/or roots. This paper demonstrates the development of mathematical concepts through the application of connections to other mathematical ideas. The developed rules and most of the employed connections are within the realm of secondary and elementary college mathematics.

• Lucky Larry #67

• The Calculus of Elasticity

  Warren B. Gordon

  Warren B. Gordon is professor and chair of the Department of Mathematics at Baruch College, and has recently been interested in integration of technology and applications to the calculus curriculum. He has just completed a text which will appear in the fall 2006, providing an integrated approach to precalculus and applied calculus, including the use of technology. E-mail: wgordon@baruch.cuny.edu

  This paper examines the elasticity of demand, and shows that geometrically, it may be interpreted as the ratio of two simple distances along the tangent line: the distance from the point on the curve to the x-intercept to the distance from the point on the curve to the y-intercept. It also shows that total revenue is maximized at the transition point from elastic to inelastic demand; when elasticity is unitary.

• Building Buildings with Triangular Numbers
  David L. Pagni

  David L. Pagni is a mathematics professor at California State University, Fullerton. His interests range across the spectrum of mathematics education from grades K-16. His teaching fields include mathematics, mathematics learning, teaching, and technology. He is currently principal investigator of a National Science Foundation Mathematics and Science Partnership called Teachers Assisting Students to Excel in Learning Mathematics (TASEL-M). E-mail: dpagni@Exchange.fullerton.edu

  Triangular numbers are used to unravel a new sequence of natural numbers here-to-fore not appearing on the Encyclopedia of Integer Sequences website. Insight is provided on the construction of the sequence using "buildings" as a viewable model of the sequence entries. A step-by-step analysis of the sequence pattern reveals a method for generating the function. Graphing calculator programs are provided for generating the sequence both recursively and explicitly for different initial "building" sizes. Finally, an explicit formula for the sequence that makes use of the "ceiling" function generalizes the results.

• Book Reviews

  FRACTALS, GRAPHICS, AND MATHEMATICS EDUCATION, Michael Frame and Benoit B. Mandelbrot, editors, The Mathematical Association of America., USA, 2002, ISBN 0-88385-169-5.

• Mathematics For Learning With Inammatory Notes for the Mortication of Educologists and the Vindication of "Just Plain Folks"

  Alain Schremmer

  In the Spring 2004 issue of The AMATYC Review, Schremmer introduced his idea for an open-source serialized text: Mathematics For Learning. The Preface to the text appeared in the Spring 2004 issue with a new chapter in each subsequent issue of The AMATYC Review. This issue contains a reorganization of the first three chapters and includes sections on Comparing Collections: Equalities and Inequalities and Specifying Collections: Equations and "Inequations."

• Software Reviews

  Edited by Brian E. Smith

  Reviewed by Tristan Denley, University of Mississippi

  Hawkes Learning Systems Courseware

  Producer and Distributor: Hawkes Learning Systems
  Address: 1023 Wappoo Road Suite 6-A, Charleston, SC 29407
  Web addresses: www.hawkeslearning.com
  Pricing Information:
  Basic Mathematics Textbook and Software Bundle $72.00
  Prealgebra Textbook and Software Bundle $72.00
  Introductory Algebra Textbook and Software Bundle $65.00
  Intermediate Algebra Textbook and Software Bundle $65.00
  College Algebra Textbook and Software Bundle $72.00
  Statistics Textbook and Software Bundle $70.00

• The Problems Section (no abstract available)

Table of Contents

• Lucky Larry #62

• Solving Triangles

  Joscelyn A. Jarrett

  Joscelyn A. Jarrett is a professor of mathematics at Gordon College in Barnesville, Georgia, He received an MS in mathematics from the University of Toronto and a PhD in mathematics education from the University of Iowa. E-mail: j_jarrett@gdn.edu

  This article discusses the four categories of triangles that are standard in most textbooks when "solving" triangles: (a) Given the lengths of two sides and the measure of an angle opposite one of the two given sides, (b) Given the lengths of two sides and the measure of the included angle, (c) Given the lengths of all three sides, d) Given the lengths of one side and the measure of two angles. It then introduces two new categories of solving triangles: (e) Given the measures of two angles and the perimeter of the triangle, and (f) Given the measures of two angles and the area of the triangle. These two new categories require the use of two non-standard theorems which are stated and proved in the article. One of the two theorems is an extension of the Law of Sines to include the perimeter. The other provides a relationship among the area, angles, and the perimeter of a triangle. Furthermore, the article gives four applications of the use of these theorems in solving problems of the two new categories introduced.

• Nurturing Mathematical Talent through Student Research

  Lisa Evered and Sofya Nayer

  Lisa Evered is a professor of mathematics at Iona College in New Rochelle, New York. She has taught mathematics at every level from elementary to graduate school and is the author of articles about education of the mathematically talented. E-mail: levered@iona.edu Sofya Nayer is an associate professor at Borough of Manhattan Community College in the City University of New York. She has a degree in engineering and a doctorate degree from Teachers College, Columbia University E-mail: yakovn@hotmail.com The tedium that characterizes many routine calculus activities necessary for average students often results in the loss of the most talented to the field of mathematics. One way to overburden teacher to nurture mathematical talent within a typical calculus class is to encourage student research. This article illustrates how student research facilitated the learning and stimulated the interest of two urban community college students. The contrast between the creative studentÕs thought and more pedestrian approaches is useful In understanding the nature of mathematical talent.

• Cramer's Rule Revisited

  Ayoub B. Ayoub

  Dr. Ayoub is a professor of mathematics at Abington College of the Pennsylvania State University. His interests include number theory, collegiate mathematics and mathematics history. He received The Pennsylvania State University's 1990 George Atherton Award for Excellence in Teaching. E-mail: aba2@psu.edu

  In 1750, the Swiss mathematician Gabriel Cramer published a well-written algebra book entitled Introduction á l'Analyse des Lignes Courbes Algèbriques. In the appendix to this book, Cramer gave, without proof, the rule named after him for solving a linear system of equations using determinants (Kosinki, 2001). Since then several derivations of this rule have appeared (Chadha, 1996; Larson et al., 2004; Shi rm & Adams, 2002). In this article, the author will introduce another proof of Cramer's rule based on the expansion in cofactors of a determinant.

• The Battle of the Zero Divisors
  Patricia Hale and Charles Hale

  Patricia Hale is an associate professor of mathematics starting her sixth year of teaching at California State Polytechnic University, Pomona. She teaches mathematics courses primarily for elementary and secondary preservice teachers. Her interests include mathematics education, group theory, and women in mathematics. E-mail: phale@csupomona.edu Charles Hale is a lecturer in the mathematics department at California State Polytechnic University, Pomona. His interests are mathematics education, mathematics history and non-Euclidian geometries. Additionally, he likes to water-ski, camp, hike and eat, not necessarily in that order. Lastly, he is the proud father of Jessica, who has been accepted to Stanford's graduate program. E-mail: crhale@csupomona.edu

  The mathematical reasons that we cannot divide by zero are not easy for most students to understand; in fact, even those students who have more than just a basic understanding of algebraic concepts still have difficulty. This is most problematic for college students who are prospective teachers since they need to develop a deep understanding of division because both national and state standards usually require these topics be taught to elementary school children. Unfortunately, many of our prospective teachers are never given the opportunity to develop a deep understanding of the reasons why division by zero is undefined; what indeterminate means in this setting; and what is meant when we say a solution does not exist because it would approach infinity. In this paper we give examples of prospective teacher's understanding of division by zero and a particular model that has helped our college students come to an understanding of why division by zero is undefined.

   

• Identifying Students' Reasons for Selecting a
  Computer-mediated or Lecture Class
  D. Patrick Kinney and Douglas F. Robertson

  Pat Kinney is a mathematics instructor at the Wisconsin Indianhead Technical College in New Richmond, Wisconsin. He has a PhD in mathematics education from the University of Minnesota. Previously, he taught mathematics at the General College of the University of Minnesota. E-mail: pkinney@witc.edu

  Douglas F. Robertson is a professor in the General College, the developmental education unit of the University of Minnesota, where he has taught mathematics and computing since 1974. E-mail: droberts@umn.edu

  Students in this study were enrolled in either an Introductory Algebra or Intermediate Algebra class taught through computer-mediated instruction or lecture. In the rst year of the study, students were asked what they believed helped them learn mathematics in the instructional format in which they were enrolled. They were also asked what they would nd di cult about learning mathematics in the instructional format in which they were not enrolled. Based on studentsÕ written responses, a set of 12 survey items was developed. The items were administered at the end of the fall semester the following year. There was a signi cant di erence at the p = 0.05 level on nine of the twelve items when comparing results from computer-mediated and lecture students who were consistent in their preference for either computer-mediated or lecture instruction. Students who selected computer-mediated instruction indicated that the software should provide step-by-step instructions and allow the students to control the pace and to navigate backwards to review. They also viewed the software as a more visual way of learning than a teacher lecturing and indicated that software holds their attention better than a teacher lecturing.

• Lucky Larry #63

• Regarding Basic College Mathematics, A
  Subversive Comment and a One Act Play

  Ben Hill

  Ben Hill is a mathematics instructor at Lane Community College in Eugene, Oregon. He has taught mathematics, statistics, or cultural anthropology at Oregon Coast Community College, the University of Maryland Asian Division, and the University of North Dakota. He holds an MS in mathematics and a PhD in curriculum and instruction from the University of Oregon. His research interests are in cultural aspects of education. E-mail: hillb@lanecc.edu

  Among math educators, it is a truism that basic college math skills are needed in every career eld. But actually this is a false professional myth. Hardly anyone makes direct use of mathematics beyond arithmetic in the course of everyday life. Moreover, math's status as an almost universally required college subject is not inevitable. Algebra and calculus could conceivably be relegated to specialists in the same way that Latin has been. This is not a research paper, but an opinion piece intended to provoke re ection and discourse. In a personal and sometimes humorous tone, the author argues that basic college math in fact deserves its position in the college curriculum, though for reasons other than those commonly asserted. By way of conclusion, ideas from the essay are explored in a ctional vignette between one teacher and one student, presented in the form of a one act play.

• Lucky Larry #64

Classroom Issues with Series Tests

Jawad Sadek and Russell Euler

• AMATYC Members Offer Their Perceptions of Interactions That Occur in Developmental Mathematics Courses
  Nancy J. Sattler

  Sattler presently teaches on-line and has taught mathematics for over 20 years at Terra State Community College in Fremont, Ohio. She chairs the AMATYC Distance Learning Committee and is past-president of OhioMATYC, presently serving as its historian, newsletter editor, and webmaster. E-mail: nsattler@terra.edu

  This study investigated teacher perception of interactions used in on-line developmental mathematics courses at two-year colleges. A total of 98 AMATYC teachers were surveyed. The following conclusions were inferred from the study's findings: (a) The teacher responding to the survey was apt to be a female between the ages of 50 and 59, had taught an average of 14 years, and had an average class size of 24 students; (b) The course taught was more likely to be a beginning or intermediate algebra class and was offered asynchronously; (c) The course management system used most often was more likely to be Blackboard, followed by textbook publishers systems, and WebCT; (d) The textbook company used was more likely to be either Prentice Hall, Addison Wesley, or Academic Systems; (e) The mean perception of Student-to-Teacher interactions varied depending on the type of interaction. There was no difference in the mean perception between the different course management systems Blackboard, WebCT, and textbook publishers. However, there was a significant difference in the mean perception between different in-person meetings; teachers had the greatest success with tests, final exams, and orientation on-line meetings; teachers had the greatest success with tests, final exams, and quizzes. tools used; teachers had the greatest success with electronic mail, audio-conferencing, and interactive video; (f) There was no significant difference in the mean perception of the different student-to-student interactions however e-mails were used the most frequently according to the on-line developmental mathematics teacher; and (g) There was a significant difference in the mean perception of the student-to-content interactions with textbooks being used the most often but instructor made video and course packet/handouts having the higher mean perceptions. General conclusions informing the field of study include: (a) on-line teachers continue to use traditional methods of assessment; (b) teachers desire some type of interaction with their students; and (c) distance classes are a blending of the old with the new.

• Book Reviews
  Edited by Sandra DeLozier Coleman

  THE MYSTERY of THE ALEPH, Mathematics, the Kabbalah, and the Search for In nity, Amir D. Aczel, Washington Square Press publication of POCKET BOOKS, a division of Simon and Schuster, Inc., New York, 2000, ISBN 0-7434-2299-6.

   

• Mathematics For Learning With Inflammatory Notes for the Mortification of Educologists and the Vindication of "Just Plain Folks"
  Alain Schremmer

  In the Spring 2004 issue of The AMATYC Review, Schremmer introduced his idea for an open-source serialized text: Mathematics For Learning. The Preface to the text appeared in the Spring 2004 issue with a new chapter in each subsequent issue of The AMATYC Review. This issue contains Chapter 3. There are six sections in this chapter: "States," "Transactions," "Usual Representations: Signed-Number-Phrases versus T-accounts," "Adding Signed-Number-Phrases," "Subtracting Signed-Number- Phrases," and "Effect of Transactions on States."

• Software Reviews

  Reviewed by Marion S. Foster, Tomball College

  Edited by Brian E. Smith

  MyMathLab/CourseCompass Course Management System

  Producer and Distributor: Pearson Education Math and Statistics
  Web addresses: www.coursecompass.com and www.mymathlab.com

  System Requirements: Speci c system requirements vary depending on your course. Most MyMathLab courses require a Windows® operating system and a supported version of Microsoft® Internet Explorer or Netscape®. However, courses for calculus and statistics also run on certain Macintosh® operating systems with supported versions of Netscape.

  Internet connection: Cable/DSL, T1, or other high-speed for multimedia content; 56k modem (minimum) for tutorials, homework, and testing. Price: No charge to instructor for course creation. Students have to obtain a student access code. This may be provided with text adoption or purchased online. Contact Pearson Education for details.

• The Problems Section (no abstract available)

• Lucky Larry #65

Editor: Barbara S. Rives, Lamar State College
Production Manager: John C. Peterson

Table of Contents

• Unexpected Constructible Numbers

  Thomas J. Osler

  Tom Osler, professor of mathematics at Rowan University, is the author of 58 mathematical papers. In addition to teaching university mathematics for 43 years, Tom has been competing in long distance running races for the past 50 consecutive years. He is the author of two books on running.
  E-mail: Osler@rowan.edu

• Teaching about Inverse Functions
  Warren Esty

  Warren Esty is a professor of mathematics in the Department of Mathematical Sciences at Montana State University in Bozeman, Montana. He has published extensively in probability theory, statistics, and mathematics education. He has written two books, The Language of Mathematics and Precalculus. In his spare time he studies ancient Rome and Greece. E-mail: westy@math.montana.edu

  In their sections on inverses most precalculus texts emphasize an algorithm for finding f -1 given f. However, inspection of precalculus and calculus texts shows that students will never again use the algorithm, which suggests the textbook emphasis may be misplaced. Inverses appear primarily when equations need to be solved, which suggests instruction about inverses should emphasize their use in solving the equation "f(x) = c." Instruction, and the algorithm used, should take advantage of the possibility of perfectly paralleling the process for solving "f(x) = y" for x (not solving "f(y) = x" for y). Switching letters after solving, rather than before solving, preserves the parallel. When f is not one-to-one (such as f(x) = x2 or f(x) = sin x), students frequently fail to find the second solution. By discussing inverses in terms of solutions to "f(x) = c," this difficulty is naturally addressed. Furthermore, the terms one-to-one and range have natural definitions in this context and the Horizontal Line Test is also natural. The algorithm for finding inverses and the associated terminology can best aid in proper conceptual development if they focus on the primary context --solving "f(x) = c" for x.

• Some Unusual Expressions For the Inradius of a Triangle
  Tom Osler & Tirupathi R Chandrupatla

  Tom Osler, professor of mathematics at Rowan University, is the author of 58 mathematical papers. In addition to teaching university mathematics for 43 years, Tom has been competing in long distance running races for the past 50 consecutive years. He is the author of two books on running. E-mail: Osler@rowan.edu Tirupathi R. Chandrupatla is professor of mechanical engineering at Rowan University. His areas of interest include finite-element analysis, design, optimization, and manufacturing engineering. He is author of the books Introduction to Finite Elements in Engineering and Optimization Concepts and Applications in Engineering, both published by Prentice Hall and used in universities throughout the world. He recently won a prestigious award for his book of Indian poetry. E-mail: chandrupatla@rowan.edu

  Several formulae for the inradius of various types of triangles are derived. Properties of the inradius and trigonometric functions of the angles of Pythagorean and Heronian triangles are also presented. The entire presentation is elementary and suitable for classes in geometry, precalculus mathematics and number theory.

• Lucky Larry #59

• Beyond Pascal's Triangle
  Darrell Minor

  Darrell Minor is a professor of mathematics at Columbus State Community College. His academic interests include number theory, game theory, the history of mathematics, and helping students discover the beauty of mathematics. Away from work, he enjoys bicycling, reading, playing softball, and spending time with his wife and three sons. E-mail: dminor@cscc.edu

  In "Beyond Pascals Triangle" the author demonstrates ways of using "Pascallike" triangles to expand polynomials raised to powers in a fairly quick and easy fashion. The recursive method could easily be implemented within a spreadsheet, or simply by using paper and pencil. An explanation of why the method works follows the several examples that are provided.

• The Power of Power Functions
  Florence S Gordon

  Florence S. Gordon is Professor of Mathematics at New York Institute of Technology. She is a co-author of Functioning in the Real World, co-author of Contemporary Statistics: A Computer Approach and co-editor of the MAA Notes volumes, Statistics for the Twenty First Century and A Fresh Start for Collegiate Mathematics: Rethinking the Courses Below Calculus. She has published extensively in mathematics and statistics education. E-mail: fgordon@nyit.edu

  Traditional college algebra courses focus almost exclusively on power functions such as y = x2 and y = x3 rather than the more general y = xp. However, it is the more general form that is the basis of the mathematical models that arise throughout the natural sciences in a host of unexpected and highly interesting, applications. This article demonstrates a variety of applications drawn from biology and other areas that lead to power functions and some of the kinds of questions that can truly motivate students to find value in the mathematics they are learning. For instance: Why can't a turkey fly? How much did a pterodactyl weigh? How many different species can the island of Puerto Rico support? How large does an island have to be to have 100 species inhabit it? How does the size of an organism relate to how fast it can run? Or swim? Or fly?

• The Importance of Introductory Statistics Students Understanding Appropriate Sampling Techniques
  Violeta C. Menil

  Violeta C. Menil is an assistant professor at the Mathematics Department of Hostos Community College of the City University of New York, (CUNY). She obtained her PhD in Mathematics and Statistics Education from New York University. Aside from her extensive academic experience, she worked as the audit statistician of the New York City Comptroller's Office for three years. Among her research interests are sampling, multidimensional scaling and univariate and multivariate data analysis. E-mail: menilv2@aol.com

  In this paper the author discusses the meaning of sampling, the reasons for sampling, the Central Limit Theorem, and the different techniques of sampling. Practical and relevant examples are given to make the appropriate sampling techniques understandable to students of Introductory Statistics courses. With a thorough knowledge of sampling techniques, students are equipped with the necessary tools to undertake basic research.

• Excellence through Mathematics Communication and Collaboration (E=mc^2): A new approach to Quality in College Algebra Gerald Marshall & Herbert Riedel

  Gerald Marshall, department head of mathematics, Tri-County Technical College, received his PhD from Illinois State University in 2000. Earlier degrees were from North Carolina State University, Florida State University, and University of Alabama in Huntsville. He has had articles published in the Mathematics Teacher, The Mathematics Educator, and the Journal of Research and Development in Education. E-mail: gmarshal@tctc.edu

  Herbert Riedel obtained a PhD in pure mathematics from the University of Waterloo, Canada. In July 2004, he left Tri-County Technical College and became Deputy Director of Nanoscience Technology at the University of Central Florida to assist with the establishment and administration of a new research center in the emerging interdisciplinary field of nanotechnology. He will continue his involvement in undergraduate education and pursue partnerships with two-year colleges designed to enhance educational opportunities and position students to benefit from careers in the new economy. E-mail: hriedel@mail.ucf.edu

  Within the context of traditional quality management principles, a program designed to enhance soft skills of College Algebra students was piloted in fall 2001. Results of the pilot project include increased retention, persistence, note-taking ability, and positive responses concerning course expectations. Quality enhancement efforts in this course have manifested significant improvements in this course and in follow-up courses, but overall success rates remain low due to a misalignment of process capability with standards. Possible solutions are presented.

• Elementary Algebraic Models in Our World: A General Education Alternative to College Algebra
  Robert Franoza & Jennifer Tyne

  Robert Franzosa is a professor of mathematics in the Department of Mathematics and Statistics at the University of Maine. He has a PhD in mathematics from the University ofWisconsin. His interests include applied topology and mathematics education. E-mail: franzosa@math.umaine.edu Jennifer Tyne is a lecturer in Department of Mathematics and Statistics at the University of Maine. She has a MS in operations research from the University of North Carolina. Her interests include mathematics education and curriculum development. E-mail: tyne@math.umaine.edu

  Elementary Algebraic Models in Our World (MAT 103) is a general education course at the University of Maine that was developed as an alternative to College Algebra. An important goal in the development of MAT 103 was the improvement of the studentsÕ attitudes about and understanding of simple algebraic models. MAT 103 was developed in conjunction with a new Masters in Science Teaching (MST) degree program at the University of Maine to provide a research laboratory for MST graduate students. In this paper we present an overview of the MAT 103 course development project, including a discussion of the background motivation, the course teaching framework, the course content framework, the course materials, and the initial evaluation of the course based on student surveys and evaluations. Appendices are included presenting a sample of class materials and a summary of the survey data.

• Lucky Larry #60

• Book Reviews
  Reviewed by Roxane Barrows
  Edited by Sandra DeLozier Coleman

  LION HUNTING & OTHER MATHEMATICAL PURSUITS, a collection of mathematics, verse, and stories by Ralph P. Boas, Jr., edited by Gerald L. Alexanderson & Dale H. Mugler, The Mathematical Association of America, Dolciani Mathematical Expositions, No. 15, United States of America, 1995, ISBN 0-88385-323-X.

• Mathematics For Learning
  Chapter 2: Accounting for Money On the Counter (II)
  Alan Schremmer

  In the Spring 2004 issue of The AMATYC Review, Schremmer introduced his idea for an open-source serialized text: Mathematics For Learning. The Preface to the text appeared in the Spring 2004 issue; Chapter 1 was in the Fall 2004 journal. This issue contains Chapter 2. There are four sections in this chapter: "(Decimal) Headings," "Adding Under A Heading," "Subtracting Under A Heading," and "(Decimal) Number-Phrase."

• Lucky Larry #61

• Software Reviews

  Reviewed by Marion S. Foster, Tomball College
  Edited by Brian E. Smith

  MyMathLab/CourseCompass Course Management System
  Producer and Distributor: Pearson Education Math and Statistics
  Web addresses: www.coursecompass.com and www.mymathlab.com
  System Requirements: Specific system requirements vary depending on your course. Most MyMathLab courses require a Windows^® operating system and a supported version of Microsoft^® Internet Explorer or Netscape^®. However, courses for calculus and statistics also run on certain Macintosh^® operating systems with supported versions of Netscape.
  Internet connection: Cable/DSL, T1, or other high-speed for multimedia content; 56k modem (minimum) for tutorials, homework, and testing. Price: No charge to instructor for course creation. Students have to obtain a student access code. This may be provided with text adoption or purchased online. Contact Pearson Education for details.

Table of Contents

• Density of Primitive Pythagorean Triples
  Duncan A. Killen

  Duncan Killen received BA and MD degrees from Vanderbilt University in the 1950's. Since retirement he has been a part time student at the Johnson County Community College, Overland Park, Kansas. He has a particular interest in mathematics. E-mail: dokbrock1@hotmail.com

  Based on the properties of a Primitive Pythagorean Triple (PPT), a computer program was written to generate, print, and count all PPTs  is an arbitrarily chosen integer. The Density of Primitive Pythagorean Triples may be defined as the ratio of the number of PPTs whose hypotenuse is less than or equal to . The PPT Density for all PPTs with a primitive hypotenuse less than or equal to , remains rather stable, even as  is increased from 5 to 1,000,000. Using a TI-83 calculator, a linear regression correlation between the number of PPTs and the value of Ix, using 36 data points distributed between  = 1,000 and  = 1,000,000 was determined and the results are as shown:

• Equivalent Vectors
  Robert Levine

  Robert Levine lived in New York City for 30 years before moving to Tucson, Arizona in 1976. He's always loved math and science, even though he never passed calculus in his youth. He went back to college at age 51 and took the four semesters of calculus. He first became acquainted with the cross-product in Calculus III which led to his discovery. E-mail: sun5down@earthlink.net

  The cross-product is a mathematical operation that is performed between two 3-dimentional vectors. The result is a vector that is orthogonal or perpendicular to both of them. Learning about this for the first time while taking Calculus-III, the class was taught that if A×B = A×C, it does not necessarily follow that B = C. This seemed baffling. The author reasoned that if this were true, there should be a way to alter the B vector in such a way that the result of the cross-product is still unchanged, but was told that this was impossible. When the course ended and there was time to think about it again, the author went to work trying to solve the impossible, and quickly succeeded. At the same time, an interesting fact was discovered about the cross-product that allowed for success. The proof was not so quick and easy though, but eventually it was accomplished as well. The proof involves an interesting twist where I present the finale, although still unproven, along with several related equations. The flow of proven equations then skips over that unproven group and eventually proves one of the equations in the group, which in turn proves the entire group.

• Student Engagement in a Quantitative Literacy Course
  William L. Briggs, Nora Sullivan, and Mitchell M. Handelsman

  William Briggs has been on the mathematics faculty at the University of Colorado at Denver for 20 years. He received his MS and PhD in applied mathematics from Harvard University. His research is in mathematical problems that arise in biology and medicine. He is a University of Colorado President's Teaching Scholar and the recipient of a Fulbright Fellowship to Ireland. E-mail: wbriggs@math.cudenver.edu Nora Sullivan received degrees from Amherst College (BA 1996) and the University of Colorado at Denver (MA 2001). She was an All- American rugby player in college. She currently works as a therapist at an Adolescent Day Treatment in Denver and is working toward her licensure. E-mail: nora sullivan@yahoo.com Mitchell M. Handelsman holds degrees from Haverford College and the University of Kansas. He is currently professor of psychology and a CU President's Teaching Scholar at the University of Colorado at Denver. In 1992 he was the Colorado Professor of the Year, named by the Council for Advancement and Support of Education. E-mail: mitchell.handelsman@cudenver.edu

  The purpose of this paper is to describe the rationale, design, objectives, and methods underlying a liberal arts mathematics course that has been taught at the University of Colorado at Denver since 1992. The course is well aligned with recent recommendations for introducing quantitative literacy into the undergraduate curriculum. Surveys administered at the beginning and end of the course revealed that student engagement takes many different forms and is related to student performance. This study provides practical insights about effective strategies for teaching such a course.

• Lucky Larry #56

• Mathematics: Assessment & Integration of Success Skills
  Roxane Barrows, and Bernita Crawford

  Roxane Barrows has 15 years of experience in the field of education. She earned a Bachelor's Degree in information systems from The Ohio State University and a Master's Degree in mathematics from Ohio University. She is currently working on a PhD in higher education at Ohio University. She has been employed at Hocking College for 12 years, first as a professor of mathematics and then as the mathematics coordinator. She currently is an Associate Dean of the School of Arts and Sciences and an adjunct mathematics professor. E-mail: barrows_r@hocking.edu Bernita Crawford has 33 years of experience in the field of education. She earned a Bachelor's Degree in education from The Ohio State University and a Master's Degree in higher education from Ohio University. She taught high school sciences for twenty-four years including physics, chemistry, biology, and general science. She has been employed at Hocking College in Nelsonville, Ohio, since 1991, first as an adjunct and then as an assistant professor in the School of Health and Nursing. Three years ago, she became the Coordinator for the Assessment of Student Academic Achievement. Among her other responsibilities are membership in the Success Skills Learning Community and Curriculum Council. She is also an active member in the Ohio Two-Year College Assessment Network. E-mail: crawford_b@hocking.edu

  Hocking College, like many institutions of higher learning, has struggled to define, document, and assess those general skills deemed necessary for success in the workplace and life. The mathematics faculty have spent many years developing appropriate tools for assessment of student academic achievement. Although the process has taken several years, it has evolved into an ongoing method utilized by faculty to improve instruction and learning. Not only are math faculty assessing student academic achievement, but they are also integrating "Success Skills" into their mathematics classes. Two Success Skills, "Communicates Effectively" and "Maintains Professional Skills and Attitudes," have been integrated into the mathematics curriculum and strategies for assessment of them have been started. The mathematics faculty also developed a test to address the Success Skill "Demonstrates Mathematics." The information from this test is shared with technology/program coordinators. Future steps involve integrating all eight Success Skills into the mathematics classes.

• Insights into the Area Model When Connecting Multiplication with Whole Numbers to Decimal Numbers
  Connie Yarema, and Carol Williams

  Connie Yarema is associate professor of mathematics at Abilene Christian University. She works with pre-service mathematics teachers as well as classroom teachers involved in Texas Teacher Quality grants. Her research interests include lesson studies designed in cooperation with classroom teachers. E-mail: connie.yarema@math.acu.edu Carol Williams is professor of mathematics at Abilene Christian University and Dean of the Graduate School. Her interests include encouraging high school girls and college women to persist in mathematics. She has been the recipient of three grants from the Mathematical Association of America in this area. E-mail: carol.williams@math.acu.edu

  This article describes a valuable lesson that university mathematics faculty members learned from fifth grade and middle school teachers in a professional development workshop. The goal of the workshop was to show how models for whole number operations could be linked to models for rational numbers and to connect the traditional algorithms to the models. While working problems, most of the teachers modeled multiplication of whole numbers, fractions, and decimals using the newly taught area model. Several presented an area model for decimals in an unanticipated way that led to an incorrect answer. The lesson learned was that to attain correct products from models, consistency in setting up the factors in an area model for multiplication is needed as the factors change from whole numbers to fractions and decimals. In the case of using base-10 blocks to represent the factors, the name of the block must be the same as the upper surface area of the block so that correct answers can be interpreted when modeling multiplication of whole numbers and decimals.

• A Mathematics Teacher's Transition toward Inquiry-Based Discourse in a Course for Prospective Elementary Teachers
  Lisa Clement

  Lisa Clement is an assistant professor of mathematics education at San Diego State University. She co-directs a Master of Arts program in Education with a concentration in K8 Mathematics Education, and trains mathematics tutors of seventh grade students in the Sweetwater Union High School District. E-mail: Lclement@mail.sdsu.edu

  Using Kazemi's and Stipek's (2001) framework of classroom practice, the discourse between students enrolled in a mathematics-for-teachers course and their instructor is examined. The teacher's practice is in transition from a focus on having students share multiple strategies toward a practice that additionally includes the mathematical justifications for those strategies, and pressing students to explore their errors. Field observations, classroom transcripts, teacher interviews, and student interviews were analyzed and triangulated for this study.

• Cartoon by Matt McClure

• Book Reviews
  Edited by Sandra DeLozier Coleman

• THE GOLDEN RATIO: THE STORY OF PHI, the World's Most Astonishing Number, Mario Livio, Broadway Books, New York, 2002, ISBN 0-7679- 0816-3.

• Mathematics For Learning With Inflammatory Notes For The Education Of Educologists Chapter 1: Counting With Number-Phrases
  Alain Schremmer

  In the Spring 2004 issue of The AMATYC Review, Schremmer introduced his idea for an open-source serialized text: Mathematics For Learning. The Preface to the text appeared in the Spring issue. This issue contains the beginning of Part 1, "Arithmetic: Numbers specified directly," and contains Chapter 1: Counting With Number-Phrases. There are two sections to this chapter: "Accounting for Money" and "Addition Leads to Large Collections."

• The Problems Section
  edited by Stephen Plett & Robert Stong

  New Problems: The AS Problem Set consists of four new problems.

  Set AQ Solutions: Solutions are given to the four problems from the AQ Problem Set that were in the Fall 2003 issue of The AMATYC Review.

• Cartoon #1 by Kenneth Kaminsky

• Cartoon #2 by Kenneth Kaminsky

Editor: Barbara S. Rives, Lamar State College
Production Manager: John C. Peterson

Table of Contents

• Problems on Divisibility of Binomial Coefficients
  Thomas J. Osler, and James Smoak

  Tom Osler, professor of mathematics at Rowan University, is the author of 58 mathematical papers. In addition to teaching university mathematics for 43 years, Tom has been competing in long distance running races for the past 50 consecutive years. He is the author of two books on running. E-mail: Osler@rowan.edu Jim Smoak is a retired mathematician with an insatiable interest in number patterns. Jim served as a ballistic meteorologist in Viet Nam from 196870, receiving a bronze star for his efforts. He maintains an active correspondence with some of America's leading mathematicians, including George Andrews, the world famous number theorist from Penn State University. E-mail: jsmoak@worldnet.att.net

  Twelve unusual problems involving divisibility of the binomial coefficients are represented in this article. The problems are listed in "The Problems" section. All twelve problems have short solutions which are listed in "The Solutions" section. These problems could be assigned to students in any course in which the binomial theorem and Pascal's triangle are presented. This includes courses in precalculus mathematics, real analysis, and number theory.

• The Centers of Similarity of Two Non-Congruent Squares
  Ayoub B. Ayoub

  Ayoub is a professor of mathematics at Abington College of the Pennsylvania State University. He received his Ph.D. in Mathematics from Temple University, Philadelphia. Ayoub's areas of interest are number theory, classical mathematics, and undergraduate mathematics education. E-mail: aba2@psu.edu

• Generating Nice Linear Systems for Matrix Gaussian Elimination
  L. James Homewood

  L. James (Jim) Homewood is a member of the fulltime mathematics faculty at the Downtown Campus of Pima Community College in Tucson, Arizona. He earned a master's degree in mathematics at Portland State University, with two additional years as a graduate associate in the doctoral program in mathematics at the University of Arizona. His major interests are analysis and functional analysis. E-mail:jhomewood@pima.edu

  In this article an augmented matrix that represents a system of linear equations is called nice if a sequence of elementary row operations that reduces the matrix to row-echelon form, through matrix Gaussian elimination, does so by restricting all entries to integers in every step. Many instructors wish to use the example of matrix Gaussian elimination to introduce their students to algorithms that are capable of handling very large linear systems. Instructors should be able to generate, if they desire, a wide variety of modestly sized nice matrices from which they may choose introductory examples and select exam questions. The formulas for generating nice 2 × 3, 3 × 4, and 4 × 5 augmented matrices are shown in this article, with emphasis on the derivation of the matrix. An instructor may use any of these formulas to generate an augmented matrix representing a "one-solution," a dependent or an inconsistent system. The author has developed TI-83 and TI-86 programs that generate nice augmented matrices.

• Finding Equations of Tangents to Conics
  George Baloglou, and Michel Helfgott

  George Baloglou is an associate professor of mathematics at SUNY Oswego, where he has been teaching since 1988. He is currently working on a book on planar crystallographic groups (wallpaper patterns), largely influenced by a symmetry course he has been teaching since 1995. His other mathematical interests include elementary inequalities, convexity, and basic number theory. E-mail: Baloglou@oswego.edu Michel Helfgott is an assistant professor of mathematics at SUNY Oswego. His main interests are the history of mathematics and its use in teaching, as well as the use of physics and chemistry as pedagogical tools in the mathematics classroom. E-mail: Helfgott@oswego.edu

  A calculus-free approach is offered for determining the equation of lines tangent to conics. Four types of problems are discussed: line tangent to a conic at a given point, line tangent to a conic passing through a given point outside the conic, line of a given slope tangent to a conic, and line tangent to two conics simultaneously; in each case, a comparison to the standard calculus method is made by way of specific examples. Extending an idea of Descartes, this calculus-free approach is based on the fact that a quadratic has a double root if and only if its discriminant is equal to zero. It should be appropriate for both precalculus and calculus students.

• Developing Simultaneous Linear Equations and Rational Equations
  Michael J. Boss´e, and N. R. Nandakumar

  Michael J. Boss´e is an Associate Professor of Mathematics Education at Morgan State University. He received his PhD from the University of Connecticut. His professional interests within the field of mathematics education include elementary and secondary mathematics education, pedagogy, epistemology, learning styles, and the use of technology in the classroom. E-mail: mbosse@moac.morgan.edu N.R. Nandakumar is a Professor of Mathematics at Delaware State University. He received his PhD in Mathematics and a Master's in Computer Science from the University of Illinois. His research interests include functional analysis, numerical analysis, and computer science. Email: nnandaku@desu.edu

  To demonstrate concepts or rapidly create quizzes, teachers commonly encounter the need to quickly create mathematical examples. Unfortunately, by producing undesirable or overly complex solutions, extemporaneously created examples can become problematic, create tense learning environments and become more confusing than they are worth. Experience reminds teachers that a moment of planning may avoid many difficult classroom scenarios. Solid mathematical understanding of a few techniques can greatly assist teachers to quickly develop appropriate examples with desired results. This paper considers techniques which will assist instructors in quickly developing appropriate examples with "nice" solutions when teaching rational equations and simultaneous linear equations.

• Lucky Larry #54

• On the Integration of Technology into the Elementary Calculus I Curriculum
  Warren B. Gordon

  Warren Gordon is Chair of the Baruch College mathematics department and has been interested, over the last ten years, in the integration of technology into the mathematics curriculum. This paper reflects the approach taken in his basic calculus course. E-mail: warren gordon@baruch.cuny.edu

  This paper suggests examples that may be used to better integrate modern technology into the calculus I curriculum, and at the same time extend the student's understanding of the underlying concepts. Examples are chosen from the usual topics considered in most courses and not limited to any specific form of the technology.

• Rotation of Axes and the Mean Value Theorem
  David Price

  David Price earned his BS and MS degrees from Southwestern University and the University of North Texas respectively. He teaches mathematics at Tarrant County College in Arlington, Texas. E-mail: david.price@tccd.edu

  This article provides a proof of the Mean Value Theorem by rotating a coordinate system through a specified angle. The use of this approach makes it easy to visualize why the Mean Value Theorem is true. An instructor can use the proof as another illustration of the rotation of axis technique in addition to the standard one of simplifying equations of conic sections.

• Examining Prospective Teachers' Understanding of Proportional Reasoning
  Richard Kitchen and Julie DePree

  Richard Kitchen is an assistant professor in the College of Education at the University of New Mexico in Albuquerque, New Mexico. His primary interests are problem solving and alternative assessment formats. E-mail: kitchen@unm.edu Julie DePree is an associate professor in the Department of Mathematics and Statistics at the University of New Mexico- Valencia Campus in Los Lunas, NM. She teaches many of the mathematics courses for teachers and also teaches statistics and algebra courses. E-mail: jdepree@unm.edu

  In this article, the authors describe their efforts to assess prospective K-8 teachers' knowledge of proportional reasoning. Based upon their analysis of prospective K-8 teachers' work on a mathematics performance task, they discuss the implications for preparing prospective teachers to teach proportional reasoning to their students. In general, the prospective teachers used good estimation strategies and were capable of engaging in proportional reasoning, but many had misconceptions about fractions, decimals, and proportions. In particular, the prospective teachers had difficulties converting between decimals and fractions. Because of errors made when working with fractions and decimals, most failed to correctly solve the task. The prospective teachers' errors and misconceptions highlighted the difficulties they had making sense of fractions and decimals. The lack of facility when converting between a decimal and a fraction illustrated that the prospective teachers need multiple experiences making sense of decimals and fractions, and converting between them. To address prospective teachers' deficiencies in making sense of fractions and decimals, converting between them, and working with proportions, the authors are placing more emphasis on these concepts in their mathematics content and methods courses for prospective K8 teachers.

• The Open Box Problem
  William B. Gearhart, and Harris S. Shultz

  William B. Gearhart received his BS degree in engineering physics and his PhD in applied mathematics from Cornell University. He is currently a professor of mathematics at California State University, Fullerton. His research interests include approximation theory, numerical analysis, optimization theory, and mathematical modeling. E-mail: wgearhart@fullerton.edu Harris S. Shultz, professor of mathematics at California State University, Fullerton, received his BA degree in mathematics from Cornell University and his PhD in mathematics from Purdue University. He has directed numerous institutes for secondary mathematics teachers and been a frequent contributer to The AMATYC Review. E-mail: hshultz@ fullerton.edu

  In a well-known calculus problem, an open top box is to be made from a rectangular piece of material by cutting equal squares from each corner and turning up the sides. The task is to find the dimensions of the box of maximum volume. Typically, the length of the sides of the corners that produces the largest volume turns out to be an irrational number. However, there are examples of non-square rectangles for which this length is a rational number. In this article we show how to generate all cases in which integer values for the dimensions of the rectangle produce rational answers. This provides calculus instructors with several rectangles for which the optimal box has "nice" dimensions.

• In Memoriam: August Zarcone

• How Mathematics Could Make Sense to Lots Of People And Why It Does Not: The Case Against Educology
  Alain Schremmer

  In the Fall 2003 issue of this Review, Schremmer brought the "Notes of The Mathematical Underground" to an end because. He is now writing a proto-textbook along the lines that he has urged all these years. This issue contains the rationale, preface, and contents for the book. Subsequent issues will contain the chapters.

• Book Reviews
  edited by Sandra DeLozier Coleman

  777 MATHEMATICAL CONVERSATION STARTERS, John dePillis,
  Spectrum Series, The Mathematical Association of America, Inc.,
  United States of America, 2002, ISBN 0-88385-540-2.

• Software Reviews 
  edited by Brian E.

• The Problems Section
  edited by Stephen Plett & Robert Strong

  The AR Problem Set consists of four new problems.
  
  Set AP Solutions
  Solutions are given to the four problems from the AP Problem Set that were in the Spring 2003 issue of The AMATYC Review.

• In Memoriam: Nelson Rich

• Lucky Larry #55

Editor: Barbara S. Rives, Lamar State College
Production Manager: John C. Peterson

Table of Contents

• The National Numeracy Network Promoting Quantitative Literacy For College Graduates
  Susan L. Ganter

  Susan L. Ganter is associate professor of mathematical sciences at Clemson University. She has directed several local and national evaluation studies, including a residency at the National Science Foundation in which she investigated the national impact of the calculus reform initiative and helped to develop the evaluation plan for several programs in the Division of Undergraduate Education. She is currently Director of the National Numeracy Network, an organization developed by the National Council for Education and the Disciplines at the Woodrow Wilson National Fellowship Foundation for the purpose of promoting quantitative literacy beyond mathematics. In addition, her work has included partnerships with industry that promote outreach to secondary mathematics students as well as professional development opportunities for secondary mathematics and science teachers. Dr. Ganter was formerly the Director of the Program for the Promotion of Institutional Change at the American Association for Higher Education and a member of the Mathematical Sciences faculty at Worcester Polytechnic Institute. sganter@clemson.edu

  In this increasingly technological age, the average citizen is confronted with a wealth of quantitative knowledge that can be overwhelming. An important part of the movement to promote quantitative literacy (QL) necessary in such a society is the design and formation of the National Numeracy Network (NNN). NNN is working to assist sites that are developing QL programs, and to create and maintain communication links between these sites and other constituencies. This article discusses educational goals for QL, as well as the activities of NNN.

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