AMATYC Review Spring 2008
From the Editor’s Keyboard
Barbara S. Rives
Winter is near, snow already came on Thanksgiving, and the holidays will soon be here. Yes, you will receive the newsletter in February, but I suspect winter will still be here, snow will still be falling (somewhere) and the spring holidays will be near.
This issue of The AMATYC Review has the last portion of articles written by Alain Schremmer. He has faithfully submitted articles for many years, first as Notes from the Underground, and more recently as Mathematics For Learning With Inflammatory Notes for the Mortification of Educologists and the Vindication of "Just Plain Folks.” He will continue writing about mathematical topics; however, as soon as his new location is available, it will be announced in the Fall 2008 issue of The AMATYC Review. A special "thank you” goes to Dr. Schremmer for all his work for AMATYC.
The articles published in this issue focus on a range of mathematics topics: developmental mathematics, symmetry, the number of real roots in cubic equations, the value of a volume of coins, the floor function and the countability of rational numbers, conditional probability and Bayes’ rule, and matching instructional strategies with student learning preferences. It is hopeful this range of topics will interest our readers.
It hardly seems possible the tenure for the current editor is almost over. Only one more issue remains (Fall 2008) and then the reins will be turned over to a new editor. Watch for the advertisement for the new editor. If you are interested in this job, please apply. The job provides a wide range of activities and a wonderful opportunity to learn what AMATYC colleagues are doing in research, classroom activities, real-world applications, and helping students become more successful in mathematics. November 2008 will soon be here.
Have a wonderful spring semester.
Barbara S. Rives, Editor
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The Number of Real Roots of a Cubic Equation
Richard Kavinoky and John B. Thoo
|Richard Kavinoky worked in the San Francisco Bay Area as a carpenter and building contractor for many years before returning to school, earning a BA in mathematics at Sonoma State, and an MA and PhD in Mathematics at U.C. Davis. He taught at U.C. Davis, Sonoma State, College of San Mateo, and now teaches at Santa Rosa Junior College. |
|John B. Thoo is professor of mathematics at Yuba College, Marysville, CA, a community college in the farming region of California’s northern Sacramento Valley, where rice fields and fruit orchards abound. Sadly, many farms today are being paved over for tract houses. John has recently taken an interest in the history of mathematics and enjoys presenting topics in the courses below calculus "through the history glass.” |
To find the number of distinct real roots of the cubic equation (1) x^3 + bx^2 + cx + d = 0,
we could attempt to solve the equation. Fortunately, it is easy to tell the number of distinct real roots of (1) without having to solve the equation. The key is the discriminant.
The discriminant of (1) appears in Cardan’s (or Cardano’s) cubic formula. However, few students today are even aware of the cubic formula, let alone have seen it. We show how a student may come up with or be led to the discriminant of (1) without appealing to Cardan’s cubic formula using ideas from a first calculus course—derivative, critical point, local extrema, and graphing—in an intuitive way. We also show how the discriminant defines a boundary in the plane across which the number of real roots of (1) changes, and apply the discriminant to determining the number of normals to the parabola y = x^2 through a given point and the number of equilibrium solutions of dx/dt = (R-Rc)x-ax^3, where Rc and a are positive constants and R is a parameter.(back to top)
Frank J. Attanucci and John Losse
|Frank J. Attanucci has served as a professor of mathematics at Scottsdale Community College in Scottsdale, Arizona, for 17 years. He received BS and MA degrees in mathematics from Arizona State University. When he is not dreaming up new ways to use two-year college mathematics and his computer algebra system to create interesting graphics or "mathematical animations,” Frank is probably hunched over an essay or book in philosophy or theology. |
|John Losse has been at Scottsdale Community College as professor of mathematics since 1975. He received his BS in mathematics from Trinity College and his MS from the University of North Carolina at Chapel Hill. He has long been interested in applications of technology to mathematics teaching, and lately spends time working with high school calculus teachers. He likes math problems which are challenging, but not too. |
|In a first calculus course, it is not unusual for students to encounter the theorems which state: If f is an even (odd) differentiable function, then its derivative is odd (even). In our paper, we prove some theorems which show how the symmetry of a continuous function f with respect to (i) the vertical line: x = a or (ii) with respect to the point: (a, 0), determines the symmetry of the antiderivative of f defined by . We conclude with an example that shows how our results lead to a "two-line proof” that the graph of any cubic function is symmetric with respect to its point of inflection.(back to top)|
$158 per Quart: The Value of a Volume of Coins
Stephen Kcenich and Michael J. Boss'e
Stephen Kcenich is an associate professor of mathematics at Montgomery College in Takoma Park, MD. He received his MS from Penn State University in mathematics. His professional interests within the field of mathematics and mathematics education are cooperative and collabartive learning, remedial mathematics education, actuarial mathematics, functional analysis, and the relationship between music and mathematics.
|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. |
The ubiquitous change jar (or any other container) is the focus of this investigation. Using random pocket change, a distribution is determined and statistical tools are employed to calculate the value of given volumes of coins. This brief investigation begins by considering money, which piques the interest of most students, and uses this foundation to carry them into increasingly deeper mathematical and statistical investigations. Real world scenarios and teaching tips are provided throughout. (back to top
Successful Developmental Mathematics Education: Programs and Students - Part II
Irene M. Duranczyk
The University of Minnesota
|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. |
This article, the second in a three-part series, outlines the qualitative research design and ndings. The qualitative study was conducted three to five years after students completed their developmental mathematics course work at a large Midwest public university. 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 first article in the series reviewed the literature for research highlighting the characteristics or successful developmental mathematics programs and students. This article summarizes the 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 discuss further the research model used and identify specific implications - what do developmental educators need to consider as they evaluate the effectiveness of their developmental mathematics programs.
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An Alternative Version of Conditional Probabilities and Bayes’ Rule: An Application of Probability Logic
Eiki Satake and Philip P. Amato
|Eiki Satake is associate professor of mathematics at Emerson College. He earned a BA in mathematics from the University of California at Berkeley, and MS, EdM, and EdD, in mathematics education and applied statistics from Columbia University. He has published numerous journal articles and authored several textbooks with Philip P. Amato in the area of mathematics, statistics, and research methods. |
Philip P. Amato is professor of mathematics at Emerson College. BA, English, MA, communication, Emerson College (‘60, ‘61); PhD communication, Michigan State University (‘63). He has published numerous journal articles in communications and authored several textbooks with Eiki Satake in the area of mathematics and statistics, two of which were selected by MAA as part of its Basic Library List.
|This paper presents an alternative version of formulas of conditional probabilities and Bayes’ rule that demonstrate how the truth table of elementary mathematical logic applies to the derivations of the conditional probabilities of various complex, compound statements. This new approach is used to calculate the prior and posterior probabilities of conditional statements by means of probability logic table along with the Bayesian principle. Unlike the more commonly used methods, such as the formula, tree diagram, and contingency table, a probability logic table approach represents a convenient, straight-forward, and useful method for calculating and teaching conditional probability and Bayes’ rule to statistical novices whose reasoning processes are fundamentally different from that of the expert. The use of a truth, or probability logic table is illustrated in comparison to the formula, tree diagram, and contingency table methods. The problem to be resolved is one frequently used in finite mathematics and elementary statistics courses, that of determining the probability of observing a family with three children. It is argued that a truth table approach is less complex and time consuming than the traditional methodologies. (back to top)|
Matching Instructional Methods with Students Learning Preferences: A Research-based Initiative for Improving Students’ Success in Mathematics
Kimberly Nolting and Paul Nolting
|Mrs. Kimberly Nolting, ABD and author, is focusing on a predictive model for student persistence through math courses based on psycho-social factors as her PhD dissertation. She has presented at national conferences and has consulted with colleges/ universities on teaching and learning as well as in program assessment and improvement. |
Paul Nolting, PhD is the math learning specialist and Intuitional Test Administer at Manatee Community College, Bradenton, FL. He is a nationally recognized, author, consultant and trainer on mathematics learning. He has presented at numerous state and national conferences, conducted PBS specials and has consulted with colleges/universities on math success.
Research supports the effectiveness of matching instructional methods with student learning preferences (Dunn et al., 1995; Pascarella and Terenzini, 2005). Several challenges exist, however, for mathematics departments to design classroom learning experiences that allow students to learn mathematics and learn how to study math through their preferred learning styles. After a research overview, this article first focuses on a learning style inventory that lends itself to designing teaching and learning strategies for math; second, focuses on a departmental plan for expanding efforts to match instructional methods with learning preferences and for helping students design study strategies that work best for them; third, presents examples of redesigning learning style-based study strategies into classroom learning experiences. Departments that move forward with these suggestions will become student-centered math departments in which students will discover that they can learn mathematics and expand their career options.
Dunn, R., Griggs, S., Olson, J., Beasley, M., & Gorman, B. (1995). A meta-analtyic validation of the Dun and Dunn learning-style model. Journal of Educational Research, 88, 353–362.
Pascarella, E., & Terenzini, P. (2005). How college affects students (Vol. 2.). San Francisco, CA: Jossey-Bass.
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Using the Floor Function to Prove the Countability of the Rationals
Jeremy Brazas and Dean B. Priest
|Jeremy Brazas is currently a second year graduate student working on his PhD in mathematics at the University of New Hampshire. He earned a Bachelor’s Degree in mathematics and a Master’s Degree in Education, both from Harding University and plans to teach college mathematics in the future. |
|Dean Priest is a Distinguished Professor of Mathematics at Harding University, Searcy, Arkansas. Some of his previous articles have appeared in the publications of NCTM and AMATYC as well as the Pacific Journal of Mathematics. He has served on the publication committee of NCTM and the project task force for AMATYC’s Crossroads in Mathematics: Standards for Introductory College Mathematics before Calculus. |
|In this paper the floor function [.] : R --> N is used to define an onto function B : N --> Q. From this it follows that Q is countable. (back to top)|
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Edited by Sandra DeLozier Coleman
LETTERS TO A YOUNG MATHEMATICIAN, Ian Stewart, Basic Books (a Member of the Perseus Books Group), New York, NY, 2007, ISBN: 9780465082322, ISBN-10: 0-465-08232-7 (pbk).
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Reviewed by Annette M. Burden, Youngstown State University
Edited by Brian E. Smith
|An Overview of Several Popular Web-Enhanced Instructional Products: Part II|
As was mentioned in Part I, a major challenge arose to develop computer assisted instructional products that were 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. In Part I, an overview of two of the more common webenhanced instructional products, ALEKS R (ALEKS Corporation, 1965) and MyMathLab R (Pearson Education, 2000) was provided. In this sequel the reader is given an overview of several other of the more common web-enhanced instructional products: Math Zone R (McGraw-Hill, 2004), Thompson NOW R (Thompson, 2005), and Eduspace (Houghton Mifflin, 2006). The most recent product, WebAssign R, introduced by ThompsonBrooks/Cole is not discussed here. Recall that 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.
MathZone is a text-specific, customizable course management system created for use with selected McGraw-Hill textbooks in mathematics. MathZone functionality for selected texts includes:
Complete textbook coverage
e-Professor (voiced-over slides)
Algorithmic interactive practice exercises and testing
MathZone has recently been upgraded to version 3.0. The enhancements include
Single screen assignment
Enhanced Communications includes
Live Classroom (NetTutor)
Students can be e-mailed by class, group, individual, or by all adjunct sections
Assignment Printout Worksheets
Student can access practice exercise sets in any order
Administrators are required to register for their course using an instructor access code. The access code is provided to instructors who adopt the MathZone product through their local sales representative. After registration and upon login, instructors must check if the required plug-in has been installed on their computer. Once this plug-in has been installed, MathZone instructors are directed to a Course Information page where they can select one of their existing courses from a list.
In the event that multiple section offerings of a course are required, an administrator would create a master course syllabus and duplicate the master course multiple times. The proper instructor permissions are set from within this master course syllabus. Once enabled, other sections may be created under this master syllabus. Only those sections that are taught by the same instructor will appear on that instructor’s Course Information page. However, all sections created will appear on the administrator’s course management page.
Assignments and announcements can be created from within the Manage Sections area. The class roster, gradebook, and the online tutor (NetTutor, a product of Link- Systems) can be accessed from the Manage Sections area as well. The online text is available via the Self Study link.
Students are required to register for their course using a purchased access code which is generally bundled with the text order. The student module interface is similar to that of the Administrator module in that the student is provided with links to assignments; announcements, gradebook, online tutor, self study, course calendar, and course management.
MathZone has a clean Administrator appearance. However, navigation from one stage of course/section development to another is rather complex and often confusing. Creation of assignments can be time consuming and complex from an administrator’s perspective as there are so many different stages or "levels” to navigate. The Administrator does not have the ability to simplify the student interface. MathZone’s e-professor is a nice feature. The interactive problems coincide with the selected text and the instructor has the ability to accept a variety of inputs. There appears to be no way to modify the Master Syllabus of a selected text.
Since ALEKS is one of the added features to MathZone, all of the inherent problems mentioned under the ALEKS section remain. The student interface appears to be easy to navigate and assignments easy to access.
In order to operate properly, MathZone requires the proper version of Java as well as a small plug-in to access the dynamic, algorithmically generated mathematics components and appropriate "viewers” to access the multimedia learning aids. In general, the overall design and functionality of this product appears to be theoretically strong in items 1, 2, 3, 6, and 7 but weak in items 4 and 5.
Thompson NOW Overview
Thompson describes this product as a "suite of services” with the following functionality:
Creation of courses
Development of course syllabi
Set up online courses and enroll students
Create assignments from
Assign tests, quizzes, tutorials, practice, and homework
View and edit assignment scores
Online communication with students and other instructors
Administrators are required to register for their course using an instructor access code. The access code is provided to instructors who adopt the product through their local sales representative. Administrators are able to do a variety of tasks, such as e-mail students, change or retrieve student passwords and/or e-mail addresses, and set tests.
Students do not need to have an access code in order to use the product. Navigation is fairly simple and straight forward.
Thompson’s NOW has a clean administrator appearance. Navigation from one stage of course/section development is misleading and not easy. It is difficult to quickly clone a course and students cannot be moved from section to section. This product does not have the multimedia help features that some of the other products have. That the gradebook can be integrated with WebCT and Blackboard might be considered a plus by some administrators.
The multimedia help features are not available. The product locks up at crucial times.
In general, the overall design and functionality of this product appears to be theoretically weak in items 1, 2, 3, 4, 5, 6, and 7.
Like MyMathLab, Eduspace is powered by Blackboard. Houghton Mifflin describes their product as online learning tool that combines the "tools of Blackboard with quality Houghton Mifflin content to help students succeed in online, traditional, and hybrid courses”. Upon closer inspection, Eduspace is a replica of MyMathLab in both appearance and functionality with a little of the functionality of MathZone and Thompson NOW thrown in.
Administrators are required to register for their course using an instructor access code. The access code is provided to instructors who adopt the product through their local sales representative. Administrator capabilities appear to mirror those of MyMathLab.
Student functionality appears to mirror that of MyMathLab and MathZone.
Since this product has just arrived on the scene, it requires further investigation. Since this product is based on Blackboard technology, the course management interface looks very much like that of MyMathLab.
The multimedia help features are not available. The product is slow to load. The interface looks eerily familiar.
It is unclear at this time whether this product contains video clips or other audio/ visual multimedia. At present, only a small number of texts are enhanced with this product functionality. The overall design and functionality of this product is difficult to determine as this product just recently surfaced.
Table 1 below provides the reader with a quick overview of all of the instructional products that were discussed in Part I and Part II. It should be noted that each of these products generally go through periodic upgrades in order to modify and enhance appearance, ease of use, and functionality. Obviously an upgrade is intended not only to keep the product on the cutting edge of technological advances but also to provide better functionality to users. Upgrade activity appears to be strongest in MyMathLab as there is generally one annual major upgrade followed by several minor upgrades throughout the year. The upgrade activity appears to be moderate in MathZone where there is generally one major upgrade once every one or two years. The upgrade activity for ALEKS appears to be less frequent. Upgrade activity for Thompson NOW and Eduspace is yet to be determined since they are relatively new on the market.
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.
Houghton Mifflin (2006). Eduspace [Online]. Available: http://college.hmco.com/CollegeCatalog/ CatalogController?cmd=Portal&subcmd=display&ProductID=12623 [2006, September 05].
Microsoft Corporation. (2006). MSDN [Online]. Avaliable: http://msdn.microsoft.com [2006, September 05].
Martin-Gay, Beginning Algebra, 4th Edition, Prentice Hall, 2005.
McGraw-Hill. (2006). MathZone [Online]. Available: http://www.mathzone.com/ [2006, September 05].
Pearson Education. (2000). CourseCompass/MyMathLab [Online]. Available: http://www. coursecompass.com/ [2006, September 01]. Thompson NOW. (2005).
Thompson NOW [Online]. Available: http://www.ilrn.com/ [2006, October 05].
Thompson-Brooks/Cole. (2006).WebAssign [Online]. Available: http://www.webassign.com/ [2006, October 05].
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 firstname.lastname@example.org.
Send reviews to:
Brian E. Smith
AMATYC Review Software Editor
Department of Management Science
1001 Sherbrooke St. West
Montreal, QC, Canada H3A 1G5
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The Problems Section
Edited by Stephen Plett and Robert Stong
The AZ Problem Set consists of four new problems.
Set AW Solutions
Solutions are given to the four problems from the AX Problem Set and corrected Problem AW-2 that were in the Spring 2007 issue of The AMATYC Review.
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Mathematics For Learning
With Inflammatory Notes for the Mortification of Educologists and the Vindication of "Just Plain Folks”
The opinions expressed are those of the author and should not be construed as representing the position of AMATYC, its officers, or anyone else.
[Editor’s note: 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 the concluding column.]
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