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The
AMATYC Review
A refereed publication of the American Mathematical
Association
of Two-Year Colleges
Editor: Barbara
S. Rives, Lamar State College
Production Manager: John
C. Peterson
Abstracts
Spring 2008 issue, Vol.
29, No.2
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From
the Editor’s Keyboard
Barbara S. Rives
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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
E-mail: ReviewEditor@amatyc.org
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The
Number of Real Roots of a Cubic Equation
Richard Kavinoky and John B. Thoo
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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.
E-mail: rkavinoky@santarosa.edu |
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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.”
E-mail: jthoo@yccd.edu |
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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
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Inherited
Symmetry
Frank J. Attanucci and John Losse
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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.
E-mail: frank.attanucci@sccmail.maricopa.edu |
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.
E-mail: john.losse@sccmail.maricopa.edu |
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
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$158
per Quart: The Value of a Volume of Coins
Stephen Kcenich and Michael J. Boss'e
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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.
E-mail: stephenkcenich@yahoo.com
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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 |
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
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Successful
Developmental Mathematics Education: Programs and
Students - Part II
Irene M. Duranczyk
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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.
E-mail: duran026@umn.edu
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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
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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.
E-mail: Eiki_Satake@emerson.edu |
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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.
E-mail: Philip_Amato@emerson.edu
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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
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Matching
Instructional Methods with Students Learning Preferences:
A Research-based Initiative for Improving Students’
Success in Mathematics
Kimberly Nolting and Paul Nolting
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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.
E-mail: kimnoltin@aol.com |
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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.
E-mail: pnolting@aol.com
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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.
References
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
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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.
E-mail: Jtv5@unh.edu |
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.
E-mail: dpriest@harding.edu |
| 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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Book
Reviews
Edited by Sandra DeLozier Coleman
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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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Software
Reviews
Reviewed by Annette M. Burden, Youngstown State
University
Edited by Brian E. Smith
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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 Overview
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:
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Complete textbook coverage
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Video lectures
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e-Professor (voiced-over slides)
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Algorithmic interactive practice
exercises and testing
MathZone has recently been upgraded to version
3.0. The enhancements include
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ALEKS integration
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Single screen assignment
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Enhanced Communications includes
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Student Lounge
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Live Classroom (NetTutor)
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Message Center
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Archive Center
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Students can be e-mailed by
class, group, individual, or by all adjunct
sections
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Assignment Printout Worksheets
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Timed assignments
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Student can access practice exercise
sets in any order
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Administrator can
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 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.
Student Module
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.
Product Functionality—Comments
Administrator Module
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.
Student Module
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.
General
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:
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Creation of courses
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Development of course syllabi
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Comprehensive gradebook
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Set up online courses and enroll
students
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Create assignments from
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course material
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test banks
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other sources
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Assign tests, quizzes, tutorials,
practice, and homework
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View and edit assignment scores
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Post messages
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Online communication with students
and other instructors
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 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.
Student Module
Students do not need to have an access code in
order to use the product. Navigation is fairly simple
and straight forward.
Product Functionality—Comments
Administrator Module
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.
Student Module
The multimedia help features are not available.
The product locks up at crucial times.
General
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.
Eduspace Overview
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.
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 product
through their local sales representative. Administrator
capabilities appear to mirror those of MyMathLab.
Student Module
Student functionality appears to mirror that of
MyMathLab and MathZone.
Product Functionality—Comments
Administrator Module
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.
Student Module
The multimedia help features are not available.
The product is slow to load. The interface looks
eerily familiar.
General
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.
Summary
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.
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.
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 aburden@as.ysu.edu.
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
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The
Problems Section
Edited by Stephen Plett and Robert Stong
New Problems
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”
Alain Schremmer
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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