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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
Fall 2006 issue, Vol.
28, No.1
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Journey
to Beyond Crossroads: A Reflection
Susan S. Wood, Philip H. Mahler, and Sadie
C. Bragg
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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 1999–2001 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
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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. (back to top)
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The
Lost Divisibility Rules for 7 and Beyond
A. J. Berry
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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 |
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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.” (back to top)
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On
the Applications of Axial Representation of Trigonometric
Functions
M. Vali Siadat
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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.(back
to top) |
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Double
Negatives
Timothy Mayo
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Tim Mayo teaches developmental
mathematics, intermediate algebra, and calculus at Mohave
Community College in Lake Havasu City, Arizona. E-mail:
TMAYO@imail.mohave.edu |
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“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 (back to top)
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People
vs. Collins: Statistics as a Two-Edged Sword
Jean McGivney-Burelle, Katherine McGivney, and
Ray McGivney
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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
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| 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. (back
to top) |
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Packing
Infnite Number of Cubes in a Finite Volume Box
Haishen Yao and Clara Wajngurt
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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
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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
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| 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. (back
to top) |
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Sketching
Curves for Normal Distributions—Geometric Connections
Michael J. Bossé
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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 |
| 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. (back to top) |
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Examining
Students’ Conceptions Using Sum Functions
Kevin Ratliff and Joe Garofalo
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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
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| 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. (back
to top) |
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What
Does Conceptual Understanding Mean?
Florence S. Gordon and Sheldon P. Gordon
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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
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| 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? (back to top) |
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. (back to top)
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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.” (back to top)
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. (back
to top)
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 cubes–identical!
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 (back to top)
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