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AMATYC Review Fall 2006
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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


Fall 2006 issue, Vol. 28, No.1

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 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:

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:

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:

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)

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:

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)

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:
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.
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Double Negatives

Timothy Mayo

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

"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!

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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:

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:

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:

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)

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:

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:

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)

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:
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)

Examining Students’ Conceptions Using Sum Functions

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:

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:

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

 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:

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:

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
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