MathAMATYC Educator
A refereed publication of the American Mathematical Association of TwoYear Colleges
Editor: Pete Wildman, Spokane Falls CC
Production Manager: George Alexander, Madison College
Volume 3, Number 2, February 2012 Issue
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This Issue’s Features
The Relationship between Computational Fluency and Student Success in General Studies Mathematics
Jennifer Hegeman and Gavin Waters, Missouri State University
The Singapore System
Fary Sami, Hartford CC
The Calculus of a Vase
Nicole Scherger, Elgin CC
How Teaching Math Is Like Improv Theater
Mike George, Borough of Manhattan CC
Ogren’s Theorem Revisited
Jose Villatoro and Travis Thompson, Harding University
A Means for Updating and Validating Mathematics Programs
Laurie A. Dunlap, University of Akron
An Integration of Math with Auto Technician Courses
Hector Valenzuela, Lake Washington Institute of Technology
MathAMATYC Educator's Departments
Use This Now
Integration by Hyperbolic Substitution
David Price, Tarrent CC
Capitalizing on the Dynamic Features of Excel to Consider Growth Rates and Limits
Daniel Taylor, Montross Middle School, VA and Deborah MooreRusso, University at Buffalo (SUNY)
Still Factorable by a Factor
James Metz, Kapi’olani Community College
Maclaurin Series for Functions with Removable Singularities
Brian Smith, McGill University
Comparing Volumes of Prisms and Pyramids
Natalya Vinogradova, Plymouth State University
Are There More Than TwentyEight Different Ways of Proving One Trigonometric Identity?
Terence Brenner, Hostos CC (CUNY)
The Problem Section
Take the Challenge
Joe Browne, Onondaga CC

Jennifer Hegeman, Ph.D., is an associate professor of mathematics at Missouri Western State University. Her current interests lie in the mathematical preparation of preservice elementary and secondary teachers, and the learning of mathematics in an online environment.
Gavin Waters is an assistant professor of mathematics at Missouri Western State University. Besides teaching, his interests include, but are not limited to economic models, differential equations, biological modeling and any other subject a student wants to contemplate.

The Relationship between Computational Fluency and Student Success in General Studies Mathematics
Jennifer Hegeman and Gavin Waters, Missouri State University
Many developmental mathematics programs emphasize computational fluency with the assumption that this is a necessary contributor to student success in general studies mathematics. In an effort to determine which skills are most essential, scores on a computational fluency test were correlated with student success in general studies mathematics at Missouri Western State University. Correlations were statistically significant but substantially lower than expected, calling into question some key assumptions about the curriculum of developmental math programs.

Fary Sami (fsami@harford.edu) is a professor of mathematics within the Science, Technology, Engineering, and Math Division at Harford Community College (HCC), Bel Air, Maryland. Professor Sami is a member of the International Education Subcommittee of the Chairs Division of AMATYC and the state of Maryland Math Group. fsami@harford.edu

The Singapore System
Fary Sami, Hartford CC
During the last 20 years, there has been an increasing awareness that the U.S. is falling behind in its mathematics education of primary and secondary school students. Deficiencies in mathematics training will eventually lead to critical shortages of future scientists and engineers. Our schools are underperforming in comparison to other developed and even underdeveloped countries with respect to mathematics education. For those who teach mathematics at the community college level, the deficiencies in mathematics education are painfully evident by the number of students requiring remedial math courses.
International comparisons of mathematics skill levels of secondary students are reported every four years in "Trends in International Mathematics and Science Study” (TIMSS, 2010), and every three years in "The Program for International Student Assessment” (PISA) (OECD, 2009). These reports show that there is a wide range of mathematics proficiencies among countries worldwide. As expected, students in developed countries performed better than those in underdeveloped countries.
During the past decade, Singapore has been among the topperforming countries in the world in mathematics education according to the TIMSS and PISA reports. However, Singapore has not always been a topranked country. Rather, it has demonstrated a marked improvement in ranking over the past two decades as the result of a total reevaluation of its mathematics instruction program in the 1980s. Because of Singapore’s success, some of our nation’s schools have adopted Singapore’s approach to teaching mathematics. Could the educational techniques used to improve Singapore’s program be applied at the community college level? The International Education subcommittee of AMATYC reasoned that lessons learned in highperforming countries might be applied to improve mathskill retention of community college students taking remedial mathematics courses in preparation for collegelevel mathematics. Although there are several factors contributing to the success of Singapore’s mathematics education system, the main focus of this article is on their primary school curriculum.

Nicole Scherger, (nscherger@elgin.edu) is an associate professor of mathematics at Elgin CC in Elgin, IL. She received her master’s degree in mathematics from Loyola University Chicago and her doctoral degree in adult and higher education from Northern Illinois University. She is currently interested in integrating technology into mathematics courses and innovative ways of teaching quantitative literacy.

The Calculus of a Vase
Nicole Scherger, Elgin CC
Of the most universal applications in integral calculus are those involved with finding volumes of solids of revolution. These profound problems are typically taught with traditional approaches of the disk and shell methods, after which most calculus curriculums will additionally cover arc length and surfaces of revolution. Even in these visibly applied areas of calculus, it can often remain a challenge to develop meaningful handson assignments; however, according to Beyond Crossroads (2006), students in mathematicsintensive courses and programs should be able to "use numerical, graphical, symbolic, and verbal representations to solve problems and communicate with others; use technology as a tool for exploring mathematical concepts; and use a variety of mathematical models, including curve fitting” (AMATYC, 2006, p. 48). Thus, activities that can address these outcomes need to be included in the calculus curriculum.
The calculus of a vase project is an activity that any calculus teacher could immediately implement with only a handful of vases and a basic understanding of curve fitting. I cannot claim this project as my own singular development, as I was first exposed to the concept of using calculus to estimate the volumes of vases when I observed my colleague, Professor Mary Ann Tuerk, engaging her calculus students in the task. Professor Tuerk and I presented this project at the 2010 AMATYC Conference in Boston. The purpose of this project is to use calculus techniques to estimate the volume, surface area, and lateral height (arc length) of a vase. (Entire article)

Michael George is an assistant professor of mathematics at Borough of Manhattan Community College. He received a MS in applied mathematics from the University of Washington and an Ed.D in mathematics education from Teachers College, Columbia University.

How Teaching Math Is Like Improv Theater
Mike George, Borough of Manhattan CC
When I first began to take classes in theater and comedy improvisation, my motives were largely creative and social, though there was some supposition that my mathematics teaching (at a community college) could benefit as well. Students are known to tout their favorite teachers as "funny.” Taking improv classes would presumably loosen me up a little at the board and make me more confident about my stage presence, thereby helping to facilitate the occasional joke or witty aside that could make for moments of shared humor between me and the class. Now, some years and many improv classes later, I have come to recognize that the art of improvising and the art of teaching share fundamental themes. In fact, I believe that they can be seen as crafts that share some fundamental rules.

Travis Thompson graduated from the University of Arkansas in 1977 (Ph.D. mathematics) and has been in higher education since that time.
Jose Villatoro graduated from Harding University with a B.S. degree in mathematics and is currently a successful businessman in Central America.

Ogren’s Theorem Revisited
Jose Villatoro and Travis Thompson, Harding University
Ogren’s Theorem (Thompson & Ogren, 1992) is a generalization of the rather unusual test for divisibility of an integer by 7. One may recall that to test an integer A for divisibility by 7, the following procedure is followed:
 The last digit of 1. A is stripped and doubled;
 This number is subtracted from the number represented by the remaining digits of the original number A;
 If the new number is divisible by 7, then the original number is divisible by 7.
For example, consider the integer 9961. Following the above rule, we would strip the last digit, 1, double it to 2, and subtract from 996, resulting in 994. A second application of the algorithm yields 91, and a final application results in 7—obviously divisible by 7. Therefore, the original number 9961 is divisible by 7.
A generalization of this technique can produce divisibility rules for other odd divisors that are not multiples of five.

Laurie Dunlap is an assistant professor in the Department of Mathematics and the Department of Curricular and Instructional Studies at the University of Akron in Akron, Ohio. Her research interests center on improving mathematics education through program reform, curriculum development, and workshops over content and teaching strategies for inservice teachers.

A Means for Updating and Validating Mathematics Programs
Laurie A. Dunlap, University of Akron
This article describes how to design program assessment for mathematics departments, in twoyear and fouryear colleges across the Midwest, based on a set of components that was generated from a Delphi survey. An example is provided to illustrate how this was done at a small fouryear college. There is an alignment between these components and a set of questions that was independently generated by the Mathematical Association of America’s Committee on the Undergraduate Program in Mathematics. This alignment serves as a form of validation for the worth of the components and the questions. The survey process that generated the components is described. Also discussed is the growing importance of program reform and how this process can be extrapolated to design program assessment for departments in other disciplines.

Hector Valenzuela, M.A. (Hector.Valenzuela@lwtech.edu) is a math faculty member at Lake Washington Institute of Technology. In addition to his work in the field of applied mathematics, he also spent 17 years in application areas of management information systems, business finance, and business development. He completed his undergraduate math education at the University of Texas at El Paso and his graduate mathematics work at Fresno State.

An Integration of Math with Auto Technician Courses
Hector Valenzuela, Lake Washington Institute of Technology
Within our research scope and design, we concentrated on developing contextualized mathematics and integrating it into the Auto Tech curriculum. All math developed was applied math and no theoretical math was taught in the integration sequence. The applied topics were developed from the auto tech curriculum. This allowed for a natural flow of understanding from the auto instructor to the math instructor to the auto tech students. The applied math curriculum was integrated into three auto tech classes: Auto Transportation Core, Auto Engines, and Auto Electrical. With integration, we had two instructors in the classroom—an auto instructor and a math instructor.
The applied math program that was developed is on the cutting edge of mathematical applications. Coteaching is quite effective when both instructors feed off of each other’s daily topics. It is clear that to have a successful integrated applied math program requires a significant amount of commitment and support. First, having a crossdivisional commitment is important. Within our college, we had excellent support from our Math Department, administration, and the Auto Department.
We helped students by (1) allowing for the student’s math development to be with contextualized math, (2) providing the students a comfortable environment where they learn math that makes sense to them, (3) reducing the time frame needed for students to complete their math requirements for their associate degree by integrating math into their auto technician classes, and (4) helping reduce the amount of money spent by the students by including the sequencing of the their math requirements within their technician program.

David Price is a professor of mathematics at Tarrant County College in Fort Worth, TX, where he has taught for thirtyfive years. He received AMATYC’s Teaching Excellence Award in 2003. His professional interests include interdisciplinary education and the history of mathematics.

Integration by Hyperbolic Substitution
David Price, Tarrent CC
Mathematics teachers constantly encourage their students to think independently. The study of integration in calculus provides an excellent opportunity to encourage inventive investigation. In contrast to differentiation, which is predominately mechanical, integration is a more creative process. One such possibility is offered by the study of the hyperbolic functions. After learning the trigonometric substitution technique of integration, students occasionally ask if the same integrals can be calculated by means of hyperbolic identities. Developing this approach provides an instructor with a variety of ways to enrich a secondsemester calculus class.

Daniel G. Taylor currently teaches middle school mathematics at Montross Middle School in Virginia. He received his master’s degree at the University at Buffalo. His research interests include utilizing technology to enhance instruction as well as brain research regarding learning.
Deborah MooreRusso (dam29@buffalo.edu) is an Assistant Professor in the Department of Learning and Instruction at the University at Buffalo (UB) where she teaches graduatelevel mathematics education courses. Before coming to UB, she taught precalculus and calculus courses for five years at St. Gregory's College and for nine years at the University of Puerto Rico at Mayag'ez.

Capitalizing on the Dynamic Features of Excel to Consider Growth Rates and Limits
Daniel Taylor, Montross Middle School, VA and Deborah MooreRusso, University at Buffalo (SUNY)
It is common for both algebra and calculus instructors to use power functions of various degrees as well as exponential functions to examine and compare rates of growth. This can be done on a chalkboard, with a graphing calculator, or with a spreadsheet. Instructors often are careful to connect the symbolic and graphical (and occasionally the tabular) representations of the functions. However, the graphs that are typically used for this are static.
The most recent versions of Microsoft® Excel® (Excel) allow instructors to illustrate the connections between the symbolic, tabular, and graphical representations of the equations through quick generation of the function graphs. This requires only minimal input including three components: the equation of the function, its starting point, and the incremental changes between independent variable values. By formatting the spreadsheet to depend on these three things, the input values (and the calculated output values) are easily manipulated, allowing for changes in scale.

James Metz is an associate professor of mathematics at Kapi'olani Community College, Honolulu, HI. Besides teaching developmental mathematics classes, he conducts workshops in South Africa with Teachers Without Borders. He just completed a sabbatical year as a volunteer teacher in Africa.

Still Factorable by a Factor
James Metz, Kap'olani Community College

Brian Smith holds a Ph.D. in mathematics from Queen’s University in Canada. He currently teaches statistics and operations research courses at McGill University’s Desautels Faculty of Management. In a former incarnation, he was professor of mathematics at Dawson College in Montreal, where he taught a wide range of twoyear college mathematics courses. Brian served as chair of the Technology in Mathematics Education Committee at AMATYC for six years and subsequently chaired the Statistics Subcommittee.

Maclaurin Series for Functions with Removable Singularities
Brian Smith, McGill University
When is a Maclaurin series not a Maclaurin series? As silly as this questions sounds, it is often confusing to students (and instructors) when a function f(x) does not satisfy the assumptions of continuity and differentiability necessary for generating a Maclaurin series using the standard formula
Yet, by a sleight of hand (substitution, arithmetic operation of both sides of an equation, etc.) we produce a new series from an existing one and claim that it is the Maclaurin series for the given function.

Natalya Vinogradova graduated from Leningrad State University and earned a Ph.D. in mathematics education at SUNY Buffalo. She currently works as an associate professor in the Mathematics Department at Plymouth State University in New Hampshire. In addition to teaching mathematics to prospective teachers, she is interested in working with in service teachers and frequently offers workshops and courses for professional development. The main goal of her work is to promote the view of mathematics as a harmonious world of logically connected ideas.

Comparing Volumes of Prisms and Pyramids
Natalya Vinogradova, Plymouth State University
Students’ experience in using formulas for volumes is often limited to substituting numbers into given formulas. An activity presented in this article may help students make connections between the formulas for volumes of prisms and volumes of pyramids. In addition, some interesting facts from number theory arise, demonstrating strong connections between different areas of mathematics. A set of identical cubical building blocks, preferably in two colors, will be needed to complete the constructions described.

Terence Brenner is an associate professor of mathematics who has taught for more than twentyfive years at Hostos Community College of the The City University of New York. He received his Ph.D. from Yeshiva University and is a contributor to both the MathAMATYC Educator and its predecessor, The AMATYC Review.

Are There More Than TwentyEight Different Ways of Proving One Trigonometric Identity?
Terence Brenner, Hostos CC (CUNY)
I have given the following problem: prove
on my precalculus test over the years. No identities were given to the students on the test, they had to know them. Here is the "obvious” solution:
Students came up with twentyseven other ways of proving it.


The Problem Section
Welcome to the Problem Section. We will strive to provide several interesting and usually challenging problems for you to consider in each issue. Content will be mathematics and puzzles connected in some way to the mathematics we teach in the twoyear college. Readers are invited (encouraged!) to submit problem proposals (with solution) for possible inclusion in this column. We also encourage readers to submit solutions to the problems posed here; we will publish the best or most interesting in a future issue.
Send all correspondence to Joe Browne at brownej@sunyocc.edu or at Mathematics Department, Onondaga Community College, Syracuse NY 13215.
The Problem Section is assembled by Fary Sami (at Harford Community College, MD) and Tracey Clancy, Kathy Cantone, Garth Tyszk, and Joe Browne (editor) (at Onondaga Community College, NY).


