MathAMATYC Educator
A refereed publication of the American Mathematical Association of TwoYear Colleges
Editor: Pete Wildman, Spokane Falls CC
Production Manager: Jim Roznowski, Delta C
Volume 2, Number 3, May, 2011 Issue
Earlier and Later Issues
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This Issue’s Features
What Community College Developmental Mathematics Students Understand about Mathematics, Part 2: The Interviews
Karen B. Givvin, James W. Stigler, and Belinda J. Thompson, University of California, Los Angeles
Capitalizing on Basic Brain Processes in Developmental Algebra – Part 2
Edward D. Laughbaum, The Ohio University
Exploring Research on Issues Impacting Mathematics in Community Colleges
June Lundy Gastón, Borough of Manhattan CC, CUNY
Grants: Genesis of Some Funded Proposal Ideas
John Pazdar, Capital Community College
MathAMATYC Educator's Departments
Use This Now
Developmental Arithmetic via KenKen
Raymond M. Houston, SUNY Westchester Community College
A Calculus Unification: Regarding Some Differentiation Rules as Sums of Special Cases
Michael W. Ecker, Pennsylvania State University's WilkesBarre Campus
Visualizing the Concept of Limits
Andrzej Sokolowski, Texas A&M University and Magnolia West High School, Magnolia, Texas
Partial Integration
Benjamin VanHavermaet and Paul Kinion, Inver Hills CC
Teaching Math with Technology
Visualizing Mathematics by Importing Images into GeoGebra
Joanne Caniglia, Kent State University; Barbara Leapard, Eastern Michigan University and Andreas Lindner
The Problem Section
Take the Challenge
Joe Browne


What Community College Developmental Mathematics Students Understand about Mathematics, Part 2: The Interviews
Karen B. Givvin, James W. Stigler, and Belinda J. Thompson, University of California, Los Angeles
Abstract
In a prior issue of MathAMATYC Educator, we reported on our efforts to find out what community college developmental mathematics students understand about mathematics (Stigler, Givvin, & Thompson, 2010). Our work painted a distressing picture of students’ mathematical knowledge. No matter what kind of mathematical question we asked, students tended to respond with computational procedures, which they often applied inappropriately and incorrectly. Their knowledge of mathematical concepts appeared to be fragile, and weakly connected to their knowledge of procedures. But we also found some reason for hope. First, we found that when students were able to provide conceptual explanations for procedures, they often produced correct answers. Second, though students rarely used reasoning on their own to solve problems, they could reason under the right conditions.

Ed is an emeritus professor from Columbus State Community College, and recently retired from The Ohio State University as the director of the Ohio Early College Mathematics Placement Testing Program and the College Short Course Program. His interests lie in teaching algebra for understanding and longterm memory with recall – using handheld technology.

Capitalizing on Basic Brain Processes in Developmental Algebra – Part 2
Edward D. Laughbaum, The Ohio University
Abstract
Basic brain function is not a mystery. Given that neuroscientists understand its basic functioning processes, one wonders what their research suggests to teachers of developmental algebra. What if we knew how to teach so as to improve understanding of the algebra taught to developmental algebra students? What if we knew how the brain processes memory of something learned, and how it recalls the memory? If we knew this, how would we change our teaching to create these outcomes in our students?
The first thing we would do is reconsider the philosophy of "explaining with examples followed by lots of homework” as causing these desired outcomes. We would question the idea that what works for physical learning also applies to the understanding of abstract ideas, and to developing longterm memory with recall.
We would implement the neural processes of associations (connections), visualizations, pattern recognition/generalizing, and meaning through contextual situations in our teaching. We would focus on the function approach as opposed to the equationsolving approach. We would use function and function behaviors to connect every concept and skill.
This paper demonstrates the process, and provides supporting evidence from the neurosciences for the process of a function approach to teaching algebra.

Dr. June Lundy Gastón is a professor in the Mathematics Department at Borough of Manhattan CC, CUNY. She currently chairs the Mathematics Department Faculty Development Committee. Dr. Gastón is a former director of the BMCC Teaching Center for Faculty & Staff Development and a former director of the CUNY Teacher Academy at BMCC.

Exploring Research on Issues Impacting Mathematics in Community Colleges
June Lundy Gastón, Borough of Manhattan CC, CUNY
Abstract
Twoyear colleges have an important role in the educational and economic advancement of the United States, particularly during the current economic crisis. They offer access to higher education for economically and/or educationally impoverished students, as well as for those who have simply returned for further study. Twoyear colleges provide academic training for those seeking to upgrade their skills to qualify for promotion or a new career. The colleges also have a crucial role in the recruitment, training, and professional development of STEM educators.
How can community college faculty best help such a diverse group of students achieve a level of mathematical proficiency that will help them meet their personal, academic and career goals? What does research suggest about coursework that will facilitate the development of abstract reasoning skills that lead to proficiency in higher level academic and careerrelated problem solving?

John Pazdar, Professor Emeritus Capital Community College, taught from 1967 until his retirement in 2002. Upon retirement, he returned to the classroom as an Adjunct Professor, Asnuntuck Community College. John is an AMATYC Charter Member and enjoys visiting with colleagues at conferences throughout the year. He also enjoys writing grants proposals and playing golf.

Grants: Genesis of Some Funded Proposal Ideas
John Pazdar, Capital Community College
Abstract
While "thinking outside the box” can be an overused phase at times, in the world of grants it can provide the genesis of ideas. The "box” is the world of academia accepted by most educators, while "thinking outside” is the process that leads to grant ideas. In the grant world, "thinking outside the box” is a process of doing something that has not been done before or of improving something that has been done before. That something may be as small as using inexpensive toy cars to better explain ratios and proportions or as large as creating an expensive national center for nanotechnology.

Raymond M. Houston is in his second year as an Instructor of Mathematics at SUNY Westchester Community College. He is also a doctoral candidate in Mathematics Education at Teachers College Columbia University. He earned his B.S. in Applied Mathematics and Chemistry from the University of Pittsburgh, and his M.A. in Curriculum and Instruction from Point Park University.
Mr. Houston has experience in mathematics instruction ranging from Prealgebra through Calculus at both twoyear and four year institutions, including time at Point Park, Robert Morris University, and Queensborough CC (CUNY). He has previously served as the Coordinator of Mathematics and Science tutoring at Point Park, Coordinator of Developmental Mathematics at Robert Morris, and as a board member for the Pennsylvania Association of Developmental Educators (PADE). His professional interests include developmental mathematics, tutoring, preparation of K12 and postsecondary mathematics educators, and curriculum development.

Developmental Arithmetic via KenKen
Raymond M. Houston, SUNY Westchester Community College
Abstract
Professional educators that teach developmental mathematics courses know that one of the most common frustrations is getting these students to practice the skills that are taught in the classroom. Since often in these courses the material is content that the students have seen repeatedly and often with minimal success in previous attempts, apprehension to practice is high. Thus, we often employ other techniques that are fun and engaging in which to incorporate practice into the educational process.

Michael W. Ecker (DrMWEcker@aol.com or MWE1@psu.edu) is an associate professor of mathematics at Pennsylvania State University's WilkesBarre Campus. Having taught college math since 1972, he received his Ph.D. in mathematics from the City University of New York in 1978 under Harry Rauch. The founder of the AMATYC Review problem section in 1981, a position he held until 1997, he has posed and solved hundreds of problems in over a dozen mathematics journals. He created such computer columns as "Mathematical Recreations" in Byte in 1984, "Recreational Computing" in Popular Computing in 1983, and the same in Creative Computing in 1985. From Jan. 1986 to Jan. 2007 he wrote, edited, and published his own newsletter REC (Recreational & Educational Computing), featuring the interplay of mathematics, computers, and recreations, along with his unifying concept of Mathemagical Black Holes. He is the author of over 500 newsletters, columns, reviews, and articles, many computerrelated, as well as five books and/or solution manuals. His other passions include racquetball, sweets, and Renee (Wife 2.0).

A Calculus Unification: Regarding Some Differentiation Rules as Sums of Special Cases
Michael W. Ecker, Pennsylvania State University's WilkesBarre Campus
Abstract
It has been a long time since I first noticed that the derivative of , which is , may be thought of as . Do you see something curious here?
Calculus II instructors have had students state the derivative of as , incorrectly citing the power rule. (That the students don't simplify their answer is another matter.) More rarely, a student reports the derivative as .

Andrzej Sokolowski is a doctoral student in the Department of Teaching Learning & Culture, Mathematics Ecuation at Texas A&M University, College Station, Texas. He is also a fulltime math and physics teacher at Magnolia West High School, Magnolia, Texas. He holds a masters degree in physics from Gdansk University, Gdansk, Poland. His research interest includes contextualization of mathematics concepts through scientific representations.

Visualizing the Concept of Limits
Andrzej Sokolowski, Texas A&M University and Magnolia West High School, Magnolia, Texas
Abstract
The concept of limits is very abstract, yet its importance for students’ comprehension of function continuity and differentiability is essential. As a growing body of research supports the assertion that understanding of mathematics is strongly related to students’ ability to use visual and analytical thinking (Zazkis, Dubinsky, & Dautermann, 1996), in this paper, I share an activity that highlights the process of visualization of the limit concept. The process of visualization is enhanced by a tool called Equation Grapher, created by the Physics Educational Team (PhET) at Colorado University at Boulder and available for free on the Internet.
This paper consists of two main segments. In the first, the concept of limits of linear and quadratic functions supported by the graphing tool is introduced. Since the concept is introduced to precalculus students, this paper focuses on the informal visual models that are used to introduce the idea of limits to students and help them embody the concepts while sketching lowerdegree polynomial functions. I purposely do not introduce the definition of limit, as students often view the limit process as neverending, and as a result, they form the misconception that a limit is never attained (Tall & Ramos, 2004). A reflection on students’ opinions about learning the process of using limits to sketch functions concludes the first part of the paper. The second part proposes extensions of the concept to evaluate limits of higherdegree polynomial and rational functions considering the dominant terms.

Benjamin VanHavermaet earned a BS in computer science with a minor in mathematics from the University of Minnesota, Minneapolis in 2010. He is just starting a career as a software engineer.
Paul Kinion received a BA in mathematics from the University of Minnesota, Morris in 1978 and a MS in mathematics from Oregon State University, Corvallis in 1982. Mr. Kinion is a math instructor in the Minnesota State Colleges and University system.

Partial Integration
Benjamin VanHavermaet and Paul Kinion, Inver Hills CC
Introduction
"What he needed was a notion, not a notation.” Karl Friedrich Gauss’ point is well taken. In mathematics, the concept or the notion is more important than the exact notation. However, notation can make a process clearer. When teaching the concept of partial derivatives, the use of the rounded dee notation, ∂x/∂y, has two advantages. It is similar to the standard derivative notation, dx/dy, so the student recognizes they are differentiating. It is slightly different though so students also appreciate there is a subtle difference to their differentiating method. When they see the rounded dee, they are trained to differentiate with respect to one variable while holding all other variables constant.

Joanne Caniglia, Ph.D. teaches mathematics education at Kent State University. She has received many professional development grants for teachers with an emphasis on technology. Joanne’s research interests focus on utilizing multiple representations to teach mathematics.
Barbara Leapard Ph.D. teaches mathematics methods at Eastern Michigan University. She has taught courses in educational technology and has spoken extensively on meaningful tasks in the middle school classroom. Barbara is the author of many articles on mathematics education and technology.
Andreas Lindner, Ph.D. is chair of the Pedagogical Academy of Lower Austria of the Austrian GeoGebra Institute. His research is to develop materials for students and teachers to use using GeoGebra. Andreas was a key organizer of the International GeoGebra Conference.

Visualizing Mathematics by Importing Images into GeoGebra
Joanne Caniglia, Kent State University; Barbara Leapard, Eastern Michigan University and Andreas Lindner
Abstract
GeoGebra is dynamic mathematics software for all levels of education that combines arithmetic, geometry, algebra and calculus. It offers multiple representations of objects in its graphics, algebra, and spreadsheet views that are all dynamically linked. By inserting images, students can not only visualize the mathematics through another representation, but also see the utility of mathematics. Geoegebra is an open source software, which allows anyone to use Geogebra for free, and because it is a Java application, it can run on any platform, making Geogebra accessible to a wide audience. In this article we illustrate how images (from multiple sources) can be inserted into Geogebra and how pictorial images demonstrate the relevance and utility of mathematics for college mathematics students.


The Problem Section
Welcome to the return of 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 Tyszka, and Joe Browne (editor) (at Onondaga Community College, NY).


