This Issue’s Features
Bridging Neuroscience and Technology for Teaching and Learning Mathematics
Alexander Y. Vaninsky, Hostos Community College
Beyond the Textbook: Getting Developmental Mathematics Students Involved in Learning
Janet E. Teeguarden, Ivy Tech Community College
Improving the Efficacy of WebBased Learning in Remedial Mathematics Instruction
Shenglan Yuan and Yelena Baishanski, LaGuardia Community College
South Korea: A Success Story in Mathematics Education
Fary Sami, Harford Community College
An Investigation of Technological Options in Developmental Mathematics
SelinaVásquezMireles, Thersa Westbrook, Debra Ward, Texas State UniversitySan Marco, and Cristella R. Diaz, Northeast Lakeview College
Flipping Three Different Mathematics Courses: Common Conclusions and Plans for the Future
Larissa Schroeder, Fei Xue, and Ray McGivney, University of Hartford
MathAMATYC Educator's Departments
Use It Now
Relating Limits and Infinite Geometric Series through Application
Christopher W. Jones, Harford Community College
Integration of Logic Circuit Boards in Teaching Truth Tables
Channa Navaratna and Yong S. Colen, Indiana University of Pennsylvania
MenakaNavaratna, Florida Gulf Coast University
Using Licorice to Reinvent Trigonometric Functions
Nicole EngelkeInfante, West Virginia University
Patrick Kimani, California State University, Fullerton
Exploring Calculus Problems with GeoGebra
Dae S. Hong and James Kennis, Hostos Community College
Jae Ki Lee, Borough of Manhattan Community College
The Problem Section
Take the Challenge
Joe Browne, Onondaga CC
Alexander Yan Vaninsky is a professor of mathematics at Hostos Community College, City University of New York. He received his masters’ degrees in electrical engineering from Moscow Power Engineering Institute, in mathematics from Moscow State University, and in education from Brooklyn College. He also holds PhD and DSc degrees in mathematical economics from Moscow Finance University. His research interests include applied mathematics and technologybased mathematics education.

Bridging Neuroscience and Technology for Teaching and Learning Mathematics
Alexander Y. Vaninsky, Hostos Community College
Teaching and learning may be viewed from two different perspectives. The conventional approach is based on the transfer of knowledge from instructor to student via verbal, visual, spatial, or other means. The neuroscience approach views teaching and learning as a sequence of processes aimed at structuring memory: it understands knowledge transfer as a direct impact on the human brain that forms domains which eventually act as knowledge centers.
This paper suggests that using technology in the mathematics classroom serves to implement the teaching tools suggested by the neuroscience approach. It presents evidence that technology contributes to the optimization of the process by which information is accumulated in working memory. In addition, technology sets the appropriate pace of information transfer from working memory to longterm memory. Technology allows for an increase in the intensity of knowledge transfer from instructor to student, while avoiding any overload on the students’ working memory. On the other hand, absence of technology could likely lead to student overload and result in mathematical anxiety It would also increase the workload for instructors trying to impart new knowledge in smaller portions with more frequent assessments. This paper considers the role of technology based on a partial implementation of this approach in a community college.

Janet Teeguarden, Professor of Mathematics at Ivy Tech Community College, was awarded both a NISOD Excellence Award and an AMATYC Teaching Excellence Award in 2011. She has taught at Ivy Tech since 2000, following 20 years on the faculty at DePauw University. Janet has taught all levels of undergraduate mathematics, but has a special love for the developmental mathematics she currently teaches. She is a recent department chair, former president of InMATYC, and former secretary and board member of ICTM. She is a frequent presenter at local, state, and national conferences and serves as a reviewer for the MathAMATYC Educator.

Beyond the Textbook: Getting Developmental Mathematics Students Involved in Learning
Janet E. Teeguarden, Ivy Tech Community College
Many college mathematics classes are taught in a traditional manner, using traditional examples from traditional textbooks. Perhaps the instructor begins by answering questions about the homework and then lectures on new material. The instructor shows various examples worked out on the board, while the students dutifully copy down these examples. In some classes, students are given the chance to try to work similar problems on their own if time permits. Students are then expected to practice these new skills by doing homework, usually problems closely resembling the examples the instructor worked in class. This homework may even be done online, with help and hints, but the problems still mimic the skills that the instructor demonstrated.
But how much learning is going on?Are students learning to merely parrot what the instructor or computer demonstrates, or do they truly understand concepts and develop the ability to apply their new knowledge to reallife situations?

Shenglan Yuan is an associate professor of Mathematics at LaGuardia Community College. She received her Ph.D. from the Graduate Center of City University of New York. In addition to her field of specialty, complex dynamics, she pursues the improvement of undergraduate STEM education and remedial mathematics.
Yelena Baishanski is an assistant professor of mathematics at LaGuardia Community College. She earned her Ph.D. in Number Theory at the Graduate Center of the City University of New York after completing degrees at Harvard and Université de Paris VII. She has presented at various conferences, most recently the 12th International Congress on Mathematical Education.

Improving the Efficacy of WebBased Learning in Remedial Mathematics Instruction
Shenglan Yuan and Yelena Baishanski, LaGuardia Community College
It is the twentyfirst century. We can decode the human genome, find ice on Mars, and discover a Higgs boson, but we are still struggling to educate our students in mathematics. Nearly half of community college enrollees require remedial instruction in mathematics; fewer than a third of these students ever move beyond it (Bailey & Thomas, 2009). The consequences on mathliteracy levels and student access to higher education are severe. At the City University of New York, according to a New York Times editorial, 57% of students could not pass the mandated algebra course, and a faculty report found that "failing math at all levels affects retention more than any other academic factor” (Hacker, 2012).
Attempting to meet the needs of remedial students, colleges have multiplied the number, variety, and format of remedial courses, notably expanding online instruction and assessment. Webbased learning systems have been praised for their flexibility, giving access to innumerable exercises, examples, and solutions at the click of a button. They provide instant evaluation, reduce grading time for instructors, allow for easy compilation of student performance data, and their costeffectiveness only adds to their appeal. But for all its advantages, webbased learning (WBL) has yet to fulfill the potential of twentyfirst century technology to meet remedial students’ needs. We outline here some of the clear successes of WBL in remedial instruction, identify drawbacks to learning that certain features of WBL can pose, and propose alterations to WBL that we feel would help our students.

Fary Sami is a professor of mathematics within the STEM Division at Harford Community College, Bel Air, MD. Fary is a cochair of the International Education Subcommittee of the Chairs Division of AMATYC. She also serves as a member of the editorial team of the problem section of the MathAMATYC Educator.

South Korea: A Success Story in Mathematics Education
Fary Sami, Harford Community College
(Complete Article)
South Korea has experienced enormous economic growth over the last 50 years and now has the 13th largest economy in the world. During the period 1961–2011, per capita gross domestic product increased from $155 to $22,424. Much of South Korea’s economic development has been attributed to improvements in its public education system. South Korean students consistently perform in the top rankings among countries according to international studies such as Trends in International Mathematics and Science Study (TIMSS), and the Program for International Student Assessment (PISA).
This article summarizes some of the highlights of the author’s learning experiences during her recent trip to South Korea. The author attended The 12th International Congress of Mathematical Education held July 815, 2012, in Seoul, South Korea. She visited the Department of Education at EwhaWomans University in Seoul, a highly rated female educational institute in South Korea. There, she met with Noh SunSook, Dean of the College of Education, and discussed the teacher preparation programs for elementary, secondary, and special education teaching.
The author’s experiences led her to the conclusion that the principal factors responsible for Korea’s success in education included equity in educational opportunity for all students regardless of their socioeconomic background and extremely active parental involvement. There is high academic rigor in Korean teachertraining programs followed by continuous teacher support and professional development and a research based pedagogy that utilizes the best teaching practices. However, the intensity of the education including afterschool programs referred to as "hagwons” has pitfalls as it leaves little time for personal life outside of school for students or teachers.
This article is based on readings, classroom visits, and interviews with Korean public school teachers, and the national presentation on Korean education system during ICME12. In this article, the author shares her experiences and presents the mathematics content of the primary and secondary schools of the most recent curriculum revision that will be implemented in 2013.

SelinaVásquezMireles is a professor in the Department of Mathematics and Director of the Center for Mathematics Readiness at Texas State University–San Marcos. She has created, implemented and evaluated the effectiveness of several models and programs in postsecondary mathematics including FOCUS, a corequisite model.
Thersa Westbrook received her Ph.D. in mathematics education from Texas State University–San Marcos. She is a lecturer/researcher at Texas State University–San Marcos, where she works in the Center for Mathematics Readiness. Her research interests are the performance and achievements of developmental mathematics students in postsecondary education, college and career readiness, and statistics education.
Debra Ward is a doctoral student at Texas State University–San Marcos, where she is studying mathematics education. While her current research is focused on mathematical problem solving and musical training, she continues to investigate the benefits of incorporating technology in the mathematics classroom.
Cristella R. Diaz is an associate professor of mathematics at Northeast Lakeview College with 20 years of postsecondary teaching experience. She is pursuing an Ed.D. at Texas State University–San Marcos in developmental education with emphasis in learning support. Her research interests include mathematics anxiety, selfefficacy, and Hispanic success in mathematics.

An Investigation of Technological Options in Developmental Mathematics
SelinaVásquezMireles, Thersa Westbrook, Debra Ward, Texas State UniversitySan Marco, and Cristella R. Diaz, Northeast Lakeview College
The purpose of this paper is to discuss an investigation of the infusion of multiple technology modalities, such as graphing calculators and the Internet, in the developmental mathematics classroom through four topics: systems of linear equations, radical expressions, radical equations, and quadratic equations. Analysis of pre and posttest scores produced statistically significant results in favor of the treatment group for the quadratic equations lesson. Moreover, with statistical significance, 100% of the students in the treatment group correctly answered the final exam question associated with the radical expressions lesson plan. Overall, students either marginally maintained or benefited from the use of multiple technology modalities. Qualitative evidence supports a technology "no harm” effect through substantial trends of students’ perceptions of technology as either calculator usage or required use of textbookrelated software.

Larissa B. Schroeder is an assistant professor of mathematics at the University of Hartford. She earned an A.B. in mathematics from the College of the Holy Cross, a M.S. in mathematics from the University of North Carolina – Chapel Hill and a Ph.D. in curriculum and instruction from the University of Connecticut. Her research interests are mathematics education and pedagogy.
Fei Xue is an assistant professor of mathematics at University of Hartford. He earned his B.S. degree from South China University of Technology, and Ph.D. from the West Virginia University. His research interests are asymptotic analysis of differential and difference systems and pedagogical calculus research.
Ray McGivney is a professor of mathematics at the University of Hartford. He earned his A.B. and M.A. in mathematics at Clark University and his Ph.D. in mathematics at Lehigh University. He has served as mathematics consultant for several school systems in Connecticut and has presented at numerous local, regional, and national meetings.

Flipping Three Different Mathematics Courses: Common Conclusions and Plans for the Future
Larissa Schroeder, Fei Xue, and Ray McGivney, University of Hartford
There are numerous models of flipped classrooms at the college
level. However, most involve science and engineering classes. In January 2012, our department began discussions of flipping our 4credit introductory calculus course (Calculus I). Independently, the authors decided to experiment with flipping three very different courses – Precalculus with Trigonometry, Calculus II, and Discrete Mathematics II. The first two courses were flipped for several weeks; the last course was flipped throughout the semester. We were particularly interested in the question of whether there were common themes and experiences in flipping our classes that occurred irrespective of the material, abilities, or mathematical maturity of our students. Answers to this question form the basis of this article.
"Flipping a course” refers to an instructional approach in which content (presentations, solved problems, definitions, etc.) is presented before class through online videos, lecture notes, and readings. The ability to pause, rewind, and fast forward these videos enables students to learn at their own pace.
At the same time "homework,” often completed in small groups, is moved into the classroom. Therefore, instructors in flipped courses are free to devote class time to assisting groups of students who are engaged in collaborative discussion and problem solving.

Chris Jones is an assistant professor at Harford Community College. He serves as the cochair for Harford’s Learning Assessment Committee and is the advisor for the Rho Beta Chapter of Phi Theta Kappa. Chris has developed and instructed a variety of courses and programs at Harford Community College and is a recipient of the NISOD award. He holds a MS in applied mathematics from Towson University.

Relating Limits and Infinite Geometric Series through Application
Christopher W. Jones, Harford Community College
One challenging aspect of introducing infinite series to a calculuslevel student is presenting applications that are interesting and promote a deeper level of thinking of the composition and behavior of infinite series. Traditional application problems of infinite series are usually introduced when instructing geometric series, since these sums are easy to compute. However, many of these applications simply find the sum and ignore the infinite limit process. The activity that follows allows students to enhance their critical thinking about infinite series and the connection between the sum and infinite process.
One of the major goals of this investigation is to facilitate student comprehension of the relationship, similarities, and differences that exist among sequences (closed forms) and infinite series. Additionally, this study may better connect the relationship between infinite limits and infinite series. Finally, the intention is for students to investigate structures that contain forms of geometric series from both a numeric and analytical view.

Channa Navaratna graduated with a PhD in Applied Mathematics from Texas Tech University in 2003. He worked as a postdoctoral research associate at the department of Biology after his graduation. He has a Bachelors degree in Electronic and Electrical Engineering, and he is very much interested in multidisciplinary research. He is currently employed as an associate professor at Indiana University of Pennsylvania.
Yong S. Colen is an associate professor of mathematics at Indiana University of Pennsylvania, Indiana, PA. He is interested in developing challenging, gifted curricula for mathematicallytalented children, and exploring ways to incorporate meaningful applications, technology, and history of mathematics into mathematics curriculum.
MenakaNavaratna is an associate professor in the Department of Chemistry and Mathematics at Florida Gulf Coast University, Fort Myers, FL. He received a M.S. and Ph.D. in Mathematics from the Texas Tech University, Lubbock, TX. Besides teaching, his interests include biological modeling using differential equations, characteristics of oscillatory dynamical systems, and mathematics education.

Integration of Logic Circuit Boards in Teaching Truth Tables
Channa Navaratna and Yong S. Colen, Indiana University of Pennsylvania
MenakaNavaratna, Florida Gulf Coast University
Truth tables are the basic elements behind the "thinking” power of electronic devices. Basic modules of semiconductor electronics are generally referred to as logic gates. These gates can make decisions based on the presented inputs (statements). To make complicated decisions, a typical processor chip contains a large collection of these gates.
Typically, presenting the concepts of truth tables is a difficult task. Introducing logic gates and implementing a class project can not only help students to understand the importance of truth tables, but it also inspires them to create their own projects. First, this article describes the basics of logic gates and truth tables. Then, we describe the logic trainer board used in our class experiments. Utilizing many pictures, we demonstrate how to implement circuits on the logic trainer board.

Nicole Engelke is an assistant professor of mathematics at West Virginia University. Her research focuses on how students learn and understand calculus concepts, particularly contextual questions. She is actively involved in the special interest group of the Mathematical Association of America on research in undergraduate mathematics education.
Patrick Kimani is an assistant professor in the Mathematics Department at California State University, Fullerton. He specializes in teaching mathematicscontent courses for elementary and middle school preservice and inservice teachers. His research investigates high school and college students’ understanding of mathematics concepts and their perceptions of the relationships among these concepts.

Using Licorice to Reinvent Trigonometric Functions
Nicole EngelkeInfante, West Virginia University
Patrick Kimani, California State University, Fullerton
Functions are a critical part of today’s mathematics. It took hundreds of years for the function concept to develop into what we understand a function to be today. The idea of functions is necessary for modeling many of the phenomena that mathematicians and scientists study. The AMATYC Crossroads standards urge mathematics educators to develop in their students an understanding of functions that allows them to use appropriate function notation, interpret functions that arise in applications, and analyze functions using different representations. In developing student thinking about functions, the standards indicate that activities should optimize learning and be accessible to diverse populations of students with differing learning styles. We present an activity in which students build their understanding of the sine function. Students start with concrete physical measurements, represent these measurements as they construct a graph, and conclude with the abstract representation of the function, f(x) = sin x. Throughout the activity, students are engaged in activities that should reach tactile, visual, and auditory learners. This activity has been used successfully in college level precalculus courses with diverse populations.

Dae S. Hong is an assistant professor in the mathematics department at Hostos Community College in Bronx, NY. He is interested in problem solving, use of technology in mathematics education, and international comparative studies.
James Kennis is an assistant professor of mathematics at Hostos Community College in Bronx, NY. His professional interests include mathematics pedagogy in the developmental classroom and increasing student motivation. His personal interests include juggling and backgammon.
Jae Ki Lee is an assistant professor at Borough of Manhattan Community College. He is interested in discovering alternative teaching and learning algorithms and teaching mathematics with technology.

Exploring Calculus Problems with GeoGebra
Dae S. Hong and James Kennis, Hostos Community College
Jae Ki Lee, Borough of Manhattan Community College
GeoGebra is a free software package that is interactive and is similar to the software the Geometers Sketchpad. This software allows students to explore numerous mathematical topics such as geometry, algebra, and transformation and also provides an excellent opportunity to demonstrate graphical representations of various calculus concepts. GeoGebra can be downloaded for free from its website (www.geogebra.org) and installed on your computer. You can either download it onto your desktop or use the Web applet version that works within your Internet browser.
In textbooks, numerous calculus topics, such as the limit of a function, the derivative of a function, and integrals are often represented algebraically, numerically, and graphically. Although multiple representations are emphasized in the teaching and learning of calculus, it often appears that students still have a limited view of the graphical representation of a derivative because they are accustomed to algebraic representations.
Because of these difficulties and such a limited view, graphing calculators and other computer algebra systems (CAS) are often used to demonstrate graphical representations of a derivative and other calculus concepts. This article explores and demonstrates graphical representations of some elementary calculus topics using GeoGebra—specifically differentiability, the mean value theorem, and Riemann sums.


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 Tyszka, and Joe Browne (editor) at Onondaga Community College, NY.

