This Issue’s Features
Binary Powering in Ancient India
J.B. Thoo, Yuba College
Characteristics of Questions that Promote Rich Mathematical Discussions
Holly Zullo, Carroll College; Jean McGivneyBurelle, University of Hartford; Kelly Cline, Carroll College; Ann Stewart, Hood College; Chris Storm, Adelphi University; David A. Huckaby, Angelo State University; Tim Melvin, University of the Pacific
Pondering, Predicting and Proving: Investigating the Sums of Powers of Sine and Cosine
Larissa B. Schroeder and Ray McGivney, University of Hartford
A Simple Alternative to Teaching Average Value
Keith Nabb, Moraine Valley Community College
Let’s Teach Engineering Notation, not Scientific Notation
Philip Mahler, Middlesex Community College
College Algebra Student Organization of Knowledge through Cheat Sheets
Woong Lim, Kennesaw State University
Bridging the Gap in Mathematics Education
Jaewoo Lee, Borough Of Manhattan Community College
MathAMATYC Educator's Departments
Use It Now
Enrichments for Teaching of Infinite Series
Michael W. Ecker, Pennsylvania State University – WilkesBarre
Factoring Quadratics: Part III
Lance Hemlow, Raritan Valley Community College
Illegally Factoring Geometric Series Proves That 0.9=1
Blane Hollingsworth, Middle Georgia State College
Walking Beyond Crossroads
Inside New Life: A Grand Vision for Developmental Mathematics
Jack Rotman, Lansing Community College
The Problem Section
Take the Challenge
Joe Browne, Onondaga Community College
John Thoo is a mathematics instructor at Yuba College in Marysville, CA, where he has been since 1995. He enjoys riding his motorcycle and plans to continue riding it until he shrinks in height and can no longer reach the ground. John has an interest in the history of mathematics and has come to admire the wealth of knowledge that historians of mathematics possess.

Binary Powering in Ancient India
J.B. Thoo, Yuba College
Mathematics is a human endeavor, and the story of mathematics is part of the larger story of mankind. There are many opportunities for us to use the history of mathematics, either as a vehicle for presenting topics or as a sidebar, to draw our students into the larger story of mathematics and, thereby, increase their interest in our subject. It works because, after all, there is nothing like a good story to get one’s attention.
We present one example from the history of mathematics, Piṅgala’s method, that can be used as a sidebar in a variety of courses, from college arithmetic to elementary or intermediate algebra to linear algebra. The problem is simple: How can we find a^{n}if ais a real or complex number without having to find n – 1 products (or, worse, A^{n} if A is a square matrix)?

Holly Zullo is an associate professor of mathematics at Carroll College in Helena, MT, and has used classroom voting in calculus, linear algebra, and differential equations for several years. She is coPI on the NSFfunded grant, Math Vote, a 3year grant to study the pedagogy of classroom voting.
Jean McGivneyBurelle is an associate professor of mathematics at the University of Hartford. She received her MS from Northeastern University and her PhD from the University of Connecticut. Her research interests involve investigating how to use technology to improve the teaching and learning of mathematics.
Ann Stewart is an assistant professor of mathematics at Hood College in Frederick, MD. She uses classroom voting in many of her courses, and she is also interested in connections between mathematics and music. She serves as senior personnel on the NSFfunded project Math Vote.
Chris Storm has been an assistant professor in the Department of Mathematics and Computer Science at Adelphi University since fall 2007. He did his graduate work at Dartmouth College He is an ExxonMobil Project NExT Sun Dot (2007) Fellow. In his free time, he enjoys playing bridge and hiking.
David A. Huckaby is an associate professor of mathematics at Angelo State University. He received his BS degree from the University of Texas at Austin and his MA and PhD in applied mathematics from UCLA.
Tim Melvin received his BS in mathematics from Carroll College, his MA in mathematics from Sacramento State University and is currently working on his PhD in mathematics at Washington State University. His professional interests include linear algebra, commutative algebra, and mathematical foundations.
Kelly Cline is an associate professor at Carroll College in Helena, MT, where he has used classroom voting in calculus, linear algebra, and differential equations since 2005. He is currently PI on Math Vote, an NSF funded threeyear project to study the pedagogy of classroom voting.

Characteristics of Questions that Promote Rich Mathematical Discussions
Holly Zullo, Carroll College; Jean McGivneyBurelle, University of Hartford; Kelly Cline, Carroll College; Ann Stewart, Hood College; Chris Storm, Adelphi University; David A. Huckaby, Angelo State University; Tim Melvin, University of the Pacific
Classroom voting is a powerful pedagogy that serves to engage students in learning mathematics and provides useful immediate feedback to students and instructors about how students are interpreting the mathematics being investigated. Classroom voting is also an effective tool for initiating rich classroom discussions. The purpose of this paper is to examine the characteristics of classroom voting questions that lead to good classroom discussions. A good classroom discussion is characterized by a high level of student engagement, a large number of students participating, students articulating mathematical concepts clearly, and students responding to their peers’ comments. As a result of analyzing questions that led to good classroom discussions, we give a classification of problem characteristics that may be useful for mathematics instructors who are interested in writing questions that foster good classroom discussions. Finally, we examine the instructor’s intent in posing a question (e.g., to introduce material, deepen understanding, review material, etc.) and how it relates to laying the groundwork for good classroom discussion.

Larissa B. Schroeder is an assistant professor of mathematics at the University of Hartford. She earned an AB in mathematics from the College of the Holy Cross, an MS in mathematics from the University of North Carolina–Chapel Hill and a PhD in Curriculum and Instruction from the University of Connecticut. Her research interests are mathematics education and pedagogy.
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 has presented at numerous local, regional, and national meetings.

Pondering, Predicting and Proving: Investigating the Sums of Powers of Sine and Cosine
Larissa B. Schroeder and Ray McGivney, University of Hartford
The goal of this article is not to add to the long list of trigonometric identities found in textbooks. Rather, our purpose is to
illustrate the dual process of discovering what we believe to be a theorem and then proving it. We strongly support efforts to
find interesting applications of mathematics that connect with students’ lives in a meaningful way. At the same time, we believe there is an important place for students to engage in tasks that reflect authentic mathematical practice (i.e., research) and believe that this investigation provides such an opportunity.
The general cosine function has the form Acos(Bx – C) + D, where A, B, C, and D are real numbers. Despite having taught trigonometric identities for years, we had not realized that f(x) = sin(x) + cos(x) is a cosine function. This caused us to wonder whether sums of powers of the sine and cosine functions were also cosine functions; for example, f(x) = sin^{2}(x) + cos^{2}(x), f(x) = sin^{3}(x) + cos^{3}(x), and so forth. More generally, we wondered if sin^{n}(x) + cos^{n}(x) has the form of a cosine function for all positive integer values of n.

Keith Nabb teaches mathematics at Moraine Valley Community College (Palos Hills, IL) and he is a graduate student in the Department of Math and Science Education at Illinois Institute of Technology (Chicago, IL). His interests include technology in mathematical learning and students’ reasoning, thinking, and beliefs.

A Simple Alternative to Teaching Average Value
Keith Nabb, Moraine Valley Community College
Finding the average value of a continuous function on an interval has several important applications (e.g., average wait time in a queue, average drug concentration in the bloodstream, and average power in a circuit). Although the relevance of the idea is unquestioned, the way it is introduced to students is often sterile and theoretical. Given this, an alternative to teaching this important idea is suggested. This approach is so intuitive that it can be done with students (not to students) and with adequate guidance, the results can be generated by the students themselves. Moreover, each of the Mean Value Theorems (MVT) from calculus—both integration and differentiation—enjoy a muchneeded reunion in class.

Phil Mahler has taught mathematics at community colleges since 1977. He has authored textbooks in trigonometry and college algebra. He is a past president of NEMATYC and AMATYC and was active in the creation of AMATYC’s Beyond Crossroads document and its Project ACCCESS. He has participated in activities at the national level on quantitative literacy and college algebra reform.

Let’s Teach Engineering Notation, not Scientific Notation
Philip Mahler, Middlesex Community College
The purpose of this article is to review some reasons why scientific notation was developed, and was very useful in the past, and to make the case that only one form of it, called engineering notation, is useful today.
We will note the following: numeric approximation has always been important; numeric results led to discoveries in pure mathematics; because numeric calculation is inherently difficult there was always a need to shorten the work involved; the development of computation by logarithmic properties helped serve this need; that computation by logarithms, including the slide rule, required what we call scientific notation.
We will then argue that with the advent of ubiquitous powerful calculating devices, although we still need scientific notation, we should teach and use it in the format known as engineering notation.

Woong Lim is an assistant professor of mathematics education in the Department of Secondary and Middle Grades Education at Kennesaw State University in Atlanta, GA. He earned his Master’s in mathematics as well as an EdD in mathematics education at the University of Houston. His primary research and teaching focus on developing teachers’ mathematics knowledge for teaching and understanding how teachers experience change.

College Algebra Student Organization of Knowledge through Cheat Sheets
Woong Lim, Kennesaw State University
Some college professors allow the use of student cheat sheets, perhaps with the hope that this can reduce test anxiety, help students organize and connect ideas, and reduce working memory strains. When students make cheat sheets, they decide which information to include and how to record it on a blank piece of paper, employing a variety of methods to express information or knowledge. This study aims to inquire about the following:
(a)In what ways do college algebra students organize their mathematical knowledge in cheat sheets?
(b)To what extent does there exist differences between strong mathematics students and weak mathematics students as to the organization of knowledge?
This article describes the efficacy of studentdeveloped cheat sheets with respect to student learning in a college algebra course taught at a community college. Some emerging patterns found in cheat sheets can help to make sense of students’ ways to select appropriate knowledge, structure, or style, which in turn can inform instruction.

Jaewoo Lee is an associate professor in the Department of Mathematics of Borough of Manhattan Community College, The City University of New York. He received his PhD in mathematics from The City University of New York, Graduate Center. He is actively mentoring students through various research projects. He is also a member of mathematics major committee, promoting mathematics to students. His research interests include combinatorial and additive number theory, combinatorics, and discrete geometry.

Bridging the Gap in Mathematics Education
Jaewoo Lee, Borough Of Manhattan Community College
Today’s students need a comprehensive and wellillustrated curriculum of mathematics. In this article, we are going to propose a classroom activity that is suitable for beginning calculus students as well as for students who are just beginning to learn to write proofs. This activity will connect the two main branches of mathematics, namely, algebra and geometry (or, more broadly interpreted, discrete mathematics and continuous mathematics). As modern mathematics shows, understanding the interaction between these two areas is not only helpful but critical for fostering the future of mathematics. This importance was understood by JosephLouis Lagrange, who said, "As long as algebra and geometry traveled separate paths, their paths were slow and their applications [were] limited. But when these two sciences joined company, they drew from each other fresh vitality and thenceforth marched on at a rapid pace toward perfection.” What follows is a suggestion on how to simultaneously introduce the concept of sequences (from discrete mathematics) and limits (from continuous mathematics), instead of presenting each as a separate subject.

Michael W. Eckeris associate professor of mathematics at Pennsylvania State University’s WilkesBarre campus. He received his PhD in mathematics from the City University of New York in 1978, founded The AMATYC Review problem section in 1981, and has posed and solved hundreds of problems in many mathematics journals. As a recreational mathematician and computer enthusiast, he created several recreational mathematics computer columns in the 1980s. From January 1986 to January 2007, he edited and published his own newsletter REC (Recreational and Educational Computing). The owner of over 100 computers, at last count, Michael is the author of 500 newsletters, columns, reviews, and articles, many computerrelated, as well as five books and solutions manuals.

Enrichments for Teaching of Infinite Series
Michael W. Ecker, Pennsylvania State University – WilkesBarre
What is the sum of the following?

Lance Hemlow has been an assistant professor of mathematics at Raritan Valley Community College, Somerville, NJ, since 1993. He received his MA in mathematics from Western Connecticut State University, and his EdM and EdS degrees in mathematics education from Rutgers University. He enjoys teaching gifted children as well as teaching in prison. Outside of mathematics, he is happily married and enjoys chess, Lincoln Center, and good pasta!

Factoring Quadratics: Part III
Lance Hemlow, Raritan Valley Community College
It is all Stephen Kaczkowski’s fault. After reading his article, Factoring Quadratics, it has been a mission to find further families of quadratics that factor under all four cases in Z[x], the ring of polynomials with integer coefficients. That is, ++, + −, − +, and − −. One set of generators was previously found in Factoring Quadratics: Part II. Now, another has been located.

Blane Hollingsworth received his PhD from Auburn University in 2008. From 2008 to the present, he has been a professor at Middle Georgia State College in Macon, GA.

Illegally Factoring Geometric Series Proves That 0.9=1
Blane Hollingsworth, Middle Georgia State College
Infinite series can be difficult for students to understand, and confusion can quickly lead to disinterest. Offering students an easy result that allows them to do something "illegal” to show the alwayspleasing result that 0.9=1can definitely pique their interest. The following "illegal” factoring formula came about during a review day in Calculus 2:
where b > 0.

Jack Rotman has been at Lansing Community College since 1973, with a focus on "developmental” mathematics. He has an MA from Michigan State University. He has been active in the state professional organizations, with multiple presentations and offices. Nationally, Jack has contributed to the AMATYC Standards (both Crossroads and Beyond Crossroads), and has chaired the AMATYC Developmental Mathematics Committee twice for a total term of 9 years. Currently, he is leading a team working on a project to reinvent developmental mathematics – the AMATYC "New Life for Developmental Mathematics” group of the Developmental Mathematics Committee. Jack was involved as an AMATYC content liaison for the "Pathways Grants” of the Carnegie Foundations for the Advancement of Teaching, and has also been involved with the work on the Dana Center New Mathways Project. Jack seeks to combine an understanding of mathematicians, of college mathematics, and of cognitive psychology to bring a new perspective on mathematics in the first two years.

Inside New Life: A Grand Vision for Developmental Mathematics
Jack Rotman, Lansing Community College
(Complete Article)
The traditional developmental mathematics model involves a sequence of algebra based courses meant to provide content similar to that found in middle school and high school mathematics classes. As a curricular design, this methodology results in sequences of developmental mathematics courses for many students; evidence exists suggesting that our current remediation does not work well enough. Through the dedication of professionals across the country, the New Life model for precollege mathematics seeks to create a positive alternative. Our basic design is based on the AMATYC standards (Beyond Crossroads), as well as previous work of MAA, NADE, and the Numeracy Network.
This article serves as a basic primer into the design of the New Life model. The New Life Project is a subcommittee of AMATYC’s Developmental Mathematics Committee; our work represents professional judgment and collaboration, rather than official AMATYC actions.

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.

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.

