MathAMATYC Educator Feb 2011
A refereed publication of the American Mathematical Association of Two-Year Colleges
Editor: Pete Wildman, Spokane Falls CC
Production Manager: Jim Roznowski, Delta C
Volume 2, Number 2, February, 2011 Issue
Earlier and Later Issues
AMATYC Members can view entire articles of this issue
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Ed Laughbaum 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 long-term memory with recall using hand-held technology.
Capitalizing on Basic Brain Processes in Developmental Algebra - Part One
Edward D. Laughbaum
The Ohio State University
Basic brain function is not a mystery. Given that neuroscientists understand the brain's 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 long-term 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 equation-solving 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.
Upon completion of her undergraduate degree in Mathematics education at BGSU, Victoria began teaching at a local high school while earning her Master's of Education Degree from Heidelberg College. She launched her collegiate career by lecturing as an adjunct instructor at a variety of local colleges and began teaching full-time at Tiffin University in the mathematics department in 2007. Dr. Ingalls completed her doctoral education from Ashland University in the fall of 2008. She lives in Tiffin with her husband and five daughters.
Mathematics Placement Test: Typical Results with Unexpected Outcomes
Based on the results of a prior case-study analysis of mathematics placement at one university, the mathematics department developed and piloted a mathematics placement test. This article describes the implementation process for a mathematics placement test and further analyzes the test results for the pilot group. As an unexpected result, the study raises awareness concerning the use of multiple measures for mathematics placement, especially as a subjective human judgment.
Douglas B. Aichele is professor and associate head of the Department of Mathematics at Oklahoma State University, Stillwater, Oklahoma. His current interests are in the content preparation of elementary/middle level teachers, mathematical learning using technology appropriately, and mathematical applications related to sled dog racing.
Cynthia Francisco is a Lecturer in the Department of Mathematics at Oklahoma State University, Stillwater, Oklahoma. She coordinates the delivery of College Algebra and supervises the Mathematics Learning Resource Center, which serves all introductory mathematics courses.
Juliana Utley is an assistant professor of mathematics education in the School of Teaching and Curriculum Leadership at Oklahoma State University, Stillwater, Oklahoma. Her research interests include pre-service teacher preparation; teacher knowledge, attitudes, and beliefs about the teaching and learning of mathematics; and issues related to supporting and retaining early career mathematics teachers.
Benjamin Wescoatt is a doctoral student in the Department of Mathematics at Oklahoma State University, Stillwater, Oklahoma. His current interests are in research in mathematics education at the collegiate level.
Computer-Aided College Algebra: Learning Components that Students Find Beneficial
Douglas B. Aichele, Cynthia Francisco, Juliana Utley, and Benjamin Wescoatt
Oklahoma State University
A mixed-method study was conducted during the Fall 2008 semester to better understand the experiences of students participating in computer-aided instruction of College Algebra using the software MyMathLab. The learning environment included a computer learning system for the majority of the instruction, a support system via focus groups (weekly class meetings), and tutorial services.
Emerging themes for the best way to learn College Algebra were (1) use of resources (45.6% indicated View an Example, Video, or Textbook); (2) soliciting help from others (44.7% indicated tutors, time in tutoring lab, or attending Focus Group); and (3) "practice, practice, practice" (approximately 30%). Least beneficial resources identified were textbooks (traditional and electronic), videos, and focus groups.
Combining the two most mentioned computer-only resources, View an Example and Help Me Solve This, accounted for 82.3% of the responses; only 14.8% identified these singly. Interestingly, combining the single effects of Help Me Solve This and tutoring, accounted for 21.0% of the responses; students appeared to value a computer resource coupled with a face-to-face resource over strictly computer resources.
The results of this study suggested that students preferred resources that directly helped them with individual homework problems, rather than those emphasizing major concepts.
Alice Welt Cunningham, Ph.D., J.D., practiced and taught tax law for many years before turning to teaching mathematics. In addition to teaching at the community college level, she has taught junior high and high school mathematics, as well as library class to four-year-olds. She is currently Assistant Professor of Mathematics at Hostos Community College, City University of New York.
Teaching Remedial Mathematics for Conceptual Understanding: Student Response
Alice Welt Cunningham
Hostos Community College
This paper describes student response, as indicated by an anonymous questionnaire, given by the author to her remedial algebra students at a community college that is part of a large East Coast open-admissions university. The course in question is aimed at the university-wide exit mathematics test necessary to graduation and college-level coursework. The author’s approach is to teach for understanding, with the goal of promoting long-term retention rather than short-term rule memorization. Of the 30 active students (those still enrolled after the date to withdraw without penalty), 19 (or close to two thirds of those students) responded to the questionnaire. With a few exceptions, their responses suggest that teaching remedial mathematics for understanding by showing the sense behind the rules facilitates both comprehension and appreciation for the subject in this difficult course.
Kunio Mitsuma was born in Tokyo, Japan. After finishing college as a mathematics major, he moved to the U.S. for his Master's (West Virginia University) and Ph.D. (Penn State) in mathematics. He is currently on the mathematics faculty at Kutztown University of Pennsylvania.
Integration of the Quadratic Function and Generalization
Kutztown University of Pennsylvania
We will first recall useful formulas in integration that simplify the calculation of certain definite integrals with the quadratic function. A main formula relies only on the coefficients of the function. We will then explore a geometric proof of one of these formulas. Finally, we will extend the formulas to more general cases.
Fary Sami is an associate professor of mathematics within the Science, Technology, Engineering, and Math Division (STEM) at Harford Community College (HCC), Bel Air, Maryland. Fary has developed several math courses and leads the Math Curriculum of the STEM division.
Course Format Effects on Learning Outcomes in an Introductory Statistics Course
Harford Community College
The purpose of this study was to determine if course format significantly impacted student learning and course completion rates in an introductory statistics course taught at Harford Community College. In addition to the traditional lecture format, the College offers an online, and a hybrid (blend of traditional and online) version of this class. The instructor and course materials, including the text, homework, and examinations, were the same for the formats compared in this study. Student learning outcomes, as measured by examination scores, were the same, regardless of format. The course completion rate was numerically lower for online than for traditional and hybrid formats. Since inclusion of lecture in both the traditional or hybrid course format did not improve student learning, it was concluded that student learning from face-to-face lectures was not as important as their learning through readings, practicing, and applying skills.
Read the complete article here.
Brian Smith holds a Ph.D. in Mathematics from Queen's University in Canada. He currently teaches statistics and optimization 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 Two-Year College mathematics courses. Brian served as chair of the Technology in Mathematics Education Committee at AMATYC for six years.
Statistics without Tears: Complex Statistics with Simple Arithmetic
One of the often overlooked aspects of modern statistics is the analysis of time series data. Modern introductory statistics courses tend to rush to probabilistic applications involving risk and confidence. Rarely does the first level course linger on such useful and fascinating topics as time series decomposition, with its practical applications of forecasting and seasonal adjustment of data. The statistical techniques involved in presenting these topics require only basic arithmetic skills and yet comprise one of the most useful applications of statistics by industry and government.
Jeganathan "Sri” Sriskandarajah has been teaching at the Madison College since 2000 and serves as the faculty adviser of the math club since its inception in 2000. He enjoys problem solving and serves as the State Director of the American Math Competitions (WI). Sri is also a member of the AMC 8 committee. From 1985-2000, he taught at the University of Wisconsin - Richland, and prior to that at the University of Delaware- Newark, University of Zambia - Kitwe and the University of Sri Lanka - Colombo. Sri has received Teaching and Employee of the Year awards from the Madison College and a service award from the MAA.
Two Mathemagical Activities That Promote Student Engagement
Mathematics doesn’t need to be a serious, dull subject that causes many students to turn away from higher courses. On the first day of classes, I engage and excite my elementary math students in some activities. I have chosen two such activities to illustrate my point.
Keith Nabb teaches mathematics at Moraine Valley Community College (Palos Hills, IL) and he is a graduate student in the Department of Math & Science Education at Illinois Institute of Technology (Chicago, IL). His interests include technology in mathematical learning and students' reasoning, thinking, and beliefs.
A New Perspective On Related Rates
Keith A. Nabb
Moraine Valley CC
A first-semester calculus student uncovers an unusual way of solving related rates problems—one that differs noticeably from the exposition found in mathematics textbooks. A classroom discussion of this new technique is described as she and her classmates grapple with the possible validity of the technique. The ways in which this discussion deepened ties across previously learned topics is emphasized.
Efraim P. Armendariz is a professor of mathematics at the University of
Texas at Austin. His mathematical interests include noncommutative ring
theory, development of educational programs addressing accessibility
issues, and development of secondary mathematics teachers. He received
the Ph.D. in Mathematics from the University of Nebraska–Lincoln in
Mark L. Daniels is a clinical associate professor of mathematics and UTeach Natural Sciences at the University of Texas at Austin. His research interests involve the preparation of preservice teachers and the incorporation of instructional methodology in mathematics courses taken by students seeking certification.
Graphing Cartesian Functions in Polar Coordinates
Efraim P. Armendariz
University of Texas at Austin
Mark L. Daniels
University of Texas at Austin
As an introduction to the graphing of functions in polar coordinates, the authors propose graphing linear and quadratic functions expressed as y = f(x) in Cartesian coordinates and making use of translated planar vectors (directed line segments) to construct the corresponding polar graph
r = f(θ), obtained by replacing y by r and x by θ.
This nonstandard approach to introducing polar graphing does not rely on a knowledge of the trigonometric functions, but may serve to enhance the understanding of polar graphs involving trigonometric functions.
David French has been teaching mathematics at Tidewater CC in Chesapeake, Virginia for over 17 years. He received his masters in mathematics from Marshall University and bachelors from Bluefield College. He keeps busy in his spare time with three children, three dogs, and a lovely wife, unless it is football season.
A Visualization of Vectors and Trigonometry
David J. French
Tidewater Community College
After introducing students to trigonometry, most precalculus courses then follow with applications, including an introduction to vectors. Most of our students are engineering majors and have seen the concept of a vector in engineering courses or physics courses. They understand the idea of using vectors to represent certain physical quantities. However, with calculus looming for these students, I like to combine three concepts into one:
- Continue to bolster students fluency in trigonometry,
- Define and demonstrate vectors and vector operations, and
- Demonstrate a mathematical proof using vectors and trigonometry.
The proofs are visual and follow the basic properties of vectors, operations with vectors, and dot product (in two dimensions of course).
Lovejoy Das is a professor of mathematics at Kent State University. He was the recipient of the Kent State University Distinguished Teacher Award in 2004 and was given the 2005 Northeast Ohio Council of Higher Education Award for Teaching Excellence. Professor Das was also recognized in 2006 by the Ohio magazine for their Outstanding Achievements in Teaching. He has published more than 50 research papers both nationally and internationally. His current interests include mathematics education and differential geometry.
Frederick Johnson received his bachelor of science from Kent State University. He holds an undergraduate degree in leisure studies with a concentration in sports management. Frederick is currently furthering his education in adolescent and young adult education in mathematics at Notre Dame College. Frederick also plans to get his master’s degree in education as well. Currently, he is a student of Professor Das’s Calculus II class at Kent State University, Tuscarawas Campus.
A New Approach for the Divergence of the Improper Integral
Lovejoy Das and Frederick Johnson
When we define a definite integral
we consider the function f, on a finite interval [a, b]. The Fundamental Theorem of Calculus states that: If f(x) is continuous on [a, b], then
where F is any function such that F'(x) = f(x) for all x in [a, b]. The Fundamental Theorem of Calculus requires that [a, b] is finite and f is bounded on [a, b], but sometimes we need [a, b] to be an infinite interval. When we consider improper integrals, our interval could be infinite. The purpose of this article is to provide another numerical approach to look at the divergence of the improper integral
because calculus students have a difficult time visualizing the divergence of this integral. This new approach will help students to comprehend the concepts of convergence and divergence of improper integrals.
Frank Wilson teaches students mathematics at Chandler-Gilbert Community College. Educators worldwide use his learning activities and textbooks focused on helping students discover practical real world applications of mathematics.
Creating Mathematical Models
Frank C. Wilson
Chandler-Gilbert Community College
Download Activity: Page 1Page 2
Three months into the semester, I asked my college algebra students, "What is the most significant thing you have learned in this class this semester?” Although some students identified a specific concept or procedure, more than 25% of the students reflected on the value of real world applications in mathematics. Consider these representative responses.
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 two-year 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 email@example.com 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).