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MathAMATYC Educator September 2012
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MathAMATYC Educator

A refereed publication of the American Mathematical Association of Two-Year Colleges

Editor: Pete Wildman, Spokane Falls CC
Production Manager: George Alexander, Madison Area Technical College

Volume 4, Number 1, September 2012 Issue
Earlier and Later Issues

AMATYC Members can view entire articles of this
issue by clicking on the button below

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This Issue’s Features

Using Group Quizzes to Engage Students in Learning Calculus
Fei Xue and Jean McGivney-Burelle, University of Hartford

The Use of Group Quizzes in Developmental Mathematics Courses
Ian Sorensen, Utah Valley University

Factors That Shape Curricular Reasoning about College Algebra Reform
Helen E. Burn, Highline CC

Moving from Anecdote to Evidence: A Proposed Research Agenda in Community College Mathematics Education (Complete Article)
Ann Sitomer, Portland CC; April Ström, Scottsdale CC; Vilma Mesa, University of Michigan; Irene Mary Duranczyk, University of Minnesota; Keith Nabb, Moraine Valley CC; John Smith, Pellissippi State CC; Mark Yannotta, Clackamas CC

Interviewing Students: An Assessment and Learning Opportunity in Mathematics
Victor Odafe, Bowling Green State University

Computer-aided Instruction: College Algebra Students’ Perceptions
Douglas B. Aichele, D. Rae Tree, Juliana Utley, and Benjamin Wescoatt, Oklahoma State University

MathAMATYC Educator's Departments

Use It Now

Unifying Results via L’Hôpital’s Rule
Michael W. Ecker, Pennsylvania State University/Wilkes-Barre

How Large Should a Statistical Sample Be?
Violeta C. Menil and Ruili Ye, Hostos CC (CUNY)

Teaching Integer Operations Using Ring Theory
Jenna Hirsch, Borough of Manhattan CC

What Do You Find? Students Investigating Patterns in Pascal’s Triangle
Samuel Obara, Texas State University

Technology

Using Websites as Resources to Supplement Instruction
Amber Rust, Anne Arundel CC, Arnold, MD

The Problem Section

Take the Challenge
Joe Browne, Onondaga CC


Fei Xue is an assistant professor of mathematics at University of Hartford. He earned his B.S. degree in applied mathematics from South China University of Technology in 2001 and his Ph.D. in mathematics, from the West Virginia University in 2006. His interests are asymptotic analysis of differential and difference systems, time scales, and pedagogical calculus research.


Jean McGivney-Burelle is an associate professor of mathematics at the University of Hartford. She received her M.S. from Northeastern University and her Ph.D. from the University of Connecticut. Her research interests involve investigating how to use technology to improve the teaching and learning of mathematics.

Using Group Quizzes to Engage Students in Learning Calculus
Fei Xue and Jean McGivney-Burelle, University of Hartford

As noted in Beyond Crossroads (AMATYC, 2006), for today’s students, learning mathematics is participatory and depends on the active involvement of students (p. 53). The National Council of Teachers of Mathematics shares the point of view that the teaching and learning of mathematics should include giving students ample opportunity to think about, write about, and discuss mathematical problems and ideas with their peers (NCTM, 2000). The point that both of these professional organizations recognize is the majority of students do not learn mathematics by simply sitting in a classroom, listening to a teacher, recording notes, memorizing assignments and regurgitating answers. Rather, they must read mathematics, reflect on it, talk about it, write about it, and relate it to their prior knowledge. Simply put, they must be actively engaged in the process of constructing their mathematical knowledge.


Michael W. Ecker is associate professor of mathematics at Pennsylvania State University’s Wilkes-Barre campus. Having taught college mathematics since 1972, he received his Ph.D. 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 numerous journals. He created several recreational math computer columns in the 1980s, and from 1986 until 2007, he edited and published his own newsletter, Recreational & Educational Computing.

Unifying Results via L’Hôpital’s Rule
Michael W. Ecker, Pennsylvania State University/Wilkes-Barre

Can the usage of L’Hôpital’s rule be stretched to unify seemingly unrelated results? Over the years, I’ve explored, discovered, and collected unusual and curious instances where this famed theorem may be applied. They unify results and yield insight into special or limiting cases as continuous extensions of general results. Some are fun, too, even if impractical.


Amber Rust has been an adjunct and a full-time mathematics instructor at community colleges since 1993. She recently earned a Ph.D. from the University of Maryland in Mathematics Education and has joined the mathematics faculty at Anne Arundel Community College in Maryland. Her interests are reading and literacy issues in mathematics and study skills. She has presented on these topics at local and AMATYC conferences.

Using Websites as Resources to Supplement Instruction
Amber Rust, Anne Arundel CC, Arnold, MD

A couple of years ago, I gave a presentation to a group of secondary-school girls who were attending a workshop at a community college. The workshop’s purpose was to encourage the girls to keep taking mathematics courses while in high school and beyond, in order to increase their future opportunities. The presentation was titled, Where Can I Find Help with Math When the Teacher or the Textbook Don’t Make Sense? Besides, It’s Really Late and I’m Tired! It was meant to make them aware, as digital natives, of the vast resources available on the Internet to help with their homework. Unexpectedly, I found that the teachers who accompanied the girls seemed to be more interested in the information presented. Those teachers and the mathematics instructors at the community college have found the resources I compiled to be very useful.


Since retiring from the Navy, Ian Sorensen has taught developmental mathematics for four years as an Assistant Professor at Utah Valley University in Orem, UT.

The Use of Group Quizzes in Developmental Mathematics Courses
Ian Sorensen, Utah Valley University

For a period of four semesters, the possibility was explored of using a "group quiz” as a learning activity that provides a collaborative learning environment, a review of the previous week’s material, and a formative assessment for both the student and the instructor. Using both quantitative (i.e., student surveys) and qualitative (i.e., student interviews) methods, this article explores the effectiveness of the learning activity in a college-level developmental mathematics course. In addition, a preliminary quantitative analysis will be discussed regarding the success of the quiz using success rates of classes that did and did not incorporate the learning activity. Moreover, since the student’s grade for the quiz is a self-assessed score, the relationship was investigated between the self-assessed score and the outcome of exam grades using a regression study. The results demonstrate that the majority of students found the "group quiz” beneficial and desirable. The preliminary qualitative analysis showed an increase in success rates suggesting that the activity is effective in knowledge retention. Finally, the self-assessed grade revealed a surprising correlation in predicting students’ average exam scores.


Helen Burn is an instructor of mathematics at Highline CC and director of the Curriculum Research Group. She has served as department coordinator and division chair at Highline. She holds an M.S. in mathematics from Western Washington University and a Ph.D. in higher education from the University of Michigan.

Factors That Shape Curricular Reasoning about College Algebra Reform
Helen E. Burn, Highline CC

This multiple case study explores factors that shaped the curricular reasoning of mathematics faculty engaged in college algebra reform in community colleges. Overall, the study found that although proposed reform of college algebra is broad in scope and influenced by the student audience, competing influences emerge that can enable or constrain faculty curricular reasoning, such as course transferability, departmental culture, and teaching norms. The findings of this study are useful to mathematics faculty and other stakeholders in the reform arena who want to understand and respond to competing influences as they engage in reform efforts in the community college context.


Violeta Menil is an associate professor at the Mathematics Department of Hostos CC of the City University of New York. Her research career started with a computer simulation (Monte Carlo study) evaluating two multidimensional scaling algorithms for fitting the weighted Euclidean distance model (Ph.D. dissertation). She has authored 12 publications and was awarded 2 research grants on developmental mathematics. Her interests focus on multidimensional scaling, univariate and multivariate statistics.


Ruili Ye is an assistant professor at Hostos CC. Her Ph.D. thesis was a first-order logic formalization of the sense and reference notions introduced by the German philosopher Gottlob Frege. Her interests include topics in mathematical and philosophical logic as well as a keen interest in Bertrand Russell.

How Large Should a Statistical Sample Be?
Violeta C. Menil and Ruili Ye, Hostos CC (CUNY)

This study serves as a teaching aid for teachers of introductory statistics. The aim of this study was limited to determining various sample sizes (n) when estimating population proportion, p. Tables on sample sizes were generated using a C++ program, which depends on population size (N), degree of precision or error level (E), and confidence level (Z). Nineteen different population sizes, five degrees of precision, and three levels of confidence were utilized. The study found out that the larger the population size, (N), the higher the degree of precision, (E) and the higher the probability/confidence level, (Z), the larger the sample size must be. Two values for the sample estimate () of the population proportion (p) were used in this study. Practical applications of randomly pulling appropriate number of samples from huge data sets were also discussed.


Ann Sitomer teaches mathematics at Portland CC in Oregon. Her research interests include students’ informal ways of reasoning about mathematics and their experience of the community college mathematics curricula.


April Ström is mathematics faculty member at Scottsdale CC in Arizona. She received her Ph.D. in mathematics education from Arizona State University and is currently the principal investigator for the NSF-funded Arizona Mathematics Partnership (AMP) project.


Vilma Mesa is assistant professor of education at the University of Michigan. She investigates the role that resources play in developing teaching expertise in undergraduate mathematics, specifically at community colleges and in inquiry-based learning classrooms.


Keith Nabb teaches mathematics at Moraine Valley CC (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.

Moving from Anecdote to Evidence: A Proposed Research Agenda in Community College Mathematics Education (Complete Article)
Ann Sitomer, Portland CC; April Ström, Scottsdale CC; Vilma Mesa, University of Michigan; Irene Mary Duranczyk, University of Minnesota; Keith Nabb, Moraine Valley CC; John Smith, Pellissippi State CC; Mark Yannotta, Clackamas CC

AMATYC’s recently adopted five-year strategic plan, Opening Doors through Mathematics (www.amatyc.org/documents/strategicplans.htm) describes five priorities. The second of these priorities highlights the need for research on student learning. In this article, we propose a four-strand research agenda, each strand illustrated with a potential research study, offering a vision for research in community college mathematics education that connects to AMATYC’s strategic plan. In particular, we  discuss the ways mathematics education research may inform our work as community college mathematics teachers.

Irene Duranczyk is an associate professor in the Department of Postsecondary Teaching and Learning at the University of Minnesota with an Ed.D. from Grambing State University, Louisiana. She has taught developmental mathematics since 1990 and is the recipient of the 2007 National Association for Developmental Education’s (NADE) Outstanding Research Conducted by a Developmental Education Practitioner Award.


John Smith is an assistant professor at Pellissippi State CC in Tennessee. He is also a doctoral candidate and lecturer in mathematics education with the Department of Theory and Practice of Teacher Education at the University of Tennessee. Research interests include affective issues in the learning and teaching of mathematics and equity in mathematics education.


Mark Yannotta is a mathematics faculty member at Clackamas CC in Oregon. He is currently working on his Ph.D. in Mathematics Education at Portland State University. His research interests include mathematics bridge courses and support systems for transferring STEM students.


Victor Odafe is an associate professor of mathematics and the chair of the Natural and Social Sciences Department at Bowling Green State University, Firelands, in Huron, OH. His professional interests include teaching and assessment strategies in mathematics and the use of mathematics cases in the classroom. Victor has been the recipient of both the Distinguished Teacher Award and the Distinguished Creative Scholar Award from BGSU Firelands.

Interviewing Students: An Assessment and Learning Opportunity in Mathematics
Victor Odafe, Bowling Green State University

The Beyond Crossroads document (AMATYC 2006) contains the Standard for Assessment of Student Learning. Its implementation standard requires that faculty use results from assessment to improve instruction. In doing this, each faculty is expected to employ multiple assessment techniques.

Interviewing students enables us to gain insight into students’ conceptual knowledge and reasoning as they engage in mathematical problem solving. Paper-and-pencil tests have the potential to mask students’ misunderstandings and misconceptions. Writing down the correct answer to a problem does not necessarily imply that students possess the correct reasoning. Advantages of using interviews include the opportunity to understand students’ level of understanding, identify misconceptions, and assess their ability to communicate mathematical knowledge. The most obvious disadvantage of the interview strategy is time. This barrier can be removed through careful planning, organization, and practice. Students’ responses on the beneficial and less than beneficial aspects of the interview assessment technique will be shared later.


Jenna Hirsch is an assistant professor of mathematics at Borough of Manhattan CC (BMCC) of the City University of New York. She received her Ph.D. from Teachers College, Columbia University. Her professional interests include remediation and student understanding of abstract algebra. In her spare time, Jenna’s daughters, husband, and pets keep her busy.

Teaching Integer Operations Using Ring Theory
Jenna Hirsch, Borough of Manhattan CC

A facility with signed numbers forms the basis for effective problem solving throughout developmental mathematics. Most developmental mathematics textbooks explain signed number operations using absolute value, a method that involves considering the problem in several cases (same sign, opposite sign), and in the case of subtraction, rewriting the problem as an addition problem. This method works, of course, but involves quite a bit of chalk to explain. It’s neither a mathematically elegant way of understanding signed numbers, nor is it the method that experienced mathematics educators use themselves.


Douglas B. Aichele is regents professor (fmr.) and associate head of the Department of Mathematics at Oklahoma State University, Stillwater, OK. 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.


D. Rae Tree is a Lecturer in the Department of Mathematics at Oklahoma State University, Stillwater, OK. She coordinates the delivery of College Algebra and supervises the Mathematics Learning Success Center, which serves all introductory Mathematics courses.


Juliana Utley is an associate professor of mathematics education in the School of Teaching and Curriculum Leadership at Oklahoma State University, Stillwater, OK. 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.

Computer-aided Instruction: College Algebra Students’ Perceptions
Douglas B. Aichele, D. Rae Tree, Juliana Utley, and Benjamin Wescoatt, Oklahoma State University

Technology permeates our daily lives; education has not been untouched. Liaw (2002) points out that "teachers, trainers, and instructional designers of computer-based or Web-based instruction would benefit by being more attentive to learners’ perceptions toward Web-based environments.” Reviewing the earlier research into student perceptions toward online or blended nonmathematical courses, studies found that students were in general satisfied with their learning experience. Additionally, students believed the courses benefited their learning, increasing their motivation and improving their self-discipline. A main reason mentioned for the satisfaction was the convenience and flexibility offered by the course structure; the courses allowed for self-pacing, learning at any place and any time, and saving time. Students did worry about the lack of traditional teachers, believing that a face-to-face component was still important for the learning experience.

 


Benjamin Wescoatt is a doctoral student in the Department of Mathematics at Oklahoma State University, Stillwater, OK. His current interests are in research in mathematics education at the collegiate level.


Samuel Obara is an associate professor of mathematics at Texas State University–San Marcos. His research focuses on curriculum reform, professional development, and mathematics modeling.

What Do You Find? Students Investigating Patterns in Pascal’s Triangle
Samuel Obara, Texas State University

In this paper, students used problem-solving skills to investigate what patterns exist in the Pascal triangle and incorporated technology using Geometer’s Sketchpad (GSP) in the process. Students came up with patterns such as natural numbers, triangular numbers, and Fibonacci numbers. Although the patterns inherent in Pascal’s triangle may seem obvious to mathematics instructors, the students in this situation had a difficult time finding them.

Problem solving is often noted to be one of the most important parts of mathematics learning, "however, solving problems is not only a goal of learning mathematics but also a major means of doing so”. Teaching problem solving should involve emphasis on the method being employed other than just solving a problem to find a specific answer.

 

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 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 brownej@sunyocc.edu or at Mathematics Department, Onondaga CC Syracuse NY 13215.
The Problem Section is assembled by Fary Sami (at Harford CC, MD) and Tracey Clancy, Kathy Cantone, Garth Tyszka, and Joe Browne (editor) (at Onondaga CC, NY).



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