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VirginiaMathematicsTeacher

A Resource Journal for Mathematics Teachers at all Levels.

Volume 37, No. 2 Spring, 2011

The Five Platonic Solids

Volume 37, No. 2 Spring, 2011

VirginiaMatheMatics teacher

The VIRGINIA MATHEMATICS TEACHER (VMT) is published twice yearly by the Virginia Council of Teach-

ers of Mathematics. Non-profit organizations are granted permission to reprint articles appearing in the VMT provided that one copy of the publication in which the material is reprinted is sent to the Editor and the VMT is cited as the original source.

EDITORIAL STAFFDavid Albig, Editor, e-mail: [email protected] Radford University

Editorial Panel Bobbye Hoffman Bartels, Christopher Newport University; David Fama, Germana Community College; Jackie Getgood, Spotsylvania County Mathematics Supervisor; Sherry Pugh, Southwest VA Governor’s School; Wendy Hageman-Smith, Longwood University; Ray Spaulding, Radford University Jonathan Schulz, Montgomery County Mathematics Supervisor

MANUSCRIPTS & CORRESPONDENCEFor manuscript, submit two copies, typed double spaced. We favor manuscripts on disk or presented electronically in Word. Drawings should be large, black line, camera ready, on separate sheets, referenced in the text. Omit author names from the text. Include a cover letter identifying author(s) with address, and professional affiliation(s).

Send correspondence to Dave Albig at: Box 6942 Radford University Radford, VA 24142

Virginia Council of Teachers of MathematicsPresident: Beth Williams, Bedford County SchoolsPast-President: Carolyn Williamson, Retired from Hanover County Public SchoolsSecretary: Debbie Delozier, Stafford County Public SchoolsNCTM Rep.: Margaret Coffey, Fairfax County Public SchoolsMath Specialist Rep.: Corinne MageeElected Board Members: Elem. Rep: Sandy Overcash, Virginia Beach City Schools; Meghann Cope, Bedford County Schools Middle School Reps: Anita Lockett, Fairfax County Public Schools; Alfreda Jornegan, Norfolk Public Schools Secondary Reps: Ian Shank, Hanover Public Schools; Cathy Shelton, Fairfax County Public Schools. 2 Yr. College Rep: Joseph Joyner, Tidewater Community College 4 Yr. College Rep: Joy Whitenack, Virginia Commonwealth; Maria Timmerman, Longwood University Membership: Ruth Harbin-Miles

Publicity: Laura Rightnour, Hanover County Public Schools

Treasurer: Diane Leighty, Powhatan County Public Schools

Webmaster: Jennifer Springer, Charlottesville City Schools

Webpage: www.vctm.org

Membership: Annual dues for individual membership in the Council are $20.00 ($10.00 for students) and include a subscription to this journal. To become a member of the Council, send a check pay-able to VCTM to: VCTM c/o Pat Gabriel; 3764A Madison Lane, Falls Church, VA 22041-3678

Printed by Wordsprint Christiansburg225 Industrial Drive, Christiansburg, VA 24073

TABLE OF CONTENTSGrade Levels Titles and Authors ................................................................. Turn to Page

General President’s Message ..........................................................................1 (Beth Williams)

General Statistical Outreach and the Census: A Summer Learning Experience. ..........................................................................2 (Gail Englert) General Teaching Time Savers: Reviewing Homework....................................3 (Jane M. Wilburne)

General Enlivening School Mathematics Through the History of Mathematics ...................................................................................4 (Martin Bartelt and Stavroula K. Gailey)

General Affiliates’ Corner .................................................................................5

General The Five Platonic Solids .....................................................................6 (Theoni Pappas)

General VCTM Awards Two Scholarships ........................................................7

General Problem Corner ................................................................................10 (Ray Spaulding)

General William C. Lowry Award Winners ......................................................34

Grades K-5 Clearing the Confusion over Calculator Use in Grades K-5 .............18 (Barbara J. Reyes and Fran Arbaugh)

Grades 2-6 Teaching Addition and Subtraction Facts: A Chinese Perspective ...22 (Wei Sun and Joanne Y. Zhang)

Grades 3-6 Dividing Fractions: Reconciling Self-Generated Solutions with Algorithmic Answers ..................................................................25 (Marcela Perlwitz)

Grades 3-6 Developing Ratio Concepts: An Asian Perspective ..........................29 (Jane-Jane L, Tad Watanabe, and Jinga Cai)

Grades 7-10 Pick a Number ..................................................................................33 (Margaret Kidd)

Grades 7-12 Those Darn Exponents: Fifty Challenging True-False Questions .....35 (Tim Tilton)

Grades 13-16 Abstractmath.org: A Web Site for Post-Calculus Math .....................36 (Charles Wells)

ABOUT THE COVER: From the book “The Joy of Mathematics” by Theoni Pappas. Reprinted by permission of Wide World Publishing (http:/www.wideworldpublishing.com) Please see the The Five Platonic Solids article on page 6.

Virginia Mathematics Teacher 1

GENERAL INTEREST

President’s MessageBeth Williams

This spring has been a busy one for our organization! If you didn’t get to attend our Annual Conference in Rich-mond, you missed an extraordinary event. Fabulous speakers led every session. Our Department of Education mathematics colleagues keynoted two sessions. Deborah Wickham and Michael Bolling shared upcoming events and happenings at the state level to keep us informed and en-ergized to move forward in our work. Many thanks to our Conference Program Chair, Lisa Hall, the VCTM Executive Board, and the Greater Richmond Council volunteers who made the “Making Mathematics Monumental” conference a wonderful affair! VCTM is an affiliate of the National Council of Teachers of Mathematics, and we support the work that they advocate. In 2010, NCTM released a position paper recommending the use of mathematics specialists to enhance teaching, learning and assessing of mathematics to improve stu-dent achievement. In Virginia, we have been blessed with many grants through the National Science foundation, the Virginia Department of Education Mathematics and Sci-ence Partnership (MSP), NCLB Title II funds and through the ExxonMobil Foundation that support developing math-ematics specialist programs. In addition, a program to provide additional leadership opportunities for 25 specialists has been implemented through Virginia Commonwealth University, University of Virginia, and Norfolk State University with funds from the National Science Foundation. During the Annual Conference, a large group of mathe-matics specialists and coaches from across the state came together for the first time to begin to formulate ways to engage in professional growth and networking opportuni-ties. This social time allowed one group of specialists that has been supported by the Leadership Program to identify and connect with other professionals who share similar job responsibilities. Our annual conference banquet Friday night was a celebration of many accomplishments. Con-tina Martin and Vickie Inge shared exciting work from this Leadership Program. If you are interested in more informa-tion about this networking fellowship, seek out the link to mathematics specialists on our website. Our 2011 Fall Journal will be dedicated to the work of Mathematics Specialists. For this special Journal under-taking to be a success, we need to have many colleagues submit articles. Please consider writing an article from your experiences as a practicing specialist, or from work-ing with one. All articles must be submitted to Dave Albig by July 1, 2011. His email address is [email protected] our last Journal, Virginia Council of Teachers of Mathematics has awarded several grants and scholar-ships. Five First Timers grant awardees and two college scholarship recipients were honored during our banquet celebration. The monies they received are given to con-tinue the work of high quality mathematics education for all. Find out more about these funds that are allocated by our organization as you read further in the Journal.

Two Virginia finalists were also recognized as Presiden-tial Awardees for Excellence in Mathematics and Science Teaching on Friday night. These colleagues have advanced to the state selection in this prestigious mathematics award. We wish Pamela Bostwick and Victoria Hugate both well as the PAEMST selection process continues.Part of being a teacher is being a lifelong learner, studying, researching, and listening to find ways to improve our in-struction and help more students succeed. To this end, my goal will be to pose a new question to you in each Journal edition. This month my question for you would be: What do you know about the Common Core Standards and their implementation? In January, NCTM released news of a joint task force made up of the Association of Mathematics Teacher Edu-cators, The Association of State Supervisors of Mathemat-ics, The National Council of Supervisors of Mathematics, and the National Teachers of Mathematics. The work of the task force is to develop actions and resources needed to help teachers implement the Common Core State Stan-dards in Mathematics (CCSSM). This task force also con-sidered ways in which the organizations can collaborate in supporting their members and other groups to advance their vision of school mathematics. The task force report identifies five priority actions to be taken as soon as pos-sible. There are also PowerPoint presentations that can in-form all stakeholders. These information opportunities are available for you to read at www.nctm.org/news/highlightsNCTM is working with other groups like NCSM to write sup-porting tasks, videos and documents for implementing the CCSSM. Dr. William G. Mccallum led the work to write the standards using the NCTM Focal Points and High School Sense Making as their foundation. His group is working on a new website, www.illustrativemathematics.org where good tasks for formative assessments and videos of good teaching strategies will be shared. Dr. Mccallum also has a blog at commoncoretools.wordpress.com Even though Virginia has not adopted the Common Core State Standards, The Virginia Board of Education has ad-opted a supplement to the Curriculum Frameworks to bring our Standards into closer alignment with the CCSSM. Our State Superintendent, Dr. Patricia Wright, wrote in a Su-perintendents Memo in February that the supplement will ensure that Virginia Standards are equal to or more rigor-ous in content and scope than the CCSSM. If you have not already done so, you should read the supplement found on the Virginia Department of Education website. The implementation of more rigorous standards requires better teaching for more learning. All of these resources will be valuable to us as we consider new ways to increase our students’ success. Your VCTM organization will continue to work to improve your learning, living and love of math-ematics.

Best wishes to you all!Beth

2 Virginia Mathematics Teacher

GENERAL INTEREST

Statistical Outreach and the Census: A SummerLearning Experience

Gail Englert

“Car A traveled 150 miles in 6 hours, and took another half hour to go the final 40 miles.” My mathematical adventure to Washington, DC to attend the American Statistical As-sociation’s Meeting Within a Meeting, a statistics workshop held August 3-4, 2009 for teachers of grades K–12, started off sounding like a badly written word problem. Lesson #1 – don’t leave Norfolk with DC as the destination on a Saturday in August without allowing extra time! Fortunately, after arriving, my plan to take advantage of our nation’s capital before the workshop began unfolded beauti-fully. At one museum, the admission fee was $19.95. When I mentioned I was a teacher, I was allowed to enter FREE!!! At another, purchases in the gift shop qualified for a teacher discount. Lesson #2 – ask for the educator discount; it may be a financially rewarding experience! The American Statistical Association (ASA), (http://www.amstat.org/) is a 170-year-old scientific and educational professional society whose goal is to enhance lives through informed decision-making by providing its members and the public with up-to-date, useful information about statistics. The ASA website contains a wealth of topics to explore…from “Making Sense of Statistical Studies” (the Student Module and accompanying Teacher’s Module includes supporting resources with 15 hands-on investigations for upper middle-school or high-school students to explore as they design and analyze data) to “Statisticians in the News”. Clicking the “Education” tab displays a welcome message from Dr. Martha Aliaga, Director of Education, and other useful classroom resources. This year’s annual Joint Statistical Meeting was held in Washington, DC. As part of an outreach to educators, the organization provided a day-long workshop focusing on the teaching and learning of data analysis, probability and statistics concepts. Workshop participants, who were divided along grade bands (k – 4, 5 – 8, 9 – 19, and BAPS [advanced placement statistics]), spent the day exploring content, classroom instruction and assessment through hands-on activities and presentations by dynamic statisticians and educators. During the workshop I learned about GAISE(Guidelines for Assessment and Instruction in Statistics Education): A Pre-K–12 Curriculum Framework. Participants in the GAISE project have created two reports of recommendations for introductory statistics courses (college level) and statistics education in Pre-K-12 years with the ultimate goal being statistical literacy. More information and materials to be used with students can be found at http://www.amstat.org/education/gaise/index.cfm. Also explored was data in a variety of contexts and representations, from Old Faithful (complete with a live link to Yellowstone to view an eruption online) to smokers vs. non-smokers (using a matrix to consider conditional and marginal probabilities). We even

looked at the mean as the balancing point of a set of data, considering how far each data point fell from the mean. In every case, the material covered was presented as a lesson on a continuum of learning, with background about the planning and assessment. Lesson #3 – realizing that statistics is all about informed decision-making. I have never presented the reason for collecting, presenting and analyzing data quite that explicitly to my students, but I will now. K – 12 teachers are encouraged to enroll in a K-12 teachers free trial membership to the ASA. The trial membership offers subscriptions to Amstat News (the ASA’s monthly membership magazine), and CHANCE (a magazine focusing on the use of statistics in everyday life). It also provides members-only access to the ASA’s top journals and discounts on all ASA meetings and products. After the trial, ASA offers a discounted annual membership ($50.00 instead of $125.00) for K-12 teachers. The following day a visit to the US Census Bureau head-quarters in nearby Suitland, MD was offered. This included interactive presentations and activities organized by Renée Jefferson-Copeland, Chief of the Census in Schools Branch. We were introduced to the 2010 Census process, Census in Schools activities, and Census’ data and on-line data access tools. The following links have wonderfully rich data that could be used for lessons in not only math, but also social studies classrooms.

Main URL: http://www.census.gov/

Census in the schools: http://www.census.gov/schools/

Fact finder:http://factfinder.census.gov/home/saff/main.html?_lang=en

Kids corner:http://factfinder.census.gov/home/en/kids/kids.html

State facts for students:http://www.census.gov/schools/facts/

Lesson #4 – there is a lot going on at the Census Bureau, even during “the other 9 years”! The visit ended with the presentation of US Census gift bags containing a treasure trove of materials any math teacher would love, state fact sliders and wheels, mugs and pencils emblazoned with the Census logo, and a huge double-sided wall map of the US with demographical information displayed. The next Joint Statistical Meeting for ASA will be held from July 31 - August 5, 2010, at the Vancouver Convention


Center in Vancouver, British Columbia, Canada. While this location isn’t as accessible for Virginia math educators as Washington, DC was, the ASA website contains a variety of great information to enhance statistics and probability instruction in my 7th grade classroom and the classrooms of my colleagues at Ruffner Academy in Norfolk. Fifth and

GENERAL INTEREST

Teaching Time Savers: Reviewing HomeworkJane Murphy Wilburne

The classroom practice of assigning homework is a ne-cessity to reinforce the topic of the day’s lesson, review skills and practice them in a variety of problems, or chal-lenge students’ thinking and application of the skills. Effec-tive mathematics teachers know how to choose worthwhile assignments that can significantly impact students’ learn-ing and understanding of the mathematics. The challenge, however, is how to manage and review the assignments in a manner that will benefit students’ learning, and use classroom time effectively. Over the years, I have tried various approaches to re-viewing and assessing students’ homework. Collecting and grading every students’ homework can be very time con-suming, especially when you have large classes and no graduate assistants to help review students’ work. On the other hand, while it is important to provide students with immediate feedback on their homework, it does not benefit them much to have the professor work out each problem in front of the class. I believe it is important for college students to take re-sponsibility for their learning. By promoting opportunities for them to communicate with and learn from each other, we can help students come to rely less on the professor to provide them with all the answers, and teach them to pose questions that enhance each other’s understanding. One technique that has been effective in my classes is to assign homework problems that vary in concept application and level of difficulty. The students were instructed to solve each problem and place a check () next to any problem they could not solve. As the students entered class the next day, they would list the page number and problem number of the problems they could not solve, on the front board in a designated area. If the problem was already listed, they placed a check ( ) next to it. Once the class started, they were not allowed to record problem numbers at the board. Other students, who were successful in solving these prob-

final lesson – even if my average speed traveling to the workshop destination was a lot slower than I expected, I am so glad I attended! Thank you, VCTM, for providing a grant to offset my travel and lodging expenses.

GAIL ENGLER, Ruffner Academy, Norfolk Public Schools

lems, immediately went to the board when they entered the class, indicated that they would solve one of the listed problems, and worked it out in detail. When they finished they signed their name to the problem. By the time I entered the classroom, students were busy solving problems at the board while others were checking their homework at their seats. If there were any questions about the problems, the student who solved the problem at the board would explain his work to the class. If there was a problem which no one was able to solve, I would provide a few details about the problem and reassign it for the next class. In a short period of time, all homework was reviewed, and I recorded notes as to which students posted solutions on the board. Rather than collecting every student’s homework, I noted the problems that gave most students difficulty and would assign similar problems in a future assignment. Students who listed the problems they had difficulty with were not penalized. Instead, those who solved the problems would receive a plus (+) in my grade book. A series of five pluses (+) would earn them a bonus point on a future exam. My classroom quizzes would always include several homework problems to help keep students accountable for completing their assignments and motivate them to review problems they had difficulty with. Those who did typically received an A! Time spent in class: approximately 5-12 minutes review-ing the homework. Time saved: abut 30 minutes per class.

JANE M. WILBURNE is assistant professor of mathematics at Penn State Harrisburg.

Reprinted with permission from FOCUS The Newsletter of the Mathematical Association of America, copyright November 2006. All rights reserved.

Statistical Outreach and the Census continued from page 2


GENERAL INTEREST

Enlivening School Mathematics Through The Historyof Mathematics

Martin Bartelt and Stavroula K. Gailey

This article describes an alternative History of Mathemat-ics course and it demonstrates how such a course can be beneficial for teachers and in turn for their students. According to the Curriculum and Evaluation Standards of the National Council of Teachers of Mathematics (NCTM) a major goal of mathematics education is to produce students who value mathematics. This goal, of valuing mathematics, requires learning about and understanding the origins of mathematics as well as appreciating the role mathematics plays in today’s society. Another goal is to create a learning environment that fosters students’ confidence in doing mathematics. In addi-tion to the NCTM Standards, the Mathematical Association of America (MAA) in its 1992 Call for Change states that mathematics teachers also need continuing experience in developing perspectives and in appreciating the historical and cultural development of mathematics. These NCTM and MAA goals have been incorporated in the Master of Arts in Teaching Mathematics program at Christopher Newport University. One of the program’s courses, MATH 573: The History of Mathematics, both fosters mathematical confidence and contributes to an ap-preciation of mathematics. The course is a survey of the origins, philosophy and development of mathematics from classical antiquity through the twentieth century. However, MATH 573 is different from the typical History of Mathematics course. In addition to problem solving, MATH 573 emphasizes how to incorporate both concepts and content in the pre-college classroom, particularly in middle school mathematics. The course is intended to enable teachers to learn about the history of mathematics and also how to apply this knowledge in their classroom. After examining some well-known texts -such as that by C. Boyer- used in standard History of Mathematics courses, it is apparent that the objectives of these courses do not emphasize how school teachers could use the material in their classrooms. And although William Dunham’s Journey Through Genius - The Great Theorems of Mathematics il-lustrates a lively approach to the history of mathematics, still the book does not refer directly to teacher-use in the class-room. In this sense, the CNU MATH 573 course is atypical. Since the students in MATH 573 are either practicing teachers or interning graduate students, they continuously ask themselves and the instructor about how they could implement what they are learning, in the History of Math-ematics course, in their own classrooms. Overview Of Course Content The text books used in MATH 573 are Great Moments in Mathematics Before 1650 and Great Moments in Mathemat-ics After 1650 by Howard Eves. Beyond standard homework and exams, students in MATH 573 are required to complete projects, biographical reports, and presentations on the

in-classroom implementation of topics from the history of mathematics. Projects are intended to illustrate various ideas associ-ated with the history of mathematics. For example, the students make simple versions of a Roman abacus, design posters of mathematical symbols and/or terms explaining how they originated, construct Moebius strips, and create a “sphere” from a collection of cylinders in order to estimate the sphere’s volume. One teacher, after canvassing local stores, found that she could buy all the materials to make a good, small version of an abacus, which she intended to use in her classroom to aid in teaching place value, for less than a dollar. The teachers, after researching, write one-page biographi-cal reports for four different mathematicians. However, these too are not typical biographies. First, the reports must include the biography of a woman, of an American, and of someone from a non-Western culture. The reports may also include mathematicians who have not been studied in class and particularly living mathematicians. Second, and more important and difficult, the biographies must contain information of interest to middle and/or high school students. The following are some examples of interesting informa-tion included in some of the biographies: Hero of Alexandria invented the first vending machine; Ada Byron Lovelace, to whom the poet Byron was married, was the first person to describe the process of computer programming; and Grace Hopper, the contemporary American mathematician, created COBOL. Referring to personal traits and events enlivens the biographies. In addition to the projects and biographies, the teachers choose two concepts/topics from the history of mathemat-ics and develop strategies and activities for incorporating these topics in their classrooms. In turn, each teacher gives a fifteen minute presentation to the others in the class so ideas from these presentations can be shared and used by the rest of them in their own classrooms. For example, one of the topics presented, which is ap-propriate for use in a general mathematics class, was that of using and writing checks. It referred to the history of count-ing by tally sticks and how the word “check” originated in England. Another interesting idea included the story of zero, and how for hundreds of years people refused to believe in it. In another presentation a teacher explained how she would have students do some important work particularly on biographies in order to learn about the disadvantaged back-grounds of some mathematicians and the effect of sociology and psychology on a mathematician’s career.

The Middle School/High School Student In addition to learning new material and the means by which it can be presented to students there is another im-portant and atypical facet of MATH 573 which relates to the


“confidence” goal for students. There is a conscious effort throughout the course to empower the middle school teach-ers to influence their students. For example, one goal of the biographies is to affect the mindset of the student. Knowing about the existence of female mathematicians can change the perspective of female students toward mathematics. Also, a physically handicapped student benefits by knowing that there are physically handicapped mathematicians. Students tend to believe that mathematics is, and always was, error-free, complete, contradiction-free, and completely logical. Since middle school and high school students usu-ally do not have these characteristics, they sometimes feel estranged from mathematics. Students will feel better when they learn that great mathematicians made mistakes; that some mathematical questions can not be answered because they are undecidable; that whole societies had trouble with the number zero; that there have been crises in mathemat-ics (e.g. the discovery of non-Euclidean geometry), and that controversy exists even now.

Conclusion This type of History of Mathematics course as part of a M.A.T. program can enable teachers to enliven their class-room teaching. It provides a way to look at material, which one may already have seen before, from a new viewpoint, to introduce and to give depth to new material, and to influ-ence the mindset of the student.

BIBLIOGRAPHYDunham, William. Journey Through Genius, The Great

Theorem of Mathematics. New York: John Wiley, 1990.Boyer, Carl, and Uta Merzbach. A History of Mathematics.

New York: John Wiley, 1989. Eves, Howard. Great Moments in Mathematics Before

1650. Washington, D.C.: Mathematical Association of America, 1983.

Eves, Howard. Great Moments in Mathematics After 1650. Washington, D.C.: Mathematical Association of America, 1983.

Edeen, Susan and John Edeen. Portraits for Classroom Bulletin Boards, Book 1. Palo Alto, California: Dale Sey-mour, 1988.

Edeen, Susan and John Edeen. Portraits for Classroom Bulletin Boards, Book 2. Palo Alto, California: Dale Sey-mour, 1988.

Leitzel, James (ed.). A Call for Change: Recommendations for the Mathematical Preparation for the Teachers of Mathematics. Washington, D.C.: Mathematical Associa-tion of America, 1992.

National Council of Teachers of Mathematics. Historical Topics for the Mathematics Classroom. Reston, VA: NCTM, 1989.

Reiner, Luetta & W. Reiner. Mathematicians Are People, Too. Palo Alto, California: Dale Seymour, 1990.

MARTIN BARTELT ([email protected]) and STAVROULA K. GAILEY ([email protected]) teach mathematics and mathematics education courses at Christopher Newport University, Newport News, Virginia.

GENERAL INTEREST

Affiliates’ CornerAffiliate Grant: VCTM awarded a $500 grant to the Battlefields Council to be used to defray expenses for a keynote speaker for their March conference.

Blue Ridge Council: Will award a $500 scholarship to a high school senior who will pursue college studies to become a mathematics teacher. Applications are due April 15. Contact: Jonathan Schulz: [email protected]<mailto:[email protected]>

Greater Richmond Council: Will award a Professional Development grant up to $1000 to a member. Applications are due May 1. Contact: Andrew Derer: [email protected]<mailto:[email protected]>

Northern Virginia Council: Annual banquet will be May 12. The guest speaker will be Albert Goetz, NCTM Journal Editor. At the banquet, NVCTM awards a scholarship to a high school senior intending to become a mathematics teacher and also recognizes top place schools in their Middle School and Junior Math Leagues. Contact: Gail Chmura: [email protected]<mailto:[email protected]>


GENERAL INTEREST

The Five Platonic SolidsTheoni Pappas

Platonic solids are convex solids whose edges form con-gruent regular plane polygons. Only five such solids exist. The word solid means any 3-dimensional object, such as a rock, a bean, a sphere, a pyramid, a box, a cube. There is a very special group of solids called regular solids that were discovered in ancient times by the Greek philosopher, Plato. A solid is regular if each of its faces is the same size and shape. So a cube is a regular solid because all its faces are the same size squares, but this box, on the right, is not

a regular solid because its faces are not all the same size rectangles. Plato proved that there were only five possible regular convex solids. These are the tetrahedron, the cube or hexahedron, the octahedron, the dodecahedron, and the icosahedron.

Here are patterns for making all five regular solids. Why not copy them, cut them out and try to fold them into their 3-dimensional forms?

tetrahedron hexahedron or cube

octahedron

icosahedron dodecahedron

tetrahedron

hexahedron

octahedron

icosahedron

dodecahedron

From the book “The Joy of Mathematics” by THEONI PAPPAS. Reprinted by permission of Wide World Publishing (http:/www.wideworldpublishing.com) Those wanting to reprint this article should contact Wide World Publishing.


GENERAL INTEREST

VCTM Awards Two $2000 Scholarships to Future Math Teachers This year, VCTM, through its Board of Directors, gave authorization to the Scholarship Committee to award a $2000 scholarship to up to two candidates that were qual-ity, prospective mathematics teachers. To receive this award, candidates must be Virginia residents that are full-time students attending a Virginia college or university with a major in mathematics and plan to teach mathematics in a Virginia school. This year’s selection process was based on the students’ academic achievements (transcripts), faculty recommendations, and personal narratives (which were required to include a field experience, class experi-ence, volunteer experience, or life experience that has in-fluenced their decisions to be a teacher of mathematics). Each candidate will also receive a complimentary, one-year student membership in VCTM. This year’s scholar-ship winners are India (Brooke) Haun, a student at Virginia Tech and Johnathon Upperman, a student at The College of William and Mary. The scholarship awards will be officially announced at this year’s VCTM Annual Conference banquet, March 12, 2011 at the Ramada Plaza Richmond West Conference Center. Both India (Brooke) Haun and Johnathon Upperman are studying to become middle and secondary teachers of mathematics. They bring to their teaching aspirations many achievements in their teacher education and math-ematics programs. India (Brooke) Haun is currently study-ing in a five-year program at Virginia Tech. She plans to complete her Master’s in Education Spring 2012. She has already begun to work in high school classrooms where

she has had great success working with teachers and their students. Before attending Virginia Tech, she graduated from Monticello High School where she received the Su-san Gilkey Award, an award that is given to a student ath-lete with the highest grade point average. She was also a Wendy’s High School Heisman nominee. Johnathon Up-perman is currently completing a four-year Bachelor’s of Science degree at the College of William and Mary. After completing this degree, he will continue his studies at Wil-liam and Mary to pursue a Master’s in Education. He has a stellar academic record and plans to draw on his suc-cesses to help inspire others to learn and appreciate math-ematics. He, too, will begin his teaching career soon after he completes his Master’s in Education. Prior to attending William and Mary, Johnathon graduated from Indian River High School. VCTM members congratulate both of these scholarship winners and wish them success in all of their teaching ex-periences to come. We also welcome India Brooke Haun and Johnathon Upperman to our profession and trust that they will share in the very important task of supporting mathematics and its teaching in Virginia. As a current VCTM member, or one who is currently thinking about being a member as your read this article, we invite you to make a contribution to the Scholarship Fund the next time you are scheduled to make your annual dues. You may also send a check to the editor made out to the VCTM Scholarship Fund. Your contribution will help VCTM to continue to support prospective teachers of mathemat-ics.


VIRGINIA COUNCIL OF TEACHERS OF MATHEMATICS

2011 SCHOLARSHIP PROGRAM

The Virginia Council of Teachers of Mathematics (VCTM) encourages those persons interested in becoming

teachers of mathematics by offering up to two (2) scholarships of $2,000 annually. VCTM’s objective in

establishing this program is to unite the efforts of its members who seek to improve the teaching of mathematics.

This year, the VCTM Board of Directors has again authorized that two scholarships may be awarded if qualified

applicants are judged deserving by the current members of the Scholarship Committee.

DESCRIPTION

Students applying for this year's scholarships must be full-time students attending a four-year Virginia College or

University, Virginia residents, currently enrolled in a degree-seeking program with a concentration in mathematics

or major in mathematics and plan to graduate in Fall 2012, Spring 2013 or Summer 2013, and plan to teach

mathematics in a Virginia school. All applicants planning to teach at elementary, middle school, high school, and

college levels are eligible. Completed application materials must be returned postmarked no later than January 1,

2012. Applicants must also submit an official transcript showing grades through the Fall 2011 term, postmarked no later than January 15, 2012.

The scholarship winner(s) will be announced at VCTM’s 33rd Annual Conference. All applicants will be notified in

writing of the Committee’s decision.

CRITERIA FOR SELECTION

Selection will be based on the applicant's potential for a successful career teaching mathematics, as indicated by

scholastic records, recommendations of two faculty members, and the applicant’s narrative statement. All applications will be reviewed and will be selected by VCTM’s Scholarship Committee, with approval of VCTM’s

Executive Board.

RECENT AWARDEES

India (Brooke) Haun, Virginia Tech and Johnathon Upperman, The College of William and Mary (2011)

Heather Sturgis, Christopher Newport University and Abby Thompson, Radford University (2010)

Jennifer Jones Trail, Averett University and Hannah Jo Joyce, Virginia Tech (2009)

Nicole Huret, Virginia Tech and Rebecca Victoria Perrigan, The College of William and Mary (2008) Kevin Bryan Jones, Radford University (2007)

Kathryn (Katie) Massey, Virginia Tech and Samantha Soukup, Longwood University (2006)

Alisa R. Mook, Virginia Tech and Allena Monique Poles, Virginia Union University (2005)

Robyn L. Brewster, Bluefield College and Jennifer McLaughlin, Virginia Tech (2004)

Amy Tribble, James Madison University and Christy Lowery, Averett University (2003)

Melissa E. Andersen, Mary Washington College (2002)

Kristy Banton, Virginia Commonwealth University (2001)

Katherine M. Sutphin, Mary Washington College (2000)

Dana N. Daniels, Longwood College and Tiana M. Taylor, Averett College (1999)

Jonathan Covel, James Madison University (1998)

Sarah E., Boyer, Mary Washington College and Katherine Elms, Virginia Tech (1997)

APPLICATION INFORMATION

All required forms can be downloaded from the website: www.vctm.org or you may request application and

recommendation forms from your Mathematics or Education Department Chair or by writing to:

Joy Whitenack, VCTM Scholarship Committee

Virginia Commonwealth University

1015 Floyd Avenue, Richmond, VA 23284-2014 804-828-5901 or [email protected]


2012 VCTM Scholarship Application for Prospective Teachers of Mathematics PLEASE PRINT Name__________________________________________________ Birth date (optional)________________ Gender (optional): M F Virginia College Now Attending:____________________________________________________________________ Other Colleges Attended (Including Community Colleges Major(s): _____________________________________ Minor(s): ___________________________________ Expected Degree:_______________________________________ Concentration *______________________ This is a (Circle): FOUR YEAR PROGRAM or FIVE YEAR PROGRAM *If you are in a special program, please describe on separate paper and attach. Expected Date of Graduation: (Circle) FALL 2012 or SPRING 2013 or SUMMER 2013 Are you a Virginia resident? ______ YES ______NO Do you plan to teach mathematics in Virginia? _____YES _____NO Level of mathematics you plan to teach (Circle all that apply): ELEMENTARY MIDDLE HIGH SCHOOL COLLEGE Current Mailing Address: ____________________________________________________________________ Permanent Home Address: ____________________________________________________________________ Current Telephone: ____________________________Permanent (Home) Telephone: ___________________________ eMail address that you regularly check: __________________________________ High School from which you graduated: _______________________________________________________________ Location (city and state): _____________________________________________________________________ High school honors; mathematics and education-related activities: ________________________________________________________________________________________________ ________________________________________________________________________________________________ I certify that the above information is correct. Signature: ___________________________________________________________Date:

Please attach a one-page statement indicating why you wish to be a mathematics teacher. In your essay, please include

a description of a field experience, class experience, volunteer experience, or life experience that has influenced your decision

to become a teacher of mathematics. Prepare your statement on 8-1/2" by 11" paper, double-spaced, using type no smaller

than 12 characters per inch (10 point). Finally, sign your statement at the bottom of the statement page.

POSTMARK DEADLINE FOR THIS APPLICATION IS JANUARY 1, 2012.

Mail all application materials to:

Joy Whitenack, VCTM Scholarship Committee Virginia Commonwealth University 1015 Floyd Avenue, Richmond, VA 23284-2014 804-828-5901 or [email protected]


GENERAL INTEREST

Problem CornerRay Spaulding


GRADES K-5

Clearing up the Confusion over Calculator Use in Grades K-5Barbara J. Reys and Fran Arbaugh

Since the publication of NCTM’s Principles and Stan-dards for School Mathematics in April 2000, considerable discussion has taken place about “key messages” of the document. The breadth of the content of Principles and Standards may hamper attempts to identify messages about particular topics. In addition, many of the fundamen-tal messages are not easily distilled into short phrases. In fact, when such messages are too succinctly articulated, the danger of oversimplification and misunderstanding arises. This misapprehension can be seen in a question that often emerges in discussions about elementary school mathematics and Principles and Standards. That is, what does Principles and Standards say about calculator use in elementary school?

Why the Interest in NCTM’s Position? Teachers, teacher educators, mathematicians, school administrators, and parents are genuinely interested in NCTM’s “official” position on calculator use in the elemen-tary grades. Why is everyone so interested? What issues surround calculator use in elementary school? Before we delve into the messages of Principles and Standards on this topic, and to support our understanding of those mes-sages, we present a short discussion of some of the issues that may be prompting the current interest in what Prin-ciples and Standards says about calculator use in elemen-tary school.

Why the Interest, and What’s the Confusion? NCTM has published several official position statements on calculators, the first in 1978 and the most recent in 1998 (see nctm.org/about/ position.htm). Public interest in cal-culator use in schools has grown steadily over the past twenty-five years, largely because of the increased avail-ability of inexpensive calculators. Through the first seven-ty-five years of the twentieth century, elementary school mathematics emphasized paper-and-pencil computation techniques out of necessity. In fact, some estimates of the amount of instructional time devoted in the elementary grades to developing hand-calculation proficiency run as high as 90 percent (Reys 1994). This emphasis was impor-tant because hand calculation was the most efficient way to compute apart from cash registers, adding machines, and expensive computers. Today, a $4 calculator can do in seconds the computations that in the past, students needed years of instruction and practice to learn and that, once learned, required significant amounts of time to ac-complish. The $4 calculator, a highly efficient and accurate com-putational tool, raises a whole set of questions that educa-tors and parents are struggling to answer. For example, what is the importance of having students learn methods for computing that their parents and grandparents learned? If we value proficiency with hand-computation techniques, do we know and accept the “cost” in terms of instructional

time and students’ motivation and interest in mathematics? If students spend less time reaching high levels of perfor-mance in hand calculation, how will the resulting “extra” time be used? Do we value instruction that develops stu-dents’ ability to think, reason, and solve problems? Can we meet both goals simultaneously—that is, develop proficien-cy with hand calculation and the abilities to think critically, reason, and solve problems? Another question relates to the perceived consequences of learning to compute by hand or with a calculator. For example, do students derive cognitive benefits from learn-ing conventional paper-and pencil algorithms for comput-ing? Does an overemphasis and reliance on conventional computation algorithms encourage students not to think or use their powers of reasoning? Does the use of calculators for computation promote students’ understanding of math-ematical situations and reasoning about solutions? For a more thorough discussion of issues that surround calcula-tor use in the elementary grades, see Ralston, Reys, and Reys (1996) and Reys and Nohda (1994). Certainly, these questions will persist until we address them. In fact, we know that the following situations prevail: • Calculators are readily available at home to children of

elementary school age, and their low cost has prompted increased access in school.

• Calculators are commonly used in the workplace to per-form simple and complex computations.

• Teachers are unsure how to use calculators to promote thinking and reasoning and whether calculators should be used as computational devices.

• Children’s own beliefs about mathematics lead some of them to view using calculators as “cheating,” or not re-ally doing mathematics.

Clearing up the Confusion: Messages in Principles and Standards Principles and Standards articulates a goal for elemen-tary school mathematics that includes computational pro-ficiency but extends well beyond that skill to the abilities to think and draw on a range of techniques and strategies to solve problems. Computation is important precisely be-cause it is necessary to solve many mathematical prob-lems. The particular method used, however, whether it in-volves mental math, paper and pencil, or a calculator, is just one part of the computation process. Students must also know what kind of computation to perform and be able to identify the appropriate numbers to use in compu-tations. Real mathematics is knowing a variety of strate-gies for solving problems and having the ability to apply them appropriately. If data from the National Assessment of Educational Progress (Silver and Kenney 2000) are any


multiples of 5.” In this example, the counting capability of the calculator allows students to focus on patterns that re-sult from adding the same number repeatedly. This type of exploration lays the groundwork for studying multiples and divisibility, important ideas in the upper elementary school grades. A more thorough discussion of this example can be found at standards.nctm.org/document/eexamples/chap4 /4.5/index.htm. Shuard (1992) writes about a classroom episode in which elementary school students “discovered” negative numbers as they were investigating subtraction with a cal-culator. One student entered 6 – 8 and was curious about the displayed result, which was –2. The teacher used the opportunity to model negative numbers by extending the number line to the left of 0. In another activity, the children started at 50 and successively subtracted the amount rolled on a number cube. One student, Jenny, continued until she had 3 left. On her next roll, she got 5. She said, “I can’t take it away. I would owe 2.” She tried 3 – 5 on a calculator and said, “It is take-away 2.” She continued to explore similar problems, making a list of those that had an answer of –1. Her list included 1 – 2, 2 – 3, and 3 – 4. When asked what number could be subtracted from 100 to give –1, Jenny said, “Easy! 101” (Shuard 1992, p. 40).

Technology should not be used as a replacement for basic understandings and intuitions; rather, it can and should be used to foster those understandings and intuitions. (NCTM 2000, p. 25)

In the upper elementary school grades, students can use the calculator to explore the relationships among vari-ous representations of rational numbers. “For example, they should count by tenths (one-tenth, two-tenths, three-tenths, . . .) verbally or use a calculator to link and relate whole numbers with decimal numbers. As students contin-ue to count orally from nine-tenths to ten-tenths to eleven-

tenths and see the display change from 0.9 to 1.0 to 1.1, they see that ten-tenths is the same as one and also how it relates to 0.9 and 1.1” (NCTM 2000, p. 150).

As students encounter problem situations in which computa-tions are more cumbersome or tedious, they should be encour-aged to use calculators to aid in problem solving. (NCTM 2000, pp. 87–88)

Guided work with calculators can enable students to explore number and pattern, focus on problem-solving processes, and investigate realistic appli-cations. (NCTM 2000, p. 77)

indication, elementary school mathematics programs have had some success in helping students perform calculations but have been largely unsuccessful in developing students’ problem-solving abilities. The writers of Principles and Standards recognized the importance of articulating a clear and research-based mes-sage about the role of calculators in elementary school mathematics. Readers will find these messages through-out the document. For example, in chapter 2, the section on the Technology Principle includes a discussion of cal-culators as tools for learning. Readers should also look at chapters 3, 4, and 5 under the Number and Operation Standard for discussions of the role of calculators in devel-oping computational fluency. The rest of this article summarizes some of the important messages from Principles and Standards about calculators in the elementary school mathematics classroom. We draw heavily from the text of the document and illustrate some of the points with examples.

Calculators are important tools for learning and doing mathematics

Electronic technologies—calculators and comput-ers—are essential tools for teaching, learning, and doing mathematics. (NCTM 2000, p. 24)

Understanding number, properties of operations, and re-lationships among numbers is central to elementary school mathematics. The calculator is a tool for exploring num-ber concepts and for generating data that can be studied for patterns. For example, students can use a calculator to skip-count by 5s (press 0, + 5, =, and so on) and color the corresponding spaces on a hundred board (see fig. 1). Students can then try the same process with other num-bers and respond to teacher prompts, such as “What pat-terns emerge?” and “Predict additional numbers that are


In addition to its use as a way to explore mathematics, the calculator is a highly efficient and accurate tool for com-puting in problem-solving contexts. With access to a calcu-lator, students can use real data and large data sets. The calculator’s efficient and accurate ability to compute frees students to think and make decisions. When a cashier uses a computer to tabulate our bill or a bank clerk uses an adding machine to total receipts, we do not think that they are “cheating.” We need to help students understand that mathematics is more than computation—that using a calculator as a tool for solving problems is not cheating. For example, students can use a calculator as a computing tool to help them answer the question “How much time would you need to count to a million or a billion?” In upper el-ementary school, students can study the effect of extreme values when computing a mean. In all grades, they can use real data to solve problems of interest to them, from tabu-lating the costs of food items in the cafeteria to gathering and summarizing data on the number of pencils in all the desks in the classroom or the number of buttons on all their clothes. Students can use calculators to help them accu-rately and efficiently solve problems to focus their attention on what calculations to perform, not just how to perform those calculations.

Calculator use is not an all-or nothing decision

Part of being able to compute fluently means making smart choices about which tools to use and when. (NCTM 2000, p. 36)

Through their experiences and with the teacher’s guidance, students should recognize when using a calculator is appropriate and when it is more efficient to compute mentally. (NCTM 2000, p. 77)

Students at this age [grades 3–5] should begin to develop good decision-making habits about when it is useful and appropriate to use other computational methods, rather than reach for the calculator. (NCTM 2000, p. 145)

Calculators should be available at appropriate times as computational tools, particularly when many or cumbersome computations are needed to solve problems. However, when teachers are working with students on developing computational algorithms, the calculator should be set aside to allow this focus. (NCTM 2000, pp. 32–33)

Throughout the document, Principles and Standards stresses that technology and, hence, calculator use is not an all-or-nothing decision. Supporting students’ under-standing of mathematical concepts helps them make good decisions about appropriate times to use a calculator. At times in their study of mathematics, students will find that other ways of computing are more appropriate than using calculators. For example, engaging students in mental-math activities supports their understanding of mathemati-

cal relationships. When teachers want to help students develop strategies that they can use to compute mentally, those strategies alone should be the focus. When students are learning ways to record computational strategies, then recording should be the focus. Elementary school teachers will certainly want students to put away their calculators at times to focus on developing other techniques and strate-gies for computing. At other times, when students are en-gaged in solving problems, formulating and applying strate-gies, and reflecting on results, a calculator is an important enabling tool. Teachers and parents must help students un-derstand that “real” mathematics is about thinking, applying strategies, reasoning, and relating ideas. Computation is a necessary tool in the process, but it is only one part of the whole process that makes up mathematics.

Good use of calculators requires teacher decision making and guidance

In the mathematics classroom envisioned in Prin-ciples and Standards, every student has access to technology to facilitate his or her mathematics learn-ing under the guidance of a skillful teacher. (NCTM 2000, p. 25) The effective use of technology in the mathematics classroom depends on the teacher. (NCTM 2000, p. 25)

Principles and Standards emphasizes the role of teachers in helping students become responsible technology us-ers. Teachers should model and explain their own ways of thinking about numbers and operations and encourage students to share their methods. The classroom envisioned by NCTM is one in which students and teachers use a vari-ety of tools, including counters, rulers, graph paper, scales, geometric shapes and solids, text books, instructional soft-ware, and calculators. Just as teachers guide and model the use of other tools, so must they help students under-stand the power and limits of a $4 calculator. The calcula-tor cannot think, and it cannot make decisions about what numbers or operations need to be used. The quality of the output of a calculator is wholly dependent on the input. Teachers must examine the instructional goals for a given unit or lesson to decide whether and how various tools, including calculators, can help students learn. In gen-eral, teachers should model and encourage calculator use when— • the focus of instruction is problem solving; • the availability of an efficient and accurate computation-

al tool is important; • the lesson involves a search for, and an exploration of,

patterns; • anxiety about computation might hinder problem solv-

ing; and • student motivation and confidence can be enhanced

through calculator use.


Summary As adults, most of us would not hesitate to pick up a cal-culator when we balance our checkbooks or do our taxes. Engineers, architects, building contractors, accountants, store clerks, and scientists readily use computing tools ev-ery day. Withholding opportunities for students to learn to use computing tools effectively and efficiently puts them at a disadvantage in today’s technological society. Principles and Standards advocates computational flu-ency as an expectation for all students. It encourages thoughtful use of calculators in elementary school class-rooms. As a society, we have always welcomed techno-logical advances that make our lives easier and our work more efficient and productive. We use word processors to write letters and prepare legal documents. We use spread-sheets to keep track of personal finances. These tools, like the $4 calculator, help us do our work more efficiently and use our results to answer questions and influence decision making. Calculators serve as efficient and accurate com-putational tools for both students and adults. Principles and Standards asserts, “Today, the calculator is a commonly used computational tool outside the classroom, and the en-vironment inside the classroom should reflect this reality” (NCTM 2000, p. 33).

Bibliography Coburn, Terrence G. “The Role of Computation in the

Changing Mathematics Curriculum.” In New Directions for Elementary School Mathematics, edited by Paul R. Trafton and Albert P. Shulte, pp. 43–56. Reston, Va.: Na-tional Council of Teachers of Mathematics, 1989.

Dick, Thomas. “The Continuing Calculator Controversy.” Arithmetic Teacher 35 (April 1988): 37–41.

Lindquist, Mary M. “It’s Time to Change.” In New Directions for Elementary School Mathematics, edited by Paul R. Trafton and Albert P. Shulte, pp. 1–13. Reston, Va.: Na-tional Council of Teachers of Mathematics, 1989.

National Council of Teachers of Mathematics (NCTM). “NCTM Position Statements.” nctm.org/about/position .htm.

———. Principles and Standards for School Mathematics. Reston, Va.: National Council of Teachers of Mathemat-ics, 2000. standards.nctm.org.

Ralston, Anthony, Barbara J. Reys, and Robert E. Reys. “Calculators and the Changing Role of Computation in Elementary School Mathematics.” Hiroshima Journal of Mathematics Education 4 (1996): 63–71.

Reys, Robert. “Computation and the Need for Change.” In Computational Alternatives for the Twenty-First Century, edited by Robert E. Reys and Nobuhiko Nohda. Reston, Va.: National Council of Teachers of Mathematics, 1994.

Reys, Robert E., and Nobuhiko Nohda, eds. Computational Alternatives for the Twenty-First Century. Reston, Va.: National Council of Teachers of Mathematics, 1994.

Shuard, Hilary. “CAN: Calculator Use in the Primary Grades in England and Wales.” In Calculators in Mathemat-ics Education, 1992 Yearbook of the National Council of Teachers of Mathematics (NCTM), edited by James T. Fey and Christian R. Hirsch, pp. 33–45. Reston, Va.: NCTM, 1992.

Silver, Edward A., and Patricia Ann Kenney, eds. Results for the Seventh Mathematics Assessment of the Nation-al Assessment of Educational Progress. Reston, Va.: National Council of Teachers of Mathematics, 2000.

Edited by JEANE JOYNER, [email protected], Depart-ment of Public Instruction, Raleigh, NC 27601, and BARBARA REYS, [email protected], University of Missouri, Columbia, MO 65211. This department is designed to give teachers informa-tion and ideas for using the NCTM’s Principles and Standards for School Mathematics (2000). Readers are encouraged to share their experiences related to Principles and Standards with Teach-ing Children Mathematics. Please send manuscripts to “Princi-ples and Standards,” TCM, 1906 Association Drive, Reston, VA 20191-9988

Reprinted with permission from Teaching Children Mathematics, copyright October 2001, by the National Council of Teachers of Mathematics. All rights reserved.


GRADES 2-6

Teaching Addition and Subtraction Facts: A ChinesePerspective

Wei Sun and Joanne Y. Zhang In its Principles and Standards for School Mathematics, the NCTM suggests that fluency with basic addition and subtraction number combinations is a goal in teaching whole-number computation (NCTM 2000, p. 84). A mastery of lower-order skills instills confidence in students and facilitates higher-order thinking. The ability to automatically recall facts strengthens mathematical ability, mental mathematics, and higher-order mathematical learning. Without this automation, students have difficulty performing advanced operations. How teachers can help children master the basic

addition

and subtraction facts is an important, long-standing issue in every country in the world. Educators in different countries have developed unique approaches to teaching basic addition and subtraction facts. This article examines how Chinese mathematics educators deal with these facts.

Differences in Language Structure Researchers have found that children’s spoken language affects how they think and, thus, can affect learning of the basic facts (Miura et al. 1994). For instance, compare the counting sequence in English with that in Chinese, as shown in table 1. Unlike the English, Chinese clearly and consistently highlights the grouping-by-ten nature of our numeration system. In Chinese, fourteen is ten-four, eighteen is ten-eight, and thirty is three-ten. The structure of the language easily leads Chinese children to view two-digit numbers as tens and ones (Cao 1994). They can readily think of 12 both as one group of ten items plus two ungrouped items and as a collection of twelve ungrouped items. English counting terms are less explicit and consistent in revealing the base ten nature of our numeration system. For example, twelve is not ten-two and twenty is not two-ten. Furthermore, Yang and Cobb (1995), in their study of children’s conception of number, found that American mothers rarely interpreted numbers in the teens as composites of one 10 and some 1’s when they interacted with their children. Instead, they usu ally initiated and guided learning activities in which children completed tasks involving numbers in the teens by counting by ones orally or with manipu latives. This practice reinforces the view that 12, for example, represents only a collection of twelve items. As a result, Chinese children are more inclined than children in the United States to use tens and ones to represent numbers and, subsequently, to use 10 as a bridge when performing addition and subtraction.

Differences in Teaching American teachers often use counting in a one-to- one correspondence to introduce addition and sub traction of whole numbers. This strategy is based on the “one more than” relationship between consecu tive whole numbers;

for example, 4 is one more than 3 and 9 is one more than 8. As a result, when two-digit addition and subtraction are introduced, American children rely heavily on counting-based and collection-based concepts; for instance, 13 is treated as a collection of thirteen objects. Chinese teachers use a three-step method to teach addition and subtraction. Children first develop an understanding of number concepts, the meanings of addition and subtraction, and the relationships between addition and subtraction. Next, children mas ter addition and subtraction facts in three substeps. First, they learn sums and related subtraction facts to 10, then they learn facts between 11 and 20, and finally, they learn facts between 20 and 100. In the third overall step, students are introduced to the addi tion and subtraction algorithms. Each step is the foun dation for the next step. Making sure that children suc cessfully complete one level before moving to the next is important to the teachers. If children acquire a solid foundation at each of these three steps, they can


easily extend the process to even larger numbers. When sums up to 10 are first introduced in Chi nese elementary schools, counting skills are empha sized to help children understand the relationships among these sums. When sums between 11 and 20 and related subtraction facts are introduced, rather than rely on counting, children are usually encour aged to create collections of tens and ones to repre sent the number; this approach is consistent with the linguistic structure of the Chinese counting sequence. For example, to teach 8 + 3, Chinese children are often asked to take two objects from a collection of three and put them together with eight to make a 10; thus, they see that the whole becomes a collection of ten and one, or eleven. The “make ten” thinking strat egy is demonstrated in the following examples:

a) 9 + 4 = ? Think: • 9 + ? = 10. • 9 + 1 = 10. • 4 / \ 1 3 • Therefore, 9 + 1 = 10; 10 + 3 = 13.

b) 8 + 7 = ? Think: • 8 + ? = 10. • 8 + 2 = 10. • 7 / \ 2 5 • Therefore, 8 + 2 = 10; 10 + 5 = 15.

The Chinese numerical language shown in table 1 plays an essential role in this strategy. Moreover, Chinese teachers believe that students should use 10 as a bridge because of its importance in the base-ten numeration system. Chinese teachers strongly emphasize using addition facts to do subtraction. By doing so, they not only encourage students to apply their previ ously learned knowledge in the new situation but also help students see how addition and subtraction are related. Consider the following examples:

a) 13 – 5 = ? Think: • 5 + ? = 13. • 5 + 8 = 13. • Therefore, 13 – 5 = 8.

b) 15 – 8 = ? Think: • 8 + ? = 15. • 8 + 7 = 15. • Therefore, 15 – 8 = 7.

Differences in Thinking Strategies Thinking strategies are emphasized in both America and China in teaching the basic facts (see, e.g., Baroody [1998]), but the way in which these facts are presented is quite different. Many American textbooks arrange the basic facts

as families, such as sums of 12, sums of 13, and so forth. The textbooks introduce a variety of strategies, such as counting up, learning doubles, or recognizing double-plus-one and double- minus-one situations (see, e.g., Addison-Wesley Mathematics [Menlo Park, California], Houghton Mifflin Math Central [Dallas, Texas], Harcourt Brace Math Advantage [Orlando, Florida]). To some extent, basic facts are viewed as associations to be memorized through hands-on activities, then recalled on demand. Chinese textbooks arrange the basic facts using fact tables (Curriculum and Teaching Materials Research Institute 1999), and the primary strat egy taught is “make 10.” Chinese teachers gen erally introduce the basic facts in units, such as the 6+ unit, the 7+ unit, and so on. These units are categorized by the known entity (addends) instead of the unknown entity (sums). This dif ference between American and Chinese teaching can be seen in table 2.

The fact families from 2 to 18 contain 153 additional facts that American students need to study; the fact table, in con-trast, contains only 81 facts. When students understand the commu tative property of whole numbers, the number of addition facts that they need to know is reduced to only 45 (see table 3). Although Chinese and American textbooks arrange addition facts differently, they both use relation-ships to mini mize the amount of information that must be memorized. When Chinese children learn the basic facts, their task involves not only memorizing but also using logical thinking and reasoning based on relationships among the numbers. Encouraging children to examine a visual aid similar to table 3 and to look for patterns and relationships can help them devise thinking strategies that can aid in mastering the basic facts (see, e.g., Baroody [1998]). Chinese teachers also teach different strategies that are not introduced in the textbooks but that can help children see the patterns among the addi tion facts. Consider the fol-lowing examples, in which n is a whole number:

• For n + 1, the sum is the next whole number, that is, the number after n in the counting sequence (Baroody 1998).

• For n + 2, the sum is the next odd or even whole number. • The sum of n + 9 can be found by adding 10 to n, then

subtracting 1. This strategy is a shortcut for the make-10 approach discussed previously.


Because they rely on such thinking strategies, Chi nese children rarely use manipulatives to figure out facts. Two other strategies for subtraction are often seen in Chinese classrooms. One is to use 10 as the bridge number in a subtraction equation. Consider these examples:

a) 14 – 9 = ? Think: • 10 = 9 + (1). • 14 – 10 = 4. • 4 + 1 = 5 (because you subtract one more, you need to add one back). • Therefore, 14 – 9 = 5.

b) 15 – 8 = ? Think: • 10 = 8 + (2). • 15 – 10 = 5. • 5 + 2 = 7 (because you subtract two more, you need to add two back). • Therefore, 15 – 8 = 7.

The other strategy also uses 10 as a bridge, but it requires students to recall simple addition facts. The following are examples:

a) 13 – 4 = ? Think: • 13 = 10 + 3. • 10 – 4 = 6. • 6 + 3 = 9. • Therefore, 13 – 4 = 9.

b) 16 – 9 = ? Think: • 16 = 10 + 6. • 10 – 9 = 1. • 1 + 6 = 7. • Therefore, 16 – 9 = 7.

Summary When using thinking strategies to perform addition and subtraction, students reinforce their under standing about the facts that they have learned by using those facts re-peatedly. By the time they fin ish learning single-digit ad-dition and related sub traction, they can easily recall the addition and sub traction facts and are more than ready to learn the formal algorithms of addition and subtraction. Chi-

nese teachers introduce these strategies as early as first grade (Curriculum and Teaching Materials Research Insti-tute 1999). Students may not be expected to master these strategies in a short time, but if the foundation is laid early, students can apply their knowledge of the basic facts and these strategies to other mathematical content that they will study later.

References Baroody, Arthur J. Fostering Children’s Mathematical Pow-

ers: An Investigative Approach to K–8 Mathematics Instruction. Mahwah, N.J.: Lawrence Erlbaum Associ-ates, 1998.

Cao, Feiyu. “Development of Pre-School Children’s Opera-tional Ability.” In Reform of Elementary Mathematics Edu-cation, 197205. China: People’s Education Press, 1994.

Curriculum and Teaching Materials Research Institute. Nine-Year Compulsory Education Elementary Math-ematics Series. China: People’s Education Press, 1999.

Miura, Iren T., Yukari Okamoto, Chungsoon C. Kim, Chih-Mei Chang, Marcia Steere, and Michel Fayol. “Comparisons of Children’s Cognitive Representation of Number: China, France, Japan, Korea, Sweden, and the United States.” International Journal of Behavioral Development 17 (Sep tember 1994): 401–11.

National Council of Teachers of Mathematics (NCTM). Princi ples and Standards for School Mathematics. Reston, Va.: NCTM, 2000.

Yang, Ma Tzu-Lin, and Paul Cobb. “A Cross-Cultural Investi-gation into the Development of Place-Value Concepts of Children in Taiwan and the United States in Educational Studies.” Educational Studies in Mathematics 28 (January 1995): 1–33.

WEI SUN, [email protected], teaches at Towson University, Towson, MD 21252. He is interested in teacher education, gifted students, curriculum development, and comparative studies. JOANNE ZHANG, [email protected],net, teaches at Hollywood Elementary School, Hollywood, MD 20636. She has a special interest in effective instruction, including mathematics teaching strategies, cross-cultural studies, and learning disabilities. The authors would like to thank Professor Arthur BAROODY for his help in revising the manuscript.

Reprinted with permission from Teaching Children Mathematics, copyright September 2001, by the National Council of Teachers of Mathematics. All rights reserved.


GRADES 3-6

Dividing Fractions: Reconciling Self-Generated Solutionswith Algorithmic Answers

Marcela D. Perlwitz In this article, I discuss some key episodes that occurred in one of my mathematics classes on basic arithmetic no-tions. The core concepts of the course included place-value numeration, whole numbers and operations, fractions and opera tions, and foundations of number theory. My instruc-tional approach focused on students’ inquiry, empha sizing their own interpretations and their explanations and justifi-cations of their answers. To support stu dents’ inquiry, the instructional tasks were open-ended and often presented within a problem-solving context. Students worked collab-oratively in small groups or pairs, then presented their so-lutions and answers to the whole group. As the teacher, I acted as a facilitator and guide for the students’ self-gener-ated solutions and their exchange of ideas and negotiation of mean ing. The episodes discussed here were selected from a sequence of lessons on the division of fractions.

Students’ Beliefs and Expectations My students brought to the classroom beliefs that were consistent with their past experi ences (Frank 1990) where learning mathematics had been characterized by the quick production of “answers.” Consequently, they thought that it was my role as the teacher to pass along proce dures and that their job was to apply the neces sary algorithms or rules. They also believed that there was only one way to solve a problem. Thus, what my students had learned re-garding expecta tions for accepted evidence of knowledge or un derstanding contradicted my emphasis on stu dent-generated solutions. Furthermore, since they were now required to think on their own, it became apparent that they placed little or no trust in their own ability to solve prob-lems and at first resisted my instructional approach. The students’ limited understanding of fractions further aggra-vated their lack of confidence.

Division with Fractions in the Context of LinearMeasurement To introduce my students to problem solving involving the division of fractions, I posed the following task. My ex-pectation was that they would solve it using self-generated methods.

In Ms. Smith’s sewing class, students are making pil-lowcases for the open house exhibit. Ms. Smith bought 10 yards of fabric for her class project. Each pillowcase requires 3/4 yard of fabric. How many pillowcases can be cut from the fabric?

Students joined their partners to find solutions to this problem. Soon thereafter, some students sug gested that this was a division problem and that if they used the invert-and-multiply rule, they would get the right answer. Howev-er, in light of the expec tation that they should explain their answers, the stu dents could not just use invert and mul-tiply without explaining and justifying how it works. Since none of the students knew the basis of the algorithm, they sought their own solution methods to find the an swer, then many used the algorithm to check the an swer. In doing so, they encountered a discrepancy between the standard-al-gorithm answer and the one they derived using their own methods. In their at tempt to reconcile the answers, sev-eral students had to come to terms with their lack of un-derstanding of the result they obtained using the standard algo rithm. Here are some of the students’ solutions. Christine: First I laid out 10 pieces of 1-yard mater ial. Then I took out 3/4 from one piece leaving 1/4 of a piece of fabric from each yard piece [see fig. 1]. Then I added up all 1/4 pieces to see how many groups of 3/4 I could


make. The final answer is 13 pillowcases with 1/4 piece leftover [pauses] or what I thought was 13 1/4. When I went to check it doing the invert-multi ply method of old days, the answer was 13 1/3 [seem ingly perplexed]. I can’t under-stand why. Next I called on David. He drew a large rectangle with 10 equal sections to represent the 10 yards and further subdi-vided the first rectangle into 4 equal rectangles (see fig. 2). David: I know we have 4 fourths in each yard; 10 x 4 = 40. In 10 yards, we have 40 fourths. Each pil lowcase needs 3/4. Thirteen times 3 is 39, so I can make 13 groups of 3 [fourths]. I have one 1/4 of a yard leftover. The answer is 13 1/4. But that’s not right. If you do 10 x 4/3 you get 13 1/3. How come? Both Christine and David doubted their self-gener ated solutions because they trusted the algorithm. Their focus on the right answer was overshadowing their activity, and they could not recognize that the numbers 13 and 1/4 re-ferred to units of a different nature. Hence, they merely jux-taposed both units with no consideration of the fact that 13 indicated the number of cuts of size 3/4 yard (or pillowcase lengths) in 10 yards and that the 1/4 indicated the 1/4-yard length of leftover material. This explana tion was further evi-dence that the students did not know the meaning of the numbers in the answer ob tained with the algorithm. The 13 1/3 in the algo rithm means 13 1/3 pillowcases, or 13 whole lengths of size 3/4 yard and 1/3 of another (or 1/3 the length of one pillowcase). Some students protested and said it was my responsibility as their teacher to explain the apparent disparity. Next, Betsy raised her hand to vol-unteer her solution. Betsy: What I did was to draw 10 squares side by side. Then I divided them into four pieces each [see fig. 3] and did the counting like this: 3/4 for one pil lowcase, another 3/4 for another pillowcase, 3/4 for another pillowcase, that’s 9/4. [As she talked, Betsy recorded her numbers in two columns while mark ing the picture accordingly, as shown in fig. 3.]

3/4 yd. 1 pillowcase 6/4 yd. 2 pillowcases 9/4 yd. 3 pillowcases 12/4 = 3 yd. 4 pillowcases

When I saw that 12/4 make 4 pillowcases and I’d used up 3 yards, I figured that with 6 yards, I can make 8 pillowcases; with 9 yards I can make 12 pillowcases.

From the last yard, I can take 3/4 and make another pil-lowcase and have 1/4 yard of fabric left [and continued recording].

6 yd. 8 pillowcases 9 yd. 12 pillowcases 9 3/4 yd. 13 pillowcases 1/4 yard leftover

I can make 13 pillowcases and there is 1/4 of a yard leftover. The answer is 13 pillowcases and the problem is solved.

Betsy’s solution involved proportional thinking as reflect-ed in her double counting of yardage and number of pil-lowcases, which she recorded in two side-by-side columns. Betsy exhibited a greater abil ity to unitize as she was able to take 3/4 as her counting unit for the length of fabric. Several students voiced their discomfort with Betsy’s an-swer. Ann spoke almost in protest. Ann: You got only 13? The answer is 13 1/3 be cause the formula is right! Betsy: The question is “How many pillowcases of 3/4 of a yard can you make?” and you can make 13 pillowcases. You can’t make another pillowcase with just 1/4 of a yard of fabric. The answer is 13, the problem has been solved! A couple of students nodded in agreement, while others insisted on the answer of 13 1/3. At this point, I reminded the class that there were three an swers to think about now: 13, 13 1/3, and 13 1/4. Several students in unison declared “the invert and multiply is the right one.” Betsy: I know how to cut fabric. The problem has been solved. The question was “How many pillow cases can you make?” Why are we arguing about the piece leftover? You either have enough to make the pillowcase using 3/4 of a yard or you don’t. Only in school you have to give answers with mixed frac tions. It doesn’t always make sense in real life. Silence followed Betsy’s comments. Then Ann volun-teered. Ann: Betsy has a good point, but I still would like to know why one gets two different answers; 13 1/4 seems right; I have counted several times and I get the same thing, 13 1/4. Betsy’s and Ann’s arguments raise two important and related pedagogical issues. On the one hand, presenting problems in context helps the learner seek solutions that


make sense, given the condi tions of the task. Betsy’s solu-tion illustrates this ability, as her answer makes the most sense in the given context. On the other hand, we want students to move beyond context and be able to generalize and work with numbers efficiently. The latter con siderations make Ann’s point a legitimate one, too. Indeed, if we were to report the measurement of 10 yards of material using a measuring stick 3/4-yard long, the answer would be 13 1/3 measuring-stick lengths. This occurs because the piece 1/4-yard long would figure in the measurement as the frac-tion 1/3 of the 3/4-yard-long measuring stick (see fig. 4). However, as a teacher, I wanted the stu dents to resolve their cognitive impasse. Teacher: OK. I would like you to think about a few things. First, what the problem is asking you. Second, think about what 13 and 1/4 stand for in your solution. Why don’t you do the measuring in our next meeting? The next class, before we began measuring, Christine opened the discussion and volunteered her thinking. She had recorded her solution in her class notebook and re-ferred to it as she talked to the class. Christine: It took me a while to understand that we were not using a yardstick as a measuring tool. We were looking at 3/4 of a yard to see how many pillowcases of that length would come out of 10 yards. When I looked at 1/4 that way, it would re ally be 1/3 of the length of a pillowcase. I figure that 1/4 is 1/3 in relation to 3/4. Several students were puzzled by Christine’s ex planation; Paula’s reaction was representative of their thoughts. Paula: I can’t understand how 1/4 can be 1/3. What is she saying? Apparently, Paula could not follow Christine’s ex-planation. She still failed to recognize that the 1/4 and the

1/3 were fractions of different units of refer ence. Paula and other students in the class did not realize that 1/4 yard is 1/3 of 3/4 yard. They could not coordinate the different units involved in the task. At this point, rather than have Christine demonstrate it, I asked students to get involved in the process of measuring. To each small group I handed a 10-yard-long unmarked white paper tape and a 3/4-yard-long unmarked colored paper tape. The task was to mea-sure and keep track of the process so they could explain the result. After the small-group work, students reported to the whole class. To facilitate their demonstrations, I taped one of the 10-yard strips on the board, and students came to the board to show how they conducted their measure-ment. The following exchanges took place. Paula: We placed the 3/4 piece on top of the 10 yard piece one time after another. We marked the point where each 3/4 piece ended and so forth. We counted 13 times and got a piece leftover. Teacher: How would you report the result of your mea-suring? Paula: Ann and I were talking about it and we are not sure. There is a 1/4 yard of fabric leftover but we still don’t know about the 1/3. Betsy: We did the same thing, but we folded the short piece [of the measuring stick] where we ran out of fabric and it’s 1/3 of the measuring stick [see fig. 5]. Kathy: We marked how many times the leftover went over the measuring stick. It was three times, so that’s where the 1/3 comes from! Other students also showed their understanding that 1/4 yard of fabric is 1/3 of a 3/4-yard piece (or 1/3 the length of the pillowcase), but some students still could not see that relationship. However, given their past experiences with


learning fractions and the fact that this was the second les-son on division of fractions, the students’ progress in their under standing of the meaning of the division by a fraction was remarkable. Discussion It is worth noting that the students, at first, did not in-terpret the pillowcase problem as being division. David was the first to suggest it, then this interpretation became widely accepted. Since these events occurred in the last quarter of the semester, my students knew that just find-ing a numerical answer was not acceptable. Still, some did not trust themselves to find their own solution methods and would have just used the standard al gorithm. As I moved around small groups, we rene gotiated the expectation that they had to find their own solutions and that if they were to use the algo rithm, they had to be able to explain how it works. I reminded them how far they had come in their un derstanding of numbers and their ability to solve prob-lems, so why would they revert to using rules they did not understand? That said, they began to generate their own solu tion methods and used the stan dard algorithm to check their an-swers. While checking, they

found a discrepancy between

t their answer and the algorithm-based answer, and their first reac tion was self-doubt. As previously shown, because of their lack of

understanding of the result of the standard

algorithm, they could not readily resolve the discrep ancy and called on me to do it. In stead, I made their conflict the

focus of the mathematical activity. To support their own resolution, I got them involved in actual mea suring. As they measured in groups, they began focusing on the re-lationship between the 1/4- yard leftover and the 1/3 in the algorithm-based an swer. They taught each other that since the measur ing stick was 3/4-yard long, then the 1/4 yard of fabric leftover was 1/3 of the 3/4-yard long measur ing stick. Exchanging ideas and supporting one an other’s learning facilitated the student’s resolution of the disparity between their self-generated solution and the algorithm-based an-swer. Some students needed further experience measur-ing and support from their classmates to relate the frac-tion of left over fabric to the corresponding fraction of the mea suring-stick length. By the end of the instructional se-quence on fractions, the students had learned the meaning of the answer they obtained using the in vert-multiply rule. However, not everybody was able to explain the rule. In the case of a whole number di vided by a fraction, students who adopted David’s solution method were able to explain how the stan dard algorithm works. However, relating their mea surements to the algorithm while dividing two frac tions proved much more difficult for the students. In fact, very few students accomplished it.

Conclusions Although the events related here occurred in a college class, the instruction and findings are pertinent to middle school instruction since the topic of dividing fractions is taught dur ing the middle school grades. In addition, the

lim ited understanding of my college students reflects the complexity of the concepts of fraction. What these experi-ences related here ultimately teach us is that unless we place more emphasis on stu dents’ understanding of num-bers and operations (NCTM 2000), we may be severely limiting our students’ chances to learn mathematics with un derstanding. Furthermore, teaching for mastery of algo-rithms will tend to perpetuate the students’ lack of confi-dence in their own ability to reason mathematically. It became apparent that my students’ experi ences had led them to believe that getting an swers was more impor-tant than the thinking in volved in the solution. At first, they greatly resisted my approach to instruction and did not want to find their own solutions. I insisted on the importance of making personal sense of mathe matics and showed them respect for their think ing and their struggle. This process of renegotia tion of mutual expectations recurred through-out the semester and informed my teaching in two ways. First, it gave me the opportunity to learn about the nature of my students’ understanding. Second, I turned my stu-dents’ current under standing into learning opportunities by guiding them to resolve their own cognitive conflicts rather than intervene to correct their misconcep tions myself. The role of context proved invalu able in the students’ efforts to make sense of the numerical answers. The instructional task I chose to introduce—the division of fractions—was em bedded in the context of linear measurement, which corresponds to the quotitive or measure ment interpretation of division that students en counter with whole numbers (Lamon 1994). The familiarity with this interpretation of di-vision and the context of measuring fabric supported the students’ efforts to reconcile their self-generated solutions with the an swer obtained using the standard al gorithm. In addition, they gained an increased confidence in their abil-ity to understand how algorithmic re sults relate to their self-generated solutions.

References Frank, Martha L. “What Myths about Mathematics Are Held

and Conveyed by Teachers?” Arithmetic Teacher 37 (January 1990): 10–12.

Lamon, Susan J. “Ratio and Proportion: Cognitive Founda-tions in Unitizing and Norming.” In The Development of Mul tiplicative Reasoning in the Learning of Mathematics, edited by Gershon Harel and Jere Confrey, pp. 89–120. Albany: N.Y.: State University of New York Press, 1994.

National Council of Teachers of Mathe matics (NCTM). Principles and Stan dards for School Mathematics. Res-ton, Va.: NCTM, 2000.

MARCELA PERLWITZ, [email protected], lives in Craw-fordsville Indiana. She is interested in algebraic thinking and the role of context in problem solving.

Reprinted with permission from Mathematics Teaching in the Mid-dle School, copyright February 2005, by the National Council of Teachers of Mathematics. All rights reserved.


GRADES 3-6

Developing Ratio Concepts: An Asian PerspectiveJane-Jane Lo, Tad Watanabe, and Jinfa Cai

The following vignette illustrates how a Taiwanese text-book series envisions introducing the concept of ratio. Textbook. There are two blocks in front of you. One is 6 cm long an the other is 2 cm. How many times as long is the 6 cm block compared with the 2 cm block? Some students use the 2 cm block as a masuring unti to figure out that 6 cm is 3 units of 2 cm. Other students reason with the two quantities directly and come up with the equation 6 ÷ 2 = 3.

Textbook. When comparing two quantities, one of them has to be used as the base quantity. There are two ways to relate the other quantity to the base quantity. The first way is to find out how much more the second quantity is than the base quantity. For example, how many cm longer is the 6 cm block than the 2 cm block?Solution. 4 cm.Textbook. The second way is to find out how many times as long is the second quantity as the base quantity. For example, 6 cm is 3 times longer than 2 cm. Another way to represent this relationship is to use the word bi. Write as 6 bi 2, and 6:2. The result of this comparison, 3, is called the “value of the ratio.”

A recent analysis of Asian curricular materials has identi-fied several key ideas that are emphasized in the introduc-tory lessons of ratio (Lo, Cai, and Watanabe 2001). These key ideas include distinguishing a multiplicative compari-son from an additive compari son; identifying a base quan-tity and measuring unit for comparison; distinguishing and relating the ratio a:b, the division a ÷ b, and the value of ratio a/b; and learning the importance of units in forming a mean ingful ratio relationship. After the introduction of ratio, two or three more lessons were devoted to the ideas of equivalent ratios, simplified integer ratios, and applications of ratio concepts. Some of these dis cussions are familiar to mathematics teachers in North America, whereas others seem to be unique to the Asian materials. In this article, we will elaborate on these key ideas and give examples from textbook series in China, Taiwan, and Japan (Division of Mathematics 1996; National Printing Office 1999; Tokyo Shoseki 1998). Our goal is not to evaluate Asian materials but rather to provide an international perspective that may help increase teachers’ experi ence and awareness when they strive to help stu dents develop ratio concepts (Cai and Sun 2002).

Introduction of Ratio ConceptsDefining ratio as being a multiplicative relationship Unlike typical U.S. textbooks that consider a:b and a/b as two different ways to represent a ratio, Asian textbooks clearly distinguish between ratio a:b as a multiplicative re-lationship between two quantities and the value of ratio as the quotient a/b of the divi sion a ÷ b. In the previous exam-

ple, the multiplica tive relationship between the 6 cm block and the 2 cm block can be represented as 6:2. The result of 6 ÷ 2 , or 3, is called the value of the ratio 6:2, where 6 is called the front term of the ratio and 2 is called the back term of the ratio. Conceptually, this idea is equivalent to saying “6 is 3 times as many as 2.” Note that the idea of using the second quantity as the base for comparison can be linked directly to measurement division (quotitive), even though the term “measurement division” is not directly used in Asian textbooks. For example, the teacher’s manual of a Japanese textbook talks about conceptualizing the value of the ratio of a:b as the relative value of a when considering b as a base quantity.

Identifying the base quantity for comparison Since the ratio is a way to compare two quantities using the division operation and since division is noncommuta-tive, the order of the two terms for a particular ratio is impor-tant. In other words, a:b and b:a describe the multiplicative relationship between quantities a and b from two perspec-tives. The value of ratio a:b is not the same as the value of b:a, unless a equals b. The Chinese teacher’s manual indi-cated the reciprocal relationship between a:b and b:a but suggested that the reciprocal relationship not be explicitly mentioned to students at the intro ductory stage to avoid possible confusion. To highlight this idea, a Taiwanese textbook posed two different questions comparing the num ber of cookies for two brothers when the younger brother has 5 cookies and the other has 2 cookies. The first question was this: “The number of cookies the younger brother has is how many times the older brother’s number?” The second question was this: “The number of cookies the older brother has is how many times the younger brother’s number?” The so-lution to the first problem was 5 ÷ 2 = 5/2 = 2 1/2 = 2.5. Students can use 5:2 to represent this ratio relationship. The solution to the second prob lem was 2 ÷ 5 = 2/5 = 0.4. Students can use 2:5 to represent this ratio relationship. A pictorial repre sentation similar to figure 1 was used to fa-cilitate understanding. Note that both fraction and decimal notations can be used for the value of ratio. We want to emphasize two cautions about forming a ra-tio relationship: 1. After the discussion of ratio definitions, the teacher’s manual in the Chinese textbook pointed out two difficulties that students may encounter when they relate ratio con-cepts to their daily experi ences. First, not all related pairs of numbers form a ratio relationship. For example, in Chinese spoken language, the phrase “5 bi 3” is used to express the scores of two teams in a sport event. However, in this context, the focus of the comparison was on the addi tive relationship (“The number of team A has so many more points than Team B”) rather than the multiplicative relation-ship (“Team A’s points are so many times the number of


discussion of a division principle: ak ÷ bk = a ÷ b when k ≠ 0, which students have learned before. Furthermore, Asian textbooks gave detailed illustrations to connect the idea of equivalent ratio with the idea of changing units. For example, a Taiwanese textbook identified a ra-tio of 20:30 as being the relationship between the width (20 cm) and the length (30 cm) of a rectangle. Then the students were asked to use 5 cm as a unit to measure the width and the length of the same rectan gle. As a re-sult, the width became 4 units (of 5 cm) and the length became 6 units (of 5 cm), thus a ratio of 4:6 can be used to represent the same width versus length relationship. Last, the students were asked to use 10 cm as a unit to measure the width and the length of the same rectangle and obtain another ratio, 2:3. Thus, the relationship 20:30 = 4:6 = 2:3 was estab-lished and illustrated by diagrams similar to figure 2.

Discussion of Simplified Integer Ratios Exercises asking students to convert a given ratio into a simplified integer ratio are another feature of ratio dis-cussion in Asian textbooks. Simplified inte ger ratios a:b

mean that both a and b are integers and that no common factor other than 1 is shared between a and b. Another way to determine if two ratios are equivalent is to convert both into simplified integer ra tios, that is, a1:b1 = a2:b2 if and only if both a1:b1 and a2:b2 are equivalent to the same simplified ratio a:b. All three textbooks include examples like the ones in figure 3 to help students apply this idea. Several significant points can be made about this type of exercise. First, it reinforces the idea that a ratio is a rela-tionship between two quantities and that those two quanti-ties can be represented in a variety of numerical forms—in-tegers, fractions, or decimals. Second, it provides another method to check the equivalence of two ratios that rein-forces the ratio concept (i.e., two ratios are equivalent if after simplifying they both equal the same simplified integer ratio). Third, it provides opportunities for students to relate numbers to each other through common multiples and factors. Lo and Watanabe (1997) have found this kind of conceptualization es sential to develop flexible proportional reasoning.

Application of Ratio Concepts After the basic concepts of ratio and equivalent ratio were established, all three Asian text book series included examples and exercises that ask students to apply the con-cepts of ratio in a variety of contexts. There were two basic types of questions: 1. The first type gave a ratio relationship between two quantities and the actual amount of one of those two quan-tities, then asked students to use the ratio relationship to find the actual amount of the second quantity. The following is an example of this type of question from the Taiwanese textbook:

The ratio between the number of boys and the number of girls in a summer camp is 4:3. There are 63 girls. How many boys are in the summer camp?

Team B’s points”). Teach ers need to be aware of the po-tential confusion that students may have about the use of language inside and outside of mathematics classrooms. A similar caution can be made about the English language, since the phrase “a to b” is used both for ratio and for a sports context in the United States. 2. The teacher’s manual indicated the impor tance of paying close attention to units when com paring two quanti-ties. In particular, at the introduc tory level, the comparisons of two like quantities should be made with the same units to make them meaningful. For example, in a Chinese text-book, the following problem was posed:

Li Ming is 1 meter tall, and his dad is 173 cm tall. Li Ming said that the ratio between his height and his dad’s height is 1:173. Is 1:173 the best way to describe the relationship between Li Ming’s height and his dad’s height?

Through discussion, students are guided to form a more meaningful ratio relationship if they either convert 1 meter to 100 cm or convert 173 cm to 1.73 meter to form the ratio 100:173, or 1:1.73. This em phasis is important when con-sidering the idea of “value of ratio” as the relative size of the second quantity when the base quantity is considered to be 1. In addition, this measurement context shows the need to define equivalent ratios.

Conceptualization of equivalent ratios Two ratios are defined as being equivalent if they rep resent the same multiplicative relationship. One nat ural implica-tion of this definition is that the values of two equivalent ratios have to be equal, that is, a:b = c:d a ÷ b = c ÷ d. In both Chinese and Japanese text books, the principle of equivalent ratios, “Multiplying or dividing the front term and the back term by the same nonzero number will create equivalent ratios,” was supported through examples and

c


This question may be classified as a missing-value-proportion problem because a proportional relationship (equivalent ratio) is involved. However, it is easier to solve than a typical proportion problem (“If a car uses 8 gallons of gasoline in traveling 160 miles, how many miles could the car travel on 30 gallons of gasoline?”) for the following two reasons: First, one major chal lenge of solving this sort of problem is to construct a ratio relationship between two dif-ferent measures: gal lons and miles. In the summer-camp problem, a ratio re lationship is stated explicitly in the ques-tion. Second, a typical proportion problem involves some “changes” in states—before and after. In these antecedent problems, the ratio and the quantities are from the same situation. 2. The second type of question in the Asian text books gave the ratio relationship between two quan tities and the

sum of the two quantities, then asked students to use the ratio relationship to find the ac tual amount of each of the two quantities. For exam ple, the following question was in-cluded in the Japanese textbook series:

Two brothers shared 1800 Yen. The ratio be tween the older brother’s money and the younger brother’s money was 3:2. How much was the older brother’s share?

To prepare students for more complex proportion prob-lems, two methods of solution for each type of problem were suggested in the student version of the textbooks. One method helped students connect ratio and fraction concepts through multiplicative comparison, thus convert-ing a ratio problem into a problem involving multiplying by a fractional amount. The other method required the direct appli cation of the principle of equivalent ratios. For the sharing-of-money problem, the Japanese text-book series ask the following sequence of ques tions to en-courage students to think about these two solution meth-ods:

1. The older brother’s money was what fraction of the total amount of money?

2. Write down a computation sentence that will de termine the older brother’s share.

3. Solve the problem using the following equation: 3:5 = x:1800. 4. What was the younger brother’s share?

The diagram in Figure 4 was used to help students con-ceptualize the first two questions. From figure 4, one could reason that if the older broth-er’s money comprised three units and the younger broth-er’s money comprised two units, then the total amount of 1800 Yen was equivalent to 5 units. So the older brother’s money was 3/5 of the total amount of money. Thus, the answer for question 2 was 1800 × 3/5, and students could figure out the older brother’s share of 1080 Yen from this computation. Question 3 above suggested a second strategy that re-quired directly applying equivalent ratios. Since the ratio between the amount of money that the older brother had (x Yen) and the amount of total money (1800 Yen) could be expressed as the ratio 3:5, one could solve this problem using the principles of equivalent ratios: Because 1800 is 360 times 5, x must be 360 times 3, which results in the an-swer of 1080 Yen. The younger brother’s share could then be solved with either approach. Using both methods helps students see how the ideas of multiplicative comparison, fractions (or decimals), ratios, simplified ratio, and equiva-lent ratios are connected.

Conclusion The concepts of ratio and proportion are among the most important topics in school mathe matics, especially at the middle school level. How ever, studies have repeatedly shown that most mid dle school students have difficulties with these concepts (NCTM 2000). This article included


ideas and examples used by Asian textbooks to teach the concepts of ratio that are fundamental to the develop ment of proportional reasoning. In Asian textbooks, the concepts were carefully introduced through an emphasis on multi-plicative comparison, the link to measurement (quotitive) division, the identification of base quantity, and the distinc-tion between ratio and nonratio pairs of quantities. The idea “value of ratio” was introduced to firmly establish the ratio’s identity as a relationship based on multiplicative compari-son rather than just another way to write a fraction. Rather than move directly into the concepts of proportion, Asian textbooks spent time developing the idea of equivalent ra-tios and simplified integer ratios and discussing how these ratio-related con cepts could be used to solve problems in everyday contexts. Typically, pictorial representations were used and multiple solution methods were discussed to help students relate ratio concepts to other previ ously learned concepts such as measurement (quoti tive) division, frac-tions, and divisors. Furthermore, exercises and examples were carefully chosen to link the ratio concepts to previous studies on fractions (including fractions greater than 1) and decimals. We believe that these approaches all aim to de-velop proportional reasoning, which is essential in solving proportion problems. In general, Asian textbook series do not include units in mathematics sentences as part of the writ ten computation. We can probably argue the advan tages and disadvan-tages of such a practice, but it goes beyond the focus of this article. Nevertheless, the Asian materials we analyzed did treat units care fully and systematically. The examples of compar ing Li Ming’s height with his father’s height as well as using the units flexibly to generate equivalent ra-tios discussed earlier in this article illustrate this emphasis. Furthermore, both the textbook series and the teacher’s manuals routinely remind stu dents to think about the mean-ings of the quantities and the units used to quantify these quantities in volved in computation. The goal is to prepare stu dents for more complex contextual problems when multiple computations are required to determine unknown quantities. The examination of curriculum and instructional practice in other nations provides a broader point of view on how topics can be treated. We hope that such an inter-national per spective can add to U.S. teachers’ background when they try to address the issues and challenges facing students’ learning of ratio and proportion.

References Cai, Jinfa, and Wen Sun. “Developing Students’ Propor-

tional Reasoning: A Chinese Perspective.” In Making Sense of Fractions, Ratios, and Proportion, 2002 Year-book of the National Council of Teachers of Mathematics (NCTM), edited by Bonnie Litwiller and George Bright, 195–206. Reston, Va.: NCTM, 2002.

Division of Mathematics. National Unified Mathe matics Textbooks in Elementary School. Beijing: People’s Edu-cation Press, 1996.

Lo, Jane-Jane, Jinfa Cai, and Tad Watanabe. “A Com-parative Study of the Selected Textbooks from China, Japan, Taiwan and the United States on the Teaching of Ratio and Proportion.” Proceed ings of the Twenty-third Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Educa tion, vol. 1, 509–20. Snowbird, Utah, 2001.

Lo, Jane-Jane, and Tad Watanabe. “Developing Ratio and Proportion Schemes: A Story of a Fifth Grader.” Jour-nal for Research in Mathematics Edu cation 28 (March 1997): 216–36.

National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics.Res-ton, Va.: NCTM, 2000.

National Printing Office. Elementary School Mathe matics, 6th ed. Taipei, Taiwan: National Printing Office, 1999.

Tokyo Shoseki. Shinhen Atarashii Sansuu (New elemen-tary school mathematics). Tokyo, Tokyo Shoseki Pub-lisher, 1998.

JANE-JANE LO, [email protected], teaches at Western Michigan University, Kalamazoo, MI 49008-5248. Lo’s special interests include studying the developing of multiplicative con-cepts and preparing future teachers. TAD WATANABE, [email protected]. teaches at Penn State University, University Park, PA 16802. His interests include children’s ultiplicative concepts and mathematics education in Japan. JINFA CAI, [email protected], teaches at the University of Delaware, Newark, DE 19716. His interests include cognitive studies of mathematical problem solving and integration of assessment into the classroom.

_________The preparation of this article was supported, in part, by a grant from the National Academy of Education. Any opinions expressed herein are those of the authors and do not necessarily represent-ed the views of the National Academy of Eduation.

Reprinted with permission from Mathematics Teaching in the Middle School, copyright March 2004, by the National Council of Teachers of Mathematics. All rights reserved.


GRADES 7-10

Pick a NumberMargaret Kidd

As a mathematics educator, middle and high school teachers frequently ask me how to motivate students. One method I have found to be effective is to engage them with a topic that personally intrigues them. Since many students are fascinated by magic, this can be used to help them learn procedures from which they normally shy away. In my experiences of teaching in various districts in the coun-try, there are three topics that give students trouble when beginning to study algebra: fractions, operations on inte-gers, and the distributive property. This article combines the motivation of mathematical magic with the difficulty of applying the distributive property and the rules for order of operations. The actively described challenges students to uncover the “magic” behind the mathematics and discover the reason we have order of operations rules. They also come to appreciate the power of using a variable, grouping symbols, and the use of the distributive property.

The Mathemagic Lesson As the “ mathemagician,” begin the lesson by asking students to think of a number from one to ten. With much intrigue, inform them that after performing some mathemat-ics magic-and with their help-you will be able to tell each student his/her starting number. Next, ask them to perform a series of calculations that end with students revealing only their final numbers. At that point you quickly tell them the numbers they first selected. The fact that you can do this with little effort seems like magic to the students and usually gets their attention. If you do this a number of times, their attention is normally riveted. At this point, they are ea-ger to learn how the answer was discovered so quickly and are more amenable to learning the process in order to un-derstand the secret of being able to discern the answer so quickly. Here are some examples to catch their imagination:

Example 1:1. Pick a number from 1 to 10.2. Multiply it by 3.3. Subtract 1.4. Multiply this by 2.5. Add 3.6. What did you get?

Example 2:1. Choose a number.2. Multiply the number by 3.3. Subtract 4.4. Multiply the result by 2.5. Add 5.6. Report your result.

Example 3: (a bit more complicated):1. Pick a number from 1 to 10.

2. Add 3.3. Multiply by 4.4. Subtract 8.5. Divide by 2.6. Add 3.7. What is your ending number?

When students begin clamoring for an explanation so that they can try this on their parents and friends, it is time to explain the mathematics behind these. If you would like to figure these out yourself, please stop reading, since the solutions will now be given.

Example 1:1. Pick a number from 1 to 10. x2. Multiply by 3 3x3. Subtract 1. 3x - 14. Multiply by 2. 2(3x - 1)5. Add 3. 2(3x - 1) + 36. What did you get?

The simplified expression becomes 6x + 1. So when the student states his ending number, you simply subtract 1 from it and divide that answer by 6.

Example 2:1. Choose a number x2. Multiply the number by 3 3x3. Subtract 4. 3x - 44. Multiply the result by 2. 2(3x - 4)5. Add 5. 2(3x - 4) + 56. Report your result.

The simplified expression becomes 6x - 3. When the stu-dent states her ending number, you add 3 to it and divide that answer by 6.

Example 3:1. Choose a number. x2. Add 3. x + 33. Multiply the result by 4. 4(x + 3)4. Subtract 8. 4(x + 3) - 85. Divide by 2. (4(x + 3) - 8) / 26. Add 3. (4(x + 3) - 8) / 2 + 37. What is your ending number?

Although Example 3 contains more and more complex operations, it still simplifies to 2x + 5.

Conclusion These number puzzles can be as short as adding 1 and subtracting the original number to as complicated as one wants to make it. Start with the simplified expression you desire. Then, add the steps in reverse order until you have


a series that is as long and as complicated as you wish. A few caveats are in order, however. Have students pick a number small enough that you can solve the last equation easily. Avoid numbers and operations that result in frac-tions or decimals. Other than that, both you and your stu-dents can have as much fun with these as you want! All of my students completed a homework assignment in which they were to create a number puzzle to impress their parent or other siblings.

MARGARET KIDD, CSU [email protected]

Reprinted with permission from The California Mathemat-ics Council ComMuniCator, September 2009.

CONGRATULATIONS

VCTM 2011

William C. Lowry Mathematics Educator of the Year

Awardees

Elementary Awardee

Anne Blevins

Pocahontas Elementary School in Powhatan, VA in Powhatan County

Math Specialist Awardee

Karen Mirkovich

Swans Creek Elementary School in Dumfries, VA in Prince William County

Middle School Awardee

Harry Holloway

Powhatan School in Boyce, VA

High School Awardee

Tammy Greer

Millbrook High School in Winchester, VA in Frederick County

College/University Awardee

Dr. Robert Q. Berry, III, Ph.D.

Associate Professor; Mathematics Education; Curry School of Education;

University of Virginia

CongratulationsVCTM 2011


GRADES 7-12

Those Darn Exponents: Fifty Challenging True-FalseQuestions

Tim Tilton

Tim Tilton is with Winton Woods City Schools in Ohio. Reprinted with permission from Ohio Journal of School Mathematics, a publication of the Ohio Council of Teachers of Mathematics. Fall 2010.


GRADES 13-16

Abstractmath.org: A Web Site for Post-Calculus MathCharles Wells

The Abstractmath web site at http://www.abstractmath.org/MM/MMIntro.htm is intended for math majors and oth-ers who are faced with learning “abstract” or “higher” math, the kind with epsilons and deltas, quotient spaces, proofs by contradiction: all those kinds of abstract things that can knock you sideways even if you got an A in calculus. I have been developing Abstractmath for a couple of years and now it is time to open it up to the wide world. Not that all is finished. There are gaps and stubs all through it. But enough is completed that it is respectable, and be-sides, I need help! Some students and math educators have already discovered the site and told me things that helped them and things that made no sense to them, as well as finding many embarrassing errors. The site needs much more help like that, and suggestions for more com-pelling examples and useful topics. Abstractmath is personal and opinionated, but it is based on research by many people in mathematics education and cognitive psychology, and on my own lexicographical re-search. It concentrates on certain types of problems. One web site can’t do everything. Mathematical English: This is a foreign language dis-guised as English. Many common logical words (notori-ously “if...then”) don’t mean quite the same thing they do in English. Common words are used with technical mean-ings, leaving the student to be confounded by their every-day connotations. Proofs: A mathematical proof has both a logical structure and a narrative structure. If you are reading a proof your major problem is to extract the logical structure from the narrative you read. Consider: “Theorem: If n is an integer and n2 is even, then n is even. Proof: Suppose n is odd...” How can a proof that n is even start out by assuming it is

odd? Abstractmath walks you through examples of proofs as a guide to how to understand them. Images and metaphors: Mathematicians use lots of com-pelling metaphors to talk and think about their topics and images to give geometric sense to them. These images and metaphors are also dangerous because they may suggest things that are incorrect. (x2 - 9 vanishes at 3.” Does that mean it doesn’t exist at 3?) When mathemat-ics start to prove something about their topic they abandon these images and metaphors and go into a rigorous mode of thinking in which all mathematical objects are inert and unchanging. Does anyone ever tell the students this (as opposed to doing it in front of them)? Abstractmath does, with examples. Mathematical objects: People new to abstract math have a great deal of trouble thinking of mathematical objects as objects rather than processes or bunches. A quotient space has elements that are sets (these sets are not substances - they are elements!). A function space has elements that are functions (not values of functions). Abstractmath dis-cusses many examples of this phenomenon. I hope you will look into abstractmath.org, whether you are a student or a teacher, and let me know how it can be improved. You can also contribute articles or examples, or publish them on your own web site and ask me to link to them.

CHARLES WELLS is Emeritus Professor of Mathematics at Case Western Reserve University.

Reprinted with permission from FOCUS The Newsletter of the Mathematical Association of America, copyright March 2007. All rights reserved.

Virginia Council of Teachers of MathematicsP.O. Box 714Annandale, VA 22003-0714 Pat Gabriel, Exec. Secretary

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rEGiONAL CONFErENCESAlgebra Readiness for Every Student

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