Virginia Mathematics Teacher

Volume 37, No. 2 Spring, 2011

The Five Platonic Solids

A Resource Journal for Mathematics Teachers at all Levels. Virginia Mathematics Teacher Volume 37, No. 2 Spring, 2011

The VIRGINIA MATHEMATICS TABLE OF CONTENTS TEACHER (VMT) is published twice yearly by the Virginia Council of Teach- Grade Levels Titles and Authors...... Turn to Page ers of Mathematics. Non-profit organizations are granted permission to reprint articles appearing in the VMT provided that one copy of the General President’s Message...... 1 publication in which the material is reprinted is sent to the Editor and (Beth Williams) the VMT is cited as the original source.

EDITORIAL STAFF General Statistical Outreach and the Census: A Summer David Albig, Editor, Learning Experience...... 2 e-mail: [email protected] (Gail Englert) Radford University

Editorial Panel General Teaching Time Savers: Reviewing Homework...... 3 Bobbye Hoffman Bartels, Christopher Newport University; (Jane M. Wilburne) David Fama, Germana Community College; Jackie Getgood, Spotsylvania County Mathematics Supervisor; Sherry Pugh, Southwest VA Governor’s School; General Enlivening School Mathematics Through the History Wendy Hageman-Smith, Longwood University; of Mathematics...... 4 Ray Spaulding, Radford University (Martin Bartelt and Stavroula K. Gailey) Jonathan Schulz, Montgomery County Mathematics Supervisor

MANUSCRIPTS & CORRESPONDENCE General Affiliates’ Corner...... 5 For manuscript, submit two copies, typed double spaced. We favor manuscripts on disk or presented electronically in Word. Drawings General The Five Platonic Solids...... 6 should be large, black line, camera ready, on separate sheets, referenced in the text. Omit author names from the text. Include a cover (Theoni Pappas) letter identifying author(s) with address, and professional affiliation(s). General VCTM Awards Two Scholarships...... 7 Send correspondence to Dave Albig at: Box 6942 Radford University General Problem Corner...... 10 Radford, VA 24142 (Ray Spaulding) Virginia Council of Teachers of Mathematics President: Beth Williams, Bedford County Schools General William C. Lowry Award Winners...... 34 Past-President: Carolyn Williamson, Retired from Hanover County Public Schools Secretary: Debbie Delozier, Stafford County Public Schools Grades K-5 Clearing the Confusion over Calculator Use in Grades K-5...... 18 NCTM Rep.: Margaret Coffey, Fairfax County Public Schools (Barbara J. Reyes and Fran Arbaugh) Math Specialist Rep.: Corinne Magee Elected Board Members: Elem. Rep: Sandy Overcash, Virginia Beach City Schools; Meghann Grades 2-6 Teaching Addition and Subtraction Facts: A Chinese Perspective....22 Cope, Bedford County Schools (Wei Sun and Joanne Y. Zhang) Middle School Reps: Anita Lockett, Fairfax County Public Schools; Alfreda Jornegan, Norfolk Public Schools Secondary Reps: Ian Shank, Hanover Public Schools; Cathy Shelton, Grades 3-6 Dividing Fractions: Reconciling Self-Generated Solutions Fairfax County Public Schools. with Algorithmic Answers...... 25 2 Yr. College Rep: Joseph Joyner, Tidewater Community College (Marcela Perlwitz) 4 Yr. College Rep: Joy Whitenack, Virginia Commonwealth; Maria Timmerman, Longwood University Grades 3-6 Developing Ratio Concepts: An Asian Perspective...... 29 Membership: Ruth Harbin-Miles (Jane-Jane L, Tad Watanabe, and Jinga Cai) Publicity: Laura Rightnour, Hanover County Public Schools Treasurer: Diane Leighty, Powhatan County Public Schools Grades 7-10 Pick a Number...... 33 Webmaster: Jennifer Springer, Charlottesville City Schools (Margaret Kidd)

Webpage: www.vctm.org Grades 7-12 Those Darn Exponents: Fifty Challenging True-False Questions...... 35 Membership: Annual dues for individual membership in the Council (Tim Tilton) 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- Grades 13-16 Abstractmath.org: A Web Site for Post-Calculus Math...... 36 able to VCTM to: VCTM c/o Pat Gabriel; 3764A Madison Lane, Falls Church, VA 22041-3678 (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) Printed by Wordsprint Christiansburg 225 Industrial Drive, Christiansburg, VA 24073 Please see the The Five Platonic Solids article on page 6. GENERAL INTEREST President’s Message Beth Williams This spring has been a busy one for our organization! Two Virginia finalists were also recognized as Presiden- If you didn’t get to attend our Annual Conference in Rich- tial Awardees for Excellence in Mathematics and Science mond, you missed an extraordinary event. Fabulous Teaching on Friday night. These colleagues have advanced speakers led every session. Our Department of Education to the state selection in this prestigious mathematics award. mathematics colleagues keynoted two sessions. Deborah We wish Pamela Bostwick and Victoria Hugate both well as Wickham and Michael Bolling shared upcoming events and the PAEMST selection process continues. happenings at the state level to keep us informed and en- Part of being a teacher is being a lifelong learner, studying, ergized to move forward in our work. Many thanks to our researching, and listening to find ways to improve our in- Conference Program Chair, Lisa Hall, the VCTM Executive struction and help more students succeed. To this end, my Board, and the Greater Richmond Council volunteers who goal will be to pose a new question to you in each Journal made the “Making Mathematics Monumental” conference edition. This month my question for you would be: What a wonderful affair! do you know about the Common Core Standards and their VCTM is an affiliate of the National Council of Teachers of implementation? Mathematics, and we support the work that they advocate. In January, NCTM released news of a joint task force In 2010, NCTM released a position paper recommending made up of the Association of Mathematics Teacher Edu- the use of mathematics specialists to enhance teaching, cators, The Association of State Supervisors of Mathemat- learning and assessing of mathematics to improve stu- ics, The National Council of Supervisors of Mathematics, dent achievement. In Virginia, we have been blessed with and the National Teachers of Mathematics. The work of many grants through the National Science foundation, the the task force is to develop actions and resources needed Virginia Department of Education Mathematics and Sci- to help teachers implement the Common Core State Stan- ence Partnership (MSP), NCLB Title II funds and through dards in Mathematics (CCSSM). This task force also con- the ExxonMobil Foundation that support developing math- sidered ways in which the organizations can collaborate ematics specialist programs. in supporting their members and other groups to advance In addition, a program to provide additional leadership their vision of school mathematics. The task force report opportunities for 25 specialists has been implemented identifies five priority actions to be taken as soon as pos- through Virginia Commonwealth University, University of sible. There are also PowerPoint presentations that can in- Virginia, and Norfolk State University with funds from the form all stakeholders. These information opportunities are National Science Foundation. available for you to read at www.nctm.org/news/highlights During the Annual Conference, a large group of mathe- NCTM is working with other groups like NCSM to write sup- matics specialists and coaches from across the state came porting tasks, videos and documents for implementing the together for the first time to begin to formulate ways to CCSSM. Dr. William G. Mccallum led the work to write the engage in professional growth and networking opportuni- standards using the NCTM Focal Points and High School ties. This social time allowed one group of specialists that Sense Making as their foundation. His group is working has been supported by the Leadership Program to identify on a new website, www.illustrativemathematics.org where and connect with other professionals who share similar job good tasks for formative assessments and videos of good responsibilities. Our annual conference banquet Friday teaching strategies will be shared. Dr. Mccallum also has night was a celebration of many accomplishments. Con- a blog at commoncoretools.wordpress.com tina Martin and Vickie Inge shared exciting work from this Even though Virginia has not adopted the Common Core Leadership Program. If you are interested in more informa- State Standards, The Virginia Board of Education has ad- tion about this networking fellowship, seek out the link to opted a supplement to the Curriculum Frameworks to bring mathematics specialists on our website. our Standards into closer alignment with the CCSSM. Our Our 2011 Fall Journal will be dedicated to the work of State Superintendent, Dr. Patricia Wright, wrote in a Su- Mathematics Specialists. For this special Journal under- perintendents Memo in February that the supplement will taking to be a success, we need to have many colleagues ensure that Virginia Standards are equal to or more rigor- submit articles. Please consider writing an article from ous in content and scope than the CCSSM. If you have not your experiences as a practicing specialist, or from work- already done so, you should read the supplement found on ing with one. All articles must be submitted to Dave Albig the Virginia Department of Education website. by July 1, 2011. His email address is [email protected] The implementation of more rigorous standards requires Since our last Journal, Virginia Council of Teachers of better teaching for more learning. All of these resources will Mathematics has awarded several grants and scholar- be valuable to us as we consider new ways to increase our ships. Five First Timers grant awardees and two college students’ success. Your VCTM organization will continue scholarship recipients were honored during our banquet to work to improve your learning, living and love of math- celebration. The monies they received are given to con- ematics. tinue the work of high quality mathematics education for all. Find out more about these funds that are allocated by our Best wishes to you all! organization as you read further in the Journal. Beth

Virginia Mathematics Teacher 1 GENERAL INTEREST Statistical Outreach and the Census: A Summer Learning Experience Gail Englert

“Car A traveled 150 miles in 6 hours, and took another half looked at the mean as the balancing point of a set of data, hour to go the final 40 miles.” My mathematical adventure considering how far each data point fell from the mean. to Washington, DC to attend the American Statistical As- In every case, the material covered was presented as a sociation’s Meeting Within a Meeting, a statistics workshop lesson on a continuum of learning, with background about held August 3-4, 2009 for teachers of grades K–12, started the planning and assessment. Lesson #3 – realizing that off sounding like a badly written word problem. Lesson statistics is all about informed decision-making. I have #1 – don’t leave Norfolk with DC as the destination never presented the reason for collecting, presenting and on a Saturday in August without allowing extra time! analyzing data quite that explicitly to my students, but I will Fortunately, after arriving, my plan to take advantage of our now. nation’s capital before the workshop began unfolded beauti- K – 12 teachers are encouraged to enroll in a K-12 fully. At one museum, the admission fee was $19.95. When teachers free trial membership to the ASA. The trial I mentioned I was a teacher, I was allowed to enter FREE!!! membership offers subscriptions to Amstat News (the ASA’s At another, purchases in the gift shop qualified for a teacher monthly membership magazine), and CHANCE (a magazine discount. Lesson #2 – ask for the educator discount; it focusing on the use of statistics in everyday life). It also may be a financially rewarding experience! provides members-only access to the ASA’s top journals The American Statistical Association (ASA), (http:// and discounts on all ASA meetings and products. After the www.amstat.org/) is a 170-year-old scientific and educational trial, ASA offers a discounted annual membership ($50.00 professional society whose goal is to enhance lives through instead of $125.00) for K-12 teachers. informed decision-making by providing its members and the The following day a visit to the US Census Bureau head- public with up-to-date, useful information about statistics. quarters in nearby Suitland, MD was offered. This included The ASA website contains a wealth of topics to explore… interactive presentations and activities organized by Renée from “Making Sense of Statistical Studies” (the Student Jefferson-Copeland, Chief of the Census in Schools Branch. Module and accompanying Teacher’s Module includes We were introduced to the 2010 Census process, Census in supporting resources with 15 hands-on investigations for Schools activities, and Census’ data and on-line data access upper middle-school or high-school students to explore as tools. The following links have wonderfully rich data that they design and analyze data) to “Statisticians in the News”. could be used for lessons in not only math, but also social Clicking the “Education” tab displays a welcome message studies classrooms. from Dr. Martha Aliaga, Director of Education, and other useful classroom resources. Main URL: This year’s annual Joint Statistical Meeting was held http://www.census.gov/ in Washington, DC. As part of an outreach to educators, the organization provided a day-long workshop focusing Census in the schools: on the teaching and learning of data analysis, probability http://www.census.gov/schools/ and statistics concepts. Workshop participants, who were divided along grade bands (k – 4, 5 – 8, 9 – 19, and Fact finder: BAPS [advanced placement statistics]), spent the day http://factfinder.census.gov/home/saff/main.html?_lang=en exploring content, classroom instruction and assessment through hands-on activities and presentations by dynamic Kids corner: statisticians and educators. http://factfinder.census.gov/home/en/kids/kids.html During the workshop I learned about GAISE(Guidelines for Assessment and Instruction in Statistics Education): A State facts for students: Pre-K–12 Curriculum Framework. Participants in the GAISE http://www.census.gov/schools/facts/ project have created two reports of recommendations for introductory statistics courses (college level) and statistics Lesson #4 – there is a lot going on at the Census education in Pre-K-12 years with the ultimate goal being Bureau, even during “the other 9 years”! The visit ended statistical literacy. More information and materials to be with the presentation of US Census gift bags containing a used with students can be found at http://www.amstat.org/ treasure trove of materials any math teacher would love, education/gaise/index.cfm. Also explored was data in a state fact sliders and wheels, mugs and pencils emblazoned variety of contexts and representations, from Old Faithful with the Census logo, and a huge double-sided wall map of (complete with a live link to Yellowstone to view an eruption the US with demographical information displayed. online) to smokers vs. non-smokers (using a matrix to The next Joint Statistical Meeting for ASA will be held consider conditional and marginal probabilities). We even from July 31 - August 5, 2010, at the Vancouver Convention

2 Virginia Mathematics Teacher GENERAL INTEREST Teaching Time Savers: Reviewing Homework Jane Murphy Wilburne The classroom practice of assigning homework is a ne- lems, immediately went to the board when they entered cessity to reinforce the topic of the day’s lesson, review the class, indicated that they would solve one of the listed skills and practice them in a variety of problems, or chal- problems, and worked it out in detail. When they finished lenge students’ thinking and application of the skills. Effec- they signed their name to the problem. tive mathematics teachers know how to choose worthwhile By the time I entered the classroom, students were busy assignments that can significantly impact students’ learn- solving problems at the board while others were checking ing and understanding of the mathematics. The challenge, their homework at their seats. If there were any questions however, is how to manage and review the assignments about the problems, the student who solved the problem in a manner that will benefit students’ learning, and use at the board would explain his work to the class. If there classroom time effectively. was a problem which no one was able to solve, I would Over the years, I have tried various approaches to re- provide a few details about the problem and reassign it viewing and assessing students’ homework. Collecting and for the next class. In a short period of time, all homework grading every students’ homework can be very time con- was reviewed, and I recorded notes as to which students suming, especially when you have large classes and no posted solutions on the board. Rather than collecting every graduate assistants to help review students’ work. On the student’s homework, I noted the problems that gave most other hand, while it is important to provide students with students difficulty and would assign similar problems in a immediate feedback on their homework, it does not benefit future assignment. Students who listed the problems they them much to have the professor work out each problem in had difficulty with were not penalized. Instead, those who front of the class. solved the problems would receive a plus (+) in my grade I believe it is important for college students to take re- book. A series of five pluses (+) would earn them a bonus sponsibility for their learning. By promoting opportunities point on a future exam. for them to communicate with and learn from each other, My classroom quizzes would always include several we can help students come to rely less on the professor to homework problems to help keep students accountable for provide them with all the answers, and teach them to pose completing their assignments and motivate them to review questions that enhance each other’s understanding. problems they had difficulty with. Those who did typically One technique that has been effective in my classes is to received an A! assign homework problems that vary in concept application Time spent in class: approximately 5-12 minutes review- and level of difficulty. The students were instructed to solve ing the homework. Time saved: abut 30 minutes per class. 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 JANE M. WILBURNE is assistant professor of mathematics at of the problems they could not solve, on the front board in Penn State Harrisburg. a designated area. If the problem was already listed, they placed a check ( ) next to it. Once the class started, they Reprinted with permission from FOCUS The Newsletter of the Mathematical Association of America, copyright November 2006. were not allowed to record problem numbers at the board. All rights reserved. Other students, who were successful in solving these prob-

Statistical Outreach and the Census continued from page 2 Center in Vancouver, British Columbia, Canada. While this final lesson – even if my average speed traveling to the location isn’t as accessible for Virginia math educators as workshop destination was a lot slower than I expected, I Washington, DC was, the ASA website contains a variety am so glad I attended! Thank you, VCTM, for providing a of great information to enhance statistics and probability grant to offset my travel and lodging expenses. instruction in my 7th grade classroom and the classrooms of my colleagues at Ruffner Academy in Norfolk. Fifth and GAIL ENGLER, Ruffner Academy, Norfolk Public Schools

Virginia Mathematics Teacher 3 GENERAL INTEREST Enlivening School Mathematics Through The History of Mathematics Martin Bartelt and Stavroula K. Gailey This article describes an alternative History of Mathemat- in-classroom implementation of topics from the history of ics course and it demonstrates how such a course can be mathematics. beneficial for teachers and in turn for their students. Projects are intended to illustrate various ideas associ- According to the Curriculum and Evaluation Standards of ated with the history of mathematics. For example, the the National Council of Teachers of Mathematics (NCTM) a students make simple versions of a Roman abacus, design major goal of mathematics education is to produce students posters of mathematical symbols and/or terms explaining who value mathematics. This goal, of valuing mathematics, how they originated, construct Moebius strips, and create a requires learning about and understanding the origins of “sphere” from a collection of cylinders in order to estimate mathematics as well as appreciating the role mathematics the sphere’s volume. One teacher, after canvassing local plays in today’s society. stores, found that she could buy all the materials to make Another goal is to create a learning environment that a good, small version of an abacus, which she intended to fosters students’ confidence in doing mathematics. In addi- use in her classroom to aid in teaching place value, for less tion to the NCTM Standards, the Mathematical Association than a dollar. of America (MAA) in its 1992 Call for Change states that The teachers, after researching, write one-page biographi- mathematics teachers also need continuing experience in cal reports for four different mathematicians. However, these developing perspectives and in appreciating the historical too are not typical biographies. First, the reports must and cultural development of mathematics. include the biography of a woman, of an American, and of These NCTM and MAA goals have been incorporated someone from a non-Western culture. The reports may in the Master of Arts in Teaching Mathematics program also include mathematicians who have not been studied in at Christopher Newport University. One of the program’s class and particularly living mathematicians. Second, and courses, MATH 573: The History of Mathematics, both more important and difficult, the biographies must contain fosters mathematical confidence and contributes to an ap- information of interest to middle and/or high school students. preciation of mathematics. The course is a survey of the The following are some examples of interesting informa- origins, philosophy and development of mathematics from tion included in some of the biographies: Hero of Alexandria classical antiquity through the twentieth century. invented the first vending machine; Ada Byron Lovelace, to However, MATH 573 is different from the typical History of whom the poet Byron was married, was the first person to Mathematics course. In addition to problem solving, MATH describe the process of computer programming; and Grace 573 emphasizes how to incorporate both concepts and Hopper, the contemporary American mathematician, created content in the pre-college classroom, particularly in middle COBOL. Referring to personal traits and events enlivens school mathematics. The course is intended to enable the biographies. teachers to learn about the history of mathematics and also In addition to the projects and biographies, the teachers how to apply this knowledge in their classroom. choose two concepts/topics from the history of mathemat- After examining some well-known texts -such as that by ics and develop strategies and activities for incorporating C. Boyer- used in standard History of Mathematics courses, these topics in their classrooms. In turn, each teacher gives it is apparent that the objectives of these courses do not a fifteen minute presentation to the others in the class so emphasize how school teachers could use the material in ideas from these presentations can be shared and used by their classrooms. And although William Dunham’s Journey the rest of them in their own classrooms. Through Genius - The Great Theorems of Mathematics il- For example, one of the topics presented, which is ap- lustrates a lively approach to the history of mathematics, still propriate for use in a general mathematics class, was that of the book does not refer directly to teacher-use in the class- using and writing checks. It referred to the history of count- room. In this sense, the CNU MATH 573 course is atypical. ing by tally sticks and how the word “check” originated in Since the students in MATH 573 are either practicing England. Another interesting idea included the story of zero, teachers or interning graduate students, they continuously and how for hundreds of years people refused to believe ask themselves and the instructor about how they could in it. In another presentation a teacher explained how she implement what they are learning, in the History of Math- would have students do some important work particularly on ematics course, in their own classrooms. biographies in order to learn about the disadvantaged back- grounds of some mathematicians and the effect of sociology Overview Of Course Content and psychology on a mathematician’s career. The text books used in MATH 573 are Great Moments in Mathematics Before 1650 and Great Moments in Mathemat- The Middle School/High School Student ics After 1650 by Howard Eves. Beyond standard homework In addition to learning new material and the means by and exams, students in MATH 573 are required to complete which it can be presented to students there is another im- projects, biographical reports, and presentations on the portant and atypical facet of MATH 573 which relates to the

4 Virginia Mathematics Teacher “confidence” goal for students. There is a conscious effort BIBLIOGRAPHY throughout the course to empower the middle school teach- Dunham, William. Journey Through Genius, The Great ers to influence their students. Theorem of Mathematics. New York: John Wiley, 1990. For example, one goal of the biographies is to affect Boyer, Carl, and Uta Merzbach. A History of Mathematics. the mindset of the student. Knowing about the existence New York: John Wiley, 1989. of female mathematicians can change the perspective of Eves, Howard. Great Moments in Mathematics Before female students toward mathematics. Also, a physically 1650. Washington, D.C.: Mathematical Association of handicapped student benefits by knowing that there are America, 1983. physically handicapped mathematicians. Eves, Howard. Great Moments in Mathematics After 1650. Students tend to believe that mathematics is, and always Washington, D.C.: Mathematical Association of America, was, error-free, complete, contradiction-free, and completely 1983. logical. Since middle school and high school students usu- Edeen, Susan and John Edeen. Portraits for Classroom ally do not have these characteristics, they sometimes feel Bulletin Boards, Book 1. Palo Alto, California: Dale Sey- estranged from mathematics. Students will feel better when mour, 1988. they learn that great mathematicians made mistakes; that Edeen, Susan and John Edeen. Portraits for Classroom some mathematical questions can not be answered because Bulletin Boards, Book 2. Palo Alto, California: Dale Sey- they are undecidable; that whole societies had trouble with mour, 1988. the number zero; that there have been crises in mathemat- Leitzel, James (ed.). A Call for Change: Recommendations ics (e.g. the discovery of non-Euclidean geometry), and that for the Mathematical Preparation for the Teachers of controversy exists even now. Mathematics. Washington, D.C.: Mathematical Associa- tion of America, 1992. Conclusion National Council of Teachers of Mathematics. Historical This type of History of Mathematics course as part of a Topics for the Mathematics Classroom. Reston, VA: M.A.T. program can enable teachers to enliven their class- NCTM, 1989. room teaching. It provides a way to look at material, which Reiner, Luetta & W. Reiner. Mathematicians Are People, one may already have seen before, from a new viewpoint, Too. Palo Alto, California: Dale Seymour, 1990. to introduce and to give depth to new material, and to influ- ence the mindset of the student. 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’ Corner

Affiliate 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]

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]

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]

Virginia Mathematics Teacher 5 GENERAL INTEREST The Five Platonic Solids Theoni 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 hexahedron 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 fivepossible regular convex solids. These are the tetrahedron, the cube or hexahedron, the octahedron, the dodecahedron, and the icosahedron.

octahedron

tetrahedron hexahedron or cube

octahedron

icosahedron dodecahedron

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

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

dodecahedron

6 Virginia Mathematics Teacher GENERAL INTEREST VCTM Awards Two $2000 Scholarships to Future Math Teachers

This year, VCTM, through its Board of Directors, gave she has had great success working with teachers and their authorization to the Scholarship Committee to award a students. Before attending Virginia Tech, she graduated $2000 scholarship to up to two candidates that were qual- from Monticello High School where she received the Su- ity, prospective mathematics teachers. To receive this san Gilkey Award, an award that is given to a student ath- award, candidates must be Virginia residents that are full- lete with the highest grade point average. She was also a time students attending a Virginia college or university with Wendy’s High School Heisman nominee. Johnathon Up- a major in mathematics and plan to teach mathematics in perman is currently completing a four-year Bachelor’s of a Virginia school. This year’s selection process was based Science degree at the College of William and Mary. After on the students’ academic achievements (transcripts), completing this degree, he will continue his studies at Wil- faculty recommendations, and personal narratives (which liam and Mary to pursue a Master’s in Education. He has were required to include a field experience, class experi- a stellar academic record and plans to draw on his suc- ence, volunteer experience, or life experience that has in- cesses to help inspire others to learn and appreciate math- fluenced their decisions to be a teacher of mathematics). ematics. He, too, will begin his teaching career soon after Each candidate will also receive a complimentary, one- he completes his Master’s in Education. Prior to attending year student membership in VCTM. This year’s scholar- William and Mary, Johnathon graduated from Indian River ship winners are India (Brooke) Haun, a student at Virginia High School. Tech and Johnathon Upperman, a student at The College VCTM members congratulate both of these scholarship of William and Mary. winners and wish them success in all of their teaching ex- The scholarship awards will be officially announced at periences to come. We also welcome India Brooke Haun this year’s VCTM Annual Conference banquet, March 12, and Johnathon Upperman to our profession and trust that 2011 at the Ramada Plaza Richmond West Conference they will share in the very important task of supporting Center. mathematics and its teaching in Virginia. Both India (Brooke) Haun and Johnathon Upperman As a current VCTM member, or one who is currently are studying to become middle and secondary teachers thinking about being a member as your read this article, we of mathematics. They bring to their teaching aspirations invite you to make a contribution to the Scholarship Fund many achievements in their teacher education and math- the next time you are scheduled to make your annual dues. ematics programs. India (Brooke) Haun is currently study- You may also send a check to the editor made out to the ing in a five-year program at Virginia Tech. She plans to VCTM Scholarship Fund. Your contribution will help VCTM complete her Master’s in Education Spring 2012. She has to continue to support prospective teachers of mathemat- already begun to work in high school classrooms where ics.

Virginia Mathematics Teacher 7

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]

8 Virginia Mathematics Teacher

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:

______

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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]

Virginia Mathematics Teacher 9 GENERAL INTEREST Problem Corner Ray Spaulding

10 Virginia Mathematics Teacher Virginia Mathematics Teacher 11 12 Virginia Mathematics Teacher Virginia Mathematics Teacher 13 14 Virginia Mathematics Teacher Virginia Mathematics Teacher 15 16 Virginia Mathematics Teacher Virginia Mathematics Teacher 17 GRADES K-5 Clearing up the Confusion over Calculator Use in Grades K-5 Barbara J. Reys and Fran Arbaugh Since the publication of NCTM’s Principles and Stan- time and students’ motivation and interest in mathematics? dards for School Mathematics in April 2000, considerable If students spend less time reaching high levels of perfor- discussion has taken place about “key messages” of the mance in hand calculation, how will the resulting “extra” document. The breadth of the content of Principles and time be used? Do we value instruction that develops stu- Standards may hamper attempts to identify messages dents’ ability to think, reason, and solve problems? Can we about particular topics. In addition, many of the fundamen- meet both goals simultaneously—that is, develop proficien- tal messages are not easily distilled into short phrases. In cy with hand calculation and the abilities to think critically, fact, when such messages are too succinctly articulated, reason, and solve problems? the danger of oversimplification and misunderstanding Another question relates to the perceived consequences arises. This misapprehension can be seen in a question of learning to compute by hand or with a calculator. For that often emerges in discussions about elementary school example, do students derive cognitive benefits from learn- mathematics and Principles and Standards. That is, what ing conventional paper-and pencil algorithms for comput- does Principles and Standards say about calculator use in ing? Does an overemphasis and reliance on conventional elementary school? computation algorithms encourage students not to think or use their powers of reasoning? Does the use of calculators Why the Interest in NCTM’s Position? for computation promote students’ understanding of math- Teachers, teacher educators, mathematicians, school ematical situations and reasoning about solutions? For a administrators, and parents are genuinely interested in more thorough discussion of issues that surround calcula- NCTM’s “official” position on calculator use in the elemen- tor use in the elementary grades, see Ralston, Reys, and tary grades. Why is everyone so interested? What issues Reys (1996) and Reys and Nohda (1994). surround calculator use in elementary school? Before we Certainly, these questions will persist until we address delve into the messages of Principles and Standards on them. In fact, we know that the following situations prevail: this topic, and to support our understanding of those mes- sages, we present a short discussion of some of the issues • Calculators are readily available at home to children of that may be prompting the current interest in what Prin- elementary school age, and their low cost has prompted ciples and Standards says about calculator use in elemen- increased access in school. tary school. • Calculators are commonly used in the workplace to per- Why the Interest, and What’s the Confusion? form simple and complex computations. NCTM has published several official position statements on calculators, the first in 1978 and the most recent in 1998 • Teachers are unsure how to use calculators to promote (see nctm.org/about/ position.htm). Public interest in cal- thinking and reasoning and whether calculators should culator use in schools has grown steadily over the past be used as computational devices. twenty-five years, largely because of the increased avail- ability of inexpensive calculators. Through the first seven- • Children’s own beliefs about mathematics lead some of ty-five years of the twentieth century, elementary school them to view using calculators as “cheating,” or not re- mathematics emphasized paper-and-pencil computation ally doing mathematics. techniques out of necessity. In fact, some estimates of the amount of instructional time devoted in the elementary Clearing up the Confusion: Messages in Principles and grades to developing hand-calculation proficiency run as Standards high as 90 percent (Reys 1994). This emphasis was impor- Principles and Standards articulates a goal for elemen- tant because hand calculation was the most efficient way tary school mathematics that includes computational pro- to compute apart from cash registers, adding machines, ficiency but extends well beyond that skill to the abilities and expensive computers. Today, a $4 calculator can do to think and draw on a range of techniques and strategies in seconds the computations that in the past, students to solve problems. Computation is important precisely be- needed years of instruction and practice to learn and that, cause it is necessary to solve many mathematical prob- once learned, required significant amounts of time to ac- lems. The particular method used, however, whether it in- complish. volves mental math, paper and pencil, or a calculator, is The $4 calculator, a highly efficient and accurate com- just one part of the computation process. Students must putational tool, raises a whole set of questions that educa- also know what kind of computation to perform and be tors and parents are struggling to answer. For example, able to identify the appropriate numbers to use in compu- what is the importance of having students learn methods tations. Real mathematics is knowing a variety of strate- for computing that their parents and grandparents learned? gies for solving problems and having the ability to apply If we value proficiency with hand-computation techniques, them appropriately. If data from the National Assessment do we know and accept the “cost” in terms of instructional of Educational Progress (Silver and Kenney 2000) are any

18 Virginia Mathematics Teacher indication, elementary school mathematics programs have multiples of 5.” In this example, the counting capability of had some success in helping students perform calculations the calculator allows students to focus on patterns that re- but have been largely unsuccessful in developing students’ sult from adding the same number repeatedly. This type of problem-solving abilities. exploration lays the groundwork for studying multiples and The writers of Principles and Standards recognized the divisibility, important ideas in the upper elementary school importance of articulating a clear and research-based mes- grades. A more thorough discussion of this example can be sage about the role of calculators in elementary school found at standards.nctm.org/document/eexamples/chap4 mathematics. Readers will find these messages through- /4.5/index.htm. out the document. For example, in chapter 2, the section Shuard (1992) writes about a classroom episode in on the Technology Principle includes a discussion of cal- which elementary school students “discovered” negative culators as tools for learning. Readers should also look numbers as they were investigating subtraction with a cal- at chapters 3, 4, and 5 under the Number and Operation culator. One student entered 6 – 8 and was curious about Standard for discussions of the role of calculators in devel- the displayed result, which was –2. The teacher used the oping computational fluency. opportunity to model negative numbers by extending the The rest of this article summarizes some of the important number line to the left of 0. In another activity, the children messages from Principles and Standards about calculators started at 50 and successively subtracted the amount rolled in the elementary school mathematics classroom. We draw on a number cube. One student, Jenny, continued until she heavily from the text of the document and illustrate some of had 3 left. On her next roll, she got 5. She said, “I can’t take the points with examples. 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 Calculators are important tools for learning and doing problems, making a list of those that had an answer of –1. mathematics Her list included 1 – 2, 2 – 3, and 3 – 4. When asked what number could be subtracted from 100 to give –1, Jenny Electronic technologies—calculators and comput- said, “Easy! 101” (Shuard 1992, p. 40). ers—are essential tools for teaching, learning, and doing mathematics. (NCTM 2000, p. 24) Technology should not be used as a replacement for basic understandings and intuitions; rather, it can and Understanding number, properties of operations, and re- should be used to foster those understandings and lationships among numbers is central to elementary school intuitions. (NCTM 2000, p. 25) mathematics. The calculator is a tool for exploring num- ber concepts and for generating data that can be studied In the upper elementary school grades, students can for patterns. For example, students can use a calculator use the calculator to explore the relationships among vari- to skip-count by 5s (press 0, + 5, =, and so on) and color ous representations of rational numbers. “For example, the corresponding spaces on a hundred board (see fig. 1). they should count by tenths (one-tenth, two-tenths, three- Students can then try the same process with other num- tenths, . . .) verbally or use a calculator to link and relate bers and respond to teacher prompts, such as “What pat- whole numbers with decimal numbers. As students contin- terns emerge?” and “Predict additional numbers that are 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)

Virginia Mathematics Teacher 19 In addition to its use as a way to explore mathematics, cal relationships. When teachers want to help students the calculator is a highly efficient and accurate tool for com- develop strategies that they can use to compute mentally, puting in problem-solving contexts. With access to a calcu- those strategies alone should be the focus. When students lator, students can use real data and large data sets. The are learning ways to record computational strategies, then calculator’s efficient and accurate ability to compute frees recording should be the focus. Elementary school teachers students to think and make decisions. When a cashier will certainly want students to put away their calculators at uses a computer to tabulate our bill or a bank clerk uses times to focus on developing other techniques and strate- an adding machine to total receipts, we do not think that gies for computing. At other times, when students are en- they are “cheating.” We need to help students understand gaged in solving problems, formulating and applying strate- that mathematics is more than computation—that using a gies, and reflecting on results, a calculator is an important calculator as a tool for solving problems is not cheating. For enabling tool. Teachers and parents must help students un- example, students can use a calculator as a computing tool derstand that “real” mathematics is about thinking, applying to help them answer the question “How much time would strategies, reasoning, and relating ideas. Computation is a you need to count to a million or a billion?” In upper el- necessary tool in the process, but it is only one part of the ementary school, students can study the effect of extreme whole process that makes up mathematics. values when computing a mean. In all grades, they can use real data to solve problems of interest to them, from tabu- Good use of calculators requires teacher decision lating the costs of food items in the cafeteria to gathering making and guidance and summarizing data on the number of pencils in all the desks in the classroom or the number of buttons on all their In the mathematics classroom envisioned in Prin- clothes. Students can use calculators to help them accu- ciples and Standards, every student has access to rately and efficiently solve problems to focus their attention technology to facilitate his or her mathematics learn- on what calculations to perform, not just how to perform ing under the guidance of a skillful teacher. (NCTM those calculations. 2000, p. 25)

Calculator use is not an all-or nothing decision The effective use of technology in the mathematics classroom depends on the teacher. (NCTM 2000, p. Part of being able to compute fluently means making 25) smart choices about which tools to use and when. (NCTM 2000, p. 36) Principles and Standards emphasizes the role of teachers in helping students become responsible technology us- Through their experiences and with the teacher’s ers. Teachers should model and explain their own ways guidance, students should recognize when using a of thinking about numbers and operations and encourage calculator is appropriate and when it is more efficient students to share their methods. The classroom envisioned to compute mentally. (NCTM 2000, p. 77) by NCTM is one in which students and teachers use a vari- ety of tools, including counters, rulers, graph paper, scales, Students at this age [grades 3–5] should begin to geometric shapes and solids, text books, instructional soft- develop good decision-making habits about when it ware, and calculators. Just as teachers guide and model is useful and appropriate to use other computational the use of other tools, so must they help students under- methods, rather than reach for the calculator. (NCTM stand the power and limits of a $4 calculator. The calcula- 2000, p. 145) tor cannot think, and it cannot make decisions about what numbers or operations need to be used. The quality of the Calculators should be available at appropriate times output of a calculator is wholly dependent on the input. as computational tools, particularly when many or Teachers must examine the instructional goals for a cumbersome computations are needed to solve given unit or lesson to decide whether and how various problems. However, when teachers are working with tools, including calculators, can help students learn. In gen- students on developing computational algorithms, eral, teachers should model and encourage calculator use the calculator should be set aside to allow this focus. when— (NCTM 2000, pp. 32–33) • the focus of instruction is problem solving; Throughout the document, Principles and Standards • the availability of an efficient and accurate computation- stresses that technology and, hence, calculator use is not al tool is important; an all-or-nothing decision. Supporting students’ under- • the lesson involves a search for, and an exploration of, standing of mathematical concepts helps them make good decisions about appropriate times to use a calculator. At patterns; times in their study of mathematics, students will find that • anxiety about computation might hinder problem solv- other ways of computing are more appropriate than using ing; and calculators. For example, engaging students in mental- • student motivation and confidence can be enhanced math activities supports their understanding of mathemati- through calculator use.

20 Virginia Mathematics Teacher Summary ———. Principles and Standards for School Mathematics. As adults, most of us would not hesitate to pick up a cal- Reston, Va.: National Council of Teachers of Mathemat- culator when we balance our checkbooks or do our taxes. ics, 2000. standards.nctm.org. Engineers, architects, building contractors, accountants, Ralston, Anthony, Barbara J. Reys, and Robert E. Reys. store clerks, and scientists readily use computing tools ev- “Calculators and the Changing Role of Computation in ery day. Withholding opportunities for students to learn to Elementary School Mathematics.” Hiroshima Journal of use computing tools effectively and efficiently puts them at Mathematics Education 4 (1996): 63–71. a disadvantage in today’s technological society. Reys, Robert. “Computation and the Need for Change.” In Principles and Standards advocates computational flu- Computational Alternatives for the Twenty-First Century, ency as an expectation for all students. It encourages edited by Robert E. Reys and Nobuhiko Nohda. Reston, thoughtful use of calculators in elementary school class- Va.: National Council of Teachers of Mathematics, 1994. rooms. As a society, we have always welcomed techno- Reys, Robert E., and Nobuhiko Nohda, eds. Computational logical advances that make our lives easier and our work Alternatives for the Twenty-First Century. Reston, Va.: more efficient and productive. We use word processors to National Council of Teachers of Mathematics, 1994. write letters and prepare legal documents. We use spread- Shuard, Hilary. “CAN: Calculator Use in the Primary Grades sheets to keep track of personal finances. These tools, like in England and Wales.” In Calculators in Mathemat- the $4 calculator, help us do our work more efficiently and ics Education, 1992 Yearbook of the National Council use our results to answer questions and influence decision of Teachers of Mathematics (NCTM), edited by James making. Calculators serve as efficient and accurate com- T. Fey and Christian R. Hirsch, pp. 33–45. Reston, Va.: putational tools for both students and adults. Principles and NCTM, 1992. Standards asserts, “Today, the calculator is a commonly Silver, Edward A., and Patricia Ann Kenney, eds. Results used computational tool outside the classroom, and the en- for the Seventh Mathematics Assessment of the Nation- vironment inside the classroom should reflect this reality” al Assessment of Educational Progress. Reston, Va.: (NCTM 2000, p. 33). National Council of Teachers of Mathematics, 2000.

Bibliography Coburn, Terrence G. “The Role of Computation in the Edited by JEANE JOYNER, [email protected], Depart- Changing Mathematics Curriculum.” In New Directions ment of Public Instruction, Raleigh, NC 27601, and BARBARA for Elementary School Mathematics, edited by Paul R. REYS, [email protected], University of Missouri, Columbia, Trafton and Albert P. Shulte, pp. 43–56. Reston, Va.: Na- MO 65211. This department is designed to give teachers informa- tion and ideas for using the NCTM’s Principles and Standards for tional Council of Teachers of Mathematics, 1989. School Mathematics (2000). Readers are encouraged to share Dick, Thomas. “The Continuing Calculator Controversy.” their experiences related to Principles and Standards with Teach- Arithmetic Teacher 35 (April 1988): 37–41. ing Children Mathematics. Please send manuscripts to “Princi- Lindquist, Mary M. “It’s Time to Change.” In New Directions ples and Standards,” TCM, 1906 Association Drive, Reston, VA for Elementary School Mathematics, edited by Paul R. 20191-9988 Trafton and Albert P. Shulte, pp. 1–13. Reston, Va.: Na- tional Council of Teachers of Mathematics, 1989. Reprinted with permission from Teaching Children Mathematics, National Council of Teachers of Mathematics (NCTM). “NCTM copyright October 2001, by the National Council of Teachers of Position Statements.” nctm.org/about/position .htm. Mathematics. All rights reserved.

Virginia Mathematics Teacher 21 GRADES 2-6 Teaching Addition and Subtraction Facts: A Chinese Perspective 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 for example, 4 is one more than 3 and 9 is one more than children’s conception of number, found that American 8. As a result, when two-digit addition and subtraction are mothers rarely interpreted numbers in the teens as introduced, American children rely heavily on counting- composites of one 10 and some 1’s when they interacted based and collection-based concepts; for instance, 13 is with their children. Instead, they usu­ally initiated and treated as a collection of thirteen objects. guided learning activities in which children completed tasks Chinese teachers use a three-step method to teach involving numbers in the teens by counting by ones orally addition and subtraction. Children first develop an or with manipu­latives. This practice reinforces the view understanding of number concepts, the meanings of that 12, for example, represents only a collection of twelve addition and subtraction, and the relationships between items. As a result, Chinese children are more inclined addition and subtraction. Next, children mas­ter addition than children in the United States to use tens and ones and subtraction facts in three substeps. First, they learn to represent numbers and, subsequently, to use 10 as a sums and related subtraction facts to 10, then they learn bridge when performing addition and subtraction. facts between 11 and 20, and finally, they learn facts between 20 and 100. In the third overall step, students are Differences in Teaching introduced to the addi­tion and subtraction algorithms. Each American teachers often use counting in a one-to-­one step is the foun­dation for the next step. Making sure that correspondence to introduce addition and sub­traction of children suc­cessfully complete one level before moving to whole numbers. This strategy is based on the “one more the next is important to the teachers. If children acquire than” relationship between consecu­tive whole numbers; a solid foundation at each of these three steps, they can

22 Virginia Mathematics Teacher easily extend the process to even larger numbers. as families, such as sums of 12, sums of 13, and so forth. When sums up to 10 are first introduced in Chi­nese The textbooks introduce a variety of strategies, such as elementary schools, counting skills are empha­sized to help counting up, learning doubles, or recognizing double-plus- children understand the relationships among these sums. one and double-­minus-one situations (see, e.g., Addison- When sums between 11 and 20 and related subtraction facts Wesley Mathematics [Menlo Park, California], Houghton are introduced, rather than rely on counting, children are Mifflin Math Central [Dallas, Texas], Harcourt Brace Math usually encour­aged to create collections of tens and ones Advantage [Orlando, Florida]). To some extent, basic facts to repre­sent the number; this approach is consistent with are viewed as associations to be memorized through hands- the linguistic structure of the Chinese counting sequence. on activities, then recalled on demand. Chinese textbooks For example, to teach 8 + 3, Chinese children are often arrange the basic facts using fact tables (Curriculum and asked to take two objects from a collection of three and put Teaching Materials Research Institute 1999), and the them together with eight to make a 10; thus, they see that primary strat­egy taught is “make 10.” Chinese teachers the whole becomes a collection of ten and one, or eleven. gen­erally introduce the basic facts in units, such as the 6+ The “make ten” thinking strat­egy is demonstrated in the unit, the 7+ unit, and so on. These units are categorized by following examples: the known entity (addends) instead of the unknown entity (sums). This dif­ference between American and Chinese a) 9 + 4 = ? Think: • 9 + ? = 10. teaching can be seen in table 2. • 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; The fact families from 2 to 18 contain 153 additional facts 10 + 5 = 15. that American students need to study; the fact table, in con- trast, contains only 81 facts. When students understand The Chinese numerical language shown in table 1 plays an the commu­tative property of whole numbers, the number of essential role in this strategy. Moreover, Chinese teachers addition facts that they need to know is reduced to only 45 believe that students should use 10 as a bridge because of (see table 3). Although Chinese and American textbooks its importance in the base-ten numeration system. arrange addition facts differently, they both use relation- Chinese teachers strongly emphasize using addition ships to mini­mize the amount of information that must be facts to do subtraction. By doing so, they not only encourage memorized. students to apply their previously­ learned knowledge in the When Chinese children learn the basic facts, their task new situation but also help students see how addition and involves not only memorizing but also using logical thinking subtraction are related. Consider the following examples: and reasoning based on relationships among the numbers. Encouraging children to examine a visual aid similar to a) 13 – 5 = ? Think: • 5 + ? = 13. table 3 and to look for patterns and relationships can help • 5 + 8 = 13. them devise thinking strategies that can aid in mastering • Therefore, the basic facts (see, e.g., Baroody [1998]). 13 – 5 = 8. Chinese teachers also teach different strategies that are not introduced in the textbooks but that can help children b) 15 – 8 = ? Think: • 8 + ? = 15. see the patterns among the addi­tion facts. Consider the fol- • 8 + 7 = 15. lowing examples, in which n is a whole number: • Therefore, 15 – 8 = 7. • For n + 1, the sum is the next whole number, that is, the number after n in the counting sequence (Baroody Differences in Thinking Strategies 1998). Thinking strategies are emphasized in both America and • For n + 2, the sum is the next odd or even whole number. China in teaching the basic facts (see, e.g., Baroody [1998]), • The sum of n + 9 can be found by adding 10 to n, then but the way in which these facts are presented is quite subtracting 1. This strategy is a shortcut for the make-10 different. Many American textbooks arrange the basic facts approach discussed previously.

Virginia Mathematics Teacher 23 Because they rely on such thinking strategies, Chi­nese nese teachers introduce these strategies as early as first children rarely use manipulatives to figure out facts. grade (Curriculum and Teaching Materials Research Insti- Two other strategies for subtraction are often seen in tute 1999). Students may not be expected to master these Chinese classrooms. One is to use 10 as the bridge number strategies in a short time, but if the foundation is laid early, in a subtraction equation. Consider these examples: students can apply their knowledge of the basic facts and these strategies to other mathematical content that they a) 14 – 9 = ? Think: • 10 = 9 + (1). will study later. • 14 – 10 = 4. • 4 + 1 = 5 (because you References subtract one more, you Baroody, Arthur J. Fostering Children’s Mathematical Pow- need to add one back). ers: An Investigative Approach to K–8 Mathematics • Therefore, Instruction. Mahwah, N.J.: Lawrence Erlbaum Associ- 14 – 9 = 5. ates, 1998. Cao, Feiyu. “Development of Pre-School Children’s Opera­ b) 15 – 8 = ? Think: • 10 = 8 + (2). tional Ability.” In Reform of Elementary Mathematics Edu­ • 15 – 10 = 5. cation, 197205. China: People’s Education Press, 1994. • 5 + 2 = 7 (because you Curriculum and Teaching Materials Research Institute. subtract two more, you Nine-Year Compulsory Education Elementary Math- need to add two back). ematics Series. China: People’s Education Press, 1999. • Therefore, Miura, Iren T., Yukari Okamoto, Chungsoon C. Kim, 15 – 8 = 7. Chih-Mei Chang, Marcia Steere, and Michel Fayol. “Comparisons of Children’s Cognitive Representation The other strategy also uses 10 as a bridge, but it requires of Number: China, France, Japan, Korea, Sweden, and students to recall simple addition facts. The following are the United States.” International Journal of Behavioral examples: Development 17 (Sep­tember 1994): 401–11. National Council of Teachers of Mathematics (NCTM). a) 13 – 4 = ? Think: • 13 = 10 + 3. Princi­ples and Standards for School Mathematics. • 10 – 4 = 6. Reston, Va.: NCTM, 2000. • 6 + 3 = 9. Yang, Ma Tzu-Lin, and Paul Cobb. “A Cross-Cultural Investi­ • Therefore, gation into the Development of Place-Value Concepts of 13 – 4 = 9. Children in Taiwan and the United States in Educational Studies.” Educational Studies in Mathematics 28 b) 16 – 9 = ? Think: • 16 = 10 + 6. (January 1995): 1–33. • 10 – 9 = 1. • 1 + 6 = 7. WEI SUN, [email protected], teaches at Towson University, • Therefore, Towson, MD 21252. He is interested in teacher education, gifted 16 – 9 = 7. students, curriculum development, and comparative studies. JOANNE ZHANG, [email protected],net, teaches at Summary Hollywood Elementary School, Hollywood, MD 20636. She has When using thinking strategies to perform addition and a special interest in effective instruction, including mathematics subtraction, students reinforce their under­standing about teaching strategies, cross-cultural studies, and learning disabilities. The authors would like to thank Professor Arthur BAROODY for his the facts that they have learned by using those facts re- help in revising the manuscript. peatedly. By the time they fin­ish learning single-digit ad- dition and related sub­traction, they can easily recall the Reprinted with permission from Teaching Children Mathematics, addition and sub­traction facts and are more than ready to copyright September 2001, by the National Council of Teachers learn the formal algorithms of addition and subtraction. Chi­ of Mathematics. All rights reserved.

24 Virginia Mathematics Teacher GRADES 3-6 Dividing Fractions: Reconciling Self-Generated Solutions with 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- Division with Fractions in the Context of Linear tions. The core concepts of the course included place-value Measurement numeration, whole numbers and operations, fractions and to introduce my students to problem solving involving opera­tions, and foundations of number theory. My instruc­ the division of fractions, I posed the following task. My ex- tional approach focused on students’ inquiry, empha­sizing pectation was that they would solve it using self-generated their own interpretations and their explanations and justifi- methods. cations of their answers. To support stu­dents’ inquiry, the instructional tasks were open-ended and often presented In Ms. Smith’s sewing class, students are making pil- within a problem-solving context. Students worked collab- lowcases for the open house exhibit. Ms. Smith bought oratively in small groups or pairs, then presented their so- 10 yards of fabric for her class project. Each pillowcase lutions and answers to the whole group. As the teacher, I requires 3/4 yard of fabric. How many pillowcases can acted as a facilitator and guide for the students’ self-gener- be cut from the fabric? ated solutions and their exchange of ideas and negotiation of meaning.­ The episodes discussed here were selected Students joined their partners to find solutions to this from a sequence of lessons on the division of fractions. problem. Soon thereafter, some students sug­gested that this was a division problem and that if they used the invert- Students’ Beliefs and Expectations and-multiply rule, they would get the right answer. Howev- My students brought to the classroom beliefs that were er, in light of the expec­tation that they should explain their consistent with their past experi­ences (Frank 1990) where answers, the stu­dents could not just use invert and mul- learning mathematics had been characterized by the quick tiply without explaining and justifying how it works. Since production of “answers.” Consequently, they thought that none of the students knew the basis of the algorithm, they it was my role as the teacher to pass along proce­dures sought their own solution methods to find the an­swer, then and that their job was to apply the neces­sary algorithms or many used the algorithm to check the an­swer. In doing so, rules. They also believed that there was only one way to they encountered a discrepancy between the standard-al- solve a problem. Thus, what my students had learned re- gorithm answer and the one they derived using their own garding expecta­tions for accepted evidence of knowledge methods. In their at­tempt to reconcile the answers, sev- or un­derstanding contradicted my emphasis on stu­dent- eral students had to come to terms with their lack of un- generated solutions. Furthermore, since they were now derstanding of the result they obtained using the standard required to think on their own, it became apparent that they algo­rithm. Here are some of the students’ solutions. placed little or no trust in their own ability to solve prob- Christine: First I laid out 10 pieces of 1-yard mater­ial. lems and at first resisted my instructional approach. The Then I took out 3/4 from one piece leaving 1/4 of a piece students’ limited understanding of fractions further aggra­ of fabric from each yard piece [see fig. 1]. Then I added vated their lack of confidence. up all 1/4 pieces to see how many groups of 3/4 I could

Virginia Mathematics Teacher 25 make. The final answer is 13 pillowcases with 1/4 piece From the last yard, I can take 3/4 and make another pil- leftover [pauses] or what I thought was 13 1/4. When I went lowcase and have 1/4 yard of fabric left [and continued to check it doing the invert-multi­ply method of old days, the recording]. answer was 13 1/3 [seem­ingly perplexed]. I can’t under- stand why. 6 yd. 8 pillowcases Next I called on David. He drew a large rectangle with 10 9 yd. 12 pillowcases equal sections to represent the 10 yards and further subdi- 9 3/4 yd. 13 pillowcases vided the first rectangle into 4 equal rectangles (see fig. 2). 1/4 yard leftover David: I know we have 4 fourths in each yard; 10 x 4 = 40. In 10 yards, we have 40 fourths. Each pil­lowcase needs I can make 13 pillowcases and there is 1/4 of a yard 3/4. Thirteen times 3 is 39, so I can make 13 groups of 3 leftover. The answer is 13 pillowcases and the problem [fourths]. I have one 1/4 of a yard leftover. The answer is is solved. 13 1/4. But that’s not right. If you do 10 x 4/3 you get 13 1/3. How come? Betsy’s solution involved proportional thinking as reflect- Both Christine and David doubted their self-gener­ated ed in her double counting of yardage and number of pil- solutions because they trusted the algorithm. Their focus lowcases, which she recorded in two side-by-side columns. on the right answer was overshadowing their activity, and Betsy exhibited a greater abil­ity to unitize as she was able they could not recognize that the numbers 13 and 1/4 re- to take 3/4 as her counting unit for the length of fabric. ferred to units of a different nature. Hence, they merely jux- Several students voiced their discomfort with Betsy’s an- taposed both units with no consideration of the fact that 13 swer. Ann spoke almost in protest. indicated the number of cuts of size 3/4 yard (or pillowcase Ann: You got only 13? The answer is 13 1/3 be­cause the lengths) in 10 yards and that the 1/4 indicated the 1/4-yard formula is right! length of leftover material. This explana­tion was further evi- Betsy: The question is “How many pillowcases of 3/4 of dence that the students did not know the meaning of the a yard can you make?” and you can make 13 pillowcases. numbers in the answer ob­tained with the algorithm. The You can’t make another pillowcase with just 1/4 of a yard 13 1/3 in the algo­rithm means 13 1/3 pillowcases, or 13 of fabric. The answer is 13, the problem has been solved! whole lengths of size 3/4 yard and 1/3 of another (or 1/3 A couple of students nodded in agreement, while others the length of one pillowcase). Some students protested insisted on the answer of 13 1/3. At this point, I reminded and said it was my responsibility as their teacher to explain the class that there were three an­swers to think about now: the apparent disparity. Next, Betsy raised her hand to vol- 13, 13 1/3, and 13 1/4. Several students in unison declared unteer her solution. “the invert and multiply is the right one.” Betsy: What I did was to draw 10 squares side by side. Betsy: I know how to cut fabric. The problem has been Then I divided them into four pieces each [see fig. 3] and solved. The question was “How many pillow­cases can you did the counting like this: 3/4 for one pil­lowcase, another make?” Why are we arguing about the piece leftover? You 3/4 for another pillowcase, 3/4 for another pillowcase, either have enough to make the pillowcase using 3/4 of a that’s 9/4. [As she talked, Betsy recorded her numbers yard or you don’t. Only in school you have to give answers in two columns while mark­ing the picture accordingly, as with mixed frac­tions. It doesn’t always make sense in real shown in fig. 3.] life. Silence followed Betsy’s comments. Then Ann volun- 3/4 yd. 1 pillowcase teered. 6/4 yd. 2 pillowcases Ann: Betsy has a good point, but I still would like to know 9/4 yd. 3 pillowcases why one gets two different answers; 13 1/4 seems right; I 12/4 = 3 yd. 4 pillowcases have counted several times and I get the same thing, 13 1/4. When I saw that 12/4 make 4 pillowcases and I’d used Betsy’s and Ann’s arguments raise two important and up 3 yards, I figured that with 6 yards, I can make 8 related pedagogical issues. On the one hand, presenting pillowcases; with 9 yards I can make 12 pillowcases. problems in context helps the learner seek solutions that

26 Virginia Mathematics Teacher make sense, given the condi­tions of the task. Betsy’s solu- 1/3 were fractions of different units of refer­ence. Paula and tion illustrates this ability, as her answer makes the most other students in the class did not realize that 1/4 yard is sense in the given context. On the other hand, we want 1/3 of 3/4 yard. They could not coordinate the different students to move beyond context and be able to generalize units involved in the task. At this point, rather than have and work with numbers efficiently. The latter con­siderations Christine demonstrate it, I asked students to get involved make Ann’s point a legitimate one, too. Indeed, if we were in the process of measuring. To each small group I handed to report the measurement of 10 yards of material using a a 10-yard-long unmarked white paper tape and a 3/4-yard- measuring stick 3/4-yard long, the answer would be 13 1/3 long unmarked colored paper tape. The task was to mea- measuring-stick lengths. This occurs because the piece sure and keep track of the process so they could explain 1/4-yard long would figure in the measurement as the frac­ the result. After the small-group work, students reported to tion 1/3 of the 3/4-yard-long measuring stick (see fig. 4). the whole class. To facilitate their demonstrations, I taped However, as a teacher, I wanted the stu­dents to resolve one of the 10-yard strips on the board, and students came their cognitive impasse. to the board to show how they conducted their measure- Teacher: OK. I would like you to think about a few things. ment. The following exchanges took place. First, what the problem is asking you. Second, think about Paula: We placed the 3/4 piece on top of the 10­yard what 13 and 1/4 stand for in your solution. Why don’t you piece one time after another. We marked the point where do the measuring in our next meeting? each 3/4 piece ended and so forth. We counted 13 times The next class, before we began measuring, Christine and got a piece leftover. opened the discussion and volunteered her thinking. She Teacher: How would you report the result of your mea- had recorded her solution in her class notebook and re- suring? ferred to it as she talked to the class. Paula: Ann and I were talking about it and we are not Christine: It took me a while to understand that we were sure. There is a 1/4 yard of fabric leftover but we still don’t not using a yardstick as a measuring tool. We were looking know about the 1/3. at 3/4 of a yard to see how many pillowcases of that length Betsy: We did the same thing, but we folded the short would come out of 10 yards. When I looked at 1/4 that way, piece [of the measuring stick] where we ran out of fabric it would re­ally be 1/3 of the length of a pillowcase. I figure and it’s 1/3 of the measuring stick [see fig. 5]. that 1/4 is 1/3 in relation to 3/4. Kathy: We marked how many times the leftover went Several students were puzzled by Christine’s ex­planation; over the measuring stick. It was three times, so that’s Paula’s reaction was representative of their thoughts. where the 1/3 comes from! Paula: I can’t understand how 1/4 can be 1/3. What is Other students also showed their understanding that 1/4 she saying? yard of fabric is 1/3 of a 3/4-yard piece (or 1/3 the length of Apparently, Paula could not follow Christine’s ex­ the pillowcase), but some students still could not see that planation. She still failed to recognize that the 1/4 and the relationship. However, given their past experiences with

Virginia Mathematics Teacher 27 learning fractions and the fact that this was the second les- lim­ited understanding of my college students reflects the son on division of fractions, the students’ progress in their complexity of the concepts of fraction. What these experi- under­standing of the meaning of the division by a fraction ences related here ultimately teach us is that unless we was remarkable. place more emphasis on stu­dents’ understanding of num- bers and operations (NCTM 2000), we may be severely Discussion limiting our students’ chances to learn mathematics with It is worth noting that the students, at first, did not in- un­derstanding. Furthermore, teaching for mastery of algo- terpret the pillowcase problem as being division. David rithms will tend to perpetuate the students’ lack of confi- was the first to suggest it, then this interpretation became dence in their own ability to reason mathematically. widely accepted. Since these events occurred in the last It became apparent that my students’ experi­ences had quarter of the semester, my students knew that just find- led them to believe that getting an­swers was more impor- ing a numerical answer was not acceptable. Still, some did tant than the thinking in­volved in the solution. At first, they not trust themselves to find their own solution methods and greatly resisted my approach to instruction and did not want would have just used the standard al­gorithm. As I moved to find their own solutions. I insisted on the importance of around small groups, we rene­gotiated the expectation that making personal sense of mathe­matics and showed them they had to find their own solutions and that if they were respect for their think­ing and their struggle. This process to use the algo­rithm, they had to be able to explain how of renegotia­tion of mutual expectations recurred through- it works. I reminded them how far they had come in their out the semester and informed my teaching in two ways. un­derstanding of numbers and their ability to solve prob- First, it gave me the opportunity to learn about the nature lems, so why would they revert to using rules they did not of my students’ understanding. Second, I turned my stu- understand? dents’ current under­standing into learning opportunities by That said, they began to generate their own solu­tion guiding them to resolve their own cognitive conflicts rather methods and used the stan­dard algorithm to check their an­ than intervene to correct their misconcep­tions myself. The swers. While checking, they found a discrepancy between role of context proved invalu­able in the students’ efforts to their answer and the algorithm-based answer, and theirt make sense of the numerical answers. The instructional first reac­tion was self-doubt. As previously shown, because task I chose to introduce—the division of fractions—was of their lack of understanding of the result of the standard em­bedded in the context of linear measurement, which algorithm, they could not readily resolve the discrep­ancy corresponds to the quotitive or measure­ment interpretation and called on me to do it. In­stead, I made their conflict the of division that students en­counter with whole numbers focus of the mathematical activity. To support their own (Lamon 1994). The familiarity with this interpretation of di- resolution, I got them involved in actual measuring.­ As vision and the context of measuring fabric supported the they measured in groups, they began focusing on the re- students’ efforts to reconcile their self-generated solutions lationship between the 1/4-­yard leftover and the 1/3 in the with the an­swer obtained using the standard al­gorithm. In algorithm-based an­swer. They taught each other that since addition, they gained an increased confidence in their abil- the measur­ing stick was 3/4-yard long, then the 1/4 yard of ity to understand how algorithmic re­sults relate to their self- fabric leftover was 1/3 of the 3/4-yard long measur­ing stick. generated solutions. Exchanging ideas and supporting one an­other’s learning facilitated the student’s resolution of the disparity between References their self-generated solution and the algorithm-based an- Frank, Martha L. “What Myths about Mathematics Are Held swer. Some students needed further experience measur- and Conveyed by Teachers?” Arithmetic Teacher 37 ing and support from their classmates to relate the frac- (January 1990): 10–12. tion of left­over fabric to the corresponding fraction of the Lamon, Susan J. “Ratio and Proportion: Cognitive Founda- mea­suring-stick length. By the end of the instructional se- tions in Unitizing and Norming.” In The Development of quence on fractions, the students had learned the meaning Mul­tiplicative Reasoning in the Learning of Mathematics, of the answer they obtained using the in­vert-multiply rule. edited by Gershon Harel and Jere Confrey, pp. 89–120. However, not everybody was able to explain the rule. In Albany: N.Y.: State University of New York Press, 1994. the case of a whole number di­vided by a fraction, students National Council of Teachers of Mathe­matics (NCTM). who adopted David’s solution method were able to explain Principles and Standards­ for School Mathematics. Res- how the stan­dard algorithm works. However, relating their ton, Va.: NCTM, 2000. mea­surements to the algorithm while dividing two frac­tions proved much more difficult for the students. In fact, very few students accomplished it. MARCELA PERLWITZ, [email protected], lives in Craw- fordsville Indiana. She is interested in algebraic thinking and the Conclusions role of context in problem solving. Although the events related here occurred in a college Reprinted with permission from Mathematics Teaching in the Mid- class, the instruction and findings are pertinent to middle dle School, copyright February 2005, by the National Council of school instruction since the topic of dividing fractions is Teachers of Mathematics. All rights reserved. taught dur­ing the middle school grades. In addition, the

28 Virginia Mathematics Teacher GRADES 3-6 Developing Ratio Concepts: An Asian Perspective Jane-Jane Lo, Tad Watanabe, and Jinfa Cai

The following vignette illustrates how a Taiwanese text- ple, the multiplica­tive relationship between the 6 cm block book series envisions introducing the concept of ratio. and the 2 cm block can be represented as 6:2. The result Textbook. There are two blocks in front of you. One is 6 of 6 ÷ 2 , or 3, is called the value of the ratio 6:2, where 6 cm long an the other is 2 cm. How many times as long is is called the front term of the ratio and 2 is called the back the 6 cm block compared with the 2 cm block? term of the ratio. Conceptually, this idea is equivalent to Some students use the 2 cm block as a masuring unti saying “6 is 3 times as many as 2.” Note that the idea of to figure out that 6 cm is 3 units of 2 cm. Other students using the second quantity as the base for comparison can reason with the two quantities directly and come up with be linked directly to measurement division (quotitive), even the equation 6 ÷ 2 = 3. though the term “measurement division” is not directly used in Asian textbooks. For example, the teacher’s manual of a Textbook. When comparing two quantities, one of them Japanese textbook talks about conceptualizing the value of has to be used as the base quantity. There are two ways the ratio of a:b as the relative value of a when considering to relate the other quantity to the base quantity. The first b as a base quantity. way is to find out how much more the second quantity is than the base quantity. For example, how many cm Identifying the base quantity for comparison longer is the 6 cm block than the 2 cm block? Since the ratio is a way to compare two quantities using Solution. 4 cm. the division operation and since division is noncommuta- Textbook. The second way is to find out how many times tive, the order of the two terms for a particular ratio is impor- as long is the second quantity as the base quantity. For tant. In other words, a:b and b:a describe the multiplicative example, 6 cm is 3 times longer than 2 cm. Another way relationship between quantities a and b from two perspec- to represent this relationship is to use the word bi. Write tives. The value of ratio a:b is not the same as the value of as 6 bi 2, and 6:2. The result of this comparison, 3, is b:a, unless a equals b. The Chinese teacher’s manual indi­ called the “value of the ratio.” cated the reciprocal relationship between a:b and b:a but suggested that the reciprocal relationship not be explicitly A recent analysis of Asian curricular materials has identi- mentioned to students at the intro­ductory stage to avoid fied several key ideas that are emphasized in the introduc- possible confusion. tory lessons of ratio (Lo, Cai, and Watanabe 2001). These To highlight this idea, a Taiwanese textbook posed two key ideas include distinguishing a multiplicative compari- different questions comparing the num­ber of cookies for son from an additive compari­son; identifying a base quan- two brothers when the younger brother has 5 cookies and tity and measuring unit for comparison; distinguishing and the other has 2 cookies. The first question was this: “The relating the ratio a:b, the division a ÷ b, and the value of number of cookies the younger brother has is how many ratio a/b; and learning the importance of units in forming a times the older brother’s number?” The second question meaningful­ ratio relationship. After the introduction of ratio, was this: “The number of cookies the older brother has is two or three more lessons were devoted to the ideas of how many times the younger brother’s number?” The so- equivalent ratios, simplified integer ratios, and applications lution to the first problem was 5 ÷ 2 = 5/2 = 2 1/2 = 2.5. of ratio concepts. Some of these dis­cussions are familiar Students can use 5:2 to represent this ratio relationship. to mathematics teachers in North America, whereas others The solution to the second prob­lem was 2 ÷ 5 = 2/5 = 0.4. seem to be unique to the Asian materials. In this article, Students can use 2:5 to represent this ratio relationship. A we will elaborate on these key ideas and give examples pictorial representation­ similar to figure 1 was used to fa- from textbook series in China, Taiwan, and Japan (Division cilitate understanding. Note that both fraction and decimal of Mathematics 1996; National Printing Office 1999; Tokyo notations can be used for the value of ratio. Shoseki 1998). Our goal is not to evaluate Asian materials We want to emphasize two cautions about forming a ra- but rather to provide an international perspective that may tio relationship: help increase teachers’ experi­ence and awareness when 1. After the discussion of ratio definitions, the teacher’s they strive to help stu­dents develop ratio concepts (Cai and manual in the Chinese textbook pointed out two difficulties Sun 2002). that students may encounter when they relate ratio con- cepts to their daily experi­ences. First, not all related pairs of Introduction of Ratio Concepts numbers form a ratio relationship. For example, in Chinese Defining ratio as being a multiplicative relationship spoken language, the phrase “5 bi 3” is used to express Unlike typical U.S. textbooks that consider a:b and a/b as the scores of two teams in a sport event. However, in this two different ways to represent a ratio, Asian textbooks context, the focus of the comparison was on the addi­tive clearly distinguish between ratio a:b as a multiplicative re- relationship (“The number of team A has so many more lationship between two quantities and the value of ratio as points than Team B”) rather than the multiplicative relation- the quotient a/b of the divi­sion a ÷ b. In the previous exam- ship (“Team A’s points are so many times the number of

Virginia Mathematics Teacher 29 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 rectan­gle. 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 inte­ger ratios a:b Team B’s points”). Teach­ers need to be aware of the po- mean that both a and b are integers and that no common tential confusion that students may have about the use of factor other than 1 is shared between a and b. Another way language inside and outside of mathematics classrooms. A to determine if two ratios are equivalent is to convert both similar caution can be made about the English language, into simplified integer ra­tios, that is, a1:b1 = a2:b2 if and only since the phrase “a to b” is used both for ratio and for a if both a1:b1 and a2:b2 are equivalent to the same simplified sports context in the United States. ratio a:b. All three textbooks include examples like the ones 2. The teacher’s manual indicated the impor­tance of in figure 3 to help students apply this idea. paying close attention to units when com­paring two quanti- Several significant points can be made about this type of ties. In particular, at the introduc­tory level, the comparisons exercise. First, it reinforces the idea that a ratio is a rela- of two like quantities should be made with the same units tionship between two quantities and that those two quanti- to make them meaningful. For example, in a Chinese text- ties can be represented in a variety of numerical forms—in- book, the following problem was posed: tegers, fractions, or decimals. Second, it provides another method to check the equivalence of two ratios that rein- Li Ming is 1 meter tall, and his dad is 173 cm tall. Li forces the ratio concept (i.e., two ratios are equivalent if Ming said that the ratio between his height and his dad’s after simplifying they both equal the same simplified integer height is 1:173. Is 1:173 the best way to describe the ratio). Third, it provides opportunities for students to relate relationship between Li Ming’s height and his dad’s numbers to each other through common multiples and height? factors. Lo and Watanabe (1997) have found this kind of conceptualization es­sential to develop flexible proportional Through discussion, students are guided to form a more reasoning. meaningful ratio relationship if they either convert 1 meter to 100 cm or convert 173 cm to 1.73 meter to form the ratio Application of Ratio Concepts 100:173, or 1:1.73. This em­phasis is important when con- After the basic concepts of ratio and equivalent ratio sidering the idea of “value of ratio” as the relative size of the were established, all three Asian text­book series included second quantity when the base quantity is considered to be examples and exercises that ask students to apply the con- 1. In addition, this measurement context shows the need to cepts of ratio in a variety of contexts. There were two basic define equivalent ratios. types of questions: 1. The first type gave a ratio relationship between two Conceptualization of equivalent ratios quantities and the actual amount of one of those two quan- Two ratios are defined as being equivalent if they rep­resent tities, then asked students to use the ratio relationship to the same multiplicative relationship. One nat­ural implica- find the actual amount of the second quantity. The following tion of this definition is that the values of two equivalent is an example of this type of question from the Taiwanese ratios have to be equal, that is, a:b = c:d c a ÷ b = c ÷ d. textbook: In both Chinese and Japanese text­books, the principle of equivalent ratios, “Multiplying or dividing the front term and The ratio between the number of boys and the number the back term by the same nonzero number will create of girls in a summer camp is 4:3. There are 63 girls. How equivalent ratios,” was supported through examples and many boys are in the summer camp?

30 Virginia Mathematics Teacher sum of the two quantities, then asked students to use the ratio relationship to find the ac­tual amount of each of the two quantities. For exam­ple, the following question was in- cluded in the Japanese textbook series:

Two brothers shared 1800 Yen. The ratio be­tween 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 appli­cation of the principle of equivalent ratios. For the sharing-of-money problem, the Japanese text- book series ask the following sequence of ques­tions 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 de­termine 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 This question may be classified as a missing-value­ (x Yen) and the amount of total money (1800 Yen) could proportion problem because a proportional relationship be expressed as the ratio 3:5, one could solve this problem (equivalent ratio) is involved. However, it is easier to solve using the principles of equivalent ratios: Because 1800 is than a typical proportion problem (“If a car uses 8 gallons of 360 times 5, x must be 360 times 3, which results in the an- gasoline in traveling 160 miles, how many miles could the swer of 1080 Yen. The younger brother’s share could then car travel on 30 gallons of gasoline?”) for the following two be solved with either approach. Using both methods helps reasons: First, one major chal­lenge of solving this sort of students see how the ideas of multiplicative comparison, problem is to construct a ratio relationship between two dif- fractions (or decimals), ratios, simplified ratio, and equiva- ferent measures: gal­lons and miles. In the summer-camp lent ratios are connected. problem, a ratio re­lationship is stated explicitly in the ques- tion. Second, a typical proportion problem involves some Conclusion “changes” in states—before and after. In these antecedent the concepts of ratio and proportion are among the most problems, the ratio and the quantities are from the same important topics in school mathe­matics, especially at the situation. middle school level. How­ever, studies have repeatedly 2. The second type of question in the Asian text­books shown that most mid­dle school students have difficulties gave the ratio relationship between two quan­tities and the with these concepts (NCTM 2000). This article included

Virginia Mathematics Teacher 31 ideas and examples used by Asian textbooks to teach the tional Reasoning: A Chinese Perspective.” In Making concepts of ratio that are fundamental to the develop­ment Sense of Fractions, Ratios, and Proportion, 2002 Year­ of proportional reasoning. In Asian textbooks, the concepts book of the National Council of Teachers of Mathematics were carefully introduced through an emphasis on multi- (NCTM), edited by Bonnie Litwiller and George Bright, plicative comparison, the link to measurement (quotitive) 195–206. Reston, Va.: NCTM, 2002. division, the identification of base quantity, and the distinc- Division of Mathematics. National Unified Mathe­matics tion between ratio and nonratio pairs of quantities. The idea Textbooks in Elementary School. Beijing: People’s Edu- “value of ratio” was introduced to firmly establish the ratio’s cation Press, 1996. identity as a relationship based on multiplicative compari- Lo, Jane-Jane, Jinfa Cai, and Tad Watanabe. “A Com­ son rather than just another way to write a fraction. Rather parative Study of the Selected Textbooks from China, than move directly into the concepts of proportion, Asian Japan, Taiwan and the United States on the Teaching of textbooks spent time developing the idea of equivalent ra- Ratio and Proportion.” Proceed­ings of the Twenty-third tios and simplified integer ratios and discussing how these Annual Meeting of the North American Chapter of the ratio-related con­cepts could be used to solve problems in International Group for the Psychology of Mathematics everyday contexts. Typically, pictorial representations were Educa­tion, vol. 1, 509–20. Snowbird, Utah, 2001. used and multiple solution methods were discussed to help Lo, Jane-Jane, and Tad Watanabe. “Developing Ratio and students relate ratio concepts to other previously­ learned Proportion Schemes: A Story of a Fifth Grader.” Jour- concepts such as measurement (quoti­tive) division, frac- nal for Research in Mathematics Edu­cation 28 (March tions, and divisors. Furthermore, exercises and examples 1997): 216–36. were carefully chosen to link the ratio concepts to previous National Council of Teachers of Mathematics (NCTM). studies on fractions (including fractions greater than 1) and Principles and Standards for School Mathematics.Res- decimals. We believe that these approaches all aim to de- ton, Va.: NCTM, 2000. velop proportional reasoning, which is essential in solving National Printing Office. Elementary School Mathe­matics, proportion problems. 6th ed. Taipei, Taiwan: National Printing Office, 1999. In general, Asian textbook series do not include units in Tokyo Shoseki. Shinhen Atarashii Sansuu (New elemen- mathematics sentences as part of the writ­ten computation. tary school mathematics). Tokyo, Tokyo Shoseki Pub- We can probably argue the advan­tages and disadvan- lisher, 1998. tages of such a practice, but it goes beyond the focus of this article. Nevertheless, the Asian materials we analyzed did treat units care­fully and systematically. The examples JANE-JANE LO, [email protected], teaches at Western of compar­ing Li Ming’s height with his father’s height as Michigan University, Kalamazoo, MI 49008-5248. Lo’s special well as using the units flexibly to generate equivalent ra­ interests include studying the developing of multiplicative con- tios discussed earlier in this article illustrate this emphasis. cepts and preparing future teachers. TAD WATANABE, txw17@ psu.edu. teaches at Penn State University, University Park, PA Furthermore, both the textbook series and the teacher’s 16802. His interests include children’s ultiplicative concepts and manuals routinely remind stu­dents to think about the mean- mathematics education in Japan. JINFA CAI, [email protected]. ings of the quantities and the units used to quantify these edu, teaches at the University of Delaware, Newark, DE 19716. quantities in­volved in computation. The goal is to prepare His interests include cognitive studies of mathematical problem stu­dents for more complex contextual problems when solving and integration of assessment into the classroom. multiple computations are required to determine unknown quantities. The examination of curriculum and instructional ______practice in other nations provides a broader point of view The preparation of this article was supported, in part, by a grant on how topics can be treated. We hope that such an inter- from the National Academy of Education. Any opinions expressed herein are those of the authors and do not necessarily represent- national per­spective can add to U.S. teachers’ background ed the views of the National Academy of Eduation. when they try to address the issues and challenges facing students’ learning of ratio and proportion. Reprinted with permission from Mathematics Teaching in the References Middle School, copyright March 2004, by the National Council of Cai, Jinfa, and Wen Sun. “Developing Students’ Propor­ Teachers of Mathematics. All rights reserved.

32 Virginia Mathematics Teacher GRADES 7-10 Pick a Number Margaret Kidd

As a mathematics educator, middle and high school 2. Add 3. teachers frequently ask me how to motivate students. One 3. Multiply by 4. method I have found to be effective is to engage them with 4. Subtract 8. a topic that personally intrigues them. Since many students 5. Divide by 2. are fascinated by magic, this can be used to help them 6. Add 3. learn procedures from which they normally shy away. In 7. What is your ending number? my experiences of teaching in various districts in the coun- try, there are three topics that give students trouble when When students begin clamoring for an explanation so beginning to study algebra: fractions, operations on inte- that they can try this on their parents and friends, it is time gers, and the distributive property. This article combines to explain the mathematics behind these. If you would like the motivation of mathematical magic with the difficulty of to figure these out yourself, please stop reading, since the applying the distributive property and the rules for order of solutions will now be given. operations. The actively described challenges students to uncover the “magic” behind the mathematics and discover Example 1: the reason we have order of operations rules. They also 1. Pick a number from 1 to 10. x come to appreciate the power of using a variable, grouping 2. Multiply by 3 3x symbols, and the use of the distributive property. 3. Subtract 1. 3x - 1 4. Multiply by 2. 2(3x - 1) The Mathemagic Lesson 5. Add 3. 2(3x - 1) + 3 As the “ mathemagician,” begin the lesson by asking 6. What did you get? students to think of a number from one to ten. With much intrigue, inform them that after performing some mathemat- The simplified expression becomes 6x + 1. So when the ics magic-and with their help-you will be able to tell each student states his ending number, you simply subtract 1 student his/her starting number. Next, ask them to perform from it and divide that answer by 6. a series of calculations that end with students revealing only their final numbers. At that point you quickly tell them Example 2: the numbers they first selected. The fact that you can do 1. Choose a number x this with little effort seems like magic to the students and 2. Multiply the number by 3 3x usually gets their attention. If you do this a number of times, 3. Subtract 4. 3x - 4 their attention is normally riveted. At this point, they are ea- 4. Multiply the result by 2. 2(3x - 4) ger to learn how the answer was discovered so quickly and 5. Add 5. 2(3x - 4) + 5 are more amenable to learning the process in order to un- 6. Report your result. derstand the secret of being able to discern the answer so quickly. The simplified expression becomes 6x - 3. When the stu- Here are some examples to catch their imagination: dent states her ending number, you add 3 to it and divide that answer by 6. Example 1: 1. Pick a number from 1 to 10. Example 3: 2. Multiply it by 3. 1. Choose a number. x 3. Subtract 1. 2. Add 3. x + 3 4. Multiply this by 2. 3. Multiply the result by 4. 4(x + 3) 5. Add 3. 4. Subtract 8. 4(x + 3) - 8 6. What did you get? 5. Divide by 2. (4(x + 3) - 8) / 2 6. Add 3. (4(x + 3) - 8) / 2 + 3 Example 2: 7. What is your ending number? 1. Choose a number. 2. Multiply the number by 3. Although Example 3 contains more and more complex 3. Subtract 4. operations, it still simplifies to 2x + 5. 4. Multiply the result by 2. 5. Add 5. Conclusion 6. Report your result. These number puzzles can be as short as adding 1 and subtracting the original number to as complicated as one Example 3: (a bit more complicated): wants to make it. Start with the simplified expression you 1. Pick a number from 1 to 10. desire. Then, add the steps in reverse order until you have

Virginia Mathematics Teacher 33 a series that is as long and as complicated as you wish. A MARGARET KIDD, CSU Fullerton few caveats are in order, however. Have students pick a [email protected] number small enough that you can solve the last equation easily. Avoid numbers and operations that result in frac- Reprinted with permission from The California Mathemat- tions or decimals. Other than that, both you and your stu- ics Council ComMuniCator, September 2009. 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.

CONGRATULATIONS

VCTM 2011 William C. LowryCongratulations Mathematics Educator of the Year VCTMAwardees 2011 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

34 Virginia Mathematics Teacher GRADES 7-12 Those Darn Exponents: Fifty Challenging True-False Questions 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.

Virginia Mathematics Teacher 35 GRADES 13-16 Abstractmath.org: A Web Site for Post-Calculus Math Charles Wells The Abstractmath web site at http://www.abstractmath. odd? Abstractmath walks you through examples of proofs org/MM/MMIntro.htm is intended for math majors and oth- as a guide to how to understand them. ers who are faced with learning “abstract” or “higher” math, Images and metaphors: Mathematicians use lots of com- the kind with epsilons and deltas, quotient spaces, proofs pelling metaphors to talk and think about their topics and by contradiction: all those kinds of abstract things that can images to give geometric sense to them. These images knock you sideways even if you got an A in calculus. and metaphors are also dangerous because they may I have been developing Abstractmath for a couple of suggest things that are incorrect. (x2 - 9 vanishes at 3.” years and now it is time to open it up to the wide world. Not Does that mean it doesn’t exist at 3?) When mathemat- that all is finished. There are gaps and stubs all through ics start to prove something about their topic they abandon it. But enough is completed that it is respectable, and be- these images and metaphors and go into a rigorous mode sides, I need help! Some students and math educators of thinking in which all mathematical objects are inert and have already discovered the site and told me things that unchanging. Does anyone ever tell the students this (as helped them and things that made no sense to them, as opposed to doing it in front of them)? Abstractmath does, well as finding many embarrassing errors. The site needs with examples. much more help like that, and suggestions for more com- Mathematical objects: People new to abstract math have pelling examples and useful topics. a great deal of trouble thinking of mathematical objects as Abstractmath is personal and opinionated, but it is based objects rather than processes or bunches. A quotient space on research by many people in mathematics education and has elements that are sets (these sets are not substances cognitive psychology, and on my own lexicographical re- - they are elements!). A function space has elements that search. It concentrates on certain types of problems. One are functions (not values of functions). Abstractmath dis- web site can’t do everything. cusses many examples of this phenomenon. Mathematical English: This is a foreign language dis- I hope you will look into abstractmath.org, whether you guised as English. Many common logical words (notori- are a student or a teacher, and let me know how it can be ously “if...then”) don’t mean quite the same thing they do improved. You can also contribute articles or examples, or in English. Common words are used with technical mean- publish them on your own web site and ask me to link to ings, leaving the student to be confounded by their every- them. day connotations. Proofs: A mathematical proof has both a logical structure and a narrative structure. If you are reading a proof your CHARLES WELLS is Emeritus Professor of Mathematics at major problem is to extract the logical structure from the Case Western Reserve University. narrative you read. Consider: “Theorem: If n is an integer Reprinted with permission from FOCUS The Newsletter of the and n2 is even, then n is even. Proof: Suppose n is odd...” Mathematical Association of America, copyright March 2007. All How can a proof that n is even start out by assuming it is rights reserved.

36 Virginia Mathematics Teacher

Virginia Council of Teachers of Mathematics Non-Profit P.O. Box 714 Organization Annandale, VA 22003-0714 U.S. Postage Pat Gabriel, Exec. Secretary PAID Blacksburg, VA Permit No. 159

DATE AND NOTE POST

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Calling Virginia Authors: Virginia residents whose articles appear in the VMT will be granted free member- ship in the VCTM for one year. To qualify, the manuscript must be at least two typewritten pages in length.