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Do You SUD0KU?

Vol. LI No. 4 Winter 2006

Vol. LI No. 4 Winter 2006

Table ofContents

SPECIAL REFLECTIONS2 Report on SAT Scores

6 NCTM Affiliate News

8 NUMB3RS

10 Rock Eagle 2005

12 Facts About Pi

14 Reading Pi

16 Measurement and Fraction Activities: Using Children’s Literature and Mathematics

18 Geometry by Tad

20 So You’ve Decided to Sponsor the Math Team . . .Now What?

22 Authentic Assessment: A Powerful Motivator in the Mathematics Classroom

27 Why Do These Work?

1 President’s Desk

3 GCTM Membership

9 Al’s Web Sites

28 Exec. Committee/Regional Reps

NCTM Press Release

NCTM Press Release

TI & NCTM Press Release

Photos by Leslie Poythress

Cheryl Hughes

Dr. Faith Wallace & Chris McSwain

Dr. Barba Patton

Tad Wanabe

Chuck Garner

Dr. Billy Lacefield

Dr. Delores Anderson

REGULAR REFLECTIONS

About the Front Cover:The sudoku is the latest craze in number puzzles. Originating in Japan, they have become very

popular in the United Kingdom, Australia, and all over the world. Check out these Web sites andwatch the magazine racks in stores for entire books of them.

www.sudoku.com.au www.edhelper.com/sudoku.htm www.sudokuforkids.com

President’s Submission

by Dottie Whitlow

GCTM President

1REFLECTIONS WINTER 2006

About this time of year, we teachers tend to get weary. We often wonder why we do whatwe do and begin to think our work is in vain. Even though we pour out our hearts andsouls, our students don’t appear to be learning. The dilemma is that the harder we teach,

the less students seem to understand. Consequently, we do not see the progress we expected tosee.

The winter days are short and the nights are long, made even longer by the endless stacks ofpapers waiting to be graded. Our family responsibilities pull us one way and duties as a teacherpull us another. With five months left in the school year, we ask ourselves, “How can we sur-vive?”

So how can we get through the rest of the year and still make a difference? The answer is thatwe have to have a plan. The secret to a successful teaching plan is to do something different. TheMaster Teacher series outlines several suggestions to give new life to your teaching, which we allneed mid-year.

• Rearrange your classroom. Turn student desks and orientate your classroom in a newdirection. Bring in a plant, put up a poster that celebrates mathematics or add a new pieceof furniture to your teaching space. Surprise students by using a different wall for youroverhead projector.

• To keep things fresh, try a new teaching strategy—even one new technique will catchyour students off guard. A new, innovative approach will be a welcomed change for bothfor you and for your pupils.

• Try looking at your students in a different light and with a new outlook. A viewpointfrom a fresh perspective might help you to better appreciate and understand your stu-dents. Empathize with your children. Remember how it was when you learned somethingnew. Remember how it felt not to understand a concept the first time it was presented.

• Say something positive to your students every day collectively as a class. Try to speakto each student individually as they come in the door. Find a point of interest they haveoutside of your class and ask about it. Are they interested in sports, arts, farming, or vol-unteering? Ask how the progress they are making.

• Find someone to help or mentor. As you advise them, you may remember somethingthat helped you at one time that you can implement again. If that’s not possible, observean older, more experienced teacher. Ask him/her for suggestions for your class or yoursituation.

• Take at least one specific action to lift your spirits personally. Do something foryourself. Mental health is as crucial to success as physical health.

After you have tried one or more of the suggestions above, then go do what you know how todo. Take it one day at a time. Don’t think, “I have five months to go.” Instead, think,

“I CAN DO THIS!”With this resolve, be the best teacher possible. We at GCTM are here to help you. If you need

us, you have only to email, and we would be happy to offer our support. Have a great secondsemester!

All-Time High in Math SAT Scoresa Result of Greater Focus on Math in Schools

REFLECTIONS WINTER 20062

Reston, VA, August 30, 2005—Thehighest math scores ever on the SATare another encouraging result of an

increased focus on mathematics education inU. S. schools, the National Council of Teachersof Mathematics (NCTM) said today.

In response to today’s release of 2005 SATscores by the College Board, NCTM PresidentCathy Seeley said, “The recent increased focuson improving the mathematics education ofour students continues to pay off. A focus onmathematics teaching and course-taking seemsto be preparing more students for the future.Today we’re seeing many students with theopportunity to take high-level courses whomight not have been encouraged to do so inthe past.”

The average SAT math scores continued astrong upward trend, increasing from 518 in2004 to 520 this year, 14 points above the markof 10 years ago and an all-time high. Mathscores for females rose by 3 points over lastyear, to 504, while scores for males rose by 1point, to 538, over the same period. TheCollege Board also reported that more highschool students are taking demanding coursessuch as precalculus, calculus, and physics. Since1995, there has been an 11 percent increase inthe number of students enrolled in precalculus(37 to 48 percent) and a 5 percent increase inthe number enrolled in calculus (22 to 27 per-cent).

“The trend of more students taking higher-level courses is encouraging,” Seeley said. “The

Council has long been a proponent of moreand better mathematics for all students. Morethan ever before, in today’s world studentsneed to take math every year of school, andthey also need to take higher-level courses inhigh school. We still have a long way to go interms of tapping the untapped potential inmany classrooms, but these scores are oneindication of the value of having more stu-dents pursuing a challenging curriculum.”Gaston Caperton, president of the College

Board, said, “I am encouraged by the improve-ment demonstrated in math, a fundamentalskill that students need to succeed in collegeand, later, in a highly competitive global mar-ketplace.” More students took the SAT thisyear than ever before, the fifteenth year in arow that the total number of test take rsincreased.The National Council of Te a ch e rs of

Mathematics was founded in 1920 and is anonprofit, nonpartisan education association.With more than 100,000 members and morethan 240 Affiliates located throughout theUnited States and Canada, NCTM is thewo rl d ’s largest orga n i z ation dedicated toimproving mathematics education for all stu-dents. The Council’s Principles and Standardsfor School Mathematics provides guidelinesfor excellence in mathematics education.

###Contact: (703) 620-9840Shannon Andrea, Communications Coordinator

Have you visited www.nctm.org lately?

Membership Article

by Susan CraigMembership Director


As the second gradingperiod ends thisweek, I find it hard to

realize that one-third of thisacademic year has passed.This time of year brings timefor reflection and thanksgiv-ing, family and friends.

Since I just returned frommy annual “retreat” at RockEagle with so many of you, itis appropriate to realize howm a ny friends I have madethrough the years at our annu-al Georgia Mat h e m at i c sConference. Each year offersan opportunity to renew thoseacquaintances and to cat chup—all while re i nv i go rat i n gmyself, and my mathematicst e a ching—and all because Iam a member of GCTM. It ist ru ly the best investment Ihave made in my professionallife.

Since this time last year wehave increased membership by120 teach e rs, about a 7%increase. We have many newstudent members, ones wehope will be members fo ryears to come and future lead-ers in GCTM. There are manyteachers who have never beenmembers. One teacher I metat Ro ck Eagle “a ve t e ra nteacher” had never heard ofthe conference before. A newdepartment chairman brought

information to her and shesays she will never miss GMCagain.

There are also many othert e a ch e rs who are membersbecause they join as part oftheir conference registration,and they attend the confer-ence only in alternate years. Sothere are many teachers for usto ask to make their biannualcommitment into an annualmembership. There are alsomany who just donít know thebenefits to be found inGCTM membership. I imag-ine that you know someone inyour school fitting one ofthese descriptions. Please takethe membership form fromthis issue or print one fromwww.gctm.org and give it tothem. Ask them to mail ittoday. We have so much tooffer mathematics teachers inour state. So get out there andencourage your colleagues. Iíllbe thankful this season for all

these re n ewals and ap p l i c a-tions!

Thanks for your commit-ment to Georgiaís studentsand to GCTM! Hap pyHolidays!

We have had several new lifemembers in the last year or so.They are:

Martha A. AllenTracy E. BeardCarl ChenardChuck GarnerGary LeMayJean LinnerPeggy PoolBarry E. ShealyDoug Wagner

Congratulations on your life-time commitment toM at h e m atics Education inGeorgia!

2005 2006 2008 Life Student TOTALNW 63 26 0 24 44 157NE 110 14 0 45 90 259

MetroW 161 42 0 81 28 312MetroE 300 41 0 39 28 408

CW 127 47 0 71 26 271CE 146 45 0 61 41 293SW 103 30 1 42 1 177SE 131 22 1 70 4 228

Out of State 4 2 0 14 0 20TOTALS 1145 269 2 447 262 2125

Membership Report November 2005

Volunteer for a Committee for the 2006 NCTM Annual Conference & Exposition!

Are you READY? The 2006 NCTM Annual Conference and Exposition will be in St. Louis during April 26-29, 2006! YOUR HELP IS NEEDED!! In fact, we’re going to need about 600-800 volunteers to host this greatprofessional opportunity for folks from around the country and the world. Make plans to attend and arrangeyour schedule so that you can volunteer some time, energy, or expertise. You can choose to volunteer for sev-eral hours, a half day, or even a full day. And to identify you as a Volunteer VIP, you will receive a distinctiveT-shirt made especially for volunteers. Share this invitation to volunteer with your colleagues as well as yourprofessional mathematics friends in other states.

—Jane T. Barnard, Publicity Co-Chair, 2006 NCTM Annual Conference and Exposition

1-Fill out the online Volunteer Form at www.nctm.org/meetings/stlouis/volform.htm

2-Please identify the committee(s) on which you would like to serve.

• Hospitality: Staffs the Hospitality Desk during the Annual Meeting hours; offers attendeesinformation on and directions to local services, sites, events, facility.

• Meeting Rooms: Checks meeting rooms between presentations; places “Session Full” signsoutside of meeting rooms; clears meeting rooms after presentation.

• Speaker Support: Staffs the speaker check-in support desk and welcome speakers.

• NCTM Bookstore: Assists staff to unpack, sort, and display NCTM materials; supports staffin assisting Bookstore customers during exhibit hours; helps box the remaining materials at theend of the meeting.

• Special Events: Assists with special entertainment events.

• Special Needs: On-site assistance with individuals with special needs; assists with special needsequipment

• Signs: Arranges for presentation signs to be placed outside each meeting room and changessigns as needed.

• Student Hosts: Recruits and trains students to work as helpers at the Annual Meeting.

• Technology: Staffs and assists in the Cyber Café and/or computer workshop labs.

• Student Exhibits: Sets up and breaks down the exhibits. Assists students during exhibit hours.

NCTM Promotes Political AdvocacyThrough Affiliates


In 2003, the Board of Directors of theN ational Council of Te a ch e rs ofMathematics (NCTM) identified political

advocacy as one of its key strategic directionsfor the future. The Council embarked on aplan to lay the groundwork for future successin advocacy, first conducting a stakeholders’audit of key external and internal audiences:news media, policymakers, third-party influ-encers (think tanks and policy organizations),NCTM leadership and members, and otherrelated organizations. Recommendations basedon the findings of the audit have guided theCouncil’s actions over the past two years.

NCTM has taken the first steps toward build-ing a culture of advocacy within its member-ship and toward more assertively contributingto the public policy process in ways that bene-fit teachers, mathematics education, and socie-ty. With a membership of nearly 100,000 mem-bers, the Council has a significant resource todraw upon for these efforts.

The NCTM Advocacy Toolkit was the firstmajor step toward engaging individual mem-bers as advocates. The toolkit, which wasintroduced at the NCTM Annual Meeting inPhiladelphia in 2004, brings together existingNCTM information and new communicationsmaterials to give individuals guidance in how tomake a difference in the public policy process.The toolkit includes the following materials:

• NCTM Legislative Platform: Board-approved NCTM statements on broadissues, the foundation of NCTM gov-ernment relations activities

• NCTM Commu n i c ations Guide:Basic how-to information for individu-

als dealing with policymakers and newsmedia

• C o n gressional Dire c t o ry : C o n t a c ti n fo rm ation and committee assign-ments for all members of Congress

• Principles and Standards ExecutiveS u m m a ry : A concise summary ofNCTM's Principles and Standards forSchool Mathematics)

• Principles and Standards FAQ s :Answers to frequently asked questionsabout the Standards

• NCTM Key Message s : C o n c i s estatements on key issues

• NCTM at a Glance: Basic back-ground information about NCTM

• NCTM Position Statements

NCTM Advocacy Toolkits are available tomembers upon request through NCTM head-q u a rt e rs or via the Web site atwww.nctm.org/advocacy. There is no chargefor the toolkits, although those who requestone are asked to provide information that willbe included in a database of advocates that willbe used for future NCTM advocacy activity.

Contact Congress Via NCTM.orgThe next phase of the advocacy effort was

the integration of software within the NCTMWeb site that allows visitors to contact mem-bers of Congress electronically. The AdvocacyWeb page can be found at


w w w. N C T M . o rg / a dvo c a cy. This Web pagemakes it simple for individuals to participate inthe policymaking process and to actively sup-port math education. In addition to providinga quick link to Congress, the page offers toolsthat help visitors stay informed about key fed-eral legislation.

It also makes visitors aware of urgent legisla-tive issues with Action Alerts. One click fromthe Action Alert box takes individuals to asample letter that can be sent immediately aswritten or that can be revised and personalized.The Advocacy page makes it easy to takeaction. It provides contact information formembers of Congress; offers general advice,including tips on how to telephone elected rep-resentatives; provides hyperlinks that connectto information about recent votes and currentlegislation; and provides easy access to nation-al and local news media sources.

Members as ConstituentsThe NCTM leadership heard from its mem-

bers that more political action was needed andexpected from the Council. In its ongoing gov-ernment relations work, NCTM works withother education organizations in Washington,

many of which have similar interests and pri-orities. The organizations that are most suc-cessful in their legislative efforts have onething in common: They are effective at gettingtheir members to act as constituents by writing,fa x i n g, e-mailing or calling Congre s s i o n a loffices when the issue and the time are right.The Council’s large membership is a greatpotential re s o u rce in the same way. Th eAdvocacy Toolkit and advocacy page of theNCTM Web site enable individuals to supportmathematics education and make their voicesheard.

I n c reasing support and invo l vement ofNCTM Affiliates is a next step in the Council'sadvocacy plans for the future. For the last twoyears, an advocacy strand has been part of theNCTM Affiliate Leaders Conferences. NCTMDirector of Communications Ken Krehbielhas presented an overview of the Council'sadvocacy activities and plans, and attendeeshave participated in a role-playing activity andreceived NCTM Advocacy Toolkits. All atten-dees also received NCTM Advocacy Toolkits.

To learn more and to support the Council'sadvocacy on behalf of mathematics education,go to www.NCTM.org/advocacy.

Calling All Teachers—Send pictures of your studentprojects, bulletin boards, oractivities, to be included in thenext issue of REFLECTIONS.

Send to: Cheryl Hughes at [email protected]

Check out the

Awards offered by GCTM

You might be eligible for one, oryou might want to nominatesomeone else. The deadlines areright around the corner.

www.gctm.org

7

NUMB3RS


Dallas and Los Angeles,September 2005—TexasInstruments (TI) will lead amath education initiative basedon the hit series NUMB3RS. Inpartnership with CBS, andworking in association with theNational Council of Teachers ofMathematics (NCTM), TI hascreated an educational outreachprogram promoting the manyuses of mathematics and sup-porting math teaching. Theprogram begins with the firstepisode of the season, premier-ing Friday, Sept. 23 at10:00–11:00 PM, ET/PT and9:00 PM CT on CBS.

The program includes TI andNCTM-developed math educa-tion activities for teachers andstudents based on theNUMB3RS TV show. The activ-ities will be based on the math-ematics presented in eachepisode, and will be availableat cbs.com/numb3rs.NUMB3RS, which premiered

on Jan. 23, 2005, focuses onFBI agent Don Eppes (RobMorrow) who recruits his math-ematical genius brother Charlie(David Krumholtz) to help theBureau solve a wide range ofchallenging crimes in LosAngeles. Inspired by actualcases, the series depicts howpolice work and mathematicsprovide unexpected revelationsand answers to the most per-plexing criminal questions.

Math teachers began using

NUMB3RS content informallyduring the show’s first seasonas a supplemental teaching toolto encourage their students toget more interested in mathwith positive results. This yearthe effort is being made officialwith financial and operationalsupport from TI, in associationwith NCTM, so that thenation’s math teachers in grades9–12 will receive high quality,engaging activities that havebeen developed by classroomteachers and leading mathe-maticians for students at grade-appropriate levels.

“TI is proud to work withNUMB3RS and NCTM to pro-mote student interest in mathe-matics,” said Melendy Lovett,president, Educational &Productivity Solutions, TexasInstruments. “TI has longworked with math and scienceeducators, and we view ourNUMB3RS program as anotherway to help classroom teachersshow students the real worldrelevancy of math.”

“It is really gratifying to knowthat NUMB3RS can serve a pur-pose beyond pure entertain-ment. We are glad that TI andNCTM see the educationalvalue in our show and we andthe cast of NUMB3RS look for-ward to doing whatever we canto spread the word that ‘Math iscool!’” stated NUMB3RSExecutive Producers and co-Creators Cheryl Heuton and

Nicolas Falacci.NCTM, the nation’s largest

and most respected associationof mathematics teachers, willwork with TI and NUMB3RS tocreate special classroom activi-ties corresponding with themath used in each episode ofNUMB3RS. Students will havethe chance to more deeplyexplore the math derived direct-ly from the concepts highlight-ed in each week’s episode.

“We’re proud to be workingin association with TI andNUMB3RS to implement thisexciting educational outreachprogram,” said Cathy Seeley,NCTM president. “By empha-sizing the real world applica-tions of math, we hope to moti-vate students to go further andlearn more challenging mathe-matics while raising theirawareness of how ‘we all usemath every day.’”

TI will also be conductingsweepstakes, commencing in2006 that will incorporate mathquestions in an “Open BookQuiz.” Two students and oneteacher will each win a trip forfour to Hollywood to meet castmembers of the show. The twowinning students will alsoreceive scholarships. The win-ning teacher will also win awalk-on part on NUMB3RS.Teachers can find additionalprogram information and ordera classroom start-up kit atcbs.com/numb3rs.

TI, in association with NCTM, Partners with NUMB3RS,the Paramount Network Television Series for CBS, to

Launch an Innovative Math Education ProgramTexas Instruments and National Council of Teachers of Mathematics to

Engage Students Using Math Concepts from Hit TV Series

Al’s Web sites


sitesforteachers.com/index.htmlLinks and descriptions of hundreds of educa-

tional sites on academic topics as well as lessonplans, clip art and other activities.

www.mathtv.org/MainMenu1.htmlA new problem is given each week for middle

school students, specifically Algebra, with videofootage explaining a sample problem. Then theactual challenge problem is viewed, and studentsmay solve it online or at their desks.

www.murderousmaths.co.ukPuzzles, games and tricks, with book reviews by

“Thag” the Mathemagician, from the UK. Theyhope to have their publications “Americanized”soon, through Scholar Publishing.

www.fi.uu.nl/wisweb/en/presentatie_enMC.html

Applets to illustrate mathematical concepts

www.unclebobpuzzles.comPuzzles, puzzles, and more puzzles! They will

even deliver them to your email address monthly.

www.schoolhousetech.com/products/mathematics/overview.htm

A site for creating worksheets. You have to pro-vide the material (text) and they provide the for-mat.

www.geocities.com/cnowlen/Cathy/Math.htm

I f you have Geometer’s Ske t chpad by KeyCurriculum Press, this site provides you withsketches and proofs that are ready to use.

JobOpportunity

GCTM seeking a new editorfor REFLECTIONS.

All interested parties please contact CherylHughes at [email protected].

NEXT ISSUEDeadline: Feb. 15, 2006

Topics:

• Math in the summer forteachers and students

• Technology in the mathclassroom

Submissions to REFLECTIONS should be sent electronically to Cheryl Hughes [email protected]. Photos and handouts should be indicated in the initial email,but sent later after acceptance. Priority is given to those articles that concern thetopics for the particular issue, but all inquiries are welcomed. Priority is also giveto Georgia teachers, as we strive to highlight excellence in teaching in our state.Typical word count is less than 800 words, but all submissions will be considered.

Writers’ Guidelines

VOLUNTEER NOW to be part of the planning and

operation of NCTM AnnualConference to be held in Atlanta,

March 21 - 24, 2007

Contact Dottie Whitlow at [email protected] to volunteer.

Rock Eagle Awards


The Georgia Council of Teachers of Mathematics presented the following awards at theGeorgia Math Conference, October 20-21, 2005.

Photo NotAvailable

Photo NotAvailable

Gladys M.Thomason Award toChristine Thomasfor her outstandingcontributions to

MathematicsEducation in Georgia

John Neff Award toDavid Stone

2005 Friend ofMathematics Award to

Paul Ohme

Teacher of PromiseAward to

Heather PriceOak Grove

Elementary SchoolPeachtree City,

Georgia

Dwight Love Awardto

Dennis Stewart

GCTM High SchoolMath Teacher of the

Year Award toJanet Davis

Starr's Mill HighSchool

GCTM MiddleSchool Math Teacherof the Year Award to

Kolleen MetarkoSouth Forsyth

Middle School

GCTM ElementaryMath Teacher of the

Year Award toSharon Pinion

Sawnee Elementary

People came from far and wide to present workshops this year at Rock Eagle. Some came from asfar away as Peru! There were workshops for Kindergarten teachers and college professors, andeveryone in between. There was a topic for everyone. On Saturday there was an actual Math Fair

set up in oneo f the bu i l d-ings, so teach-e rs from allover Georg i acould see onein progress.


Teachers and administrators listened to new ideas, tried theirhand at making clinometers, created songs about math top-ics, and heard many new ideas. Participation was the key, and

the ideas and information was there for the taking. Some even got PLUs by bringing a prior approval formand having their session form signed at each session.

Rock Eagle Memories

Facts About Pi


Because pi (the ratio of the circumference ofa circle to its diameter) is an important conceptin mathematics and science, March 14 (fromthe date 3 - 14) has been proclaimed as a spe-cial “Pi Day.” Scientists have explored the con-cept of pi, an irrational number, beginning per-haps as far back as 4,000 years ago and contin-uing right up to the present. With the adventof the computer, pi has been calculated tomore than 51 billion decimal places! Let yourstudents create games about pi, research thehistory of pi, memorize pi to 50 places, ordevelop ways to understand pi. These can be inthe form of group or individual projects, to becompleted during class or to take home.

The symbol for pi was introduced by theEnglish mathematician William Jones in 1706,who wrote: 3.14159 = !

So for this special birthday year of the symbol“pi,” the sky is the limit on the possibilities forcelebrating “Pi-Day.”

Here are just a few suggestions for makingthe day special:

• Celebrate pi day by having studentscreate and display pi day cards, pi daypoems, or pi day posters.

• Celebrate with food. The food connec-tion is punningly obvious—studentscan share a pizza pie or apple pie butfirst must figure out the circumferenceand the diameter.

• Celebrate with problems about the areaand circumference of circles. Use non-standard methods of measuring, theneat them in the end!

Web Resources

• The Pi Pages: http://www.cecm.sfu.ca/pi/pi.html

• A History of Pi: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pi_through_the_ages.html

• Slices of Pi: Rounding Up Ideas for Celebrating Pi Day by Larry Lesserhttp://www.tenet.edu/tctm/downloads/TMT_Fall_04.pdf

Ready-to-Go Activities

• Buffon’s Needle: An Analysis and Simulation: http://www.mste.uiuc.edu/reese/buffon/buffon.html

• Making a Pi Necklace: http://mathforum.org/teachers/middle/activities/pi_day.html

• Activities for Pi Mathematics: http://archive.ncsa.uiuc.edu/edu/RSE/RSEorange/Piactivities.html

• Pi Day Songs: http://www.winternet.com/~mchristi/piday.html

• Archimedes and the Computation of Pi: http://www.math.utah.edu/~alfeld/Archimedes/Archimedes.html

• Graphics for the Calculus Classroom: Archimedes’ Calculation of Pi:http://www.ima.umn.edu/~arnold/graphics.html#archimedes

• The Pi Trivia Game: http://eveander.com/trivia/

• Famous Problems in the History of Mathematics: Finding the Value of Pi:http://mathforum.org/isaac/problems/pi1.html

• Monte Carlo Estimation for Pi: http://polymer.bu.edu/java/java/montepi/MontePi.html

The following page is an activity submitted by Laura Davis of Westside High School in Bibb County.

Worksheet


by

Faith H. Wallace, Ph.D.

& Chris McSwain

Kennesaw State

University

Reading π: A Book List for Middle & Upper Grades


With more emphasis on reading inthe content areas, mathematicsteachers have an opportunity to

tap into the world of trade books that fea-ture compelling details and exciting adven-tures centered around mathematical con-cepts. Trade books are books written andintended for general public use, and are notwritten specifically for school and academics(Harris & Hodges, 1995). That means that theauthor is generally passionate about the con-tent and has a message for the reader. Tradebooks can be informational and present other-wise technical information in ways that grabthe reader's attention. This can be achievedthrough captivating illustrations, little knowndetails, and links to everyday life (Tunnell &Jacobs, 2000). Novels and short stories canprovide a familiar context for presenting com-plex content, spark excitement for learning,and naturally introduce concepts, vocabulary,and ab s t ract thought (Ducolon, 2 0 0 0 ;Hunsader, 2004; Moyer, 2000; Murphy, 1999).

With Pi Day steadily approaching, we havecompiled a list of resources that will help stu-dents understand the concept of Pi. This listcontains informational trade books as well as apicture book and a screenplay, and in somecases just a chapter or two are actually devotedto Pi. All can help to expand student knowl-edge of the concept of Pi and build apprecia-

tion of the use of Pi within story develop-ment. We include an annotation of each selec-tion, but we encourage you to read each text sothat you can determine the appropriateness foryour students. All titles are easily availablet h rough online book wa rehouses such asAmazon.com or BarnesandNoble.com andwill make fantastic additions to any Pi Day cel-ebration.

Pi by Darren Aronofsky Can the workings of the universe be

tied to the digits in !? In this screenplay, MaxCohen, a mathematical genius, has createdEuclid, a fast and strong computer in his apart-ment. He believes that Euclid will help himidentify certain mathematical patterns that willexplain everything around us, including theworkings of the universe. While being pursuedby a Wall Street firm and the Kabbalah for dif-ferent reasons, Max drives himself crazy tryingto find the patterns within !. Is he able to findthe answer he is seeking? Does he end upworking for the Wall Street firm? Or does hehelp the Kabbalah? All this and more is thefocus of this sophisticated and intriguingscreenplay. This complex screenplay is suitablefor very mature audiences only.

The Joy of π by David Blatner In a world filled with exceedingly advanced

technology and a history of genius minds, arethere limits to the knowledge humans canaspire to? Despite the persistence and bril-

“Probably no symbol in mathematics has

evoked as much mystery, romanticism, miscon-

ception and human interest as the number pi.”

—William L. Schaaf, Nature and History of Pi

“The computation of pi is virtually the

only topic from the most ancient stratum

of mathematics that is still of serious inter-

est to modern mathematical research.”

—Len Berggren, Jonathan Borwein,

Peter Borwein, Pi: A Source Book

liance of mankind, ! has, thus far, proven to beunconquerable. Why has the study of ! causedmany people to dedicate their lives to journey-ing into its' depths? Is there any value in theuniqueness of the number? These questionsany many more are addressed in The Joy of !.B l atner engages the reader by tracing theknown history of this unique number. This isa fabulous re fe r-ence for teachersto who have yet tosearch the depthsof !. Carefully selected sections can augmentany classroom instruction.

Sir Cumference and the Dragon ofPi: A Math Adventure by CindyNeuschwander

Who ever thought figuring out ! could be amatter of life and death? What does ! have todo with dragons and potions? Find out in thispicture book as a young boy goes on a journeyto figure out ! in order to save the life of hisfather. This book would serve as a wonderfulread aloud within any classroom studying theconcept of !, and it can be used effectively ingrade 3 and up.

Math Talk: Mathematical Ideas inPoems for Two Voices by TheoniPappas

What's better than a poem about mathemati-cal concepts? An entire book of poetry devot-ed to mathematics! In this book, Pappas writespoems designed to be read aloud by two orthree students. Two poems in the collectiondeal with the concept of !. These are greatread-alouds and inspire dramatics.

Fractals, Googols and OtherMathematical Tales by TheoniPappas

Have you ever thought about the psychologi-cal effects of never knowing your exact value?In this short story, ! is humorously personi-fied. He suffers ridicule from other numbers asa result of their confidence in their position on

the nu m b e rline. Don't fret! love rs! Hebuilds his

indefinite self back up by telling of the adven-tures mathematicians have been through whilestudying him. This selection is a perfect read-ing assignment to introduce the concept.

Math Stuff by Theoni Pappas Is there any practical application to come out

of the study of !? How far has computing thedigits of this infinitely long number gottenover the years? All of this is answered in thisinformational trade book written in an engag-ing manner. This is a great reference providinganswers to the question commonly asked bystudents, “What does this have to do withme?”

GCTM does not itself endorse or specifically recom -mend any of the above publications. All quotes takenfrom The Joy of Pi, by David Blatner.


“The mysterious and wonderful pi is reduced to a gargle

that helps computing machines clear their throats.”

—Philip J. Davis, The Lore of Large Numbers

“Exploring pi is like exploring the universe.”

—David Chudnovsky

“It's like exploring underwater. You're in

the mud and everything looks the same.”

—Gregory Chudnovsky

by Dr. Barba Patton

Measurement and Fraction Activities: Linking Children’s Literature and Mathematics


Ch i l d re n’s literat u re and mat h e m at i c sare a favorable partnership for today’scurriculum. Simply stated, children’s

picture books and students should be broughttogether to make learning more purposeful inall content areas, not just reading and languagearts. Many textbooks become a dry collectionof information packed with huge amounts offacts and details such that the content matter isunbearable and un-enjoyable for the learners.A c c o rding to Rich ge l s, To m l i n s o n , a n dTunnell (1993), unflattering representations ofincomprehensible textbooks overwhelm manyof today’s classrooms. Teaching and learningdo not need to be this way. Options exist forteachers who are enthusiastic to use supple-mentary text materials. One type of supple-mentary material would be children’s literature.Children’s picture books provide students ofall ages and reading levels with opportunitiesto interact with a human element that tradi-tional textbooks just do not have (Ravitch andFinn, 1987). Children’s literature is not meantto be the only solution to reading and learning,but it can be a source of truncated and mean-ingful text that is not academically insulting.Children’s picture story books are a vehicle form at h e m atics as well as language skills tomature while students speak, listen, read andwrite in the mathematics classroom (Hellwig etal, 2000). This type of literature is teachingmaterial for sparking children’s interest in read-ing while providing the foundation for workingwithin the various content areas.

A c c o rding to Guiett (1999), students arebrowbeaten by the rules and regulations thatmathematics levies upon them. However, thisfeeling of fright need not be cause for alarmbecause many possibilities exist for alleviating

this fear be it real orimagined by the stu-d e n t s. One suchpossibility is thei n clusion of ch i l-dren’s literature intothe mat h e m at i c scontent are a . I no rder for studentsto connect the

abstract concepts of math, they must be ableto engage with this subject in the most naturalof means; their personal lives (Whitin andWhitin, 1996). When mathematical conceptsare included in useful, real-world situations,students can decode these mathematical con-cepts and experience a sense of success withthose concepts (NCTM, 2000). Children’s pic-ture story books are the device to make thispersonal, real-life connection.

Book: What Pete Ate From A-ZAuthor: Maira KlamanPublisher: Puffin, New York

Summary:What Pete Ate is an example of children’s lit-

erature that is both educational and humorous.Pete is the author’s dog, and the bookdescribes the many adventures Pete had as heate various valuable and sentimental belong-ings of his family. The illustrations fully sup-port the text with bright colors and whimsicaldrawings.

Objectives:The students will be able to divide groups of

objects into groups to illustrate ?, ?, ?, 1/3, 2/31/5, 2/5, 3/5 and 4/5.


The students will be able to measure objectswith a standard and/or broken ruler

Pete ate parts of many things. One of thethings that he could have eaten was his master’sruler. With this activity, the students will try touse their ruler after Pete ate a part of it.

Activity 1One of the first activities is to measure Pete’s

bones (treats).Often children only read the last number on

a ruler and do not fully understand that theyare trying to determine the number of unitsthat the object covers. This activity just placesthe paper bones at random places on the floorch a rt andhas the chil-d ren deter-mine thenumber ofunits it cov-ers.

The floor chart is prepared by using the backa piece of vinyl wallpaper approximately 8” by13’. Mark one foot intervals on the chart as 0,1,2,3, … using a permanent marker.

The paper dog bones where prepared by fold-ing large pieces on paper in fourths and thencutting in the shape of a dog bone. One wasabout 2’ and the other was about 3’.

Activity 2Measuring leashes is fun and will provide

practice in measuring for the children. For thisactivity you willneed seve ra ldifferent leash-es (or ones pre-pared from rib-

bon) if the real ones are not available andrulers. The children used both the broken andthe standard ones successfully.

The broken rulers can be prepared by simplyplacing rulers on a copier, make a copy onheavy paper i.e. index paper, cut out then tearoff the end. Be sure to make a number ofrulers and have the children to use a differentruler for each task in order to become moreproficient with the broken ruler.

Activity 3Measuring the collar and

tag can be a challenge forsome children. Of course,c o l l a rs ra n ge in size.Using either real collarsor simple ones made withr i bbon and bu ck l e s, l e tthe ch i l d ren pra c t i c eusing their broken ruleras well as standard onesto gain more accura cywith the task.

Activity 4The children were

able to measure thelength of each pieceof cheese snack tothe nearest 1/4inch.

Activity 5For a different approach to fractions, the stu-

dents were able to count the number of cheesesnack pieces in a bag and then divide them intotwo groups to determine what is one half.

CONTINUED ON PG. 26

by Tad Wanabe

Penn State University

Geometry by Tad


As the new Georgia Performance Standards were introduced, elementary school geome-try was recognized as one area where additional attentions are needed. It is in this areathat teachers need to deepen their understandings of both content and pedagogy.

Furthermore, additional materials may be needed to supplement the existing curricular materialsto help students meet the new standards.

As the GCTM members are well aware, the new GPS were modeled after the Japanese nation-al standards. One features of the Japanese standards in the area of elementary school geometryis the emphasis on students actually constructing geometric figures as a way to study geometry.Here “construction” is not about the traditional high school (unmarked) ruler and compass con-struction, rather, the emphasis is students actually constructing geometric figures by foldingpapers, drawing with rulers, compass and other instruments, working with concrete materials,and otherwise physically manipulating geometric figures. Thus, Japanese textbooks includefamiliar activities such as making colorful designs using colored triangular tiles or colored sticksof various lengths. Activities such as tangram puzzles, tessellation with regular polygons and cut-ting out “snowflakes” from folded papers are also commonly seen. However, Japanese text-books also include some activities that are not commonly seen in the U.S. textbooks. Here arethree such examples.

Making Designs with CompassMaterials: compass, grid papers, colored pencils/markersDirections: Using a compass, draw the designs shown below. Encourage students to think

about where they have placed the needle (i.e., the centers of the circles). Once students com-plete the designs, you can encourage them to make other designs. Students may want to colorthe completed designs.

Making Designs with a Ruler and a ProtractorMaterials: plain papers, ruler, protractorDirections: First draw a 4 cm segment, OA. Then, draw a 2cm segment at Point A forming

45 degree angle. Now draw a 4 cm segment forming 45 degree at the end point of the second


segment. Draw another 2 cm segment at the end point of thisnew segment. Repeat this process to see what design you get.Figure below shows what the end result looks like.

Extension: You can use explore what happens to the design ifyou change the lengths of the segments or the angle measure-ments. For example, will the combination, 3cm - 36°- 6cm - 36°work?

Making Quadrilaterals by Overlapping Quadrilaterals and/or TrianglesMaterials: 'stencils' with rectangles and triangles of various types and shapesDirections: Have students draw different quadrilaterals by overlapping two stencils. Sort the

quadrilaterals, paying attention to what stencils were used to draw each quadrilateral. Here aretwo possible quadrilaterals:

No activity automatically leads to students' understanding, and these activities are not excep-tions. Although these activities might be interesting to us, we must carefully analyze what spe-cific mathematical understandings might be developed through these activities and what priorknowledge may be necessary for such an understanding. Furthermore, teachers may want tocarefully think about what tools they might provide to their students. None of the Japanese ele-mentary mathematics textbooks I have seen incorporate dynamic geometry software. Theseactivities can be adopted into such an environment, but before we do so, we should carefully ana-lyze what we gain/lose by having students engage in these (and other) activities in such an envi-ronment. Textbooks can only provide some suggestions. How lessons actually happen dependson teachers.

by Dr. Chuck Garner Rockdale MagnetSchoolConyers, GA

So You’ve Decided to Sponsor the Math Team...Now What?


SEPT. 27, 2005—So, you’ve decided tosponsor the Math Team. Or maybe yourprincipal asked you to Sponsor the Math

Team. Or possibly some students approachedyou about a Math Team, and you agreed.Regardless how it happened, you are now theMath Team Sponsor! Now What?

What’s AvailableThe first thing you need to know is that the

Math Team “season” is as long as you wish itto be. The first contest is in September, withthe last tournament in April. This can be ayear-round job, if you so choose. You maysign-up for as many contests and tournamentsas you and your students can handle; the moreyou do, the better your team will become.

There are two types of competitions: con-tests (done at your school—you send in theresults) and tournaments (not done at yourschool—you travel to another site).

Contests are a good way to involve many stu-dents at relatively little cost and effort, andprovide a good introduction to many types ofproblemsolving. Recommended contests foryour Math Team include the following.

Math League A series of six six-questiontimed tests given at four-week intervalsthroughout the year, this contest givesstudents a challenge and keeps themcoming back! www.mathleague.com

Mandelbrot A challenging series of fours even-question timed tests, with atimes team-round component ofproof-based questions, this provides anexcellent way to build mathematicalproblem-solving in a team environ-ment. www.mandelbrot.org

AMC The American Mat h e m at i c sCompetition is the premier one-time

event for Math Teams across the coun-t ry. This 25-question timed test isoffered in three levels, one for middleschool (AMC8), one for 9th and 10thgraders (AMC10), and one for 11thand 12th (AMC12). The AMC is thefirst step to the American InvitationalMathematics Exam, and from there onto the United States Mat h e m at i c sOlympiad. www.unl.edu/amc

Tournaments, although more expensive dueto travel, are much more fun! At tournaments,your students get to meet hundreds of otherteenagers that are also interested in mathemat-ics. If your school has an student environmentsuch that liking math is not “cool,” a tourna-ment can be a great place for students to real-ize that they are not alone in liking math, andthat even math can be “cool.”Tournaments usually consist of two rounds: a

timed written test and a speed round calledciphering. The tests range anywhere from anhour to two hours in length and from 25 to 50q u e s t i o n s — u s u a l ly mu l t i p l e - ch o i c e. Th eciphering round usually consists of series ofquestions, given one-at-a-time, that each stu-dent must solve in under two minutes (thenumber of questions and the time limits vary).Some tournaments also offer a team cipheringround, where groups of four students mustwo rk quick ly to solve a pro bl e m .Recommended tournaments for your MathTeam include the following.

UGA This tournament, held in Octoberin Athens, offers a 90-minute, 25-ques-tion multiple-choice test, and both anindividual and team ciphering round.www.math.uga.edu/mathmeet.html


Mercer Held in November on the Maconcampus, this tourament offers a 90-m i nu t e, 45-question mu l t i p l e - ch o i c etest and an individual ciphering round.www.mercer.edu/math/math contest/contest.htm

Georgia Southwestern This is the fir sttournament established in the state (31years ago), and still one of the best!Held in Americus in February, GSWgives students a 90-minute, 40-ques-tion multiplechoice test and a cipheringround in which students work in pairs.www.gsw.edu/~math/mathtourn.html

Armstrong Atlantic Students work a 40-q u e s t i o n , 6 0 - m i nute mu l t i p l e ch o i c etest at the Savannah campus inFebruary, along with a team cipheringround. www.math.armstrong.edu/tournaments

Other universities and high schools through-out the state hold tournaments. This includesthe Junior Va rsity State Mat h e m at i c sTournament, for 10th graders and younger,hosted by Woodward Academy. The GCTMsponsors the Middle School State MathematicsTournament and the invitation-only VarsityS t ate Mat h e m atics To u rn a m e n t . C h e ck theGCTM website for more details.

Getting Students InvolvedHow do you attract students to a Math Team?

Many people suggest bulletin board displays,announcements over the school’s P.A. system,or a newsletter. Those are all great suggestions,and may work for your school, but what I

found works for me is a short 5-minute pres-entation in each math class at my school.Although time-consuming, this approach hasadvantages: the students know who I am, theycan ask me questions, and I can show them aquick problem-solving strategy to give them aidea of what Math Team is like.

Okay—How do you keep them once you’vegot them? Start with the basic staple of life:food! Offer cookies, cokes, chips, fruit, water,and other goodies to attract them to the meet-ings. I gradually phase-out food offerings overthe year so that the ones who only attendmeetings for food eventually drop off.

Offer incentives to keep the students comingback. Tophies and books are common prizesfor high scoring students, but there are moreoptions: Math Team t-shirts, Math Team pen-cils, and a recognition ceremony at the end ofthe year are excellent options. Offer the chancefor someone to earn the leadership title of“Math Team Captain.” My school allows thestudents to earn a Math Team letter for theirletter jackets, and we have Math Team recogni-tion included as part of our school’s end-of-the-year awards ceremony.

But the best way to keep students comingback is to make it fun! Challenge them, teachthem, and laugh with them. If the students seethat you are interested in this as much as theyare, and that you find it fun, they will respondpositively to that. (That’s pretty good advicefor the classroom, too!)

Honor a mentor, a professor, a fellowteacher. Give to GMET.

Georgia Mathematics Education Trust

by William O. Lacefield,

III, Ed.D.

Mercer University

Tift College of

Education, Atlanta

Authentic Assessment: A Powerful Motivator in the Mathematics Classroom


Mo t ivating mat h e-matics students isnot always easy.

Many components of schoolculture may affect motivation-al levels of s t u d e n t s ; t h e s ecomponents include disposi-tions displayed by the teacherand fe l l ow students, t h eadopted curriculum, teachingmethodologies, and the physi-cal env i ronments of cl a s s-rooms. Perhaps, though, oneo f the gre atest infl u e n c e supon student motivation isassessment. The manners inwh i ch teach e rs choose toassess their students' achieve-ment levels and performanceabilities cannot help but shapehow students approach math-ematics learning. If studentsa re fearful of or anxiousabout assessment proceduresin the mathematics classroom,they may fail to reach theirpotential abilities. Because wewant the best for our students,it is crucial that we reflect onhow mathematics students areassessed.

Assessment is one of the sixPrinciples for Sch o o lMathematics embraced by theNational Council of Teachersof Mathematics (2000). Thekeenest teachers are constant-ly assessing their students'u n d e rstanding and ach i eve-

ment, in informal and formalways, through the use of avariety of assessment strate-gies. They observe their stu-dents carefully, ask meaningfulquestions to stimulate criticalthinking, and design a varietyof formative and summativeassessments that provide use-ful information regarding stu-dents' progress and under-standing. Paper and pencilassessments have stood thetest of time, particularly inmathematics, but by incorpo-rating many types of assess-ment into our teaching, weappeal to diffe rent learn i n gstyles and intelligences.

Standardized TestingStandardized testing is a real-

ity that we all must fa c e.Although standard i zed testshave been used for decades,many people have expresseddispleasure with such assess-ments. A variety of reasonsaccount for the disapproval,including the question of howwell standard i zed tests pro-vide info rm ation re l ated toholistic student growth andachievement levels. There arefa c t o rs of m at h e m at i c sachievement that may not beable to be measured in a stricttesting atmosphere, particular-ly when those being tested are

made tofeel thatt h e i rp e r -f o r -m a n c eis of extreme importance totheir future accomplishments.I n d e e d , testing situat i o n soften raise anxiety levels ins t u d e n t s. A dd i t i o n a l ly,because many standard i ze dassessment instruments con-tain test items that ask for dis-crete details, often in a multi-ple choice fo rm at , s t u d e n t sare forced into linear thinkingpatterns. Standardized testings i t u ations may hinder stu-dents' demonstration of criti-cal thinking abilities, and rele-vance to life outside of schoolis often lacking (Schoenfeld,2002).

Authentic Forms of Assessment

Fo rt u n at e ly, s t a n d a rd i ze dtests rep resent but one ofmany opportunities to assessstudent achievement in math-ematics. Many mathematicst e a ch e rs, h aving decided toexplore and implement cre-at ive methods of a s s e s s i n gstudent learning and perform-ance, are finding that authen-tic forms of assessment canlead to desirable re s u l t s.


Th rough authentic assess-ment, students perform real-wo rld tasks in order tod e m o n s t rate meaningfulap p l i c ation of e s s e n t i a lk n ow l e d ge and skills.Mathematics teachers who useauthentic assessment methodsfind that there are crucial dif-ferences between the natureof traditional testing proce-d u res and the nat u re ofauthentic assessments. Whenstudents are allowed todemonstrate or explain whatthey know and what they areable to do in manners thatinterest them and pique theircuriosity, their motivation lev-els tend to increase. Authenticassessment procedures stimu-l ate students' interp re t at i o nskills, creativity levels, mean-ingful reflection, and oral andwritten communication. Notonly do students find enjoy-able opportunities to highlighttheir abilities, but teachers area l l owed to assess students'mathematical competence lev-els using their own reflectivep rocesses and pro fe s s i o n a ljudgment, rather than simplymarking objective test itemscorrect or incorrect (Burke,1999).

One special benefit of theuse of authentic assessmentsis the resulting fl ex i b i l i t y

granted to teachers. Whent e a ch e rs decide to ex p l o reauthentic assessment pro c e-dures, they are presented witha wide variety of assessmentopportunities from which tochoose. In fact, the numberof assessment options avail-able is only limited by teach-ers' own imaginative thinking.The National Council ofTe a ch e rs of M at h e m at i c sadvocates the use of a richvariety of assessment types in

multiple learning situations, invarious group configurations,and at many diffe rent timeperiods within the school year.

Once a mathematics teacherbecomes enthused ab o u tauthentic assessment, he orshe should find that increasedprofessional knowledge leadsto more fruitful studentassessment. Teachers have aresponsibility to re m a i nabreast of the latest researchand practical developments inthe fields of c u rr i c u l u m ,i n s t ru c t i o n , and assessment.In addition to expanding their

levels of professional knowl-edge, teachers should embracethe idea that students' culturalb a ck grounds will affect theassessment measures to whicht h ey most re a d i ly re s p o n d .Certainly, the very nature ofauthentic assessments leads toeasy understanding of whym a ny mat h e m atics teach e rshave become personally deter-mined to use assessment pro-cedures that are most useful totheir students, most re a d i lyhighlight students' strengths,and reveal possible areas inwhich growth may be needed(Popham, 2005).

Representative Types of AuthenticAssessment

While there are many typesof authentic assessment thatare used successfully in math-ematics classrooms, some areused ex t e n s ive ly. Th e s ei n clude teacher observat i o n ,portfolio assessment, and per-formance assessment.

Teacher ObservationM at h e m atics teach e rs are

able to gain a tre m e n d o u samount of information aboustudents' strengths, weakness-es, and developmental levelsthrough focused and mean-ingful observation. As stu-


dents complete assignments,make choices among activities,and interact with other stu-dents, they not only performthe actions invo l ved in thetask at hand, but they alsoreveal rich aspects of theirp e rsonalities and cog n i t i o nlevels. Teacher observationshould never be underestimat-ed as an assessment proce-dure.

E ven info rmal observat i o nof students is valuable, butteacher observation is perhapsmost effective when accompa-nied by systematic documen-tation of what teachers haveseen and heard. Reflectinghuman nat u re, t e a ch e rs arem o re like ly to re m e m b e revents accurately if they arere c o rded in written fo rm .Checklists and anecdotal evi-

dence aretwo effec-tive meth-ods ofd o c u -m e n t i n go b s e r va-tions. In

a ddition to fa c i l i t ating there m e m b rance of details ofhappenings, written documen-tation also allows teachers torevisit mathematics classroomactivities and reflect upon thepossible meanings of student

a c t i o n s, b e h av i o rs, a n dresponses.

Portfolio Assessment A number of mathematics

teachers are finding value inthe use of student portfolios,which are collections of stu-dent work samples that high-light growth in mathematicsu n d e rstanding and ach i eve-ment levels. Often, a portfo-lio, designed to “tell a story”about a student, consists of abox, binder, or crate that con-tains a variety of samples ofstudent perfo rmance thathave been produced in numer-ous contexts across time.Students are often allowed todecorate their portfolio con-tainers according to their owntastes and interests and aregenerally allowed some choicewith regard to what will beplaced in portfolios. Amongitems that might be includedin mathematics portfolios ares t u d e n t - p roduced ex p l a n a-tions of mathematical prob-lem solving, critical thinkingap p l i c at i o n s, p rojects thati n t egrate mat h e m atics withother subjects, reflective jour-n a l s, a u d i o t ap e s, v i d e o t ap e s,DV D s, rep re s e n t ations ofmathematical models, photo-graphs of students usingm a n i p u l at ive s, CDs contain-

ing spreadsheets and othertechnology applications, anda rt wo rk of various types.S u ch eclectic collections ofstudent work can be extreme-ly helpful to mat h e m at i c st e a ch e rs in re c ognizing stu-dent ach i eve m e n t , c og n i t ived eve l o p m e n t , and socialgrowth over time.

While standardized tests mayreveal how students performwhen presented with linearquestions within constra i n t sof time and place, portfolioassessment can help to revealhow students perform whent h ey are unre s t ra i n e d .Naturally, students need mate-rials, encouragement, support,and sufficient time if they areto create documents and otherworks that rev eal importantinformation about themselvesas students. In addition torequiring individual portfolios,some mat h e m atics teach e rsbuild cl a s s room port fo l i o sthat focus not only on individ-ual students but also on theclass as a whole. Classroomportfolios might include someof the same types of artifactscontained in individual port-folios as well as collaborativeefforts that reveal the effec-tiveness of the group dynam-ics within the class. Otheritems that might enhance


cl a s s room port fo-lios include video-t ap e s / DVD's ofclassroom activitiesor dra m atic pro-d u c t i o n s, b ra i n-storming activities,p h o t ograp h s, cl a s s - p ro d u c e db o o k s, and scrapbooks ofmemorabilia. It is hoped thatwhen students know that theirc o n t r i butions to indiv i d u a land classroom portfolios aregoing to be important parts ofcollections, they are motivatedto put forth their best effortsin completing such contribu-tions. Just like anyone else,m at h e m atics students enjoyknowing that their efforts arevalued.

PerformanceAssessment

A major goal of mathemat-ics education should be top rovide students with skillsthat will be useful to themt h roughout their live s. I norder to assess students' abili-ties, many mathematics teach-ers have striven to implementp e r fo rmance assessment.Pe r fo rmance assessmenti n c o rp o rates active invo l ve-ment on the part of the stu-dent. Rather than passivelycompleting items on a multi-ple choice or true/false test,

students who aree n gaging in per-fo rmance assess-ment are given theo p p o rtunity tod e m o n s t r a t ea ch i evement by

a c t u a l ly “doing something.”Examples of what studentsmay be asked to do includeparticipating in a mathemati-cal experiment, describing am at h e m atical pro c e s s, u s i n gt e ch n o l ogy to rep resent am at h e m atical model, a n ddemonstrating the solution ofa mathematics problem withmanipulatives. Performanceassessment allows students tohighlight their talents and theactivities they enjoy, while giv-ing teachers the opportunityto assess learners in meaning-ful contexts.

Optimal Use ofAuthentic Assessment

In order for authentic assess-ment procedures to be mostbeneficial and motivational tot e a ch e rs and students, t h eymust be used thoughtfully andap p ro p r i at e ly. Rather thanemploying the exclusive use ofone type of authentic assess-ment, teachers should ensurethat they include a variety oftypes of assessment in theircl a s s room activ i t i e s. Wh i l e

some students respond well tocertain forms of assessment,others may not. A diversity ofassessments adds richness tothe learning environment andallows the teacher to observestudents in multiple contextsover time.

Certainly, if authentic assess-ments are well planned andare used in creative manners,they can empower students topursue knowledge accordingto their own desires and inter-e s t s. When students areassessed in multiple way sacross authentic contexts andt h roughout wide spans oftime, they begin to recognizepatterns in their mathematicalabilities. Such awareness ofp e rsonal talents and skillshelps students to use theirareas of expertise in thought-ful and fruitful manners.

Authentic Assessmentas a Motivator forTeachers and Students

Authentic forms of assess-ment can be motivating tomathematics teachers in thatthey allow for many differentrepresentations of their stu-dents' achievements. Teachersshould find the classroom amore exciting place for them-selves and for their studentswhen they encourage students


Children’s Literature CONTINUED FROM PG. 17

They were also ableto determine 1/4and 3/4 of a bag aswell as 1/3, 2/3 and1/5, 2/5, 3/5 and4/5. Of course, it was not possible for intro-ducing factors and divisors.

References:Guiett, D. (1999). From alphabet to zebras: Using children’s

l i t e r a t u re in the Mathematics classroom. Ohio MediaSpectrum, 51, 32-36.

Klages, C., Patton, B., & Nagel, A. (2005) Gone to theDogs presentation presented at the Hawaii International

Conference on Education, Jan 6, 2005, Honolulu, HA.Patton, Klages and Nagel, (in press). Measuring Penny.

Texas Mathematics Teacher.Ravitch, D., & Finn, C.E. (1987). What do our 17-year-

olds know?: A report on the first national assessment of historyand literature. New York: Harper and Row.

Richgel, D.J., Tomlinson, C. & Tunnell, M.O. (1993).Comparision of elementary students’ history textbooks and tradebooks. Journal of Educational Research, 86,161-171.

Whitin, D.J. and Whitin, P.E. (1996). Fostering metaphor -ical thinking through Children’s literature. In P.C. Elliott (Ed.),Communication in mathematics: K-12 and beyond,1996 yearbook of the National Council of Teachers ofMathematics (pp.60-65). Reston, VA: National Councilof Teachers of Mathematics.

to demonstrate their achievements in meaning-ful contexts. Furthermore, teachers who makeconsistent use of authentic assessments areunlikely to experience the tedium that oftenaccompanies grading seemingly endless stacksof papers. It stands to reason that teachers areprobably better at teaching when they aree n j oying the processes of i n s t ruction andassessment. Authentic assessment can add tosuch enjoyment.

Authentic assessment also serves to be verymotivating to students, who feel empoweredand successful when they are able to demon-strate what they have learned in manners thatare comfortable, enriching, and engaging for

them. Mathematics stu-dents have a desire toknow that their contribu-tions are appreciated andvalued. Learners want tofeel capable, creative, resourceful, and success-ful. Authentic forms of assessment can makeit easier for mathematics teachers to foster suchfeelings. Perhaps the most important advan-tage of authentic assessment is that it facilitatesplacing the focus on students' strengths and tal-ents, rather than their weaknesses--and there isprecious little in education that could be moremotivational than emphasizing the positive!

GCTM Summer AcademyJune 13-16, 2006

Macon College, Macon, GeorgiaWatch the GCTM Web site for details as to grade bands and topics offered.

Why Do These Work?

Dr. Delores Anderson

Piedmont College


In the previous issue (Fall 2005), it was implied that the author, Mr. E. J. Wilson, didn’t under-stand why his summation works. He advises he understands his summation, and wanted to shareit with us. Being a lay person, he simply needed help with the correct mathematical notation.

The following response was submitted by:Delores Anderson, Part-Time Instructor

School of Education, Piedmont CollegeDemorest, Georgia 30535

The sum of the integers between 2 integers, p. 18

Since F > 0 and L > 0, then. F and L are counting numbers.

Sum = (L squared – F squared + F + L) / 2

= (L squared + L – F squared + F) / 2

= [L(L+ 1) – F(F – 1)] / 2

= (L(L+ 1)) / 2 – (F(F – 1)) / 2

When the famous German mathematician, Carl Friedrich Gauss, was eleven years old, histeacher challenged the class to add the first 100 counting numbers. It did not take young Carllong to find the pattern and present the answer to his teacher 1 + 2 + … + 99 + 100 = 5050. Ingeneral, the sum of the first n counting numbers is (n(n+1)) / 2 .

So, (L(L+ 1)) / 2 is the sum of the counting numbers from 1 to L.

Let n = F – 1. Then, [(F – 1)((F – 1) + 1)] / 2 = ((F – 1)F) / 2, which is the sum of thecounting numbers from 1 to F – 1.

Subtracting the sum of the counting numbers from 1 to (F – 1) from the sum of the countingnumbers from 1 to L leaves the sum of the counting numbers from F to L, which is (L(L+1)) / 2 – (F(F – 1)) / 2 .

GCTM Executive Committe 2006


PresidentDottie Whitlow235 Peachtree StreetNorth Tower, Suite 1700Atlanta, GA [email protected]

Executive DirectorBecky [email protected]

V.P. for Constitution & PolicyDon Slater2406 Woodbridge DriveMarietta, GA [email protected]

SecretaryPatti [email protected]

Council Publications EditorCheryl Hughes50 W Broad StreetFairburn, GA [email protected]

President ElectBarbara [email protected]

2006 Conference ChairCindy [email protected]

V.P. for Awards/HonorsEllice [email protected]

V.P. for Regional ServicesMissy Walker115 Carrington ParkJonesboro, GA [email protected]

TreasurerDan Funsch2819 Peach Orchard RoadAugusta, GA [email protected]

NCTM RepresentativeChristine ThomasCollege of EducationGeorgia State University30 Pryor Street

Atlanta, GA [email protected]

V.P. for CompetitionsDebbie Poss2406 Woodbridge DriveMarietta, GA [email protected]

V.P. for AdvocacyJacquie Allison2262 Overton RoadAugusta, GA [email protected]

WebmasterBryson Payne5489 Highway 19NDahlonega, GA [email protected]

Membership DirectorSusan Craig1011 Stewart AvenueAugusta, GA [email protected]

Regional Representatives

Central East:Amber [email protected]

Central West:Cathy [email protected]

Metro EastLenisera [email protected]

Metro West:(vacant)

South East:Mike [email protected]

South WestSilvana [email protected]

North East:Bradley [email protected]

North West:Jeni [email protected]

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GCTM AnnualConference

Georgia MathConferenceOct. 19 - 21,

2006

GCTMSummerAcademy

Macon CollegeJune 13-16,

2006

NCTM AnnualConference

St. Louis,Missouri

April 26-29,2006

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