A Phi Delta Kappa Professional Development Institute

MAKING ALGEBRA CHILD'S PLAY!® The One-Day Hands-On Equations® Workshop

(for teachers of grades 3 to adult)

~ This one-day workshop empowers teachers to understand and use the visuaJ and kinesthetic HANDS-ON EQUATIONS patented teaching methodology for successfully presenting essential algebraic concepts to students in grades 3 to adult. Through this methodology, practically all upper elementary and middle school students can experience success with basic algebraic concepts, thus raising student mathematical aspirations.

Participants in this workshop will:

• Observe a live demonstration lesson with young children

• Experience the "teacher-as-the-coach" mode of instruction

• Understand the rationale for introducing algebraic concepts early and concretely

• Understand effective pedagogy for use in the HANDS-ON EQUATIONS learning environment

• Use the manipulatives to understand and solve such algebraic equations as: 4x + 2 = 2x + 1 o, 2(x+ 4)+ x = 2x + 10, and 2x + (-x) + 3 = 2(-x)+l5

• Use the manipulatives to set up and solve verbal problems

• Receive an individual set of HANDS-ON EQUATIONS (Complete program for use with one student)

This workshop is recommended for upper elementary teachers, middle and junior high math teachers, math coordinators, school principals, and curriculum directors. The workshop will also be of interest to teachers of the gifted (grades 2-5) and teachers of LD students (grades 4-12). All workshops are conducted by certified instructional staff.

For additional infonnation contact· Sorenson and Associates. (800) 993·6284, http:// www.Borenson.com, or Center for Professional Development and Services. Pbi Delta Kappa, (800) 766-1156.

r---------------------, I A Phi Delta Kappa Professional Development Institute I MAKJNG AlGEBRA CHILD'S PLAY!®: A Hands-On Equations® Workshop

I I I

(lcx:ation) (dale)

I Distri ct/Organization -------------

1 Name _________________ __

I Address --------------------------1 I City ----------------------

1 State/Zip ----------------

1 Work Telephone (_) ------------

1 I Home Telephone (_) ----------­

Workshop lime: 9:00a.m. to 4:00 p.m.

Regismnion Fee: $150 (Includes an individual set of HANDS-ON EQUATIONS and lunch)

Make check(s) payable lo: Phi Della Kappa

Mail to: Phillip Harris Phi Della Kappa P.O. Box 789 Bloomington, IN 47402.(}789

TCM-10/98 L---------------------J

October 15 October 15 October 16 October 16 October 19 October 19 October20 October20 October20 October 21 October21 October21 October22 October22 October22 October 23 October 23 October 23 October26 October26 October26 Oetober27 October 27 October27 October 28 October28 October28 October29 October29 November4 November 4 November 4 November 5 November 5 November 5 November 5 November6 November6 November6 November9 November9 November9 November 10 November 10 November 10 November 12 November 12 November 12 November 12

Hands-On Equations Inventor Dr. Henry Borenson

1998 FALL SCHEDULE

St. Cloud. MN Memphis, TN Fayetteville. AR Topeka, KS Oklahoma City. OK Odessa, TX Tucson, AZ Albany. GA Muskogee. OK Mesa, AZ Watertown. NY Richmond. VA Louisville. K Y Brockton. MA Charlottesville, VA Boise. ID Bangor. ME Harrisonburg, VA Aint.MI Parsippany. NJ Johnstown, PA Duluth, MN Asheville, NC State College, P A Denver. CO Tallahassee. FL Winston-Salem. NC Pueblo. CO St. Louis. MO Charlotte. NC Tulsa, OK El Paso, TX Glen Burnie, MD Greenville. MS Buffalo, NY Madison, WI Pine Bluff. AR Cleveland. OH Waukesha. WI Rome,GA Cedar Rapids. lA Norfolk, VA Wichita. KS Grantville, PA Chattanooga, TN Elk Grove Village, IL Bowling Green. KY Aorence. SC S. Burlington, VT

November 13 November 13 November 13 November 13 November 16 November 16 November 16 November 16 November 17 November 17 November 17 November 17 November 18 November 18 November 18 November 18 November 19 November 19 November 19 November 19 November20 December I December l December I December 2 December 2 December 3 December 3 December 4 December4 December 4 December 4 December 7 December? December 7 December 7 December 8 December9 December 9 December 10 December 10 December 10 December 10 December 11 December II December 11 December II December II

Lexington, KY Farmington Hills, M1 Plattsburgh. NY Portland, OR Ithaca, NY Nashville, TN San Antonio. TX Seattle, WA Binningham, AL Kalamazoo, Ml Plymouth Meeting, P)!

Spokane. WA Montgomery, AL Decatur, IL Billings. MT Columbia. SC Huntsville. AL Hartford. CT Cincinnati. OH Salt Lake City, UT Columbus, OH Savannah. GA Frederick. MD Irving. TX Texarkana, TX Alexandria, VA Spartanburg, SC Houston, TX Sacramento. CA South Bend. IN Anderson, SC Beaumont, TX West Palm Beach. FL Columbus. LN NewBern, NC Austin, TX Carmel, IN Altamonte Springs, FL Columbus, GA Ontario, CA St. Petersburg. FL Atlanta. GA Meridian. MS Oceanside, CA Jacksonville. FL Peoria, IL Jackson. MS Jackson, TN

A new twist on nines Katie Morrison, a student in Joanna Pose's third­grade class at Park Avenue School in Des Moines, Iowa. recently discovered the following twist on the strategy for learning the multiplication facts for nines. She wrote, " I have been multiplying by 9s and I found an easier way. For the multiplication sentence 9 x 7, l think of 9 + 7 = 16. The 'six' in sixteen tells me that a 6 will be in the tens place when I multiply 9 x 7. We tried this with all the nine , My teachers had never thought of that before. r made a chart:

9 X I = Q _; think 9 + l = l Q 9 X 2 = l_: think 9 + 2 = I 1 9 X 3 = 2 _; think 9 + 3 = I 2

9 X 8 = 1_; think 9 + 8 = I 1"

Katie has not studied algebra yet, but the reason her method works can be shown algebraically. When we multiply any one-digit variable by 9, we get the following:

9n=( l0-l)n= lOn - 11 = I On - 11 + J 0 - I 0 = ( l On - l 0) + I 0 - n = 10(11 -1) + ( 10 - II)

When we add this same var iable to 9, we get

9 + n = ( I 0 - l ) + n = I 0 + (11 - I ).

The variable expression in boldface type is the same in both ca es, which shows that her method will work for any one-digit number.

I try to nurture this type of exploration among my students, but am especially pleased by Katie's ability to communicate her conjecture so c learly.

Thanks!

Cheryl L. Arevalo Des Moines Independent

Community School District Des Moines. lA 50309-3382

The following NCTM members, who were selected randomly from the readers of the journal, were gen­erous enough to complete an extensive evaluation of the March 1998 issue of Teachi11g Children Mathe-

OCTOBER 1998

matics. The comments of these professionals will help in the continual improvement of the joumal.

Joan Barle Anna Corbett Beverly Faust Linda C. Fowler Clark E. Gardener Leah Gamer Cathy T. Hale Susan Heidemann Kathy M. Higgins Carolyn Hoyt Marsha Ingrao Garry Katz Sally Livingston Shauna Lund Judy McLeod

Denise P. Masullo Tawana Miller Susan Nowosad Linda Pazos Yvonne Rothe Cherie Schafer Kathy Schanbacher Martha Poole Simmons Helen Ca erez Smith Tamsy Sneed Janet Steele Mary Jane Stewart Peggy Taylor Theodora A. Wieland Judith Beebe Zoeller

Congratulations to Cherie Schafer. who won the $25 gift certificate from NCTM. Thank you to all again for your participation.

In NCTM journals Readers of Teachi11g Children Mathematics might enjoy the following articles and deprutments in the October 1998 issue of Mathematics Teaching in the Middle School:

• "Roll the Dice-an Introduction to Probability," Andrew Freda

• "Cartoon Comer: Working on Study Habits; And the Date Is Approximately ... , .. Julie A. Fisher

• "Reflections on Practice: Exploring How One Problem Contributes to Student Learning," Susan N. Friel

For a complete listing of the contents of this and other NCTM journal , ee the NCTM Web site at www.nctm. org . .A.

The Editorial Pmwl appreciates the imerest and I'OI11es the views of those who take the time to send 11s tlleirt·om­me/1/s. Readers wlto are commellling on articles are encouraged to send copies of their correspondence to the cwthors. Because of space limiwtions, lellet:\ and rejoin­ders from tmtlwrs an• limited to 250 wonts each. Leiters are also edited for style and comem. Please double­space all leiters that are to be considered for p11blicllli<1n.

SEEKING K-12 AUTHORS We publish workbooks that help teachers teach. Write

for our free manuscript guidelines and complete catalog. .............. '. Authors, Fducators Publishing Service, Inc.

31 Smith Place, Cambridge, MA 02138 800-435-7728, x 252 • www.epsbooks.com

TEACHING CHILDREN

VOLUME 5 , NUMBER 2 OCTOBER 1998

ON THE COVER

In "A Problem Worth Revisiting," Linda Schulman and Rebeka Eston explore how to present a classic problem to students, add new ingredients along the way, and return to it without students' remarking, "Leftovers again?" Photo­graph by Sue Collum; all rights reserved. Readers are encouraged to submit color photographs or color slides of children actively involved in exploring mathematical ideas for possible use on the cover, accompanied by a brief explanation of the activity. Please send such submissions to the "Readers' Exchange" department, Teaching Children Mathematics, 1906 Association Drive, Reston, VA 20191-1593. Include a statement from the parents of the children pictured, indicating their permission for NCTM to publish the photograph, and include an appropriate credit line for the photographer.

The mission of the National Council ofTcochcl"\ of Malhcmaucs is 10 provide

'·~•on and leadership '" impro' mg lhe 1cachmg and learnmg of mathematiC\ \0

1ha1 C\'Cry Student i~ ensured an equitable S1111ttfards-bascd malhemai iC\ edu­

cation and every lcncher of mathematics ~~ cn;ured the opponunity 10 grow profe>sionally.

Teaching Children Mathematics •s an ofTic13l JOUrnal of the Natrona! Council of Teachers

of Malhematic>. It 1s a forum for the c'changc of ideas and a ~ourcc of ac1iv1tics and peda­

gogical sLmtegiC\ for malhemniiCS education prc-K-6. II prc>Cnl~ new developments in cur­

nculum. m~lrucuon. learnmg. and teacher cducat1on; interpretS the re>uh' of research: and

m general prov1de' infonna1ion on any aspt.'CI of lhe broad 'pectrum of malhcmauc' educa­

tion appropriate for prcservice and in-service lc;&chcrs. The publ ication~ of lhe Council pre­

>ent a ,·ariel) of newpoml\. The vie";, e~prc~ or unphcd in lh1<, publication. unlc'~

other\\~ noted. ~hould not be mterprelcd 3'> oOicial po"uons of the Counc1l.

66

c 0 N T E N

Readers' Exchange

Standards 2000: Refining Our Efforts Glenda Lappan

A Problem Worth Revisiting Linda Schulman and Rebeka Eston

Exploring Interplanetary Algebra to Understand Earthly Mathematics Robert M. Berkman

Early Childhood Corner Kindergarten Is More Than Counting Kate Kline

Feisty Females: Using Children's Literature with Strong Female Characters Karen Karp, Candy Allen, Linda G. Allen, and Elizabeth Todd Brown

T s

TEACHING CHILDREN MATHEMATICS

Math by the Month Using Mathematics to Tell Stories Stuart J. Murphy

Math Storybooks Virginia Vogel Zanger

Problem Solvers Guess the Weight!

Solution to the "How Much Film?" Problem Judith Olson

Kids + Conjecture = Mathematics Power Danise Cantlon

News from the Net Cut-the-Knot Beth Lazerick

Guide to Advertisers

Reviewing and Viewing Computer Materials, Paul G.

Becher and Douglas H. Clements: New Books, David J. Whitin: Etcetera

Authors Needed for the 2001 NCTM Yearbook

OCTOBER 1998

Journal Staff JOHN A. THORPE, £tecu/llt Dm·ctor HARRY B. TUNIS, Otrt.'clor of Pu/JiicnlltJIIf ANDY REEVES, Dtrecwr of Eduarwl Sen·ice< JOAN ARMISTEAD, Semnr Jaunwl Editor DANIEL H. BREIDENBACH, Jtmnwl &litor KATHLEEN CHAPMAN, Jounwll:.'tlimr ANN M. BUTTERFIELD, Joumal Pmtluctimt Manager NANCY K. GREEN, PAMELA A. HALONEN, Editors BETH HAHN, Keybomrling SpecialiM LYNN S. GATES, Supervt.wr of Rc•view St!n•ices SHEILA J . BARKER, Revu•w Srrvirr.1 A w.<ta/11 ANN E. JENKINS, Re1•iew Srn'k<'< Ani11a111 ROSEMARIE ROLLO, Revie11 Sen ICI'\ A Hi.ftalll

Marketing Staff CYNTHIA C. ROSSO, Dtrutor of Marketing ServiC'es TOM PEARSON, Ad1·emsmgl£fltilmr Mmwger PATTY MARKUSSON, Adw•nlllnJI/&Itibits Manager SANDRA S . BELSLEY, Adl·emlln!i/Exhtbits Assisltlnt ROSA Q, SNIECHOSKI, Markrtmg Anistnnr

NCTM Board of Directors GLENDA LAPPAN, Michig:m SlniC Universily; Presitlem. GAIL BURRILL,

Univcrsi1y of Wisconsin-M:~dhon; PttSI President. JOHN A. THORPE, NCTM; &emrive Director. PATRICIA F. CAMPBELL, Univcrsily of Maryland 31 College Park. ANN CARLYLE, Ellwood Elcmcnlary School. California. LORINO (TERRY) COES Ill, Rocky Hill School. Rhode hland. DWIGHT A. COOLEY, Fore>l Oak Mid­dle School, Texas. LINDA M. GOJAK, Ha" l..cn School Ohio. RITA C. JANES, Newfound Educational As\OCtale\, Nc" foundland. RICHARD KOPAN, Calgary. Albena. STEVEN J. LEINWAND, ConnccltCUI Depan:mcm of EducaltOn. TOM LEWlS, Jane Addams Elemenlllr) School. lllinoi<. KA.REN A. LONGHART, Flal· head High School. Momana. JOHNNY W. LOTT, Universi1y of Momana. JOHN VAN DE WALLE, Virginia Common"callh Umvcrsily.

Editorial Panel DEANN HUINKER, Univcrsily of Wiscon<in-Milw:tukee: Chair ANGELA G IGLIO ANDREWS, Scoll Elcmenlary School, lllinoi> JAMES BARTA, U1ah S1a1e Univcr,ily LILLIAN (NORDIE) DEAL, Chrislma School Districl. Delaware DAVID FUYS, Brooklyn College, New York CHARLES P. GEER, Texa> Tech Unhcr-;tty RICHARD KOPAN, Calgary. Albena: Boord of Directors Unison DANIEL H. BREIDENBACH, NCTM: Staff liaison

Correspondence should be addrc;~ 10 Tt'nclting Children Mnrltenwtu:.r. 1906 A\­<octauon Drive, Res1on. VA 20191-1593. Man115Cnp1S should not e~cced len page\ of 1ex1 and ;hould be typewnnen on nne \Ide onl). double-spaced wilh wide margm' Ftg­urc; 'hould appear on M!parale 'hcc". M.mu-.cripl• ,hould be prepared .. ccordmglo 1he Chicago Manual of St)le and lhc Umrt•d Srureo M<'tric Assc>cimion's Gwd~ ta tit<' U.re of thf Merrie System. No au1hor tdcnttlicnuon should appear on 1he manuscnp1: 1hc JOUrnal U>CS a blind-review process. Five copic' nrc required . Three di;ks for compulcr pro· gmms wi1h more 1han tiflcen Mnlcmcnl' 'hould be furnished. Permission to phoiOcopy malcrial from Teaching Childrtm Mmhemalirs 1\ gramcd 10 persons who wish 10 dt>lrib­ulc iiCms individually (nol in combinn1ion wilh other anicles or works). for cducmionnl purpo>CS. in limited quanliliC>. and free of charge or a1 cost 10 libmrian; who wbh 10 place a limi1cd number of copic..' till rc\cnc; 10 aulhors of scholarly paper:.; and 10 any pan) wishing 10 make one copy for pcr,onnl u\c. Penni;sion mus1 be obw.incd 10 use JOUrnal material for course packet>. commercial works. adverthing. or profc,_.onal-de­velopmenl purpose<>. U!oe> of JOUrnalmmcrinl be)ond !hose outlined above may 'iolalc U.S. copyright law and mu\1 be brought 10 1he anenuon of lhc Nauonal Council of Teache~ of Mathemancs For a complclc \lalcn\Cnl of 'CTM's copyright polic). -.ce the

'CTM Web sile. www.nclm.org. For mfonnauon oo reprints or back iS~>ues. "nlc to lhe Cus1omer Service depanmenl 111 lhc llcadquancl'\ Office. lnfonnation 1< a'atlable from lhe Headquaners Office rcgardtng lhe 1hrec other official j ournals, Marlttmanc• Ttadting in rite Middle School. 1hc Mmltt'fttllltc.t Te11cher, and 1he Jounwl for Resl'nrrh m Mmhemarics Educarion. Special raiC' for >tudenl~. instilutions, bulk >Ub,cribel" .• md rc1ircd members arc ah.o n1ailablc. The index for each volume appear> in 1he M<~y i>>uc. Teaching Children Mmltemtlll('> " indexed in Academic Index, Biography lmlc•. Con­tell/.\ l'nges in Etlucarwn. Current /mit·~ /tl Jrmnwls m Educaricm, £dtwatitm lmlrr. Ex­ccptianol Child Educmian Re;oun·es. i.llerawre Analysis of Micmcompmer l'ull/t('(l· tion<. Metlia Revtew Di11est. and lcmralbltllt fllr Ditlaktik tier Mathemarik.

Teaching Children Mathematic.! (ISSN 1073· 5836) ( I PM 1124463) '' pubh~hcd monthly except June. July, and Augu'l a1 1906 A>\ocialion Drive. Re~10n. VA 20191 -1593 Due, for iJidhidual member,htp 111 lhc Council are S57 (U.S.). whtch mcludc< $18 for a Teaching Cltildrm Matltemtmt·s 'ub,criplion and S4 for a 1en-1<\UC NCTM Nn• r Bulletin >Ub>eriplion. Forctgn ,ub,cnber' add S8 (U.S.) for 1hc lirsl JOUrnal .md Sol (U.S.) for each addillonal JOUrnal Pcnodtcab po~tage paid a1 Herndon. Virgmta, und a1 addiuonal maihng office; I'OST lASTER: Send addres< change;, 10 Tfadt· mg Cltildre11 Marltemtllrcs. 1906 A"nctallon Drive. Res1on. VA 20191-1593. Tclc· phone: (703) 620-9840: order' (800) 235· 7566. fax: (703) 476·2970: fa;>. on demand

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67

76

Two students' recordings with invented code

(a) There are 6 sea creatures in the tide pool. Some are stars and some are hermit crabs. How many sea stars and hermit crabs are in the tide pool?

(b) I see 8 pieces of fruit in a basket. I see some pineapples and some pears. How many pineapples and pears do I see in the basket?

One student's use of invented code and conventional symbols

(a) We planted 8 sunflower seeds in a cup. Some have germinated. Some have not. What might the cups look like now?

quickly. The recordings shown in figure 4 were completed by the same child on the same day. The recording io figure 4(a) shows the child's invented system. Eston asked the child to describe the sys­tem. The child responded, "The number tells how many. These [pointing to a number in a circle] are the seeds that have not germinated, and these [pointing to a number within arcs] have. There are lots of opposites. When you put them together [trac­ing the horizontal lines within the symbol between the numbers], you get a total of eight seeds." Eston asked if he could see another way to record this information. The child's eyes grew wide. "I've got to do thi over," he exclaimed. The recording shown in figure 4(b) is the result. It is interesting to note that even this child, when asked if he had found all the ways, responded, "I think so."

1.\ 7-;.8 2 f6.:: .r lZ-t.i ~G> 6-% J-;. 8 y~L\ ~8 ~

~ , .......... ~~5 ;:;3 '5-t ~--;. g o~g;;.g 1~7-:-9 _.;;..

\~tO==- 8 7·d ... ::- 5I

(b) We planted 8 sunflower seeds in a cup. Some have germinated. Some have not. What might the cups look like now?

Conclusion Young children can sustain interest in significant mathematical questions when those questions relate to their personal and classroom experiences and allow a variety of entry poiJ1ts. Presenting chil­dren with opportunities to share and record find­ings encourages the invention of terms and sym­bols that have meaning. Such inventions can lead to the use of the conventional forms.

Revisiting a problem throughout the school year affords opportunities for children to build important generalizations. The revisits allow mathematical ideas. strategies, and skills to unfold for both teachers and children. Through recording sheets and annotated observations, these problems yield important assess­ment data as changes are noted easily across revisits.

For most children, this type of problem may

TEACHING CHILDREN MATHEMATICS

continue to be explored during the first and second grades as well. Older children may work with larg­er numbers and record their thinking in more sophisticated ways. They may also explain how they know that they have named all solutions. Over the course of three years, only one child at the kindergarten level ended the year by finding all the possible combinations of eight in a systematic sequence and announced proudly that sbe knew that she had found all the ways.

References Atkinson. Sue. "A New Approach to Maths." In Mathematics

with Reason, edited by Sue Atkinson. Portsmouth, N.H.: Heinemann Educational Books, 1992.

Brooks. Jacqueline G .. and Martin G. Brooks. The Case for Constructivist Classrooms. Alexandria. Va.: Association for Supervision and Curriculum Developmem, 1993.

Ginsberg, Herbert P. Children's Arithmetic: How They Learn It and How You Teach lt. Austin: ProEd, 1989.

Hughes. Martin. Children and Number: Difficulties in Leanring Mathematics. Oxford: Basil Blackwell , 1986.

Mills. Heidi, Timothy O'Keefe. and David Whirin. Mathemat­ics in the Making. Portsmouth. N.H.: Heinemann Educa­tional Books, 1996.

Nelson, Doyal. and Joan Worth. How to Choose and Create Good Problems for Prima1y Children. Reston, Va.: Nation­al Council of Teachers of Mathematics, 1983.

Polacco, Patricia. Jusr Plain Fancy. New York: Bantam Books, 1990. Ward, Leila. I Am Eyes: Ni macho. New York: Greenwi1Jow

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Robert M. Berkman

Ro/Jerr Berkman, Tanglukitm@aol.com. teaches mathematics and science at the Salk School for Sciences. New York, NY 10003, a public school. His imeresrs include concept develop­me/If. reclmology in education, and school-based managemem.

78 TEACHING CHILDREN MATHEMATICS

t the beginning of class with my

fifth graders one day, l draw a dia-

(fig. I ) on the chalkboard. I

tell a story about a visitor from another galaxy who

spots a group of Earthlings sitting in an unusual

formation in a field somewhere on Earth.

Excited by her discovery, I continue, the space alien takes a picture and reports the following to her cohorts waiting on the mother ship, which is parked at a meter in the nearby Andromeda galaxy: "1Uxo*!3 ... )()(4 JU*x ¢oo¢oo¢!" Any­body _who has studied foreign languages in high school knows that this exclamation roughly trans­lates into "Yo, dudes, I just caught an excellent look at these Earth folks sitting in a field, and boy, are they funny looking!" The official intergalactic census taker looks at the picture and writes down a number. Did he write "13," "7," or "23"?

Earthlings look at this picture and scratch their heads. If they counted using their fingers, they would find 13; and if they used the formation , 7. But why should "23" be considered a possibility? When prompted for reasons that the answer might be 23, children can be very creative. One child explained that the space alien added 13 to the number 7 to make 20, then added another 3 because it was part of the 13, which became 23.

Perhaps a closer look at the visitor who made this observation would help, I suggest. As luck would have it, I happen to have a photograph that she left behind (fig. 2).

Still confused? A hint is that like us Earthlings, our space travelers first organized their number systems by using their fingers. This hint immedi­ately leads students to jump out of their seats. ''She counts '23' because there is a two on one hand and a three on the other !'' one student explains. Everybody nods their heads in agree­ment. l counter with a question: "So, if I took one of these people out of the field, how many would the alien count now?" The students scratch their heads. I fo llow up with another question: "Why do we count the way we do?" One child suggests grouping the people in twos and threes but cannot explain how doing so would result in the number "23." 1 suggest that they look at how their own fingers are organized. The students hold their fin­gers in front of them, wiggling them back and forth. A minute or two later, the lights go on.

One of my students explains that since this alien has three fingers on one hand and two on the

OCTOBER 1998

What the space visitor saw

The five-fingered space traveler

Earthly and extraterrestrial counting systems

How the Earthling counts:

1 group of ten and 31eft over= 13

How the extraterrestrial counts:

2 groups of five and 3 left over = "23"

79

80

other, they probably count in groups of five. instead of ten like us. This extraterre trial groups her people differently (fig. 3).

The student's explanation provokes a big "aha;· and when I suggest that they can try this system on their parents, and perhaps win some money in the process, pencils and notebooks sud­denly appear out of nowhere. The students are hooked, and J am ready to dig deeper and stretch their thinking further.

Suppose that the traveler shown in figure 4 gazed at the . arne field? How would he group the people he sees? How many ''leftovers .. would there be?

"If you said '15,' " 1 explain to my students, "you arc well on your way to a lucrative career with the Galaxywide Census Service. Of course, since you have lO report your findings to hundreds of different civilizations, you should be prepared to be able to supervise your census takers. For example, suppo e that the previously mentioned space traveler with eight fingers was employed by you and . aid that he had counted '26' aliens. How would you explain to another Earthling how many aliens there are?" My students write out their explanations in their notebooks and share their ideas with one another.

Russel writes. "The alien in the problem saw '26' aliens. If you want to find out what the real an wer is. I have to translate it. 2 is the first digit, so there are 2 groups of something. 2 groups of eight is ixteen, and ixteen plus six i twenty­two." As I look over his explanation, l a k him to tell us about the 6 that was added to the 16. He adds to his explanation, "l got the '6' from the second digit of '26.' These are the people who couldn't make up another group, so they're left over."

A different extraterrestrial counting system

When I am sati lied that the group understands these types of problems, I give them an assign­ment. "Make up your own alien braintea er, and put the answer to it on the back of your paper." We talk about things that they should consider when creating their teaser. For example, does the alien always have to count people? Does the alien have to have five or eight tingers? What number of lin­gers would be very difficult for counting large groups? The students start their problem!. in class, doing a rough draft of what the alien will look like, the number of fingers on its hands, and the answers. T look over their teasers before the s tu­dents go home to finish them for homework.

The following day. I have the children switch problems with one another. T hey work on the problems and discuss them. I then pop the ques­tion, Is there some way of solving this problem so that it works with all kinds of aliens? Some stu­dents are confused until I clarify the question, Is there some sort of rule or recipe for changing alien number to Earth numbers? After working on several of these problems, the students recog­nize a method for solving them; the goal then i to have them put it into words. This act of general­ization is the foundation for algebraic thinking.

The children work in groups to discuss their ideas, each member having a job to perform. l walk around and monitor the groups' progress. checking to see that they have explained their ideas clearly. A typical explanation by a tudent reporter ounds like this: "You would count the number of fingers and multiply it by the first digit, and then add the remainder which didn't make a group." At the same time, I ask students whether they can abbreviate their explanations. I "seed'' each group with examples of other abbre­viations that we use, like "x" for "multiplied by" and '·+." which can mean "plus" or "added to.'' The formula "Groups Digit x Fingers+ Remain­der= Human Number" is a typical response. As the group arrive at thi level of symbolism. l wonder aloud whether it i possible to abbreviate the words. I give examples like .. Mr." for Mister, and "em" for "centimeters." The groups come up with formulas like "GD x F + R =£#,"where GD is Groups Digit, F is the number of fingers on an alien's hands, R is the "remainder" of people not put in a group. and £#is the "Earth number." As I monitor each group's progress, I ask whether other people will know what these symbols mean, which inspires students to create a key to the dif­ferent letters. As the class concludes. we look at the explanation. created by each group and com­pare the notations they used, noting which ones are clear and which ones may confuse the reader.

The next day, a new problem (fig. 5) appears.

TEACHING CHILDREN MATHEMATICS

one that is related to the previous situation but requires children to manipulate their thinking into a new context. This problem was easily under­stood by the children: they recognized that for this group to be called "30," the fifteen people must be divided into three equal groups, with nothing left over. Therefore, the alien must have five fingers. When I told them that another alien had reported " 17 ," they solved the problem by subtracting the seven "remainders" and saw that the eight left over was the "group" of 1. This vari­ation set them up for a more difficult problem that combines the first two problems: Suppose the alien reported "23" people? How many fingers did she have on her hands?

I bad the children work together in groups on this problem, emphasizing that I would expect a clear _explanation of how they figured it out dur­ing our wrapping-up session at the conclusion of the lesson. One group stated that "we took away the '3,' because that was the leftover; and we knew that the ones that were left over were in two groups, so we divided it by '2' and we knew the alien had six fingers. We knew we were right because if we had two groups of six, it would be twelve, plus the three would be fifteen, and that's how many people there were." Other groups employed different strategies. One group divided the number of people by two and got seven with a remainder of one, which they knew was inconect. They knew that the answer could not be five fin­gers because that amount would make three groups of five with no remainder, so they tried six and checked to see if it would yield the correct answer. Theirs was a guess-and-check strategy in which incorrect answers provided clues about the correct one.

But Is It Algebra? I designed this activity several years ago to intro­duce fifth graders to the concept of base systems, but while analyzing ways of solving the problems with my students, I realized that a lot of informal algebraic thinking is taking place. When I pointed out this notion to my students, they were quite horrified: "How can this be algebra?" one young lady challenged. "Isn ' t algebra supposed to use a lot of letters and be really hard? That's what my brother told me."

Introductory concepts in algebra include the study of unknowns, and in the context of taking a census of space aliens, these unknowns, or vari­ables, are very "friendly." When children look for generalizable rules and express these relation­ships using symbols to represent operations and variables, they are creating their own kind of alge-

OCTOBER 1998

A variation of the problem

Draw a picture of the space alien who reported that the group of people shown here was "30:'

bra. However, just as children are encow·aged to invent their own spellings in the "whole lan­guage" approach to writing, I encourage them to invent their own symbols. My goal is not to have children learn the formal notation of algebra but rather to understand that their equations are a sys­tem of communication that others can use for a practical purpose. My goal is for these children to make the connections when they see the same relationship expressed in a more conventional way further down the road.

This informal algebra is not limited to creating formulas for changing alien numbers to Earth numbers; it can also be applied to setting up and solving equations. For example. when the chil­dren are told that the alien numeral "23" is equal to Earth number fifteen, they can reason that two "full groups" plus three " leftovers" equals fifteen, or, in the more conventional algebraic notation, 2x + 3 = 15. Many students realize that if two groups plus three is fifteen, then two groups alone is twelve, and one group is six; therefore, the alien had six fingers. The connection to solving alge­braic equations should be clear. Students are incorporating "inverse operations" to solve this problem; that is, when the equation says to add three, they need to subtract three to "undo" it; when they divide the remaining pieces by two, they are ·'undoing" the multiplication. Again, algebra is not being taught directly, but the instruction incorporates algebraic thinking as a vehicle for problem solving.

Starting Out Simple Figure 6 gives examples of three kinds of "alien" problems. The example that opens this article is a complex problem that is difficult for children to solve on their own, but it is a good motivational tool because it presents children with a mystery to be solved. Once they "see" the way space visitors group things, an "aha" expe-

81

82

Examples of alien problems

Type A: The Earth number is known, and the alien's hand is known. Find the alien number.

. ~ . • • • • Suppose a four-fingered alien saw this

illustration of these planets; what number would she write down? ("22")

Type B: The alien number is known, and the number of fingers on the alien's hand is

known. Find out the Earth number .

A space traveler with the hands shown here wrote down "42:' How many objects did he

actually see? (26)

Type C: The Earth number is known, and the alien number is known. Find how many fingers are on the alien's hand.

A space alien saw this group of stars and wrote down "20." How many fingers did this alien have

on his hand? (8)

rience usually takes place. Many of my students take the problem home and present it to their parents as a brain teaser. From there, we learn how to count in a variety of " planetary" number systems. One example involves making a set of " handprints" of three different aliens and match­ing the number of items each would count given the same number of objects (see fig. 7). The hands shou ld be drawn large enough for children to count the fingers. Student should al o be given fifteen ·'asteroid counters" that can be !-.pread out and matched up with the hands. Some Mudents may want to place the counters on top of the hand to find the solution . whereas other may want to u e loops of trings to show the groupings. Through thi activity the children develop an informal, intuitive approach to solv­ing algebraic equations, which can be written out in sentences using unknowns to represent the missing information. Thus, to match ·'23," they need to interpret it to mean "2 sets of alien hands + 3 = fifteen, .. whereas "17'' means '·I set of alien hands plus 7 more= fifteen."

A variation of this problem involves figuring

out how many objects were counted by comparing the alien handprint with the alien number. The same materials shown in figure 7 can be used, but the correct numbers are matched with the hand­prints and the students are asked how many aster­oids or spaceships each alien counted. Suppose that an eight-fingered space creature wrote down ''24." What number would she be wri ting in earth numbers?

The most complex problem involves compar­ing alien numbers with Earth numbers and figur­ing out the appearance of the alien hands. The opening problem in this article is an example of thi kind of problem. and children will struggle to olve problems like it. It is important. therefore.

that the students have many opportunities to work with the first two type of problems before mov­ing on to the third type. Again, using the concrete materials is important for students to develop and master the concepts involved. lt is important to support the children as they develop and master their strategies and share them with others. Alge­bra is more than solving equations correctly; it is understanding the various " attack'' methods.

TEACHING CHILDREN MATHEMATICS

An activity that combines all three types of problems

Handprint "A" Handprint "8" Handprint "C"

Fifteen asteroids are circling a planet. Match the numerals each alien wrote out below with their handprints above.

·~· ·c· ·;· •••••

G 0 EJ (8)

I give the children ample time to master the three basic types of problems: converting from Earth numbers to alien numbers, converting from alien numbers to Earth numbers, and figuring out the appearance of a space alien after comparing an Earth number with an alien number. After that, I throw it open to them. For example, l have stu­dents make up their own "alien puzzlers" that use these types of problems. In one instance, a student wrote out the question, showed the numbers, illustrated the problem, and hid the alien hands under a flap of paper. Another child made an over­Jay from a piece of transparent film, which could be placed over the picture to show the groupings that the alien would make as she counted the number of relatives in a photograph. This activity taps into children's creativity and Jove of the fantastic.

As stated previously, my students never believe me when I tell them that what they are doing involves algebra. They are so wrapped up in envi­sioning space creatures with different types of hands zooming around the universe counting up objects that it never occurs to them that what they are doing involves such serious mathematical concepts as base systems, inverse operations, and working with variables. Perhaps the moral of this story is that by Laking our students beyond the stars, we can help them better understand mathe­matics here on Earth. A

OCTOBER 1998

(C) (A)

The A IMS Education Foundation is now

accepting appl ications to host A Week with AIMS

workshops for 1999.

83

Kindergarten Is More Than Counting

T eachers of primary-grade children realize the importance of helping their students develop an understanding of number rela­

tionships. It is important to encourage the kind of thinking that allows children to readily decompose numbers into parts and know how to put parts together to make a whole. This thinking sets the foundation for working with larger numbers, using reasoning to approach computation, and devel.op­ing sophisticated mental strategies. Parker (1998) describes the importance of building what she calls "fluency with small numbers." She defines fluency as being able to take apart and put back together numbers without even thinking, or with automatic­ity, and believes that before the third grade, chil­dren should be fluent with numbers to I 0.

So how does fluency develop? The NCTM's curriculum standards for grades K-4 describe the need to develop number relationships by compos­ing and decomposing sets of objects (1989, 39). Mathematics Their Way (Baratta-Lorton 1976), one of the most popular K- 2 mathematics programs, conta.ins a multitude of activities focusing on parts and wholes with manipulatives. Three activities among the many with which readers may be famil­

iar- The Hand Game, Lift the Bowl, and Peek through the Wall-encourage children to think about number combinations and how smaller parts can be put together to make the whole. In Peek through the Wall ,

Kare Kline. kare.kline@wmich.edu, teaches mathematics education courses at West em Michi­gan Universiry, Kalamazoo, MI 49008. She is imeresred in investigating ways to develop young children's understanding of number.

Tlris departmem addresses the early childhood reacher's need to support young children's emerging mathematics understandings and skills in pre-K through second-grade classrooms in a com ext rhar conforms with currem knowledge about tire way that young children/eam math­ematics. Readers are encouraged to submit manuscripts ro editors Kate Kline, Departmem of Mathematics. Western Michigan Universi~y. Kalamazoo. M/49008. or Sally Rober ts, College of Education, Wayne State Universiry. Detroil. M/48202.

for example, children use a see-through wall to divide a set of objects into two parts, such as three and one for a set of four, and respond out loud, "three and one." This task is repeated for the remaining combinations for four (two and two, four and zero).

The philosophy underlying these activities is that with repeated exposure, children will eventual­ly begin to remember these combinations. How­ever, in my experiences teaching primary-grade children, I found that many students did not remember these combinations. As long as the man.ipulatives were available, they were more like­ly simply to count by ones to figure out the proper response. Even in the Peek through the Wall activ­ity, many children continued to count those three objects on the other side of the wall at the end of the year! Many children continued to rely on counting by ones as their primary strategy in ftrst or second grade and even beyond.

I began, then, to ask myself some serious ques­tions. Why do some children seem to get stuck in counting by ones and never develop fluency with numbers? Were all the concrete objects I was using to promote fluency actually encouraging children to count by ones? How could I help all my students become more fluent?

As I began talking to colleagues and looking for information to answer these questions, 1 encoun­tered many stimulating ideas. [ found a wealth of literature on developing young children's mental imagery with numbers (Baroody and Standifer 1993; Payne and Huinker 1993; Van de Walle 1990; Wirtz 1980). The essential element of these recommendations was to use specially patterned arrangements of numbers, such as dot patterns on dice and dominoes and ten-frames, to encourage children to develop mental imagery of those num­bers (see fig. 1).

The recommendation was made that these mod­els be used in a quick-image format. To do so, flash

TEACHING CHILDREN MATHEMATICS

Examples of dot patterns and ten-frame models

• •

• • •

• •

One of the dot patterns for 7

one particular image on the overhead projector screen for three seconds and then hide it from view. Ask s.tudents to tell how many dots were shown and to describe what they saw. This tactic will encourage them to think about the parts of the images. You may want to flash the image a second time for three seconds, then hide it again, to give students a chance to adapt their visual images. As students become better at recognizing the patterns instantaneously, second looks are rarely required. It is important to flash the image f or only three sec­onds. lf you show it for too long, students will work from the picture rather than with a mental image, and i f you show it too briefly, students will not have time to form a mental image.

Using Quick Images These activi ties can serve well as five-to-ten-minute warm-up activities throughout the year. ln addition, a variety of ways can be used to al ter the quick­image activities, making i t possible to present new ideas throughout the year as well. For example, cllil­dren may draw picwres of the quick images rather than describe them orally, hold up a numeral card to represent the number they saw, or show on their fin­gers the number they saw. Showing responses on fingers leads to interesting discussions when you ask two or tJu·ee chil dren to compare their representa­tions. For a quick i mage for 6, some children will hold up three fingers on each hand; some, four on one hand and two on the other; and sri ll others, all five fingers on one hand and one on the other.

Fluency with 8 Encouraging children to find different ways to see and describe the quick images is ex tremely valu­able. I t creates an environment that promotes flex­ible thinking and sharing of ideas. Figure 2 shows a sampl.e conversation that I had with students about the ten-frame model for 8. I often asked questions such as " How many would there be i f l

OCTOBER 1998

Ten-frame for 6 Ten-frames are filled from left to right, and the top row must be filled before filling the bottom row.

Classroom dialogue about the ten-frame model for 8

Students discuss how many dots they saw altogether and how they saw them, after the following was displayed on the overhead projector for three seconds and then hidden from view.

• • • •

• • • Teacher: How many dots did you see altogether? Ping: Eight.

•

Teacher: What helped you remember that there were eight dots? Ping: Well, I saw that the whole top row was filled, and that makes five. And there were three on the bottom row, and five and three make eight. Teacher: Did anyone see it in a different way? Mohammed: I saw two empty boxes in the bottom row, and ten take away two is eight. Teacher: How else did others know that there were eight? Aeysha: I saw a [group of] six. I thought of the three dots on top and the three dots on the bottom row as six. And then there were two more on the top, and six and two is eight.

(Display the ten-frame again, ask the following question, wait three sec­onds, and hide from view.]

Teacher: If I take away two dots from the ten-frame, how many will be left? Celia: Six. Teacher. Did anyone get a different number? [Pause] All right, so, Celia, how did you think about this problem? Celia: Well, if you take two away from the top row, that would leave three on the top and three on the bottom. And three and three is six. Teacher: Who else would like to explain their thinking? Cherise: I knew it was six left, because three take away two is one. Teacher: And how did you use that information to find the answer? Cherise: Well, if you have one left, just add it to the five on top, and five and one makes six!

85

----------

---------

took two away (or added two)?" to encourage thinking based on the identified parts of a number. Cherise's response was typical of students who had developed tluency with a particular number.

As l attempted to explain Cherise's reasoning in numerical form to another teacher, I wrote the following:

8=5+3 8 - 2=5+3-2 8 - 2 = 5 + (3 - 2) 8 - 2=5+1 8 - 2=6

We both realized the complexity of the reasoning that Cherise displayed and the flexibil ity with which this would allow her to work with numbers. I do not claim that Cherise was thinking in this numerical way. However, she was able to decom­pose the 8 into two pruts-5 and 3-and operate on only one of those parts. This flex ibility in working with numbers will provide a strong foundation for mental mathematics with larger numbers and for developing computation strategies.

Numbers between 10 and 20 Although I would not expect kindergarten students to be fluent with numbers in the teens, it is impor­tant to help them develop visual imagery for these numbers using the ten-frame. I found that the most effective model was two ten-frames in a horizontal orientation with the ones in the top frame and the ten in the bottom frame (see fig. 3). The reader may wonder why it would not be more helpful to use a vertical orientation for the ten-frames, with a full ten-frame on the left and the ones on the right, since this configuration matches the way we write numbers in the teens. For example, we would write a I below the full ten-frame to represent one 10 and then write a 6 below the other ten-frame to repre-

Horizontal ten-frame model for 16; 6 and 10 make 16.

• • • • • • • • • • • • • • • •

86

sent six 1 's. Although this approach may make sense to adults who already know how to write the symbols to represent the numbers in the teens, my kindergarten students did not find it very useful.

The names for the numbers in the teens fo!Jow a pattern in that we say the ones fu·st and then we say "teen" to represent 10 (except for the number names for II and 12). J found that it was much more helpful to work on the names of the numbers first before working on their symbols. And show­ing the ones in the first ten-frame followed by the ten, as in figure 3, matches more closely with our names for those numbers.

Once my students knew the names for the teen numbers and easily recognized them on the ten­frame model, they were much more flexible in using different orientations for the ten-frames. They were able to recognize that 10 and 7 make 17 and that viewing the frames in a ve11ical orientation does not change the number.

Assessment Ideas ln addition to observing students during quick­image sessions, 1 also use two tasks to assess their fluency with numbers. One is to connt out objects into your hand, hide some, and leave the rest show­ing. When working on fluency for 5, for example, count out five objects into your hand. Ask the child, "How many do I have?" lf the child is able to respond, "Five," then you may proceed. Take two away and leave three visible in your hand. Ask, "I took some away. If I have three showing, how many did I take away?" Repeat for other combina­tions for five. If a child is able to respond correctly and with ease for all combinations for a pa1ticular number, then the child is fluent with that number.

A second assessment task is to use story problems to identify those students who are transferring what they know about numbers to real situations. One example is to provide a list of toys at various prices (see fig. 4). Several story problems can be created from this list. such as "Imagine you have ten cents to shop in this toy store. What would you buy?" Alex's exceptional response to this problem is shown in fig­ure 5. I knew from his response that he had a good understanding of how to combine numbers to make 10. He ftrst drew pictures of all toys possible then checked those that he had decided to buy. Interest­ingly, he chose to buy two bowling balls, a car, and a truck and added 1 to the cost of the car (I + 3) and to the cost of the truck ( l + 5) to get 4 and 6. I asked him if he had spent aU of his money, and he replied, "Yes, because 1 know six and four make ten."

In the years I taught kindergartners and worked with other teachers using quick-image activities, I saw dramatic increases in students' fluency with

TEACHING CHILDREN MATHEMATICS

numbers to J 0. Previously, very few students would have been successful on the tasks desctibed here, especially the hand-hiding task. But after repeated experiences with quick-image activities throughout the year, the majority of students were confident in their ability to think about numbers and could put them together and take them apart with flexibility and ease.

Concluding Comments ft is important to understand that quick-image activities are nor a replacement for concrete activi­ties. Opportunities to work with concrete manipu­latives are essential as children develop fluency with numbers. However, equally important are quick-image activities to encourage children to move beyond a counting-by-ones strategy and think about number combinations.

Fortunately, some new curricula, including reform curricula based on the NCTM's Standards, use dot-pattern and ten-frame images to develop number sense. However, the only program that uses these models in a quick-image format on a continual basis is !nvestigations in Numbet; Data, and Space (Kliman and Russell 1998), which uses them throughout first and second grade as warm-up routines to extended investigations. I invite readers to use some of the quick-image activities described in that program or in this article and to share with colleagues the increased fluency that students will undoubtedly display.

References Baratta-Lonon. Mary. Mathemmics Their Way. New York:

Addison-Wesley Publishing Co .. 1976. Baroody. Arthur, and Dorothy Standifer. "Addition and Subtrac­

tion in the Primary Grades." In Research Ideas for the Classroom: Early Childhood Mathematics, edited by Robert Jensen. New York: Macmillan, 1993.

Kliman. Marlene. and Susan Jo Russell. ·'Building Number Sense." Ln Investigations in Numbet: Data, and Space, Grade /.White Plains, N.Y.: Dale Seymour Publications, 1998.

Parker, Ruth. "Building Pacility wi th Numbers: Grade Level Appropriate Expectmions for Mental Computation." Hand­out at community session, Watertown, Mass., 1998.

Payne, Joseph, and DeAnn Huinker. "Early Numbers and Numeration." In Research Ideas for the Classroom: Early Childhood Mathematics, edited by Robert Jensen. New York, Macmillan, 1993.

Van de Walle, John. "Concepts of Number." In Mathematics for the Young Child. edited by Joseph Payne. Reston, Va.: National Council of Teachers of Mathematics. 1990.

Van de Walle, John, and Karen Bowman Watkins. "Early Devel­opment of Number Sense." ln Research Ideas for the Class­room: Early Childhood Mathematics. edited by Robert Jensen. New York: Macmillan, 1993.

Wirtz. Robert. New Beginnings. A Guide to the Think, Talk, Read Math Cemer for Beginners. Monterey. Calif.: Curricu­lum Development Associates. 1980. A

OCTOBER 1998

List of toys and prices for story problems

Alex's solution to toy-store word problem

What would you buy with ten cents?

87

I

I

Using Children's terature witl

G iris enter school more mathematics

ready than boys. By the time they

graduate from high school, however,

females have been outdistanced by males in the

number of higher-level mathematics courses taken

and in the results of crucial tests, such as the math-

ematics portion of the Scholastic Achievement

Test (American Association of University Women

I 99 1 ). They are also much less likely to pursue

majors and careers that relate to mathematics.

Why? What happens to girls as they age from

eight to sixteen that puts them at a disadvantage in

mathematics?

The change in girls' interest and confidence in mathematics during this developmental period is part of a larger phenomenon. As girls progress through the late el.ementary school years and enter middle school, they are frequently seen as losing

Karen Karp, Candy Allen, Linda G. Allen, and Elizabeth Todd Brown

their daring nature (Brown and Gilligan 1992). They change from outspoken ten­year-aids into young adolescents who respond, " I don' t know," to most ques­tions. Specifically, researchers Ouellette

Karen Karp. kskarpOI@ulkyvm.louisville.edu, reaches marhemarics educarion ar rhe Univer­siry of Louisville. Louisville, KY 40292. Her professional imeresrs focus 011 equiry issues in marhemarics educario11. Ca11dy Alle11. cca33 / @aol.com. and Linda Alle11 teach eight- and nine-year-old childre11 ar Goshen Elememary School, Goshen. KY 40026. They share a11 imer· est ;, connecti11g children's lirera111re to mtllhematics and scie11ce. Todd Brown. etb50@aol.com. reaches fourrh- a11d fifth-grade s111de111s ar Wheeler Elementary School. Louis1•ille, KY 40291. Her pro[essio11al imeresrs are ;, children's literature, particularly books wirh strong female proragonists.

88

and Pacelli ( 1983) report that adolescent females often lack a "hardy personality," which they define as the propensity to look forward to changes and challenges, feel in control of one's life, be respon­sible for one's owD actions, aDd survive unfavor­able conditions.

Interestingly, Ouelle tte and Pacelli also connect the characteristics of a hardy personality to suc­cessful problem-solving skills. Al­though the problem-solving skills that they discuss are not exclusively math­ematical, their findings could connect with young females' loss of interest in mathematics at this same age. When female students lack confi­dence in their ability to prevail in novel or chall.enging situations, then approaching mathematics problems with risk-takj ng behaviors seems unLikely. Mathematics educators suggest that the kinds of mathematical problem-solving strategies that we teach should directly relate to solving the problems that students face in the real world. The reverse seems true, as well. Girls who do not develop hardy inner personalities, or what we describe as the characteristics of a "feisty female," may not be prepared to tackle problem-solving situa-tions. How can we build this strength and nur-ture these students to mature mathematically?

One strategy is to present girls with models of hardy female personalities through examples of young problem solvers found in children's literature. These characters can act as springboards to math­ematics lessons as teachers link these

TEACHING CHILDREN MATHEMATICS

• •

5trong Female Characters "feisty females'' to mathematical actiVIIIC . . Chil­dren's literature is a way to supply a powerful con­text in which to build mathematical tasks and is a strong inlluence in the development of children's perceptions about their world. Through books we can find young and feisty female who face adven­ture bravely; make hard decisions; solve problems

OCTOBER 1998

of their own and of others; and u e their connection. to. and relationships with, other people as ways to both develop a sense of spirit and face new chal­lenges. Walkerdi nc states that stories are "one of the powerful ways in which constructions of gender are authorized and regulated" ( 1994. 128). Gilbert notes that "through constant repetition and layering, story

89