Recent  research supports Vygotsky\u27s  zone of proximal development  theory;  children receiving peer assistance can stretch their learning beyond  their individual accomplishment. A study of a multiage classroom  revealed three strategies used by children working together to solve  math problems--modeling, tutoring, and pairing/sharing activities. The  multiage setting invites collaboration and sharing, while helping  students meet mathematics learning standards. (MLH

Dever, Martha Taylor

Mansano, Noel N.

Zila, Randy

English

DigitalCommons@USU

MULTIAGE CLASSROOMS: A New Way to Learn M'ath A recent study reveals how children work together to solve math problems in multiage groups. . MARTHA TAYLOR DEVER , RANDY ZILA , AND NOEL N. MANZANO " ow did you co unt those blex:ks so f~l?" seven-year-old Sara asked tcn-year -o ld Dan as she examined the design they had constru..:ted together. Each of the four groups of blocks had one red. onc white. and one blue block arranged in a symmetrical pattern . - I didn ', coun l," Dan responded. ,, ( j ust saw fouli groups of three. and I knew that was twelve," This is nOI a scenario from iJ. onc-room schoolhouse at the lurn of the century. Sara and Dan are learning in a contempo-rary multiage classroom where children of different ages often work together. Whalls a Multiago CllSSIGOm? A mul tiage classroom contains children who are at least one year apan in age. and the classroom structure is theoreticall y linked to a concept put fo rward by VygOlsky (l978). who suggests that there are two developmental levfls at which children learn. At one level. children can do things on thei r own: at the Olher leve l. Manha Taylor [kver is a.~sis tant professor ur early I:hildhood edocalioo al Ulah Slale Universily In Logan. Randy Zlta IS pnnclpal of Marx Blai r Elemenlary School In I ovelaod. Colorddo. Noel N. Manzano is a graduale .)tudcnl Oil the Unlversny of Nonhcm ColorJdo In GrL'f!ley . 22 they need guidance. Between these levels is what Vygotsky ca ll s the "zone o f prox-ima l development." where children who receive assistance can stretch their learn-ing beyond what they are able to do alone. Recent research appears 10 suppo r! Vygo tsky 's .theory. and advocatcs fo r mUliiage classroom grouping compare the s tructure 10 thai of a famil y, whe re YOU nger members have many opportuni -ties to observe and imitate the competen-c ies of o lder members (Katz. Evange lou. and Hartman 1990). Teachers in mulliage classrooms are more likely than those in same-age class-rooms to accept a wide range of student abi lities. The multiage structure also is bet-ter suited to address developmental disc.rep-ancics in individual children. For example. an o lder child who is developmentally de-layed in a particular area has the opportu-nity to enhance his or her self-confidence by interacting with younger peers (Katz. Evangelou, and Hartman. 1990). Learning Math In a Multiage Setting We recenlly conducted a two-week study in fo ur multiage classrooms during which we made daily observations to de-tennine how mathematics is taught in a mu ltiage selling. We were awed by (he amount of co llaboration and sharing mat took place among children of different ages. Children working in small groups were busy sharing materials. ideas. and time. Sometimes the groups consisted en· tirel y of novices, sometimes entirely of more experienced students. and at other ~ times both types were included. Our observat ion{ ;cveal cd three types o f strateg ies the Children used as they worked together: modeling. tutoring. and pairing/sharing. Modeling. Just as Dan taught Sara how to calculate the number o f blocks in their design. students with more experti se often modeled for others in multiage groups. While Dan was probably unaware that he was se rving as a model for Sara when he j multiplied fo ur ti mes three. and Sara did not immediately grasp the concept o f mul-ti plication, she was introduced to a new way of thinking about numbers. We observed another example of mod-eling when Andy. an older child . was asked by the teacher to create a bi laterall y symmetri cal pattern wi th blocks. " How do you know ~our pattern is bi· laterall y symmctrical?"lhe teac her asked him as he began constructing his pattern . • " Because I make one side and then I copy it to make the other side ." Andy re-sponded. "Then I know the s ides are ex-actl yal ike ." In language even the youngest chi ldren in the group could understand . Andy had no t only defined bilateral sym metry but PRINCIPAL • accumtely described onc way to construct such a figure. When the teache r suS· gesled 10 the OIhers that they also make bilaterally symmetrical pallem s. Andy assisted a younger classmate with the task. moving from the role of model to that of tutor. teach David how to do il. In the process, Leslie nOI onl y got needed prac tice but the sat isfac tion of knowing thaI she had helped David learn something new. (ColI(illlled Oil p . 26) Math Standard.s and Multlage learning TU1Dring. We often observed children who had mastered a conc;pt helping those who had nol. For example. in a group of children working with coins. the ·teacher" s goal was to introduce the various coins and the ir denominations to some. whi le those already fami li ar wi th the va lue of each coin learned how to make change. The national matbemati06 standards for K-12 schools, developed by th~ National Council of Teachers of Mathematics (1989), eSlabli~ five general goals: • I. That all students learn to value mathemalic.s; 2. That they become confident in their ability to do mathematics; . 3. That they become mathematical problem ,6Iv=; 4. That they learn 10 communicate mathematically; and 5. That they learn t6 reason mathematically. David, to whom the concept of coin vaJue was new, caught on quickly . After crc:ating different coi n combinat ions to equal 25 cent~. he became vcry interested in making change. Les lie . who had been practicing mak ing c hange . began to According ta the Council. "students should be exposed (0 numerous and varied in-terrelated experiences that encourage them (0 value (he mathema(icaJ enterprise, to develop mathe8latical habits·of mind, and to understand and appreciate the role of ~matics in human affairs." Students are also "encouraged,.o explare, to guess, and even to make and correct errors so that they gain confidetlce in their ability to solve complex problems." I National Teachers' Day is May 3rd Recognize your staff in a positive, dynamic way! Plan now for spring recognition! 1. "I make a difference. I TEACH." bullon. - $1 . regularly $1.50 (824) magnets .• $2.50 each. regularly $3 (M24) bumper stickers •• $1 each (82488) 2. "Teachers make a world of difference." buttons - $1 each (regularly $1 .50 each) (831) bumper stickers .. $1 each (83188) magnets •. $2.50 each. regularly $3 (M31) 3. · You make a difference" button. - $1 each (regularly 1.50 each) (82) greeting cards .• $1 each. regularly $1.50 (GC02) 4. · Special Person" button. - $1 each (You save 50 cenls eaCh). (820) stickers - packet of 250. $22 (820) 5. "I make a difference." buttons ·· $1 each (You save 50 cents each). (830) magnet.·· $2.50 each (regularly $~) M30 stickers .. pac et of 250, $22 (830) SPECIAL MIXED STICKER PACKAGE 100 of each of the above 5 designs .• 1 3/4· round slickers 500 stickers ' -- onl $49 13 e Ilteet> . Call us toll-free to place your order today! 1-800-536-2363(BEME) '" P.O. Box 3202 (Dept. PM2) Portland, OR 97208 1.800·S36·2363(BEME) • FAX (503) 287·0771 VisaIMC accepted by phone and mail. Purchase orders also accepted. Please add $4.95 shipping and handling. 24 PRINCIPAL f C tJllluuu'd };'om p. :!.J J Pa;n·ng a"d Shan·" g. Wt: ob~erved ont: muilluge clj.:.sroom Ihal makt::. a daily pr:.ll:l ict: of ,h;,tring ~ l udelll -(Tl·'.IIed math probkm:.. For C'i..:.l lllpil: . hl'n: wa:. Dana 's problem; Hl'r parent!> had told her she could buy a T - ~hln thai ~hl' wanled ir she saved he r money. bUI she needcci'tu de lennine how much il would COSI after adding Ihe sales IH . Juli e. onl y 7. was fasc inated by D;,tnj. ·~ prob lem. Although she had no kno", ledge of how 10 compute frarlion :o.. :o.he paired with Dana to fi nd a ~olution . ·· M) dad ,>ay~ Ihattht: lady wi ll add " I X celll ~ for eve: r)' dollar Ihal 111)' :o. hin co:o. t ~:· ~ana said . ·'How much doc:~ the , hln l.:O:o. tT asked Juli!;.· . " 11 cost~ 59.99 '-' replied Dana. "That mean!'! we ha ve 10 add up ~ ix cents ninl' time~ and we'lI have 99 ('enls le ft over. I s o L u T o 26 don'l know how much tax Iht:y chargu for 99 Ct:nb," A~ Dana ve rbalized her Ih in king. Ju lie bc-gan writing Ih..: number 6 nine time~. Dana added Ihl'lll 4ui rkly. gOI 5-l. then added 1ha\ number to $95)9 10 gel $ 10.53. With the teacher's ass istance . they thl'n added anothe r six cent s to come up wilh Ihl' answer 10 Dana '!) problem; The T· shin would l:OSI $ 10.59. Whik ~ uch l"oope rat ivt: ~ trate gic~ migh t a lso be obse rved in s ingle-age d U.'>sroullu;, Ihe: l1lulti age selling seems to i l/I 'il l' co ll aboration and sharing- neating ;,tn environment whi c h SUppOTl S the goa ls of the math standards established by the National Council o f Teachers of M:llh· ematics (Si' f' box. p . 1 .. 1). As we observed children o f different ages sharing their thinking about solut ions N s F o R I to math problems, it was apparent that mult iage groupi l1g providt:s stude nts w"h opportun ities to Icum from each other. 0 REFEItENCI:S K;,Hl , Li li;,m G. ; EvangcloLJ, Dcme tra : and ~Ianman , Jeancltc Alli son . Tilt, Cll.H' for M i.\nl-tl.t.: l' Groll(Jillg III Early E~Jl/nll i(Jl/. Washlllglun, D.C.: National Associm ion for Ih~' EdUl:alion of Young Children, 1990. N:llional Council of Teachers of M<llhcmalics. CIIITi, '1I11I1II mill Emlllwiflll SWI/dards Jill ' SduJO/ 1"'1(1/11(' 11/(1110 . Rc slOn . V.I. : The Council. 1989. Vygolsky, L S. in M illd ill SII!"i l,ty : Tile O(' r('/ofUlIl' lII IIf J-l i.t.:lic' " P.~yd/(lJogklll P ro, ·t'S.H'.1 (cd. M . Cole: V. John -Ste iner; S. Snibner: ;lnd E. Soubcnnun). Cambridge. Mass.: Harvard U lli \'Cr_~ ily Press. IlJ78. r , L E A o E R s THE SElF· DIRECTED SCHOOL: Empowering the Stakeholders By Ronald G. Mcintire and John T. Fessenden. Integrated into the present system of t most school districts, this strategy builds on the concept of advisory site councils, giving decision·making power to leaders who are devoted to real change and improvement. Examples show how this approach helps in meeting goals and moti-vating students. ISBN 49267-5 $29.95 SEVEN STEPS TO EFFECTIVE INSTRUCTIONAL LEADERSHIP By Elaine K. McEwan. The positive approach of this research-backed guidebook will help you give clear goals to both new and experienced princ ipals , An ideal tool for action and self-appraisal, it includes a checklist of 30-6pecific lead-erstiip standards, Instructional leaders from the National Distinguished PrinCi pals and the Illinois Principals Association provide helpful examples. ISBN 49467·8 $24.95 IMPLEMENTING CONTlNUOUS IMPROVEMENT MANAGEMENT (elM) IN THE SCHOOLS By Bill Borgers and Tommy Thompson. Written for schools or 'districts that are involved in, or considering , Continuous Improvement Management, this book focuses on a Texas school's actual experience in implementing e lM, including the systematic techniques, as well as the quatity and cost benefits. II gives you a real-world affirmation of the breakthrough teaChings of Edwards Deming, William Glasser and J.M. Juran. ISBN 49502-X 5;!9.95 PRINCIPAL 

Multiage Classrooms: A New Way to Learn Math

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Multiage Classrooms: A New Way to Learn Math

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