[Shelly Frei] Teaching Mathematics Today

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Teaching

MathematicsToday

Shelly Frei



Shell Education

5301 Oceanus Drive

Huntington Beach, CA 92649-1030

http://www.shelleducation.com

ISBN 978-1-4258-0376-6

© 2008 Shell Education

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Sara Johnson

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The classroom teacher may reproduce copies of materials in this book for classroom use

only. The reproduction of any part for an entire school or school system is strictly prohibited.

No part of this publication may be transmitted, stored, or recorded in any form without written

permission from the publisher.

Publisher

Corinne Burton, M.A.Ed.

Teaching Mathematics

Today



Table of ContentsChapter 1—Introduction: A Balanced Approach 7

Overview o Reserch 7

The Ntionl Council o Techers oMthetics Stndrds/Focl Points 21

How to Use This Book 23

Post-Reding Reection 24

Chapter 2—Planning Instruction 25

Te Building 25

Aligning Instruction with MtheticsStndrds 29

Develoing Mthetics Curriculu Tieline 32

Accelerting nd Decelerting theIntroduction o New Concets 36


Chapter 3—Managing the Mathematics Classroom 43

Involving Prents nd Students to IroveProgr Eiccy 43

Lesson Delivery Strtegies 50Coon Mthetics Clssroo

Mngeent Issues 59

Mxiizing Instructionl Minutes 63


Chapter 4—Dierentiating Instruction 71

The Iortnce o Dierentited Instruction 71

Dierentited Lerning 72

Dierentition Strtegies 76Grouing Strtegies or the Clssroo 78

Grou Activities 82




Chapter 5—Developing Mathematical Vocabulary 8 5

Secilized Mtheticl Vocbulry

or All Lerners 85Vocbulry Develoent or EnglishLnguge Lerners 86

Levels o Lnguge Acquisition 88

Integrting Vocbulry Develoent intoInstruction 90


Chapter 6—Building Conceptual Understandingwith Manipulatives 101

Mngeent o Mthetics Mniultives 101Tyes o Mthetics Mniultives nd

How They Are Used 104

Preventing Mniultive Deendency 107


Chapter 7—Teaching the Procedure 111

Building on Understnding 111

Teching More Thn One Wy 113

Tking Notes 115Cretive Wys to Prctice 116


Chapter 8—Teaching Problem Solving 121

Why Tech Proble Solving? 121

Stes or the Proble-Solving Process 122

Proble-Solving Strtegies 125

How to Crete Word Probles 141


Chapter 9—Integrating Mathematics Acrossthe Curriculum 145

Integrting Mthetics nd Litercy 145

Integrting Mthetics nd Science 149

Integrting Mthetics nd Socil Studies 151

Mthetics in the Rel World 153

Table o Contents



Integrting Mthetics nd Technology 155

Using Sredsheets in Mthetics

Clssroo 156Using Alets in Mthetics Clssroo 158

Using Grhing Clcultors in MtheticsClssroo 164


Chapter 10—Assessing Students 169

Aligning Assessent with Instruction 169

When to Assess 170

Inorl Assessents 171

Forl Assessents 173Dt-Driven Instruction 180

Using the Dt Collected 185


Chapter 11—Conclusions 191

Develoing n Intervention Curriculu 191

Wht Should Mthetics InstructionLook Like? 194

Post-Reding Reection 197Appendix: Reerences Cited 198

Table o Contents



Teaching Mathematics Today



ChapterOne

Introduction:A BalancedApproach

Overview of ResearchTeching thetics in tody’s diverse clssroos cnbe chllenging, but it lso rovides techers with nyexciting oortunities to ss on lie skills s well stheticl knowledge Mthetics is subject thtwill directly ect every single student who enters theclssroo Profciency in thetics trnscends si-ly succeeding in school or scoring well on stte stn-

drdized tests Mthetics techers directly inuencehow students will roch robles nd exeriencesin lie: going to the grocery store, esuring wlls toint the house, borrowing oney ro riend, cookingels, driving cr, blncing checkbooks, ying txes,buying houses, king investents, nd so ny otherdily tsks




There re ny roches to teching theticsThis roessionl develoent guide is built round

blnced roch to the instruction o theticlconcets

Common Approaches in Mathematics Instruction

Oten techers eel coortble teching the wy theywere tught It is wht they reeber nd wht theyknow, so it becoes the wy they tech, regrdless owhether they believe it is the correct wy to tech

A coon ige o “tyicl” thetics clssroohs the techer stnding t n overhed rojector show-ing the equtions nd the oruls while the studentstke notes tht will theoreticlly hel the colete thessigned textbook robles Mny thetics educ-tors ocus on skills nd oer ostly rocedurl rcticeStudents lern the oruls nd the rocedures involvedin the vrious thetics discilines This or oinstruction ocuses on lot o eoriztion nd skill-nd-drill rctice Techers oer lecture tye instruction nd

then students colete the ges in the texts during clsstie Then they tke hoe ore rctice worksheets orhoework, with no urther suort or those who do notunderstnd the theticl rocedures involved Thistye o instruction is hening in eleentry schools swell s iddle schools nd high schools

Textbooks re lso lrge rt o “tyicl” theticsclssroo However, the use o textbooks lone cn cre-te reduced use o eective instructionl tie becuse

the textbooks oten lck relevnt guidnce regrding howto ddress dierent lerning styles, engge students, inte-grte niultives, nd dierentite instruction bsed onthe ny individulized lerning needs tht theticsinstructors see in their clssroos Oten textbooks reflled with distrcting ictures nd designs tht do notdd to the theticl corehension o the key con-cets The textbooks lso include ny ore toics thn



Introduction: A Balanced Approach

the verge student cn ossibly lern in school yerTechers who lck guidnce or exerience y think they

re suosed to oen the textbook on dy one nd techs r s they cn by the end o the yer Thereore nyeductors re not teching soe o the iortnt ound-tionl concets tht students need in order to continue onto subsequent thetics courses

Another tye o thetics rogr lens ore towrdexlortion o theticl concets through concetu-l investigtion These rogrs re oten very oulrin eleentry schools Students use concrete terils,

such s niultives, nd rticite in exerientsnd kinesthetic deonstrtions tht exhibit theti-cl concets However, these tyes o lessons soetieslck the connection bridging the “un” ctivities to thectul theticl concets nd bstrct or o therobles Soeties this roch coes ro text-book-bsed rogr tht oers little ssocition betweenthe lerning ctivity with colorul ictures on one sideo the textbook nd the er nd encil robles listedon the other side

Oten techers re orced to ollow strict district cingchrts or course outlines tht delinete wht concetsneed to be covered during the school yer Becuse o theressure techers eel to cover ll the toics, ech toicis tught, rcticed, nd ssessed nd then the clssoves on to the next toic There is oten little regrdo whether students re ctully ble to truly lern ndbsorb ech toic (Mrzno, 2003) Consequently, only subset o the students reches level o stery o

ny given skill

No one denies tht there re indeed ny studentswho succeed in these tyes o thetics clssroosThey esily build on theticl concets nd suc-ceed in higher levels o thetics courses Yetthere re lso countless students who re struggling tochieve ssing grdes in their thetics clsses



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Mny students re iling district nd stte thet-ics ssessents I techers only exlin the rules nd

evlute correct or incorrect nswers, then the studentscoe wy with liited view o theticl exer-tise (Lert, 1990)

While no one ethod o instruction hs been rovens the single best wy to tech thetics, usingreserch-bsed designs nd rocedures hve heled edu-ctors recognize the best etures or roching lern-ing gols in thetics (Hiebert nd Grouws, 2007)Tody’s thetics techer needs to develo scoe

nd sequence tht integrtes ultile oortunities orrctice rior to ssessent, nd tie to retech ndrevisit those skills tht students hve not stered terthe ssessent

A Balanced Approach in Mathematics Instruction

Reserch hs estblished tht students need both ro-cedurl nd concetul knowledge in order to lern ndunderstnd thetics (NCTM, 2000) Knowledge o

the rocedures nd oruls re criticl to overll rof-ciency in thetics Also, exlortion o the concetsthrough concrete exerients nd nul niultiono theticl concets is vitl to the overll under-stnding o the “why” in thetics instruction

Thereore, blnced roch cn link these two dis-tinct roches nd oer connections tht cn ledstudents to higher rofciency nd understnding otheticl concets It is ineective to ehsize

high degree o rocedurl rofciency without develo-ing concetul knowledge It is necessry to rovideocused instruction tht oves the student ro theconcrete to the bstrct nd then to the liction othe concet (Mrzno, 2003; Sutton nd Krueger, 2002)Focusing on only the concetul knowledge is notenough to hel students chieve in the clssroo ndin rel-world situtions



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Techers integrte lterntive teching ethods, ni-ultives, nd dditionl rctice into stndrd clssroo

lectures with blnced roch Techers re givenssistnce in how to ln instruction so tht lessons willlign with stte stndrds nd ddress students’ needsTechers eloy strtegies tht ddress dierent lern-ing styles, engge students, nd dierentite instructionThere re ultile oortunities or rctice rior tossessent, nd tie to retech nd revisit those skillstht students hve not stered ter the ssessentWhen students re given suicient rctice, they cnroch being ble to use the newly lerned skill in

new situtions with ccurcy so tht tht skill will beretined (Sous, 2006)

A blnced roch clls or chnge in the clssic roleo the techer in thetics clssroo Techersbecoe cilittors, heling to ove students’ under-stnding o given concet ro the concrete to thebstrct, nd fnlly to the concetul liction otheticl concets Reserch by Fillroy nd Rolnohs shown tht the trnsition ro concrete odels to

the lgebric equtions cn be diicult or students tochieve (Kiern, 1999) Techers need to be ccessibleto students nd understnd the rocess through whichstudents need to rogress in order or the to ke theju ro the concrete hse to the concetul hseFollowing the concrete hse, students ove to thebstrct hse in which they lern the lgorith, usingthe understnding they gined ro their exerienceswith the niultives At this oint, the concetulunderstnding becoes iortnt It hels orgnize

theticl rocedures nd lso hels students under-stnd the rorite rocedures to ly in dierentsitutions (Kiltrick, et l, 2001) Ultitely studentsneed to ove to rocedurl uency, which is “the bili-ty to coute, clculte, nd use rules nd oruls cor-rectly, quickly, nd with ssurnce” (Den nd Florin,2001) Without this, students will struggle to deeen




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corehension o theticl ides nd the bility tosolve thetics robles (Kiltrick, et l, 2001)

Reserch suggests tht the trditionl dichotoies oexlining the best ctegories or teching skill eicien-cy nd concetul understnding re no longer helulThe etures o teching tht cilitte skill eiciencynd concetul understnding do not ll netly into ct-egories requently used to contrst ethods o teching,such s exository versus discovery, direct instructionversus inquiry-bsed teching, student-centered versustecher-centered teching, nd trditionl versus reor-

bsed teching (Hiebert nd Grouws, 2007)

Thereore, the blnced roch suggested in Teaching Mathematics Today crosses beyond these coonethod lbels to crete collection o gret strtegies,sle chrts or recording clssroo ngeentinortion, nd ssessent inortion or techers toesily nd eectively use to eet students’ needs Thesuggestions in this book will ssist techers in develo-ing their own reserch-bsed, best teching strtegies to

ddress dierent lerning styles, engge students, nddierentite instruction

Components o a Balanced Approach inMathematics Instruction

Standards-Based Instruction

The Ntionl Council o Techers o Mthetics(NCTM) believes tht ll students should lern ior-

tnt theticl concets nd rocesses with under-stnding (NCTM, 2000) In n eort to hel tech-ers eet higher stndrds nd the diverse needs ostudents, the NCTM Bord o Directors hs oredseverl docuents designed s ids to nyone kingdecisions regrding thetics eduction o re-kindergrten through grde 12 students Originllythere were three serte docuents One ocused



1

on curriculu, one on roessionl stndrds, nd thelst on ssessent stndrds These reresented the

frst ttet to give extensive thetics-reltedgols or the eductionl feld, nd were well receivedOver tie, these docuents were revisited, reviewed,nd revised The result ws tht in 2000, NCTMublished one docuent to encoss ll the golsor teching thetics in eduction It is clledPrinciples and Standards or School MathematicsIt serves s the bsis or ny sttes’ thet-ics stndrds nd s suort or decisions regrdingthetics in schools nd wht should be tught

t ech grde level The teching strtegies or bl-nced roch to thetics instruction tht redescribed in Teaching Mathematics Today ddressech o the six rinciles within the revised NCTMdocuent: equity, curriculu, teching, lerning,technology nd ssessent By ligning with stn-drds, districts cn work towrd the NCTM chllengetht everyone deserves to understnd thetics Itis not only or select ew (NCTM, 2000)

Integrated Curriculum

Rther thn working on subjects in isoltion ro

one nother, studying reding rt ro writing,

nd rt ro th, science, socil studies, nd

other curriculr res, children lern best when they

re engged in inquiries tht involve using lnguge

to lern, nd tht nturlly incororte content

ro vriety o subject res (NCTE, 1993)

It is iortnt or students to understnd tht educ-tion is not series o cortentlized subjects ththve nothing to do with one nother Rther, studentsneed to relize tht lerning is ore like rug, wherell subjects re woven together to crete brod scoeo understnding tht is ultitely ost useul whenll the strnds ft together




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Techers o thetics ust ut concets into rel-lie context or students in order or the to under-

stnd the concets nd ke the ersonl “Whenthetics evolves nturlly ro roble situ-tions tht hve ening to children nd re regulrlyrelted to their environent, it becoes relevnt ndhels children link their knowledge to ny kindso situtions” (NCTM, 1989) This rel-lie context,which is necessry or develoing student understnd-ing, coes ro integrting other subjects into th-etics instruction

Lnguge skills re ost coonly nd esily integrt-ed into thetics instruction Fro n erly ge, student is exosed to literture nd develos levelcoort using nd discussing books “Oortunitiesor discourse in both reding nd thetics instruc-tion roote children’s orl lnguge skills s well stheir bility to think nd counicte theti-clly” (Moyer, 2000) Literture lso rovides il-ir context through which theticl concets,roble solving, tterns, nd dt cn be exlored

nd understood Students do not see the need to lernabout thetics until they cn lern bout rel liethrough thetics

The nture o inquiry ebedded in science nd socilstudies lends itsel to the use o thetics s tool or understnding nd extension Mtheticlconcets such s dt collection, corison, ndnlysis; tterns; robbility; nd grhicl reresen-ttions cn ll be lerned nd understood using science

nd socil studies toics

Technology is n ever-growing nd chnging feld ineduction It is iortnt or thetics techersto hel students understnd tht technology cn beused or ore thn lying ges nd text essgingriends Mtheticl concets cn be enhnced ndexlored through the use o the Internet, couter



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sotwre, grhing clcultors, nd other technologyroducts

Teaching Mathematics Today rovides suggestionsnd strtegies or techers to integrte theticscross the curriculu It rovides inortion boutreching ll lerners nd brodening students’ under-stnding o theticl concets

Student Engagement

The strtegies in Teaching Mathematics Today regered towrd engging students nd creting oti-

vtion or their lerning rocesses Cthy L Seeley,2004 resident o the Ntionl Council o Techerso Mthetics (NCTM), discussed in her essge“Enggeent s Tool o Equity” how students’ctive enggeent in their own lerning icts theirchieveents She sttes, “Student enggeent iserhs our ost iortnt tool in our bttle or equi-ty” When students re ctively otivted nd busyreching lerning gols, they re lso ctively con-

structing knowledge nd oving towrd successulstery o key concets The techer is not the onlyindictor o student success in this odel Rther, thestudents hve oortunities to hve ownershi nd greter understnding o the ides nd concets thtthey re intercting with When students re ctivelyinvolved in writing, odeling, exloring, nd discuss-ing thetics versus sily wtching the techerdo these things, students re ore likely to be suc-cessul (Seeley, 2004b) Using niultives, tking

notes, resenting PowerPoint slide shows, nd hvingstudents odel robles re exles o strtegiestht ctively engge students in the lerning rocessTeaching Mathematics Today resents vrious strte-gies or student enggeent in thetics lessons




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Dierentiated Instruction

As students ove ro the concrete, to the bstrct,

to the liction hse o lerning, they re exosedto concet or skill nuerous ties Students shouldhve ultile exeriences with toics, llowing the tointegrte the toics into their knowledge bse (Mrzno,2003) However, not ll students rocess the new inor-tion in the se wys or bring the se skill sets tothe lerning exerience Soe students need extr tieto rocess concets nd look t robles in dierentwys (Sutton nd Krueger, 2002) Other students needurther teching or teching resented in ultile wysTeaching Mathematics Today rovides chrts, strtegies,nd tis or identiying individul student needs nd howto dierentite instruction to eet those needs withinthe clssroo

Cooperative Learning

Cooertive lerning tsks re encourged nd describedin this book Cooertive lerning oers the oortu-nity or students to lern ro one nother (Sutton ndKrueger, 2002) In ddition, students cn be ctivelyinvolved in their success nd ssue resonsibility ortheir own lerning with the suort o other studentsAll students cn beneft ro cooertive lerningIt enbles high-eroring students to stretch theirunderstnding o concet nd ke new connectionsbetween teril “Results re quite roising or usingeer-ssisted lerning with low-eroring students ”(Gersten nd Clrke, 2007) These students beneft ro

seeing the teril resented in ultile wys wherethey cn be ctively involved in lerning theticlconcets Teaching Mathematics Today rovides tech-ers with n understnding o how to incororte thisethod o instruction into the concets nd lessons theyre lredy teching



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Problem Solving

Reserch shows tht students who re not successully

stering theticl concets tend to deonstrteslow or inccurte retrievl o bsic theticl cts,len towrd iulsivity when solving robles, nd hvediiculty oring entl reresenttions o theti-cl concets or keeing inortion in working eo-ries (Gersten nd Clrke, 2007b) One study ound thtchildren iroved in overll theticl rofciencywhen they were tught thetics through roble-solving strtegies Not only were they chieving bettertest scores, but lso incresing in the bility to cou-nicte their understnding o the theticl concetsorlly nd in writing The conclusions ollowed thtthe roble-solving roch to thetics showedstudents nd techers tht the two were connected ndtht the roble-solving strtegies heled with overlltheticl rofciency (Hrtweg nd Heisler, 2007)

Reserch hs shown tht rel-lie lied ctivities ndroble-solving ctivities estblish contextul settingor ny lessons, roviding otivtion nd encourg-ing curiosity (Hiebert nd Crenter, 1992) Overll,the chllenging nd interesting tsks ound in lic-tion robles hel techers engge students in lern-ing (Seeley, 2004) Integrting roble solving s onesect o the curriculu ollows the blnced rocho thetics instruction

Teaching Mathematics Today oers ste-by-ste ro-cess to tech students roble-solving strtegies Theset ctivity exles re ent to crete indeendent,coetent student roble solvers

Guided Practice

In guided rctice, techers tke “we do” roch tohel their students understnd the concet being tughtThe couniction nd interction between the techer




1


nd student hve to be ore signifcnt thn just solv-ing robles on worksheet “Prctice does not ke

erect, it kes ernent” (Sous, 2006) Throughthe guided-rctice ethods encourged in this book,techers onitor students’ erly rctice nd ke sureit is ccurte They rovide tiely eedbck so tht theskills re lerned ernently nd correctly Guidedrctice hels reduce initil errors nd inors studentso the criticl stes in lying new skills (Sous, 2006)As lredy entioned, Teaching Mathematics Today incorortes instructionl strtegies tht blnce roce-durl rofciency nd concetul understnding, while

ctively engging students in rctice exeriences thtre designed to deeen their understnding nd connecttheir theticl knowledge to rel lie

Manipulatives, Games, and Calculators

Mniultives re essentil to heling students under-stnd theticl concets Using niultivesregulrly rovides hnds-on exerience nd hels stu-dents construct useul enings or the theti-

cl concets they re lerning (Grouws nd Cebull,2000)

The use o niultives hs becoe coon in theriry grdes nd hs roven to be n eective toolor illustrting eleentry theticl concetsWhen students use concrete objects to reresent th-eticl ides, they lern to orgnize their thinking ndreect on concrete reresenttions (Den nd Florin,2001) These se tools cn be very eective in iddle

nd high school thetics clssroos For exle,niultives, such s lgebr tiles, hve extended thishysicl reresenttion into Algebr I nd rovide bsis or develoing lgebric concets (Shr, 1995)

Students need le oortunities to rctice in order tobe ble to execute rocedures utoticlly without con-



1

scious thought (Kiltrick, et l, 2001) Plying geswith the gol o reinorcing skills, rehersing inor-

tion, nd building retention o theticl concetsis one wy to llow students the rctice tie necessryor skill to be stered Students in tody’s clssroosre very otivted by entertinent Mtheticlges cn ique their interests nd give the sense oun while they re lerning

The use o grhing clcultors cn be helul s wellAccording to reserch (1997) coiled by Dr Bert Wits,coounder o the develoentl rogr Teching with

Technology, nd Heidi Poerntz, roessor o th-etics t Ohio Stte University, “grhing clcultorscn irove clssroo dynics, boost students’ conf-dence levels, nd roote understnding o theti-cl concets nd unctions” Further reserch showstht students who hd ccess to clcultors were betterwith entl clcultions nd estitions, s well s bet-ter ble to solve rel-lie robles Student chieveentin generl ws higher (Kiltrick, et l, 2001; Heller, etl, 2006) With incresed use o grhing clcultors

during instruction, higher test scores were chieved eveni students did not hve ccess to the grhing clcul-tors during the test (Heller, et l, 2006)

Teaching Mathematics Today discusses how niul-tives, ges, nd clcultors serve s incredible toolsor engging students s well s ddressing the needs okinesthetic, visul, nd English lnguge lerners Thisbook rovides ngeent techniques nd strtegies orusing these terils in the clssroo

Vocabulary Development

Teaching Mathematics Today lso rovides techerswith resources or develoing the secilized vocbu-lry necessry or theticl-concet corehen-sion Mtheticl lnguge is very recise cored




0


with the English used in coon discourse Thiskes the study o theticl vocbulry dierent

ro ost other content res students study Vriousvocbulry-develoent ctivities re vilble so thtstudents, including English lnguge lerners, cn trulyunderstnd the cdeic vocbulry tht will hel theunlock the theticl concets

Intervention

Rther thn witing to fnd out which students willrequire intervention nd dditionl instruction in order

to ss the required thetics clsses, there is nincresing need or innovtive rogrs tht rerestudents to corehend theticl concets ndfll in chieveent gs “We ust exect ll o ourstudents to lern thetics well beyond wht wereviously exected We need ll students to be orerofcient thn in the st, nd we need ny orestudents to ursue creers bsed on thetics ndscience” (Seeley, 2005) In order to rech these gols,thetics rogrs re needed tht oer ound-

tionl concets with blnce o couttionl ndrocedurl skills, concetul corehension, nd rob-le-solving rctice so tht students cn build on gen-erl thetics rofciency Eective interventionrogrs tht rere students or college nd highereduction should ocus on rediness rther thn justreedition (Oesterreich, 2000) Soe o the osteective teching rctices suggested or low-chiev-ing students s well s secil eduction students revisul nd grhic deictions o robles, systetic

nd exlicit instruction, sll-grou instruction, stu-dent think-louds, eer-ssisted lerning ctivities, ndortive ssessent dt (Gersten nd Clrke, 2007)These teching rctices re inused in ny o thestrtegies oered in this book



Assessment and Data-Driven Instruction

Stndrds-bsed instruction begins with the gol o ll

students stering the given curriculu with ro-rite instruction, terils, nd suort In orderor this gol to be chieved, techers ust hve frgrs o where students re in their rocess o lerning theticl concet, wht they need to ccolishto chieve stery, nd how they will rech the setgols (Wili, 2007) Techers ust then use orlnd inorl ssessent strtegies “inute by inutend dy by dy, to djust their instruction to eet theirstudents’ lerning needs” (Wili, 2007) Assessentsrovide techers with the necessry dt to understndwhich students re struggling in secifc res o the cur-riculu

This book rovides strtegies nd chrts or orl ndinorl ssessents, s well s wys to use dt todrive urther instruction within the clssroo

The National Council of Teachers of

Mathematics Standards/Focal PointsThe Curriculu Focl Points re the ostiortnt theticl toics or echgrde level They corise relted ides,concets, skills, nd rocedures tht or theoundtion or understnding nd lsting lerning(htt://wwwNCTMorg)

The Ntionl Council o Techers o Mthetics(NCTM), one o the leding ntionwide uthorities in

teching thetics, hs been roviding stndrdsnd suggestions or school thetics courses ordecdes The content stndrds ound in the Principlesand Standards or School Mathematics (NCTM, 2000)re used either directly or s corison stndrd orschool districts cross the ntion Mny sttes reer to





this guide s odel o how to develo nd cultivtethetics corehension or students who rogress

through ech grde level towrd grdution ro highschool These content stndrds re divided by grdebnds—rekindergrten through second grde, thirdthrough fth grde, sixth through eighth grde, nd ninththrough twelth grde Ech grde bnd encosses thevrious discilines o thetics: nuber nd oer-tions; geoetry; esureent; lgebr; nd dt nly-sis, robbility, nd sttistics

Following the uthoring o this corehensive collection

o stndrds, NCTM creted the Curriculum Focal Points or Prekindergarten through Grade 8 Mathematics: AQuest or Coherence This docuent encossed thejor ocus res or which to rovide ehsis or theincluded grde levels The gol ws to eet the needso incresing ccountbility, students nd techers whoove oten, nd the cost o continully develoingthetics curriculu By ulling out only the ostvitl concets tht re necessry in tht grde level, thisdocuent ws creted to describe bseline stndrds or

student knowledge In this docuent, there re descri-tions o the theticl concets nd skills, rtherthn lists o the gols, stndrds, objectives, nd lerningexecttions, s re ound in the stndrds docuentThe collection o descritions ws creted to insiretechers to discuss with one nother the direction o thethetics courses in the school They were cretedto guide the ortion o thetics curriculu ndto insire the strtegies nd lesson lns used to techtheticl concets “This work y ssist in the

cretion nd eventul develoent o new odels ordefning curriculu, orgnizing instruction, develoingterils, nd creting eningul ssessents tht cnhel students lern criticl theticl skills, ro-cesses, nd wys o thinking nd cn esure nd co-unicte wht students know bout the theticstht we exect the to lern” (NCTM, 2006)




How to Use This BookTeaching Mathematics Today is ent to be guide or

thetics techers The book is designed to sn llthe grde levels ro kindergrten through grde 12, ndlso cn be dted or the vrious discilines o th-etics

• This book oers reserch-bsed exlntionso the teching strtegies tht re ost criticlnd highly eective or thetics techers toinclude in teching theticl concets

• Ech chter ocuses on dierent sect oteching in thetics clssroo

• A school or thetics dertent ightchoose to work through the entire book sthey streline nd erect their theticsrogr

• A school or thetics dertent ightchoose to use this book s the bsis orintervention rogrs being ileented

• An individul techer ight choose to usethe book to irove the eectiveness othetics instruction in his or her clssroo

• Techers could kee the book s resource toreer to or the secifc res s they relte to thetics clssroo

• New techers cn red the extensive exlntionso the strtegies nd eloy the in theirlessons

• Vetern nd new techers cn red nd ly

the techniques tht re described in the chtersto their current instruction o theticlconcets

• Techers cn lern extensively bout eetingthe dierent needs nd oering ccess to corecurriculu or struggling students nd Englishlnguge lerners




Post-Reading Reflection 1. Wht ws your defnition o blnced roch to thetics

instruction t the beginning o this chter?

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2. Would you revise your initil defnition? I so, how?

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3. Reect on two coonents o blnced roch in theticsinstruction nd exlin why these re iortnt or techers tounderstnd

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ChapterTwo

Planning Instruction

Team BuildingOne o the keys to n eective thetics rogr is

te building A strong te lys the oundtion orconstructive curriculu lnning, qulity instructionlcourses, nd eeting the students’ totl theticsneeds A te should be ut in lce or eductors nddinistrtors to work srter, not hrder Both sllnd lrge decisions hve to be de regrding thet-ics instruction When te is king those decisions,it is less likely tht best teching rctices will be co-roised nd stes will be tken wy ro the over-rching gol o stndrds-bsed instruction

Deending on the school, district, county, nd stte,thetics tes cn tke ny ors nd unctionsBut regrdless o the tye or unction o the te, itis necessry to hve dinistrtive suort nd well-trined, enthusistic techers who re coitted toincresing student chieveent nd ileenting well-structured curriculu




Team Building in Mathematics Instruction

This tye o te would ost oten be utilized t the

school level nd would consist o techers within r-ticulr grde level or thetics dertent, or tech-er reresenttives ro ll grde levels or theticsdertents This tye o te cn ke decisions orschool-wide thetics instruction or or secifcgrde level’s thetics instruction

When lnning the curriculu or the school, the teneeds to ensure tht its curriculu ligns with the sttestndrds nd uses dt to trget students’ needs nd

show cdeic growth Eective ileenttion o thecurriculu will be the driving orce behind the overllthetics rogr

Prior to the strt o ileenting the thetics ro-gr, the school thetics lnning te should dis-cuss edgogy nd te consensus It is recoendedtht the te eet regulrly to review student rogressnd the tieline or ll theticl concets Theyshould discuss ucoing lessons, reect on best tech-

ing rctices, nd discuss the students’ needs This tecn evlute the ow o content throughout the yer orro one course to the next, nd the consistency o oli-cies cross school thetics courses

In order or these eetings to be successul, the schoolthetics lnning te y wnt to ke deci-sions regrding the ollowing issues:

• how decisions will be de within the te

• how to ke the students’ needs the riority orll rogr-relted decisions

• develoing lesson ln ides or ucoingconcets

• estblishing curriculu tielines nddjustents bsed on students understndings




• how course lceents cn be exible enoughto eet chnging student needs

• eeting to discuss ssessent trends

• eeting to discuss the eectiveness o thedecisions de on the school-wide theticsrogr ileenttion

• how to ileent orl nd inorlssessents in the clssroo

• how to interret dt s wy to drive utureinstruction

• lnning how to hndle those students whoiss clsses

• lnning how to hndle those students who rerequently trdy to clss

• i nd how hoework will be ssigned

• i nd how hoework will be grded ndreviewed

• deciding wht dinistrtive suort isnecessry to ensure tht techers re eectivelyileenting the school-wide theticsrogr

Cross-Curricular Team Building

This tye o te building would tke lce ostly tthe secondry level In generl, eleentry techerstech cross the content res They understnd thtbest rctices involve integrting the content res nd

giving thetics unction outside o direct th-etics instruction However, with secondry techers,this concet is not ut into lce s oten This tye ote could be de u o dertent reresenttivescross the content res or ech grde level or school-wide




In order or eetings to be successul, tes y wntto ke decisions regrding the ollowing issues:

• coon curriculu strnds between contentres

• which rojects or ctivities cn be coleted orcredit in ultile courses utilizing instructionlskills necessry in ech course

• eective wys to illustrte connections betweencontent res to students

• wht dinistrtive suort is necessry

to ensure tht techers re eectivelyileenting cross-curriculr instruction

Vertical Team Building

This tye o te would be used to discuss vrious to-ics o iortnce to thetics instruction on lrgescle The urose o this tye o te would be to keverticl connections cross thetics curriculund llow schools to kee in ind the brod sectru

o sirling instruction Reresenttives would then beble to reort bck to their hoe schools to udte theircollegues nd dinistrtors on the toics discussedduring the eeting For sll school districts, this tecould be corised o thetics leders ro echschool In lrger districts, it ight be helul to onlyinclude reresenttives ro schools within ech eederttern

In order or eetings to be successul, tes y wnt

to discuss the ollowing issues:

• how concets re tught in ech grde level

• how to eectively tech rticulr concets

• wht curriculu resources re being used tech grde level



• wht holes exist in student knowledge, how tobridge the g in student knowledge, nd how

to correct instruction so tht uture clsses ostudents do not hve the se holes

• how to nlyze the schools’ thetics dtwith ttention to successes s well s chngesor revisions tht need to be de in order orstudents to be successul

Aligning Instruction with MathematicsStandards

The Need or Mathematics Standards

Stndrds-bsed instruction is the bsis o eductiontody, s well s the center o techer nd studentccountbility There is signifcnt need or con-tent stndrds in thetics instruction (Den ndFlorin, 2001) Stndrds-bsed instruction is one wyto eet the criticl need or ll students to be on theoreront o worldwide technology, scientifc nd th-

eticl dvnces, trde, nd develoent Stndrdsrovide ccountbility so tht students receive thenecessry skills to continue their lerning rocesses inurther grde levels, nd higher thetics in collegend beyond (Den nd Florin, 2001) Furtherore,the stndrds build on ech other For exle, in stndrds-bsed syste, fth grde techer cn beginto instruct students with resonble execttion thtthe students coe with soe understnding o thetheticl concets tht build u to the fth grde

content stndrds in the lower grdes While therere lwys going to be students who struggle with thevrious concets, stndrds-bsed instruction ensurestht students hve t lest been introduced to echiortnt theticl concet or their grde levelsnd ssessed on the levels o stery o tht concet




0


While there is soe vrition, ost sttes hve estb-lished the required content tht student needs to

know by the end o ech grde level in ech contentre Mthetics lessons should be bsed on thestte content stndrds so tht students cn be equl-ly nd dequtely rered or uture theticscourses s well s ny stte-ndted theticsssessents

Dissecting the Mathematics Standards

Whether sttes hve their own stndrds or use those

bsed on NCTM stndrds, there is soe level o grdesertion There is lso distinction ong the stn-drds or ech o the discilines o thetics

NCTM uses the fve content stndrds res o nubersnd oertions, lgebr, geoetry, esureent, dtnlysis nd robbility; nd the fve rocess stndrdsres o roble solving, resoning nd roo, couni-ction, connections, nd reresenttions (NCTM, 2000)Ech o these res encosses gols tht exnd cross

ll the grde levels The gols led to execttions thtdier bsed on ech grde-level bnd These exect-tions urther seciy wht the students in tht grde-level bnd should be ble to do NCTM uses the grdebnds Prek–2, 3–5, 6–8, nd 9–12

Soe sttes choose to urther distinguish in echindividul grde, rther thn using the grde bndsRegrdless o how stte’s stndrds re rrnged, it isiertive or techers to creully study nd understnd

the content stndrds in their sttes or their ssigneddiscilines nd grde levels It is lso iortnt to hvesoe knowledge o the content stndrds o the grdessurrounding their own in order to know where the stu-dents theoreticlly hve been nd where they need to goin thetics develoent



Once techer hs good understnding o the requiredstndrds, ln or cing chrt needs to be creted or

the yer tht will hel the techer cover the necessryskills nd concets It is best to do this in te toensure tht ll concets re tught, roer tie llot-ent given, dierentition strtegies nd cretive les-son lns utilized, nd revisions de bsed on studentssessent re included

It is no longer best teching rctice to oen the dotedthetics textbook to ge one nd work s r sossible by the end o the school yer There re oten

sueruous ctivities throughout the textbook, ndny o the necessry stte-ndted concets y becovered in the bck o the book, which the clss ightnot even get through Rther, techer ust:

1. Exine the content stndrds

2. Mke long-ter, but exible, ln to coverthe

3. Decide on stndrd nd design lessons tht

will lign with the stndrd 4. Finlly, decide which ortions o the

thetics textbook, suleentryterils, books, niultives, nd lessonstrtegies will hel the students lern therequired theticl concets

Any ln or cing chrt should llow or soe exibili-ty The techer will continully ssess students, orl-ly nd inorlly, to see i ore tie is needed to rech

stery o content stndrd Soeties techer willneed to send ore tie on one oundtionl concet inorder to begin to introduce new concet tht will builduon it Also, ssessents will hel techers lern whenstudents lredy know certin concets Then, less tiecn be sent on these in order to rovide ore instruc-tionl tie or unknown concets





Developing a Mathematics CurriculumTimelineOne resource tht techers cn use or instructionllnning is the Timeline or Mathematics Curriculum (ge 35) This chrt cn hel techers ensure tht llthe necessry objectives re covered during ny giventhetics course by oering long-rnge lnning bythe content stndrds

Choosing Resources to Teach MathematicalConcepts

Becuse it is necessry to tech to the content stndrdsnd not “to the textbook,” use o tielines reindstechers to fnd the necessry resources to cover therequired content stndrds This cn involve ickingthe rorite lessons ro the doted theticstextbooks As rule, ollowing textbooks ro ge oneto the fnl ge is not the best use o instructionl tieOten textbooks hve lessons tht re not required orgiven grde levels in thetics course Soeties

textbook lessons re redundnt or unnecessry bsed onstudent re-ssessents Thereore, the tieline-ln-ning rocess requires tht techers fnd suleentrythetics resources in order to rech the objectives

Soeties techers need to ln urther rctice gesor niultive ctivities in order to give students therequired rctice o concet Tielines re lso use-ul or llowing exibility in eeting student needs orenough tie to lern concet beore oving on to

new concet Techers cn use these resources, longwith district or school-site cing chrts, to record rog-ress towrd teching to the content stndrds Techerscn lso use tielines to deonstrte to dinistrtors,rents, nd other techers tht they re teching therequired stndrds



First nd oreost, techers should give students dig-nostic tests to fnd out wht they lredy know Bsed

on student results, the techer cn ke curriculudecisions nd ocus lessons tht correlte with the itesor which students did not deonstrte stery Then,techers cn colete the Timeline or MathematicsCurriculum

Guidelines or Planning the MathematicsCurriculum Timeline

1. Alwys strt by lotting out the content

stndrds nd thetic objectives tht needto be covered

2. Choose the terils, textbooks, lessons,niultives, rctice ges, ctivities, ndresources tht will best ddress the stndrdsnd objectives

3. Frequent ssessent beore, during, nd terunits o study will hel the techer to decidehow uch tie is needed or ech concet

covered

4. I licble, use idyer benchrk/qurtertests s guides or when to introduce concetsduring the yer





Directions or the Mathematics CurriculumTimeline

Write the dy o the week or dte in the frst colunThe second colun is where you cn record the contentstndrd or curriculu objective being ddressed Writethe ne o the lesson in the third colun nd the th-etics resource or rogr title in the ourth colunThen, write the ge nubers in the fth colun In thesixth colun, write which ortions o the lesson will becovered In the lst colun, write ny suggested dt-tions or notes tht techers y need or ech lesson



T i m e l i n e f o r M

a t h e m a t i c s C u r r

i c u l u m

C o u r s e :_________

_______________________

_______________________

_______________________

_____

T i m e s a n d D a y s o f I n s t r u c t i o n :__________

_______________________

_______________________

_____

D a y /

D a t e

C o n t e n t S t a n d a r d

o r C u r r i c u l u m

O b j e c t i v e

L e s s o n

T i t l e

M a t h e m a t i c s

R e s o u r c e

P a g e s

L e s s o n

C o m p o n e n t s

t o B e C o v e r e d

A d a p t a

t i o n s

o r N o t e s





Accelerating and Decelerating theIntroduction of New ConceptsThere is gret del o ressure on techers in thet-ics clssroos tody Techers re exected to eductetheir students in the rorite content stndrds sotht they all cn deonstrte stery Soeties whentechers re trying to tech the required theticlconcets or their grde levels, they fnd tht the stu-dents lck the rorite oundtionl skills necessryto lern new inortion Techers need to rrnge theyer’s required theticl concets so tht the les-

sons nd ctivities build uon ech other in strightor-wrd wy tht ost students will be ble to ollow

Oten, techers re given strict cing chrts to uhold ndre held ccountble with the requireent o turning inrequent ssessents o rogress Continuous cdeicrogress or every student is gol tht ll techers hveBut, inevitbly, ech clssroo o students contins wide rry o bilities nd the ressure to show continulgrowth or ech student becoes incresingly chlleng-

ing Techers y not eel coortble odiying cur-riculu to eet the needs o struggling students But,it y not be rorite or every student to coleteech lesson s it is lid out Soe students need extrtie or rctice nd others do not Additionlly, tech-ers y likewise eel uncoortble with the best wysto djust the curriculu or gited students However,djusting the ce o the curriculu is necessry whenstudents vry in bility levels

Guidelines or Accelerating and Decelerating theIntroduction o New Concepts

With the use o theTimeline or Mathematics Curriculum,in ddition to the tis nd techniques entioned in thedierentition section o this book, techer should eelwell equied to then ke djustents to lessons so



tht new concets nd skills re being introduced t rte rorite to the student needs in the clssroo

Beginning lceent ssessents should be used todignosticlly show which concets the students hvelredy stered nd which concets need to be s-tered The techer cn use this ssessent inortion,tken t the beginning, iddle, nd end o the course,s well s throughout vrious units o study, to choosewhich lessons to cover nd which lessons y not be snecessry

A techer ight fnd tht certin skills coe esily tothe jority o the students in the clssroo When thisis true, the techer cn llow less tie or the rcticend liction o those skills nd choose insted to con-tinue to the next lnned lesson Siilrly, i techerfnds tht the concets in rticulr lesson re verychllenging to the students, this techer ight llowore tie or ech coonent o given lesson—theodeling, guided rctice, indeendent rctice, ndliction ges nd ctivities

The techer could lso decelerte the ce o the lessonwith ore ir or grou ctivities to urther llow oor-tunity or stery o the lesson concets beore ovingon to new lesson I tie llows, the techer ightoer n dditionl dy o liction ctivities beoreoving on to new lesson

Accelerating the Introduction o New Concepts

A techer y dt vrious lesson coonents orccelerting single lesson in order to go through theconcets ster These re otions tht the techerwould use i the students lredy understood uch othe teril At ties, it y even be rorite toski rticulr lesson i the whole clss hs reviouslystered the concet This cn be deterined by thelceent test





The ollowing sections show how techer cn tke vri-ous coonly included coonents o thetics les-

sons nd llow or ccelertion o the lesson concets

Daily Problems to Solve—The techer cn dislythe roble(s) or students to solve on trnsren-cy s they enter the roo This cn lso be oereds extr credit or work to colete when the rc-tice robles re fnished

Vocabulary Activities—The techer y sk the

students to do quick-write o the necessry terst the beginning o the week in order to ocus ononly the unknown vocbulry

Skill Practice—This coonent ocuses ore onindeendent rctice nd could be given s hoe-work The techer cn decide which robles needto be rcticed in the clssroo nd which shouldbe coleted t hoe It is not lwys necessry tocolete every roble in order to show stery o

certin concet

Application Learning—The techer y choose onedy during week or the ges nd extr ctivitiesin which students ly their concet lerning Thetecher ight lso reduce the ount o tie thtis dedicted to these ctivities

Assessment o the Objective—Quizzes cn be givens tke-hoe ssessents I skill is ssessed

ore thn once, or i students hve stered secifc skill, the techer y choose to eliinte quiz



Necessary ComponentsComponents Adaptable for

Acceleration

Vocabulary Activities—These ctivities re

vitl or English lnguge lerners

Teaching the Objective—Students ust

hve well-lnned lesson with odeling

nd rctice i they hve not lredy

deonstrted stery o the lesson toic

Skill Practice—This should never be

coletely eliinted Students lwys

need to rctice wht they hve lerned in lesson

Daily Problems to Solve—Dily robles

cn be coleted s n entry ctivity or or

extr credit

Vocabulary Activities—Vocbulry

ctivities cn be coleted s grou or

cn only ocus on unknown ters

Skill Practice—Prctice robles cn be

given s hoework or ewer robles cn

be ssigned in clss

Application Learning—It is ossible to

reduce the tie llotted ech dy or

liction ctivities or only colete

the once--week

Assessment o the Objective—The

students cn colete tke-hoe

ssessents.

Decelerating the Introduction o New Concepts

When students need ore instructionl tie to ster concet, it is necessry to decelerte the ce o thecurriculu This is oten necessry when students didnot receive the roer ount o instruction on the skillin the revious yer or they re new to the curriculund hve hd inil exosure to the skill This cn be

deterined by the lceent test




0


The ollowing section shows how techer cn tke vri-ous coonly included coonents o thetics les-sons nd llow or decelertion o the lesson concets

Daily Problems to Solve—Students cn solve thedily robles nd then work on revious dy’sroble s review

Vocabulary Activities—The techer cn ln ul-tile ctivities, extending into orl rctice nd

written rctice, to rctice the lesson vocbulryTeaching the Objective—The techer cn oer ul-tile oortunities or cooertive lerning ctivi-ties s students work on guided rctice o lessonconcets The techer cn lso oer ore guided-rctice robles to work on with rtners

Skill Practice—The techer cn give ore roblesThe techer ight choose to hve the students workon the robles with rtners in clss beore beingsked to indeendently work on the t hoe

Application Learning—The students cn senddditionl tie on the liction ctivities Theycn work on liction ctivities with rtners, insll grous, or indeendently

Assessment o the Objective—The techer cn usethe se ssessents s re- nd ost-tests or les-son concets



Post-Reading Reflection1. Wht tes exist or lnning thetics instruction in your grde

level, school, or district, nd wht unction do they serve? I therere no tes in lce, which tye o te would you like to seeileented, nd why?

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2. How do you ln your curriculu tieline ech yer? Wht will youdd or odiy on your current tieline bsed on the reding?

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3. Wht tyes o odifctions hve you de in your clssroo toccelerte or decelerte the ce o curriculu introduction?

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Managing theMathematics

Classroom

Involving Parents and Students toImprove Program EfficacyIt is iortnt to kee ositive nd oen lines o

couniction with rents regrding the eductiono their children s they rogress in understndingtheticl concets (Seeley, 2004b) The ollowingsuggestions will contribute to success or ll studentsin ny thetics course curriculu

ChapterThree




Parent Involvement

It is the techer’s resonsibility to hel rents under-

stnd secifclly how they cn ssist their childrenin succeeding in thetics course It would bevery benefcil or ll thetics techers to involverents in the rocess erly in the yer nd in ositivewys

1. Estblish good rort with rents by usingeective couniction skills

2. Enlist the hel o rents s inuentil llies

3. The Parent Letter (ge 46) cn be used ordted s resource or exlining thesignifcnce o succeeding in the theticscourse nd the necessity or students to stertheticl concets

4. Show rents how they cn be vitl rtners inencourging their children to rein ocused onlerning the necessry concets

5. Mke rent couniction ccessible bytrnslting docuents tht re sent hoe irents sek lnguge other thn English

Once ositive contct hs been de, rents will beore willing to hel techer i behvior or cdeicrobles rise I lerning nd behvior robles reinvolved, they re best resolved when techers ndrents work together to exine the context o therobles nd devise solution As soon s robleoccurs, it is iortnt to cll the rents to discusscourses o ction to reedy the sitution When rob-les re discussed, lwys ocus the converstion onthe behvior nd the secifc wys in which the rentcn ssist the techer in resolving the sitution



DocumentationDocuenttion is iortnt Thereore, the techer

should kee log o ech tie the rents re contctedso tht the techer cn reer to it i dditionl contctor dinistrtion involveent is needed The log lsoserves s esure o ccountbility or the techer, stu-dents, rents, nd dinistrtion The log will reecttht the techer notifed rents s reventtive courseo ction rior to giving iling grde or tking disci-linry ction The log cn lso show when the techercontcted rents or ositive rise nd eedbck TheTeacher/Parent Contact Log (ge 49) cn be used or

this urose The techer cn quickly record the dteo the contct in the frst colun nd the tie o thecontct in the second colun, check the toics tht werediscussed, nd record to who the techer soke Thetecher should dd ny fnl notes regrding ction stesor ollow-u in the lst colun By keeing one log orech student, the techer is ble to quickly reerence theresons, requency, nd dtes o contct

Student InvolvementCreting contrct with the students is deocrticrocess tht llows students to hve ownershi o theirroles nd their behvior in the clssroo Contrcts cnbe develoed ter greeing on the clssroo rules ndrocedures with students Furtherore, discussions cnbe initited regrding both techer nd student exect-tions Contrcts cn be used in the clssroo setting ors school-wide olicy The Social Contract (ge 48)cn be used or this urose, or school y develo

its own contrct The Student Letter (ge 47) cn lsobe used to exlin to students the signifcnce o r-ticiting nd succeeding in the thetics course ndthe uroses or stering the concets For youngerstudents, the student letter nd socil contrct cn bedjusted to ft their grde levels nd silifed or theirreding levels nd understnding

Managing the Mathematics Classroom




Parent LetterDer Prents,

Your child is beginning new thetics course The course is titled:

______________________________________________________________________

The rogr, texts, nd resources tht I will use re:

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________In this clss, your child will work on theticl concets guided by the contentstndrds Your child will study these concets using hnds-on lerning tools,rticiting in thetics rctice nd liction ctivities, nd lerning newvocbulry secifc to the concets I will ssess ll students requently to deter-ine whether concets need to be introduced, reviewed, or retught

Plese encourge your child s he or she lerns the oundtionl concets, nd helhi or her estblish quiet lce to study nd fnish ny necessry hoework Iyou re not ble to hel your child with thetics robles, lese hel yourchild write down which res he or she is struggling with so tht I cn direct les-sons towrd those needs

Feel ree to inquire bout your child’s rogress or let e know o ny robles sthey rise

Plese red the ttched rules or the clssroo, s well s the ositive nd neg-tive consequences or behvior Plese sign this letter nd return it

Sincerely,

____________________________________________

Teacher signature

____________________________________________

Parent signature




Student LetterDer Student,

You re strting new thetics course The course is titled:

______________________________________________________________________

The books nd resources tht you will use re:

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________This course will hel you build on wht you lredy know how to do in thet-ics You will rticite in the lessons in order to lern the iortnt theti-cl ides, skills, nd vocbulry you will need to succeed in the uture

It is iortnt or you to understnd the concets nd the vocbulry so tht youknow how to solve wide vriety o robles in your th books, tests, uturethetics courses, nd rel-lie situtions

You re resonsible or letting e know i you re struggling with soe o the con-

cets nd need ore rctice Also, lese fll out the ttched student contrctThis will hel e better understnd the wys tht you best lern in clssroo

Plese red the ttched rules or the clssroo, s well s the ositive nd neg-tive consequences or behvior Plese sign this letter nd return it

Sincerely,

____________________________________________

Teacher signature

____________________________________________

Student signature




Social ContractStudent Name: ____________________________________________

Teacher Name: ____________________________________________

Class/Period/Section: ______________________

1. How would you like the techer to tret you?

___________________________________________________________________

___________________________________________________________________

2. How would you like other students to tret you?___________________________________________________________________

___________________________________________________________________

3. How do you think the techer would like to be treted?

___________________________________________________________________

___________________________________________________________________

4. How do you wnt to resolve robles between you nd the techer?

___________________________________________________________________

___________________________________________________________________

5. Exlin wht the clssroo needs to be like in order or you to lern best

___________________________________________________________________

___________________________________________________________________

6. Exlin wht you cn do to hel ke sure the clssroo is goodlerning environent

___________________________________________________________________

___________________________________________________________________




T e a c h e r / P a r e

n t C o n t a c t L o g

S t u d e n t N a m e :___

_________________________

______

C l a s s / P e r i o d / S e c t i o n :___________________________

P h o n e N u m b e r :__

_____________

P a r e n t / G u a

r d i a n N a m e :_____________________________________________________

T o p i c s : ( C h e c k a l l t h

a t a p p l y . )

D a t e

T i m e

T y p e o f

C o r r e s p o n d e n c e

A b s e n c e s

G r a d e s

C o n d u

c t

T a r d i e s

O t h e r

S p o k e t o

C o m m e n t s / N o t e s



0


Lesson Delivery Strategies

The Aective FilterThe ective flter is best illustrted s hidden “screen”tht rises when stressul or eotionl situtions occurA techer cnnot see tht student’s “screen” is risedConsequently, ll the teching or tht dy is blocked tothe student becuse o his or her nxiety Oten distrct-ing ctors ro hoe get in the wy o the student’slerning rocesses However, t ties, the roble yhve to do with the thetics clss itsel Perhs

student is nxious bout his or her wek couttionor roble-solving skills Perhs n English lngugelerner fnds it diicult to corehend the techer’s stnner o seking A student with secil needs ybe nxious bout understnding the lesson concets ndbecoe discourged with the ce nd terils o theclss These issues directly ect the students’ ectiveflters

When the ective flter is u, it is not esy to lern

Thereore, techers need to work on lowering ectiveflters to tech the dy’s lessons by exining currentrctices tht y inhibit students ro lerning thenecessry content The ollowing tis y hel techerslower ective flters:

• Provide students with sentence res orresonding to questions or or exlining theirthinking This hels students exress theirthoughts using rorite cdeic lnguge

• Give students suicient tie to think, rocess,nd reherse with others beore sking questionor requiring n ctivity to be coleted

• Eloy ctive lesson ctivities such s odeling,guided rctice, ste-by-ste exlntions, rtnerwork, sll-grou ctivities, niultivectivities, nd the liction ges




• Utilize the strtegies oered in the dierentitioncoonent o this book

Every techer needs to consider nonthretening ndencourging wys to lower student’s ective flter inorder to roote lerning Lowering the ective flterleds to incresed bility to lern the lnned contentnd ly lerning towrd success in the theticscurriculu

Questioning

In ny thetics clssroos cross the ntion, n

observer ight fnd tht the techers re oten doing thejority o the tlking nd lso sking the jority othe questions However, students lerning theticsneed to be ctively thinking, engging their inds, ndsolving rel-lie robles This hens when techerslern good questioning strtegies

In thetics clssroo, questions should:

• kee students dyniclly engged in the lesson

• llow tie or students to exress nd rticulteides

• let students her lterntive solutions ndresoning ro eers

• erit the techer to check or understndingthroughout the lesson

• llow tie or the techer to ressess the ce nddirection o the lesson

• odel or students the roble, solution, nd

resoning rocess• rovide students with ny oortunities to

rctice

• otivte their individul lerning o theticlconcets




In order to roch higher levels o thinking, techerscn look to Bloo’s Txonoy In the lst fve decdes,

ny techers hve used this s sort o hierrchy oquestions tht dvnce ro less to ore colex lev-els o cognition Benjin Bloo, long with grouo eductionl sychologists, creted design o skillsto orgnize levels o cognitive thinking The lnws ublished in the book Taxonomy o EducationalObjectives, Handbook I: The Cognitive Domain, (Bloo& Krthwohl, 1956) with wht hs becoe oulrlyrecognized s Bloo’s Txonoy

The ollowing chrt gives exles o eective ques-tions using the fve NCTM strnds

Strand Goal o the Question Examples

Nuber nd

Oertions

Understnd dierent wys

nubers re reresented

nd used in rel lie

• Wht is the o ?

• Drw icture to exlin your

nswer

Nuber nd

Oertions

Understnd nuber

systes

• Exlin wht re in

• Estite your totl nd

core your nswer

Nuber nd

Oertions

Understnd the eects

nd reltionshis o

oertions

• How did you deterine ?

• Wht re ll the wys ?

Nuber nd

Oertions

Understnd estition in

roble solving nd

couttion

• Wht would hen i ?

• Is your nswer vlid, nd why?

Nuber nd

Oertions

Aly theories relted to

nubers

• Is your nswer resonble?

• How dierent is ro ?





Mesureent Core, contrst, nd

convert systes o

esureent


• Which unit o esureent would

you use? Exlin

Mesureent Understnd the need to

esure quntities in the

rel world nd use the

esures to solve

robles

• Exlin wht is in

• Wht reltionshi does hve

to ?

Mesureent Estite esureents inrel-world situtions

• In wht other sitution could ?• Is your nswer resonble?

Mesureent Understnd use nd selec-

tion o rorite units

o esureents nd

tools in rel-world

situtions


• How would hve been dierent

i it were sller? lrger? stronger?

Geoetry ndStil Sense

Visulize nd illustrtewys in which shes cn

be cobined, subdivided,

nd chnged

• Wht i were ?• Exlin why you coleted

the wy you did

Geoetry nd

Stil Sense

Describe, drw, identiy,

nd nlyze 2–D nd 3–D

shes

• How is like ?

• Core nd contrst

Geoetry nd

Stil Sense

Understnd the use o

coordinte geoetry tolocte objects in both two

nd three diensions, nd

to describe objects

lgebriclly


• Cn you construct odel thtwould chnge ?





Algebric

Thinking

Use exressions,

equtions, inequlities,

grhs, nd oruls to

reresent nd interret

situtions

• Wht would hen i we took

soething wy ro nd

relced it with ?

• Which exression shows ?

Exlin

Algebric

Thinking

Describe, nlyze nd

generlize vriety o

tterns, reltions, nd

unctions

• Why is grh useul to show

the kind o dt given in chrt?

• Cn you guess y “rule” nd

exlin?

Dt Anlysis

nd Probbility

Understnd nd use the

tools o dt nlysis or

nging inortion

• Wht i were ?


Dt Anlysis

nd Probbility

Identiy tterns nd

ke redictions ro

n orderly disly o dtusing concets o rob-

bility nd sttistics

• How would you orgnize to

show ?

• Wht would result i ?

Dt Anlysis

nd Probbility

Use sttisticl ethods to

ke inerences nd vlid

rguents bout

rel-world situtions

• Wht conclusions cn you interret

ro your grh (dt)?

• Is your nswer resonble?

Chart adapted rom Miami-Dade County Public Schools’ MathematicsQuestion Task Cards (http://www.doe.org)




Eective Questioning Strategies

The coon rctices o “techer sks question, stu-

dents rise hnds, techer clls on one student to nswerquestion” or “techer clls on student, techer sksquestion, student nswers correctly or incorrectly” si-ly do not rovide ll the benefts o good questioningin thetics clssroo In these situtions, thetecher only hers n nswer ro one student while therest o the students brethe sighs o relie tht they renot being clled on Oten, n observer notes tht it iscoon or techers to indvertently cll on the sestudents reetedly Students struggling with the con-tent or with the lnguge o instruction will be hesitntto rise their hnds or even resond to the techer’s ques-tions Even their ers o being clled on cn kee thero rocessing the concets involved in successullynswering the questions Most iortntly, these tech-ing rctices do not llow ll students to deonstrteknowledge nd engge in theticl resoning ndroble solving The ollowing questioning strtegiescn llow or ech o the benefts o good thetics

clssroo questioning listed bove

“Everyone Involved” Questioning Strategy

1. The techer sks question

2. The techer llows the students “think tie”to rocess their nswers to the question Inecessry, students re llowed to use er tosolve equtions nd rocess their own thinking

3. The techer directs the students to shre nswerswith rtners or sll grous The techerencourges the to coe to consensus on thecorrect nswer nd to be ble to ke cse orwhy their nswer is correct

4. The techer clls on soeone ro one o thegrous to give the grou’s nswer




The “Quick Check or Understanding”Cooperative-Questioning Strategy

1. The techer sks question nd dislys vriousresonses on the bord, one o which is correct

2. The students work with rtners to solve theroble nd deterine which nswer on thebord is correct

3. The techer hs the rtners disly their nswers

The “Quick Check or Understanding”

Individual-Questioning Strategy 1. The techer sks question

2. The techer llows the students “think tie”to rocess their nswers to the question Inecessry, students re llowed to use er tosolve equtions nd rocess their own thinking

3. The students record their nswers on sllwhitebord or iece o er nd hold the u so

the techer cn see the resonses

The “Leveled" Questioning Strategy

One urther issue to consider when questioning during thetics lesson is which level ech student is on Atecher should know the levels o theticl under-stnding o ech o his or her students In ddition, i nyo the students re English lnguge lerners, the techershould know ech o their bilities to sek, listen to,

red, nd write in English When the levels re roerlyssessed nd known, techers cn use this or everyone’sbeneft in lesson with the ollowing strtegies:

1. The techer sks question nd clls on studentwhose content-bility level or lnguge cquisitionlevel tches tht o the question being sked



Students o lower levels would be sked to nswerlower-level questions nd students o higher levels

would be sked to nswer higher-level questions 2. Ech question sked tches the level or is

slightly bove the level o the student sked

The entire clss benefts ro this dierentition strt-egy s the students re working with ech other to fndnswers nd rticiting in the lesson ccording to theircontent knowledge or lnguge levels

Pacing a Mathematics Lesson

A well-ced thetics lesson is very iortntEective techers ln lessons tht ctively enggestudents during instructionl tie s they exlore newtheticl concets This rocess begins with well-lnned nd ced thetics lesson tht includes theollowing:

Beore the lesson, time is not wasted due to student con-usion or poor planning. The teacher can “jump into”

the lesson concepts without delay.• The content stndrd nd/or lesson objectives re

visible These cn be either in “oicil” lngugeor in “student-riendly” lnguge Studentsshould begin ech lesson knowing wht they relerning nd why they re lerning it

• The techer lnned relistic lesson objectivestht cn be et by ost o the students during thegiven clss eriod or content block

• The techer gthered nd rered lessonterils or ll the lesson ctivities beore thestudents entered the clssroo

• Mterils re suitble or the ge nd bilitylevels, or odifed nd dted so tht they willbe corehensible





During the lesson, the teacher incorporates eective les-son components where students are on task and engaged

in learning new concepts.• The lesson begins with nticitory ctivities

tht connect students’ rior knowledge to thetheticl concets they will be lerningThe techer begins to crete enthusis ndhels students understnd how rior lessonconcets will build on wht they will lern inthe new lesson Iortnt vocbulry cn beintroduced

• The techer uses direct instruction nd techerodeling to introduce new theticlconcets

• The techer blnces techer-directed ctivities,note tking, guided rctice, nd interctivectivities s students lern bout new concets

• The techer continully checks or studentunderstnding during the lesson The techer

djusts the ce nd the length o tie necessryor the vrious lesson coonents By checkingor understnding, the techer cn be exiblewith the tie needed or reviewing or retechingconcets

• The students get the chnce to rctice skillsnd ly theticl resoning The techerguides student rctice

• Students re given the chnce to rctice witheers nd to use the our doins o lnguge—listening, seking, reding, nd writing—s theyrctice new theticl concets



At the end o the lesson, the teacher brings closure andties key mathematical concepts to the overall oundation

that is built upon as students approach mastery in thespecic content standards. The teacher makes decisionson whether students can continue with new concepts orwhether more review, practice, or instruction is neces-sary.

• Indeendent-rctice ctivities, inorl or orlssessent ctivities, or rojects tht tch thelnned objective, s well s the guided rcticetht the students engged in during the lesson re

used to deterine the students' understnding othe lesson

Common Mathematics ClassroomManagement Issues

The Physical Environment

While this y see like etty considertion, thehysicl environent in clssroo is oten the out-coe o techer’s outlook nd eductionl edgogyThe clssroo setu lso hs n eect on the wy stu-dents, rents, nd others eel s they enter the rooTechers need to exine the gols they hve or studentlerning nd then set out to rrnge the clssroo in wy tht will cilitte the tyes o lerning ctivitiestht will be ost otivtionl or student success




0


Walk through the classroom you have set up.

Consider these questions. • Cn students esily ccess their desks,

terils, textbooks, nd niultives? Howwill they get er, encils, or other suliesbeore or during thetics lessons?

• Is the clssroo set u or techer lecture only,or is there soe llownce or cooertivelerning nd interctive ctivities?

• Does every desk hve esy visibility o thebord or overhed rojector?

• Are the wlls cluttered? Cn students fndinortion they need to hel the reviewjor theticl concets?

• Are the wlls bre? Is there ny suort sstudents try to reeber key vocbulry,oruls, nd ides secifc to thetics?

• Wht colors hve been chosen in theclssroo? Wht generl eotion do theyconvey?

• Is there lce to clerly disly clssroorules, consequences (ositive nd negtive), nddily lesson objectives?

• Is there lce where student work nd eortsre celebrted?

• Is the generl look o the clssroo cluttered

nd rone to distrcting students, or clen ndorgnized?

• Is the hysicl sce esily dtble or directclssroo lecturing, cooertive ctivities,centers or grou ctivities, nd quietindeendent rctice o skills?



Interactive Learning Activities

Mny techers hesitte to use interctive lerning ctivi-

ties becuse they er the chos tht y ensue Otherslove to throw grous together to work nd then sit bckt their desks nd use the tie to correct ers, withoutonitoring the work tht is being done These re twoextrees to roching interctive lerning in th-etics clssroo Neither o the is recoended

The ollowing guidelines will help teachers ensure suc-cessully incorporating interactive learning activitiesinto a mathematics lesson.

1. The lnning nd rertion or interctivectivities needs to be done beore the studentsenter the clssroo All terils need to bessebled nd redy to use

2. Pln sile lesson or the frst tie tht ninterctive ctivity is done in clss Model ndrole-ly the correct nd incorrect roceduresexected o the students or this tye o ctivity

As the students becoe ore ilir withthe rocedures o the ctivity, ore in-deththetics content cn be introduced

3. The techer ust tell or reind the students othe ollowing every tie tht they will rticitein n interctive ctivity:

• the ctivity’s objective nd urose

• the execttions or student conduct

• the consequences or not eeting theexecttions

• the rocedures or the interctive ctivity





4. Follow through on the consequences Thestudents will soon lern tht they enjoy

interctive ctivities ore thn stright lecturingnd will robbly be ore t to ollow theconduct execttions

5. The techer is the ost ctive erson in theroo during n interctive ctivity The techeris constntly wlking round the roo in closeroxiity to ll the students, onitoring theresonses nd the student work, nsweringquestions, clering u ny isconcetions thtrise, nd giving eedbck

6. I the frst tie is not successul, try gin withtht same ctivity on nother dy Go throughll o the bove reinders gin nd give thenother chnce It tkes tie or students tobecoe ilir with the rocedures o newctivity

Mathematics Classroom Procedures

Every successul clssroo needs estblished rules,rocedures, nd consequences Every techer hs di-erent wy to eectively set these u in the clssrooClssroo rules should be estblished erly in the schoolyer nd clerly osted or students Consequences, bothnegtive nd ositive, re lso de cler to studentserly in the school yer However, wht oten is not socler re the rocedures The block o tie designtedor thetics instruction ech dy (or ech the-tics eriod) hs its own rocedures Every thet-

ics techer needs to ke decisions on the roceduressecifc to the thetics instructionl tie nd kethe execttions cler to students



• Wht will students do when they re fnished withn ssignent during indeendent rctice?

• Are students llowed out o their sets or nyurose during indeendent rctice or duringcooertive ctivities?

• Wht noise level is ccetble or the clssroo or rticulr tye o ctivity?

• How will students get er or shrened encilsi they need the?

• How will the techer distribute niultives orclcultors? How will these things be collected

ter their use?• Is restroo use or drinking wter ro wter

ountin eritted during instructionl tie?

• Wht should student do i he or she needsssistnce solving roble?

• During grou work, y grou discuss ides ndquestions with other grous?

• How will hoework be ssigned, collected, ndgrded?

• How will students ccess correct nswers tohoework or indeendent-rctice robles?

Maximizing Instructional MinutesIn tody’s eductionl environent o high ccountbil-ity nd high stkes, it is the techer’s resonsibility toxiize every ossible inute vilble or studentlerning Techers hve to throw out ny sueruousctivities tht do not directly relte to the content stn-

drds nd objectives, even i they hve lwys used theextr ctivities





For each day’s lesson:

• Choose the content stndrd

• Pln the lesson objectives to be covered in one

lesson eriod• Pln lesson ctivities—re-ssessent,

review, instruction, odeling, guided rctice,checking or understnding, indeendentrctice, cooertive rctice ctivities,liction ctivities, nd ssessent o lessonobjective

• Gther the terils nd resources

• Give students the big icture by writing the

lesson’s gols clerly or students to see

Be Prepared

The frst wy to chieve the gol o xiizing instruc-

tionl tie is or the techer to be well rered Itis lso helul to be well cquinted with the contentstndrds nd the course concets tht need to be s-tered by the end o the yer Tlk to techers in the yerhed to see wht skills they eel re ost iortnt orstudents to hve stered or the curriculu they willcover in the ollowing yer

Techers should send tie creting their dily lessonlns so tht instructionl tie is not wsted with re-

rtion during clss tie or with the ileenttion ounrcticed rocedures



1. I the techer hs the lesson plan ready and thematerials prepared beore the students enter the

roo, there is less wsted tie in which thestudents hve to wit or the techer to gther thedy’s instructionl resources

2. In thetics lesson tht involves ultilecoonents (re-ssessent, review, instruction,odeling, guided rctice, checking orunderstnding, indeendent rctice, cooertiverctice ctivities, liction ctivities,ssessent o lesson objective), the techer shouldlook over the various components beorehand tosee i ny dditionl terils re needed

3. The techer should plan or extra activities i thereis extr tie t the end o the clss eriod or orstudents who fnish erly

4. The answers to any lesson problems should besolved and accessible beore the techer begins thelesson This will hel the techer quickly ssessthe student’s nswers in order to inoint those

who still need hel

Utilizing Additional Instructional Time

Soeties lesson goes ster thn the techer ntici-ted The techer should lwys hve ew ctivitieslnned to ke the ost o dditionl instructionltie in the clssroo

• Explain the Concept/Summarize the Lesson—Hve the students work with rtners to exlin

the dy’s concet to one nother This givesdditionl rctice nd llows students the chnceto reherse the cdeic vocbulry essentil tothe concet Most students understnd concetbetter once they hve hd to exlin it to soeoneelse This is very benefcil or English lngugelerners nd struggling students





• Personal Agendas—Ech student hs chrt with list o ctivities tht re rorite or hi or

her to colete indeendently These ctivitiescn be bsed on skills needed or reedition orskills or ccelertion, deending on the studentWhen student hs coleted n indeendentctivity on his or her ersonl gend, he or sheust obtin the techer’s initils next to thectivity’s ne nd descrition on the chrt inorder to ove to new ctivity

• Solve to Earn a Pass—Ask ech student to solve sile roble orlly, or give n exlntion or

defnition s the “ss” to get into line in order toleve the clss

• Numbered Heads Together—Hve the studentsquickly or “Nubered Heds Together” grousby getting into grous o our nd nubering oone to our They ut their heds together tosolve roble given by the techer tht is bsedon teril covered on revious dy Theyust ke sure everyone understnds the nswer,

becuse the techer then rolls die or sins sinner to deterine which nuber o studentro ech grou will give the grou’s nswer

• Standardized Test Preparation—Give ech grouo our students set o nswer crds lbeled A,B, C, nd D Then the techer cn give the testrertion-tye robles to solve Ech ossiblenswer is lbeled A, B, C, or D It is esiest tohve these questions rewritten on trnsrencyIn their grous, ech student solves the roble

individully The grou then discusses theirsolutions in order to coe to consensus on thenswer The techer then directs ech grou tohold u the nswer crd it chose The wholegrou is resonsible or knowing the nswer swell s the exlntion or why the other ossiblenswers do not work



• Problem Solving Journal—Hve students writebout their understndings o vocbulry,

concets, or rocedures in journls• Real-lie Application—Students rctice rel-lie

liction ctivities nd ges ro reviouslessons in order to urther rctice with steredor nerly stered concets

Managing Classroom Time with EicientTransitions

In tyicl thetics lesson, the techer y direct

the students through vrious odes o ctivity A lessony include whole-clss discussion, ir work, sll-grou discussion, cooertive ctivities, ges ndliction rctice o skills, nd indeendent work

During given lesson, the techer will wnt to xi-ize instructionl tie o theticl concets ndiniize trnsitions ro one lesson ctivity to thenext Without cler execttions, rules, nd rocedures,trnsitions cn cuse conusion tht tkes wy ro

lerningTrnsition tie is coonly the tie when behviorrobles rise This is esecilly true i the roer ro-cedures hve not been tught nd regulrly rcticedEvery techer should strive to eiciently estblish trnsi-tion rocedures When trnsition tie in theticsclssroo is liited, tie or lerning nd lying con-cets nd skills re incresed Tking extr tie t thebeginning o the yer to odel nd role-ly correct nd

incorrect trnsitions will sve instructionl tie lter inthe yer This is true even or older students

When initilly rcticing trnsitions, s well s through-out the yer, the techer needs to rere the students toke eortless trnsitions





1. Be clear about expectations Tell the studentswht they will be doing nd check or

understnding beore llowing the to trnsitionto new ctivity

2. Reherse trnsitions ro vrious tyes oclssroo ctivities oten The students will needample opportunities to practice

3. Choose an auditory or visual signal to helstudents know when to chnge ctivities Whenusing signl or trnsitions, the studentsneed tie to think beore they re required to

ct The techer should give the signl, whichinstructs the to be silent, inor the o theexecttions, nd then give the signl ginTht gives the erission to ove to the nextctivity

Possible Signals to Use During Transitions:

• turn lights on nd o quickly

• squeky toy

• clssroo chnt

• counting bckwrds

• hold hnd u in ir

• wind chies or xylohone

• thubs u

• trin whistle

• rin stick• bell or gong



Post-Reading Reflection1 Reect on your teching style Wht teching rctices or behviors do

you hve tht y cuse students to ut u their ective flters? Howcn you work to chnge those behviors or teching rctices?

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2. Choose lesson tht you enjoy teching Crete t lest fve questionsusing the guidelines nd strtegies rovided or eective questioningMke sure tht you hve t lest one question or ech level o studentin your clssroo

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3. Choose n interctive ctivity you hve used in the st or crete new ctivity or concet you re teching this yer Mke list o therocedurl stes, terils, nd trnsition techniques you will use tollow the ctivity to run soothly

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ChapterFour

DifferentiatingInstruction

The Importance of DifferentiatedInstructionNot too long go, it ws thought tht students in these grde level shring the se clss lerned in sii-lr wys Tody, it is cler tht this viewoint is wrongStudents hve dierent lerning styles, coe ro di-erent cultures, hve dierent levels o lnguge bili-

ties, exerience vriety o eotions, nd hve vriedinterests Becuse students dier in cdeic rediness,techers hve relized tht they ust dierentite theirteching to better eet these diverse needs

Dierentition hs ny ces deending on the r-ticulr students nd techers involved, the outcoes




o these lerners, nd the structure o the clssrooenvironent (Pettig, 2000) Dierentition enco-

sses wht is tught, how it is tught, nd the roductsstudents crete to show wht they hve lerned Thesethree ctegories re oten reerred to s content, rocess,nd roduct Techers should dierentite content, ro-cess, nd roduct ccording to students’ rediness, theirlerning styles, nd their interests (Ntionl ReserchCouncil, 1990) I lerning exerience tches closelywith student’s skills nd rior knowledge (rediness),he or she will lern better Creting ssignents thtllow students to colete work ccording to their re-

erences (lerning styles) will hel lerning exeriencesbecoe ore eningul I toic srks exciteentin the students (interests), then they will becoe oreinvolved in lerning nd will better reeber wht theylerned To ke the thetics lesson ctivitiesost eective, techers should dierentite the lessonsNot ll students need to be engged in exctly the sectivity t exctly the se tie

Differentiated LearningAll students lern dierently nd struggle with dier-ent theticl concets Even the level o strugglingtht student exeriences vries Becuse o this, nyo the se reserchers who creted the Reding Firstinititive develoed syste o identifction knowns Resonse to Intervention (RTI) The RTI odel su-orts the ide tht techers should look or curriculrintervention designed to bring child bck u to seed ssoon s he or she begins hving robles “RTI hs theotentil then to llow disbilities to be identifed nddefned bsed on the resonse child hs to the inter-ventions tht re tried” (Cruey, 2006) Deending on thelevels o diiculty they re hving with the theticscurriculu, students re clssifed s Tier I, II, or III



Dierentiating Instruction

Techers who dierentite instruction need to:

• Allow n rorite ount o tie or lnning

nd rertion o instruction nd ctivities• Assess students oten nd use the results to guide

instruction nd rorite intervention strtegies,i necessry

• Mke secifc lns to eet vried student needsnd counicte those lns clerly to rents

• Utilize ny exible grouing strtegiesthroughout clssroo ctivities

• Vry the tyes o ctivities nd odes oinstruction used in the clssroo to eet thelerning needs o ll students

• Continully reect on ersonl teching strtegiesnd odiy the ethods o theticsinstruction i students re not resondingositively to the delivery ethods

• Ask or hel ro secilists, other theticstechers, rents, dinistrtors, nd nyone elsewho cn hel resond to student needs

Dierentiation by Specic Need

Tier I Students

Tier I students re generlly king good rogresstowrds the stndrds, but y be exeriencing teo-rry or inor diiculties These students y struggleonly in ew o the overll res o theticl con-cets They usully beneft ro eer work nd rentl

involveent They would lso beneft ro confdenceboosters when they re succeeding Although they reoving hed, ny robles tht do rise should be dig-nosed nd ddressed quickly in order to ensure tht thesestudents continue to succeed nd do not ll behind




Tier II Students

Tier II students y be one or two stndrd devitions

below the en on stndrdized tests These studentsre struggling in vrious res nd these struggles reecting their overll success in thetics clss-roo These students cn usully resond to in-clssdierentition strtegies nd do not oten need the helo student study tes

Tier III Students

Tier III students re seriously t risk o iling to eet

the stndrds s indicted by their extreely nd chroni-clly low erornce on one or ore esures o thestndrdized test These students re oten the oneswho re being nlyzed by soe tye o in-house studentssistnce te in order to look or overll interventionsnd solutions In the clssroo, these would be the stu-dents who re hving diiculties in ost o the ssign-ents nd iling ost o the ssessents

English Language Learners

Students who re English language learners re lern-ing concets nd lnguge siultneously They needto hve context dded to the lnguge While they yhve cquired socil lnguge skills, the lnguge othetics is very cdeic in nture This is oneo the ost iortnt keys to success with Englishlnguge lerners, s these students cquire the neces-sry vocbulry or greter corehension o the coursecontent

Below-Grade-Level Students

The below-grade-level students will robbly need theconcets to be de ore concrete or the They yneed ore work with niultives nd lictionges Furtherore, students who re struggling y



eel discourged by the nuber o rctice robles on tyicl rctice ge By giving the extr suort nd

understnding, these students will eel ore secure ndhve greter success

Above-Grade-Level Students

Mny o the objectives covered in thetics clssre new to the students All students need fr oun-dtion in the core knowledge o the curriculu Evenabove-grade-level students y not know uch o thisinortion beore lesson begins However, the ctivi-

ties nd end roducts cn be dted in order to be ro-rite to those students’ individul levels High-chiev-ing students cn crete dierent end roduct in orderto deonstrte stery or be given dierent ounts owork i stery o concet coes quickly





Differentiation Strategies

Strategies or EnglishLanguage Learners Strategies or Below-Grade-Level Students Strategies or Above-Grade-Level Students

• Always do the vocabulary

development component

and allow extra practice

to apply and use the

vocabulary with the

concepts

• Allow more time to

simultaneously process thelanguage and the content

• Use visual displays,

illustrations, and

kinesthetic activities

• Oer notes that are

partially flled in so that

students can ocus on

necessary inormation

• Start with concreteexamples and use

manipulatives

• Reduce the total number

o problems

• Plan or oral rehearsal with

partners o the academic

language behind the

mathematical concepts

• Evaluate the use o wordproblems Read them

aloud and emphasize

key words that indicate

procedural action

• Allow or partner work

• Allow partner work

or oral rehearsal o

solutions

• Allocate extra time or

teacher-guided practice

• Model oten, showing

them step-by-step how

to solve the problems

• Allow or kinesthetic

activities where they

organize the step-by-step

processes on ash cards

beore they actually use

the inormation to solve

problems

• Shorten the number o

practice problems in asingle work session

• Have easy-to-ollow

notes o the most

important procedural

inormation already

made up or these

students to add to

• Use activities centered

on students’ interests• Reerence the chart

on the next page,

depending on the degree

o diiculty a student is

having

• Oer accelerated

processing activities or

allow or these students

to skip practice

activities that they

already have mastered

• Shorten the number o

practice problems• Assign only the most

diicult problems

• Assign step-by-step

explanations o the

solution process

• Have the students

create notes and

procedural steps to

guide the rest o theclass

• Request oral

presentations o the

concepts, which will

beneft all students in

the classroom

• Have students create

games or practicing

concepts and skills



The chrt on this ge gives suggestions or dierentition bsed on the degree towhich student is struggling with theticl concets

Strategies or Tier 1Students

Strategies or Tier IIStudents

Strategies or Tier IIIStudents

• Use assessments to

identiy the areas in

which these students are

not at mastery

• These students would

beneft rom pair work

in which sometimes astudent is the teacher

(in the areas in which

this student excels) and

sometimes the student is

the peer learner

• Reteach concepts in a

dierent way

• Allow small groups to

study concepts together• I possible, ask or

parental involvement in

keeping these students on

task in assignments

• Oer additional practice

in the areas in which this

student is experiencing

diiculty

• Allow partner work orstudents to check the

work and build confdence

that they are on the right

track

• Use assessments to

identiy the areas in

which these students are

not at mastery

• Follow the suggestions or

below-grade-level students

listed on the previouspage

• These students would

beneft rom a beore-

school or ater-school

intervention program

• Reteach concepts in a

dierent way

• Oer extra practice in

struggling areas withstudy groups or peers

• Allow time or peer

tutoring

• Extended mathematics

instruction would beneft

these students

• These students require

extra intervention

programs

• Consider beore-

school or ater-school

programs, or extended

mathematics periodsto combat the risk o

ailure

• The school's student

assistance team can

determine i students

in this category might

need testing or special

education needs and

an individualized

education program

(IEP) or modifcations

during assessment





Grouping Strategies for the ClassroomWhen considering ech dy o thetics lesson ctiv-

ities, the techer needs to decide the best wys to grouthe students The techer should consider whether thecontent cn be best lerned nd rcticed in:

• whole-grou setting

• sll grous

• rtners

• individully

Small-Group LearningWith the sll-grou ortion, the techer needs toconsider whether to lce the students together withsiilr bilities (hoogeneous grous) or with vryingbilities (heterogeneous grous) It is generlly greedtht it is not eective to hve the students work in these grous or the entire rogr Rther, fexiblegrouping, where students re groued dierently orvrious ctivities nd ssignents throughout the dy,

llows students to get dierent eedbck nd suortro clsstes (Rdencich & McKy, 1995) It lsokees students ro singling out or lbeling the “slowgrou” or the “srt grou”

Paired Learning/Working with Partners

Giving students chnce to work with rtners llowsthe to hve ore tie to rocess new content nd rc-tice lying new knowledge in less stressul situtions

There re ny instnces when ired lerning is use-ul grouing strtegy

• Oten, when niultives re being used, twostudents cn sit side by side nd shre terilsso tht ech hs n oortunity to niulte theterils



• When students need to reherse inortion boutconcets they hve lerned or solve roble

together, they cn discuss the stes necessrybeore shring the work with the clss

• English lnguge lerners requently need torctice using the necessry vocbulry or echtheticl concet They need tie to sy thevocbulry words, use the in context, nd getiedite eedbck bout wht they sid

Independent Work

Working cooertively with others is instructionllysound, but it should not be viewed s the only wy toeectively tech students They lso need oortunitiesto exlore on their own nd to exine, question, ndhyothesize bout their own lerning Lerning on one’sown develos students’ confdence nd nturl curios-ity Students will hve higher success with indeendentwork i it directly reltes to the skills they were work-ing on during guided-rctice ctivities When studentsre engged in indeendent work, the riry gol o

the techer is to onitor their rocesses How closely techer onitors students’ work during indeendentrctice is usully good indiction o how well the stu-dents will ctully retin the inortion This is lso good tie or techers to give lower-bility studentsone-on-one or sll-grou instruction

Cooperative Learning

At ties it is rorite or techers to orgnize stu-

dents into sll grous in order to ccolish n c-deic tsk Within ech grou, the students re giveninstructions or coleting tsk This is clled coo-ertive lerning It is dierent ro coon “grouwork” in which students just work in grou together,without individul ccountbility nd roles In order or




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the grou to be considered “cooertive,” ech studentneeds to be ssigned role Thereore, no tter wht is

being sked o the, ll students re rticitingTo ke cooertive grouing eective, techers needto consider the students’ chieveent levels Thestudents cn be groued into heterogeneous groups inwhich students’ bility levels re vried, homogeneousgroups in which students hve the se bility levels, orfexogeneous groups tht cobine both heterogeneousnd hoogenous grous within one lesson Within thetics clssroo, ll three tyes o grouing tech-

niques cn be used Techers should evlute lessonsto deterine which technique will work best or echindividul lesson

Soe techers worry tht not ll students will rticitein cooertive lerning grous There re stes tht tech-ers cn tke in order to ensure cooertive rticition

1. Teach appropriate social skills to students bydeonstrting these behviors nd role-lyingwith students

2. Plan activities where ll students re

• working concurrently,

• ositively encourged to hel ech other,

• given roles tht re codeendent on the otherstudents, nd

• individully ccountble or the work tht isturned in

3. Use a student-led group evaluation where studentscn orlly or inorlly evlute

• how they worked together,

• wht their grou did well, nd

• wht their grou could do to irove



Choosing a Grouping Strategy

The chrt below nlyzes soe o the ost eective

grouing strtegies to eloy bsed on the techer’sgol or instruction Plese note tht techer couldchoose one grouing strtegy or rticulr gol in thelesson or use ultile strtegies to eectively rech thesingle gol

Goal or Instruction Grouping Strategy to Employ

Deliver lesson instruction to llstudents

Whole-grou instruction

Check or understnding during lesson Pir, sll grou, or indeendentctivity

Hve students think bout nd rcticegiving n nswer to question beoreresonding to techer

Pir shring ctivity

Hve students rctice concet Pir, sll grou, or indeendentctivity

Oer rctice or using cdeic

lnguge o thetic vocbulry

Pir shring ctivity or sll-grou

ctivity

Hve students work in grous whiletecher identifes jor needs or re-teching

Hoogenous sll grous

Hve students tech nd lern roech other while rcticing concet

Heterogeneous sll grous

Hve students orlly ly concet Pir or sll-grou ctivity

Use niultives Pir or sll-grou ctivity

Hve students ly concet Pir, sll-grou, or indeendentctivity

Oer sel-suorting rctice tie Indeendent ctivity

Hve students deonstrte stery o lesson concet

Indeendent ssessent





Group ActivitiesThere re vriety o grou ctivities tht engge lern-

ers nd llow techers to dierentite the cing, rc-tice work, nd ctul instruction during lessons

• Students re oten very socil creturesCooertive grouing cters to this socilside o their ersonlities It lso gives theoortunities to rctice the cdeic vocbulrysurrounding the concets Furtherore, grouctivities oten boost the confdence nd risk-tking tht will lter ect individul student

work

• Physicl ges hel kinesthetic lerners reeberthe key concets better nd get students out otheir sets

• Plying bord ges or crd ges chllengesthe students nd encourges lerning Plyingges s whole clss on the bord or overhedcn serve s or o inorl ssessent s thetecher checks or student understnding These

ges re un nd eductionl nd lso helensure tht ll students re successul

• Lrge-grou ges rovide se environent orlower-level students to tke risks These studentsre suorted by their tetes nd cn stretchtheir thinking outside o their coort zones

• Creting ge bsed on the concet o thetecher "coeting" ginst the clss llowsstudents to rticite nd eel good bout beting

the techer This ge works well using hngns odel nd dting the skills the students reworking on to ft tht odel I the clss nswersthe question or solves the roble beore thehngn is colete, they get oint I theycn’t nswer the question or solve the roble,the techer gets oint The clss lwys wins



Mny techers cringe t the suosed chos o grouctivities However, techers cn ollow these guide-

lines to hel structure grou ctivities nd rootective lerning

1. Exlin the proper expectations and group-work rules. These ctivities cn rovide thenecessry rctice nd rocessing tie orcorehension o lesson concets

2. Students need practice with the procedures involved in grou-work ctivities

3. It is iortnt to provide the appropriate timelimit or students to work in sll grous orirs to colete roble sets

4. Grou work llows students to serve as peerteachers by heling one nother lern theconcets Soeties the techer cn evenlern ro the students by noting how thestudents exlin the ides to one nother

5. Allow the students to evaluate how eectivelythe grou work ided their corehension othe theticl concets

I grou ctivity did not work the wy the techerintended, it should be coleted gin on the ollow-ing dy This y hel students understnd the rules,execttions, nd consequences ore clerly Oten,students re excited bout the chnce to work in grous

nd will cooerte nd ollow directions when they ullyunderstnd the execttions nd consequences





Post-Reading Reflection1. How re dierentition strtegies or English lnguge lerners siilr

to dierentition strtegies or below-grde-level students?

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2. List three wys you cn dierentite thetics instruction orstudents in your clss who re eroring bove grde level

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3. Choose one lesson tht you ln to tech this yer Decide on grouing strtegy tht will enhnce the lesson nd rovide eningullerning oortunities or ll students

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ChapterFive

DevelopingMathematical

Vocabulary

Specialized Mathematical Vocabularyfor All LearnersIn thetics, vocbulry is highly secilizedThese words re oten not encountered in everydy lieThereore, ll students need n exlicit introduction nd

exlntion o these vocbulry words in order to be bleto ly the to their understnding o theticlconcets The tsk is even ore diicult or Englishlnguge lerners These vocbulry words re not tyi-clly the words tht English lnguge lerners will lernduring their structured English Lnguge Develoentclss eriod Thereore, it is u to the content techer




who teches thetics to ke certin tht Englishlnguge lerners lern the necessry vocbulry in order

to chieve corehension o theticl concets ndcurriculu

Furtherore, the dierent res o thetics (eg,nuber sense nd theticl resoning) nd thevrious discilines (eg, lgebr, trigonoetry, geoetry,nd clculus) hve dierent coiltions o secilizedvocbulry words Soeties there is overl withwords cross theticl res nd discilines, butoten there re new words tht re secifc to just one

theticl re or disciline It is vitl to understndthe vocbulry or secifc disciline in theticsbecuse this knowledge ids in ccess to the core cur-riculu All students re required to deonstrte s-tery o the concets, nd this will only be ossible i theyfrst chieve understnding o the vocbulry words thtexlin, describe, nd cilitte ech o the theticlconcets

Vocabulary Development for EnglishLanguage LearnersIt is iortnt or ll students lerning theticsto be ilir with the secilized vocbulry ebed-ded within the rctice nd liction o the con-cets English lnguge lerners esecilly need consis-tent, structured instruction in lerning English (EnglishLnguge Develoent) They lso need well-lnned,sheltered instruction throughout content lessons ndeective ctivities to develo thetics vocbulry(Den nd Florin, 2001)

It is not enough to give the students list o words ndhve the look u the defnitions in dictionries or text-book glossries Students who re struggling with lern-ing lnguge re not going to fnd the rocess esier bysily being given ore words to sort through Wht



Developing Mathematical Vocabulary

English lnguge lerners need re context-ebeddedlesson ctivities tht cquint the with the necessry

words or corehension o the content nd llow theto rctice the use o the words in ctivities tht snlistening, seking, reding, nd writing ctions

Mthetics techers need to be cogniznt o thelnguge diiculties students hve who re lerningEnglish Mny thetics techers believe Englishinstruction is the job o the English techer However,the English techer is not ocusing on the secilizedthetics vocbulry nd the contexts rorite to

it during English clss There re other necessry ln-guge coonents or the students to lern t tht tieThereore, it is necessry or the thetics techer tooer the scolding students need or ccess to the-ticl concets By knowing the lnguge level o echindividul student, the techer cn ln rorite les-sons tht blnce vocbulry develoent, instruction,odeling, interctive ctivities, nd suort

Types o Language Prociency

One jor concet thetics techers need to recog-nize is the dierence between the two tyes o lngugerofciency or English lnguge lerners Ji Cuinscoined the two tyes o lnguge Bsic InterersonlCouniction Skills (coonly reerred to s BICS)nd Cognitive Acdeic Lnguge Profciency (co-only reerred to s CALP) (Crword, 2004) BICS reersto student’s social language Profciency in socil ln-guge requires no secifc instruction nd tyiclly tkes

s little s three yers to cquire This knowledge cn becquired through edi sturtion, usic, nd socil sit-utions Students cn esily see very cble in socillnguge becuse they need it to survive A techer cnoten be tricked by student’s level o BICS The techery her student chtting with riends nd conversewith tht student beore or ter clss These convers-




tions y led the techer to believe tht the student hs fr grs o the English lnguge However, tht se

student is iling ssessents, struggling to kee u withssignents, nd unble to write well bout theti-cl content This student lcks CALP

CALP, or academic language, tkes seven or ore yersto cquire CALP is rofciency in the lnguge o thecontent res nd o the clssroo A student who hsstrong CALP hs cond o the use o English withincontent res In thetics, student with stronglevel o CALP is ble to understnd key vocbulry, use

it in correct context, nd write well bout his or herunderstnding o theticl concets nd roceduresThis level o cdeic lnguge is not lerned esily ndintuitively, like BICS This lnguge rofciency onlycoes with exlicit instruction nd lnned objectivesby the content techers Tht is one reson why vocbu-lry-develoent lessons re so iortnt or techers oEnglish lnguge lerners to incororte into thet-ics lessons

Levels of Language AcquisitionEective thetics techers o English lngugelerners lso need to know the levels o lnguge cquisi-tion or ech English lnguge lerner in the clssrooThe rorite lesson or student who hs just ovedinto the country is going to look very dierent ro therorite lesson or n English lnguge lerner who isclose to being considered uent in the English lnguge

Mny sttes hve oicil ssessents ent to deter-ine the level t which student is ble to use EnglishThese ssessents cover the res o listening, seking,reding, nd writing Soe o the ssessents hve serte score or ech doin o lnguge nd then coosite score tht cobines the overll level t whichthe student is eroring in English



This chrt gives inortion bout the levels o lngugecquisition o English lnguge lerners nd suggestions

to techers or how to eet those students' needs


BeginningEarly Intermediate/

IntermediateEarly Advanced/

Advanced

These students all into awide range o limited Englishcomprehension Theyhave minimal or limitedcomprehension with noverbal production Some

beginning students are justable to give one- or two-word responses Some arebeginning to comprehendhighly contextualizedinormation and are ableto speak in very simplesentences

These students have goodcomprehension o inormationin context They mayexhibit restricted ability tocommunicate ideas, but theycan usually reproduce amiliar

phrases in simple sentences Asthey improve in profciency,they improve in the ability tocommunicate ideas, althoughthey may exhibit errors inproduction, especially whenwriting or speaking about highlyspecialized content

These students may “trick”teachers into thinking thatthey are uent in EnglishBut they oten struggle whenthey have to explain theirunderstanding o an answer

or write out the procedureso a concept They lack theability to ully communicatehigher levels o thinking incontent-specifc academiclanguage

Teachers should:

• Provide a lot o context

or mathematicalconcepts

• Use physical movementand visuals to explainmathematical vocabulary

• Use sentence ramesto help students placemathematical conceptsinto context

• Ask yes/no questionsor questions where theanswers are embedded inthe questions

• Always includevocabulary-developmentactivities

Teachers should:

• Provide visuals and context

or mathematical concepts• Encourage cooperative and

interactive activities inorder to make mathematicalcontent comprehensible

• Ask questions that requiresimple sentences withknown vocabulary

• Elicit simple explanationsand summaries

• Support writing and readingtasks

• Oten include vocabulary-development activitiesand the proper ways tocommunicate, using themathematics vocabulary

Teachers should:

• Provide structured

group discussion oconcepts beore requiringindividual practiceand writing aboutmathematical reasoning

• Elicit explanationsthat analyze andsynthesize mathematicalinormation

• Model the higher levelso thinking with use o

specialized vocabulary• Regularly practice

vocabulary-developmentactivities and then takethe students to thenext levels o higher-level thinking using thevocabulary



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Integrating Vocabulary Developmentinto InstructionInterctive vocbulry-develoent ctivities should beregulrly integrted in thetics lessons in ll clss-roos These tyes o ctivities re esecilly necessryor clssroos with English lnguge lerners, studentsstruggling with theticl concets, or ny studentswho hve not shown stery o the vocbulry

Techers should ollow these guidelines beore beginningto tech the vocbulry ctivities deonstrted in this

resource• Decide how long to use one vocbulry ctivity

beore introducing new one

• Pln or extr teching tie when newvocbulry ctivity is being introduced

• Choose n rorite ctivity in order to eetthe llotted clssroo tie or the rticulrlesson

• “Frontlod” the lesson with vocbulry words

beore the students need to ly the duringrctice ctivities nd robles

• Revisit st vocbulry words in ddition tocurrent words, i lesson requires the

• Reet the ctivity with the se words or newwords i it needs to be rcticed ew ties beorethe students cn correctly eror the ctivity

• Clerly stte the urose or n ctivity, thebehvior execttions, nd the consequences or

not ollowing the execttions i students will beout o their sets

Vocabulary-Development Activities

These ctivities oten cover the our doins o ln-guge—listening, seking, reding, nd writing It is




iortnt or students to hve rctice using new vocb-ulry in ultitude o wys in order or the to tke

ownershi o word nd be ble to use it indeendentlyReetition is s iortnt s exosing students to wordsin ultile wys

During these ctivities, it is iertive tht the techer isctively onitoring the rocess The role o the techeris to enorce rocedures (or exle, in n orl rcticectivity, the techer needs to ke sure the students reorlly rcticing) nd to cler u ny vocbulry iscon-cetions or isunderstndings

These ctivities cn be used s “rontloding” ctivities,s rctice centers or students to use with rtners orsll grous, or s reedil or enrichent ctivitiesduring extr instructionl tie

Activity 1

Chart and Match

1. The techer writes the vocbulry words secifcto the dy’s lesson on the bord

2. The students need to hve three-colun gridlbeled with the ollowing hedings: Word;Illustration or Example; nd Defnition orDescription The students write the vocbulrywords down the let side o the grid One wordgoes in ech row

3. Next, the techer introduces the words nd

leds whole-clss discussion The techeruses exles nd drws ictures to show thevocbulry words

4. The techer then directs the students to drw icture or to write n exle o the vocbulryword in the iddle colun o the grid, next to thecorresonding vocbulry word




5. In the fnl colun, the clss decides on wy todescribe or defne the word This should not be

dictionry defnition Rther, ter the discussion,the students cn write their understndings o theword or the techer cn hel the write student-riendly exlntions

6. Ater reviewing the fnished grid, the studentscut u the squres With rtners, they cnwork together to lce the three sections orech vocbulry word (the word, the ictureor exle, nd the defnition or descrition)together

7. The techer wlks round to onitor rogress ndcler u ny isconcetions With extr tie, thetecher cn use one grid or whole-clss ctivity

8. The techer hnds ech student one iece roone cut-u coleted grid I there re not enoughieces or everyone, the techer cn substituterevious vocbulry words or reet vocbulrywords

9. The students wlk round, red their ieces toother students, nd trde crds

10. Then the techer directs the students to sto,red their fnl crds, nd fnd the other twostudents who hve their tching coonentsEch grou o three (one student with the wordcrd, one student with the illustrtion or exlecrd, nd one student with the descrition ordefnition crd) will stnd together nd resent the

vocbulry word to the clss




Activity 2

Vocabulary BingoIn this ctivity the students crete their own bords,which iniizes rertion or techers

1. The techer writes the vocbulry words secifcto the dy’s lesson on the bord nd gives echstudent blnk three-by-three grid

2. The students re directed to write one word inech squre Ech student will hve his or her

words written in dierent squres, in dierentorder With ety squres, the techer cn chooseto hve the students write revious theticlvocbulry words, write the vocbulry wordstwice in two dierent squres, or llow thoseety squres to be “ree” sces

3. The students re encourged to drw sllictures or exles next to the loction wherethey lced the word on their bingo chrts i therere ny words they know

4. Then the techer leds clss discussion o theenings nd exles o the dy’s vocbulrywords The students get ore tie to writeor drw exles to hel the understnd thevocbulry words they re not ilir with

5. Next the techer strts the bingo lerning ctivityThe techer reds descrition o the vocbulryword or shows n exle or reresenttive

icture6. The students locte the word nd cover it with

rker Cut-u ieces o er or sllthetics niultives cn be used I thestudents wrote the se vocbulry word twice,they cn cover both sces




7. The techer should wlk round to onitortht students re covering the correct words

Deending on tie, the techer cn decidewhether students re working to cover singlerow, two rows, or “blck out” their bingo bords

8. With extr tie, the “winner” cn review thewords nd defnitions out loud or the benefto the whole clss Also, the techer cn hvestudents review the defnitions o their coveredwords with rtners

Activity 3

Which Statement Is Inaccurate?

In this ctivity, ech vocbulry word secifc to thedy’s lesson will be used in our written sentences Thetecher cn do this hed o tie, or sk the students towrite the sentences once they hve been introduced tothe vocbulry For ech vocbulry word, three ccu-rte theticl sentences will be written nd oneinccurte thetics sentence will be written on

trnsrency (See the exle below)

1. The techer dislys our sentences on noverhed rojector Three re ccurte nd one isinccurte The sentences re nubered one toour

2. The students will work in grous o our Thegrou will red the sentences loud

3. Ech individul will decide which sentence is

inccurte They will ech hold the nuber withtheir fngers while hiding it ro their grou (eg,holding three fngers u—but hidden ro thegrou—i nuber 3 is the inccurte sentence)

4. When everyone hs decided, ech individulwill show his or her conclusion to the teThe te will discuss the nswer in order to




rech consensus When the te hs reched consensus, one student will write down the

nswer on sll whitebord or iece o er5. When ll tes hve reched consensuses, the

techer will sk ech te to disly its nswerto the rest o the clss The clss cn discuss echte’s results

6. With extr tie, the techer cn hve the tesconvert the inccurte sentence to n ccurtesentence by king rorite chnges

Sle sentences using the vocbulry word raction:

1. The nuber is raction

2. The nuber ¾ is raction

3. The nuber 65 is raction

4. The nuber 7 5 is raction

Answer: #3 is the inaccurate statement.

Activity 4

Sharing Markers

This is shring ctivity Oten, in grou-shring situ-tion, there re students who love to do ll the tlkingnd students who do not rticite t ll In this ctiv-ity, everyone shres vocbulry sentences eqully

1. The techer shres the vocbulry words secifcto the dy’s lesson The techer gives three toour rkers to ech student These cn be sllthetics niultives, sll ieces o er,sll cndies, or ny other sll objects Besure the techer checks ll the students' ediclors or llergies beore bringing ood into theclssroo




2. The students or sll grous o three to fvestudents

3. The techer reviews the rules:

• Every tie student sys sentence with vocbulry word in it, he or she gives wy rker

• The students hve to use ll o their rkers

• Once student's rkers re “sent” he orshe is not llowed to sy nything ore

• Students need to be resectul nd listen to

the vocbulry sentences ro ll o theirclsstes

4. The students generte sentences using the lessonvocbulry words until ll students hve “sent”their rkers

The sentence cn consist o defnition or n exlesentence The techer y wnt to shre execttions ounccetble sentences (eg, I like rctions)

A student could choose to sy ll three sentences withthree vocbulry words t once Then tht studentwould be fnished seking nd would be directed tolisten to the other students in the grou Or, studentcould choose to wit until ll o the other students refnished beore shring his or her sentences The techershould exlin tht, idelly, the students will tke turns,shring one sentence t tie This ctivity ensurestht those who need ore suort cn tke little ore

tie to or their sentences In this ctivity however,the students should not go round in circle becuseEnglish lnguge lerners or struggling students yneed ore tie listening beore they re redy to shre




Activity 5

Sentence Frames or VocabularyThis ctivity will vry deending on the vocbulrybeing introduced It is best exlined using secifcexle, but cn be dted to ny set o theticsvocbulry

1. First, the techer shres the vocbulry wordssecifc to the dy’s lesson

2. Then, the techer shres sile sentence tht

res the vocbulry in roer theticlcontext

3. The sentence re hs blnks in which thestudents will substitute inortion The techershould write the sentence re on the bord oron sentence stri The techer odels coletesentences using the vocbulry

This exle uses the sle vocbulry

words equation nd equivalentWhen techer is teching these words, coon wy to exress the concet is tosy:

The equation _____ is equivalent to theequation _____

4. The techer then writes sle nswers to ut inthe blnks

For this exle, the techer writes: 3 + 5 = 8;6 – 2 = 4; 4 + 0 = 4; 9 – 1 = 8




5. The students work in irs to rctice orllyrehersing the vocbulry with the right

substitutions Younger students would beneftro hving ctul slis o er to nullylce into the blnks on their sentence strisThen, they could orlly reherse the inortion

In the exle, the students would rc-tice sying:

The equation 3 + 5 = 8 is equivalent to the

equation 9 – 1 = 8The equation 6 – 2 = 4 is equivalent to theequation 4 + 0 = 4

6. The techer directs irs o students to shre thenswers they ce u with

7. The rtners then work to coe u with othersubstitutions to correctly use with the sentence

re They rctice together nd then shre withthe clss




Post-Reading Reflection1. Why is it iortnt or students to develo thetics vocbulry?

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2. How re English lnguge lerners nd below-grde-level studentssiilr in the re o vocbulry develoent?

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3. Identiy two wys you cn integrte vocbulry develoent into yourinstruction

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10

ChapterSix

Building ConceptualUnderstanding with

Manipulatives

Management of MathematicsManipulativesReserch reetedly shows tht students gin oreconcetul understnding nd re ore successul indeonstrting stery o concets when they hve hd chnce to concretely exerience theticl con-cets using niultives In ddition, when studentsuse the niultives, they eror better cdeicllynd hve ore ositive ttitudes towrd thet-ics (Leinenbch nd Ryond, 1996) However, nytechers, esecilly iddle school nd high school tech-ers, shy wy ro using niultives



10


Oten techers er tht students will isbehve ndly with the ites, rther thn ocus on the lesson

concets Techers do not lwys understnd how touse niultives eectively in order to ke bstrctconcets concrete nd they y dred the extr tienecessry to rere, ss out, nd collect the niul-tives However, s students gin ore exerience usingniultives, they will coe to recite the rcticend liction tht niultives dd to their th-eticl lerning exerience Thereore, it is iortntor techers to consider the use o niultives

The ollowing section oers generl suggestions or n-ging niultives in the clssroo

• First nd oreost, be clear about expectations while using niultives Mke sure thtstudents understnd the consequences orisusing the niultives (eg, tking thewy)

• Extra time will be needed the rst time thtthe clss exeriences using new tye o

niultive The techer should ln orthis extr tie in order to oer roriteexlntions o how to use nd hndle the newniultives The techer lso needs to lnextr tie or the students to be introduced to theniultive nd to rctice using it

• Beore working with niultives, give studentstime or ree exploration o the terils One totwo inutes is lenty o tie Children’s nturlinclintion is to ly nd exlore This willgive the tie to do so beore ocused work orinstruction needs to begin

• Use resealable bags or plastic bins to grou ndorgnize the niultives together Lbel thebgs nd bins nd lce the on shel Mke



10

sure requently used niultives re within therech o the students This will llow the to

gther their terils indeendently i rorite• I niultives re coonly used in small-

group settings, the techer should prepare bags oseparated manipulatives beore the students arrive

• I niultives re used or whole-group instruction, hve t lest one set o niultivesor ech student or or ech ir o students

• Middle school nd high school students shouldbe exected to clean up and reorganize the

niultives in the bgs or bins or the nexteriod or the next tie they re to be used Thetecher will need to llocte clss tie or this

• Crete trnsrency set o niultivesor modeling on the overhead I n overhedrojector is unvilble, ke lrge set o theniultives nd bck the with elt, Velcro®,or gnets to disly nd odel with on bord

• Use labeled, colored pocket olders to keectivities with ultile coonents togetherStore olders in crdbord boxes or lsticcontiners

• Laminate the games or instructions on how to usemanipulatives to reserve the or ultile uses

• Display charts and instructions on a bulletinboard in the clssroo These chrts should becreted beore the students enter the roo orinstructionl tie

• Occsionlly n ctivity requires the use o nyites (niultives, er, rkers, encils,etc) or the niultives re very loud whenbeing used on tble surce Give the students sll iece o elt to use s mat or theirmanipulatives This will reduce the overllclssroo noise nd enble students to styorgnized when working with their terils

Building Conceptual Understanding with Manipulatives



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Types of Mathematics Manipulativesand How They Are UsedMniultives y vry with the thetics course,but those in the ollowing chrt re coonly used

Type o Manipulative Possible Uses in a Mathematics Classroom

Rulers etric nd stndrd

• esureent

• length

• stndrd nd nonstndrd units

• selecting rorite tools o esureent

• coring etric nd stndrd esureents

Calculators • clcultions

• nuber nd oertions concets

• lgebr equtions

• roble solving

• th cts

• ddition, subtrction, division, nd ultiliction

• re

Computers • nuber nd oertions concets• equtions sredsheets

• grhing

• roble solving

• lgebr equtions

Geoboards • geoetry

• sorting nd clssiying

• describing

• drwing

• syetry• stil resoning



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Number Lines • nuber nd oertions concets• ositive nd negtive nubers

• rctions

• ordinl nd crdinl nubers

• ddition nd subtrction

Number Cards • nubers nd oertions concets

• ositive nd negtive nubers

• ordinl nd crdinl nubers


Dice • liction ges

• th-cts ges

• ddition, subtrction, division, ultiliction

• nuber sense

• rediction nd sttistics

Scales etric nd stndrd

• esureent

• stndrd nd nonstndrd units

• weight

• selecting rorite tools o esureent• coring etric nd stndrd esureents

Pattern Blocks • nuber sense

• whole nubers

• lgebr

• rctions

• stil visuliztion/estition

• tterns

• reresenttion o dt nd concrete objects

Base 10 Blocks • nuber sense

• lce vlue


• re nd volue



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2-D Shapes • geoetry• sorting nd clssiying

• coring nd ordering

• recognizing shes

• describing

• drwing

• re nd erieter

3-D Shapes • geoetry

• sorting nd clssiying

• coring nd ordering• recognizing shes

• describing

• drwing

• re, erieter, volue

Play Money • equtions with oney

• rel-lie situtionl word robles

• roble solving

FlashCards

• nubers nd oertions concets• th cts


Linking Cubes • nuber sense


• coring nd ordering

• re nd volue

Fraction Bars • coring nd ordering

• ddition, subtrction, division, nd ultiliction• nuber sense

• fnding equivlent rts

• rctions



10

Preventing Manipulative DependencyAnother coon er ong techers using niul-

tives is tht students will becoe deendent on their useStudents need to hve uent rocedurl bilities Theyneed to coute nd clculte equtions correctly nduse oruls nd theticl rules switly However,this rocess usully coes ore redily when studentsre exosed to ultile reresenttions o concets byfrst exloring the concretely nd then lerning howto roceed using the exeriences they gined ro usingniultives

Mniultives re ent to hel students use concreteobjects to corresond to theticl ides Techersshould devise lessons in which students lern to org-nize their thinking in concrete wys However, techerslso need to ove students st the concrete into icto-ril nd bstrct thinking

Mniultives cn be gret tool or engging studentsin lerning new concets They re esecilly helulwhen techers re trying to rech kinesthetic, visul, nd

English lnguge lerners

Sily lcing the niultives in ront o students isnot enough to gurntee tht they will lern ro theAlwys roviding the, without instructing students inhow to begin to solve the se robles without the,will lso rove detrientl to students when they relter ssessed without hving ccess to the niul-tives Techers need to work with students over tieto hel the connect the object, the sybol, nd the

theticl concet tht is reresented

The inortion on the next ge highlights the rogres-sion techers need to ollow in order to revent niu-ltive deendency




10


1. Introduce niultives Explain the role o

niultives, how they connect to n overlltheticl concet, nd the execttions orstudent use

2. Give students practice in using theniultives to exlore the theticlconcet

3. Model the theticl concet withpictures tht relce the niultives Mkeconnections between the niultives nd the

ictures

4. Give students practice in using pictures (s substitution or the niultives) to exlorethe theticl concet

5. Tech the abstract qualities o the mathematicalconcept Mke connections between theictures nd the equtions or oruls

6. Provide le oortunities to practice solving

the equtions without pictures or manipulatives 7. Return to niultive use when needed, but

repeat the entire process to ove students tobstrct thinking nd roble solving



10


Post-Reading Reflection1. Wht ws your view o niultive use in the clssroo rior to

reding this chter? I it chnged ter reding this chter, exlinhow

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2. Choose niultive you hve ccess to in your clssroo Write word roble tht involves using tht niultive to solve the

roble

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1

ChapterSeven

Teaching theProcedure

Building on UnderstandingBy its very nture, thetics instruction ilieslerning skills tht ust build on one nother (Dennd Florin, 2001) Students ust use wht they lredyknow in order to ly it to ny new lerning Thereore,it is vitl tht techers consider the process o buildingon understnding s they roch their theticscurriculu ech yer

1. Start with the mathematics content standards, notthe textbook.

• The next grde level’s techer will exectstudents to coe in with these skills coveredin the content stndrds These concets ressessed t the end o the school yer



11


• A techer cnnot oen theticstextbook nd tech cover-to-cover becuse

textbooks y cover nonrequired concetsnd oten give only little concetintroduction, which ight not be dequteor struggling students

• The content stndrds oten introduce concet in one grde nd then revisit it t ore in-deth level the ollowing yer(s)

2. Find out what they already know.

• Techers should look over the concets thtwere suosed to be covered in reviousyers, s well s wht students will berequired to ster in the ollowing yer

• Techers ust not ssue tht the studentshve stery in wht they were suosed tocover during the revious yer

• Beore introducing concet, ssess riorknowledge (Do they know wht they re

suosed to know ro lst yer? At whtlevel will you be ble to begin the lesson?),nd give the necessry bckground knowledge(Is there soething they need to know beorethey cn strt lerning bout this concet?)

3. Build understanding or every new procedure beingtaught.

• Directly tech nd odel the new concetGive ny exles tht build on the

students' rior knowledge• Post nd exlin the rocedure Drw

ictures, when helul

• Give students suicient guided rctice

• Allow the students to verblize rocedures



11

• Monitor their rogress Ask questions ndllow the to sk questions so they cn

clriy understnding nd build o wht theylredy know

• Allow students to rctice with nd exlinconcets to eers

• Mke sure tht the indeendent rcticetches wht they were rcticing in guidedrctice, rther thn giving the coletelydierent or hrder robles to solve oncethey re reoved ro ny suort

• Review the rocedure beore oving on tothe next sect o tht theticl concetor beore dding ore chllenging level

Teaching More Than One WayStudents need to lern lot in single cdeic yer ndsoeties they don’t lern it ll the frst tie Whenolder students il thetics course, they re otenrequired to reet it, cing the se textbooks, techingstrtegies, routines, curriculu terils, nd soe-ties even the se techers When younger studentsil to grs theticl concet, they re oten givenextr rctice work tht resents the teril in these wy they did not understnd in the frst lce

Teaching the Procedure

Students who experience “the same—but a secondtime” are basically being set up to ail once again. It

is detrientl cycle tht denies students the rightkind o intervention nd dierentition to eet theirneeds



11


1. Consider the multiple intelligences o thestudents, introduced by Howrd Grdner in

Frames o Mind (1993) When techers llowstudents to use their own strengths towrdlerning bout thetics, students will hveore success Beyond the thetic/logiclintelligence, techers cn exlore ore linguistic,kinesthetic, usicl, or socil wys to investigtetheticl concets

2. Consider the students’ learning styles I only orlinstructions re given, then the visul lernersight be struggling Oer uditory, visul, ndkinesthetic wys o direct rocedure nd guidedrctice

3. Change your teaching techniques I you usullyuse overheds, try hving students crete ostersUse cooertive grous in which students frstsolve robles individully nd then show theirnswers to the other three grou ebers, coeto grou consensus, nd fnlly resent thegreed-uon solutions to the rest o the clss Try

dierent niultives thn you used the frsttie

4. Experiment with partner activities Alternteong eer rctice, techer-directed rtnerctivities, guided concet exlntion, eertutoring, fnding ultile rtners to shreconcets with, nd rndo ixing

5. Reeber tht lerning new theticlconcet is rocess Scold esy to hrd, ndstrong techer suort to indeendent workStart with the concrete, move to the abstract,and nally move to the application o themathematical concept.



11


Taking NotesTking eective notes is not skill tht students lern

indeendently Rther, they need to be explicitly taughthow to take notes in an organized manner Good th-etics notes re going to look very dierent ro goodnotes in nother subject Thereore, it is the resonsibil-ity o thetics techers to tech their students howto best orgnize theticl inortion

• Mth journls usully hve n eleent o studentreedo nd choice Students cn coy exleso thetics robles, colete indeendent

rctice robles, write down their understndingo rocedures, nd note questions to sk thetecher Techers cn eriodiclly collect thjournls nd quickly check students’ rogress

• Mth lerning logs re techer-directed notestht students cn lter reer to when they reworking indeendently or studying or ucoingtests Usully, there is set ort tht studentsuse or ech new jor theticl concet

Techers should hold students ccountble or thenet orgniztion nd colete inclusion o vitlinortion Techers should collect lerninglogs oten to check or ccurcy o recordedinortion, net orgniztion, nd inclusion othe centrl inortion Beore test, techersshould give students rctice in clss t redingnotes nd using the to reherse nd reeberkey inortion tht they will need to knowThey cn even quiz rtners bout inortion in

the notes



11


Consider the ollowing skills tht re involved in well-written math learninglog notes

• The crucil vocbulry needs to be identifed nd defned

• The rocedure should be written correctly nd concisely

• The rctice robles should be ket serte ro the orgnized key-ide notes

• Soe theticl concets re best described with illustrtions

• Students should be tught how to identiy the centrl ide in ech geo notes

• There should be t lest one correct exle o how to solve roble• No extrneous, unhelul inortion should be included

• The notes should be legible nd well orgnized

• Color coding or highlighting skills should be tught to students ndlied to good note tking

Creative Ways to PracticeStudents need le oortunities to think bout th-eticl concets, relte the to their everydy lives,nd successully eror theticl unctions

Techers get rustrted when they send whole eriodteching concet nd then get bck oorly coletedhoework Oten, techers re sending too uch tietlking nd not llowing enough tie or eective stu-dent rctice Eective rctice does not en students

colete the even nubered robles in clss nd theodd nubered robles or hoework This strtegydoes not llow students to receive dequte rctice orsuccessul liction nd stery o thetics



11


The ost iortnt thing techers need to do is toallot suicient time or practice Techers need to

structure practice time with guided-rctice ctivitiesso tht students re held ccountble or rcticing

The ollowing cretive ctivities llow students interest-ing wys to rctice theticl concets

Peer Explanation Activity

Ater students hve been introduced to new the-

ticl concet, the techer cn ir the with rtnersnd give ech ir dierent roble to solve Then,the techer cn direct ech ir to join nother ir ndexlin their solution to one nother This ctivity cnhel the reinorce rocedure nd concet understnd-ing

Whiteboard Cooperative Group Activity

Students work in grous o our, ech with nuber roone to our Ech grou hs one sll whitebord ndone en The techer dislys roble nd directs llthe students to solve the roble individully on erThen, the grou ebers core their solutions Isoeone hs dierent nswer, the grou cn discussthe rocedure Once everyone hs the correct solution,the techer chooses nuber ro one to our Thegrou eber whose nuber corresonds to the nu-ber given by the techer records the grou solution on

the sll whitebord Tht erson dislys the nswerto the clss beore receiving new roble to solve inthe se wy I ny grou hs dierent nswer, theclss discusses the solution together with the techer todecide which nswer is correct



11


Oral Rehearsal Activity

The “lnguge o thetics” requires very secifc

ortion o English syntx nd new ters Thereorestudents need chnces to rctice “seking thet-ics” In the tyicl question/nswer clssroo dynic,only the confdent students rise their hnds to nswerquestions Techers need to structure orl rctice sotht every student is thinking o nswers nd beingllowed to shre the with rtner In nother orlctivity, the techer cn disly roble on the bord,sk or volunteer to red it, nd then continue withtht same exle (rther thn writing out nother one)so tht each student cn choose to red it An exlewould be reding lce vlue nuber, such s 253,467correctly Thus, ech student gets the oortunity toorlly rctice, nd ll students cn rticite withintheir own coort levels

Graphic Organizer Review Activity

Beore test or t the end o unit, the techer divides

the clss into irs The techer then gives ech iro students tching colored er No two irs cnreceive the se color Ech ir grhiclly dislysthe iortnt inortion ro tht unit on its erss review or study guide The ir should decide on wy to disly the inortion tht kes sense toech o the nd clerly shows the inortion the irwnts to include The irs should be encourged towrite exlntions or stes, drw ictures or digrs,nd show sle robles tht hel the review ny

theticl concets nd robles they hve beenlerning Ech eber o ir should hve the seinortion on his or her er

When ll the irs re fnished creting their grhicorgnizers, the students slit ro their irs to fndnew rtners with dierent colored ers They ech



11


shre their grhic orgnizers nd verblly exlin theinortion they included Next they dd ny new

inortion lerned ro the other students tht wsnot lredy included on their ers Then they echfnd new rtners nd reet the se shring rocessThe techer cn decide to set tie liit on this ctiv-ity or require ech student to shre with set nubero other students Using this ctivity llows studentsultile oortunities to review nd reinorce crucilinortion

Working as a Team Cooperative Activity

Students cn work in irs or in sll grous Echerson on the te needs dierent colored encil Thetecher dislys roble The students rotte, echwriting one ste nd ssing the er While the otherstudents y oer suggestions, hel, or dvice, they ynot tke the colored encil ro the student whose turnit is The students continue to rotte until the robleis solved As the techer onitors the work, he or shecn quickly ssess who contributed which ortion o the

roble solution by looking t the colors




Post-Reading Reflection1. Wht sects o instruction re iortnt in teching the rocedure?

___________________________________________________________________

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2. Describe two new wys to resent inortion tht you lerned rothis chter nd why it is iortnt to do so

___________________________________________________________________

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___________________________________________________________________

3. Think bck to how or i you were ever tught to tke notes Whtstrtegies would hve been helul to you when you were in school?Why is note tking vluble skill to tech your students?

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10



1

ChapterEight

Teaching ProblemSolving

Why Teach Problem Solving?Tody’s world is chnging ridly Mny o the chngesen tht rofciency in bsic theticl concetswill becoe ore nd ore criticl

Furtherore, theticl resoning nd roble solv-ing will be crucil to the success o tody’s students sthey work to fnd solutions to robles in everydy lie

Just knowing the bsic cts nd oruls is not enoughto solve the wide rnge o robles nd situtions thtrise in lie It is no surrise, then, tht roble solv-ing is n iortnt, current thetics ocus in theclssroo



1


While students ust be skilled t eroring cout-tions required to fnd theticl solutions, this is only

rt o the rocess Beore students begin to niultethe inortion in roble, they should understnd itsening nd ln wy to solve it

Students, thereore, need to lern bout roble solv-ing s rocess nd the strtegies they cn ly tofnd solutions (Kiltrick, et l, 2001) The rocess oroble solving goes beyond fnding sile solutions;it encourges students to reect on the solutions, kegenerliztions, nd extend robles to include new

ossibilities or investigtion Once students lern therocess o roble solving, they cn use theticlroches to solve rel-lie robles

The ges tht ollow rovide exlntions nd exleso 12 roble-solving strtegies tht cn be dted toeet students' needs Inortion bout ech strtegywill rovide insight into wys rticulr strtegy cnbe used in the clssroo Exles re given or echstrtegy They deonstrte the liction o the strt-

egy to the solution o the roble These exles renot rorite cross ll grde levels They re onlyused to deonstrte the use o the strtegy

Steps for the Problem-Solving ProcessIt is iortnt tht students ollow logicl nd syste-tic roch to their roble solving These our steswill enble students to tckle robles in structurednd eningul wy These stes re not intuitive or

lerners Thereore, techers will need to ln instruc-tionl tie to exlicitly tech the rocess, odel it, ndfnlly llow or le oortunities or guided nd indi-vidul rctice s students roch the roble-solvingstrtegies



1

Teaching Problem Solving

Step One

Understanding the ProblemEncourge students to red the roble creully nu-ber o ties until they ully understnd wht it sks Asstudents re lerning this ste nd rogressing towrdinternlizing it, the techer will llow tie or studentsto discuss the roble with eers or rewrite the roblein their own words Students should sk internl ques-tions such s, “Wht is the roble sking e to do?”nd “Wht inortion is relevnt nd necessry orsolving the roble?” (This will need to be reetedlyodeled or the students in the lerning rocess)

Next, students should underline ny unilir wordsnd fnd their enings Selecting the inortion theyknow nd deciding wht is unknown will hel thebegin too see how to solve the roble They shouldlso see i there is ny unnecessry inortion It willbe helul or techers to odel these rocesses untilstudents understnd how to colete the on theirown

Step Two

Planning and Communicating a Solution

Students should decide how they would solve the rob-le by thinking bout the dierent strtegies tht couldbe used Soeties it will be necessry or students touse ore thn one strtegy to solve roble

They could try to ke redictions, or guesses, boutthe roble Oten these guesses result in generliz-tions, which hel to solve robles Students should bediscourged ro king wild guesses, but they shouldbe encourged to tke risks They should lwys thinkin ters o how this roble reltes to other roblestht they hve solved



1


As they ttet dierent strtegies, they should kee record o those they hve tried so tht they do not reet

theThe 12 strtegies in this book include:

• drwing digr

• drwing tble

• cting it out or using concrete terils

• guessing nd checking

• creting n orgnized list

• looking or ttern

• creting tree digr

• working bckwrds

• using siler nubers

• oen-ended roble solving

• nlyzing nd investigting

• using logicl resoning

Other strtegies include:

• breking down ides into sller ieces• writing nuber sentence

• writing down ides s work rogresses so studentsdo not orget how the roble ws roched

• roching the roble systeticlly

• rereding the roble in order to rethinkstrtegies i the student becoes "stuck"

• orlly deonstrting nd exlining how nnswer ws reched

Step Three

Refecting and Generalizing

Mny ties the solution nd strtegies in one roblecn hel students know how to solve nother roble



1


Thereore students need to lern the iortnce oreecting on the work they hve done Techers need

to tech students the criticl rocess o reection Thisrocess should be odeled s techers show roblesolutions Techers cn even solve robles incorrectlyin order to go through the reection rocess nd “ctch”istkes Students need to decide i their nswers kesense nd i they hve nswered the question tht wssked They should illustrte nd write their thinkingrocesses, estitions, nd roches This gives thetie to reect on their rctices nd grow in the use oroble-solving strtegies When they hve n nswer,

they should exlin the rocess to soeone else

Step Four

Extension

This ste lso needs to be exlicitly tught nd odeledby techers becuse students need rctice in internl-izing it Students need to lern how to sk theselves“wht i” questions to link this roble to nother This

will tke their exlortions to deeer levels nd encour-ge their use o logicl thought rocesses Studentsshould lso decide i it is ossible to solve the roblein siler wy

Problem-Solving Strategies

Strategy One—Drawing a Diagram

This strtegy oten revels sects o the roble tht

y not be rent t frst A digr tht uses silesybols or ictures y enble students to see thesitution ore esily nd cn hel the kee trck othe stges o roble in which there re ny stesStudents need to develo the skills nd understnding touse digrs eectively



1


• Using sile line drwing to sybolize nobject will hel student visulize sitution

Example: How ny rkers would be needed i youlced rker t every two-eter oint on ten-eterroe?

In resonse, students y clculte entlly 10 ÷ 2 = 5,so fve osts re needed However, i students drw theroe nd rkers, they will see tht six rkers reneeded becuse n dditionl one is needed or the strt-ing oint

• Using tie/distnce line to disly theinortion hels to show distnce or oveentro one oint to nother

Example: A signost is lced on the highwy It systht the city is 30 kiloeters to the west nd the ocen is65 kiloeters to the est How r re you ro the citywhen you re 17 kiloeters ro the ocen?

Students should drw line nd write the distnces onit

30 k + (65 k – 17 k) = 78 k

• Students will need to scle down digrs i lrgeres need to be drwn Show students how to usescled-down esureents to solve roblend then convert the solution to the ctulesureents

Example: In drwing, one centieter could hve thevlue o one kiloeter Alterntively, one-centieterline could reresent 10 kiloeters or even 500 kiloe-ters, deending on the scle o the drwing



1


• Students y need to or show directionswhen drwing digrs, or use s s the ocus

o roble• They need to understnd how to lot course by

oving u, down, right, or let on grid, or usethe coss oints to direct theselves—north,south, est, west, northest, southwest, nd so on

Example: Plot our dierent routes ro Byee toGuy without ssing through ny town twice erroute

• Students will fnd it helul to drw digrsnd use sybols to show the reltionshis ongthings

Example: John, Jck, nd Fred love nils John'svorite nils re fsh nd horses Jck's vorite ni-ls re horses nd rbbits Fred's vorite nils refsh Which boys hve fsh s their vorite nils?

John HorseJack RabbitFred Fish

• Drwing ictures cn hel students orgnize theirthoughts nd siliy roble

Example: Orgnize the our doino ieces in squreshe with ech side o the squre dding u to totlo 10



1


Strategy Two—Drawing a Table

This strtegy hels students to orgnize inortion

so tht it cn be esily understood nd reltionshisbetween one set o nubers nd nother becoe clerA tble kes it esy to see the known nd unknowninortion The inortion oten shows ttern orrt o solution, which cn then be coleted It lsocn hel reduce the ossibility o istkes or reetitionsStudents need to develo the skills nd understnding tocrete nd use tbles eectively

• Techers y hve to hel students decide on the

nuber o coluns or rows to ft the vribles

• First decide how to clssiy nd divide theinortion Estblish the nuber o vriblesto be included in the tble Then decide on thenuber o rows, coluns, nd hedings

Example: There re 18 nils t the r Soe rechickens nd others re cows There re 70 legs thtre visible How ny o ech tye o nils cn beseen?

Students will need to drw tble tht hs three col-uns

Number oChickens

Number oCows

Number oLegs

• Oten ttern becoes obvious when creting tble Students y leve gs in the tblesnd colete the tterns entlly or ollow the

ttern in the tble to fnd the inortion theroble is sking or

Example: Two eole re being cored in this rob-le: Mrs Crword is 32 yers old nd her dughter Lisis eight yers old How old will Lis be when she is hls old s her other?



1


A two-colun tble is drwn By leving gs nd cl-culting entlly, we estblished tht when Lis is 24

yers old, her other will be 48 yers old

Lisa Mrs. Craword

8 32

9 33

10 34

11 35

12 36

13 37

24 48

This second exle shows how ttern cn be estb-lished

Example: A child is lying ge o bsketbll by hi-

sel in the rk Then, t regulr intervls, other grouso students begin to rrive t the rk Fro ech newgrou, two children decide to join the bsketbll geThe frst grou hs three children, the second grou hsfve children, nd the third grou hs seven childrenHow ny grous will hve ered by the tie therere 64 eole in the rk?



10


Three coluns re needed or the tble The colunsshould be heded Groups, People, nd Total

Groups People Total

1 1

1 3 4

2 5 9

3 7 16

4 9 25

5 11 36

6 13 49

7 15 64

Seven grous will hve ered

Strategy Three—Acting it Out or Using ConcreteMaterials

This strtegy uses objects or terils to reresent eole

or things in the roble This hels students visulizethe roble in concrete wy nd fnd the solution

• A vriety o objects such s bens, counters,blocks, toys, or ersers cn be used to sybolizeeole or lces These objects cn be ovedthrough the stes o the roble It is iortntto chrt this oveent to kee trck o therocess

• Students cn lso ct out the roles o the dierentrticints deending on the sitution in theroble

Example: Their grndrents sent Leslie, Rndll, ndCrl $190, in totl, or their birthdys Their rentshd to divide the oney u so tht Rndll ws given$20 ore thn Leslie, nd $30 ore thn Crl Howuch oney were they ech given?



1


Ask three students to ct out the rts o the childrennd use $190 ly oney or the exercise

Strt by giving Leslie n estited ount

Give Leslie $50

Rndll should be given $20 ore: $50 + $20 = $70

Crl should be given $30 less thn Rndll:$70 – $30 = $40

Totl $50 + $70 + $40 = $160 This totl is too low

Strt with giving higher ount

Give Leslie $60

Rndll should be given $20 ore: $60 + $20 = $80

Crl should be given $30 less thn Rndll:$80 – $30 = $50

Totl $60 + $80 + $50 = $190 This is correct

Strategy Four—Guessing and Checking

This strtegy llows student to ke n eductedguess nd check the guess ginst the roble I it isnot correct solution, the student revises the guess ndchecks until correct solution is ound

• Ech student begins by fnding the iortnt ctsin the roble nd king resonble guessbsed on the inortion Techers will wntto hel students when they frst strt using thisstrtegy Students will lern ro their ulty

guesses• Students then check their guesses ginst the

roble nd revise their guesses until they fndthe correct nswer Creting tble is good wyto ke sure ll guesses re checked



1


Example: Aln is fve yers older thn Sul Aln’s gelus Sul’s ge totls 25 Wht re their ges?

Students should frst note the iortnt inortiongiven to the in the setu o the roble In this rob-le, the iortnt inortion is tht:

Aln is fve yers older thn Sul

Aln’s ge lus Sul’s ge totls 25

Students should crete tbles to hel the solve theroble

Guess Alana’s Saul’s Total age age

Students should now test their guesses


1 12 7 19TooLow


1 12 7 19 2 18 13 31

TooHigh


1 12 7 19 2 18 13 31 3 15 10 25

C o r r e

c t



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Strategy Five—Creating an Organized List

This strtegy is used insted o tble when greter

ount o inortion is vilble Students need toollow rocedure or sequence to fnd the solution tothe roble nd ke sure no inortion is let out orreeted

• When creting the list, one ite should reinconstnt while the others chnge Students needto write down the rocesses they re using to styon trck

• Students cn work sequentilly, using theinortion ro the roble or flling in the gso ttern once it is creted Systetic work iskey to being successul in this strtegy

Example: Shun hs three toy crs tht he kees on hisbookcse One is red, one is blue, nd the third is greenHe likes to chnge the order in which he dislys theHow ny dierent wys cn he do this?

To solve this roble, lce the red cr (r) frst, nd then

lce the other crs in their two ossible ositions

r

b

g

b

r

r

g

g

b

r

b

g

g

g

b

b

r

r

There re six dierent ossibilities



1


Strategy Six—Looking or a Pattern

This strtegy is n extension o drwing tble nd cre-

ting n orgnized list It is oten used becuse th-eticl tterns cn be ound everywhere

There re ny wys to check or ttern

• Find the dierence between two consecutivenubers

• Find out i the nubers re rising or lling in regulr sequence

• Decide whether the nubers hve been ultiliedor divided by ny given nuber

• Once ttern is ound, it cn be extended orcontinued

Soe tterns include two oertions

Example: 6 9 8 11 10 __ __

This ttern includes two oertions: + 3, – 1

Example: When Jcklyn went strwberry icking, one

out o every six strwberries hd worholes How nygood strwberries were there out o 84?

Look t this tble We cn see tht ttern hs beenestblished The “Good” colun is incresing in ul-tiles o 5, nd the “Bd” colun is incresing by 1s Soi there is totl o 84 strwberries icked, 70 will begood nd 14 will be bd

Good Bad Numberof

strawberries

5 1 6 10 2 12 15 3 18 20 4 24 25 5 30 30 6 36 35 7 42



1


Strategy Seven—Creating a Tree Diagram

This strtegy uses digr with dierent brnches to

reresent reltionshis between dierent ctors in roble A tree digr enbles the students to visul-ize the dierent ctors in the roble nd ensures thtno ctors re reeted or issed

• First the students need to identiy the iortntctors in the roble nd list the Thenthe ctors need to be connected using lines orrentheses

• When the roble is colete, it should bechecked to ke sure tht ll ctors re roerlyconnected nd no inortion hs been let out

Example: A clss o students ws sked to list its voriteruits In the list the students included les, ornges,nd bnns Once the initil list hd been coiled,students were sked to orgnize the three tyes o ruitin order, ro their lest vorite to their ost voriteThe students in the clss covered every ossible cobi-ntion How ny cobintions were there?

Most vorite Lest vorite

ornges bnnsAles

bnns ornges

les bnnsOrnges

bnns les

ornges lesBnns

les ornges



1


Strategy Eight—Working Backwards

This strtegy is used to solve robles tht include

nuber o linked ctors or events, where soe o theinortion hs not been rovided The object is todeterine the unknown inortion The events occurone ter the other nd ech stge or iece o inortionis ected by wht coes next

• To solve the roble, begin with the endinginortion nd work bckwrds until theroble is solved It is iortnt to reebertht theticl oertions will hve to be

reversed

Example: Arnold bked cuckes over the weekendEch dy during the week he took three cuckes toschool to shre with his riends On Sturdy, when hecounted, there were 18 let How ny cuckes hd hebked?

Begin with the inortion you know—the nuber ocuckes Arnold ended with—nd work bckwrds

Arnold has 18 cupcakes at the end o the week.

On Friday he brought three cupcakes to school: 18 + 3 = 21

On Thursday he brought three cupcakes to school: 21 + 3 = 24

On Wednesday he brought three cupcakes to school: 24 + 3 = 27

On Tuesday he brought three cupcakes to school: 27 + 3 = 30

On Monday he brought three cupcakes to school: 30 + 3 = 33

Arnold made 33 cupcakes over the weekend



1


Strategy Nine—Using Simpler Numbers

This strtegy cn be used to solve diicult or coli-

cted roble in order to siliy the nubers

• Begin by substituting sller nubers or lrgernubers to ke the clcultions esier Thenuse the se stes to solve the originl roble

• Or solve series o siler robles to see i ttern eerges nd ly the ttern to the orecolicted robles

Example: It took 16 rtists 10 hours to int hl url

Only 4 o the rtists cn sty to fnish the other hlHow long will ech rtist work i there re only 4 rtiststo colete the other hl?

Strt with siler exle

I it tkes 4 rtists 8 hours to int hl url, howlong will it tke 2 rtists to int the other hl?

First, fnd out how long it would tke 1 rtist to inthl the url ll by hisel

He would hve to work our ties longer to do the job o4 rtists, so he would tke 32 hours

4 x 8 = 32

I 2 rtists work on the other hl, ech will only hve towork hl s uch tie s one rtist, so it would tkethe only 16 hours

32 ÷ 2 = 16

Now solve the originl roble

We know tht 16 rtists tke 10 hours, so 1 rtist wouldhve to work 16 ties longer to do the job o 16 rtists,so he would tke 160 hours

16 x 10 = 160 hours



1


I 4 rtists work on the other hl, they would ech hveto work only one ourth s uch tie, so they would

ech work 40 hours160 ÷ 4 = 40

Strategy Ten—Open-Ended Problem Solving

This strtegy is used to exlore robles tht ightbe nswered in nuber o wys Although fndingthe correct nswer is iortnt, techers should vluethe student’s rocess or solving the roble nd gininsight into the student’s develoentl understnding

• When structuring robles, words such s create,make, design, explore, nd investigate should beused

• Students should use rocesses such s lbelingcounters to visulize the roble, trying dierentcobintions o nubers, nd working to fnd sny solutions s ossible

Example: Investigte which cobintions o the digits 3,

4, 6, nd 7 will crete ddition robles tht hve sulling between 100 nd 120

4 3 3 4 4 3 3 4+ 7 6 + 6 7 + 6 7 + 7 6

1 1 9 1 0 1 1 1 0 1 1 0

3 7 3 6+ 6 4 + 7 4

1 0 1 1 1 0



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Strategy Eleven—Analyzing and Investigating

When using this strtegy, nlyze wht is known nd

wht needs to be known Use the known inortion toinvestigte the roble nd collect dt

• Begin by king n estite Estiting is neective ethod or guging the resonbleness on nswer

• Ater estiting, ln n roch to solvethe roble Deterine wht is involved inthe tsk nd wht strtegies should be used togther inortion Decide how the inortiongthered should be resented

Example: Ech weekdy, crs continully stre st theschool How ny weekdys will it tke or illioncrs to ss the school?

One possible solution

Students believe there re two ek tric ties,8:00–9:00 am nd 3:00–4:00 pm During the reindero the dy, crs do not ss oten

Plan: Survey the tric during the ek hours nd t theother ties during the dy Tlly the nuber o crs thtss during 10-inute eriod, nd ultily by six toget the nuber o crs er hour

Peak Hour

8:00–9:00 am 250 crs in 10-inute eriod,250 x 6 = 1,500 crs/hour

3:00–4:00 pm Assue it is the se, 1,500 crs/hour

Other Times

1:00–2:00 pm 10 crs in 10-inute eriod,10 x 6 = 60 crs/hour



10


The eriod between idnight nd 6:00 am would bevery quiet—estite 5 crs er hour during this tie

The ow o tric in one 24-hour eriod cn be recordeds ollows:

Pek Hours = 3,000 crs

Midnight to 6:00 am = 5 crs/hour x 6 hours = 30 crs

All Other Ties = 60 crs/hour x 16 hours = 960 crs

Totl Crs in One weekdy = 3,000 + 30 + 960 = 3,990

1,000,000 divided by 3,990 crs/dy = 2506 weekdysuntil one illion crs ss the school

Strategy Twelve—Using Logical Reasoning

This strtegy is used when the roble gives inor-tion s ieces o uzzle Ech iece o inortion isiortnt to solve the roble Process o eliintionis one roch in this strtegy, where ech iece oinortion builds to the solution Drwing grid toorgnize given inortion is lso n roch tht cnbe used

• Students strt by reding ech clue creully ndthoroughly Oten the clues need to be deltwith in n order dierent ro how they wereresented Then the students tke the stesnecessry to solve the roble

Example: Julie, Yushiko, nd S re ech bout to et

sndwich or lunch On the lte there is toto snd-wich, honey sndwich, nd enut butter sndwichUse the ollowing clues to work out which sndwichbelongs to ech erson



1


By drwing grid, students cn use the inortion theylerned in the roble to cross out inortion nd visu-lize the correct nswers

S cnnot hve the enut butter sndwich i he isllergic to nuts Yushiko will not et the honey snd-

wich becuse she does not like sweet things Julie’ssndwich hs slt nd eer on it, which ens it isnot the enut butter or the honey sndwich Fro thisinortion you cn fll in the grid to show who getswhich sndwich

tomato honey peanutbutter

Julie ✔ ✘ ✘

Yushiko ✘ ✘ ✔

Sam ✘ ✔ ✘

How to Create Word ProblemsWhen techers re creting word robles to use withtheir students, it is iortnt tht they kee severlthings in ind

• Decide the roble-solving strtegy being utilizedbeore writing the roble

• Use rorite vocbulry or the students in theclssroo

• Kee the sentences sile

• Only include s ny robles s the studentscn colete in one sitting

Julie’s sndwich hs slt nd eer on it

S is llergic to nuts

Yushiko dislikes sweet things



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• Work the roble t lest twice beore giving it tothe students

• Hve nother techer work the roble to checkor clrity, vocbulry/word choice, nd correctthetics

Here re soe coon words ound in word roblesor techers to use when they re creting word rob-les These words cn lso be used with students togive the clues or which oertion to use when reding word roble

Adding Subtracting Multiplying Dividingincresed

ore thn

cobined

together

totl

su

dded to

lus

dditionl

together

joined

ltogether

ginrise

both

in ll

decresed

sller

reduce

inus

less

dierence

between/o

less thn

ewer thn

loss

tke wy

let

reins

o

ties

ultilied

er

ech

roduct o

ctors

twice (2)

double (2)

trile (3)

rction o

t this rte

into

slit

hl

verge

cut

er,

out o

rtio o

quotient o

ercent

divided by

divided eqully

echir shre



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Post-Reading Reflection1. Choose roble-solving strtegy to use with this roble I ore

thn one strtegy cn be used, ke notes bout how you could helyour students deterine which strtegy is ost eicient to use Adoll's dress cn be creted with ⅘ o yrd o teril How nydresses cn be creted i you hve 22 yrds o teril?

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

2. Choose one roble-solving strtegy Write word roble tht

utilizes tht strtegy to solve the roble___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________




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1

ChapterNine

IntegratingMathematics Across

the Curriculum

Techers o thetics re bound one dy to be skedthe ge-old question, “But when I ever going to usethis?” By integrting thetics cross the contentres, students will relize the iortnce o thet-ics in their dily lives nd how oten it is used in subjectsother thn thetics

Integrating Mathematics and LiteracyMthetics is oten inccurtely erceived s “strictlynubers” While it is true tht thetics is hevy innubers, orders o oertions, nd rocedurl oruls,there re lso lenty o oortunities to exine theossibilities or connecting reding nd writing to keytheticl concets



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The Language o Mathematics

There is ctully lnguge o thetics tht students

hve to lern to red, interret, sek, nd write s theytckle theticl concets The linguistic eleento thetics vries by culture, s well For exle,the wy rctions re red in English is dierent syntcti-clly ro reding rctions in other lnguges Englishstrts with the nuertor ( crdinl nuber) nd thenthe denointor (n ordinl nuber) There re exce-tions: students do not sy 12 s “one second,” but rther“one hl” In Jnese, to exress rction, the studentfrst sttes the denointor nd then the nuertor“One ourth” literlly trnslted ro Jnese is “oursections o which we re reerring to one section”

Reading in a Mathematics Class

Content res such s thetics lend theselves tothetic odes o instruction Within these thees,students re given cognitive net o vocbulry, bck-ground, nd concets with which to connect the new

lerning nd understnding (Moyer, 2000) This is ese-cilly iortnt or English lnguge lerners who relerning lnguge nd content concets siultneouslyA student ust be ble to red concet introductions,exlntions, word robles, exles, nd instruc-tions in thetics book or within n ctivityAdditionlly, students need to lern certin ters ndhrses to understnd wht oruls, rocedures, ndorders o oertion to ly to word robles These resecifc to the theticl content re A lnguge

rts techer will not tech students how to nvigte ndcorehend the reding tht is necessry in th-etics textbook It is u to the thetics techerto odel nd exlicitly deonstrte the necessry red-ing skills or the successul reding tht is necessry inthetics



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Integrating Mathematics Across the Curriculum

How to Integrate Fiction and Nonction

A techer y ly books to lessons to hel contex-

tulize theticl concets Soe o the skills thtstudents need or reding re lso necessry in the-ticl thinking These include surizing, sequenc-ing, nd fnding the in ide Integrting reding ndthetics cn irove students’ generl lngugeskills nd their bilities to counicte nd exresstheselves theticlly (Moyer, 2000)

Mny students enjoy reding nd listening to storiesbecuse fction text is eling nd interesting There

re ny fction texts vilble tht directly exlintheticl concets nd ose secifc theticlrobles to solve in un nd exciting wy Additionlly,fction books oten resent oortunities to tech logicthrough interesting robles tht re estblished by thechrcters Fiction books oten resent thetics inn unintiidting wy or students to understnd ndcn be used with students o ll ges

Reding corehension skills re essentil or c-

deic success throughout ll grde levels Students reoten tught strtegies or corehending fction textsHowever, "[i]t's essentil tht students lern ro theerliest grdes through high school how to red nonfc-tion, i they re to survive nd thrive in n dult worldcred with inortion" (Collier, 2006) Nonfctiontexts nturlly rovide rel-lie situtions nd inor-tion tht students cn use to understnd nd lytheticl concets s well s reding corehen-

sion strtegiesNonction and ction texts can be used or discussionsabout the ollowing mathematics concepts:

• counting/generl nuber sense

• estition




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• rctions, decils, nd ercentges

• esureent

• sile lgebric unctions nd inequlities• logic nd robbility

• reserching, collecting, nd coring inortionnd dt to nlyze nd disly in grhs

Writing

As students discuss nd red bout theticl con-cets, techers cn lso guide the in their develoento writing the lnguge o thetics The techer

cn use the ollowing to cilitte roer writing boutthetics

• Allot tie or th lerning logs

In lerning logs, the techer guides the studentsto write the ost iortnt vocbulry, concetides, rocedures, oruls, nd exles in theirnotes

• Use sentence res

Sentence res hel students write thet-ics in the roer context The techer scoldsthe roer syntx nd vocbulry o iortnttheticl concets This rocess is esecillyiortnt or English lnguge lerners

Example: The ______ esures ______ inches

• Allot tie or th journls

In th journls, students hve oortunities toexlin, reect, rctice, nd review theticlconcets During this rocess they exnd theirtheticl thinking nd internlize the con-cets



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Techers cn direct students to write:

• bout their understnding o how to solve

robles

• n exlntion o how they solved roble

• bout their level o corehension on certintheticl skill

• redictions o how they think certin roblesshould be solved

• questions they hve or the techer

• review o the stes or rocedures involved in

solving robles• sle word robles or th-relted stories

where they cn ly the theticl concetstht they re lerning

• reection on new concets lerned tht dy

• sel-ssessent o how well they eel they did onhoework or quiz

For eleentry school techers, writing is core subjecttht oten is neglected when they hve to ocus on other

subjects These techers will fnd tht students cn giniortnt skills when they lern to write bout th-eticl resoning Secondry techers ight fnd thteven when students understnd theticl concets,they y not be ble to exress their understndingWhen these techers cn eectively cobine writ-ing instruction nd theticl-resoning instruction,they re building eective couniction bout th-eticl concets

Integrating Mathematics and ScienceMthetics nd science re discilines tht cn otenbe tught together quite nturlly When techers userel-lie science exerients to rctice theticlskills, students re ble to understnd the science nd



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thetics ore coletely becuse they develoownershi in their exerients Lerning is reinorced

nd bstrct concets becoe eningul, leding tobetter retention in both content res Here re soeo the wys tht thetics nd science cn be esilyconnected

Measurement—Science exerients re idel orlying theticl esureent concets to rel-lie situtions For exle, students develo orecolete sense o esuring length when they rehnded ruler nd wiggly wor to esure

Geometry—The nturl world is bursting with shesto exlore Tking the geoetry lesson into lie sci-ence is nturl wy or students to exlore geoetricconcets

Analyze scientic experiment data using mathemati-cal ormulas—Oten, students hve hrd tie under-stnding how the oruls they re lerning in th-etics relte to the rel world I they re given chnce to ly the to robles they re solving in

exerients, it becoes ore eningul

Finding patterns in naturally occurring scienticevents—By exloring rel-lie tterns tht occur innture, the techer vlidtes tterns studied in th-etics clss

Algebra ormulas in real-lie scientic situations—Students oten struggle with lgebric concets Ithey re ble to see lgebr oruls “in ction” while

conducting exerients, they will hve better core-hension o the concets

Probability—Both thetics nd science exlorethe nture o robbility As students conduct exeri-ents, they redict, test, nd record the results, ro-viding rel-lie exerience with the concet o rob-bility



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Integrating Mathematics and SocialStudiesThere re ny wys to integrte theticl con-cets with socil studies nd history Socil studieskes students ble to connect with the st, lernbout lces nd eole, nd ly this lerning to theresent Students cn use theticl ethods tonlyze historicl nd socil tterns When techerssuccessully integrte dierent content-re concets,students cn ore successully retin inortion ndke connections within their own lerning

Word Problems

The ost obvious nd erhs esiest wy to integrtesocil studies with thetics is to crete word rob-les tht cobine socil studies toics with theti-cl concets tht the students re lerning To kesense o theticl concets, students oten need toly the to rel-world questions

Here is an example o how to take two Mid-continentResearch or Education and Learning (McREL) standardobjectives and combine them to create a word problem.

Grades 3–5 Mathematics Standard: Stndrd 3: Usesbsic nd dvnced rocedures while eroring the ro-cesses o couttion

Level II, Benchmark 1: Multilies nd divides wholenubers

Knowledge/Skill Statements: 2 Multilies whole nu-bers; 4 Divides whole nubers

Grades 3–5 History Standard: Stndrd 3: Understndsthe eole, events, robles, nd ides tht were signif-cnt in creting the history o their sttes



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Topics: 1 Exlorers nd settlers o the stte nd region;2 Culturl diusion, dttion, nd interction

Level II, Benchmark 3: Understnds the interctions thtoccurred between the Ntive Aericns or Hwiinsnd the frst Euroen, Aricn, nd Asin-Pcifc exlor-ers nd settlers in the stte or region

Example Problem: In 1850, there were 92,000 eolewho lived in Cliorni, including Ntive Aericnsnd erly settlers In 1900, there were t lest 12 tiestht nuber o new rrivls About how ny rriv-

ls hd coe to Cliorni in those 50 yers?

Connections to Historical Dates

It is oten diicult or students to ully understnd theconcet o the st Yers nd dtes hve little rel en-ing Young children re still develoentlly rocessingthe concets o st, resent, nd uture Students o allages need multiple opportunities to process inormationabout dierent eras in order to begin to understand theidea o history. Mthetics cn hel to clriy theseconcets

Example Problem: The Jnese ttcked Perl Hrboron Deceber 7, 1941 I US Nvy Junior Oicerws 25 yers old on tht dy nd survived the ttck,how old is he now?

Connecting to Historical DataAs students study the lces nd the eole tht they relerning bout in socil studies, they cn use thet-ics to connect these lces to their own lives They cncore ncient civiliztions to the size o their resentsttes nd use census dt to core the nuber oeole living in one re to the nuber o eole living



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in their cities They cn use s to clculte nd e-sure dierences between loctions


Example Problem: On the , there re 1,300 ilesbetween Boston nd Chrleston using the King’sHighwy, n erly Aericn tril It took losttwo onths or settlers to wlk between those lcesbecuse they could only trvel 20–25 iles er dywith their wgons I you were to wlk 25 iles royour city, where could you ossibly end u?

Mathematics in the Real WorldI techers re stressing sily the “skills nd drills” otheticl concets without heling students lythese rocedures nd oruls to rel-world situtions,they re doing students gret disservice When stu-dents study theticl concet s ebodied by rel event or lce, they understnd nd reeber itAliction o bstrct concets to rel lie is owerul

instructionl tool In ech o the ollowing exles, note how the rel-lierobles ush students to broden their theticlcorehension

Primary Skill: Addition

Additional Skill: Clssifction

How ny boys re in the clss? How ny girlsre in the clss? How ny totl students re in theclss?

13 + 14 = 27



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Primary Skill: Division

Additional Skills: Multiliction, subtrction, nd order

o oertion

Jon’s o brought 43 cookies to clss There re 25students in the clss I everyone tries to tke 2 cook-ies ech, how ny students wouldn’t be ble to hve second cookie?

43 ÷ 2 = 21 R125 – 21 = 4 students who don't get 2 cookies

How ny cookies would we need or everyone to beble to hve 3 cookies ech?

25 x 3 = 75 cookies

Wht i the techer wnted cookie, s well?

25 x 3 + 1 = 76 cookies

Primary Skill: Perieter

Additional Skills: Multiliction nd oney

As und-riser, our clss wlks ls round the blockGtes Street is 165 eet long Johnson Avenue is 100eet long Brooks Boulevrd is 200 eet long CrterStreet is 143 eet long I we circle round the block 1tie, how r hve we gone?

165 t + 100 t + 200 t + 143 t = 608 t

I you get ledge or $300 or ech l, how ny

ties would you hve to go to ern $2100?

$2100 ÷ $300 = 7 ls

How r will you hve wlked?

608 t x 7 = 4,256 t



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Integrating Mathematics andTechnologyMthetics nd technology re nturlly ssocited,but hve not lwys been eectively integrted into theclssroo Students y hve been llowed to lycouter ges tht incororte thetics drills,but then would be orbidden to “chet” on ctivities ortests by using clcultors Now eductors understndtht technology, such s couters nd clcultors, cnirove students’ eduction (Den nd Florin, 2001)Rther thn using couters s drill-rctice chines,

techers now encourge students to use sredsheets,online lets, nd ore to exlore, ly, nd dislytheticl lerning Students cn receive onlinethetics tutoring nd hoework hel ro nu-ber o websites Couters nd clcultors hve becoetools tht students utilize in order to reserch, gtherdt, orgnize their notes on theticl concets, ndnlyze rel-world robles nd situtions

Using technology cn rovide students with oortuni-

ties to develo nd use their theticl higher-levelthinking skills to solve robles tht re relevnt totheir dily lives

Dierentiation within Technology-EmbeddedMathematics

Inevitbly, techers fnd rnge o theticl ndtechnicl coetence in ny grou o students Whentechers incororte technology into thetics les-

sons, there re vrious wys to dierentite instructionin order to eet the needs o ll students

• Use vried grouing strtegies when ssigninggrou thetics rojects

• Vry the level o suort given to students s theyuse couters nd clcultors to solve roblesnd disly dt



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• Be exible when setting tie liits or work thtrequires technology tools

• Allow students to use extr technology tools orscolding uroses, when necessry

• Use ultile technologicl reresenttions otheticl concets

Using Spreadsheets in a MathematicsClassroomSredsheet rogrs such s Miscrosot Excel® cn bevluble resources or the thetics techer There

re infnite ossibilities or use Unlike word rocessors,sredsheets llow the organization and manipulation odata in specied column and row locations In sred-sheet, there re ny wys to disly nd ly orulunctions to nubers While techers o ll subjects yuse sredsheets to kee records o grdes, clssroobudgets, ttendnce, nd checklists, the theticstecher cn lso use the or instruction

A B

1

2

3

4

Students cn use sredsheet rogrs to:

• niulte dt relted to clssroo studies

• build nlyticl, theticl, interretive, ndtechnicl skills

• coute theticl oruls

• show theticl tterns

• crete grhs tht show lgebr nd trigonoetryreltions nd unctions



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Creating, Reading, and Interpreting GraphInormation

Mny stte content stndrds require tht students lernto crete nd interret these tyes o grhs Studentscn use sredsheets to redict chnges in nubers,build dierent kinds o grhs, nd core nd niu-lte ny kinds o dt

Formulaic Manipulation o Data

With sredsheet, students cn lern bout, ly, ndexerient with ny dierent orders o oertion

Soe o the unctions include:


• dislying vlue s currency, ercent, or decil

• clculting ens nd verges

• clendr oruls nd conversions

• lying secifc lgebric unctions to dt (suchs x + 2 in certin colun)

• clculting running blnces

• creting ultiliction tbles

• clculting ercentges nd decils

• converting esureents

• rounding nubers

Money

Sredsheets cn be used to study wide vriety ofnncil toics Students cn use sredsheets to cretend nge budgets Sredsheets will disly negtiveblnces in n lternte color, such s red Students cncrete oruls, enter ny tyes o fnncil dt, ndkee long-ter records




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Research Products

Students cn use sredsheets to disly inortion tht

they hve reserched For exle, they ight reserchnutritionl recoendtions, crete chrts to dislythe inortion, conduct surveys o the ctul nutri-tionl intkes o their clsstes, nd crete corisonchrts Students cn lso use sredsheets to orgnizescience nd theticl dt in the se wy

Using Applets in a MathematicsClassroom

What Are Applets?

Alets re sotwre-coonent rogrs tht erorrecise unctions on web ges or in couter rogrsThese rogrs cn be used s tools (such s clcul-tor) or or entertainment value (such s sll o-uge to ly) or the eole who visit the web ges oruse the couter rogrs However, unlike coletecouter rogr, lets do not run indeendently A

sredsheet rogr ight contin n let tht unc-tions s clcultor or grh ker A word-rocessingrogr ight contin n let text editor or synonyfnder An eil nger ight ccess n let thtdislys video or hotogrh fles An let is usullydesigned to eror single ction nd hve built-inrestrictions tht revent it ro hring the user’s co-uter

There re ny eductionl websites tht rovide ree

lets or downlod Soe re sily or un, such sking n ige wve bck nd orth, nd others hveore rcticl, cdeic lictions such s cretinggrhs There re ny websites tht rovide instruc-tions on selecting, instlling, nd using lets



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Strategies or Using Applets in a MathematicsClassroom

A thetics clssroo should rovide ccess to vri-ous websites tht oer let tools tht students cn uses they re rocessing nd lerning theticl con-cets Soe o these sites y hve virtual manipula-tives lets A techer nd students ight fnd lgebrtiles to neuver, logic uzzles to solve, oney to kechnge with, or blnk br grhs, line grhs, nd iegrhs tht llow students to enter nd disly dtThese let tools ssist dierentition in the th-

etics clssroo becuse students cn eloy thelet tht best tches the skill they re building ndrogress t their own seed

Computer Lab Use

In order to use let tools in thetics clssroo,students idelly hve ccess to their own couters In couter lb, the techer cn odel their use on lrge disly ro ster couter, nd then students

cn rctice t their own ce t individul couters sthe techer onitors their rogress

Classroom Use

I couter lb is not vilble, techer cn use noverhed rojector or other disly to odel the vriouslet tools to the whole clss during the theticslesson Certin lets cn then be designted or indi-vidul use during rctice tie or irs o students

Here re three sle lets or eleentry, iddleschool, nd high school levels o thetics Theysuggest the endless ossibilities or theticl skillsnd concet rctice tht let tools oer to students



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Examples o Applet Use in an Elementary Classroom

Enter the website or The Ntionl Librry o

Virtul Mniultives ro Uth Stte University thttp://nlvm.usu.edu/en/nav/vlibrary.html

Click on the chrt Numbers and Operations, Grades 3–5

Find nd click Money in the chrt rovided

Tech the students how to rctice the three ctivities

How Much Money?

The let shows set o bills nd coins The studentstye in the ount o oney shown nd click on theword check to fnd out i they re correct Then theyclick on the words New Problem to request new rob-le



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Pay Exact Amount

The let shows sets o bills nd coins There is nount o oney shown bove the box The studentsclick nd drg the correct set o bills nd coins to the boxThen they click on the word check to fnd out i they recorrect Then they click on the words New Problem torequest new roble

Make a Dollar

The let shows set o coins The students ust drgthe correct coins to the box to ke dollr nd click onthe word check to fnd out i they re correct They clickon the words New Problem to request new roble




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Example o Applet Use in a Middle School Classroom

Enter the website or The Ntionl Librry o

Virtul Mniultives ro Uth Stte University thttp://nlvm.usu.edu/en/nav/vlibrary.html

Click on the chrt Geometry, Grades 6–8.

Find nd click Congruent Triangles ro the chrtrovided

Tech the students how to rctice king congruenttringles using dierent cobintions o sides ndngles

Instructions or ech cobintion re given long theto o the let When the blue nd red tringles retched together, they turn green A essge ers ithey re congruent



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Example o Applet Use in a High School Classroom

Enter the website or The Ntionl Librry oVirtul Mniultives ro Uth Stte University thttp://nlvm.usu.edu/en/nav/vlibrary.html

Click on the chrt Algebra, Grades 9–12.

Find nd click Algebra Tiles ro the chrt rovided

Tech the students how to niulte the virtul x - nd y -xes nd tiles in order to solve lgebr robles




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Using Graphing Calculators in aMathematics ClassroomThe grhing clcultor cn be n iortnt tool in llres o thetics

Graphing calculators can help elementary throughhigh school students use higher-level thinking andapply mathematical concepts to specic problems.

Graphing calculators alone cannot achieve the goalsthat educators have or student success. Teachersneed to show students how to use graphing calcula-

tors as a tool or mathematical-concept comprehen-sion and application.

A study by Grouws nd Cebull (2000) notes tht “tech-ers sk ore high-level questions when clcultors reresent, nd students becoe ore ctively involvedthrough sking questions, conjecturing, nd exloringwhen they use clcultors”

Classroom Management o Calculator UseMny techers dred clss-wide clcultor use becuseit requires creul ngeent However, with roerorgniztion, clcultors cn llow techers to sendeven ore tie develoing theticl understnd-ing, resoning, nuber sense, nd liction

1. All the clcultors should be nubered beorestudents receive the Ech student shouldbe ssigned clcultor nuber This wy,

the techer cn esily kee trck o whetherclcultors re dged or not returned

2. All clss sets o clcultors should be distinctlyrked with bright int or ernent rkeron the outside cover so tht they re esilyidentifble by techers, dinistrtors, or otherstudents i they re tken out o the clssroo



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3. The clcultors should be stored in lsticshoeboxes or in n over-the-door shoe rck with

the corresonding nubers rinted on the slots4. Suicient clss tie should be llotted or the

orgnized retrievl nd return o the clcultors Istudents will need the grhing clcultors t thebeginning o the clss eriod, instructions shouldbe dislyed s the students rrive

5. The techer should lwys check or return nddge ter clcultors hve been used Thetecher cn use check-o list to quickly note

which nubers re returned without dge

Teaching Graphing Calculator Skills

Students cn lern, reinorce, nd review theti-cl concets using grhing clcultors Prctice withthe clcultors will solidiy students' understnding osuch concets s nuber sense, lgebric thinking, dtnlysis, stil resoning, roble solving, nd unitso esureent (Grouws nd Cebull, 2000) Techers

ust integrte grhing clcultors into their lessonswith this rincile in ind

1. Once students receive the clcultors, the techershould deonstrte keying nd dt entry skills

2. To tech skill, the techer should request thtthe students locte the keys nd unctions onthe clcultor The techer should ilirizestudents with the enus nd screens tht thesekeys nd unctions ccess Students need tie to

sk questions nd exlore the unctions

3. I ultile stes re needed to colete thectivity, the techer cn list the stes on osterdisly, overhed rojector, or ocus rojector sotht students cn reerence the stes during thelesson



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4. The techer cn designte students who re oreilir with clcultor use to ssist those who

re not 5. To kee students on tsk, they should exect to

shre their work with rtners, the techer, or thewhole clss t ny tie

The “I Do, We Do, You Do” Approach

Grhing clcultors re cble o roviding

ultile reresenttions o theticl concets

By building tbles, trcing long curves, nd zooing

in on criticl oints, students y be ble to rocessinortion in ore vried nd eningul wy

(Sith, 1998)

Grhing clcultors cn build on concetul under-stnding by llowing students to rctice nuerousreresenttions o concets nd exeriences in wytht is not ossible by using er nd encil lone As result o these ethods, techers re ble to enggestudents ore eectively by ddressing dierent lern-

ing styles nd develoing understnding tht leds tohigher-level thinking Techers don’t oten ssocite theuse o grhing clcultors with the concetul rocessGrhing clcultor ctivities cn engge students inbuilding concetul understnding while giving the rc-tice necessry or rocedurl rofciency in clcultoruse As students ove through ech hse o lerning,they re exosed to concet or skill nuerous ties byutilizing the ollowing roch in ech lesson

1. I do—The techer odels the colete rocedureusing disly roduct such s couterrogr or rojector

2. We do—While the techer reets ech ste, thestudents engge in the ste s clss

3. You do—The students rctice the coleterocedure indeendently



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Post-Reading Reflection1. Why is it iortnt to integrte thetics cross the content res?

___________________________________________________________________

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___________________________________________________________________

2. Describe how students cn use th journl in ech o the

ollowing content res: reding, writing, socil studies, science, ndthetics

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3. Exlore the Ntionl Librry o Virtul Mniultives site Find nlet tht goes with lesson you need to tech this yer Write bouthow you will use the let in tht lesson

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Assessing Students

Assessents re lerning oortunities

or students nd or techers Studentscn lern ore thetics, nd tech-ers cn lern ore bout their students—nd soeties ore thetics, swell—ro ll tyes o ssessents(Long, 2000)

Aligning Assessment with InstructionThe gol o stndrds-bsed instruction is to hve seto concets tht ll students cn ster by the end othe yer In order to ccolish tht gol, techersust ssess their students roritely throughout theyer Assessents cn tke ny ors nd should givetechers the inortion they need to ke inoreddecisions bout wht to tech nd how to tech it

ChapterTen



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In order or techers to roritely use the inortiongthered ro ssessents, the ssessents ust lign

closely with the theticl concet being tughtI not, techers will not know whether students trulyunderstnd the concet

Techers ust consider severl things when cretingnew ssessents or reviewing reviously creted ssess-ents to use with their students

• The ssessent should cover teril tht wstught

• The thetics should be relevnt nd enggingto the students

• The execttions or the fnished roduct shouldbe cler

• The ssessent should clerly show students’theticl knowledge, understnding, ndthinking rocesses

• The ctivity used or ssessent should hve cler urose or either orl or inorl

ssessentWhen ssessent ligns with instruction, both techersnd students beneft Students re ore likely to gin dee understnding o the curriculu when instructionis ocused nd they re ssessed on wht they re tughtInstruction-ligned ssessents re lso tie eectiveor techers becuse they onitor lerning nd cn beintegrted into dily instruction nd clssroo ctivi-ties

When to AssessAssessent is long-ter, ongoing rocess Techerscn gin insight into students’ lerning through inorlobservtions, listening to grous counicte, ctivi-ties nd rojects, nd orl tests nd quizzes



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Assessing Students

There re severl oints throughout lesson where use-ul ssessent cn be de Deending on the results,

techers cn decide i they should continue with the les-son s lnned or chnge gers i student understndingis low Those oints o lesson include:

• ter ccessing rior knowledge

• during guided rctice

• during indeendent rctice

Informal Assessments

Inorl ssessent is vluble wy or techers to get quick understnding o how students re rogressingin understnding theticl concet However,given tht these ssessents re inorl, techers otendo not record the nd hve no evidence tht indictesstudent growth It is iortnt to hve esy ethodso recording inorl ssessents to use or grdingreorts, rent conerences, nd to show dinistrtors

Recording Methods

Mxiizing instructionl tie is n iortnt skill thttechers need in order to cover the necessry terilthroughout the yer Although ssessent is one o theost iortnt sects o eduction, oten techers donot llot rorite tie or eningul ssessentTechers cn use esy ethods or recording inorlssessents tht do not tke dditionl clss tie tocolete Using clibord or sll binder is silewy to llow or ortbility o the recording sheets

• Use blnk grid to record notes or ech studentduring discussion, ir ctivity, or sll-grouwork

• Design check-o sheet with students’ nesdown the side colun nd list o severlobjectives or concets being tught in the lesson



1


cross the to Give students check in echre in which you observe the showing stery

throughout the lesson• Post schedule o students who will individully

eet with the techer ech dy o the week Hven ctivity or quick tsk they cn colete whilet their eeting Use dt sheet to record nyisconcetions nd observtions while echstudent is working

• Use generl rubric to guge student understndingWlk round the roo to observe nd recordunderstnding or severl students ech dy to seehow they re rogressing

Check or Understanding

Another wy to xiize instructionl tie is to checkoten or understnding during the lesson When thisis done, the techer cn use this knowledge to decidewhether to roceed with urther lesson concets, reetinstruction or soe lesson concets, oer ore rcticewith the concet, or ski ortion o the lesson tht the

students lredy understnd The techer y use theknowledge to dierentite instruction by deciding howny rctice robles nd ctivities re necessry orby llowing soe students to work indeendently whilethe techer works with sll grou o students whoneed ore instruction Overll, checking or under-stnding guides the techer in how to djust instructionso tht the lessons directly eet students’ instructionlneeds (Wili, 2007)



1

Assessing Students

Strategies or Checking or Understanding

Goal Strategy Summary o Strategy

Students re co-ortble with thece o the lessonThey re coortblewith the concetsthey re lerning

Dislythubsignl

The techer sks the students to show signl under their chins (or rivcy) Theylce their thubs u, down, or wveringin the iddle to deonstrte the ollowing:“Yes, I totlly understnd this,” “No, I donot understnd this,” or “I think I under-stnd this”

The students under-stnd how to solven exle roble

Pir shring The students ech solve roble ndshre it with their rtner When bothrtners gree on the solution, they showthe techer “thubs u” signl

The students cnsolve roble cor-rectly nd get theright nswer

Usingwhitebords

The students colete roble on white-bords nd show their nswers to the tech-er The techer cn quickly guge whetherstudents re solving the robles rori-tely nd cn then ke lesson decisions

ccordingly

The students under-stnd the concetwell

Fst writingssignent

The students cn quickly write two to threesentences exlining the concet or sequenc-ing the rocedure or solving roble

Formal Assessments

Tests and Quizzes

Written tests nd quizzes re oten the ost coontye o orl ssessent used in clssroos Thesetyes o ssessents re useul when techers wnt tosee wht student understnds within secifc th-eticl concet They cn lso be helul to guge



1


long-ter understnding o concets nd how studentsre lying the theticl skills lerned throughout

the yer There re ny wys in which tests nd quiz-zes cn be utilized in thetics

• Techers y use short dignostic tests whenbeginning unit o study to discover students’rior knowledge

• Techers y use sll quiz s quick-checkor understnding within unit o study to see istudents re king rogress

• Techers y use unit test tht cobines

severl theticl concets s culintion tothe unit

• Techers y llow students to crete their owntests nd nswer keys s ens or checkingtheir understnding o concet

Rubrics

The Eectiveness o Using Rubrics

The use o rubrics cn be n eective ssessent strt-egy in thetics clssroo A techer y wntto crete generl rubrics to id in the rocess o usingssessents to direct urther instruction With rubrics,techers cn crete nd convey relistic execttions orstudent work s well s their own teching strtegies ndlesson lnning in order to refne their objectives or theclss

When techer counictes cler ssignent direc-

tions nd then rovides rubric tht delinetes theexecttions o the coleted ssignent, the studentsre sulied with ll the necessry inortion they needin order to be successul with n ssignent Rubricscn be esecilly useul or ssignents where studentshve to exlin their resoning or their corehension



1

Assessing Students

o rticulr lesson concet In this cse, the techercould crete one generl rubric tht could be used ny

ties throughout the yerA well-written rubric contins the objectives tht thestudent will chieve when he or she coletes thessignent nd erornce indictors tht esure hisor her rogress Techers need to tech students howto exine rubrics beore beginning ssignents, checktheir work throughout the ssignents, nd fnlly usethe rubric guidelines to evlute the coleteness otheir work once they re fnished Rubrics llow tech-

ers to defne wht kes n ssignent colete beorestudents begin work, so they cn nticite how theywill be evluted nd ccet their fnl grdes

Creating a Mathematics Rubric

1. Beore creting thetics rubric, defne thessignent’s objectives Stte stndrds re gret lce to begin when creting rubric becusethey show exctly wht needs to be stered

2. Generte list o the concets the students reexected to ster s well s nother list o wysthey cn show tht they corehend the teril

3. Highlight the ost iortnt ites on both lists

4. Describe the erornce criteri in detil Theseshould clriy ech hse o the work

It is iortnt or techers to decide how ny crite-ri to include The ore detiled the rubric, the ore

involved the scoring rocess Mny techers fnd thtour to fve ites re idel Too ew, nd the techer willbe unble to eectively evlute the erornce o stu-dents with vried lerning bilities Too ny, nd therubric will be too extensive nd tie consuing to useThe rubric should be esily corehensible or studentsso they cn evlute their own erornces



1


A General Overview o Rubric Guidelines

Lowest Level Middle Level Highest Level• Student shows

inilcorehension othe concet nddoes not coletethe ssignent ortsk

• Mny o thessignent's ortsk’s requiredcoonents reissing

• The student ustcontinue to workon the ssignentor tsk in order tocorehend theconcet

• Avergechieveent; lowestccetble score

• The student showsbsic understndingo the concetbut not coletestery

• Best ossiblecoletion o thessignent or tsk

• A score t this leveldeonstrtes totlcorehension othe concet

Once the techer hs identifed the lowest, iddle, ndhighest levels o erornce, he or she cn fll in theinteredite levels

Sle rubrics cn be ound on ges 177 nd 178There is lso General Rubric (ge 179) tht shows gen-erl criteri or vrious stes o n ssignent, includingcoletion o robles, correct clcultions, nswers

tht relte to the toic, nd logicl resoning



1

Assessing Students

T w o - P o i n t R u b r i c

P o i n t s

C r i t e r i a

2 •

T h e s o l u t i o n i s c o r r e c t

a n d t h e s t u d e n t h a s d e m

o n s t r a t e d a t h o r o u g h

u n d e r s t a n d i n g o t h e c o

n c e p t a n d p r o c e d u r e

•

T h e t a s k h a s b e e n u l l y

c o m p l e t e d u s i n g s o u n d

m a t h e m a t i c a l m e t h o d s

•

T h e r e s p o n s e m a y c o n t

a i n m i n o r f a w s , b u t i t i s

t h o r o u g h a n d t h e

u n d e r s t a n d i n g i s e v i d e n

t

1 •

T h e r e s p o n s e i s o n l y p a

r t i a l l y c o r r e c t

•

T h e s o l u t i o n m a y b e c o

r r e c t , b u t t h e r e s p o n s e d

e m o n s t r a t e s o n l y a p a r t i a l

u n d e r s t a n d i n g o u n d e r l y i n g m a t h e m a t i c a l c o n c

e p t s o r p r o c e d u r e s

•

O r , t h e s o l u t i o n i s w r o n g ,

b u t t h e s t u d e n t d e m o

n s t r a t e s u n d e r s t a n d n g o t h e

c o n c e p t o p r o c e d u r e s

0 •

T h e s o l u t i o n i s i n c o r r e c t

•

T h e r e s p o n s e i s i n c o m p

r e h e n s i b l e a n d / o r d e m o n

s t r a t e s n o u n d e r s t a n d i n g o

t h e c o n c e p t

R u b r i c a d a p t e d r o m F

C A T 2 0 0 4 S a m p l e T e s t M a t e r i

a l s ( 2 0 0 3 ,

F l o r i d a D e p a r t m e n t

o E d u c a t i o n )



1


F o u r - P o i n t R u b r i c

P o i n t s

C r i t e r i a

4

•

T h e s o l u t i o n i s c o r r e c t a n d t h e s t u d e n t d e m o n s t

r a t e s a t h o r o u g h

u n d e r s t a n d i n g o t h e c o n c e p t o r p r o c e d u r e

•

T h e t a s k h a s b e e n u l l y

c o m p l e t e d u s i n g s o u n d m a t h e m a t i c a l m e t h o d s

•

T h e r e s p o n s e m a y c o n t a i n m i n o r f a w s , b u t i t i s

t h o r o u g h a n d t h e

u n d e r s t a n d i n g i s e v i d e n

t

3

•

T h e s o l u t i o n d e m o n s t r a

t e s a n u n d e r s t a n d i n g o m a t h e m a t i c a l c o n c e p t s o

r

p r o c e d u r e s i n v o l v e d i n t h e t a s k

•

T h e r e s p o n s e i s m o s t l y

c o r r e c t b u t c o n t a i n s s m a

l l e r r o r s i n e x e c u t i o n o

m a t h e m a t i c a l p r o c e d u r e s o r d e m o n s t r a t i o n o c o

n c e p t u a l u n d e r s t a n d i n g

2

•

T h e r e s p o n s e i s o n l y p a r t i a l l y c o r r e c t

•

T h e s o l u t i o n o r m e t h o d

o r s o l v i n g t h e p r o b l e m m

a y b e c o r r e c t , b u t

t h e r e s p o n s e d e m o n s t r a t e s o n l y a p a r t i a l u n d e r s t a n d i n g o u n d e r l y i n g

m a t h e m a t i c a l c o n c e p t s

o r p r o c e d u r e s

•

E r r o r s s h o w m i s u n d e r s t

a n d i n g o p a r t s o t h e t a s

k o r a u l t y c o n c l u s i o n s

1

•

T h e r e s p o n s e s h o w s l i m

i t e d u n d e r s t a n d i n g o m a t h e m a t i c a l c o n c e p t s a n d

p r o c e d u r e s

•


l e t e a n d t h e p a r t s o t h e p r o b l e m t h

a t h a v e b e e n

s o l v e d a n d / o r e x p l a i n e d

c o n t a i n s e r i o u s f a w s o r

i n c o m p l e t e c o n c l u s i o n s

0

•

T h e s o l u t i o n i s c o m p l e t

e l y i n c o r r e c t

•


r e h e n s i b l e a n d / o r d e m o n

s t r a t e s n o u n d e r s t a n d i n g

o

t h e c o n c e p t

R u b r i c a d a p t e d r o m F

C A T 2 0 0 4 S a m p l e T e s t m a t e r i

a l s ( 2 0 0 3 ,

F l o r i d a D e p a r t m e n t

o E d u c a t i o n )



1

Assessing Students

G e n

e r a l R u b r i c

D i r e c t i o n s : T h i s r u b

r i c i n c l u d e s g e n e r a l c r i t e r i a

o r g r a d i n g m u l t i s t e p a s s i g n m e n t s t h a t i n v o l v e w r i t t e n

e x p l a -

n a t i o n s t o q u e s t i o n s

I n e a c h o t h e c o l u m n s , s

p e c i y c r i t e r i a a n d e x p l a i n h o w t h e y r e l a t e t o t h e a c t i v

i t y a n d

t h e l e v e l s o p e r o r m

a n c e t h a t c a n b e a c h i e v e d

T o e v a l u a t e a n a c t i v i t y , c i r

c l e a l e v e l o p e r o r m a n c e

o r e a c h

c r i t e r i o n a n d a s s i g n

a n u m b e r o p o i n t s T o t a l t h e p o i n t s a n d r e c o r d t h e m

C r i t e r i a

L e v e l I

( 0 – 4 p t s . )

L e v e l I I

( 5 – 8 p t s . )

L e v e l I I I

( 9 – 1 0 p t s . )

S e l f -

S c o r e

P e e r

S c o r e

T e a c h e r

S c o r e

S t e p s i n t h e

a c t i v i t y h a v e

b e e n c o m -

p l e t e d

Q u e s t i o n ( s )

h a v e b e e n

a n s w e r e d

R e s p o n s e s

r e l a t e t o t h e

q u e s t i o n s b e i n g

a s k e d

C a l c u l a t i o n s

a r e s h o w n a n d /

o r e x p l a i n e d

I d e a s a r e s u p -

p o r t e d w i t h

l o g i c a l r e a s o n -

i n g a n d / o r e v i -

d e n c e



10


Data-Driven InstructionDt-driven instruction reers to the rocess o designing

curriculu nd instructionl strtegies to tch dtro student ssessents This dt cn be collectedro vrious dily ctivities such s student-techerinterction nd observtions, guided nd indeendentrctice, nd orl nd inorl ssessents The keyto dt-driven instruction lies in gthering nd interret-ing the dt nd understnding how to use it

Dt nlysis is eningless i it does not resultin eningul instructionl chnge Dt-driven

eductors re ble to use sutive nd or-tive ssessent dt together to ileentstrtegic, trgeted, ocused instructionl inter-ventions to irove student lerning Theseinterventions should be ligned with stte stn-drds nd district curricul s well s content-se-cifc, develoentlly rorite best rctices(McLeod, 2005)

Once the dt is collected, it needs to be nlyzed Dt

cn be nlyzed by student or by skill/concet/stndrdWhen looking t dt ro rticulr student, it isiortnt to see wht skills they re not stering ndny coonlities those skills ossess For exle,i the student is hving diicult tie ultilyingnd dividing, he or she is robbly wek in dditionnd subtrction becuse those re the skills built ro cler understnding o ultiliction nd divisionIdentiying the theticl concets with which the

student is struggling will guide uture intervention ortht rticulr child When looking t dt by concet,skill, or stndrd, techer cn see where grou o stu-dents is hving diiculty Deending on the nuber ostudents hving diiculty, techer cn crete n inter-vention with just tht rticulr grou o students, orretech the concet to the whole clss (Den nd Florin,2001)



1

Assessing Students

Dt nlysis is ost eicient when the roer tools reused or orgnizing the dt The tbles in the ollowing

ges will id in nlyzing clss dt

Assessment Item Analysis

This dt orgnizer (ge 182) cn be used to evlutehow ech student nswered rticulr questions on nssessent o ultile skills I students did oorly onone tye o question, it y show tht skill hs notbeen stered or tht they do not hve the underlyingskills necessry to solve tht roble This dt cn hel

identiy skills nd concets tht need to be retught I techer is using rubric to evlute erornce-bsedor inorl tsk, oint vlues should be ssigned to therubric nd then the techer should utilize this chrt sinstructed

To colete the chrt:

• Correct students’ tests nd deterine the nubero oints ech ite is worth

• Record the students’ nes in the frst colunnd the ite nubers in the frst row o the tble

• Write the nuber o oints student receivedon ech ite underneth the corresonding itenuber For exle, i ech ite on test isworth one oint, record 1 or correct nswernd 0 or n incorrect nswer

• At the botto o the tble, record the totl ointserned in the clss nd the ercentge correct or

ech ite First, dd the totl oints tht clsserned on n ite, collectively, nd divide it bythe totl ossible oints tht the clss could hveerned Then, ultily the decil by 100



1


Assessment Item AnalysisUnit: _________________________________________________________________________

Assessment Title: _____________________________________________________________

Total Points

Percentage Correct

Item Numbers

Students’ Names



1

Assessment Item Analysis by Standard

This dt orgnizer (ge 184) cn be used to gin insight

into the rogress being de or given stndrd by r-ticulr student or the whole clss I techer is using rubric to evlute erornce-bsed or inorl tsk,oint vlues should be ssigned to the rubric nd thenthe techer should utilize this chrt s instructed

To colete this chrt:

• Write the students’ nes in the frst colun othe tble

• Record the nuber o the stndrds ssessed inthe digonl coluns

• In the row below the stndrds, record the itenuber(s) ro the ssessent tht correlte toech stndrd

• In the second row, record the totl nuber oossible oints tht students could ern on thoseites

• Ater grding the ssessent, fnd the totlnuber o oints students erned or echstndrd by counting the oints or the itesdesignted or tht stndrd

• Ater nlyzing the ssessent by stndrd,deterine the ercent o stery er stndrdFirst, clculte the totl nuber o ossible ointstht students could ern on given stndrdThen, dd u the totl nuber o oints tht they

ctully erned Finlly, divide the totl nubero oints the students erned collectively by thetotl nuber o oints they could hve ernedcollectively nd ultily the decil by 100

Assessing Students



1


Assessment Item Analysis by StandardUnit: _________________________________________________________________________

Assessment Title: _____________________________________________________________

Item Numbers

Points Possibleper Standard

Total Points

Percentage ofPoints Earned

Standards

S t u d e n t s ’ N a m e s

Standards



1

Assessing Students

Using the Data CollectedDt by itsel is not inherently useul It is not until

techers ut the dt into workble docuent nd n-lyze the results tht the dt becoes soething theycn use in the clssroo Techers cn use the dt tocounicte with students s well s to identiy ndddress student isconcetions to guide uture instruc-tion Dt cn be used to convey inortion to rents,dinistrtors, nd the counity, s well

Communicating with Students

It is iortnt to show students dt in order to helthe understnd their rogress throughout the yer Atties, letter grdes cn becoe dunting nd ulling u grde sees iossible When students hve dt thtesures their erornces on sll tsk, they cnsee secifc wys to irove their theticl under-stnding This gives the workble, eningul golsWhen students gin stery o theticl skill, itincreses their confdence nd otivtion to continue to

chieve in other res o thetics (Wili, 2007)

Strategies or Identiying Student Misconceptions

It is ct tht there re ny students who il orstruggle in thetics clsses Furtherore, studentsdo not lwys understnd why they re hving roblesin thetics clss They ight relize tht they restruggling, but eel owerless to “ctch u” once theyll behind They ight cli tht they “do not like

th” when they re relly hiding behind the relitytht they do not understnd ny theticl con-cets Soeties students eel discourged nd rectngrily in eorts to hide their senses o ilure While techer ight fnd soe students who ctively seek th or understnding, the techer will lso hve stu-dents who do not know how to do so



1


Here re soe rcticl strtegies or identiying studentisconcetions nd understnding

1. The techer uses requent orl nd inorlssessents to check or understnding duringlessons

2 The techer creully wtches ssessents oindividul students or evidence o stery orconusion

3. When the results o the whole clss re nlyzed,the techer uses these dignostic tools to

hel identiy which concets need the ostinstructionl tie

4. The techer kees ccurte records o student ndclss rogress nd uses the to see where studentsre struggling

5. The techer creully onitors clssroo ndhoework ssignents While these do notlwys need to be grded, the techer uses theor signs o how well the students re grsing the

contents6. The techer reviews the students’ notes tken in

clss The techer cn eriodiclly sk studentsto exlin their corehension o vocbulry,rocedures, or concets in order to ddress studentisconcetions in urther lessons or decide ireteching the lesson is rorite

7. The techer uses rubrics With rubric, thetecher cn identiy the skills evident or lcking

in student work The techer cn shre rubricswith students, or hve the students identiy therubric scores on their own ers or the erso eers, with the nes covered This will helstudents ocus on the ost iortnt concets o lesson



1

Assessing Students

Addressing Student Misconceptions

Once the techer hs given the ssessents, grded

the, collected nd nlyzed the dt, nd identifedthe res in which the students re ltering, the techerchooses n ction ln or ddressing student iscon-cetions nd reteching terils It is lwys oreeective to chnge the strtegies nd instructionl lnswhen reteching concet tht the students did notunderstnd the frst tie

Suggestions or Reteaching the Concepts

The techer y not hve tught ll the coonents o the lesson ln Thetecher cn decide to teach the skipped components to retech the concet

The techer cn choose a strategy rom the dierentiation chart to use orreteching the concet The techer should choose the rorite level rothe chrt in order to ick n eective strtegy or ech grou o students

The techer cn llow students opportunities to talk about the concept with

rtners or within sll grous

The techer cn choose an activity such s ge or roject tht urtherenhnces students’ understnding but resents the inortion in new wy

The techer cn llow those who successully stered the content to reviewdierent liction ctivities while meeting separately with a small group othose who did not ster the content

The techer cn review the vocabulary with activities i the techer susectstht the students still do not hve the cdeic vocbulry necessry to core-hend the concets nd rctice the skills



1


Communicating with Parents

Dt cn be used s n eective tool in rent-techer

conerences nd counictions It llows rentsto see secifclly where their children re strugglingor excelling, nd rovides useul eedbck or wys inwhich rents cn hel t hoe The dt will clriywht is exected o students nd where their children rectully eroring ccording to the stte stndrds

Communicating with Administrators

Dt cn be used to show dinistrtors how students

re rogressing throughout the yer in reltion to s-tery o stte stndrds This will give dinistrtors nide o how students y eror on ny stte-ndt-ed, stndrdized ssessents The dt will lso indictei or when during the yer to begin n intervention ro-gr beore or ter school Adinistrtors would lsobe interested in the dt to nticite which individulstudents nd how ny in totl will likely be retinedt their schools

Communicating with the Community

In ny sttes nd districts, dt is esure o ccount-bility or students, rents, techers, nd dinistr-tors When the dt revels n overll trend in ro-gr, school, or district, it rovides vluble inortionto show wht is working nd wht is not working in thecounity’s schools



1

Assessing Students

Post-Reading Reflection1. Why is continul ssessent integrl to the success o your students?

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

2. How cn you use dt to understnd wht concets need to be tughtin the uture?

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

3. Crete rubric tht ssesses n inorl roject or ctivity tht youlredy use in your clssroo

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________




10



1

ChapterEleven

Conclusions

Developing an Intervention CurriculumFor students struggling in thetics, it is iortntto receive soe tye o intervention s soon s ossibleso they do not ll behind According to the Resonseto Intervention (RTI) ethod, not ll students need these tye o intervention Tier I students will onlyneed intervention s rt o dily clssroo thet-ics ctivities Techers o these students should be bleto colete this tye o intervention on their own TierII nd Tier III students need ore hel thn clssroo

techer cn do on his or her own These students needoutside interventions These interventions cn tkelce in vrious settings nd require thoughtul ln-ning

A curriculu te should eet requently to designintervention rogrs These eetings need to begin t



1


the end o the revious yer, during the suer, or tthe very beginning o the current school yer The soon-

er decisions re de regrding the rogrs tht will beileented, the ore tie there will be to develo thetieline, ln instruction to eet students’ needs, ndconduct roessionl develoent Decisions tht needto be de include:

• deterining which tye(s) o intervention will beoered

• deterining the ount o instructionl tie orech tye o intervention

• deciding the length o ech rogr• looking t the secifc curriculu tht hs been

chosen nd creting tieline or techers toollow

• ssigning nd trining ersonnel

Instructional Intervention Settings

The curriculu te cn choose between ny dier-ent tyes o intervention settings deending on the needs

o the students, ssessent results, resources vilble,nd tie liittions

Small-Group Instruction

This intervention setting includes but is not liitedto ull-out rogrs or struggling students, secileduction students, nd/or English lnguge lernersIn sll-grou instruction, students receive dditionlinstruction on the theticl concets with which

they hve diiculty, nd ore concrete rctice o bsictheticl skills nd trgeted stndrds

Beore-/Ater-School Program

In beore-school nd ter-school rogrs, studentsreceive quiet instructionl tie wy ro the distrc-



1

Conclusions

tions o the regulr school dy They re not issingny ctivities or clssroo instruction during this tie

Students cn receive reedil hel or rere or utureclss lessons through extr review nd rctice

Saturday School

Siilr to beore-school or ter-school rogr,Sturdy school rovides tie or students who nor-lly hve outside counity coitents such ssorts rctice, religious ctivities, or volunteer oortu-nities, to coe or dditionl thetics instruction

Students tend to rrive to Sturdy school ocused ndredy to lern thetics becuse they do not hve theother subjects nd distrctions tht re rt o the regulrschool dy

Summer School

A district y oer suer school or students in thesuer beore they begin new course, or s ree-dil tie or students who need to strengthen theirunderstnding o theticl skills introduced duringthe revious yer Becuse suer school is tyicllyscheduled or our to six weeks, students cn irovetheir skills t slower ce nd in ore ocused n-ner

Keys to Remember When Using an Intervention Program

• Include ultile reresenttions o concets

• Allow students to ove through the stges o

theticl develoent (concrete, bstrct,liction) slowly so tht sound connections rede between ech ste

• Crete n environent in which students eelcoortble sking questions nd discussingconcets when they do not understnd soething



1


• Use cretive instructionl strtegies or reteching

• Kee oen couniction ong techers,

dinistrtors, students, nd rents so thteveryone understnds wht rogress is being dend wht lerning still needs to be develoed

• Hel students choose lerning gols to kee theotivted throughout the intervention rogr

• Be ositive nd encourging with students t llties

• Hve other relevnt, thetics-bsed ctivitiesor ges students cn turn to s brek ro the

intervention curriculu i their rustrtion levelsget too high

What Should Mathematics InstructionLook Like?At this oint, techers y be eeling overwheled byll the strtegies nd best rctices dvocted withinthis book It is essentil to cobine these strtegies ndke sense o how they work together This section will

show techers how to ut it ll together

Mthetics instruction ust dt to the needs othe students in ech clssroo However, there resoe generl guidelines to ollow in every ctivity ndlesson

1. Introduce the standards-based mathematicalconcept.

• Mke sure the students re engged nd

excited• Access students’ rior knowledge

• Colete vocbulry-building ctivity

2. Teach the lesson.

• Use dierentited instruction

• Use niultives



1

Conclusions

• Begin with concrete concets nd eventullyove to the liction stge (this will tke

ore thn one lesson)3. Inormally assess understanding.

• I there is high understnding, ove on to thenext ste

• I there is low understnding, retech theconcet beore oving to the next ste

4. Begin guided practice.

• Use dierentited instruction

• Use niultives• Utilize sll grous nd eer-ssisted

lerning

• Be very involved with students’ lerning ndrovide continul eedbck to the bouttheir rogress

5. Inormally assess understanding.

• I there is high understnding, ove on to thenext ste

• I there is low understnding, retech theconcet beore oving to the next ste

6. Begin independent practice.

• Use dierentited ctivities

• Use ges or ctivities to reinorce guidedrctice

• Utilize cooertive lerning, grou work,sll grous, or indeendent work

• Intervention tie here is key or studentswho re struggling or need extr scolding

7. Assess the nal concept.

• Use orl or inorl ssessent, deendingon the lesson

• Use results to decide on uture lessons



1


Key Strategies or Implementing a SuccessulMathematics Program

• Techers need to lern the theticscurriculu theselves so they cn xiizestudent understnding nd be knowledgeblewhen students sk questions

• Techers should lwys kee their students inind when lnning curriculu nd ressessingtheir curriculu tielines I students re notking the rogress they need to truly lern thetheticl concets, djustents need to be

de• Techers need to be rered or the lessons they

re teching, with ll sulies nd resourcesredily vilble

• Techers should lwys instruct students in theconcrete stge o concet beore oving to thebstrct nd liction stges

• Techers need to dierentite instruction dily toeet the needs o ll students in the clss

• Techers need to be systetic nd orgnized toutilize ssessents eectively nd in tielynner

• Techers need to utilize the suort odinistrtors nd rents in order to xiizethe resources vilble in the clssroo



1

Conclusions

Post-Reading Reflection1. Wht re three keys to creting successul thetics intervention

rogr?

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

2. How hs your ide o thetics instruction chnged ter redingthis book?

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

3. Mke list o the to three things you wnt to chnge or begin doing inyour clssroo bsed on the inortion you hve red

___________________________________________________________________

___________________________________________________________________

______________________________________________________________________________________________________________________________________

___________________________________________________________________

___________________________________________________________________



1


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1

Appendix

References Cited (cont.)

Hiebert, J, & Crenter, T (1992) Lerning nd teching with understnding In Handbook o research on mathematics teaching and learning (Grouws, D A Ed) New York: McillnPublishing Co

Hiebert, J, & Grouws, DA (2007) Eective teaching or the development o skill andconceptual understanding o number: What is most eective? Reserch Brie retrievedMrch 22, 2007, ro htt://wwwnctorg/news/contentsx?id=8468

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Ntionl Council o Techers o Mthetics (1989) Curriculum and evaluation standards orschool mathematics Author




References Cited (cont.)

Ntionl Council o Techers o Mthetics (2000) Principles and standards or schoolmathematics Author

Ntionl Council o Techers o Mthetics (2006) Curriculum ocal points or prekindergarten through grade 8 mathematics Author

Ntionl Reserch Council (1990) How people learn: Brain, mind, experience, and schoolWshington, DC: Ntionl Acdey Press

Oesterreich, H (2000) The technical, cultural, and political actors in college preparation programs or urban and minority youth New York: ERIC Publictions(ERIC Docuent Reroduction Service No ED448243)

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[Shelly Frei] Teaching Mathematics Today