Enhancing Core Mathematics Instruction for Students At Risk for Mathematics Disabilities

Findings from experimental interven-

tion research support the notion of

making core mathematics instruction

more systematic and explicit for stu-

dents with or at risk for mathematics

disabilities. Core mathematics pro-

grams that utilize an explicit and sys-

tematic instructional approach provide

in-depth coverage of the most critical

content areas of mathematics and

reflect current research on effective

mathematics instruction. One such core

kindergarten curriculum is the Early

Learning in Mathematics (Davis &

Jungjohann, 2009) program. Recent

curricular reviews suggest, however,

that many core math programs fail to

include the instructional design princi-

ples that have been empirically validat-

ed to increase student mathematics

achievement. Teachers can follow eight

specific guidelines to increase the

instructional intensity of their existing

curriculum and thus enhance the quali-

ty of core mathematics instruction.

Promising outcomes are emerging fromstudies that target instructionalapproaches for students with or at riskfor mathematics disabilities (MD). Todate, much of this research has focusedon the effects of small-group interven-tions (Newman-Gonchar, Clarke, &Gersten, 2009), although some recentstudies have begun to investigate theimpact of core math instruction on stu-dent achievement under the day-to-dayconditions typically found in generaleducation classrooms. Findings indi-cate that core math instruction canhelp prevent learning difficulties andpromote critical math outcomes(Agondi & Harris, 2010; Chard et al.,2008; Clarke, Smolkowski et al., 2011).According to the National Mathe-

matics Advisory Panel (NMAP, 2008),core programs should play an integralpart in the design and delivery of coremath instruction. Math programs influ-ence the ease and manner in whichteachers deliver effective core instruc-tion. They provide teachers with aninstructional foundation when modifi-cations are needed to increase instruc-

tional intensity for struggling learners(Baker, Fien, & Baker, 2010). On a dailybasis, core programs largely define thetype of math content taught and arelikely to represent the main source ofmath instruction that students receivein a given school year (Clarke, Doabler,et al., 2011). Thus, core instructionmust keep typically achieving studentson track to develop math proficiencywhile also addressing the learningneeds of students with or at risk forMD (see box, “What Does the ResearchSay About Math Proficiency?”). For thegreatest impact, core instruction mustprovide in-depth coverage of the criti-cal content of school mathematics andreflect current research on effectivemath instruction (National Center forEducation Evaluation and RegionalAssistance, NCEERA, 2009; NMAP,2008).Although few would argue with the

idea of using high-quality math pro-grams, research suggests that manycommercially available programs arenot explicit enough to meet the needsof students at risk for math failure.

48 COUNCIL FOR EXCEPTIONAL CHILDREN

TEACHINGExceptionalChildren,Vol.44,No.4,pp.48-57.Copyright2012CEC.

Enhancing CoreMathematics Instructionfor Students At Risk forMathematics DisabilitiesChristian T. Doabler � Mari Strand Cary

Kathleen Jungjohann � Ben Clarke � Hank Fien

Scott Baker � Keith Smolkowski � David Chard

Inclusion

TEACHING EXCEPTIONAL CHILDREN � MAR/APR 2012 49

1814

Recent curricular reviews indicate thatmany math programs fail to offer (a)demonstrations of target content, (b)frequent and structured student prac-tice, and (c) procedures for academicfeedback. Elementary math programsreviewed by Bryant and colleagues(2008), for example, showed lessonslacked sufficient teacher demonstra-tions and opportunities for studentpractice to build math proficiency. Sim-ilarly, Doabler, Fien, Nelson-Walker, &Baker (in press), in an investigation ofsecond- and fourth-grade textbooks,found few opportunities for students toverbalize their mathematical thinking.Teachers, therefore, need guidelines forhow to address these weaknesses ofcurriculum design and enhance coremath instruction for students with or atrisk for MD.There are some practical guidelines

that teachers can follow to make theircore math instruction more explicit andsystematic for students with or at riskfor MD. To do so, however, teachersneed to understand the construct ofexplicit and systematic instruction, asexemplified by such programs as EarlyLearning in Mathematics (ELM; Davis& Jungjohann, 2009). The eight guide-lines we present for making core mathinstruction more explicit and systemat-ic reflect the instructional design prin-ciples of ELM and the growing knowl-edge base of effective math instruction(Gersten et al., 2009; NMAP, 2008).

The guidelines we describe alsohave implications for special educators.First, because special educators regu-larly work with students who haveexperienced multiple years of failure incore program instruction, they have avested interest in supporting generaleducators to prevent MD. Also, schoolsoften rely on special educators’ knowl-edge of evidence-based practices todesign effective math instruction. Thisdemand is increasing given the pushfor the delivery of evidence-based mathinstruction in general education class-rooms (Hoover & Love, 2011). Specialeducators can assist schools in provid-ing an enhanced core program to meetthe needs of students struggling toreach proficiency (Baker, et al. 2010;Cummings, Atkins, Allison, & Cole,2008).

What Is Explicit andSystematic Instruction?

Math intervention studies consistentlydemonstrate that students with or atrisk for MD learn better in classroomsthat provide explicit instruction com-pared to classrooms that use othertypes of instructional approaches(Baker, Gersten, & Lee, 2002; Gerstenet al., 2009; NMAP, 2008). Gersten etal., for example, analyzed 11 studiestargeting interventions for teaching stu-dents with MD. Findings indicated alarge and meaningful effect (d = 1.22)for explicit instruction on student mathachievement.

Explicit instruction is a method forteaching “essential skills in the mosteffective and efficient manner possible”(Carnine, Silbert, Kame’enui, & Tarver,2004, p. 5). In a recent practice guide,the Institute of Education Sciences(IES) recommended that math inter-ventions should provide explicit andsystematic instruction when teachingstruggling learners (NCEERA, 2009).The practice guide indicated that thelevel of empirical evidence supportingthis recommendation was strong forraising mean achievement levels of at-risk learners. Explicit and systematicinstruction incorporates (a) unambigu-ous teacher models, (b) carefullysequenced instructional examples, (c)instructional scaffolding, (d) timelyacademic feedback, and (e) cumulativereview (NCEERA, 2009). Although theIES practice guide recommends thesefeatures in the context of small-groupinterventions, we contend that manylearners would benefit if core mathinstruction also provided explicit andsystematic instruction.

Practical Guidelinesto Examine and EnhanceCore Math Instruction

To make core math instruction moresystematic and explicit, we suggest fol-lowing eight guidelines extracted fromthe ELM program (Davis & Jungjo-hann, 2009). ELM is a comprehensive,core kindergarten mathematics curricu-lum specifically designed to promotestudents’ conceptual understandingand procedural fluency in the criticalcontent of kindergarten mathematics(see http://ctl.uoregon.edu/research/projects/elm/lesson_sampler) TheELM program consists of 120 lessonsthat are 45-minutes long and comprisesthree content areas: (a) whole numberand number operations, (b) measure-ment, and (c) geometry. ELM incorpo-rates research-based principles of mathinstruction specifically targeted to stu-dents struggling in mathematics:

• Regular use of teacher modelingand demonstrations.

• Visual representations of mathideas.

50 COUNCIL FOR EXCEPTIONAL CHILDREN

What Does the Research Say About Math Proficiency?

A consistent finding of research is that students with math disabilities often havedifficulty developing math proficiency (Kilpatrick, Swafford, & Findell, 2001).Math proficiency comprises both conceptual and procedural knowledge; they aredependent upon each other in that one cannot “acquire the former without thelatter” (Wu, 1999, p. 14).Conceptual knowledge involves understanding the relationship between

representations of math concepts and abstract symbols. For instance, a studentwith conceptual understanding of whole numbers is able to recognize that thenumber 5 can be represented with five blocks, five tally marks, or the writtensymbol of 5.Procedural knowledge is the ability to perform math procedures fluently and

effortlessly. A student with procedural knowledge can retrieve answers to basicmath facts by memory. In contrast, a student who lacks procedural fluency willrely on less sophisticated strategies, such as finger counting, to solve basic mathproblems.

• Frequent opportunities for studentpractice.

• Instructional scaffolding.

We recently tested the efficacy ofthe ELM program (Davis & Jungjo-hann, 2009) in 65 general educationkindergarten classrooms from threeschool districts in Oregon (Clarke,Smolkowski, et al., 2011). Results sug-gest that ELM was beneficial for allstudents in general and students at riskfor MD in particular. Students whowere typically achieving remained ontrack (i.e., made expected gains acrossthe year) and at-risk students in ELMclassrooms reduced the achievementgap with their typically achievingpeers. Based on these encouragingresults and the fact that ELM isanchored in the converging knowledgebase of effective math instruction, wecontend the guidelines presented inthis article will serve as useful tools forteaching students struggling withmathematics.

Guideline 1: Prioritize InstructionAround Critical Content

Because instruction should target themost essential information pertainingto a domain (Carnine, et al., 2004),teachers need ways to determine thecritical concepts and skills that stu-dents must master in a given schoolyear. This is especially true for teachersin the United States who use textbooksthat have been criticized for coveringtoo many topics (Kilpatrick, Swafford,& Findell, 2001; NMAP, 2008). By pri-oritizing instruction around criticalcontent, teachers can focus more

instructional time on the concepts andskills that promote a sound base formore advanced understanding. Forexample, if students develop a funda-mental understanding of whole num-bers, that knowledge is likely to trans-

fer to and lay the foundation for work-ing with rational numbers (Wu, 2009).To identify “what” to teach, teachersshould consult the content standardsrecognized by their respective stateeducational agency and national bodies(i.e., Common Core State StandardsInitiative, 2010; National Council ofTeachers of Mathematics, 2006). Suchstandards in the primary grades tend totarget number sense and particularaspects of measurement and geometry.In kindergarten, a primary focus of

mathematics instruction should be ondeveloping early number sense (Kil-patrick et al., 2001). Students who pos-sess a sense of number can, for exam-ple, make numerical magnitude com-parisons (e.g., 5 is 2 more than 3),count using efficient strategies (e.g.,counting on instead of always begin-ning with 1), and compose (i.e., com-bine numbers to form larger numbers),and decompose (i.e., break down larg-er numbers into smaller numbers)numbers. Many children acquire asense of number through informallearning experiences prior to enteringkindergarten. However, a considerablenumber of children receive insufficientexposure to such informal learningexperiences, or otherwise fail to learnfrom these experiences, and conse-quently enter school with a poorunderstanding of number sense(NMAP, 2008).Additional content areas of kinder-

garten mathematics are geometry andmeasurement. The Common CoreStandards for Mathematics (CommonCore State Standards Initiative, 2010)recommend that kindergarten children

use direct measurement, represent andinterpret data, and understand shapesand spatial reasoning. Geometric andspatial thinking allow children to repre-sent figures and better understand theirspatial environment. Children also use

measurement when they compare anddescribe the attributes of geometricshapes, money, and everyday objects.Comprehensive math instructionaround geometry and measurementcan improve the outcomes of childrenat risk for math failure (Sarama, Clem-ents, Starkey, Klein, & Wakeley, 2008).

Guideline 2: Preteach RequisiteSkills to Ensure Success WithNew Material

Students with or at risk for MD oftenhave difficulty connecting new contentwith what they already have learned.Teachers should bear in mind the back-ground knowledge that students bringto the classroom and explicitly addressthe prerequisite skills necessary forlearning more difficult content.Teachers can use simple warm-upactivities to help jumpstart students’background knowledge. These activi-ties will allow students to make theconnection between previously learnedcontent and new material. For exam-ple, a teacher might use a countingactivity (count by fives to 60) to pre-pare students for telling time in 5-minute increments.

Guideline 3: Carefully Select andSequence Instructional Examples

The selection and use of instructionalexamples plays a major role in teachingstudents with or at risk for MD. Chardand Jungjohann (2006) noted that stu-dents are more likely to become con-fused when initial teaching examplesare complex. “Real-world” problemsoften pose difficulty for strugglinglearners because these problem typestypically combine different math con-cepts and require multiple steps tocomplete (NMAP, 2008). Teachers canpromote success with new math con-tent by using simpler examples at thestart of instruction and increasing thedifficulty of teaching examples as stu-dents gain a general understanding ofthe concept. It is also important toinclude several positive teaching exam-ples and be selective with negative ornonexamples (Coyne, Kame’enui, &Carnine, 2007). For example, whenteaching students to identify basicthree-dimensional shapes, use three-

TEACHING EXCEPTIONAL CHILDREN � MAR/APR 2012 51

By prioritizing instruction around critical content, teacherscan focus more instructional time on the concepts and skillsthat promote a sound base for more advanced understanding.

dimensional shapes of varying colorand size to provide a range of positiveexamples. For each three-dimensionalshape (e.g., cone), show its two-dimen-sional counterpart (e.g., triangle) tohelp students learn how to differentiatebetween shape types.

Guideline 4: Scaffold Instructionto Promote Learner Independence

Instructional scaffolding is support thatteachers provide to facilitate students’development of math proficiency. Asstudents become more independent intheir learning, the scaffolding is gradu-ally withdrawn. Instructional scaffold-ing is essential for teaching studentswith or at risk for MD. Teachers canuse the “I do it. We do it. You do it.”approach (Archer & Hughes, 2010) toscaffold math instruction when intro-ducing struggling learners to new anddifficult content. In the “I do it” phase,the teacher explicitly models the les-son’s target content (e.g., solving wordproblems). How long a teacher worksin the “I do it” phase depends on howquickly students grasp the material.One or two instructional examplesmight suffice if the material is relative-ly easy for the majority of students, butmultiple demonstrations are moreappropriate if the content is more diffi-cult and students require more sup-port. For the “We do it” phase, theteacher guides the students in practiceopportunities (e.g., solving word prob-lems together). In the “You do it”phase, students engage in independentpractice opportunities (e.g., studentscompleting word problems on theirown).

Guideline 5: Model andDemonstrate Instructional TasksThat Students Will Learn

Instruction becomes more explicitwhen teachers model what they wantstudents to learn. Teacher modelingconsists of unambiguous explanationsand clear demonstrations. Models cantake various forms, including statingsimple facts (e.g., “This is a 6.”), defin-ing math vocabulary (e.g., “A fractionis a point on the number line.”), think-ing aloud about math procedures (e.g.,“I solved this problem by adding the

numbers in the ones column first.”), ordemonstrating a math skill (e.g., show-ing how to align two numbers accord-ing to their place-value positions).Archer and Hughes (2010) suggestedthree ways to improve the quality ofteacher models:

1. Use clear and concise language;this helps clarify the target skill orconcept.

2. Provide several models—but notso many so that instruction getsbogged down with a lot of teachertalk.

3. Allow students to actively participatein the models, such as answeringquestions.

Figure 1 presents an activity fromthe ELM program (Davis & Jungjo-hann, 2009) that incorporates Archerand Hughes’s (2010) recommendations.In the example, students learn how toadd 1 to a number. Instruction beginswith five students standing in front ofthe class. As students are “added” tothe line (up to 10), the teacher explicit-ly states how adding 1 is the same assaying the next number on the numberline.

Guideline 6: Provide Frequentand Meaningful Practice andReview Opportunities

To help students develop math profi-ciency, it is imperative that teachersprovide effective practice opportunities,including both guided and independentpractice (Archer & Hughes, 2010; Hud-son & Miller, 2006). The purpose ofguided practice is to help support initiallearning of math concepts. For exam-ple, a teacher might guide a rationalcount activity by helping an entire

class or individual students count out12 pennies. Independent practice, onthe other hand, occurs without teacherassistance and helps students increaseautomaticity of new and previouslylearned content. For independent prac-

tice, students might complete comput-erized practice on their own to buildfluency with basic math facts. Suchpractice opportunities should be usedto (a) engage students in mathematicaldiscourse and (b) review previouslylearned material.Engaging Students in Mathematical

Discourse. There is substantial variabil-ity among students entering school inthe number of opportunities they haveto use mathematical language outsideof school (NMAP, 2008). Therefore,teachers should provide frequentopportunities for students to verbalizetheir mathematical thinking. Teacherscan effectively promote math discourseby having students answer questionsthat directly relate to the target con-tent. These questions, which can beposed to specific individuals or thegroup at large, might require one-wordanswers (e.g., “Ruby, what shape isthis?”) or more detailed explanations(e.g., “Camille, please explain how youmeasured the perimeter of the class-room.”). An efficient way to organizequestions posed to the group at large isto use some type of response signal.Response signals allow all students in agroup to answer together. For example,a teacher might use a snap of the fin-gers, a tap on the board, or a verbalcue (“get ready”) to indicate when stu-dents should respond. This allowsteachers to control the pace of instruc-tion and provide appropriate “thinktime” for those students who needadditional time to answer.Reviewing Previously Learned

Material. Finally, to maintain new andpreviously learned material, studentsmust be given cumulative reviewopportunities (Kilpatrick et al., 2001).Cumulative review helps verify

whether students understand the mate-rial. Teachers can incorporate reviewopportunities through paper-pencilactivities (i.e., worksheets) and end-of-lesson discussions. Review activitiesshould include opportunities for stu-

52 COUNCIL FOR EXCEPTIONAL CHILDREN

The purpose of guided practice is to helpsupport initial learning of math concepts.

dents to discriminate when and whennot to apply newly learned skills(Stein, Silbert, & Carnine, 2006). Forexample, after having learned how tosolve multidigit addition problems withregrouping (carrying ones to tens), stu-dents might complete problems that doand do not require regrouping; this willhelp students recognize that regroupingis only necessary for particular multi-digit addition problems.

Guideline 7: Use VisualRepresentations of Math Ideas

Students struggling with mathematicsoften have difficulty grasping the rela-tionship between math representationsand abstract symbols (NCEERA, 2009).Visual models allow students to under-stand this important relationship acrossmath concepts and ideas. A recentmeta-analysis of math interventionshas empirically supported the use of

visual models by teachers and students(Gersten et al., 2009). In kindergarten,visual models typically involve numberlines, hundreds charts, base-10 blocks,and ten-frames. Teachers also userulers and two- and three-dimensionalshapes. Figure 2 shows how visualmodels are utilized within the ELMprogram to build proficiency in wholenumbers. An effective way to incorpo-rate math models in instruction is by

TEACHING EXCEPTIONAL CHILDREN � MAR/APR 2012 53

Figure 1. Example Activity: Adding 1 to a Number

Note. Reprinted with permission from Early Learning in Mathematics by K. Davis and K. Jungjohann, Lesson 81, pp. 1–2.Copyright 2009 by the Center on Teaching and Learning.

Teacher wording Say, “Today we are going to learn how to add 1 to a number.”

Ask 5 children to stand up in the front of the class.

“Everybody, how many children are standing in the front of the room?” (“Five.”) “Yes, there are 5 children.”

“Let’s look at the number line as we add or plus 1. We’ll start with 5 because that’s how many children we have.”

Touch the numeral 5 on a number line or the hundreds chart. “What numeral?” (“Five.”) “Yes, five.

“Each time we add 1 child, I’ll move to the next numeral. Remember, when we add or plus 1, it’s just the same as counting by ones.”

“Now, we’re going to add or plus 1 more.” Quickly, ask another child to stand in front of the room.

“Since we added 1 more child, I’ll touch the next numeral on the number line.” Touch the next numeral on the number line. “Everybody, what numeral?” (“Six.”) “Yes, six.”

“There are 6 children standing in the front. We’re going to add or plus 1 more.” Quickly, ask another child to stand up.

“Since we added 1 more child, I’ll touch the next numeral on the number line.” Touch the next numeral on the number line. “Everybody, what numeral?” (“Seven.”)

Repeat the preceding 2 steps until there are 10 children standing up. “We added 1 with children and on the number line.”

”

using a concrete-representational-abstract (CRA) approach (Hudson &Miller, 2006). As part of the CRAapproach, a teacher might begininstruction with concrete examples(e.g., counting blocks) and then inter-weave pictorial representations (e.g.,tally marks) and abstract symbols (e.g.,numbers) as students grasp conceptualunderstanding.

Guideline 8: Deliver TimelyAcademic Feedback, BothCorrective and Confirmatory

Academic feedback is a teacher’s verbalreply or physical demonstration to astudent response. It can take the formof an error correction or a response

affirmation. Correcting student errors isan important aspect of math instruc-tion because unattended errors are like-ly to lead to later misconceptions(Hudson & Miller, 2006; Stein et al.,2006). An effective and timely way tocorrect an error is to state the correctanswer and then restate the questionto the student. For example, a teachermight say, “This is a hexagon. Whatshape is this?” If the student continuesto demonstrate difficulty with the ques-tion, the teacher might have the stu-dent say the shape’s name in unison(“Owen, say it with me. This is a hexa-gon.”). Response affirmations aredelivered after a student provides acorrect response. By simply repeating

the student’s answer (e.g., “Yes, thisnumber is 16”), teachers can use thistype of feedback to extend studentlearning.

Using the Guidelines in Practice

Figure 3, which presents a kinder-garten lesson introducing teen num-bers, illustrates how teachers can usethese guidelines to enhance core mathinstruction.The lesson’s primary instructional

objective is for students to understandthat teen numbers (i.e., 11–19), in par-ticular irregular teens, are composed ofone 10 and some more ones. Second-ary objectives include: (a) identifyingteen numbers and (b) counting on

54 COUNCIL FOR EXCEPTIONAL CHILDREN

Figure 2. Using Visual Representations of Math Ideas

Visual representations How are they used in ELM?

The number line is introducedwhen students begin working withthe number 1. Numbers are addedto the number line after theirintroduction.

Finger models, tally marks, andten frames serve as concrete andpictorial representations of number.Students are introduced to thesemodels as they begin to workwith numbers 0–10.

Base-10 blocks allow studentsto develop a conceptualunderstanding of place value.Instruction utilizes these modelswhen introducing numbersgreater than 10.

The place value chart is used tocount the number of school days,starting with the first day. Everyten days, students regroup 10 ones(straws) for 1 ten (a bundle ofstraws). The hundreds chartprovides a reference for counting,number identification, and quantitydiscrimination.

Note. ELM = Early Learning in Mathematics (Davis & Jungjohann, 2009). Graphics reprinted with permission from EarlyLearning in Mathematics by K. Davis and K. Jungjohann. Copyright 2009 by the Center on Teaching and Learning.

1 2 3 4

• • • ••

• • •

Finger models Tally marks Ten frames

Base-10 blocks

Place value chart Hundreds chart

Number line

Figure 3. Enhancing Mathematics Instruction

TEACHING EXCEPTIONAL CHILDREN � MAR/APR 2012 55

Everyone, let’s review some numbers.(Show the number card for 18.)

When I tap the card, tell me the name ofthis number. Ready, what number?

(Tap card.)Yes, 18. Eighteen is 10 and eight more.Eighteen is 10 and how many more?

(Wait for response.)Yes: eight more.

(Repeat with the numbers 14, 5, 11, 8,and 19. Provide a few individual turns.Confirm all responses.)

Today, we’re going to learn a new teennumber.

(Show the number card for 13.)This number is 13. What number?

(Tap card.)Yes, 13. This is how to write 13.

(Write on board.)Thirteen is a tricky number. It ends with“teen” but does not begin with “three.”Listen, THIR-teen. Say it with me.

(Wait for response.)Yes: 13.

(Ask a few individuals to identify 13.For incorrect responses say, “This is13. What number?”)

Thirteen is 10 and three more. Thirteen is10 and how many more?

(Wait for response.)Yes: three. Watch as I use the cubes andten-sticks to “make” 13. A ten stick ismade up of 10 cubes. Count with me to makesure the ten-stick has 10.

(Count in unison.)So 10 cubes and 1 ten-stick are equal orthe same. How many cubes make up a tenstick?

(Wait for response.)Yes: 10.

To make 13, I need one ten-stick and threecubes.

(Count out 3 cubes.)Now I’m going to count on from 10 to 13.

(Point to ten-stick and touch each cubewhen counting.)

Listen. 10, 11, 12, 13. Count with me.TEN, 11, 12, 13. Your turn! Count on from10 to 13.

(Provide individual turns. Confirm allresponses.)

(Hand out a set of base-10 models to eachstudent. Tell the class that they willuse the base-10 models to “make”up different numbers. Provide a rationalefor why it is useful to work with base-10models; that is, base-10 models allowstudents to compose teen numbers into oneten and some more ones. Have number cardsfor 3, 7, 11, 12, 13, 14, 16, and 18.Show each card and have students identi-fy. Then, have students “make” numbersand count the base-10 models to confirm.For numbers greater than 10, havestudents “count on” from 10.)

Now it’s your turn to use the base-10models.

(Show the number card for 16.)What number?

(Tap card.)Yes, 16. Sixteen is 10 and six more.Sixteen is 10 and how many more?

(Wait for response.)Yes: six. To make 16, we need one ten-stickand six cubes. Use your base-10 models tomake 16.

(Monitor student responses.)Let’s make sure we have 16 by counting onfrom 10. TEN, 11, 12, 13, 14, 15, 16. Yourturn! Count on from 10 to 16.

(Provide individual turns. Confirm allresponses.)

(After students have represented eachnumber, arrange number cards in randomorder and have students identify inunison; tap each card so that studentssay the answers together. At the les-son’s conclusion, remind students thatteen numbers are made up of one ten andsome more ones. This will help studentsreflect on the base-10 system.)

1613

18

from the number ten. The irregularteens, which are 11, 12, 13, and 15,often pose difficulties because of theirunique pronunciations (e.g., 13 is pro-nounced thirteen instead of threeteen).Regular teens (i.e., 14, 16, 17, 18, and19) are less challenging because theyare pronounced by first stating thenumber in the ones position and then“teen” (e.g., sixteen). Given the dis-tinctive qualities of teen numbers,Stein et al. (2006) recommend thatteachers introduce regular teens beforeirregular teens.Note that the sample lesson incor-

porates teacher modeling (“I do it”),instructional scaffolding, and struc-tured practice opportunities. It alsoincludes frequent opportunities for stu-

dents to respond (i.e., one for eachnumber) as well as occasions for groupresponses. Enhancements such asthese, based on our guidelines and theexisting literature on effective mathe-matics instruction, require modesteffort to implement and are likely topromote more promising outcomes. Forexample, to help students develop afirm understanding of teen numbersand the base-10 system, the lesson pri-oritizes instruction around one irregu-lar teen number: 13. Assuming thatstudents have already mastered theregular teens as well as numbers 11and 12, the teacher will separatelyintroduce the remaining teen number,15, in a later lesson. A general strategyfor introducing new teen numbers is

about once every two or three lessons;this helps promote content masteryand minimize student confusion(Hudson & Miller, 2006).In the sample lesson, the teacher

initiates instruction with a brief reviewof previously learned material andthen provides an overview of the les-son’s target content. Clear expectationsat the start of instruction help studentsunderstand what is required of them.Then the teacher overtly models howto pronounce, write, and represent thetarget number. As demonstrated in thisexample, teachers can strengthen stu-dents’ conceptual understanding bylinking the abstract (i.e., numbercards) and visual representations (i.e.,base-10 models) of numbers (NCEERA,2009). The lesson presented in Figure 3demonstrates how to support initiallearning of new concepts (e.g., teennumbers) by leading off with a simplerinstructional example. Because the pos-sibility of student confusion increaseswhen instruction starts with complexproblems (Chard & Jungjohann, 2006),more effective instruction begins witheasier examples to introduce conceptsand teaches generalizable strategies—such as the “count on” strategy used inthe sample lesson. This lesson planalso shows how teachers can extendstudent learning through timely aca-demic feedback and incorporate signalresponse techniques (e.g., clappingwhen counting or tapping the numbercard) to manage responses from theentire class.

Final Thoughts

Many students experience persistentdifficulties in mathematics, and thechallenges that these students face aregreat. Only through concentratedefforts can schools hope to meet themajority of their students’ learningneeds, including both those studentson-track for success and those strug-gling to learn the basics of early math-ematics. Core mathematics instructionshould sufficiently cover the most criti-cal math content and adhere to theprinciples of instruction that are effec-tive for teaching students strugglingwith math. Effective, evidence-basedcore math programs such as ELM

56 COUNCIL FOR EXCEPTIONAL CHILDREN

(Davis & Jungjohann, 2009) facilitatesuch systematic and explicit instruc-tion. Teachers who do not have accessto such programs can enhance andimprove their math instruction by fol-lowing the eight guidelines presentedhere.

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Christian T. Doabler (Oregon CEC),Research Associate; Mari Strand Cary,Research Associate; Kathleen Jungjohann(Oregon CEC), Senior Instructor andResearch Assistant; Ben Clarke (OregonCEC), Research Associate; Hank Fien,Research Associate; and Scott Baker(Oregon CEC), Associate Director, Center onTeaching and Learning, University ofOregon, Eugene. Keith Smolkowski,Associate Research Scientist, Oregon ResearchInstitute, Eugene. David Chard (Texas CEC),Dean, Annette Caldwell Simmons School ofEducation and Human Development,Southern Methodist University, Dallas, Texas.

The research reported here was supported bythe Institute of Education Sciences, U.S.Department of Education, through GrantsR305K040081 and R305A080114. The opin-ions expressed are those of the authors anddo not represent views of the Institute or theU.S. Department of Education.

Address correspondence concerning this arti-cle to Christian Doabler, University ofOregon Center on Teaching and Learning,1600 Millrace Dr., Suite 108, Eugene, OR97403 (e-mail: [email protected]).

TEACHING Exceptional Children, Vol. 44,No. 4, pp. 48–57.

Copyright 2012 CEC.

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