11/24-Milller S. & Hudson, P. (2006)

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Techniques for Program Support

Helping Students WithDisabilit ies Understand What

Mathematics Means

Susan P. Miller

Pamela J. Hudson

Joey is a new second-grade student inMs. Abemathy's class. According toJoey's IEP, one of his instructional goalsis to master basic addition and subtrac-tion facts. Ms. Abemathy decided toassess Joey's perfonnance on these skillsso that she could plan appropriateinstruction.

The first pan of the assessmentinvolved a written assignment that con-sisted of 10 addition facts (with sums 0to 9) and 10 subtraction facts (with dif-ferences 0 to 9). Joey scored 70% on thisassignment. He knew all the additionfacts and four of the subtraction facts.

In the second part of the assessment.Joey used plastic cubes to represent andsolve five basic-fact addition problem sand five basic-fact subtraction problems.Joey correctly represented the additionproblemshe counted cubesto representthe first number, counted cubes to repre-sent the second number, and counted allthe cubes together to determine the

answer. When it came to subtraction.

however, Joey did all five problems incor-rectly. Instead of counting cubes to rep-resent the first numb er (that is. the totalnumber) and then removing the numberof cubes represented by the second num-ber. Joey used the same process todemonstrate subtraction facts that hehad used to demonstrate addition facts.Ms. Abemathy realized that Joey neededfurther instruction to promote his con-ceptual understanding of subtraction.

Many students with disabilities contin-ue to struggle with understanding whatmathematics means. Like Joey, theymemorize basic facts or step-by-stepmathemat ica l p rocedures wi thoutunderstanding the underlying conceptsrelated to the problems. Thus, instruc-tion designed to help students under-stand the meaning of the mathematicsthat they are learning in school is veryimportant (see box "Why Do StudentsNeed a Conceptual Understanding ofMathematics?"] .

This article will share five evidencebased guidelines for implement ingmathematics instruct ion designed topromote conceptual understanding(seebox "Guidelines for Implement ingMathematics Instruction Designed toPromote Conceptual Understanding"}Th e use of these guidelines can facilitatethe acquisition, retention, and general-ization of many mathematics objectivesand ultimately enhances students' abilities to see and understand the relation-ships and connections among importantmathematical concepts .

Guideline 1 : Use VariousModes of RepresentorionMathematics instruct ion intended topromote conceptua l unders tandingneeds to include a variety of modes ofrepresentation. Educators should represent concepts in multiple ways toensure meaningfulness and generaliza-tion of the concept {Cathcart, Pothier,Vance, & Bezuk, 2000; NCTM, 2000;Tlicker, Single ton, & Weav er,2002}.

The concre te - represen ta t iona l -abstract instruct ional process is aneffective method for promoting concep-tual understanding of mathematics byusing various modes of representation(Butler, Miller, Crehan, Babbitt, &Pierce, 2003; Gagnon & Maccini, 2001;Harris, Miller, & Mercer, 1995; Mercer &

28 COUNCIL FOR EXCEPTIONAL CHILDREN


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Why Do Students Need aConceptual Understandingof Mathematics?

Developing conceptual understand-ing is one of the primary goals ofmathematics instruction. Accordingto the National Council of Teachers

of Mathematics [NCTM, 2000, p. 4),the "need to understand and be ableto use mathematics in everyday lifeand in the workplace has never beengreater and will continue toincrease." Students with conceptualknowledge understand the deepmeaning of abstract mathematicalsymbols and operations. Conse-quently, they are much more likely tomeet the mathematical demands thatthey encounter during their school

years. Such deman ds include passingmathematics courses and passingminimum competency examsrequired for high school graduation.Students with conceptual knowledgealso are more likely to be successfulwhen they use mathematics in suchpostsecondary settings as jobs, insti-tutions of higher education, and var-ious facets of independent living.

Mathematics educators agree thatall studentsincluding those with

disabilitiesbenefit from rich con-ceptual understanding and subse-quent abilities to apply mathematicsconcepts in flexible ways when solv-ing complex problems (Butler,Beckingham , & Lauscher, 2005).Thus, carefully crafted instructionthat emphasizes conceptual under-standing is very important.

Miller, 1991-1994). Figure1 is a diagramof the concrete, representational, andabstract levels.

The teacher begins instruction at theconcrete level. He or she may demon-strate specific mathematical concepts byhaving students dramatize the conceptor by using three-dimensional objects.Dramatization involves having studentsrole-play or physically experience,through movement in space, mathemat-ics problems and related concepts. It isa powerful medium that can help stu-

FIgura 1. Concrele-R*prMnlatlonal-AbBtract

Concrete

3 a c

1 "

3 n

1

Pictorial

6 /

4

/ / /

Abstract

5zA

1

dents understand a problem situationand become actively engaged in experi-encing the concept (Bley & Thornton,1995; Cathcart et al., 2000). For exam-ple, in a study that involved teachingthe concept of perimeter to middle andhigh school students with learning dis-abilities, the teacher used dramatization(that is, students walking around theclassroom close to the walls) to demon-strate and reinforce the concept. Thisconcrete demonstration showed stu-dents that perimeter m eans the distancearound the boundaries of a room (Cass,Cates, Smith, & Jackson, 2003). Observ-ing and participating in math-relateddramatization motivates students andhelps them visualize important con-cepts. The physical actions help clarifythe relationships among problem com-ponents.

Developing conceptual

understanding is one of the primary

goals of mathematics instruction.

After the class has represented amathematics problem through dramati-zation, the teacher and students can usemanipulative devices to demonstrateand model the concept. In the previousexample, after students had dramatizedthe concept of perimeter by walkingaround the classroom, they usedgeoboards to find the perimeter of vari-ous shapes.

Unifix cubes (plastic cubes that linktogether) or base-10 blocks can repre-sent problems and their m eaning duringconceptual instruction that focuses onteaching the basic operations of addi-tion, subtraction, multiplication, and

division. Figure 2 shows how Unifixcubes can represent a variety of opera-tional concepts.

When students have mastered a con-cept at the concrete level, the teachercan move instruction to the representa-tional level. At this level, instructioninvolves using two-dimensional pic-tures, drawings, or tallies to demon-strate the same mathematical conceptthat students began to learn by usingmanipulative devices. The teachereither constructs the drawings or usesinstructional materials that provide thedrawings. Additionally, students learnto draw their own pictures or tallies torepresent and solve problems, as shownin Figure 3.

After students have mastered a con-cept at the concrete and representation-

al levels, the teacher can graduallyremove manipulative devices and picto-rial representations so that studentslearn to solve the problem at theabstract level, that is, by using numbersonly. During instruction at the abstractlevel, the teacher expects students tomemorize procedures and facts with

Guidelines for implementingMathematics InstructionDesigned to PromoteConceptual Understanding

Use various modes of representa-tion.

Consider appropriate structuresfor teaching specific concepts.

Consider the language of mathe-matics.

Integrate real-world applications.

Provide explicit instruction.

TEACHING EXCEPTIONAL CHILDREN SEPT/OCT 2006 29


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Figure 3. Using Unfflx Cubes for Concrete-Level liutructton

3 CI, a2 a a

D D

10-5-

fluency to promote efficient problemsolving.

The concrete-representational-abstract instructional process ensuresthat educators integrate various modesof representation into mathematicsinstruction for teaching important con-cepts. Regardless of the mode of repre-sentation that the teacher selects for aparticular lesson (manipulative models,pictures, or tallies), he or she also dis-plays written numb er symbols to ensurethat students make a connectionbetween the conceptual and abstractrepresentations. For example, if ateacher is using Unifix cubes to demon-strate the concept of subtraction, stu-dents should see the numerical repre-sentation of the problem (e.g., 8 - 3= ), as well as the cubes. To ensurethat students pay attention to thenumerical representations in addition tothe manipulative or pictorial models,students should read the problem aloudbefore and after they build the concreteor pictorial model to represent and solvethe problem.

Guideline 2: ConsiderAp prop riate Sflructures forTecKhing Specific Concepts

One of the most important aspects ofplanning and implementing instruction

intended to teach mathematical con-cepts is determining the appropriate les-son structures to use with a specificconcept. The lesson structure is theframework, or the way that the teacherwill demonstrate the concept and theway that the students will practice it toincrease their conceptual understand-ing. Although educators agree thatusing manipulative devices, pictures,and diagrams to represent m athematical

concepts is beneficial (Miller, Butler, &Lee, 1998), teachers must also considerthe best way to use these materials.Different concepts seem to lend them-selves to particular conceptual lessonstructures. Included among these struc-tures are comparing and contrasting,examples and nonexamples, and step-by-step processes (Hudson & Miller,2006).

Figure 3. Using i>t


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characteristics in the examples, theteacher begins to intersperse nonexam-ples, that is, shapes that are not poly-gons, as shown in Figure 5.

The Step-by-Step Lesson Structure

Teachers can use the step-by-step lessonstructure to teach many concepts inmathematics. The explicitness of follow-ing specified steps in sequence is partic-ularly helpful for students with memoryor organizational-thinking deficits.

When the teacher uses this structureto leach the concept of multiplication,he or she explains, for example, that3 x 2 means three groupsof two. Theteacher can use paper plates to repre-sent the groups and cubes to representthe number of objects in each group(Mercer & Miller, 1994). The first step insolving an equation such as 3x 2 is torepresent the first number, or group,with three p aper plates. The second stepis to represent the objects in each groupby counting out two cubes onto eachpaper plate. The third and final stepinvolves counting the total numberofcubes and writing the answer to theequation, as shown in Figure 6. Theteacher can also use the step-by-steplesson structure when teaching suchconcepts as division, regrouping, sim-plifying fractions, and measurement.

Regardless of the lesson structurethat the teacher selects, he or she firstprovides a definition of the conceptbeing taught and then discusses thecharacteristics of the concept. He or shefollows this introduction by using one ofthe structures to develop the students'further understanding of the concept.

Guideline 3: Consider theLanguage of MathematicsThe language that teachers use duringmathematics instruction is very impor-tant, especially when they teach newconcepts. Moreover, language is animportant part of the thinking processthat studen ts use when they solve math-ematics problems. The language ofmathematics is unique and sometimescreates problems for students. Studentswith language disabilities or Englishlanguage learners may have particulardifficulties interpreting Ihe lang uage ofmathematics and may consequently

Figure 4. Compare-and-Contrast Lesson Structure

&

JL .2

= _ 2 _4

_L4

= 12

have severe difficulties acquiring,retaining, and generalizing importantconcepts (Bielenberg & Fillmore,2004/2005; Williams, 2006).

When a teacher plans instructionintended to teach mathematicalcon-cepts, he or she should consider the stu-dents' current vocabulary knowledgeand their language abilities (Bley&Thornton, 1995). The teacher may needto plan learning activities that relatetomathematics vocabulary before begin-ning the concept instruction. Addition-

ally, teachers should select terms that

they will use in a consistent manner

during instructional demonstrations.

Examples of vocabulary choices include

the following:

Timesor multiplied by.

Regroup or rename.

Minus, subtract, or take away.

Sum, total,or answer.

Ones place or ones column. .

Figure 5. Exomple-and-Nonexampie Lessen Structure

This is\a

polygon.

\This is apolygon.

This is nota polygon.

Point to the polygon:

TEACHING EXCEPTIONAL CHILDREN S E P T / O C T 2006 31


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Figure 6. Step-by-Step-Concept Lesson Structure

1. Look at the first number and countthat many plates to represent the groups.

2. Look at the second number and putthat many cubes oneach plate.

3. Count all the cubes to find the to ta land write th e answer.

The teacher must be careful to avoidlanguage that is beyond the vocabularylevel or cognitive deveiopment of thelearner (Kame'enui& Simmons, 1990).

The complexity of concept acquisitionnecessitates clear explanations in lan-guage that students can understand.When students understand the lan-guage that the teacher uses, they canfocus their attention on understandingthe concept without the interference ofunknown terminology.

Another important aspect of mathe-matical language involves the students'abihties to communicate their under-standing of the concepts being taught.

Such communication strengthens thestudents' abilities to link the new learn-ing to previous knowledge and subse-quently construct new understandings

Figure 7. Language Dialogue Bex

that they cati apply in multiple settings.Many teachers use classroom discus-sions to facilitate communication aboutmathematics. They give students oppor-

tunities to articulate their ideas, explaintheir solutions, and comment on otherstudents' ideas (Bielenberg& Fillmore,2 004/2005; Chazan& Ball, 1999).

Language is an important part

of the thinking process that

students use when they solve

mathematics problems.

Unfortunately, tbese opportunitiesfor oral discourse are sometimes diffi-

This shape has__

All theeidesare .This shape i s a _

eldee.

This shape has_Al l t he s ides a re .T h is eh a pe l s a _

sides.

cult for students with learning and language disabilities who hesitate to participate in class discussions (BaxterWoodward, & Olson, 2001; BaxterWoodward, Voorhies, & Wong, 2002)Mathematics journals allow these students to communicate their understanding of concepts through writing rathethan through talking (Baxter, Woodward, & Olson. 2 005). A dditionally, language dialogue boxes can provide thsupport that many students need focomm unicating their thoughts (AIlsoppKyger, & L ovin, 2006J. The dialogue boincludes some information, along withblank spaces in which students placeadditional information related to theconcept being taught. These dialoguboxes reduce the language demand andallow students to focus their attentionon the concept. Figure 7 shows a sample dialogue bo x.

Guideline 4: IntegrofeReal-World ApplicationsThe fourth guideline for planning andimplementing mathematics instructiondesigned to promote conceptual understanding is to integrate real-world applications of the concept being taughtHelping students see the relevance ofmathematics concepts in their dailylives increases their motivation andattention to the lesson, in addition topromoting generalization (GoldmanHasselbring , & the Cognition an dTechnology Group at Vanderbilt, 1997).

Teachers can provide applications tothe real world in a number of ways.Showing the connections betweenmathematics and the world of work isone way to make important linkagesThe teacher can invite workers from avariety of professions as guest speakers

to help highlight the importance ofmathematics in specific careers or jobs.Field trips into the community also givestudents opportunities to directly applymathematics concepts and skills in real-world situations. In addition, vignettes,word problems, and special schoolproj-ects can deal witb topics of interest tostudents. For example, the teacher canask students to identify two items thatthey would like to purchase from theconcession stand at the local movie the-

ater and determine the amount of


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money that they would need to coverthe cost of both items, including salestax (see Figure 8).

Finally, the class can use technologyto make important linkages betweenmathematics concepts and real-worldapplications. Problem-solving softwareprograms simulate real-life or imaginarysituations so that students can experi-ence complex situations and have theopportunity to apply their mathematicsskills. These virtual-reality learningexperiences allow students to feel as ifthey are actually participating in theevent (Kozma & Schank, 1998).

Similarly, problem-solving softwarecan give students practice in identifyingproblems, selecting strategies, findingsolutions, and evaluating the resuhs of

important decisions (Matbews, Pracek,& Olson, 2000). Video-based problemsolving has been particularly effective ingiving students opportunities to applymathematics concepts and skills toauthentic situations presented in thevideo (Bottge & Hasselbring, 1993;Bottge, Heinrichs, C han,& Serlin, 2001).For example, one vignette presented inthe Jasper Woodbury series (Cognitionand Technology Group at Vanderbilt,1992, 1993, 1997; Williams et al., 1998)

shows students in a school developingan idea for raising money and then sys-tematically planning and implementingthe fundraiser. Students work in groupsto solve the mathematics problems thatoccur within tbe context of the videovignette.

Guideline 5: ProvideExplicH InslructionTeaching students to understand mathe-matics concepts requires carefullydesigned lessons with clear and explicitteacher instruction. The literature relat-ed to mathematics instruction, as wellas other content area instruction for stu-dents with disabilities, indicates thatstuden ts need explicit instruction to pro-mote their understanding of difficuhconcepts and related skills (Fuchs,Fuchs, Hamlett, & Appleton, 2002;Greenwood, Arreaga-Mayer, & Carta,1994; Miller et al., 1998; Montague,1997; Swanson, 1999; Swanson &

Hoskyn. 1998, 2001). Rivera (1996) rec-

ommends the following sequence forexplicit instruction:

1. Advance organizer.

2. Demonstration.

3. Guided practice.

4. Independem practice.

This sequence ensures that studentshave the motivation and prerequisiteskills to be successful in the lesson, andit provides a gradual shift of responsi-bility from the teacher's demonstrationto the student's independent perform-ance.

Advance OrganizerThe advance organizer typicallyincludes a review of needed prerequisite

knowledge, a statement related to thepurpose or objectives of the lesson, anda rationale for learning the new content(Hudson, 1996; Lenz, Ellis, & Scanlon,1996). Students with and without dis-abilities improve their learning andretention of new skills and conceptswhen educators use advance organizers(Kooy, Skok, & McLaughlin, 1992).

Demonstration: "I Do"

After the advance organizer, the teacherprovides the de monstration ("1 do"). Intbe "I do" phase of the lesson, theteacher models the concept and solvesthe problem while saying aloud wbatthe students should be thinking whilethey solve similar problems. Althoughthis phase of the lesson emphasizesteacher demonstration, maximizing tbestudents' engagement and participationis very im portant. The teacher can facil-itate such participation by encouragingstudents to answer questions and givechoral responses, as well as by usinginstructional materials. Such participa-tion helps students stay on task andreinforces learning. These activities alsohelp the teacher monitor the students'understanding of the concept. Theteacher needs this information to deter-mine how many examples and prob-lems he or she should model. Teacherdemonstrations must be exphcit, con-

cise, repetitive, and organized.

Figure 8. Concession-StandExampie

Guided Practice: "We Do"When the demonstration is complete,the teacher moves to guided practice("we do"). This "we do" phase beginswith a high level of teacher support(step-by-step prompts), progresses to amedium level of support (multiple stepsperformed with fewer prompts), movesto low-level support [that is, the teacherreduces prompts further or groups sev-eral steps together so that students takeon more responsibility), and ends withno prompts (that is, students completethe task on their own, but can requestand receive assistance from theteacher). This gradual progression of

TEACHING EXCEPTIONAL CHILDREN SEPT/OCT 2006 33


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decreased teacher support sets studentsup for success with the new concept.

Independent Practice: "You Do"The remaining phase of the explicitteaching sequence involves independ-ent practice ("you do"). During this"you do" phase, students demonstrate

their understanding of the new conceptwithout teacher assistance. The inde-pendent practice activity should alignclosely with the teacher demonstrationand the guided practice phase of the les-son. For example, if the teacher and stu-dents used manipulative devices duringthe teacher demonstration and guidedpractice, students should be allowed touse the same devices during independ-ent practice. Mastery criteria for inde-pendent practice should be set high

(e.g., 90%) to ensure that the studentsunderstand the concept.

Final ThoughtsDesigning instruction to develop stu-dents' conceptual understanding ofmathematics is an important goal.Helping students with disabilitiesunderstand the underlying meaning ofmathematics promotes acquisition,retention, and generalization of manymathematics objectives. It also helps

them recognize mathematical relation-ships and connections. Without a focuson teaching conceptual understanding,mathematics instruction becomes mem-orization of meaningless facts and pro-cedures.

Without a focus on teachingconceptual understanding,

mathematics instruction becomes

memorization of meaningless

facts and procedures.

Instruction that the teacher plansand implements by using the five evi-dence-based guidelines (i.e., use vari-ous modes of representation, considerappropriate structures for teaching spe-cific concepts, consider the language ofmathematics, integrate real-world appli-cations, and provide explicit instruc-

tion) supports the acquisition of highlevels of conceptual understanding. Ahigh level of understanding will servestudents well while they progressthrough the mathematics curriculumwithin school and when they translatethe use of mathematics to environmentsoutside school.

References

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Baxter, J. A., Woodward, J., & Olson, D.(2001). Effects of reform-based mathemat-ics instruction in five third grade class-rooms. Elementary School Journal, Wl,529-548,

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Bielenbe rg, B., & Fillmore, L. W. (200 4/2005). The English they need for the test.Educational Leadership, 62(4), 45-49.

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special needs (3rd ed.; pp. 322-346) .Upper Saddle River,NJ: Merrill.Mercer, C. D., & Miller. S. P. (1991-1994).

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Mercer, C. D., & Miller, S. P. (1994) .Multiplication facts 0 to 8L Lawrence,KS:Edge Enterprises.

Miller, S. P., Butler, F., & Lee, K. (1998).Validated practices for teaching mathe-n:iatics to students with learning disabili-ties: A review of literature. Fbcus onE x c e p t i o n a l C h i l d r e n , 3 1 ( 1 ] ,1 - 2 4 .

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instruction in mathematics for studentswith learning disabilit ies. Journal ofLearning Disabilities, 30, 164-177.

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Swanson, H. L. (1999). Instructional compo-nents that predict treatment outcomesforstudents with learning disabilit ies:Support for a combined strategyanddirect instruction model. LearningDisabilities Research & Practice, 14,129-140.

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(2002). Teaching mathematics to alt chil-

dren. Upper Saddle River, NJ: PearsonEducation.

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Susan P. Miller, Professor, Depariment ofSpecial Education, University of Nevada LasVegas. Pamela J. Hudson, Associate Pro-fessor, Department of Special Education andRehabilitation, Utah State University, Logan.

Address correspondence to Susan P. Miller,Department of Special Education, Box453014, University of Nevada Las Vegas, LasVegas. NV 89154 (e-mail: millersp@unlu.

nevada.edu).

TEACHING Exceptional Children.Voi 39,M o . l,pp. 28-35.

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