I n s l i i i i l i o i i

t

GoingBeyond

The MathWars"

We are living and teaching at a timewhen more students with disabilitiesare being included in general educationclassrooms and heing held to the samestandards as their general educationpeers than ever before (van Garderen,2008). Yet we know that many stu-dents continue to struggle with math.For example, recent test results indicatethat only 32% of all eighth graders areat or above proñcient in mathematics(Lee, Grigg, & Dion, 2007). Further, itis postulated that 5% to 8% of allK-12 students have disabilities inmathematics (Kunsch, Jitendra, &.Sood, 2007). Only 8% of students withdisabilities scored at or above profi-cient on a national assessment ofmathematical proficiency (Lee et al.,2007). Clearly, improving mathematicsinstruction for this group of students isof great importance.

Now we are presented with twoseemingly conflicting ways to structuremathematics teaching and mathematicsclassrooms—direct instruction, tradi-tiotîally seen as belonging to special

education, and inquiry-based teaching,seen as belonging to general education.Both sides are convinced they havefound the "right way" to teach mathe-matics to all students. Worse yet, manygeneral and special education teachersare being taught completely differently,at a time when collaboration is moreimportant than ever before (DeSimone&. Farmar, 2006). What can we as spe-cial education teachers do to under-stand inquiry-based teaching? Whatcan we do to collaborate with generaleducation teachers to help maximizeall students' learning? And finally.

the kinds of activities and strategiessupported by the national mathematicsstandards (Maccini & Gagnon, 2006).However, they are hindered by a num-ber of obstacles. More than half of thespecial education teachers surveyed inthis study were unfamiliar vv̂ ith thestandards. This statistic compares unfa-vorably with the 95% of mathematicsteachers who are familiar with thestandards (Maccini & Gagnon, 2002).Special educators continue to report alack of materials, a lack of support,and perhaps most important, a lack ofconfidence in teaching mathematics

Special educators continue to report a lack of

materials, a lack of support, and perhaps most

important, a lack of confidence in teaching mathematics.

what can we do to bridge the divideover our differing conceptions of math-ematics teaching and learning?

One study has shown that specialeducation teachers do believe that theirstudents wotild respond positively to

(Maccini & Gagnon, 2006). We knowthat these issues are quite commonamong special educators. This articlewill help to improve teachers' under-standing in this area and enable ourconceptions of teaching and leaming

14 COUNai. FOR EXCEmONAL CHILDREN

A SpecialEducator's Guideto Understanding

and AssistingWith Inquiry-Based

Teaching inMathematics

Jane E. Cole and Leah H. Wasbum-Moses

mathematics to be complementaryrather than conflicting.

What Is lite ConflictSurrounding McrthemciticsInstruction?

We know that students with disabilitiesstruggle with basic facts as well aswith tasks such as generalization,applying metacogtiitlve strategies, dis-criminating key points from inelevantinformation, and solving multistepproblems (Macclni & Gagnon, 2006).When confronted with more complexproblems, studems often show lack ofpersistence, low self-confidence, andnegative attitudes toward problem solv-ing in general (Woodward & Montague,2002). The response of special educa-tion traditionally has been behavioral.We use task analysis, breaking upproblem solving into steps. We usemnemonics and flash cards to aid withfact memorization and retrieval. Weteach algorithms for solving each typeof problem and then use various drill-and-practice methods. We emphasize

skills instruction, starting with themost basic skills and steps and pro-gress to the more complex (Woodward& Montague, 2002). Several prominentreviews of the literature in learning dis-abilities (LD) support the use of thesestrategies (e.g., Vaughn, Gersten, &Chard, 2000).

However, in recent years the con-sensus among general education math-ematics professionals has been a movetoward more inquiry-based teachingand away from direct instruction. Thismovement aligns with the NationalCouncil of Tfeachers of Mathematics(NCTM) standards (National Council ofTeachers of Mathematics, 2000). Inbrief, inquiry-based teaching is a stu-dent-centered approach that ofteninvolves students in solving problemsin groups, problems that require themto apply a variety of mathematicalskills in real-world contexts. Theemphasis is on conceptual understand-ing rather than the ability to memorizefacts and apply algorithms (NationalResearch Council, 2001a; Stein, Smith,

Henningsen, & Silver, 2000). Under thismethod, "teachers help students make,refine, and explore conjectures on thebasis of evidence, and use a variety ofreasoning and proof techniques to con-firm or disprove those conjectures.Students are flexible and resourcefulproblem solvers" (National Council ofTeachers of Mathematics, 2000, p. 3).

The goal of this type of teachingand learning is to promote motivationfor and active engagement in mathe-matical thinking (National Council ofTeachers of Mathematics, 2000). orachieving "mathematical proficiency"(National Research Council. 20Dlb).Exposure to higher-level thinkingprocesses is a must in this environ-ment. Stein and Smith (1998) offer atask analysis guide that can help usunderstand the difference between tra-ditional mathematical demands (lowerlevel demands) and higher leveldemands tliat are recommended bymath educators today. Lower leveldemands consist of memorization (e.g.,math facts) and procedural tasks (e.g..

TEACHING EXCEPTIONAL CHILDREN ' MAR/APR 2010 15

using an algorithm without beingrequired to understand and explain theunderlying concepts). Higher leveldemands include procedures with con-nections tasks, which focus on proce-dure with the intent of understandingunderlying concepts, and encouragestudents to represent ideas in tnultipleways; and doing mathematical tasks,which requires new and complexthinking processes in order to btiildrelationships among concepts. It isimportant to note that classrooms thatemploy inquiry-based teaching andlearning look very different from tradi-tional classrooms. The teacher's role isto share essential information andshape the classrootn culture into one inwhich differing mathematical ideas andarguments are sliared and valued.Mathematical tools are constructed andused to guide thinking as studentssolve problems. Classroom tasks aremore collective in nature and areaccessible to all students (Hiebert etal.. 1997).

More recently, the National Mathe-matics Advisory Panel, convened byPresident Bush in 2003. was chargedwith the task of reviewing research onschool mathematics atid determiningwhat works in teaching and learning inK-12 mathematics. Major findingsincluded an increased focus on algebra,additional practice opportunities incomputation, and the strengthening ofmathematics preparation of teachers.The Panel was quite clear about recom-mending a balanced approach to teach-ing mathematics, indicating that nei-ther a fully inquiry-based teachingapproach nor a fully direct instructionapproach was effective. They alsonoted that "explicit instruction withstudents who have mathematical diffi-culties has shown consistently positiveeffects on performance with wordproblems and computation" (NationalMathematics Advisory Panel, 2008, p.425).

Thus, it appears that the two teach-ing approaches can and should becombined in order to benefit all stu-dents (Bottge, Rueda, LaRoque, Serlin,& Kwon, 2007). However, this mergerof theoretical frameworks must beginwith special and general educators

FIgmw 1 . Addftlenal Resources

Schema-based instruction

http;//www.teachingld.org/pdf/teaching_how-tos/joürnal_articles/Article_5.pdf

CogniUve strategies

http://www.unl.edu/csi/math.shtniJ

Scaifolding

http://www.teacherweb.com/CT/CUntotiPierson/ParetitResources/TCM2006-MultiageClassroom-08-19a.pdf

http://fcit.usf.edu/mathvids/strategies/si

Peer-mediated instruction

http://www.cast.org/püblications/ncac/ncac_peermii.html

http://www. promisingpractices, net/program, asp? program id = 143

Concrete-represeatational-abstract sequence

http://w w w. k8accesscenter. org/traitiing.resotirces/CRAJnstructionaLApproach. asp

http://www.specialconnections.kit.edu/cgi-bin/cgi wrap/speccomi/main.phpicat^instruction&subseciion = math/era

Mnemonics

htq)://faculty.kutztown.edu/schaeffe/Mnemonics/mnemonics.html

http://www.ldonline.org/article/Using_MnemonicJnstruction_To_Teach_Mathwww.onlinemathleaming-com

working together to improve instruc-tion (DeSimone & Parmar, 2006). Initia-tion of the collaborative effort shouldinvolve an awareness of effectiveinstructional strategies tor teachingstruggling learners in mathematics.

What Works in Teaching Mathto Students WHh Disabilities?

The ptofessiuiial literature in teachingmathematics to struggling learnersoffers several evidence-based strat^iesthat can be used in general education

ing we summarize the strategies, pro-vide links for further study, anddescribe how special educators mightwork with their general education col-leagues to implement each. Figure 1provides additional resources for eachstrategy.

Schema-Based Instruction

Schema-based instruction is a methodof solving mathematical word prob-lems that helps learners understandthe structure of the problems they are

lljt appears that the two teaching approaches can andshould he comhined in order to heneät all students.

settings. These strategies include (a)schema-based instruction, (b) cogni-tive strategies, (c) scaffolding, (d)peer-mediated instruction, (e) con-crete-representational-abstract (CRA)sequence, and (f) mnemonics (Kunschet a].. 20O7; Maccini. Mulcahy. &Wilson, 2007; Witzel. 2005). U isimportant for both general and spedaleducation teachers to understand howto implement these strategies in themathematics classroom. In the follow-

given. Students are expected to breakthe problem down and diagram specif-ic parts in order to understand how toproceed. Students also use backwardchaining [turning problems into anequation tbat needs to be solved) todetermine the goal of the problem andhow to locate missing information. Asstudents diagram varied problems,they learn to recognize specific schemarelated to different problem types(Maccini et al., 2007). Figure 2 pro-

16 CouNQL FOR EXCEPTIONAL CHILDREN

Figur« a. Example of a Schematic Representation of a TMnafer WordProblem

Word Problem; Timmy has $5 dollars. His uncle gove him $7 more.How much money does Timmy have now?

T

$ $ $ $ $

U$ $ $ $ $ $ $

vides an example of a schematic repre-sentation of a transfer word problem.

Teachers can collaborate to imple-ment schema-based instruction by pro-viding both large- and small-groupexplicit instruction to supplementinquiry-based activities. For example,many inquiry-based activities involvestudents in solving problems in multi-ple ways. Students essentially developtheir own schémas to solve variousproblems. After these activities, teach-ers can make explicit these variousschémas and related diagrams to solidi-fy the skills learned through theinquiry-based tasks. Guided and inde-pendent practice can reinforce theseskills as students apply what they havelearned.

Cognitive Strategies

Cognitive strategies in mathematicsallow students to focus on the neces-sary steps for solving word problemssuccessfully. For example, "Say, Ask,Check" is a very simple cognitive strat-egy that upper elementary students canuse to ensure that they are thinkingthrough the problem and checkingtheir work (Montague, 2005). Anothersuch strategy that has been research-validated is STAR, whose steps includeSearch the word problem, TYanslate theword problem. Answer the problem,and Review the solution (Maccini &Gagnon, 2005J. In brief, these strate-gies can be seen as a way to look sys-tematically across different types ofproblem solving. They differ fromschema-based instruction in that stu-

dents are not necessarily required todiagram the problem steps. Also, stu-dents may use a checklist to workthrough each step of the strategy.

Ghecklists are one way to integratethese strategies into inquiry-based les-sons. Checklists can help guide stu-dents as they attempt to solve moreopen-ended problems. For example,students could be provided a full sheetof paper or a sticky note listing thesteps of the STAR strategy describedearlier. They then check off each stepas they complete it, and/or record theirthinking at each step along the way.The use of checklists helps studentsstay on task, both academically andbehaviorally. The teacher can usechecklists before inquiry-based lessonsto remind students how to solve prob-lems and/or to ensure that individualstudents are following along and keep-ing up with their group as the activityprogresses.

Scaffolding

Scaffolding is a way of building newinstruction onto previously taughtskills, which aiiows students to usewhat they have already learned to mas-ter new knowledge. These connectionsare particularly important in mathe-matics, as the field has shifted from astep-by-step acquisition of facts andalgorithms to a more integratedapproach that makes connectionsamong mathematical disciplines thatwere traditionally seen as separate(e.g., fractions, algebra, geometry;National Council of Teacbers of Mathe-

matics, 2000), Scaffolds are supportsthat help the students move from priorknowledge to new knowledge. Theycan include tools such as a checklistand techniques such as explicit model-ing and demonstration by the teacher(Rosenshine & Meister, 1992). It isimportant for teachers to note thatscaffolds are not intended to be perma-nent supports for students; they areintended to fade out over time as thestudents demonstrate proficiency in thespecific skills.

Scaffolding can take place through-out the mathematics lesson. For exam-ple, it can be used during a structuredreview of previously learned informa-tion prior to the implementation ofinquiry-based activities. It can also beused as a teachable moment duringthese activities as students draw ontheir background knowledge to addressthe problem at hand. Scaffolding pro-vides a rationale when students ask,"Why do we have to learn this?" asteachers can refer to future learningneeds. Finally, it provides a frameworkfor teachers to connect the end of onelesson to the beginning of another andthus assists with planning.

When scaffolding is used in theclassroom, teachers start with the foun-dational skills that have been previous-ly mastered by students and explicitlyteach how the prior knowledge is nec-essary for the new skill. Teachers usemodeling, guided practice, independentpractice, and immediate correctivefeedback as they teach the new contentto the students. Figure 3 provides anexample of scaffolding when teachingstudents to use a cognitive strategy tosolve a word problem.

Peer-Mediated Instruction

Peer-mediated instruction is defined as"paire of students working collabora-tively on structured, individualizedactivities" (Kunsch et al., 2007, p. 1). Itprovides an opportunity for students toengage in the kind of mathematicalcommunication recommended by theNCTM Standards (National Councilof Teachers of Mathematics, 2000).Students are paired by ability level-higher performing students are pairedwith lower performing students.

TEACHING EXCEPTIONAL CHILDREN | MAR/APR 2010 17

Typically a teacher will rank all stu-dents in the class and pair studentsacross the median. For example, thehighest performing student in the tophalf would be paired with the highestperforming student in the lower half ofthe class. Both students take turns act-ing as both tutor and tutee with thehigher performing student acting astutor initially in each session to pro-vide a model. This structtire encour-ages acquisition of concepts by allow-ing each student to take a leadershiprole, resulting in increased ownershipover his or her own learning (Kunschet al., 2007J. It is important to note thenecessity of training for students priorto the implementation of the model inthe classroom. Because all studentswill act in the role of tutor, all studentsshould participate in the training ses-sions to leaxn how to engage the tutee,provide immediate corrective feedback,and interact appropriately with thetutee (Fuchs, Fuchs, Mathes. &Martinez. 2002),

Fuchs, Fuchs, Hamiett, et al. (1997)studied the use of peer-mediatedinstruction within the mathematicsclassroom. The student dyads incorpo-rated four key components to the tutor-ing sessions. First, the tutor modeledand gradually faded a verbal routinethat explicitly stated each step in theprocedure to complete a specific prob-lem type. The tutor then provided step-by-step feedback to either reinforce acorrect response or remediate an incor-rect response. The final two compo-nents included frequent interactionsbetween the tutor and tutee and reci-procity of roles within the dyads.

The use of this model requires pre-planning on the part of both specialand general educator, in terms of pair-ings of the students and how themodel will fit into the inquiry-basedclassroom. The pairings can assist stu-dents in moving from a mastered prob-lem type to learning a slightly morecomplex problem. It offers more struc-ture than traditional inquiry-basedgroup work, but does not rely heavilyon teacher-led direct instruction. Inaddition, collaborative instruction byboth general and special educatore canbe used to model the kind of coUabora-

Ftgure 3. Example off Scoffffoldtng to Teach a Cognitive Word Problem-Solving Strategy

1. Specify steps in the strategy,

(a) Teacher identifies the name of each step.

(b) Teacher describes each step including how and when to conduct it.

(c) Students identify the name of each step and can describe how and whento apply them.

2. Teacher modeling of strategy steps.

»(a) Teacher models each step in the strategy, verbalizing his/her thoughtprocesses as each step is completed,

(b) Teacher models the strategy as the students verbalize each step to becompleted.

3. Students apply strategy steps.

(a) Students apply each step ofthe strategy while verbalizing aloud.

(b) Students apply each step of tbe strategy while whispering.

jm (c) Students apply each step of the strategy without verbal seli-prompts.

4. Practice opponunities.

(a] Teacher provides guided practice where students may practice using thestrategy with immediate corrective feedback provided.

Cb) Teacher provides independent practice where students practice the strategyand receive delayed feedback.

5. Teacher provides immediate corrective feedback.

_ ta) During teacher modeling, teacher remediates the whole class if incorrect,

f steps are verbalized by the class,(b) During student application of steps, teacher monitors student progress and

p iinmediately corrects observed enors.

(c) During guided practice, teacher monitors student progress and immediatelycorrects observed errors.

money, counters, Cuisinaire rods, andso forth. Secondary teachers may findmanipulatives sucb as algebra tiles oran algebra balance scale to help teachmore advanced algebra concepts.

This strategy begins with studentslearning to use manipulatives, physical-ly maneuvering the item to solve anytype of problem. Once they master theskill using manipulatives. they moveinto the representational phase, wbichinvolves them in using tally marks, pic-tures, or other semiconcrete items thatcannot be maneuvered in order tosolve the problem. Finally, studentsmove into the abstract phase, in whichthey use traditional numbers and sym-bols to solve the problem (Witzel,2005).

This strategy lends itself well toinquiry-based instruction, in that stu-dents can use the manipulatives towork through inquiry-based activities.

tive activities desired of the students.The use of peer-mediated instructionwas found to increase student achieve-ment and growth when compared to atraditional teacher-led instructionalmodel (Fuchs et al.. 1997).

Cone rete- Representa liona I-Abstract(CRA) Sequence

This method is a way to approachleaming in mathematics through theprogressive use of concrete manipula-tives, representational pictures, andabstract symbols (Maccini et al., 2007:Witzel. 2005). The National Mathe-matics Advisory Panel (2008) has rec-ommended the use of the technique forstudents at all levels of math instruc-tion. Al the elementary level, the CRAsequence can be used to teach the fourbasic operations, money, time, frac-tions, and beginning algebra throughthe use of base-10 blocks, reaJisüc play

18 CotJNGL FOR EXCEPTIONAL CHILDREN

It facilitates the use of these activitiesby allowing students with disabilitiesto participate on their instructionallevel. It also provides a more struc-tured framework for teachers to use toassess their students" progress.Teachers can then guide students tomove through the sequence as theymaster the skills acquired through eachlesson.

Mnemonics

.Mnemonic strategy "refers to a word,sentence, or picture device or tech-nique for improving or strengtheningmemory" (Test & Ellis, 2005, p. 12).Mnemonics can be either teacher- orstudent-created. The three main typesof mnemonic strategies are keyword,pegword, and letter. Keyword mnemon-ics involve choosing words that soundlike the word or words to be remem-bered and creating a visual representa-tion. For example, students can picturea skateboard with two sets of twowheels each to recall the math fact 2 x2 (Wood & Frank, 2000). A pegwordmnemonic uses words that are associ-ated with specific numbers in order tohelp students recall facts. An example

would be "shoe" for "two" and "door"for "four," and create a story connect-ing two shoes with a door to assist stu-dents in remembering this faa. A lettermnemonic Is perhaps the most com-mon mnemonic strategy, in which oneword or phrase that starts with thesame letter is made into an acronym.An example of this strategy is PEMDAS{Please Excuse My Dear Aunt Sally),for order of operations.

It is arguable to say that mnemonicsare most difficult to blend withinquiry-based instruction because ofthe reliance of this strategy on memo-

skill being taught. Collaboration isessential here: special and general edu-cation teachers must work together todetermine what skills are necessary formemorization and how students willbe assessed for understanding.

Conclusion

It is clear that both general and specialeducators are needed to improve stu-dent ouicomes in mathematics. In fact,one study reponed that middle schoolmath teachers who taught in inclusionclassrooms indicated that their "mostvaluable resource . . . was other peo-

It is clear that both general and special educators areneeded to improve student outcomes in mathematics.

rization rather than conceptual under-standing. Mnemonics do appear toassist students in leaming basic mathfacts, which remain one of the greateststumbling blocks of students with dis-abilities (van Garderen, 2008). Further,teachers must ensure that studentswith whom they use these strategiesunderstand the concept behind the

pie—mainly special education teach-ers, aides, guidance counselors, and/orschool psychologists" (DeSimone &Parmar, 2006, p. 107). As discussedearlier, inquiry-based and directinstruction approaches to teachingmathematics need not be incompati-ble. In fact, a well-designed combina-tion is essential to student growth inmathematics (National MathematicsAdvisory Panel, 2008J, and researchhas shown that special educators canbe successful at integrating theseapproaches (Bottge et al.. 2007).Communication and planning is key;making time to communicate aroundissues of curriculum and instruction inmathematics will support the learningof all students.

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Fuchs, L. S., Fuchs, D.. Hamlett, C. L..Phillips. N. B.. Karns. K.. & Dutka, S.(1997). Enhancing students' helpingbehavior during peer-mediated instruc-tion with conceptual mathematical expla-nations. The Elemeniary Scliool Journal,9?(3), 223-249.

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TEACHING Exceptional Children, Vol. 42.No. 4. pp. ¡4-20.

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