investigating elementary mathematics curricula: focus on students with learning disabilities

Investigating Elementary Mathematics Curricula:Focus on Students With Learning Disabilities

Casey HordUniversity of Cincinnati

Jill A. NewtonPurdue University

The purpose of this study was to investigate three elementary mathematics curricula to examine the accessibility forstudents with learning disabilities (LD) with regards to challenges associated with working memory. We chose to focuson students’ experiences when finding the area of composite shapes due to the multiple steps involved for solving theseproblems and the potential for these problems to tax working memory. We conducted a qualitative analysis of how eachcurriculum provided opportunities for students with LD to engage with these problems. During our analysis, we focusedon instruction that emphasized visual representations (e.g., manipulatives, drawings, and diagrams), facilitated math-ematical conversations, and developed cognitive and metacognitive skills. Our findings indicated a need for practitio-ners to consider how each curriculum provides instruction for storage and organization of information as well as howeach curriculum develops students’ thinking processes and conceptual understanding of mathematics. We concludedthat all three curricula provide potentially effective strategies for teaching students with LD to solve multi-step problems,such as area of composite shapes problems, but teachers using any of these curricula will likely need to supplement thecurriculum to meet the needs of students with LD.

Students with learning disabilities (LD) are facing risingexpectations in mathematics due to the Individuals withDisabilities Education Act (United States Department ofEducation, 2004), the No Child Left Behind Act (2002),and the adoption of the Common Core State Standards forMathematics (Council of Chief State School Officers andNational Governors Association, 2010). As mandated bythese educational policies, students with LD are expectedto succeed with the general education curriculum, as dem-onstrated by proficiency at grade level on state assess-ments, and learn mathematics in preparation for success incollege and in future careers. Raising achievement levelsof students with LD to meet these higher expectations islikely to be a significant challenge for primary and sec-ondary schools (Powell, Fuchs, & Fuchs, 2013).

Challenges for Students With LD in MathematicsStudents with LD tend to struggle with memory and

processing of academic content (Geary, 2004). These stu-dents also often exhibit low levels of academic achieve-ment compared with their scores on intelligence tests(i.e., low achievement compared with IQ score) or arenon-responsive to progressively more intense forms ofremediation (i.e., response to intervention) (see Gresham& Vellutino, 2010). Students with LD display low levels ofacademic achievement in mathematics in areas such ascounting, fact retrieval, and word problems, even more sofor students with LD in both mathematics and reading(Fuchs & Fuchs, 2002; Geary, 2004). Students with LDmay also struggle with computation especially when mul-

tiple steps are involved (Geary, 2004; Woodward &Montague, 2002). Multi-step problems, in general, tend tobe particularly difficult for students with LD (Swanson &Beebe-Frankenberger, 2004).

Students with LD struggle with working memory(Swanson & Siegel, 2001). Working memory is a keyfactor in solving challenging, multi-step mathematicsproblems for all students; in general, as the number and/orthe complexity of steps increases, the demand on workingmemory also increases (Ayres, 2001). It is common forstudents, who struggle with working memory, to have dif-ficulties with storing important information from one stepof a multi-step problem while trying to process and storeinformation from another step of the problem (Keeler &Swanson, 2001; Swanson & Siegel, 2001). Students withLD also struggle with working memory when they arerequired to combine information from different parts ofmulti-step problems to find a solution (Swanson &Beebe-Frankenberger, 2004).

Students with LD tend to struggle with multi-step prob-lems most often when they have low levels of conceptualunderstanding about the mathematical topic in theproblem (Geary, 2004; Woodward & Montague, 2002).In general, low levels of understanding of a conceptor process tend to compound problems with workingmemory (Barrouillet, Bernardin, Portrat, Vergauwe, &Camos, 2007). For instance, when processing ideas thatrequire significant effort and time (in part due to a lowlevel of conceptual understanding of the topic), studentsare likely to struggle more with storing old information

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while thinking about new information; conversely, whenstudents with LD can more quickly and easily processinformation because they understand the concept, they aremore likely to succeed in multi-step situations (Barrouilletet al., 2007).

Teaching Students With LD to Solve Challenging,Multi-step Problems

For teaching students with LD to solve multi-stepproblems, special education research supports utilizingvisual representations, teaching diagramming of math-ematical relationships, and development of cognitiveand metacognitive strategies (e.g., Cass, Cates, Smith, &Jackson, 2003; van Garderen, 2007; Scheuermann,Deshler, & Schumaker, 2009). In a study by van Garderen(2007), eighth-grade students with LD received diagram-ming instruction and demonstrated success at solvingmulti-step word problems. The participants were taughtthe structure and purpose of diagrams and how to createschematic diagrams for solving word problems with twounknowns. Additionally, the participants were explicitlytaught cognitive and metacognitive skills such as learningto ask themselves if they made reasonable choices ofproblem-solving methods based on the information in theproblem and the answer they found. The participants weretaught to identify the overall task and use that informationto determine the necessary subtasks for solving theproblem. Then, students were instructed to create two-partdiagrams for word problems with two unknowns; the firstsection of the diagram was created to solve for the firstunknown and the second section was for helping studentsapply the value of the first unknown (i.e., the answer to thesubtask) to relevant information in the problem to solve forthe second unknown (i.e., the answer to the problem) (vanGarderen, 2007).

Research supports that other important strategies forhelping students with LD succeed with multi-step prob-lems involve exposure to concrete and semiconcrete rep-resentations (Cass et al., 2003; Scheuermann et al., 2009).In a study by Cass et al. (2003), secondary school studentswith LD were successful with perimeter and area problemsafter receiving instruction emphasizing the concrete—semiconcrete—abstract (CSA) sequence. After calculatingthe perimeter and area of concrete items (e.g., class-rooms), the participants transitioned to semiconcrete rep-resentations (e.g., geo-boards) and eventually progressedto using only abstract mathematical symbols to solve forperimeter and area (Cass et al., 2003). The participantswere also successful in calculating the area of compositeshapes (ACS) (e.g., L-shaped polygons with right angles

only) after the CSA intervention. ACS problems involvingL-shaped figures require multiple steps during which stu-dents may choose to divide an L-shaped figure into tworectangles, find the length of an unlabeled side (with someACS problems), calculate the area of each rectangle, andadd the areas of each rectangle to find a total area of thecomposite shape. An alternative strategy for solvingL-shaped, ACS problems is for students to choose to com-plete a series of steps in which they add area to a compos-ite shape (e.g., to change an L-shaped figure to a rectangle)to allow them to use formulas and, eventually, subtract thearea they added to find the area of the composite shape.

In a study by Scheuermann et al. (2009), middle-schoolstudents with LD were also instructed using a CSAsequence. These students first worked with concrete,three-dimensional items (e.g., cubes and buttons) to solveproblems. Then, the teacher transitioned to usingsemiconcrete, two-dimensional items (e.g., drawings andtally marks). Eventually, the students used only abstractmathematical symbols to solve word problems and equa-tions. Also, in the context of guided inquiry, the studentsdemonstrated, explained, and justified their thinkingprocesses. During these discussions, the teacher helpedstudents utilize concrete, semiconcrete, and abstract rep-resentations to support their learning and participation inmathematical discussions about the topic (using equa-tions to solve word problems). The teacher also providedstructure to the discussions by prompting students toexplain their thinking processes and solutions to theteacher and their peers, and, after these discussions, toreevaluate their processes and solutions. After the inter-vention, many of the participants made improvements inword problem solving including problems that involvedmulti-step equations (e.g., 5n − 6 = 14).

The studies by Cass et al. (2003), van Garderen (2007),and Scheuermann et al. (2009) provide instances in whichstudents with LD were able to succeed in solving multi-step problems. The diagramming and cognitive andmetacognitive instruction provided to the participants inthe study by van Garderen may have addressed some of theproblems students with LD have with working memory.Difficulties with storing and organizing information inmemory are reduced when students have effective strate-gies for storing and organizing information in diagrams onpaper (Keeler & Swanson, 2001) and develop thinkingskills to overcome difficulties with integrating information(Swanson & Beebe-Frankenberger, 2004). Aside frominformation storage, which is often provided by concreteand semiconcrete representations in a CSA sequence, theutilization of CSA and mathematical discussions with

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students (Cass et al., 2003; Scheuermann et al., 2009)likely developed conceptual understanding among the par-ticipants in these studies and, as a result, faster processingof the material which may have led to less difficultiesassociated with working memory (Barrouillet et al., 2007;National Council of Teachers of Mathematics, 2000).

Mathematics Textbooks and Students With LDWhile teachers often rely on their experiences and

beliefs about the importance of a topic when planninginstruction, textbooks also have a significant impact onteachers’ lesson planning (Banilower et al., 2013). Specialeducation researchers have expressed some concern withthe extent to which mathematics textbooks supportstudents with LD; these researchers have noted a lackof instruction for teaching students how to effectivelydiagram problems, developing students’ thinking skillsneeded for solving problems, and fostering the skills stu-dents need for effective participation in mathematical dis-cussions (van Garderen, Scheuermann, & Jackson, 2012;Jitendra et al., 2005). The results of several textbookstudies conducted by special education researchers sug-gested that traditional mathematics textbooks (i.e., text-books with lessons involving primarily direct instructionand opportunities for students to practice the skillslearned) are often missing key instructional componentsfor students with LD (e.g., Bryant et al., 2008; Jitendraet al., 2005; Sood & Jitendra, 2007; Xin, 2007; Xin, Liu,& Zheng, 2011), including lessons designed to (a) improvestudents’ reasoning about mathematics (NCTM, 2000),(b) develop cognitive and metacognitive skills about math-ematical problems (van Garderen, 2007), and (c) teachstudents to utilize representations of mathematical ideas(Fuchs, Fuchs, Finelli, Courey, & Hamlett, 2004; Griffin &Jitendra, 2008; Xin, 2008). However, explicit instructionhas often been recommended for students with LD(Gersten et al., 2009) and is common in traditional text-books (Bryant et al., 2008; Jitendra et al., 2005).

Analyses of standards-based curricula (i.e., curriculabased on NCTM’s vision of school mathematics) haveyielded different results. Studies by special educationresearchers (e.g., Sood & Jitendra, 2007; Xin et al., 2011)suggest that in comparison to traditional mathematics text-books, standards-based mathematics textbooks give stu-dents more opportunities to think about connectionsbetween mathematical ideas and real-world contexts andto work with concrete and semiconcrete representations.For example, Sood and Jitendra (2007) found EverydayMathematics (EM) (University of Chicago SchoolMathematics Project, 2004) provided opportunities for

students to work with a variety of representations todevelop number sense compared with the traditional text-books they studied. However, special education research-ers have expressed concern that both traditional andstandards-based mathematics curricula do not provide stu-dents with LD enough instruction for learning to createtheir own representations and to effectively use multiplerepresentations for solving challenging mathematics prob-lems (van Garderen et al., 2012; Jitendra et al., 2005).

Rationale and Research QuestionsWhile textbooks analysis studies have been conducted

with attention to the ways in which the texts serve studentswith LD (e.g., Bouck & Kulkarni, 2009; Bryant et al.,2008; van Garderen et al., 2012; Jitendra et al., 2005; Sood& Jitendra, 2007; Woodward & Brown, 2006; Xin et al.,2011), sufficient attention has not been paid, specifically,to how mathematics curricula support the workingmemory needs of students with LD. There is a need forspecial education researchers to investigate how curriculaprovide support in situations potentially problematic forworking memory (e.g., multi-step problems) in which stu-dents with LD could be taught skills for diagramming andcognitive and metacognitive strategies or students with LDcould be provided with opportunities to learn fromsequential instruction (e.g., CSA) and from mathematicalconversations (Cass et al., 2003; van Garderen, 2007;NCTM, 2000; Scheuermann et al., 2009). An in-depthanalysis of how textbooks approach multi-step problems,such as ACS problems, will provide insight into how math-ematics curricula address the needs of students with LD inchallenging educational situations for both teachers andstudents. The specific research question under investiga-tion is: To what extent are the needs of students with LDrelated to working memory addressed in current elemen-tary mathematics curricula for learning challenging,multi-step problems such as finding areas of compositeshapes?

MethodsWe examined all grade levels of three K-5 mathematics

curricula: Investigations in Number, Data, and Space(Investigations) (TERC, 2008), Everyday Mathematics(University of Chicago School Mathematics Project,2004), and Math Connects (MC) (McGraw-Hill, 2011).These curricula were chosen based on the range of learn-ing experiences they offer to students (e.g., example-driven instruction, collaborative learning, and contextualproblems). Rather than analyzing the textbooks in a quan-titative manner (e.g., recording the frequency of problem



types), we sought to provide a detailed description of howACS problems were addressed and connect these findingsto extant literature about effective instruction related tomulti-step problems and challenges related to workingmemory for students with LD. We searched for instancesin which students were provided with opportunities tobuild conceptual understanding (e.g., working with con-crete and semiconcrete representations or participating inmathematical discussions), and how the curricula providedstrategies for students to utilize visual representations(e.g., manipulatives, drawings, and diagrams) and todevelop cognitive and metacognitive skills.

We examined each section of the curricula for students’opportunities to learn how to find the ACS or acquirefoundational knowledge for solving these problems (e.g.,learning about area and making connections between areaand other mathematical ideas). We also searched relevantsections of other supplemental teacher materials (e.g.,guidebooks for differentiating instruction and re-teaching)for any instructional guidelines for teachers or activitiesfor students related to finding ACS or related foundationalskills. We described each instance that we interpretedas being related to the teaching of ACS in relation to howthe presentation of the mathematical content addressedworking memory deficits among students with LD.

We actively searched the curricula for situations inwhich representations of concepts were likely to benefitstudents with LD as a method for storing and organizinginformation for minimizing difficulties with workingmemory. For example, we searched for opportunities forstudents to make lists or diagrams to store information asthey proceeded through multi-step problems. We also con-sidered the benefits of concrete and semiconcrete repre-sentations, rather than written text which could potentiallycause difficulties for students with LD related to readingdifficulties and working memory. We assumed that prob-lems can arise when a struggling reader is required toprocess difficult information, such as written text, whilealso attempting to process and store mathematical ideas(Barrouillet et al., 2007; Fuchs & Fuchs, 2002).

Step-by-step instruction for breaking the multi-step,ACS problem into more manageable pieces was also afocus of this study. We worked under the assumption thatdividing multi-step problems into temporarily isolatedtasks is likely to improve the probability of success forstudents with LD. We based this decision on the tendencyof these students to struggle with multi-step problems(Geary, 2004; Swanson & Beebe-Frankenberger, 2004;Woodward & Montague, 2002) and the tendency for manylearners to have less difficulties related to working

memory with simpler, shorter tasks rather than morecomplex and lengthy tasks (Ayres, 2001; Barrouillet et al.,2007). Therefore, we targeted situations such as whenstudents were taught how to decompose multi-step prob-lems into a series of steps and received cognitive andmetacognitive instruction on how to process the informa-tion in each step (see van Garderen, 2007).

FindingsThe instructional approaches for each curriculum will

be explained in detail in the next three sections, includingpotential benefits of instructional strategies for studentswith LD. We will describe how each curriculum empha-sized visual representations, cognitive and metacognitiveinstruction, and opportunities for building conceptualunderstanding. More specific information about instruc-tional strategies, regarding when and how certain skillswere addressed, is included in the following sections andin Tables 1 and 2.Instructional Approaches in Investigations

Visual representations are common in Investigationsrather than lengthy text to describe mathematical ideas orpresent problem information. Students work frequentlywith concrete and semiconcrete representations beforesolving problems using only abstract representations (suchas mathematical symbols). We noticed foundationalaspects of ACS beginning in first, second, and third gradewhen students are given multiple opportunities to makelarger two-dimensional shapes out of smaller shapes usingboth concrete and semiconcrete representations. Forexample, in the first and second grade, students are pre-sented with opportunities to cover shapes on paper withsmaller shapes made of plastic to illustrate how a largeshape can be covered with many smaller shapes, howsmaller shapes can fit inside of larger shapes, and howsmall shapes can be combined to make larger shapes. Infourth grade, students are expected to find the area ofL-shaped figures that are covered with a grid. At one point,rather than using squares to determine area, fourth gradersare asked to use triangular units to determine ACS. Fifthgraders are asked to create hexagons with concrete, two-dimensional shapes (e.g., triangles, smaller hexagons, andparallelograms); this could serve as a foundational skillwhen students add area to composite shapes (to later besubtracted) to allow them to use formulas to solve ACSproblems.

A foundation of understanding is thoroughly developedbefore abstract conventions are taught to the students. Forexample, although fifth graders are explicitly taught thatmultiplying length and width will lead to finding the area


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of a rectangle, students are not explicitly taught theformula for finding the area of rectangles at any point inthe K-5 Investigations curriculum. In general, students arenot directly taught procedures for finding area until afterthey have received multiple opportunities to gain a thor-ough understanding of the concept of area.Instructional Approaches in EM

EM also extensively uses concrete and semiconcreterepresentations to build foundational knowledge before,but also during, the teaching of the use of abstract repre-sentations such as formulas. The EM differentiation hand-book frequently suggests that teachers direct students torepresent mathematical concepts concretely. The firstinstance of teaching of foundational knowledge for ACS

problems was found in second grade when students areexpected to create larger shapes with smaller shapes suchas making a parallelogram with two triangles or rectanglesmade up of several triangles and parallelograms. This typeof task is also presented in third and fourth grade whenstudents are asked to cover large rectangles with smallershapes such as squares, parallelograms, triangles, andrhombuses.

Even though formulas for area of rectangle and triangleproblems are introduced in the fourth grade, EM continuesto emphasize concrete and semiconcrete representationsafter this point in the curriculum. For example, EMemploys concrete representations of ACS problemswhen fifth-grade teachers are encouraged to ask students

Table 1Timeline and Description of Instruction With the Curricula

Curriculum 1st Grade 2nd Grade 3rd Grade 4th Grade 5th Grade

Investigations Covering largeshapes withsmall shapesand discussionof this conceptin quilt context

Covering largetrapezoids withsmall trianglesColoring smalltriangles andtrapezoidsdifferent colorsthat are a withinlarge triangle

Composing rectanglesusing smaller shapes(e.g., tetronimoes)Finding the area ofunfolded boxescovered with gridsand determiningACS represented ondrawings ofpegboardsDiscussing whydifferently shapedcomposite shapes(which are coveredwith grids) can havethe same area

Composing anddecomposing shapesusing semiconcreterepresentationsFinding the area ofL-shaped figures thatare covered with agridUsing triangles(rather than squares)to determine ACS ondrawings ofpegboards

Explicit teaching thatmultiplying lengthand width will leadto finding the area ofa rectangle eventhough the formulasare never presentedsymbolically (e.g.,l × w = a for area ofa rectangle)Composinghexagons withconcrete 2-D shapes(e.g., triangles,smaller hexagons,and parallelograms)

EverydayMathematics

NA Composing shapessuch as makingparallelogramswith twotriangles orlarge rectanglesusing trianglesandparallelograms

Covering largerectangles withsmaller shapes suchas squares,parallelograms,triangles, andrhombuses

Direct teaching ofcombining anddecomposing shapes,area formulas, andformula rationale(e.g., linking thearea of rectangleformula to arrays)Solving ACSproblems with gridscovering the figures

Composing anddecomposing ofshapes andadding/subtractingareas of subsectionsSolving ACSproblems with gridscovering the figures

Math Connects NA Direct teaching ofprocedures forcomposing anddecomposingshapes usingconcrete andsemiconcreterepresentations

Direct teaching ofstrategies for solvingACS problemstaught in isolationusing concrete,semiconcrete, andabstractrepresentations

Solving ACS problemswith unlabeled sidesand no grid presentDirect instruction forfinding the area ofsmall rectangles(placed within alarge rectangle) andadding the area ofeach small rectangleto find the area ofthe large rectangle

Explicit teaching forestimating the areaof composite shapeswith rounded edgeswhen the figures arecovered with gridsDirect instruction forsolving for thesurface area of arectangular prismwhich is presentedas a net



questions about cutting a long, narrow rectangle into twopieces and reconnecting the two pieces to make a broaderrectangle in the context of laying carpet (decomposing andcombining shapes). Semiconcrete representations are alsoutilized in the fifth grade in the context of nets (or draw-ings of unfolded rectangular prisms); fifth graders solvefor the surface area of nets by finding the area of each faceand adding the areas of the faces together to find the totalsurface area.

Explicit instruction is common in EM at the abstractlevel, and thinking skills (e.g., metacognitive skills) oftenneeded for solving ACS problems, such as combining and

decomposing shapes and area formulas, are explicitlytaught to students. The curriculum often includes explana-tions for students of the rationale behind formulas bylinking the formulas to concrete and semiconcrete repre-sentations. For example, fourth graders are directly taughtthe connection between the formula for the area of a rect-angle and arrays. In fourth and fifth grade, after the for-mulas for the area of rectangles and triangles are presentedsymbolically, students combine and decompose shapes (byworking with paper cut-outs of shapes) and are directlytaught how the process is related to the formulas for areaof a triangle and area of a rectangle. Also, while learning

Table 2Sample Activities for Teaching Area of Composite Shapes (4th grade)

Investigations Connections Between Ideas and Discussion of Problem SolvingMethods

The students are asked to decompose and find the area of thered figure (which was represented by a picture of ageoboard).

Teachers are encouraged to “Call on a few students to explaintheir reasoning. Keep stressing that it does not matter howthe shape is decomposed; if the area of the smaller shapesis counted correctly and combined, it will always equal thearea of the hexagon” (p. 136 of fourth grade studenttextbook, TERC, 2008).

Teachers are provided with this hypothetical student answerand justification:“I divided the shape into three pieces. First, I made 2 by 3rectangle, that’s 6 square units. There’s two small trianglesand so that’s another square unit. I made a rectanglearound the larger triangle—that rectangle had an area of 2,so the triangle is 1 square unit. So it’s 8 square unitsaltogether” (p. 137 of fourth grade student textbook,TERC, 2008).

Everyday Mathematics Students Building Composite Shapes and Finding AreaStudents were asked to put four shapes together to make a square on dot paper. Then, students were asked for the area of each

of the small figures and the area of the square. They were then asked to use the same small shapes to form a triangle and findthe area of that triangle

The teacher is asked to “have students explore different waysof combining various 2-dimensional shapes to form newshapes” (p. 692 of the fourth grade teachers’ edition,University of Chicago School Mathematics Project, 2004).Teachers are provided with diagrams for possible solutionssimilar to these:

Math Connects Direct Teaching of Procedures and Organization of InformationAs a part of a contextual problem about painting backdrops,

the students were asked to first “find the area of onesection of the backdrop” and then “multiply by 3 to findthe area of the entire backdrop” (p. 466 of fourth gradestudent text, McGraw-Hill, 2011).

Note. All figures are re-creations of problems in the textbooks. The only variations are minor changes in dimensions or visualappearances of figures.


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the rationale behind the formula for area of parallelo-grams, students learn how they can temporarily decom-pose a shape when they break parallelograms into twotriangles, find the area of the triangles, and add the areas.Instructional Approaches in MC

MC provides explicit instruction for teaching skillsneeded for each step of solving ACS problems and theseskills are directly taught in isolation. Foundational skillsneeded for each step for solving ACS problems are oftentaught utilizing a combination of explicit teaching andconcrete and semiconcrete instruction. The first instanceof foundational teaching for ACS problems is in thirdgrade, following CSA instruction, when the curriculumprovides multiple practice problems in which students areasked to find missing side lengths for L-shaped figureswith some side lengths labeled. Finding missing sidelengths is a skill typically presented in perimeter prob-lems, but this skill is also often needed to solve one stepwithin multi-step, ACS problems. For instance, in fourthgrade, students are asked to solve for the ACS, such asL-shaped figures, with missing sides and no grid present.

Students are directly taught the procedures (e.g., cogni-tive processes) to either divide composite shapes intomanageable pieces (e.g., rectangles and triangles) ortemporarily add area to a composite shape to make it easierto solve for the area of that shape. Students have multipleopportunities to develop the skill of combining and divid-ing shapes beginning in second grade. Second graders aredirected to divide larger shapes into smaller shapes bydrawing lines across hexagons to divide them into smallershapes, such as triangles, trapezoids, and parallelograms,or cutting paper with scissors to decompose large tri-angles, trapezoids, and hexagons into smaller triangles,parallelograms, and trapezoids. Pattern block activities areused to show second graders that smaller shapes can becombined to make larger shapes such as rectangles, tri-angles, and composite shapes. Fourth graders are directlyguided to find the area of the smaller, equally sized rect-angles (for which length and width are already provided),which are placed within a larger rectangle, and add thearea of each small rectangle to find the area of the largerrectangle.

Semiconcrete instruction (e.g., pictures of compositeshapes often covered by grids) is heavily emphasized. Inthird grade, composite shapes are covered with grids andeach unit or half-unit is labeled. Explicit teaching of con-cepts is common at this point when students are instructedto count the units and half-units to find the ACS and areeven directed to mark squares (such as with an X or dot or1, 2, 3. . . as they count) so they do not lose their place. In

fifth grade, students are exposed to multiple compositeshapes covered with grids where parts of the shapes do notfollow the gridlines because the shapes have somerounded edges and the students are explicitly taught stepsfor solving these problems. Students are directed to countthe completely covered squares as one square unit, countthe partially covered squares as one-half unit, and add allof these units and one-half units to find an estimate of thearea. Fifth graders are given explicit instruction for how tofind the surface area of a rectangular prism (which ispresented as a “net”) when the students are shown how tostore semiconcrete and abstract information in a table(drawings of each face and numbers representing the areaof each face) and how to solve for the area of this com-posite shape by finding and adding the area of each face.

DiscussionThe goal of this study was to describe the types of

support that each set of curricular materials provided forstudents with LD, particularly related to finding the ACS.We found that each curriculum offers benefits for studentswith LD, albeit in different ways. For example, potentialstrategies for reducing difficulties with working memorysuch as use of visual representations, activities for devel-oping conceptual understanding, information storagestrategies, and opportunities to develop cognitive andmetacognitive skills were found in all three curricula. Inthis discussion section, we will link the findings for eachcurriculum to previous studies in which students with LDsolved multi-step problems which were potentially prob-lematic regarding working memory.Instructional Approaches in Investigations

The use of visual representations in Investigations pro-vides accessibility for students with LD related to diffi-culties associated with working memory. In Investigations,students with LD have opportunities to use critical think-ing skills and are challenged to reason mathematically;yet, students with LD are put in a position to potentiallysucceed due to the support provided by visual representa-tions of the concepts. When students with LD experiencecognitive overload, while thinking critically about math-ematical concepts and during mathematical discussions, itis likely to benefit these students to refer to visual repre-sentations of mathematical concepts (e.g., manipulativesand diagrams). As a result of information being storedwithin visual representations, students can devote moreattention to processing the mathematical ideas rather thantrying to remember information as they attempt to decodeand process new information (see Keeler & Swanson,2001; Swanson & Beebe-Frankenberger, 2004). Students



with LD may also benefit from this emphasis on visualrepresentations rather than written text due to possibledifficulties they may have with decoding written text inmathematical situations (e.g., word problems) due toreading or processing difficulties (Fuchs et al., 2004;Geary, 2004).

Through the use of CSA and activities designed toprompt critical thinking and mathematical discussions, asolid foundation in conceptual understanding of area isdeveloped as students progress through the curriculum.These teaching methods are consistent with recommenda-tions from NCTM (2000) and special education research-ers (e.g., Miller & Mercer, 1993; Scheuermann et al.,2009) for improving mathematical understanding andproblem solving success rates. Ideally, the intended levelof conceptual understanding developed by this curriculumcould also serve as an important foundation for helpingstudents with LD overcome difficulties with workingmemory. When students have a solid understanding of aconcept, they are likely to process information morequickly and have less chance of losing important informa-tion from memory storage (Barrouillet et al., 2007; Keeler& Swanson, 2001).

Investigations also includes a balance between concrete,semiconcrete, and abstract instruction. Information is fre-quently presented using concrete or semiconcrete repre-sentations in multiple situations before students areexpected to use abstract representations. This sequence ofinstruction is likely to help students with LD succeed witha variety of mathematics problems (see Miller & Mercer,1993). In general, explicit instruction is often not presentin Investigations until after the students have had multipleopportunities to develop a solid understanding of the topicthrough inquiry-based activities. The use of explicitinstruction raises important considerations regardingwhen and how students with LD should receive explicitinstruction as mentioned by Gersten et al. (2009). Theseresearchers recommended consideration that students withLD may benefit from a blend of implicit and explicitinstruction (e.g., using manipulatives to build foundationalunderstanding, frequent in Investigations, in combinationwith explicit instruction for organizing information in atable as seen in MC). The introduction of explicit instruc-tion later in the instructional sequence also raisesquestions about the development of cognitive andmetacognitive skills that have been beneficial for studentswith LD in special education research (van Garderen,2007). Investigations does not emphasize direct instruc-tion of thinking processes early in the learning process, butstudents are provided with multiple opportunities to par-

ticipate in mathematical discussions with visual supportspresent. The questions of whether students with LD coulddevelop cognitive and metacognitive skills while partici-pating in the mathematical discussions, which scaffoldscould be necessary during these discussions, and whenthese scaffolds should be introduced, deserve further con-sideration by researchers.

Teachers using the Investigations curriculum may needto consider if activities are too open-ended and howfrequently and thoroughly students with LD need to besupported with methods for storing and organizing infor-mation (e.g., learning to create and use a table of informa-tion) during mathematical discussions. The challenge ofwhen to introduce explicit instruction may be magnifiedwith students with LD during longer, more challengingmulti-step problems. Teachers may need to introduceexplicit instruction sooner to provide working memorysupport for students with LD. However, some studentswith LD, with the benefits of a solid foundation in con-ceptual understanding, frequently presented visual repre-sentations, and practice solving longer, more challengingproblems (and valuable mathematical discussions aboutthese problems), may have success with Investigations andmay not need as much scaffolding as other students withLD who have not had this set of experiences (see Boaler &Humphreys, 2005; Clements & Sarama, 2009; NCTM,2000).Instructional Approaches in EM

Foundational knowledge is heavily emphasized in EM;however, compared with Investigations, EM provides moredirect, explicit instruction overall, and earlier in the teach-ing process. Visual representations are also emphasizedheavily in EM, but connections are made more explicitlythan in Investigations. Students are given more directguidance while working with manipulatives and diagrams.Connections between formulas and the mathematical ideasbehind the formulas are explicitly stated and taught inways that may develop foundational skills for solving ACSproblems.

EM relies heavily upon the CSA teaching sequencewhich is often combined with explicit instruction. Thiscombination is consistent with researchers’ recommenda-tions for CSA in general as well as recommendations forconsideration of how explicit instruction can be combinedeffectively with other teaching techniques (Gersten et al.,2009; Miller & Mercer, 1993). The semiconcrete phase isfrequently used to help students see connections betweenideas, and potentially, build conceptual understanding. Theabstract phase is strongly linked to explicit instruction forteaching the rationale behind formulas. Instruction that


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could facilitate the development of students’ skills forindividual steps within the multiple steps involved insolving ACS problems also takes place during thesemiconcrete and abstract phases. For example, studentshave opportunities to develop skills such as decomposingcomposite shapes, using formulas to solve for the area ofeach subsection, and adding the areas of the subsections tofind a total area. These thinking processes may help stu-dents develop cognitive and metacognitive skills neededfor solving multi-step problems in which they need tosolve different subtasks and combine the results fromthose subtasks to find the overall solution to the problem(see van Garderen, 2007).

As with Investigations, students with LD using EM mayexperiences difficulties with working memory whenworking on longer, open-ended problems. However, EMprovides explicit instruction and scaffolding earlier thanInvestigations which may be helpful for students with LDin these situations. Yet, teachers using EM may also needto consider the potentially negative impact of introducingexplicit instruction early during instruction as it is intro-duced in this curriculum. Some students with LD mayneed more practice with manipulatives and developingfoundational skills, before formulas are taught, to helpthem better understand the rationale behind the formulas(see Boaler & Humphreys, 2005; Clements & Sarama,2009; NCTM, 2000). In the short term, explicit instructionis potentially effective to help students solve problemsmore quickly; however, this earlier introduction of explicitinstruction may slow the progress of students with LD inbecoming resilient, persistent problem solvers and devel-oping deep conceptual understanding of topics (NCTM,2000; Woodward & Brown, 2006).Instructional Approaches in MC

Explicit instruction is heavily emphasized in MC. Evenwhen concrete and semiconcrete representations are usedto illustrate concepts, instruction is explicit and step-by-step. The prominence of step-by-step instruction in MC ispotentially useful for students with LD to address thechallenges associated with working memory which areexacerbated for students with LD when they need to solvemulti-step problems (Swanson & Siegel, 2001). By teach-ing students with LD to break multi-step problems into aseries of one-step problems, MC may be reducing thedifficulties students with LD would normally face (seeAyres, 2001). MC provides explicit instruction for how tobreak multi-step, ACS problems into one-step tasks, howto solve each task, and how to combine the informationfrom each task and other important information in theproblems. Direct teaching of these processes is consistent

to some degree with the intervention employed by vanGarderen (2007) with regards to breaking multi-stepproblems into manageable pieces and developing themetacognitive skills needed to combine information fromsubtasks to find the overall answer in the problem.

We noticed that the authors of MC placed emphasis onteaching students directly how to store and organize infor-mation. Students are directly taught how to mark parts ofthe problems they have counted or solved (e.g., markingsquares in a grid and writing down answers to subtasks) soinformation can be stored effectively on paper rather thanstudents with LD trying to hold information in memorystorage while processing other parts of multi-step prob-lems. Learning these organizational skills may help stu-dents overcome challenges associated with workingmemory by helping them devote more attention to pro-cessing rather than having to split attention betweenstorage and processing of information (see Barrouilletet al., 2007). While students may not have opportunities todevelop their own systems for representing information(as found to be lacking in traditional curricula by Jitendraet al., 2005), students are explicitly taught how to createsystems for representing information in ACS problems.

Explicit instruction and scaffolding are common in MC;however, teachers may need to use extra caution, whenimplementing explicit instruction in this curriculum, tomonitor students’ understanding of the steps required inthe problem. Potentially, due to a lack of experiencesworking with manipulatives and participating in math-ematical discussions, students may not develop the neces-sary level of conceptual understanding needed for successwith ACS problems. Teachers may need to supplement MCwith experiences for students designed to facilitate con-nections between length, area, parallel lines within rect-angles, and relationships between different shapes (e.g.,rectangles and triangles) (see Boaler & Humphreys,2005). Overall, with MC, some important organization andproblem solving skills are emphasized, but teachers mayneed to include more activities to develop foundationalunderstanding for students.Limitations of Study and Recommendations forFuture Research

Our conclusions about how working memory-relatedneeds of students with LD are being met by mathematicscurricula and how teachers may need to supplement par-ticular curricula should be interpreted with caution. Whilethere are many K-5 curricula on the market, we analyzedonly three curricula; further research should be conductedto examine how other curricula meet the needs of studentswith LD. Also, while we believe studying how curricula



approach ACS problems can be informative for specialeducation researchers and practitioners, other problemtypes should also be targeted by researchers. In addition,we sometimes found it challenging to identify situationswhen foundational skills indirectly related to ACS prob-lems were developed by the curricula and how relevantthese lessons were to ACS. Due to the many relationshipsbetween mathematical ideas, determination of the mostimportant connections between related mathematical ideasand ACS problems was left to our interpretation. Weattempted to focus on activities we deemed foundationalconcepts for ACS problems such as composition anddecomposition of shapes. Other researchers may havedeemed other related activities more important.

While textbooks play a significant role in determiningclassroom instruction (Banilower et al., 2013), teacherscan provide supplemental instruction to the enactment ofthe curriculum when needed. The findings in this study donot fully represent the experiences of students with LD inclassrooms in which each of these curricula are enacted.Future studies should be targeted toward the enacted expe-riences of students with LD with various mathematicscurricula and the adjustments teachers make to instruction(based the characteristics of different curricula) to supportstudents with LD. Researchers should especially considerhow curricula and teachers’ adjustments support studentswith LD as they develop the skills necessary for solvingthe multi-step problems that are emphasized in theCommon Core State Standards for Mathematics (Councilof Chief State School Officers and National GovernorsAssociation, 2010).

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