as understanding the fundamental Conceptual isdefined knowledge Pennsylvania standards, asoutlinedby the andmasterlearn mathematics to ensure allstudentsimportant ceptual understanding, whichis on procedural andcon- learning focusa combined instructional Panel,Advisory 2008). This requires computational problems in of standard to algorithms solve well asdevelop, accurate execution should conceptually understand, as matics (National Mathematics S special education services, special education services, students whoreceive tudents, including Academic Standards. Instructional Sequence for Sequence MathematicsInstructional Concrete-Representational- mathe

- Mathematics Advisory Council Mathematics and Advisory the goalsoutlined by the National vides theframework for meeting (CRA) pro- sequence ofinstruction concrete-representational-abstract the mathematics classroom, the prepare students for success in Hinton, andStrozier, 2014). To lations to solve aproblem (Flores, sequence ofmathematical manipu- to executeis definedastheability a cal contexts. Procedural knowledge applied across multiplemathemati- isfluidandcanbe knowledge and Strozier, 2014). of This type within adomain(Flores, Hinton, between conceptsconnections tenets ofadomain,aswell asthe

Abstract: approach provides agraduated sequence/ The instructional CRA of Representation:” throughdefined learning “Stages Bruner andKenney(1965),who hasitsrootsCRA inthework of What isCRA? Academic Standards. the requirements ofthePennsylvania • • •

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conceptually supported line of work Representational. The “seeing” secondary level. When using to create meaningful connections stage uses representations of the CRA, the teacher should provide among concrete, representational, objects to model problems. In this multiple opportunities for and abstract levels of understand- stage, the teacher transforms the practice and demonstration to ing. CRA is an intervention for concrete model into a representa- help students achieve mastery mathematics instruction that tional (semiconcrete) level, which of the mathematical concept. research suggests can enhance may involve drawing pictures; Explicit instruction that involves the mathematics performance using circles, dots, and tallies; or, the use of manipulatives should using stamps to imprint pictures of students in a classroom. It is a also include the presentation of for counting. three-stage learning process where the numerical problem (Miller, students learn through physical Abstract. The “symbolic” stage Stringfellow, Kaffar, Ferreira, & manipulation of concrete objects, uses abstract symbols to model Mancl, 2011). followed by learning through problems. At this stage, the teacher pictorial representations of the After multiple teacher demon- models the mathematical concept concrete manipulations, and strations, students are given at a symbolic level, using numbers, ending with solving problems opportunities to practice the notation, and mathematical sym- using abstract notation. mathematical model and use bols to represent the number verbalization while using the Concrete. The “doing” stage uses algorithm. The teacher uses opera- CRA sequence. concrete objects to model problems. tion symbols (+, × ,÷, –) to indicate In the concrete stage, the teacher addition, subtraction, multiplica- What materials do I need to begins instruction by modeling each tion, or division. engage in CRA? mathematical concept with concrete How can I use CRA? Concrete manipulatives such as materials (e.g. red and yellow chips, algebra tiles, cups, sticks, or base cubes, base ten blocks, pattern CRA may be implemented at ten blocks can be used in the CRA blocks, fraction bars, geometric all grade levels individually, in sequence. The teacher should use figures). small groups, or for the entire the manipulative and movements class. It can be used with chil- that match the conceptual under- dren at the elementary or standing behind the algorithm.

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Why should I use CRA students on all fractions assess- Examples methods? ments. They also outperformed There is a positive impact for general education students on students: word problem subtests that had fractions and equivalency • Interaction with concrete embedded in the problem Base Ten Blocks for Place Value materials enhances student (Butler, Miller, Crehan, Babbitt, retention of stepwise proce- Concrete Representational Abstract & Pierce, 2003). dural options in mathematical problem solving. Concrete • Students who rely on 342 materials allow students to memorizing procedural 300+4+2 encode and retrieve information steps and lack the concep- in a variety of sensory options: tual understanding related visual, auditory, tactile, and to foundational operations kinesthetic (Witzel, 2005). will not understand why • Students learning secondary steps are used (Mancl, Miller, level math benefit from engag- & Kennedy, 2012). Chips for Equivalent Fractions ing with properly designed Newman Studio.stock.adobe.com Newman • CRA methods to teach place Concrete Representational Abstract © concrete materials to support value, fractions, addition, the learning of math concepts subtraction, multiplication, (Witzel, 2005). algebra, and word problems 2 x 3 = • Under the instruction of a middle are supported by research 3 x 3 9 school math teacher, results (Mancl, Miller, & Kennedy, 2012). indicated that students who • Early quantitative competen- learned with CRA methods to cies that children must possess solve algebra transformation include the relation between equations outperformed peers number words, numerals and receiving traditional instruction the quantities they represent, Algebra Tiles for Adding (Witzel, Mercer, & Miller, 2003). fluently manipulating those Integers • CRA provides an opportunity representations, knowledge Concrete Representational Abstract for increased interaction with of the number line, and content and increased frequency basic arithmetic (Geary, 2011). of response for all students -3+5=2 (Witzel, 2005). Students with disabilities struggle to understand the core concepts • After receiving CRA instruction, that underlie operations and algo- students significantly improved rithms used to solve problems that their conceptual understanding involve whole numbers and ratios of equivalent fractions. While (Butler, Miller, Crehan, Babbitt, & engaging in a post-test, students Pierce, 2003). These deficiencies would fall back on the represen- become even more critical with tational drawings to guide their the National Council of Teachers way through a problem (Butler, of Mathematics (NCTM) charge to Miller, Crehan, Babbitt, & Pierce, focus on conceptual understand- 2003). ing and problem solving rather • In a controlled study, students than rule-driven knowledge. CRA with disabilities receiving CRA methods can bridge the concep- instruction performed as well as tual understanding to operational general education 8th grade procedure gap for students. Wild Orchid.stock.adobe.com Wild

3 © References: Butler, F.M., Miller, S.P., Crehan, K., Babbitt, B., and Pierce, T. (2003). Fraction Mancl, D.B., Miller, S.P., & Kennedy, M. (2012). Using the Concrete –Representational– Instruction for Students With Mathematics Disabilities: Comparing Two Teaching Abstract Sequence With Integrated Strategy Instruction to Teach Subtraction With Sequences. Learning Disabilities Research & Practice. 18(2), 99-111. Regrouping to Students With Learning Disabilities. Learning Disabilities Research & Practice. 27(4), 152-166. Bruner & Kenney. (1965). Monographs of Society for Research in Child Development. Wiley, 1-75. Miller, S. P., Stringfellow, B.K., Ferreira, D., Mancl, D.B. (2011). Developing Computation Competence Among Students Who Struggle With Mathematics. Flores, M.M., Hinton, V., Strozier, S. (2014). Teaching Subtraction and Multiplication Teaching Exceptional Children. 44(2), 36-46. With Regrouping Using the Concrete-Representational-Abstract Sequence and Strategic Instruction Model. Learning Disabilities Research & Practice. 29(2), 75-88. Witzel. B.S. (2005). Using CRA to Teach Algebra To Students With Math Difficulties in Inclusive Settings. Learning Disabilities: A Contemporary Journal. 3(2), 49-60. Geary, D. C. (2011). Cognitive predictors of individual differences in achievement growth in mathematics: A five year longitudinal study. Developmental Psychology, Witzel, B., Mercer, C. & Miller, D. (2003). Teaching Algebra to Students With Learning 47, 1539-1552. Difficulties: An Investigation of an Explicit Instruction Model. Learning Disabilities Research & Practice. 18(2), 121-131. National Mathematics Advisory Panel Report (2008). Retrieved from U. S. Department of Education website: http://www2.ed.gov

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