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## Concrete, Representational, and Abstract:Building Fluency from Conceptual Understanding

Robert Berry III and Kateri Thunder

Introduction
The National Council of Teachers of Mathematics (NCTM) stated, “
Effective mathematics teaching focuses on the development of both conceptual understanding and procedural fluency” (NCTM, 2014; p. 42).  Conceptual understanding is comprehending the meaning of mathematical concepts and reasoning about the relationships between concepts. A student with conceptual understanding is able to explain why values can be described using part-whole relationships and why values can be equivalent.  Procedural fluency consists of accuracy, efficiency and flexibility (Russell, 2000).  A student with procedural fluency is able to select an efficient strategy that fits the given numbers to accurately solve the problem.  In this way, procedural fluency is closely related to number sense and requires students to do more than just memorizing facts (Baroody 2006).  Procedural fluency reduces cognitive load while problem solving, which allows the student to focus on the mathematical relationships of the problem at hand.  There are three phases of procedural fluency development: (1) exploration and discussion of number concepts, (2) use of informal reasoning strategies based on properties of the operations, and (3) the eventual development of automaticity (NCTM, 2014).  Effective mathematics teaching develops procedural fluency with number sense while engaging students in thinking deeply about the relationships among values.
Research across grade levels and with all populations of students (including students with special needs and English Language Learners) has demonstrated that CRA is an effective teaching strategy for developing procedural fluency built on conceptual understanding (Crawford & Ketterlin-Geller, 2008; Hudson, Miller, & Butler, 2006; Paulsen, 2005; Witzel, 2005). CRA (sometimes called CSA or CPA) is a three-phase instructional approach with each phase building on and explicitly connecting to the previous instruction.  Concrete is the first phase, often referred to as “the doing stage”, when instruction focuses on using manipulatives or concrete objects.  Representation (semi-concrete or pictoral) is the second phase, often referred to as “the seeing stage”, when instruction connects the concrete manipulatives to drawing, pictures, and other visual representations of concrete objects.  Abstract is the third phase, often referred to as “the symbolic stage”, when instruction connects the concrete and semi-concrete representations to using only numbers and mathematical symbols or to mentally solving problems.  The three phases are flexible and reflective of students’ readiness to explain concepts and to fluently apply strategies with different levels of representation.  At every level, there should be parallel modeling of each representation with mathematical vocabulary and numbers.

Rekenreks and Part-Whole Bar Models

Rekenreks and part-whole bar models are tools that support developing procedural fluency while deepening and exploring conceptual understanding.  They also make connections among concrete (rekenrek), representational (part-whole bar model), and abstract (number sentences) (CRA) modes of representations.

The rekenrek consists of 20 beads in two rows of ten, each broken into two sets of five by color (i.e., in each row the first five beads are red and the next five are white) (see Figure 1).  Adrian Treffers, a mathematics curriculum researcher at the Freudenthal Institute in Holland, is credited with developing the manipulative (Fosnot & Dolk, 2001). Directly translated, rekenrek means counting rack. The rekenrek is sometimes mistaken as an abacus at first glance, but it is not based on place value columns, and it is not used in the same manner as the abacus.  Its main characteristic is that it has a five-structure with analogous representations of the five fingers on each of our hands and the five toes on each of our feet. It is often used to develop strategies for basic facts such as working with the five-structure, making ten, one more or less than, compensation and stretching children toward using these strategies with the support of or in place of counting (Fosnot & Dolk, 2001).

Figure 1:Rekenrek

The five-structure of a rekenrek offers visual support to quickly see the quantity of five without counting. This is a form of subitizing, the ability to recognize the number of objects in a set without actually counting them (Clements, 1999; MacDonald & Shuway, 2016).  Subitizing with a rekenrek is a concrete representation to flexibly decomposing and composing numbers based on part-whole relationships.  For example, on a rekenrek, students can see that seven is composed of five (five red beads) and two (two white beads) and eight is composed of five (five red beads) and three (three white beads). Figure 2 shows a rekenrek representing  7 + 8.  Students may subitize and see 7 + 8 is the same as 5 + 5 + 2 + 3 or 10 + 2 + 3 or 10 + 5.  Students can also see a concrete representations of the commutative property of addition (7 + 8 = 8 + 7).  Using rekenreks, students can purposefully practice building combinations of numbers and develop fluency based on their understanding of part-whole relationships.

Figure 2: 7 + 8 on the Rekenrek

Part-whole bar models are a visual representation that show the magnitude of a number as well as the relationships among the whole number and its parts.  In Singapore, the bar model is used as a tool for problem solving because it communicates graphically complex relationships, such as comparisons, part-whole calculations, ratios, proportions, and rates of change (Hoven & Garelick, 2007).  For example, in a bar model, 15 can be decomposed visually into its parts and represented proportionally.  Figure 3 shows part-whole bar models representing 7+8, 5+5+2+3, and 10+5.

As students visually represent their rekenrek work with part-whole bar models and then symbolically represent the models with number sentences, they can connect concrete, representational, and abstract models of basic addition facts.  These three levels of representation also provide students with an opportunity to begin an exploration of the concept of equality

Figure 3: 7+8, 5+5+2+3, and 10+5

using part-whole bar models

 15 7 8
 15 5 5 2 3
 15 10 5

CRA in Action

In the classroom vignette that follows, Mr. Dominguez, a first-grade teacher, is working with students using rekenreks along with part-whole bar models to build fluency of basic addition facts based on number sense (Virginia Mathematics Standards of Learning (SOL) 1.7) and to explore the concept of equality (SOL 1.15).  Mr. Dominguez’s class regularly uses manipulatives to represent and describe their thinking.  In a previous lesson, the class used ten-frames to develop basic addition facts to ten.

Figure 4:  Jaxson’s representation of seven on rekenrek

Mr. Dominguez says to his students, “I am thinking of a way to show seven on my rekenrek. Can you show me a way?”  All of the students use their rekenreks to show seven. Mr. Dominguez takes note of the different ways students represented seven on the rekenreks.  For example, he notes that some students show five beads on the top and two on the bottom, some show four on the top and three on the bottom, and some show six on the top and one on the bottom.  As the students work, he asks them to find more than one way to represent seven on the rekenrek.

After some time, Mr. Dominguez invites Jaxson to describe his representation of seven.  Jaxson describes that he has four beads on top and three on the bottom (see Figure 4).

Mr. Dominguez says, “Let’s show Jaxson’s thinking another way by writing it on our part-whole bar model on the board (see Figure 5).  So, boys and girls, if seven is the whole, what are the parts Jaxson described?”  He calls on Donovan who responds by saying the parts are 4 and 3.  Mr. Dominguez asks Jaxson, “Can you write that as a number sentence on the board?”  Jaxson writes 4 + 3 = 7.

 7 4 3
 7
 4 3

Figure 5: Jaxson’s representation using part-whole bar model

Mr. Dominguez engages the class in a discussion of the connections among Jaxson’s concrete representation on the rekenrek, his semi-concrete representation of the part-whole bar model, and his abstract representation of the number sentence.

Mr. Dominguez sees Aisha’s representation on her rekenrek and asks her to describe her representation (see Figure 6).

Figure 6:  Aisha’s representation of seven on rekenrek

After Aisha’s explanation, Mr. Dominguez engages the class to represent Aisha’s representation using the part-whole bar model (see Figure 7) and then writing a number sentence (6 + 1 = 7)

Again, the class discusses the relationship among Aisha’s concrete representation on the rekenrek, her semi-concrete representation of the part-whole bar model, and her abstract representation of the number sentence.

Mr. Dominguez uses Jaxon and Aisha’s thinking to write the equation 4 + 3 = 6 + 1 on the board.  He then asks the class what the number sentence means and how can they use Jaxson’s and Aisha’s thinking to make sense of the number sentence.

Figure 7: Aisha’s representation using part-whole bar model

Marcel says, “If you do 4 + 3 first you will get 7 and then you do 6 + 1 you will get 7…so it means 7 is equal to 7.”

Many students nod in agreement.  Mr. Dominquez challenges the students to find other number sentences that can be written similar to 4 + 3 = 6 + 1.  They can use the rekenreks, part-whole bar models, and number sentences to problem solve.  The students work independently to develop their own number sentences while Mr. Dominguez circulates and confers with students.

Jamal writes the number sentence 5 + 2 = 7 + 2.  Mr. Dominguez confers with Jamal to learn about his thinking.  Then Mr. Dominguez asks Jamal to create two part-whole bar models in which the whole on both models is 7.  Jamal makes the first bar model for 5+2 quickly. After thinking a bit and checking with his rekenrek, Jamal creates a second bar model for 7+0.  Jamal then writes a new number sentence 5 + 2 = 7 + 0.

To bring closure to the lesson the students share their equations with partners.  Then Mr. Dominguez engages them in whole class discussion to share and compare equations.  Jamal is able to share his mistake, his strategy to fix his mistake, and what he learned from his mistake.

Discussion and Implications

Time is a critical factor in teachers’ instructional decisions about both what and how to teach mathematics.  Effective teaching practices improve student achievement as well as student retention and application of concepts and skills over time. Choosing to implement evidence-based, effective teaching practices maximizes instructional time.

Effective mathematics teaching focuses on the development of both conceptual understanding and procedural fluency” (NCTM, 2014; p. 42).  Procedural fluency and conceptual understanding are both important components of developing mathematical proficiency.  One should not be sacrificed for the other; in fact, Mr. Dominguez’s lesson is an example of developing procedural fluency built on conceptual understanding.  As the students problem-solved to become more efficient and flexible with basic addition facts, they relied on their understanding of part-whole relationships.  Their fluency worked in tandem with their number sense for composing and decomposing numbers.  The tasks, materials, and discussion within Mr. Dominguez’s lesson built upon their conceptual understanding and continued to deepen this understanding by allowing students to extend their understanding to a new context, new representations, and further mathematical discourse.  Additionally, students began to explore the new concept of equality with an emphasis on meaning-making..

CRA (also known as CSA or CPA) is an effective teaching strategy for all populations of students (including students with learning disabilities and English Language Learners).  Rekenreks and part-whole bar models are effective tools for developing procedural fluency from conceptual understanding.  Mr. Dominguez’s lesson is an example of making the CRA connection across all levels in one lesson.  It is not necessary that these connections are made within one lesson; in fact, this not often the case.  Rather, connections are made across a unit or series of lessons.  As teachers plan instruction, they should engage students in using meaningful concrete, representational, and abstract representations and explicitly making connections among them.  CRA is also an effective teaching practice for differentiation.  Students may be ready for problem solving and explaining their reasoning with different levels of representation (concrete, representational, abstract).  Students can choose to use the level of representation for which they are ready or teachers can assign students to use one or more levels of representation based on their readiness.

Teachers need to develop procedural fluency that is built upon and reinforced by conceptual understanding using concrete, representational, and abstract representations for all students.  Two powerful tools are rekenreks and part-whole bar models.  These effective teaching practices will maximize instructional time and grow mathematically proficient learners who are deeply engaged in the complex, rigorous problems of mathematics.

References

Baroody, A. J. (2006).  Mastering the Basic Number Combinations. Teaching Children Mathematics 13(1), 23–31.

Clements, D. H. (1999). Subitizing: What is it? Why teach it? Teaching Children Mathematics 5(7), 400-405.

Crawford, L. & Ketterlin-Geller, L.R. (2008). Improving Math Programming for Students at Risk: Introduction to the Special Topic Issue. Remedial and Special Education, 29(1), 5-8.

Fosnot, C. T., & Dolk, M. (2001). Addition and subtraction facts on the horizon. In V.Merecki& L. Peake (Eds.), Young mathematicians at work: Vol. 1. Constructing number sense, addition and subtraction (pp. 97-113). Portsmouth, NH: Heinemann.

Hauser, J. (2004). Concrete-representational-abstract instructional approach. Retrieved from http://165.139.150.129/intervention/ConcreteRepresentationalAbstractInstructionalApproach.pdf on February 25, 2017.

Hoven, J. & Garelick, B. (2007). Singapore Math: Simple or Complex? Educational Leadership, 65(3), 28-31.

Hudson, P., Miller, S.P., & Butler, F. (2006). Adapting and Merging Explicit Instruction Within Reform Based Mathematics Classrooms. American Secondary Education, 35(1), 19-32.

MacDonald, B. L., & Shumway, J. F. (2016). Subitizing Games: Assessing Preschoolers' Number Understanding: Reflect and Discuss. Teaching Children Mathematics, 22(6), 340-348.

Paulsen, K.J. (2005). Infusing Evidence-Based Practices into the Special Education Preparation Curriculum. Teacher Education and Special Education, 28 (1).

Russell, S.J. (2000). Developing computational fluency with whole numbers. Teaching Children Mathematics, 154-158.

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.

Robert Berry
NCTM President elect
University of Virginia
rqb3e@eservices.virginia.edu

Kateri Thunder
Faculty