Virginia Mathematics Teacher Spring 2017

Concrete, Representational, and Abstract: Building Fluency from Conceptual Understanding Robert Berry III and Kateri Thunder

Introduction

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

The National Council of Teachers of “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 Mathematics (NCTM) stated,

Virginia Mathematics Teacher vol. 43, no. 2

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