Teaching Algebra with the Concrete-Representational-Abstract

National Council of Teachers of Mathematics (NCTM) Annual

Meeting and Exposition, San Antonio, TX

April 8, 2017

Joseph Sencibaugh, Ph.D., Brooke Callan, and Brennen Almus

Webster University

jsencibaugh77@webster.edu, brookecallan59@webster.edu,

brennenealmus84@webster.edu

Using the Concrete-Representational-Abstract

Technique to Teach Algebra to Students Who

Are Struggling

What Works Clearinghouse:

Best Evidence Encyclopedias

 Improving Mathematical Problem Solving Recommendations

 Prepare problems and use them in whole-class instruction—

MINIMAL EVIDENCE

 Assist students in monitoring and reflecting on the

problem-solving process—STRONG EVIDENCE

 Teach students how to use visual representations—

STRONG EVIDENCE

 Expose students to multiple problem-solving strategies—

MODERATE EVIDENCE

 Help students recognize and articulate mathematical concepts and

notation—MODERATE EVIDENCE

 http://ies.ed.gov/ncee/wwc/PracticeGuide.aspx?sid=16

Early Predictors of High School Math

 Mastery of fractions and early division is a predictor of students' later success

with algebra and other higher-level mathematics.

 Need to focus on whole number division and fractions and teaching them

better.

 Math scores on national standardized tests among U.S. high school students

have not improved in three decades and are significantly behind those in

countries such as China, Japan, Finland, the Netherlands and Canada at a time

when math proficiency is a requirement for many jobs.

 Noted students who "start ahead in math generally stay ahead" and that those

who "start behind generally stay behind" and it looked to find the reason.

 http://www.post-gazette.com/stories/news/education/formula-written-for-

math-success-640962/?p=0

Concrete-Representational-Abstract

Approach

 Concrete (Doing Stage). In the concrete stage, the teacher begins

instruction by modeling each mathematical concept with concrete

materials (e.g., red and yellow chips, cubes, base-ten blocks, pattern

blocks, fraction bars, and geometric figures).

 Representational (Seeing Stage). In this stage, the teacher transforms the

concrete model into a representational (semi-concrete) level, which may

involve drawing pictures; using circles, dots, and tallies; or using stamps

to imprint pictures for counting.

 Abstract (Symbolic Stage). At this stage, the teacher models the

mathematics concept at a symbolic level, using only numbers, notation,

and mathematical symbols to represent the number of circles or groups

of circles. The teacher uses operation symbols (+, –, x, /) to indicate

addition, subtraction, multiplication, or division.

Concrete Level

 The CONCRETE LEVEL involves the manipulation of

objects. At this level the learner concentrates on both the

manipulated objects and the symbolic processes.

 See the following example (Metz, 2011):

Representational Level

 The REPRESENTATIONAL LEVEL involves working with

illustrations of items in performing math tasks.

 Items include dots, lines, pictures of objects, or nonsense items.

 The emphasis is on developing associations between visual

models and the numerical equations.

 See the following example (Metz, 2011):

Abstract Level

 The ABSTRACT LEVEL involves the use of numerals, which

is using only numbers to solve math problems.

 See the following example (Metz, 2011):

Programs Using CRA Approach

Applied Math Made Easy

Solving Equations: An Algebra Intervention

Hands On Equations

Singapore Math

Math-U-See

Algebra Tiles

Applied Math Made Easy

Solving Equations

Applied Math Made Easy

Solving Equations