Posted by: degarcia | November 24, 2008

Concrete and Abstract Representations (Using Mathematical Tools)

Concrete-Representational-Abstract Instructional Approach

 

What is the Concrete-Representational-Abstract (CRA) Instructional Approach?

The CRA Instructional Approach is “an intervention for mathematics instruction that research suggests can enhance the mathematics performance of students.” (Hauser)  The Approach is a “three-part instructional strategy, with each part building on the previous instruction to promote student learning and retention and to address conceptual knowledge.” (Hauser)  The three parts are as follows:

  • Concrete: In this stage, the teacher begins instruction by modeling each mathematical concept with concrete materials. In other words, this stage is the “doing” stage, using concrete objects to model problems.
  • Representational: 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.  In other words, this is the “seeing” stage, using representations of the objects to model problems.
  • Abstract: In 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, multiplication, or division.  This is the “symbolic” stage, where students are able to use abstract symbols to model problems (Hauser).

In the classroom, this approach is a facilitating framework for students to create meaningful connections between concrete, representational, and abstract levels of thinking and understanding.  Students’ learning starts out with visual, tangible, and kinesthetic experiences to establish basic understanding, and then students are able to extend their knowledge through pictorial representations (drawings, diagrams, or sketches) and then finally are able to move to the abstract level of thinking, where students are exclusively using mathematical symbols to represent and model problems (Hauser).

 

Studies have shown that “students who use concrete materials develop more precise and more comprehensive mental representations, often show more motivation and on-task behavior, understand mathematical ideas, and better apply these ideas to life situations,” (Hauser).

 

What is the Purpose of the CRA Approach?

 The overarching purpose of the CRA instructional approach is to “ensure students develop a tangible understanding of the math concepts/skills they learn.” (Special Connections, 2005) Using their concrete level of understanding of mathematics concepts and skills, students are able to later use this foundation and add/link their conceptual understanding to abstract problems and learning.  Having students go through these three steps provides students with a deeper understanding of mathematical concepts and ideas and provides an excellent foundational strategy for problem solving in other areas in the future. (Special Connections, 2005).

 

How Do I Implement the CRA Approach in My Classroom?

One of the first and most important steps to implementing the CRA approach in the classroom is to “use appropriate concrete objects to teach particular math concepts/skills. Discrete materials (e.g. counting objects such as beans, chips, unifix cubes, popsicle sticks, etc.) are especially helpful since students can see and feel the attributes of the objects they are using.” (Special Connections, 2005).

 

 

Once students have mastered the concrete level of performance, introduce appropriate drawing procedures where students problem solve through drawing simple representations of the concrete objects they previously used (e.g. tallies, dots, and circles). “By replicating the movements students previously used with concrete materials, drawing simple representations of those objects supports students’ evolving abstract understanding of the concept/skill,” (Special Connections, 2005).

 

 

Finally, after a student demonstrates a thorough understanding of the representative level, use appropriate strategies to help students move from that representational level to the more abstract level.  If students have trouble moving to the abstract, “re-teach the mathematics concept/skill using appropriate concrete materials and then explicitly show the relationship between the concrete materials and the abstract representation of the materials.” (Special Connections, 2005)  If students already have a concrete level of understanding of that concept/skill, “provide students opportunities to use their language to describe their solutions and their understandings of the mathematics concept/skill they are learning,” (Special Connections, 2005).

CRA Problem Example

This picture shows how to use the CRA instructional approach with the problem 4+5=9.

 

Teacher Tips for Using Mathematical Manipulative Tools in the Classroom

What are Mathematical Manipulative Tools?

 Manipulative materials are concrete models or objects that involve mathematics concepts.  The most effective tools are ones that appeal to several senses, and that can be touched and moved around by the students (not demonstrations of materials by the teacher).  The manipulative materials should relate to the students’ real world (Heddens, 1997).

 

List/Uses of Suggested Mathematical Tools (Manipulatives)

Picture courtesy of Lisa Ann de Garcia

Some common tools used in elementary mathematics instruction.

Photo Credit: Lisa de Garcia 

Base 10 Blocks

·        Base 10 Blocks come with units (one cube), longs (made up of 10 units), flats (made up of 10 longs or 100 units), and cubes (made up of 10 flats or 1000 units).

·        Base 10 blocks can be used for many math procedures:

o       Introducing the concept of place value

o       Reading and writing numbers

o       Zero as a place holder

o       Addition

o       Subtraction

o       Multiplication

o       Division

o       Fractions

o       Decimals/operations with decimals

o       Expanded numeration

o       Counting/skip counting

o       Geometry (area and perimeter)

o       Probability, ratios, proportions

·        See: http://www.susancanthony.com/Resources/base10ideas.html for more detail and ideas

 

Cuisenaire Rods

·        Cuisenaire Rods are colored wooden or plastic rods that have values from one to ten and are colored by the number they represent:

o       White rod = 1 cm.
Red rod = 2 cm.
Light green rod = 3 cm.
Lavender rod = 4 cm.
Yellow rod = 5 cm.
Dark green rod = 6 cm.
Black rod = 7 cm.
Brown rod = 8 cm.
Blue rod = 9 cm.
Orange rod = 10 cm.

·        Cuisenaire Rods can be used for:

o       Probability and statistics

o       Ratios, proportions

o       Fractions

o       Addition

o       Subtraction

 

Pattern Blocks

·        Pattern Blocks are one centimeter thick multicolored blocks that come in six shapes: hexagons, squares, trapezoids, triangles, parallelograms, and rhombi. Each shape is a different color.

·        Pattern blocks can be used for:

o       Fractions

o       Patterns

o       Geometry

o       Symmetry

o       Addition

o       Multiplication

o       Equations

o       Ratios

·        See the following website for more information and detail: http://www.netrox.net/~labush/nctm.htm#Pattern

 

Unifix Cubes

·        Unifix Cubes are colorful, interlocking cubes that link in only one way.  They come in  ten solid colors that make them quite visual for demonstrations and easily allow for patterning and sorting.

·        Unifix cubes can be used for:

o       Counting/skip counting

o       Addition

o       Subtraction

o        Multiplication

o        Division

o       Patterns

o       Number sense

o       Equality

o       Fact families

o       Perfect squares

o       Graphing

o       Place value

o       Fractions

o       Measurement

o       Probability

·        See the following website for more detail: http://www.netrox.net/~labush/nctm.htm#Unifix

 

Hundred Chart

·        A Hundred Chart is a series of 100 squares in a 10 x10 arrangement that forms a larger square.  The small squares are labeled from 1 to 100.

·        A hundred Chart can be used for:

o       Visual counting/skip counting

o       Visual addition and subtraction

o       Fractions

o       Decimals

o       Percents

·        See the following website for more detail: http://www.christiancottage.com/articles/HundredChart.html

 

Snap Cubes

·        Snap Cubes are similar to Unifix Cubes but they link on all sides (six ways).  They come in a variety of colors

·        Snap cubes can be used for:

o       Counting/skip counting

o       Addition

o       Subtraction

o        Multiplication

o        Division

o       Patterns

o       Number sense

o       Equality

o       Fact families

o       Perfect squares

o       Graphing

o       Place value

o       Fractions

o       Measurement

o       Probability

o       Geometry

o       3D shapes

 

Fraction Bars

·        Fraction Bars are colored cubes or tiles that are proportional and represent whole, halves, thirds, fourths, fifths, sixths, eighths, tenths and twelfths.

·        A set usually includes:

o       one red whole

o        two pink halves

o        three orange thirds

o        four yellow fourths

o        five green fifths

o       six teal (blue-green) sixths

o        eight blue eighths

o        ten purple tenths

o        twelve black twelfths

·        They can be used for

o       Fractions

§ Fraction number sense

§ Equivalent fractions

§ Operations with fractions

§ Mixed numbers

o       Proportion

o       Ratios

 

Color 1” Tiles

·        These are 1” x1” square color tiles.  They usually can be purchased with a mix of four colors.

·        They can be used for:

o       Counting and skip counting

o       Patterns

o       Multiplication

o       Division

o       Addition

o       subtraction

o       Geometry (perimeter and area)

o       Place value

o       Measurement

o       Graphing and probability

·        See the following for more detail and ideas: http://www.learningresources.com/text/pdf/2218book.pdf

 

Two-Sided Counters

·        Two-Sided Counters are usually circular chips with different colors on each side.

·        The can be used for:

o       Integers

o       Addition

o       Subtraction

o       Multiplication

o       Division

o       Probability

o       Ratios

o       Percentages

 

10 Frames

·        Ten frames are a rectangle made up of ten squares (5 by 2).  They can have dots on them representing values 1-10 or can be blank.

·        They can be used for:

o       Addition

o       Subtraction

o       Grouping

o       Counting

o       Place value

 

100 Beads

·        100 Beads are made by stringing 100 beads on a string.  They are made up of 2 colors that alternate every ten.

·        They can be used for

o       Counting and Skip Counting

o       Basic math operations

 

Beans and Cup

·        Beans and Cup comprise of a cup and beans.  The teacher will decide how many beans to put in a cup for designated problems.

·        They can be used for:

o       Counting

o       Addition

o       Subtraction

o       Multiplication

o       Division

  

Other Useful Mathematics Tools that can be implemented in the classroom are: 

  • Number Line
  • Dice
  • Calculators
  • Decks of Cards
  • Tangrams
  • Geared Clocks
  • Fraction Circles
  • 24 Game
  • Place Value Practice Boards

Places to Buy Mathematical Tools

http://www.eaieducation.com/?gclid=COb69aTk35kCFRwwawodjSVyWA

 

http://www.enasco.com

 

http://lakeshorelearning.com

 

http://gamblersgifts.com (cards and dice)

 

 

eManipulatives

 

http://www.wiley.com/college/musser/CL_0471263796_S/emanipulatives/nav/met-index.html

 

http://www.mathplayground.com/math_manipulatives.html

 

http://nlvm.usu.edu/

 

 

Useful Links

 

Concrete Level Instruction: http://coe.jmu.edu/mathvidsr

 

Q & A About Manipulatives With Marilyn Burns: http://teacher.scholastic.com/products/instructor/burnsqa.htm

 

7 Musts for Using Manipulatives: http://content.scholastic.com/browse/article.jsp?id=4003

 

More on Math Manipulatives: http://www.iched.org/cms/scripts/page.php?site_id=iched&item_id=math_manipulatives

 

 

 

References

Hauser , Jane. Concrete-representational-abstract instructional approach.  Retrieved April 9, 2009, from the Access Center: Improving Outcomes for all Students K-8. Web site:            http://www.k8accesscenter.org/training_resources/CRA_Instructional_Approach.asp

Special Connections, (2005). From concrete to representational to abstract. Retrieved April 9, 2009, from the Special Connections Web site: http://www.specialconnections.ku.edu/cgi-bin/cgiwrap/specconn/main.php?cat=instruction&subsection=math/cra.

 Heddens, James W., (1997). Improving mathematics teaching by using manipulatives.       Retrieved April 9, 2009, from Edumath Web site:http://www.fed.cuhk.edu.hk/~fllee/mathfor/edumath/9706/13hedden.html.

 

 

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