CRA:
CONCRETE-TOREPRESENTATIONAL-TO-ABSTRACT
HOW DOES THE NEW COMMON CORE AS WELL AS BIG
IDEAS SUPPORT THIS TYPE OF INSTRUCTION?
By Mindy Robison
Email: Mindy.Robison@CanyonsDistrict.org
WEBSITE:
http://robisonlovesmath.wikispaces.com/
 On the right hand side, click on
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CRA Training – New Core
LEARNING TASK:
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What is ½ ÷ ¼ ?
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What is ¼ ÷ ½ ?
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Explain how you got your answers?
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Draw a picture to prove your answer.
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Can you draw another picture to show your answer?
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How can you concretely explain this to a student?
THE SESSION OBJECTIVES:
Find ways to effectively and efficiently implement CRA
using the Common Core and Big Ideas Learning.
 Learn how to and why it is important implement
Instructional Shift #3:
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Picture It, Draw It!
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Use multiple representation of mathematical entities.
THE PLAN!
Review CRA
 Review Instructional Shift #3
 Review the basics of Manipulatives
 Look at specific lessons from Big Ideas Learning that use
manipulatives.
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Look at virtual manipulatives.
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Specifically look at Dynamic Classroom and the Interactive Activities.
Look at algebra tiles and counters.
Discuss ways to improve our math classes through the
use of manipulatives.
COMMON CORE STANDARDS FOR MATHEMATICAL
PRACTICE
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#4 Model with mathematics
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#7 Look for and make use of structure.
QUICK REVIEW
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Why CRA?
“ The purpose of teaching through a concrete-to-representationalto-abstract sequence of instruction is to ensure students develop a
tangible understanding of the math concepts/skills they learn.
When students are supported to first develop a concrete level of
understanding for any mathematics concept/skill, they can use this
foundation to later link their conceptual understanding to abstract
mathematics learning activities. Having students represent their
concrete understandings…by drawing simple pictures that replicate
or mimic their use of concrete materials, students are supported in
the process for transferring their concrete understanding to the
abstract level.”
 http://www.specialconnections.ku.edu/cgibin/cgiwrap/specconn/main.php?cat=instruction&subsection=math
/cra
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CONCRETE:
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“The concrete level of understanding is the most basic
level of mathematical understanding. It is also the most
crucial level for developing conceptual understanding of
math concepts/skills. Concrete learning occurs when
students have ample opportunities to manipulate
concrete objects to problem-solve. For students who
have math learning problems, explicit teacher modeling
of the use of specific concrete objects to solve specific
math problems is needed.
REPRESENTATIONAL
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“At the representational
level of understanding,
students learn to
problem-solve by
drawing pictures. The
pictures students draw
represent the concrete
objects students
manipulated when
problem-solving at the
concrete level.
ABSTRACT

“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 to indicate addition, multiplication,
or division.”
INSTRUCTIONAL SHIFT #3:
PICTURE IT, DRAW IT!
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“The failure to capture the mathematics being taught with a
picture that helps students visualize what is going on is one
of the most serious missed opportunities I observe.” p.19
“The simple reality every teacher faces . . . is that very rarely
do more than ½ of the students process the math being
taught, see the math being taught, or feel the math being
taught in the same way their teacher is seeing it.” p. 21
“Valuing of alternative approaches enriches our instruction
and provides new levels of access to mathematical
understanding” p. 25.
SO WHAT SHOULD WE SEE IN AN EFFECTIVE
MATHEMATICS CLASSROOM (P.25)
Frequent use of pictorial representations to help
students visualize the mathematics they are learning
 Frequent use of the number line and bar models (see
www.thesingaporemaths.com )to represent numbers
and word problems.
 Frequent opportunities for students to draw or show
and then describe what is drawn or shown.
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MANIPULATIVES?
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Discussion:
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How do you define a math manipulative.
ADVANTAGES & DISADVANTAGES TO MANIPULATIVES IN
PARTNERS OR SMALL GROUPS:
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Please list at least 2 advantages to using manipulatives
in your math classroom.
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Please list at least 2 disadvantages to using
manipulatives in your math classroom.
SUGGESTIONS FOR USING MANIPULATIVES:
Talk with your students about how manipulatives help to
learn math.
 Set ground rules for using manipulatives, reiterate those
rules every time you use manipulatives.
 Develop a system for storing manipulatives.
 Allow time for your students to explore manipulatives
before beginning instruction.
 Encourage students to learn names of the manipulatives
they use.
 Provide students time to describe the manipulatives
they use orally or in writing. Model this as appropriate.
 Introduce manipulatives to parents.
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“MANIPULATIVES IN MIDDLE GRADES MATHEMATICS”,
RESEARCH SUMMARY.”
“Although manipulatives have an important place in
learning, their physicality does not carry the meaning of
the mathematical idea. They often can be used in a rote
manner . . . Students may require concrete materials to
build meaning initially, but they must reflect on their
actions with manipulatives to do so.”
 http://www.nmsa.org/Research/ResearchSummaries/M
athematics/tabid/1832/Default.aspx
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“MANIPULATIVES IN MIDDLE GRADES MATHEMATICS”,
RESEARCH SUMMARY.”
Student performance with manipulatives may exceed
student performances without manipulatives.
 Student achievement levels are related to teachers’
experience and expertise with manipulatives.
 The “relation between manipulatives and their intended
referents may not be transparent to children.”
 Time of interaction with manipulatives affects success
for elementary and for middle students as they assume
responsibility for their use.
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“MANIPULATIVES IN MIDDLE GRADES MATHEMATICS”,
RESEARCH SUMMARY.”
Teacher use is related to teachers’ prior experience with
manipulatives.
 Children’s comprehensions of manipulatives depend on
instruction.
 Manipulative use alone cannot be expected to improve
mathematics education.
 Teachers reported manipulatives were used for
demonstrations, problem solving, change of pace,
rewards, “fun,” and better understanding.
 Computer manipulatives and physical manipulatives
have different affordances, and both types should be
used in middle grades classrooms.
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7TH GRADE BOOK: LESSON 1.2 – LET’S FIGURE OUT
HOW TO TEACH IT!
Turn to page 8 in your textbook and read through page
8.
 Your goal with your group is to create a basic lesson plan
that you could teach.
 Questions to consider:
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What do you and your students need to know about integer
counters to understand this section?
 How do you help the students understand the relationship
between the counters and the concept?
 What examples will best help your students understand this
concept?
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7TH GRADE (RED BOOK) INTERACTIVE COMPONENT
Now go to www.bigideaslearning.com and login.
 Click on the Teachers, and then click on the Red Book.
 On the left side click on Dynamic Classroom.
 Click on the 7th Grade Book.
 Click on Lesson 1.2.
 As a group investigate the Algebra Tiles application.
 Create at least 5 problems that you could use to help
teach section 1.2.
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SUMMARIZE LESSON 1.2
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Why does the book using integer counters?
What do you and your students need to know about integer
counters to understand this section?
Does the order of your examples matter?
What problems could arise from using integer counters?
What problems could arise if a student doesn’t understand
the integer counters?
What did you learn from talking and sharing with other
teachers?
SMARTBOARD
How could you use (if available) a SmartBoard to help
strengthen lesson 1.2?
 Examples from my
classroom.
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7TH GRADE BOOK: LESSON 2.4. SOLVING EQUATIONS
USING ADDITION OR SUBTRACTION.
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Now that we have figured out 1.2 – lets look at page 70
and figure out how we would teach Lesson 2.4.
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Read through the section and then create a basic lesson
plan.
You may use the online resources.
 Come up with at least 5 examples you could use to help your
class understand the basics of solving equations.
 Carefully consider the examples and the order that you
chose to present them.
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SUMMARIZE LESSON 2.4
How does the textbook use manipulatives?
 Will using the tiles guarantee concrete understanding?
 After sharing, do you want to make any changes to the
lesson you created?
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SMARTBOARD
How could you use (if available) a SmartBoard to help
strengthen lesson 2.4?
 Examples from my
classroom.
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NOW DO WHAT WILL HELP YOU THE MOST!
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In the 7th grade book look at the following sections:
Lesson 1.3 (Subtracting Integers)
 Lesson 1.5 (Dividing Integers)
 Lesson 2.4 (Solving Equations Using Addition or Subtracting)
 Lesson 2.6 (Solving 2-Step Equations)
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In the 8th grade book look at section 1.1. What do students
already have to know in order to fully understand this
section?
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Look at the red book Chapter 2 to help answer the above question.
Spend time looking at Dynamic Classroom and the interactive
activities that are available to you.
SUMMARY & GOALS
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Understand CRA and the purpose it has in your
classroom.
Understand Instructional Shift #3 and find at least 1
specific way you can use it this upcoming school
year.
Recognize that Big Ideas Learning supports the use
of manipulatives.
Be familiar and use at least a few of the virtual
manipulatives or Interactive Activities (online resources)
from Dynamic Classroom.
 Try algebra tiles and counters and see what happens.
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Use what is available for you- don’t create more
work for yourself!
ADDITIONAL RESOURCES:
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National Library of Virtual Manipulatives:
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http://nlvm.usu.edu/