Supporting Mathematics Instruction
Paraprofessional Training
Mimi Velázquez
October 29, 2015
Norms
 Understand
that those who work, learn.
 Phrase
questions for the benefit of
everyone.
 Recognize
that everyone has expertise.
 Challenge
ideas, not people.
 Share
talk time.
 Limit
use of electronic devices.
Parking Lot

Burning Issues

Questions

Comments

Ideas to Share
Learner Outcomes
At the end of this training you will be able to

describe the components of concrete-representational-abstract (CRA)
instruction

relate the benefits of CRA
Let’s do math!
An ice-cream shop uses 6 bananas to make 4
identical sundaes. How much banana is used
in each sundae? Use a tape diagram to show
your work.
Let’s do math!
Concrete-to-Representational-toAbstract (C-R-A)
The purpose of teaching through a concrete-torepresentational-to-abstract sequence of instruction is
to ensure students truly have a thorough understanding
of the math concepts/skills they are learning. When
students who have math learning problems are allowed
to first develop a concrete understanding of the math
concept/skill, then they are much more likely to
perform that math skill and truly understand math
concepts at the abstract level.
fcit.usf.edu/mathvids/strategies/cra.html, 2014
CRA Instructional Sequence

CRA is a three-part instructional strategy

Each subsequent part builds on previous
instruction

The sequence promotes student learning and
retention

Students should be explicitly and directly
instructed to make connections through these
stages.
2010 Region 3 Education Service Center / Texas A&M University
CRA Sequence
C
• Concrete or
hands-on
instruction that
involves the
manipulation of
objects
R
A
• Representational
• Abstract stage,
stage, with different
involving the use
levels including
of numerals and
pictures, technology, operational
or tally marks
symbols to
represent the
previous levels
2010 Region 3 Education Service Center / Texas A&M University
Definition of Concrete

The use of manipulatives or models that
represent the concept being studied is
critical.

It is the “doing” step of instruction.

Students must be fluent in manipulation of
the concrete materials and must be able to
explain the connection to the skill.

Students should verbalize what they did to
solve the problem and why.

Students should be taught to mastery level
for each step before moving to the next step.
2010 Region 3 Education Service Center / Texas A&M University
Definition of Representational

Students learn to problem-solve by drawing
pictures.

Replacing concrete manipulative objects with
pictures or drawings

It is appropriate for students to begin drawing
solutions to problems as soon as they
demonstrate they have mastered a particular
math concept/skill at the concrete level.
2010 Region 3 Education Service Center / Texas A&M University
Definition of Representational
Definition of Abstract
 The “symbolic” step of instruction.
 Written mathematical numbers are used to
represent concepts or skills.
2010 Region 3 Education Service Center / Texas A&M University
Definition of Abstract
Algorithms
Algorithms without understanding
 Students learn the steps of the algorithm,
but they don’t understand why they work
 Students may develop some inaccurate ideas
that will then need to be unlearned before
they can learn the correct idea.
 Errors practiced and hard to break
 Limited retention
Algorithms
Algorithms with understanding
 Balanced conceptual understanding and
procedural skills
 Students are able to perform the operation
fluently but also understand what they’re
doing.
 Students develop number sense.
 Reduction in practice time
 Extended retention and application
Deborah Ball, Secretary’s summit on Mathematics, Washington, D.C., 2003,
http://www.ed.gov/rschstat/progs/mathscience/ball.html (accessed November 12, 2006
CRA Instructional Sequence
1.
3.
4.
2.
5.
Read Write Draw (RDW)
RDW or Read, Draw, Write (a Number Sentence and a Statement)
Mathematicians and teachers suggest a simple process applicable to all
grades:
1) Read.
2) Draw and label.
3) Write a number sentence (equation).
4) Write a word sentence (statement).

This could explanation of the process used.
Modeling with
Interactive
Questioning
Guided
Practice
Independent
Practice
Read Draw Write (RDW)
A loaf of bread was cut into 6 equal slices. Each of the 6
slices was cut in half to make thinner slices for sandwiches.
Mr. Beach used 4 slices. His daughter said, “Wow, you used
2
4
of the loaf!” His son said, “No, he used .” Work with a
6
12
partner to explain who was correct using a tape diagram.
Read Draw Write (RDW)