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Concrete
- Representational - Abstract
Sequence
of Instruction
Purpose
The purpose of teaching through
a concrete-to-representational-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.
What is it?
- Each math concept/skill is first modeled with
concrete materials (e.g. chips, unifix cubes, base ten blocks,
beans and bean sticks, pattern blocks).
- Students are provided many opportunities to
practice and demonstrate mastery using concrete materials
- The math concept/skill is next modeled at
the representational (semi-concrete) level which involves
drawing pictures that represent the concrete objects previously
used (e.g. tallies, dots, circles, stamps that imprint pictures
for counting)
- Students are provided many opportunities to
practice and demonstrate mastery by drawing solutions
- The math concept/skill is finally modeled
at the abstract level (using only numbers and mathematical
symbols)
- Students are provided many opportunities to
practice and demonstrate mastery at the abstract level before
moving to a new math concept/skill.
- As a teacher moves through a concrete-to-representational-to-abstract
sequence of instruction, the abstract numbers and/or symbols
should be used in conjunction with the concrete materials
and representational drawings (promotes association of abstract
symbols with concrete & representational understanding)
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What are the critical elements of
this strategy?
- Use appropriate concrete objects to teach
particular math concept/skill (see Concrete Level of Understanding/Understanding
Manipulatives-Examples of manipulatives by math concept area).
Teach concrete understanding first.
- Use appropriate drawing techniques or appropriate
picture representations of concrete objects (see Representational
Level of Understanding/Examples of drawing solutions by math
concept area). Teach representational understanding second.
- Use appropriate strategies for assisting students
to move to the abstract level of understanding for a particular
math concept/skill (see Abstract Level of Understanding/Potential
barriers to abstract understanding for students who have
learning problems and how to manage these barriers).
- When teaching at each level of understanding,
use explicit teaching methods (see the instruction strategy
Explicit Teacher Modeling).
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How do I implement the strategy?
- When initially teaching a math concept/skill,
describe & model it using concrete objects (concrete
level of understanding).
- Provide students many practice opportunities
using concrete objects.
- When students demonstrate mastery of skill
by using concrete objects, describe & model how to perform
the skill by drawing or with pictures that represent concrete
objects (representational level of understanding).
- Provide many practice opportunities where
students draw their solutions or use pictures to problem-solve.
- When students demonstrate mastery drawing
solutions, describe and model how to perform the skill using
only numbers and math symbols (abstract level of understanding).
- Provide many opportunities for students to
practice performing the skill using only numbers and symbols.
- After students master performing the skill
at the abstract level of understanding, ensure students maintain
their skill level by providing periodic practice opportunities
for the math skills.
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How Does This Instructional Strategy
Positively Impact Students Who Have Learning Problems?
- Helps passive learner to make meaningful connections
- Teaches conceptual understanding by connecting
concrete understanding to abstract math process
- By linking learning experiences from concrete-to-representational-to-abstract
levels of understanding, the teacher provides a graduated
framework for students to make meaningful connections.
- Blends conceptual and procedural understanding
in structured way
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Additional Information
Research
Support For The Instructional Features Of This Instructional
Strategy: Allsopp
(1999); Baroody (1987); Butler, Miller, Crehan, Babbit, & Pierce
(2003); Harris, Miller, & Mercer (1993); Kennedy
and Tips (1998); Mercer, Jordan, & Miller (1996); Mercer
and Mercer (2005); Miller, Butler, & Lee (1998); Miller
and Mercer, 1995; Miller, Mercer, & Dillon (1992);
Peterson, Mercer, & O'Shea. (1988); Van De Walle (2005);
Witzel, Mercer, & Miller (2003).
Create seperate pages
Concrete
What is it?
Understanding manipulatives
Examples of manipulatives (concrete objects) by math concept
level
Suggestions for using manipulatives (concrete objects)
What is it?
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.
Understanding manipulatives (concrete
objects)
To use math manipulatives effectively, it is
important that you understand several basic characteristics
of different types of math manipulatives and how these specific
characteristics impact students who have learning problems.
As you read about the different types of manipulatives, click
on the numbers beside each description to view pictures of
these different types of manipulatives.
General types of math manipulatives:
Discrete -
those materials that can be counted (e.g. cookies, children,
counting blocks, toy cars, etc.).
See examples - 1 |
2 | 3 | 4 |
Continuous -
materials that are not used for counting but are used for
measurement (e.g. ruler, measuring cup, weight scale, trundle
wheel). See example - 1
Suggestions for using
Discrete & Continuous materials with students who have
learning problems:
Students who have learning problems need to have
abundant experiences using discrete materials before they will
benefit from the use of continuous materials. This is because
discrete materials have defining characteristics that students
can easily discriminate through sight and touch. As students
master an understanding of specific readiness concepts for
specific measurement concepts/skills through the use of discrete
materials (e.g. counting skills), then continuous materials
can be used.
Types of manipulatives used to teach
the Base-10 System/place-value (Smith, 1997):
Proportional - show relationships
by size (e.g. ten counting blocks grouped together is ten
times the size of one counting block; a beanstick with ten
beans glued to a popsicle stick is ten times bigger than
one bean).
- Non-linked proportional -
single units are independent of each other, but can be "bundled
together (e.g. popsicle sticks can be "bundled together
in groups of 'tens' with rubber bands; individual unifix
cubes can be attached in rows of ten unifix cubes each). See
examples - 1 | 2 | 3 |
- Linked
proportional - comes in single units as well
as "already bundled" tens units, hundreds units, & thousands
units (e.g. base ten cubes/blocks; beans & beansticks). Se
examples - 1 | 2 | 3 | 4 | 5
Non-proportional -
use units where size is not indicative of value while other
characteristics indicate value (e.g. money, where one dime
is worth ten times the value of one penny; poker chips where
color indicates value of chip; an abacus where location of
the row indicates value). A specified number of units representing
one value are exchanged for one unit of greater value (e.g.
ten pennies for one dime; ten white poker chips for one blue
poker chip, ten beads in the first row of an abacus for one
bead in the second row). See example - 1
Suggestions for using
proportional and non-proportional manipulatives with students
who have learning problems:
Students who have learning problems are more
likely to learn place value when using proportional manipulatives
because differences between ones units, tens units, & hundreds
units are easy to see and feel. Due to the very nature of non-proportional
manipulatives, students who have learning problems have more
difficulty seeing and feeling the differences in unit values.
Examples of manipulatives (concrete objects)
Suggested manipulatives are listed according
to math concept/skill area. Descriptions of manipulatives are
provided as appropriate. A brief description of how each set
of manipulatives may be used to teach the math concept/skill
is provided at the bottom of the list for each math concept
area. Picture examples of some of the manipulatives for each
math concept area can be accessed by clicking on the numbers
found underneath the title of each math concept area. This
is not meant to be an exhaustive list, but this list does include
a variety of common manipulatives. The list includes examples
of "teacher-made" manipulatives as well "commercially-made" ones.
Counting/Basic Addition & Subtraction
Place Value
Multiplication/Division
Positive & Negative Integers
Fractions
Geometry
Beginning Algebra
Counting/Basic
Addition & Subtraction Pictures
See examples -
1 | 2 | 3 |
- Colored chips
- Beans
- Unifix cubes
- Golf tees
- Skittles or other candy pieces
- Packaging popcorn
- Popsicle sticks/tongue depressors
Description of use: Students
can use these concrete materials to count, to add, and to
subtract. Students can count by pointing to objects and counting
aloud. Students can add by counting objects, putting them
in one group and then counting the total. Students can subtract
by removing objects from a group and then counting how many
are left.
Place Value
Pictures
See examples -
1 | 2 | 3 | 4 | 5 | 6 | 7
- Base 10 cubes/blocks
- Beans and bean sticks
- Popsicle sticks & rubber bands for bundling
- Unifix cubes (individual cubes can be combined
to represent "tens")
- Place value mat (a piece of tag board or other
surface that has columns representing the "ones," "tens," and "hundreds" place
values)
Description
of use: Students are first taught to represent 1-9
objects in the "ones" column. They are then taught
to represent "10" by trading in ten single counting
objects for one object that contains the ten counting objects
on it (e.g. ten separate beans are traded in for one "beanstick" -
a popsicle stick with ten beans glued on one side. Students
then begin representing different values 1-99. At this point,
students repeat the same trading process for "hundreds."
Multiplication/Division
Pictures
See examples -
1 | 2 | 3 |
- Containers & counting objects (paper dessert
plates & beans, paper or plastic cups and candy pieces,
playing cards & chips, cutout tag board circles & golf
tees, etc.). Containers represent the "groups" and
counting objects represent the number of objects in each
group. (e.g. 2 x 4 = 8: two containers with four counting
objects on each container)
Counting objects arranged in arrays (arranged in rows and
columns). Color-code the "outside" vertical column
and horizontal row helps emphasize the multipliers
.
Positive & Negative Integers Pictures
See examples -
1
- Counting objects, one set light colored and
one set dark colored (e.g. light & dark colored beans;
yellow & blue counting chips; circles cut out of tag
board with one side colored, etc.).
Description of use: Light colored objects
represent positive integers and dark colored objects represent
negative integers. When adding positive and negative integers,
the student matches pairs of dark and light colored objects.
The color and number of objects remaining represent the solution.
Fractions Pictures
See examples -
1 | 2 | 3 |
- Fraction pieces (circles, half-circles, quarter-circles,
etc.)
- Fraction strips (strips of tag board one foot
in length and one inch wide, divided into wholes, ½'s,
1/3's, ¼'s, etc.
- Fraction blocks or stacks. Blocks/cubes that
represent fractional parts by proportion (e.g. a "1/2" block
is twice the height as a "1/4" block).
Description of use: Teacher models how to
compare fractional parts using one type of manipulative.
Students then compare fractional parts. As students gain
understanding of fractional parts and their relationships
with a variety of manipulatives, teacher models and then
students begin to add, subtract, multiply, and divide using
fraction pieces.
Geometry Pictures
See examples -
1
- Geoboards (square platforms that have raised
notches or rods that are formed in a array). Rubber bands
or string can be used to form various shapes around the raised
notches or rods.
Description of use: Concepts such as area
and perimeter can be demonstrated by counting the number
of notch or rod "units" inside the shape or around
the perimeter of the shape.
Beginning Algebra
Pictures
See examples -
1 | 2 |
- Containers (representing the variable of "unknown")
and counting objects (representing integers) -e.g. paper
dessert plates & beans, small clear plastic beverage
cups 7 counting chips, playing cards & candy pieces,
etc.
Description of use: The algebraic
expression, "4x = 8," can be represented with four
plates ("4x"). Eight beans can be distributed evenly
among the four plates. The number of beans on one plate represent
the solution ("x" = 2).
Suggestions for using manipulatives (Burns, 1996)
- Talk with your students about how manipulatives
help to learn math.
- Set ground rules for using 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
Representational
What is it?
Examples of drawing solutions by math concept level
What is it?
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. 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. While not all students need to draw
solutions to problems before moving from a concrete level of
understanding to an abstract level of understanding, students
who have learning problems in particular typically need practice
solving problems through drawing. When they learn to draw solutions,
students are provided an intermediate step where they begin
transferring their concrete understanding toward an abstract
level of understanding. When students learn to draw solutions,
they gain the ability to solve problems independently. Through
multiple independent problem-solving practice opportunities,
students gain confidence as they experience success. Multiple
practice opportunities also assist students to begin to "internalize" the
particular problem-solving process. Additionally, students'
concrete understanding of the concept/skill is reinforced because
of the similarity of their drawings to the manipulatives they
used previously at the concrete level.
Drawing is not a "crutch" for students
that they will use forever. It simply provides students an
effective way to practice problem solving independently until
they develop fluency at the abstract level.
Examples of drawing solutions by math concept
level
The following drawing examples are categorized
by the type of drawings ("Lines, Tallies, & Circles," or "Circles/Boxes").
In each category there are a variety of examples demonstrating
how to use these drawings to solve different types of computation
problems. Click on the numbers below to view these examples.
1 | 2
Abstract
What is it?
Potential barriers to abstract understanding for students who
have learning problems and how to manage these barriers
What is it?
A student who problem-solves at the abstract
level, does so without the use of concrete objects or without
drawing pictures. Understanding math concepts and performing
math skills at the abstract level requires students to do this
with numbers and math symbols only. Abstract understanding
is often referred to as, "doing math in your head." Completing
math problems where math problems are written and students
solve these problems using paper and pencil is a common example
of abstract level problem solving.
Potential barriers to abstract understanding
for students who have learning problems and how to manage these
barriers
Students who are not successful solving problems
at the abstract level may:
- Not understand the concept behind the skill.
Suggestions:
- Re-teach the concept/skill at the concrete
level using appropriate concrete objects (see Concrete
Level of Understanding).
- Re-teach concept/skill at representational
level and provide opportunities for student to practice
concept/skill by drawing solutions (see Representational
Level of Understanding).
- Provide opportunities for students to use
language to explain their solutions and how they got them
(see instructional strategy Structured Language Experiences).
- Have difficulty with basic facts/memory
problems
Suggestions:
- Regularly provide student with a variety of
practice activities focusing on basic facts. Facilitate independent
practice by encouraging students to draw solutions when needed
(see the student practice strategies Instructional Games,
Self-correcting Materials, Structured Cooperative Learning
Groups, and Structured Peer Tutoring).
- Conduct regular one-minute timings and chart
student performance. Set goals with student and frequently
review chart with student to emphasize progress. Focus on
particular fact families that are most problematic first,
then slowly incorporate a variety of facts as the student
demonstrates competence (see evaluation strategy Continuous
Monitoring & Charting of Student Performance).
- Teach student regular patterns that occur
throughout addition, subtraction, multiplication, & division
facts (e.g. "doubles" in multiplication, 9's rule
- add 10 & subtract one, etc.)
- Provide student a calculator or table when
they are solving multiple-step problems.
- Repeat procedural mistakes.
Suggestions:
- Provide fewer #'s of problems per page.
- Provide fewer numbers of problems when assigning
paper & pencil practice/homework.
- Provide ample space for student writing, cueing, & drawing.
Provide problems that are already written on learning sheets
rather than requiring students to copy problems from board
or textbook.
- Provide structure: turn lined paper sideways
to create straight columns; allow student to use dry-erase
boards/lap chalkboards that allow mistakes to be wiped away
cleanly; color cue symbols; for multi-step problems, draw
color-cued lines that signal students where to write and
what operation to use; provide boxes that represent where
numerals should be placed; provide visual directional cues
in a sample problem; provide a sample problem, completed
step by step at top of learning sheet.
- Provide strategy cue cards that student can
use to recall the correct procedure for solving problem.
- Provide a variety of practice activities that
require modes of expression other than only writing
*Key Idea
Student learning & mastery greatly depends
on the number of opportunities a student has to respond!! The
more opportunities for successful practice that you provide
(i.e. practice that doesn't negatively impact student learning
characteristics), the more likely it is that your student will
develop mastery of that skill.
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Videos
There are no videos for this Strategy. See
Explicit Teacher Modeling
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