Place Value Progressions:
Make an array to show 4.3 x 2.4
Implementing CRA Math
Interventions
Fifth grade “Add, subtract, multiply, and divide
decimals to hundredths, using concrete models
or drawings and strategies based on place value,
properties of operations, and/or the relationship
between addition and subtraction; relate the
strategy to a written method and explain the
reasoning used.”
2014 UMTSS
Bradley S Witzel, Ph.D.
Professor, Winthrop University
[email protected]
@BradWitzel
© Witzel, 2014
1
• Fifth grade “Add, subtract, multiply, and divide decimals to
hundredths, using concrete models or drawings and strategies
based on place value, properties of operations, and/or the
relationship between addition and subtraction; relate the
strategy to a written method and explain the reasoning used.”
• 4.3 x 2.4 = ?
multiply
4
.3
2
8
0.6
.4
1.6
.12
© Witzel, 2014
Intervention Aspect of Place Value
• Language focused for conceptual building and
to aid think alouds
• Physical models to show place value
– http://ntnmath.com/video%20index/Index%20Videos/Polynomials/lesson%20108.htm
• Graphic Organizers for procedural memory
and conceptual memory
• Possibility of a graduated sequence of
instruction (CRA)
8 + 0.6 + 1.6 + 0.12 = 18.24
© Witzel, 2014
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© Witzel, 2014
(4.3)(2.4) using CRA
Ones times
tenths are
tenths.
There are
16 tenths.
Total = 8 ones; 22 tenths; 12
hundredths
8.0_
2.2_
0.12
© Witzel,
2014
10.32_
4
(4.3)(2.4) using CRA
Ones times
ones are
ones. There
are 8 ones.
Tenths
times ones
are tenths.
There are 6
tenths.
Ones times
ones are
ones. There
are 8 ones.
2
Ones times
tenths are
tenths.
There are
16 tenths.
Tenths times
tenths are
hundredths.
There are 12
hundredths.
5
X
X
X
X
X
X
X
X
X
x
x
x
x
x
X
X
X
X
X
x
X
x
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
x
x
x
X
x
X
x
Total = 8 ones; 22 tenths; 12
hundredths
8.0_
2.2_
0.12
10.32_
© Witzel, 2014
Tenths
times ones
tenths are
tenths.
There are 6
tenths.
Tenths times
tenths are
hundredths.
There are 12
hundredths.
6
1
CRA as effective instruction
(4.3)(2.4) using CRA
multi
4
.3
2
8
0.6
.4
1.6
0.12
(Gersten et al, 2009; NMP, 2008; Riccomini & Witzel, 2010; Witzel, 2005)
Total = 8 ones; 22
tenths; 12 hundredths
8.0_
2.2_
0.12
10.32_
© Witzel, 2014
Concrete to Representational to Abstract Sequence of
Instruction (CRA)
• Concrete (expeditious use of manipulatives)
• Representations (pictorial)
• Abstract procedures
Excellent for teaching accuracy and understanding
7
© Witzel, 2014
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Using Arrays for Decimals
(1.3)(0.6)
X X X X X X X X X X
X X X
X X X X X X X X X X
X X X
X X X X X X X X X X
X X X
X X X X X X X X X X
X X X
X X X X X X X X X X
X X X
X X X X X X X X X X
X X X
0.60
0.18
Total = 0.78
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© Witzel, 2014
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Concrete Misconceptions
Today’s Agenda
(Witzel & Allsopp, 2007)
• Manipulative objects teach children
• Any concrete object transfers to abstract
understanding
• All students will achieve higher gains in math
scores when taught using concrete objects
• All math can be taught through the use of
concrete objects
• Math teachers are united in their belief in
manipulative instruction
• CRA Introduction
• Tier 1
– Whole Numbers
– Fractions
• Tier 2 and 3
– Place Value
– Rational Numbers
• Application
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© Witzel, 2014
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2
USDOE on the
Use of visual representations
Stepwise Approaches
“Studies showed that when faced with
multistep problems, students frequently
attempted to solve the problems by randomly
combining numbers instead of implementing
a solution strategy step by step. The process
of encouraging students to verbalize their
thinking—by talking, writing, or drawing the
steps they used in solving a problem— was
consistently effective” (NCTM Research Brief p. 2).
• IES Practice Guide on RtI Math:
– “Recommendation 5. Interventions should include opportunities
for students to work with visual representations of mathematical
ideas and interventions should be proficient in the use of visual
representations of mathematical ideas” (2009).
• IES Practice Guide on Fractions:
– “Recommendation 2. Help students recognize that fractions are
numbers and that they expand the number system beyond whole
numbers. Use number lines as a central representational tool in
teaching this and other fraction concepts from the early grades
onward” (2010)
• IES Practice Guide on Word Problems:
– “Teach students how to use visual representations” (2012)
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Algebra Intervention Meta-analysis
(Hughes, Fries, Riccomini, & Witzel, in-review)
• Cognitive and model-based problem solving
© Witzel, 2014
#5
14
Include visual representations of
mathematical ideas (Gersten et al., 2009)
• Visual representations used in conjunction
with Tier 1 level instruction
• CRA interventions used to support the core
instruction and student growth
ES= 0.693
• Concrete-Representation-Abstract
ES=0.431
• Peer Tutoring
ES=0.102
• Graphic Organizers alone
ES=0.106
• Technology
ES=0.890
• Single-sex instruction
ES=0.090
© Witzel, 2014
CRA
15
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CCSS 1.NBT.4 Add within 100 using models and
strategies based on place value
26 + 18
(Gersten et al, p. 35)
(Witzel, et al, 2013)
© Witzel, 2013
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© Witzel, 2014
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3
CCSS 2.NBT.7 Add and subtract within 100 using models
and strategies based on place value
33 - 18
© Witzel, 2014
(Witzel, et al, 2013)
Concrete
19
20
Number Line Progression
Count to five using blocks (Witzel, 2013)
While
accuracy is
important, it
is the
deliberate use
of counting
that should be
assessed.
5
7
Abstract
Witzel, 2013
Computation:
Addition and Subtraction
6
Representational
5
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Witzel, 2013
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© Witzel, 2014
Number Line Progressions
Number Line Progressions
Is red greater than or less than green?
Is A greater than or less than B?
Is red greater than or less than green? or
Is A greater than or less than B?
A
A
B
B
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23
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4
Progressions:
(2/3)(1/2)
Progressions:
Identify 1/3 on a number line
Think aloud.
Verbally say, “2/3rds of 1/2”
1/2
1/3
X
(2/3) (1/2) = 2/6
(Witzel & Little)
Elementary Math
How could this be translated to 1/3?
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Progressions:
¾÷½
© Witzel, 2014
Complex fractions start early
Verbally say, “How many one-halves go into threefourths?” What is the logical answer?
Consider the algebra expression
Why do students erroneously
1𝑥−4𝑦
rename it as
?
¾
X
Rather than
© Witzel, 2014
(Witzel & Little)
Elementary Math
3𝑥 2
3𝑥
−
4y
3𝑥
This instruction starts early
27
Mixed to improper and back: concrete
• 8÷2
=
28
+ 1/2
=
+
(1 + 1 + 1)
+
(3 + 3)
29
𝟑
=3𝟒
(Witzel & Riccomini, 2009)
(2 + 2)
• 13 ÷ 4
𝟏𝟐+𝟑
𝟒
Intervention with Fractions procedures
+
• 8÷3
𝟏𝟓
𝟒
© Witzel, 2014
2/
3
© Witzel, 2014
3𝑥 2 −4𝑦
3𝑥
1
Answer: 1+ 1/2 = 3/2
Check: 1(1/2) + 1/2(1/2) = 3/4
26
+
+
7
=
(2 + 2 + 2)
© Witzel, 2013
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30
5
Aim interventions at
procedural processes (Witzel & Riccomini, 2009)
1/
3
+
Find new ways to reach the students
– 2/3
-
-
=
© Witzel, 2013
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• Choose the math topic to be taught;
• Review procedures to solve the problem;
• Adjust the steps to eliminate notation or calculation
tricks;
• Match the abstract steps with an appropriate
concrete manipulative;
• Arrange concrete and representational lessons;
• Teach each concrete, representational, and abstract
lesson to student mastery; and
• Help students generalize what they learn through
word problems.
• Choose the math topic to be taught
• Review abstract steps to solve the problem
• Adjust the steps to eliminate notation or calculation
tricks
• Match the abstract steps with an appropriate concrete
manipulative
• Arrange concrete and representational lessons
• Teach each concrete, representational, and abstract
lesson to student mastery (accuracy without hesitation)
• Help students generalize learning through word problems
and problem solving events
33
Choose a math topic
•
•
•
•
32
Implement CRA instruction in your
classroom. Here’s how:
CRA delivery
(From Witzel, Riccomini, & Schneider, 2008)
© Witzel, 2014
© Witzel, 2014
© Witzel, 2014
34
Example: Long division
1) Leftmost digit of the dividend is divided by the
divisor.
2) Multiply, subtract, bring down and maintain
place value
3) Divide the difference by the divisor and place
the answer according to place value
4) Repeat steps 2 and 3 until all of the digits in the
dividend are subtracted
Plan what is to be taught ahead of time.
Group lessons according to the big idea as
determined by your state standards
Sequence the lessons so they start basic and
gradually introduce new topics
Any math topic can be examined for CRA
linkage
Warning – rational numbers will require additional
steps to clarify the answer
© Witzel, 2014
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© Witzel, 2014
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6
Review the abstract steps used to solve
the problem
•
•
•
•
Example: Long division
To maintain place value we must change our
typical language.
What is the desired math outcome of the group
of lessons?
Determine the procedural goal of the
combination of math skills
List out the steps or procedures
Remember, not all math skills require abstract
knowledge.
015
8 126
-8
46
- 40
a) 8 into 1 hundred can’t be done (see alternate algorithms)
b) 8 into 12 tens is clear, 1 set of 8 tens
c) 8 into 46 ones is clear, 5 sets of 8 ones
d) there are 6 remaining,
e) The answer is 15
© Witzel, 2014
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•
015
126
- 80
46
- 40
6
a) There aren’t 8 hundreds in the dividend’s 100 place
b) There are 8 tens in 120, 1 set of 8tens = 80
c) There are 8 ones into 46, 5 sets of 8ones = 40
d) there are 6 remaining,
e) The answer is 15
39
•
•
•
𝟔
𝟖
𝟖
40
Example: Long division
Initial understanding of content will be based on
interactions with concrete objects, so be careful which
ones you choose.
The conceptual effectiveness of the manipulative object
should be noted in accordance to the math skill being
taught.
Avoid concrete objects that only cover a few skills.
You may have to teach two stages of concrete knowledge
Also, not all concrete objects are appropriate for CRA
instruction. Some materials are effective for conceptual
growth while others are useful for procedural work.
Base ten blocks would match the place value needs in this
algorithm
8
015
126
- 80
46
- 40
6
a) There aren’t 8 hundreds in the dividend’s 100 place
b) There are 8 tens in 120, 1 set of 8tens = 80
c) There are 8 ones into 46, 5 sets of 8ones = 40
d) there are 6 remaining,
e) The answer is 15
http://nlvm.usu.edu/en/nav/vlibrary.html
© Witzel, 2014
𝟔
© Witzel, 2014
Match the abstract steps with an
appropriate concrete manipulative
•
38
Example: Long division
8
•
8
a) a is an error. It doesn’t work in this algorithm.
b) Sets of tens and ones might be confusing. Let’s
restate the multiplication that gets us there.
Change or modify steps to create the most logical
and sequential set of procedures
Take a child’s point of view when reviewing steps
© Witzel, 2014
8
6
© Witzel, 2014
Adjust the steps to eliminate notation
or calculation tricks
•
6
41
𝟔
𝟔
𝟖
𝟖
© Witzel, 2014
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7
Arrange concrete and representational
lessons
•
Example: Long division
Practice concrete manipulations. The same questions
that you encounter you can be certain your students
will as well.
Practice how to mark pictorial representations that
appear similar to concrete manipulations.
Be Consistent in your use of math language. Make
certain that your language throughout instruction
matches the language required for the desired
outcome.
•
•
8 126
a) There aren’t 8 hundreds
b) There is 8 sets of 10= 80
4 tens are left over.
c) There 46 ones total left over
d) There are 8 sets of 5 = 40
e) The remainder is 6 ones
f) The answer is 10 + 5 = 15 r. 6
Roadblock. Choose smaller numbers for C and R activities. These steps
teach the process. A can be used for larger numbers.
© Witzel, 2014
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© Witzel, 2014
Help students generalize what they
learn through word problems
Teach each CRA lesson to mastery
•
•
•
•
Model and guide students in their use of manipulative objects
and pictorial representations.
Teach students step by step gradually introducing
mathematical vocabulary. Allow students to name or invent
their stepwise procedures within instruction.
Move from concrete to representational to abstract learning
levels only after students show accuracy without hesitations
in manipulations or drawings.
Assess each level of learning according to stepwise
procedures. Take account of students who created different
procedures.
© Witzel, 2014
•
•
45
subtraction
Line up
place value
Remainder
•
√
√
√
√
√
Tarek
√
X
√
√
X
Miguel
√
√
√
√
√
Manuel
√
√
X
X
X
Jose
√
√
√
√
√
Pam
√
√
√
√
X
Michele
√
√
√
√
√
Brandon
√
√
√
√
√
Stan
√
√
√
√
X
© Witzel, 2014
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Teach each CRA lesson to mastery
Answer
Mike
Incorporate word problems throughout a
lesson to help show social relevance as to
why a math skill is important to learn
Use language experiences through the
learning process to help prepare for word
problem and problem solving application
© Witzel, 2014
Grading procedures
Multi fact
44
•
•
•
47
Model and guide students in their use of manipulative
objects and pictorial representations.
Teach students step by step gradually introducing
mathematical vocabulary. Allow students to name or
invent their stepwise procedures within instruction.
Move from concrete to representational to abstract
learning levels only after students show accuracy
without hesitations in manipulations or drawings.
Assess each level of learning according to stepwise
procedures. Take account of students who created
different procedures.
© Witzel, 2014
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8
Summary
• Math visuals are important to making sense of
problems
• CRA is a highly successful intervention for
students who may otherwise struggle in math
• The goal of CRA is abstract success
© Witzel, 2014
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“ Positive results can be achieved in a reasonable
time at accessible cost, but a consistent, wise
community-wide effort will be required.”
National Math Panel (2008, p.12)
© Witzel, 2013
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9