Key Strategies for
Mathematics Interventions
You have 8 bags of cookies. Each bag has 4
cookies in it. How many cookies do you have in
all?
• Solve it. Show all your work. Write a reason for
each step.
• Make a drawing that helps solve it.
• What kind of problem is this?
• Make up another problem with the same
underlying structure.
You have 32 cookies to sell at a bake sale. You
want to put them into bags with 4 in each bag.
How many bags of cookies will you make?
• Solve it. Show all your work. Write a reason for
each step.
• Make a drawing that helps solve it.
• What kind of problem is this?
• Make up another problem with the same
underlying structure.
Dual Role of Interventionists
Being an interventionist requires all of the
knowledge and skill of being a classroom
teacher, plus more:
Interventionists need to know where each child
is on each learning progression.
The Common Core Standards provide learning
progressions.
Instructional Strategies
Along with in-depth content knowledge, both
classroom teachers and interventionists need to
be skillful at using proven instructional
strategies:
• Visual representations (C-R-A framework)
• Common underlying structure of word problems
• Explicit instruction including verbalization of thought
processes and descriptive feedback
• Systematic curriculum and cumulative review
Agenda
1. Review the Common Core Standards and look at
learning progressions
2. Consider the key research-based instructional
strategies as outlined in the IES Practice Guide
In Grade 3, instructional time should focus
on four critical areas:
(1) developing understanding of multiplication and
division and strategies for multiplication and
division within 100;
(2) developing understanding of fractions, especially
unit fractions (fractions with numerator 1);
(3) developing understanding of the structure of
rectangular arrays and of area;
(4) describing and analyzing two-dimensional shapes.
In Grade 4, instructional time should focus
on three critical areas:
(1) developing understanding and fluency with multidigit multiplication and division;
(2) developing an understanding of fraction
equivalence, addition and subtraction of fractions
with like denominators, and multiplication of
fractions by whole numbers;
(3) understanding that geometric figures can be
analyzed and classified based on their properties.
Learning Progression
Multiplying and dividing begins with
repeated addition:
know that the concept of multiplication is
repeated adding or skip counting – finding the
total number of objects in a set of equal size
groups
• be able to represent situations involving
groups of equal size with objects, words and
symbols
•
Learning Progression
• Use strategies to multiply, eventually learn the
multiplication combinations fluently
• Know how to multiply by 10 and 100
• Use number sense to estimate the result of
multiplying
• Use area and array models to represent
multiplication and to simplify calculations.
Learning Progression
• Understand how the distributive property
works and use it to simplify calculations
15 x 8 = (10 x 8) + (5 x 8)
• Use alternative algorithms like the partial
product method (based on the distributive
property) and the lattice method
• Be able to identify typical errors that occur
when using the standard algorithm.
Learning Progression
Learning Progression
Types of Knowledge
Understanding concepts
Skillful performance with procedures (fluency)
Generalizations that support problem solving
Examples
Understanding what multiplication means;
seeing it in the area model.
Skillful performance of multi-digit multiplication
Generalization of concepts and skills to
advanced mental math. (Number Talks video)
To diagnose
If a student isn’t sure how to start with 12 x 18, they
probably don’t know the underlying concept. Use base
ten blocks, area models, etc.
If a student can solve 12 x 23 but not
35 x 48, guided practice is needed,
perhaps with the partial product method.
If a student is having difficulty with 356 x 27,
they need more insight into the procedure
in order to generalize it to larger numbers.
To diagnose
Common Core for your grade.
Learning progressions across grades.
Types of knowledge to guide diagnosis and
intervention.
Key Strategies
1. Visual representations (C-R-A framework)
2. Common underlying structure of word
problems
3. Explicit instruction including verbalization of
thought processes and descriptive feedback
4. Systematic curriculum and cumulative review
Visual Representations
Intervention materials should include opportunities
for students to work with visual representations of
mathematical ideas and interventionists should be
proficient in the use of visual representations of
mathematical ideas.
• Use visual representations such as number lines,
arrays, and strip diagrams.
• If visuals are not sufficient for developing accurate
abstract thought and answers, use concrete
manipulatives first. (C-R-A)
Visual Representations
What visual representations are often used in
3rd and 4th grade?
Area model
Fraction models
Base ten blocks (also concrete models)
C-R-A
• The point of visual representations is to help
students see the underlying concepts.
• A typical teaching progression starts with
concrete objects, moves into visual
representations (pictures), and then
generalizes or abstracts the method of the
visual representation into symbols.
Objects – Pictures – Symbols
C-R-A for multiplying
Bugs have 6 legs. Ashley found 5 bugs. How
many legs are on all 5 bugs.
C: Model this with unifix cubes and count or
skip-count to get the answer.
R: Make a drawing or use 6 five-frame cards.
A: 6 x 5 = 30
Objects – Pictures – Symbols
• Young children follow this pattern in their
early learning when they count with objects.
• Your job as teacher is to move them from
objects, to pictures, to symbols.
You have 12 cookies and want to put them
into 4 bags to sell at a bake sale. How many
cookies would go in each bag?
Objects:
Pictures:
Symbols:
There are 21 hamsters and 32 kittens at the
pet store. How many more kittens are at the
pet store than hamsters?
Objects:
Pictures:
Symbols:
32
21
?
• Elisa has 37 dollars. How many more dollars
does she have to earn to have 53 dollars? (Try
it with mental math.)
37 + ___ = 53
C-R-A
53 ducks are swimming on a pond. 38 ducks fly
away. How many ducks are left on the pond?
First, try this with mental math.
Next, model it with unifix cubes. (see the C-R-A)
C-R-A
53 ducks are swimming on a pond. 38 ducks fly
away. How many ducks are left on the pond?
Then use symbols to record what we did.
4 13
53
-38
15
 18 candy bars are packed into one box. A
school bought 23 boxes. How many candy
bars did they buy altogether?
 Objects: Model it with base ten blocks
 Pictures: Use an
area model
nlvm.usu.edu
Symbols:
You create the C-R-A
• Your class is having a party. When the party is
over, 3/4 of one pan of brownies is left over
and 2/4 of another pan of brownies is left
over. How much is left over altogether?
• Students will be at different places in the CRA
learning progression.
Next Intervention Strategy:
Common Underlying Structure of
Word Problems
Interventions should include instruction on
solving word problems that is based on
common underlying structures.
• Teach students about the structure of various
problem types and how to determine appropriate
solutions for each problem type.
• Teach students to transfer known solution
methods from familiar to unfamiliar problems of
the same type.
Multiplication
• How many cookies would you have if you had
7 bags of cookies with 8 cookies in each bag?
Equal number of groups
• This year on your 11th birthday your mother
tells you that she is exactly 3 times as old as
you are. How old is she?
Multiplicative comparison
Division
• Ashley wants to share 56 cookies with 7
friends. How many cookies will each friend
get? Partitive division: sharing equally to find
how many are in each group
• Ashley baked 56 cookies for a bake sale. She
puts 8 cookies on each plate. How many plates
of cookies will she have? Measurement
division: with a given group size, finding how
many groups
• Multiplication and division situations differ
only by what part is unknown. Any
multiplication problem has a corresponding
division problem.
7 ∙ 8 = ___
7 ∙ ___ = 56
Multiplication
• A giraffe in the zoo is 3 times as tall as a
kangaroo. The kangaroo is 6 feet tall. How tall
is the giraffe? (write the equation)
• The giraffe is 18 feet tall. The kangaroo is 6
feet tall. The giraffe is how many times taller
than the kangaroo?
• The giraffe is 18 feet tall. She is 3 times as tall
as the kangaroo. How tall is the kangaroo?
Kangaroo
Scale factor =
Giraffe
6 feet
6 feet
3 times
?
?
18 feet
?
3 times
18 feet
•
6 ∙ 3 = ___
6 ∙ ___ = 18
___ ∙ 3 = 18
Transfer to problems
of the same type
Length ∙
Width =
Area
5
5
?
8
?
8
?
40
40
Multiplication/division problems
Grouping problems
How many peanuts would the monkey eat if she ate 4 groups
of peanuts with 3 peanuts in each group?
The monkey ate 4 bags of peanuts. Each bag had the same
number of peanuts in it. If the monkey ate 12 peanuts all
together, how many peanuts were in each bag? (how many in
each group?)
The monkey ate some bags of peanuts. Each bag had 3
peanuts in it. Altogether the monkey ate 12 peanuts. How
many bags of peanuts did the monkey eat? (how many
groups?)
Rate problems
A baby elephant gains 4 pounds each day. How many
pounds will the baby elephant gain in 8 days?
A baby elephant gains 4 pounds each day. How many
days will it take the baby elephant to gain 32 pounds?
A baby elephant gained 32 pounds in 8 days. If she
gained the same amount of weight each day, how much
did she gain in one day?
Price problems
How much would 5 pieces of bubble gum cost if each
piece costs 4 cents?
If you bought 5 pieces of bubble gum for 20 cents, how
much would each piece cost?
If one piece of bubble gum costs 4 cents, how many
can you buy for 20 cents?
Array and Area problems (symmetric problems)
For the second grade play, the chairs have been put
into 4 rows with 6 chairs in each row. How many chairs
have been put out for the play?
A baker has a pan of fudge that measures 8 inches on
one side and 9 inches on another side. If the fudge is
cut into square pieces 1 inch on each side, how many
pieces of fudge does the pan hold?
Combination problems
The Friendly Old Ice Cream Shop has 3 types of ice
cream cones. They also have 4 flavors of ice cream.
How many different combinations of an ice cream
flavor and cone type can you get at the Friendly Old Ice
Cream Shop?
Next Intervention Strategy:
Explicit Instruction
Instruction during the intervention should be
explicit and systematic. This includes
•providing models of proficient problem solving,
•verbalization of thought processes,
•guided practice,
•corrective feedback, and
•frequent cumulative review.
The National Mathematics Advisory Panel
defines explicit instruction as:
• “Teachers provide clear models for solving a problem
type using an array of examples.”
• “Students receive extensive practice in use of newly
learned strategies and skills.”
• “Students are provided with opportunities to think
aloud (i.e., talk through the decisions they make and
the steps they take).”
• “Students are provided with extensive feedback.”
Explicit Instruction
The NMAP notes that this does not mean that
all mathematics instruction should be explicit.
But it does recommend that struggling students
receive some explicit instruction regularly and
that some of the explicit instruction ensure that
students possess the foundational skills and
conceptual knowledge necessary for
understanding their grade-level mathematics.
Example 1
The boys swim team and the girls swim team held a car
wash. They made $210 altogether. There were twice as
many girls as boys, so they decided to give the girls’
team twice as much money as the boys’ team. How
much did each team get?
First, work this out yourself in any way that you can. If
you can draw a picture, do that also.
Here’s how I would solve this
The boys swim team and the girls swim team held a car wash.
They made $210 altogether. There were twice as many girls as
boys, so they decided to give the girls’ team twice as much
money as the boys’ team. How much did each team get?
If the boys get $50, then the girls get $100. Does that add up to
$210?
If the boys get $60, then the girls get how much? ($120). Does
that add up to $210?
What would you try next?
Student Thinking
• Remember that an important part of explicit
instruction is that students also need to
verbalize their thinking.
• “Provide students with opportunities to solve
problems in a group and communicate
problem-solving strategies.”
Example 2
Which is larger, 3/8 or 3/4 ?
How would you help a struggling student with this?
Use fraction circles to represent this problem and find a
solution. Explain your solution to your partner.
Then try these:
1/2 __ 1/8
7/8 __ 3/4
3/4 __ 5/8
(Let the partner explain their thinking on these.)
See the article on fraction representations
Conclusions about explicit teaching
It is appropriate when…
• Some important way of looking at a problem
is not evident in the situation (decomposing
one ten into ten ones)
• A useful representation needs to be presented
(circle fractions; the area model)
Conclusions about explicit teaching
It may be more appropriate to let students
figure things out when…
• The goal is about making connections rather
than becoming proficient with skills
• Remembering requires deep thought (how to
find equivalent fractions)
Example 3
How many eggs are in 15 cartons, if there are 12
in each carton?
What are the students doing in this video?
How did they learn to do this?
Create a similar problem and ask two others to
solve it.
This seems like explicit teaching, but is it?
http://www.khanacademy.org/video/multiplication-6--multiple-digitnumbers?playlist=Arithmetic
Which characteristics does it address, which
does it not address?
Always ask your students to explain how they got their
answer. Knowing this gives you insight into how to help
them move to the next step in their understanding and
skill.
“Guided practice” doesn’t mean that you do the work
for the student, it’s a form of coaching. They are
developing skills and understanding simultaneously;
think of your job as helping establish their
understanding, and their job as developing the skill.
Explicit and Systematic
Operations with fractions packet:
• Equivalent fractions
• Adding and subtracting fractions with the same denominator
• Adding and subtracting fractions with different denominators
(a multiple, not multiples)
• Multiplying a fraction and a whole number by repeated
addition
• Finding a fraction of a whole number
• Multiplying a fraction times a fraction
One More Intervention Strategy:
Fluent retrieval of basic facts
Interventions at all grade levels should devote about
10 minutes in each session to building fluent retrieval
of basic arithmetic facts.
• Provide about 10 minutes per session of instruction to
build quick retrieval of basic arithmetic facts. Consider
using technology, flash cards, and other materials for
extensive practice to facilitate automatic retrieval.
See Math Facts packet
• For students in kindergarten through grade 2,
explicitly teach strategies for efficient counting to
improve the retrieval of mathematics facts.
• Teach students in grades 2 through 8 how to use
their knowledge of properties, such as commutative,
associative, and distributive law, to derive facts in
their heads.
Website Resources
• Nothing Basic about Basic Facts
• Nine Ways to Catch Kids Up