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Key Strategies for Mathematics Interventions

Heather has 8 shells. She finds 5 more shells at the beach. How many shells does she have now? • Solve it. Show all your work. Explain how you

solved it. • Make a drawing that helps solve it. • What kind of problem is this? • Make up another problem with the same

underlying structure.

Heather has 8 shells. She finds some more shells at the beach. Now she has 12 shells. How many shells did she find at the beach? • Solve it. Show all your work. Explain how you

solved it. • Make a drawing that helps solve it. • Is this the same underlying structure as the

first problem?

5 apples cost $2.75. How much do 12 apples cost? • 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


In a bag of 40 M&Ms we found 9 red ones. How many red M&Ms would be in a bag of 100? • 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


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.

Classroom teachers need to know the content standards in detail. Interventionists also need to understand how learning builds within each topic area.

Content Knowledge

• Common Core Standards – Critical areas at each grade (focus on number and operations)

• Common Core Standards – Learning progression framework: Understanding, skillful performance, generalization

Instructional Strategies

Along with in-depth content knowledge, both classroom teachers and interventionists need to be skillful at using proven instructional strategies.

Agenda

1. Review the Common Core Standards for the areas you teach

2. Consider the key research-based instructional strategies as outlined in the IES Practice Guide• 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

Content Knowledge – Standards

In Kindergarten, instructional time should focus on two critical areas: (1) representing and comparing whole numbers,

initially with sets of objects; (2) describing shapes and space. More learning time in Kindergarten should be devoted to number than to other topics.


In Grade 1, instructional time should focus on four critical areas: (1) developing understanding of addition, subtraction, and

strategies for addition and subtraction within 20; (2) developing understanding of whole number

relationships and place value, including grouping in tens and ones;

(3) developing understanding of linear measurement and measuring lengths as iterating length units;

(4) reasoning about attributes of, and composing and decomposing geometric shapes.


In Grade 2, instructional time should focus on four critical areas: (1) extending understanding of base-ten notation; (2) building fluency with addition and subtraction; (3) using standard units of measure; (4) describing and analyzing shapes.

Look through the shortened versions of the standards and group them into main topics. Look for learning progressions within the topics.


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.

Content Knowledge – StandardsIn Grade 4, instructional time should focus on three critical areas: (1) developing understanding and fluency with multi-digit

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.


Look through the shortened versions of the standards and group them into main topics. Look for learning progressions within the topics.

Learning Progression

Adding and subtracting begins with basic understanding of number relationships:• Counting on, counting back• Making five, making ten (knowing

combinations to the anchor numbers of 5 and 10)

Children solve problems with clear underlying structures by using strategies that eventually grow into fluency: • Solving simple joining and separating problems, first

by counting objects, then by using strategies such as counting on, making ten, using doubles, etc., then developing automaticity.

• Writing number sentences to represent problems, including ones with missing addends or subtrahends.

• Fluently adding and subtracting within 5 (kindergarten), 10 (1st grade), 20 (2nd grade).

Adding and subtracting with two or more digits is based on an understanding of place value:• Adding tens and tens and ones and ones.

Children continue to use objects, drawings and strategies (mental math) to solve multi-digit joining, separating and comparing problems as they develop proficiency with the symbolic procedures.• For example, what’s 520 + 215 ? (use mental math)• Fluently adding and subtracting within 100 (2nd

grade) and 1000 (3rd grade).


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

• know multiplication combinations fluently (which may mean some flexible use of derived strategies)

• 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.


• 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.



Types of Knowledge

Understanding concepts

Skillful performance with procedures (fluency)

Generalizations that support further learning

Examples

Understanding what addition and subtraction mean; understanding the concept of place value

Skillful performance of single-digit addition

Generalization of skip counting to multiplication

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)

Take a moment to think about the visual representations of math that are used in your curriculum. Sketch as many as you can think of. These can be drawings made by children or visuals that are used for teaching. Anything that isn’t symbols or words.

Visual Representations

• The point of visual representations is to help students see the underlying concepts.

• A typical learning progression starts with concrete objects, moves into visual representations (pictures), and then generalizes or abstracts the method of the visual representation into symbols.

C – R – A

CRA for decomposing 5

C: How many are in this group? How many in that group? How many are there altogether?

R: How many dots do you see? How many more are needed to make 5?

A: 3 + ___ = 5

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 -3815

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

Symbols:

nlvm.usu.edu

You create the C-R-A

• Your class is having a party. When the party is over, ¾ of one pan of brownies is left over and ½ 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.

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.

Joining and Separating Problems

• Lauren has 3 shells. Her brother gives her 5 more shells. Now how many shells does Lauren have? (joining 3 shells and 5 shells; 3 + 5 = ___)

• Pete has 6 cookies. He eats 3 of them. How many cookies does Pete have then? (separating 3 cookies from 6 cookies; 6 - 3 = ___)

• 8 birds are sitting on a tree. Some more fly up to the tree. Now there are 12 birds in the tree. How many flew up? (joining, where the change is unknown)

Comparing and Part-Whole

• Lauren has 3 shells. Ryan has 8 shells. How many more shells does Ryan have than Lauren?

• 8 boys and 9 girls are playing soccer. How many boys and girls are playing soccer?

• 8 boys and some girls are playing soccer. There are 17 children altogether. How many girls are playing?

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

• Addition and subtraction situations differ only by what part is unknown. Any addition problem has a corresponding subtraction problem.15 + 12 = ___ 15 + ___ = 27

• The same is true for multiplication and division.10 8 = ___∙ 10 ___ = 80∙

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 3 times ?6 feet ? 18 feet? 3 times 18 feet

6 3 = ___∙6 ___ = 18∙

___ 3 = 18∙

Transfer to problems of the same type

Length ∙ Width = Area

5 8 ?5 ? 40? 8 40

Multiplication/division problems

Grouping problemsHow 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 problemsA 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 problemsHow 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 problemsThe 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?

Explicit Instruction

Recommendation 3: 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 thisThe 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?

Step 1: Label an unknown amount

Let b = the amount of money for the boys. Since the girls get twice as much, they get 2b.

Step 2: Write an equation, then solve it.

b + 2b = 210 3b = 210 b = 70. The boys team gets $70. The girls team gets twice that, or $140. To check our thinking, does $70 + $140 = $210 ?

Step 1: Draw a picture to represent what you know.

Step 2: Write an equation, then solve it.b equals 1/3 of $210, b = 1/3 210∙

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 21/2 + 1/4 Create a real-world problem that corresponds to this.Use fraction circles to represent this problem and find a solution. Explain your solution to your partner.What did you learn about equivalent fractions?

1/2 + 1/8 3/8 + 1/4 3/4 + 3/8 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 (the use of base ten blocks; the bar model)

• A heuristic is helpful (Label an unknown, write an equation, solve it)

Conclusions about explicit teaching

It may be more appropriate to let students figure things out when…• Remembering requires deep thought (how to

find equivalent fractions)• The goal is about making connections rather

than becoming proficient with skills

What is NOT explicit teaching? http://www.khanacademy.org/video/multiplication-6--multiple-digit-numbers?playlist=Arithmetic

Which characteristics does it address, which does it not address?

http://www.khanacademy.org/video/multiplication-6--multiple-digit-numbers?playlist=Arithmetic

http://www.khanacademy.org/video/multiplication-6--multiple-digit-numbers?playlist=Arithmetic

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.

Example 3

Lucy has 8 fish. She buys 6 more fish. How many fish does she have then?What are the students doing in this video?

How did they learn to do this?Create a similar problem that can be solved using the ten-frame cards and ask two others to solve them in different ways.

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.

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

Explicit and Systematic

Let’s Count to 10On the NCTM Illuminations website

Learning progressions are systematic

Learning progressions are systematic

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.

Nothing Basic about Basic Facts

Number Talks

A classroom method for developing understanding, skillful performance and generalization

Components of Mathematical Proficiency

• Conceptual Understanding - Comprehension of mathematical concepts, operations, and relations.

• Procedural Fluency - Skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.

• Strategic Competence - Ability to formulate, represent, and solve mathematical problems.

• Adaptive Reasoning - Capacity for logical thought, reflection, explanation, and justification.

• Productive Disposition - Habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

Summarize and Apply

Learning progressionUnderstanding… skillful performance… generalization

Summarize and ApplyContent Area Teaching Methods

Early number sense Ten frame cardsBasic addition and subtraction CGI problems with manipulatives to develop

strategies; number talks

Place value Place value cards used with base 10 blocks

Basic multiplication and division CGI grouping, sharing and measuring problems to develop strategies; arrays leading to area model; number talks

Addition and subtraction with regrouping

C-R-A

Multi-digit multiplication Area model, distributive property, partial productsFraction equivalence, addition and subtraction

Fraction circle manipulatives with C-R-A; scaling up and down

Fraction multiplication Area modelFraction division “Measuring” problems; bar modelIntegers Number line; additive inverse (chip model)Ratios and proportions Scaling up and down

• Nine Ways to Catch Kids Up