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?

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 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.

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 52 + 26 ? (use mental math)• Fluently adding and subtracting within 100 (2nd grade) and 1000 (3rd grade).

Learning Progression

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 feedback4. 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)

Consider a child in kindergarten who is having difficulty knowing whether 18 is more or less than 15 (for example).

Which has more? Which has less? How many more?

Visual Representations

Visual Representations

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 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 easy 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.

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?

Is the teacher using a form of C-R-A?

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

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.

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?

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

Next Intervention Strategy:

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.

Explicit Instruction

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?

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.”

What are some of your students struggling with?

Move to tables with the same grade. Decide what is difficult right now for your struggling students.How can you use visual representations and explicit instruction to help them understand?

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

Let’s Count to 10On the NCTM Illuminations website

Website Resources

• Nothing Basic about Basic Facts• Nine Ways to Catch Kids Up