Key Strategies for Mathematics Interventions

You have 8 bags of cookies. Each bag has 4 cookies in it 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 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 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.

In Grade 5, instructional time should focus on three critical areas: (1) developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions); (2) extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations; (3) developing understanding of volume.

In Grade 6, instructional time should focus on four critical areas: (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; (4) developing understanding of statistical thinking.

In Grade 7, instructional time should focus on four critical areas: (1) developing understanding of and applying proportional relationships; (2) developing understanding of operations with rational numbers and working with expressions and linear equations; (3) solving problems involving scale drawings and informal geometric constructions, and working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume; (4) drawing inferences about populations based on samples.

In Grade 8, instructional time should focus on three critical areas: (1) formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.

Multiplying and dividing begins with repeated addition: 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. Look at the common core for 3rd-4th: Do you see all of these things there? All of the concepts and skills needed for instruction on the common core standards are not always explicit.

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) Show 5th grade Number Talks video on associative property.

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. Show 5th grade Number Talks video on associative property.

To diagnose Common Core for your grade. Learning progressions across grades. Types of knowledge to guide diagnosis and intervention. Show 5th grade Number Talks video on associative property.

Key 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 What does it mean to focus “intensely”? And “systematically.”

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) Using the “math problems,” draw a picture to represent each one.

Visual Representations What visual representations are often used in 3rd and 4th grade? Area model Fraction models Base ten blocks (also concrete models)

Objects – Pictures – Symbols 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 Using the “math problems,” draw a picture to represent each one. 25

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 Show Mr. Meyer video

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 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 ? Do CRA’s on paper

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) Show Number Talks 2nd grade multi-digit

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 Show Number Talks 2nd grade multi-digit 15

18 candy bars are packed into one box. A school bought 23 boxes 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. Use blank C-R-A form

Next Intervention Strategy: 36

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 See the “math facts” packet for more problem types.

Multiplication and division situations differ only by what part is unknown. Any multiplication problem has a corresponding division problem. 7 ∙ 8 = ___ 7 ∙ ___ = 56 This idea is extremely important for students to understand. 40

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?

6 ∙ 3 = ___ 6 ∙ ___ = 18 ___ ∙ 3 = 18 6 feet 3 times ? 18 feet Kangaroo • Scale factor = Giraffe 6 feet 3 times ? 18 feet 6 ∙ 3 = ___ 6 ∙ ___ = 18 ___ ∙ 3 = 18

Transfer to problems of the same type Length ∙ Width = Area 5 8 ? 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 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. How would you solve this? How would you help a struggling learner to find an answer?

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.”

See the article on fraction representations 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. Use Number Talks video

This seems like explicit teaching, but is it? http://www.khanacademy.org/video/multiplication-6--multiple-digit-numbers?playlist=Arithmetic Which characteristics does it address, which does it not address?

Always ask your students to explain how they got their answer 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