Key Strategies for Mathematics Interventions

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?

Let b = the amount of money for the boys. Since the girls get twice as much, they get 2b. b + 2b = 210

Step 1: Label an unknown amount

Step 2: Write an equation, then solve it.

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∙

Example 2

Leroy paid a total of $23.95 for a pair of pants. That included the sales tax of 6%. What was the price of the pants before the sales tax?

• What can you explain about your own thinking that would help a struggling learner?

• What methods can you teach explicitly that a student might not figure out on their own?

Label a variable: Let c = cost of the pants.

Understand that 6% is not of the total cost, but 6% of the cost of the pants: 6% of c (.06) c∙Write an equation: The cost of the pants c plus the sales tax (.06) c∙ equals the TOTAL COST _________________

This is where your professional judgment comes in. If you tell the student what equation to write, they’ll come to depend on you to always tell them.

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 3

1/2 + 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.)

Example 4

3/4 ÷ 1/8Work this problem any way you can, then compare your answers at your table. Discuss your method for solving.What problems do students typically have with your method? Create a real-world situation that is represented by this.

Division by a fraction has a specific meaning, which most people need to be taught explicitly. It is the “measurement” definition of division: How many 1/8’s are there in 3/4? (or how many times does 1/8 go into 3/4?)(Recommendation 4, about the underlying structure of word problems, ties in here.)

A visual representation is the best way to teach about the concept of division by a fraction:

Try making visual representations for ⅘ ÷ ⅕ ⅘ ÷ ⅖ ⅔ ÷ ⅙ 1½ ÷ ¼ ¾ ÷ ⅜ ⅛ ÷ ¼

Conclusions about explicit teaching

It is appropriate when…• Some important way of looking at a problem is

not evident in the situation (what division by a fraction means)

• A useful representation needs to be presented (the bar model)

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

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?

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

Example 5

Lucy has 8 fish. She buys 5 more fish. How many fish does she have then?What are the students doing in these video clips?How did they learn to do this?Practice with each other using the ten-frame cards.

Explicit and Systematic

• Work through the section on multiplication of the Origo Math student workbook. In what ways does this represent a systematic approach?

Recommendation 2

Instructional materials for students receiving interventions should focus intensely on in-depth treatment of whole numbers in kindergarten through grade 5 and on rational numbers in grades 4 through 8. These materials should be selected by committee.

What would this include?

5th Gr. Common Core Standards

• Summarize the whole number and rational number goals for 5th grade.

Recommendation 4

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∙

• You’re driving on a vacation. You drive at 50 mph for 7 hours. How far have you driven?

• You drive 300 miles in 6 hours. How fast were you driving, on average? (how many miles do you go in each hour)

• How long does it take you to drive 400 miles at 50 mph?

• You’re driving on a vacation. You drive at 50 mph for 7 hours. How far have you driven?

Counting out 7 groups of 50: d = 50 · 7 This procedure generates the formula d = r · t

• You drive 300 miles in 6 hours. How fast were you driving, on average? (how many miles do you go in each hour)

Dividing 300 into 6 groups (partitive division): r = 300/6

• How long does it take you to drive 400 miles at 50 mph?

Counting how many 50’s in 400 (measurement division). 400/50 = t

Distance = Rate ∙ Time

50 mph 7 hrs300 miles = x 6 hrs400 miles = 50 mph x

Transfer to problems of the same type

Area = Length ∙ Width

5 840 540 8

Visual Representations

• The point of visual representations is to help students see the underlying concepts. - Draw a picture for each of the problems.

• 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 to doing math using pictures, and then 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?

37 + ___ = 53

53 ducks are swimming on a pond. 27 ducks fly away. How many ducks are left on the pond? First, try this with mental math.Next, model it with unifix cubes. Then use symbols to record what we did. 4 13 53 -2726

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:

• 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.

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.