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NCTM 2010

Math Fact Fluency: How and Why

We Teach for Flexible ThinkingWe Teach for Flexible Thinking11:00-12:00

Sam Strother

IDMT

Boise State University

samstrother@boisestate.edu

Agenda

• Opening Task

• Fluency History

• Theory Into Practice Overview

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• Theory Into Practice Overview

• Focus on Addition

• Focus on Multiplication

• DMT Research: Multiplication Fact Fluency

© CDMT

Opening Task

• What are some related facts that you could use to help you solve this fact if you either didn’t know the answer or you forgot?

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know the answer or you forgot?

6 x 8

© CDMT

Common Derived Facts Strategies

for 6 x 8

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4

6 x 5 = 30

6 x 3 = 18

6 x 4 = 24

6 x 4 = 24

6 x 6 = 36

6 x 2 = 12

Using 5 Halving and Doubling Using a Square

5 x 8 = 40

1 x 8 = 8

3 x 8 = 24

3 x 8 = 24

6 x 10 = 60

-(6 x 2) = -12

Using 10

2 x 8 = 16

2 x 8 = 16

2 x 8 = 16

Using 2

Take Students’

Ideas Seriously

Press Focus on the

Developing Mathematical Thinking

Project (DMT)

Building

Mathematical

Understanding

Press

Students

Conceptually

Encourage

Multiple

Strategies

Address

Misconceptions

Focus on the

Structure of

the

Mathematics

©CDMT 2008

Fluency through Flexibility

� Fluency and flexibility are closely linked.

� Fluency is apparent when you can solve problems, answer questions, and extend patterns in

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questions, and extend patterns in a quick and efficient way.

� Flexibility is the ability to solve problems in a variety of ways, use information already known to solve unknown problems, and the capability to determine the most efficient method to use when confronted with a challenging problem.

(Fuson, 2003; Star, 2005; Steffe, 1979; Van Amerom, 2003)© CDMT

Fluency History� 1920’s-50’s: Debate between drill and meaningtheorists

� 1960’s-90’s: Research and curriculum development

� Today: Reform and Traditional

Current and past findings are conclusive:

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Current and past findings are conclusive:Drill is extremely limited in terms of developing fluency. Building meaning through mental strategies, practice with using efficient mental strategies, and making connections between various derived fact strategies has consistently shown to increase fluency. More importantly, students using derived fact strategies are able to transfer and retain their knowledge long-term more effectively than students using memorization and drill.

(Baroody, 1985; Brownell, 1935; Dawson & Ruddell, 1955; Fuson, 1992; Henry & Brown, 2008); Thornton, 1978)

© CDMT

Why does drill seem to work for some

students?

• ‘Bonding’ does occur for a small number of facts if they’re repeated often enough.

• Students with strong memories will likely retain

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• Students with strong memories will likely retain some facts through any process. This is much like their ability to remember phone numbers, addresses, schedules, and trivia facts more effectively than other people.

• Many people are actually using derived facts strategies without knowing they are doing so.

(Van de Walle, 2007)© CDMT

Addition Table: Memorization

+ 0 1 2 3 4 5 6 7 8 9 10

0 0 1 2 3 4 5 6 7 8 9 10

1 1 2 3 4 5 6 7 8 9 10 11

2 2 3 4 5 6 7 8 9 10 11 12

•122 ‘different’ facts

•9+6 is unrelated to 6+9

•9+6 is unrelated to 9+5

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3 3 4 5 6 7 8 9 10 11 12 13

4 4 5 6 7 8 9 10 11 12 13 14

5 5 6 7 8 9 10 11 12 13 14 15

6 6 7 8 9 10 11 12 13 14 15 16

7 7 8 9 10 11 12 13 14 15 16 17

8 8 9 10 11 12 13 14 15 16 17 18

9 9 10 11 12 13 14 15 16 17 18 19

10 10 11 12 13 14 15 16 17 18 19 20© CDMT

Addition Table: Flexible Thinking

(Derived Facts Strategies)+ 0 1 2 3 4 5 6 7 8 9 10

0 0 1 2 3 4 5 6 7 8 9 10

1 1 2 3 4 5 6 7 8 9 10 11

2 2 3 4 5 6 7 8 9 10 11 12

•Two common strategies:

Doubles variations (+1/-1)

and Make-10

•9+6 is related to 6+9

•9+6 and 9+5 are related and

10

3 3 4 5 6 7 8 9 10 11 12 13

4 4 5 6 7 8 9 10 11 12 13 14

5 5 6 7 8 9 10 11 12 13 14 15

6 6 7 8 9 10 11 12 13 14 15 16

7 7 8 9 10 11 12 13 14 15 16 17

8 8 9 10 11 12 13 14 15 16 17 18

9 9 10 11 12 13 14 15 16 17 18 19

10 10 11 12 13 14 15 16 17 18 19 20

•9+6 and 9+5 are related and

can be solved using the

same strategy (e.g. Make-10)

•Remaining facts (40) are

generally ‘easy’ facts to

remember (e.g. + 1, + 2)

•Students who learn to

compensate ( e.g. 6+3 is

6+4-1, etc.) further reduce

the number of facts that

need to be memorized in

isolation.

© CDMT

x 1 2 3 4 5 6 7 8 9 10 11 12

1 1 2 3 4 5 6 7 8 9 10 11 12

2 2 4 6 8 10 12 14 16 18 20 22 24

3 3 6 9 12 15 18 21 24 27 30 33 36

4 4 8 12 16 20 24 28 32 36 40 44 48

144 different

facts to be

memorized

Commutative

property is often

‘ignored’

Multiplication Table: Memorization

4 4 8 12 16 20 24 28 32 36 40 44 48

5 5 10 15 20 25 30 35 40 45 50 55 60

6 6 12 18 24 30 36 42 48 54 60 66 72

7 7 14 21 28 35 42 49 56 63 70 77 84

8 8 16 24 32 40 48 56 64 72 80 88 96

9 9 18 27 36 45 54 63 72 81 90 99 108

10 10 20 30 40 50 60 70 80 90 100 110 120

11 11 22 33 44 55 66 77 88 99 110 121 132

12 12 24 36 48 60 72 84 96 108 120 132 144

‘ignored’

© CDMT

x 1 2 3 4 5 6 7 8 9 10 11 12

1 1 2 3 4 5 6 7 8 9 10 11 12

2 2 4 6 8 10 12 14 16 18 20 22 24

3 3 6 9 12 15 18 21 24 27 30 33 36

4 4 8 12 16 20 24 28 32 36 40 44 48

5 5 10 15 20 25 30 35 40 45 50 55 60

Doubles

x 5

x 10

Multiplication Table: Anchor Facts

5 5 10 15 20 25 30 35 40 45 50 55 60

6 6 12 18 24 30 36 42 48 54 60 66 72

7 7 14 21 28 35 42 49 56 63 70 77 84

8 8 16 24 32 40 48 56 64 72 80 88 96

9 9 18 27 36 45 54 63 72 81 90 99 108

10 10 20 30 40 50 60 70 80 90 100 110 120

11 11 22 33 44 55 66 77 88 99 110 121 132

12 12 24 36 48 60 72 84 96 108 120 132 144

x 10

Squares

4 anchor facts (78 facts)

Leaving 66 facts

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x 1 2 3 4 5 6 7 8 9 10 11 12

1 1 2 3 4 5 6 7 8 9 10 11 12

2 2 4 6 8 10 12 14 16 18 20 22 24

3 3 6 9 12 15 18 21 24 27 30 33 36

4 4 8 12 16 20 24 28 32 36 40 44 48

5 5 10 15 20 25 30 35 40 45 50 55 60

Doubles

x 5

+/- (n)

x 10

+/-(n)

Multiplication Table: Derived Facts

5 5 10 15 20 25 30 35 40 45 50 55 60

6 6 12 18 24 30 36 42 48 54 60 66 72

7 7 14 21 28 35 42 49 56 63 70 77 84

8 8 16 24 32 40 48 56 64 72 80 88 96

9 9 18 27 36 45 54 63 72 81 90 99 108

10 10 20 30 40 50 60 70 80 90 100 110 120

11 11 22 33 44 55 66 77 88 99 110 121 132

12 12 24 36 48 60 72 84 96 108 120 132 144

Square

+/-(n)

4 anchor facts with

simple derived facts

strategies

(126 facts)

Leaving 18 facts© CDMT

What about timed tests?• The use of timed tests as fluency assessments is ideal. One timed test every 2-3 weeks should be adequate but many factors may increase/decrease that frequency.

• Using timed tests as instructional tools is at best

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• Using timed tests as instructional tools is at best inefficient, and at worst, damaging to student learning.

• Drill is only advisable after students have developed mental strategies, shared and analyzed their strategies with peers, and practiced those strategies in a variety of settings.

© CDMT

(Baroody, 1985; Brownell, 1935; Dawson & Ruddell, 1955; Fuson, 1992; Henry & Brown, 2008; Van de Walle, 2007; Whalen, 2000)

Fact Fluency: Theory into

Practice

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Practice

© CDMT

Special Education and Math Education

Research: Two Different Perspectives

Special Education Math Education

• Explicit instruction

• Repetition

• Problem-solving approach

• Connections

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• Repetition

• Reducing complexity and streamlining curriculum

• Focus on process and skills

• Connections

• Modifications and accommodations

• Streamlining curriculum

• Focus on concepts and understanding

(Gersten, Fuchs, Williams & Baker, 2001; Baroody & Dowker, 2004; NCTM, 2000; Xin & Jitendra, 1999)

© CDMT

Jerome Bruner’s ideas (1996)

“Of central importance is viewing education as more than curriculum and instructional strategies. Rather, one must consider the broader context in how culture shapes the mind and provides the toolkit by context in how culture shapes the mind and provides the toolkit by which individuals construct worlds and their conceptions of themselves and their powers.”

Representational Progression: How Learners

Understand Formal Symbolism and Abstract Concepts

Enactive: tangible, experiential, “real”

Iconic: direct representations of ‘reality’

Symbolic: formal signs and symbols with culturally mediated meanings

(Bruner, 1966; Driscoll, 2004)

Anchor Facts: Encouraging

Common Strategies

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Addition Subtraction Multiplication Division

•Doubles Variations

6+6 or 7+7 to solve

•Reverse Doubles Variations

2 (Doubles)8x2 to solve 8x358x5 to solve 8x6

•All Multiplication anchors

•Missing factor

© CDMT (Baroody & Dowker, 2004; Driscoll, 2004; Gray & Tall, 1994; Thornton, 1978,)

6+6 or 7+7 to solve 6+7

•Make 10

8+2 to solve 8+4

•Reverse Make 10

8x5 to solve 8x6108x10 to solve 8x11 or 8x9Squares8x8 to solve 8x9 or 7x8Halving and Doubling8x3 to solve 8x6

•Missing factor multiplication

� “How did you know you would __________to find the answer?”

� “Would that same idea work on this new

Questioning to Promote DFS

� “Would that same idea work on this new problem? Why or why not?”

� “Were there any steps to your method that could have been combined to make your method even faster/easier?”

© CDMT (Brendefur & Frykholm, 2000; Kazemi, 1998)

Addition Fluency

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Addition FluencyEncouraging Derived Facts Strategies

© CDMT

SUBITIZING

How many dots do you see?

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Subitizing

• Subitizing

▫ instantly seeing how many.

▫ Latin for ‘suddenly’

©CDMT

▫ Latin for ‘suddenly’

▫ Quick image tasks can be used to imply derived facts strategies

▫ The direct perceptual apprehension of the numerosity of a group.

(Clements, 1999)

1.1

1.2©CDMT

1.3

1.4©CDMT

Games and Activities: Addition Table

Patterns

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“Find all of the doubles.”

+ 0 1 2 3 4 5 6 7 8 9

0 0

1 2

2 4

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2 4

3 6

4 8

5 10

6 12

7 14

8 16

9 18

(Van de Walle, 2007; Wright, 1997)

Games and Activities: Addition Table

Patterns

• “Find what happens when you add 2 to any number.”

• “Add a number to another number that is one

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• “Add a number to another number that is one more.” (e.g. 1+2, 2+3)

• “How many combinations can make 1o.”

• “How many sums are more than 10?”

© CDMT (Van de Walle, 2007; Wright, 1997)

Games and Activities: Number Cards

� “Why do these numbers belong together?”

� “Why is one number circled?”

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13

85

3

7

4

� “Which number is missing?”

� “How can you figure out what it is? Are there other numbers it could be?”

� Have students create their own and test you or a peer.

© CDMT

13

8? 3

?

4

(Van de Walle, 2007; Wright, 1997)

Multiplication Fact Fluency

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Multiplication Fact FluencyTheory Into Practice

© CDMT

Multiplication Fact Fluency Unit5 weeks

• Arrays▫ Tiles, grid paper, and pictures

• Multiplication Tables▫ Investigating patterns▫ Practicing fluency (5 min. to fill in the table)▫ Investigating patterns▫ Practicing fluency (5 min. to fill in the table)▫ Finding troublesome facts▫ Connections to arrays

• Number Talks▫ Problem Strings▫ Derived Facts Practice

• Strategy Cards▫ Like traditional flash cards▫ Use 2-3 related facts

© CDMT

Arrays

41

6

© CDMT

7

6

5 x 6 = 30

2 x 6 = 12

Task 1: Arrays

• On your grid paper, draw a few 7 x 12 arrays.

• Think of some easy, related facts that would help you solve 7 x 12 if you didn’t know the answer.

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you solve 7 x 12 if you didn’t know the answer.

• Use your grid paper arrays to draw what these facts strategies would look like if you ‘sliced’ or ‘split’ or ‘added to’ your arrays.

© CDMT

(Van de Walle, 2007)

Task 2: Multiplication Tables

• You have 2 minutes to fill in as many facts as you can.

• Before you start, try to think of some methods

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• Before you start, try to think of some methods that will help you fill in as many facts as possible in the 2 minutes.

© CDMT

Multiplication Tables and Arrays:

Iconic Connections

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© CDMT

10-Rule• When you multiply a number by 10, the number becomes greater by one place value.

• 4th grade students in the Netherlands spend most of the first semester developing an understanding of the 10-rule. Japanese students investigate a variation of the 10-rule from 3-5th grade.

• How do we address the 10-rule in the U.S.?

• Use the 10-rule (or a variation/extension of the rule) to explain:

5 x 10 14 x 100

7 x 20 23 x 30

2.5 x 20 4.8 ÷ 2.4 150 ÷ .1

(Fuson, 2003; Van Amerom, 2003; Watanabe et al., 2006)©CDMT

Number Talks: Two Variations

• Derived Facts Practice

Students generate various related and useful facts to solve a single fact they are presented. These related facts (strategies) are used to solve a

© CDMT

related facts (strategies) are used to solve a ‘follow-up’ fact.

• Problem Strings

Teachers use a sequence of facts to imply/encourage a desired strategy

6x2 6x5 6x7

Strategy Cards

• Similar to flash cards

• Use these to increase mental fluency and increase the number of related facts students are comfortable using.

© CDMT

comfortable using.

• “What is the fastest, most efficient way to remember the more difficult fact?”

• Two-step solutions are most efficient