What is FLUENCY? - · PDF filewhat is fluency? collaborate to craft a working definition of mathematical fluency. write your definition on a blank sheet to share with the whole

What is FLUENCY?

COLLABORATE TO CRAFT A WORKING DEFINITION OF

MATHEMATICAL FLUENCY. WRITE YOUR DEFINITION

ON A BLANK SHEET TO SHARE WITH THE WHOLE

GROUP.

A BALANCED APPROACH TO

ASSESSING FACT FLUENCY IN GR. 1-2

DA NIELLE PA LM

KRISTINE GETTELMAN

M ILWA UKEE PUBLIC SCHOOLS

Going Beyond Fast Facts

Compare to Reading…

Think about how you assess a student’s reading ability:

• Do you time them to see how many words they can read correctly in a specified amount of time?

• Do you listen and observe as they read?

• Do you ask questions to see if they understand what they’re reading?

Compare to Reading…

Imagine only using timed tests to assess reading ability…

• Do you time them to see how many words they can read correctly in a specified amount of time?

• Do you listen and observe as they read?

• Do you ask questions to see if they understand what they’re reading?

• Would you be confident in your assessment?

Conceptual Understanding of Basic Facts

hair sat chair with red girl curly a in little the brown

I am going to give you 10 seconds to memorize a list of 12 words.

Write down as many words as you can remember.

I am going to give you 10 seconds to memorize a list of 12 words.

Write down as many words as you can remember.

A girl with curly red hair sat in the little brown chair.

Conceptual Understanding of Basic Facts

Components of Mathematical Fluency

Accuracy

Efficiency

Flexibility

Accuracy Efficiency

correct-ness

concern about double-checking results

can easily & quickly carry out a strategy

doesn’t get “lost” in too many steps or lose track of the logic of the strategy


Flexible Thinking comfortable with more

than one approach

use knowledge of number relationships and operations to solve unknown problems

higher levels of thinking (appropriate to age & grade)


Flexible Thinking comfortable with more

than one approach

use knowledge of number relationships and operations to solve unknown problems

higher levels of thinking (appropriate to age & grade)


"Math fact fluency leads to higher order mathematics.

Through automaticity students free up their working memory and can devote it to problem solving and learning new

concepts and skills (Geary, 1994).”

Why focus on fluency?

Common Core State Standards—Mathematics

Instructional Shift:

Rigor Conceptual understanding

Procedural skill and fluency

Application of skills in problem solving situations

Common Core State Standards—Mathematics

Required Fluencies:

How can we get information about students’ mathematical fluency?

Assessing Fluency: A Case Study

Timed Tests

Using the timed test, what can you tell about this student’s fluency?

What are the next instructional steps for this student based on this assessment?

Journaling

o Meaningful writing prompts

o Assess flexibility & understanding of strategy selection

o Engage students in self-reflection and self-monitoring

Using the journal responses, what can you tell about this student’s fluency?


Interviews o Opportunity to listen

to children explain what they know

o Teacher is able to ask follow-up and clarifying questions

o As a formative assessment, can inform instruction for individual or class

o Formal (1:1 interviews) or informal (during instruction)











Using the fluency interview portfolio, what can you tell about this student’s fluency?


Observation

o Focused observations of students at work provide insights into students’ strategies

o Can be documented on a clipboard as students work

Using observational data, what can you tell about students’ fluency?

Student 1 more Combinations

of 10 Doubles

Applies

commutativity

Sam + +

Elijah + + +

Summer +

WHAT ARE THE ADVANTAGES AND DISADVANTAGES OF EACH OF THE ASSESSMENT METHODS

THAT WE STUDIED TODAY?

Debrief

Instructional Implications

Fluency is an endpoint

Balanced Assessment

Purposeful Instruction

Consistent Practice Opportunities

The Impact of Instruction

Considerations…

Think about your current assessments. Is there a balance of each of the components—accuracy, efficiency, and flexibility? If not, how might you alter your assessment practices to have a better balance?

As you reflect on your students’ fact fluency, what would you like to know more about? Which of the tools discussed today might help you gain this information?

Discuss your thoughts on the issue of timed tests. What might you do as a teacher to avoid potential negative impacts with your students?

How might we help parents better understand the four components of fluency?

References:

Thank you for attending our session!

Common Core Standards Writing Team (2011). Progressions for the Common Core State Standards in Mathematics, K-2 Operations and Algebraic Thinking. Tucson, AZ: Institute for Mathematics and Education, University of Arizona.

Geary, D.C. (1994). Children's mathematical development: Research and practical applications. Washington, DC: American Psychological Association.

Gojak, L. (2012). Fluency: Simply fast and accurate? I think not! NCTM Summing Up, November 1, 2012.

Kling, G. & Bay-Williams, J.M. (2014). Assessing basic fact fluency. Teaching Children Mathematics, 20(8), p. 489-497.

O’Connell, S. & SanGiovanni, J. (2011). Mastering the basic math facts in addition and subtraction. Portsmouth, NH: Heinemann.

Assessing Fluency Case Study

Type of

Assessment

Using the assessment, what can you tell about the

student’s fluency ability in the following categories:

Next Instructional Steps Accuracy

Correct answer Efficiency Solves quickly

Flexibility and

Appropriateness Uses relationships and

properties

Timed

Assessment

Student

interview

portfolio

Journal

Prompt

Created by Danielle Palm. Not for distribution or reproduction.

Student Name: _______________________________ Date: ______________

K5 Fluency Interview: +1

Strategy

Level 3:

Recompose or Convert to an

Easier Equivalent Problem. Level 2:

Count on/back Level 1: Count all

2 + 1 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

1 + 1 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

4 + 1 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

1 + 3 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

1 + □ = 5 Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

3 + □ = 4 Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

K5 Fluency Interview: -1

Strategy

Level 3:




5 - 1 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

3 - 2 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

4 - 1 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

4 - 3 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

3 - □ = 2 Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

2 - □ = 1 Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

□ Continue to work on fluency

flexibility using missing addends and

subtrahends

Fluency Codes R:c R:√ EE:c EE:√ CO:c CO:√ T:c T:√ CO-F:c CO-F:√ CA-F:c CA-F:√

Recall Easier

Equivalent

Count On Used Tool Count on with

fingers

Count all with

fingers

Strategy Level 3 Level 2 Level 1

Demonstrates fluency within 10 using this strategy:

Su

btr

ac

t

+/-

1

+/-

2

+/-

0

Do

ub

les

Ma

ke

te

n

Use

Do

ub

les

Use

te

n

Ad

d




Strategy

Level 3:




2 + 1 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

0 + 2 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

3 + 2 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

2 + 2 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

2 + □ = 5 Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

2 + □ = 3 Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:


Strategy

Level 3:




5 - 2 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

3 - 2 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

4 - 2 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

5 - 3 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

3 - □ = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

2 - □ = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:



subtrahends


Recall Easier

Equivalent


fingers

Count all with

fingers



Su

btr

ac

t

+/-

1

+/-

2

+/-

0

Do

ub

les

Ma

ke

te

n

Use

Do

ub

les

Use

te

n

Ad

d




2nd Grade Fluency Interview: +1 / -1

Strategy

Level 3:




6 + 1 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

4 + 1 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

15 + 1 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

20 – 1 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

8 – 1 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

18 - 1 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

1 + 18 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

6 - 1 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

1 + □ = 12 Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

□ + 7 = 8 Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

□ – 1 = 6 Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

16 – □ = 15 Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:


Recall Easier

Equivalent


fingers

Count all with

fingers


□ Continue to work on fluency flexibility using missing addends and subtrahends


Su

btr

ac

t

With

in

20

+/-

1

+/-

2

+/-

0

Do

ub

les

+/-

10

Ma

ke

te

n

Use

Do

ub

les

Use

te

n

Exte

nd

ed

Fa

cts

With

in

10

Ad

d W

ith

in

20

With

in

10



2nd Grade Fluency Interview: +2 / -2

Strategy

Level 3:




8 + 2 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

7 - 2 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

2 + 5 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

8 - 6 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

11 + 2 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

13 – 11 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

2 + 18 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

11 – 2 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

7 + □ =9 Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

12 + □ = 14 Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

□ – 4 = 2 Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

□ – 2 = 14 Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:


Recall Easier

Equivalent


fingers

Count all with

fingers




Su

btr

ac

t

With

in

20

+/-

1

+/-

2

+/-

0

Do

ub

les

+/-

10

Ma

ke

te

n

Use

Do

ub

les

Use

te

n

Exte

nd

ed

Fa

cts

With

in

10

Ad

d W

ith

in

20

With

in

10



2nd Grade Fluency Interview: Derived Facts Using Making Ten

Strategy

Level 3:




6 + 3 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

4 + 8 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

5 + 11 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

9 – 6 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

12 + 7 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

17 – 5 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

11 - 6 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

16 – 12 = Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

9 + □ = 15 Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

11 + □ = 18 Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

13 - □ = 9 Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

□ – 7 = 12 Recall

Make Ten

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:


Recall Easier

Equivalent


fingers

Count all with

fingers




Su

btr

ac

t

With

in

20

+/-

1

+/-

2

+/-

0

Do

ub

les

+/-

10

Ma

ke

te

n

Use

Do

ub

les

Use

te

n

Exte

nd

ed

Fa

cts

With

in

10

Ad

d W

ith

in

20

With

in

10



3rd Grade Fluency Interview: X1 / ÷1

Strategy

Level 3:

Recompose or Convert to an Easier Equivalent

Problem. Level 2: Skip Count

Level 1: Direct Model

(Draw a picture, array, fingers,

hundred chart, etc.) 6 X 1 = Recall Strategy: Mentally Fingers Tool:

4 X 1 = Recall Strategy: Mentally Fingers Tool:

8 ÷ 1 = Recall Strategy: Mentally Fingers Tool:


□ X 7 = 7 Recall Strategy: Mentally Fingers Tool:

□ ÷ 1 = 6 Recall Strategy: Mentally Fingers Tool:

3rd Grade Fluency Interview: X 2 / ÷2

Strategy

Level 3:









7 X □ =14 Recall Strategy: Mentally Fingers Tool:

□ ÷ 4 = 2 Recall Strategy: Mentally Fingers Tool:

Demonstrates Multiplication/Division fluency using this strategy:

Fo

un

da

tio

na

l

Fa

cts

X1

÷1

X2

÷2

X5

÷5

X10

÷10

X0

÷0

De

rive

d

Fa

cts

X3

÷3

X4

÷4

X9

÷9

X6

÷6

X7

÷7

X8

÷8

Fluency Codes R:c R:√ EE:c EE:√ SC:c SC:√ SC-F:c SC-F:√ T:c T:√ P:c P:√

□ Continue to work on fluency flexibility using missing factors and dividends




Strategy

Level 3:









5 X □ = 20 Recall Strategy: Mentally Fingers Tool:

40 ÷ □ = 8 Recall Strategy: Mentally Fingers Tool:


Strategy

Level 3:


Problem. Level 2:






3 X □ = 30 Recall Strategy: Mentally Fingers Tool:

20 ÷ □ = 10 Recall Strategy: Mentally Fingers Tool:


Fo

un

da

tio

na

l

Fa

cts

X1

÷1

X2

÷2

X5

÷5

X10

÷10

X0

÷0

De

rive

d

Fa

cts

X3

÷3

X4

÷4

X9

÷9

X6

÷6

X7

÷7

X8

÷8


□ Continue to work on fluency flexibility using missing factors and dividend


3rd Grade Mathematical Fluency Portfolio

Student Name:

Date:


Fo

un

da

tio

na

l

Fa

cts

X1

÷1

X2

÷2

X5

÷5

X10

÷10

X0

÷0

De

rive

d

Fa

cts

X3

÷3

X4

÷4

X9

÷9

X6

÷6

X7

÷7

X8

÷8


Date:


Fo

un

da

tio

na

l

Fa

cts

X1

÷1

X2

÷2

X5

÷5

X10

÷10

X0

÷0

De

rive

d

Fa

cts

X3

÷3

X4

÷4

X9

÷9

X6

÷6

X7

÷7

X8

÷8


Date:


Fo

un

da

tio

na

l

Fa

cts

X1

÷1

X2

÷2

X5

÷5

X10

÷10

X0

÷0

De

rive

d

Fa

cts

X3

÷3

X4

÷4

X9

÷9

X6

÷6

X7

÷7

X8

÷8






Strategy

Level 3:




2 + 1 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

1 + 1 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

4 + 1 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

1 + 3 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

1 + □ = 5 Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

3 + □ = 4 Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:


Strategy

Level 3:




5 - 1 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

3 - 2 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

4 - 1 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

4 - 3 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

3 - □ = 2 Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

2 - □ = 1 Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:



subtrahends


Recall Easier

Equivalent


fingers

Count all with

fingers



Su

btr

ac

t

+/-

1

+/-

2

+/-

0

Do

ub

les

Ma

ke

te

n

Use

Do

ub

les

Use

te

n

Ad

d




Strategy

Level 3:




2 + 1 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

0 + 2 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

3 + 2 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

2 + 2 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

2 + □ = 5 Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

2 + □ = 3 Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:


Strategy

Level 3:




5 - 2 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

3 - 2 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

4 - 2 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

5 - 3 = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

3 - □ = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:

2 - □ = Recall

Doubles

Used Doubles

Mentally

Fingers

Tool:

Mentally

Fingers

Tool:



subtrahends


Recall Easier

Equivalent


fingers

Count all with

fingers



Su

btr

ac

t

+/-

1

+/-

2

+/-

0

Do

ub

les

Ma

ke

te

n

Use

Do

ub

les

Use

te

n

Ad

d


Monitoring 1st Grade Fluency


Recall Easier

Equivalent


fingers

Count all with fingers

Strategy

Progression

Level 3 Level 2 Level 1


September

□ Continue to work on fluency flexibility using

missing addends and subtrahends

October



November/December



January/February



March/April



May/June




Su

btr

ac

t

+/-

1

+/-

2

+/-

0

Do

ub

les

Ma

ke

te

n

Use

Do

ub

les

Use

te

n

Ad

d


Su

btr

ac

t

+/-

1

+/-

2

+/-

0

Do

ub

les

Ma

ke

te

n

Use

Do

ub

les

Use

te

n

Ad

d


Su

btr

ac

t

+/-

1

+/-

2

+/-

0

Do

ub

les

Ma

ke

te

n

Use

Do

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les

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Monitoring 1st Grade Fluency


Recall Easier

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fingers

Count all with fingers

Strategy

Progression

Level 3 Level 2 Level 1


Fluency: Simply Fast and Accurate? I Think Not!

By NCTM President Linda M. Gojak NCTM Summing Up, November 1, 2012

As mathematics educators at all levels consider effective implementation and instruction related to state or Common Core standards, a frequently asked question is, “What does it mean to be fluent in mathematics?” The answer, more often than not, is, “Fast and accurate.” Building fluency should involve more than speed and accuracy. It must reach beyond procedures and computation.

Principles and Standards for School Mathematics states, “Computational fluency refers to having efficient and accurate methods for computing. Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently. The computational methods that a student uses should be based on mathematical ideas that the student understands well, including the structure of the base-ten number system, properties of multiplication and division, and number relationships” (p. 152). What a wonderful description of fluency! It reminds us that a student cannot be fluent without conceptual understanding and flexible thinking.

Focusing on efficiency rather than speed means valuing students’ ability to use strategic thinking to carry out a computation without being hindered by many unnecessary or confusing steps in the solution process. Accuracy extends beyond just getting the correct answer. It involves considering the meaning of an operation, recording work carefully, and asking oneself whether the solution is reasonable.

Fluency encompasses more than memorizing facts and procedures. In fact, I believe memorization is one of the least effective ways to reach fluency. Anyone who has spent time teaching in the elementary grades realizes how many students are unsuccessful at rote memorization and how often they revert to counting on their fingers. We would agree that third or fourth graders who are counting on their fingers certainly have not reached a level of fluency, even though they may do it pretty quickly and accurately!

How do we help students progress from the early stages of counting to mathematical fluency? Let me give you a personal example. At the beginning of the school year, I gave a class of third-grade students a sheet with 10 addition facts. Under each fact was the word “explain,” followed by a line. I asked one of the students the sum of the first fact, 8 + 9, and she immediately began to count on her fingers—certainly not the action of a student who is fluent with addition facts. Before she reached the sum I asked her, “What do you know that would help you find the sum of 8 and 9?” She thought for a brief time and replied, “Oh, it’s 17.” When I asked her how she had gotten that without counting, she looked at me and said, “I just took 1 off the 8 and gave it to the 9. That made it 7 + 10. That’s easy—it’s 17.”

One might argue that child was not fluent. I believe, however, that she demonstrated fluency and more. She was able to use her understanding of place value, addition, and the associative property to arrive at a correct response. She was efficient, accurate, and flexible in her thinking—all in a matter of seconds. What made the difference between her fumbling first attempt and her successful second one? It was being provided with the chance to stop and think about what she already knew and apply that understanding to 8 + 9.

Do we give students the opportunity to think about what they know and understand and use it in ways that make sense to them? Do we model questions that students should be asking themselves as they strive to reach fluency in mathematics? As the student completed that assignment, she didn’t need much more prompting. She continued to work on the rest of the facts efficiently and flexibly. She no longer needed to count on her fingers to complete the assignment.

It is interesting to note that fluency isn’t mentioned in the high school Common Core Standards. The standards for grades K–8 refer to fluency in relation to mastery of basic facts and computational skills. As we think about fluency, we should realize that it is more than procedural. Are there mathematical topics in which we want students’ thinking to be flexible, efficient, and accurate beyond computation and procedures? Can a student reach fluency in areas of geometric thinking, algebraic thinking, statistical reasoning, or measurement? What does geometric fluency look like? What are the characteristics of a student who is fluent in algebra? What areas of fluency in the K–12 curriculum reach beyond procedures and calculations but are not mentioned in the standards?

Our students enter school with the misconception that the goal in math is to do it fast and get it right. Do we promote that thinking in our teaching without realizing it? Do we praise students who get the right answer quickly? Do we become impatient with students who need a little more time to think? As we strive for a balance between conceptual understanding and procedural skill with mathematical practices, we must remember that there is a very strong link between the two. Our planning, our instruction, and our assessments must build on and value that connection. Fluency entails so much more than being fast and accurate!

488 April2014•teaching children mathematics | Vol. 20, No. 8 www.nctm.org

Copyright © 2014 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

www.nctm.org Vol. 20, No. 8 | teaching children mathematics • April 2014 489

Assessing

Basic FactFluency

By Gina Kling and Jennifer M. Bay-Williams

Think about how you assess reading fl uency. Does your assessment plan involve listen-ing and observing as children read as well as asking reading comprehension questions? Now imagine what you might learn about

students’ reading fl uency if you used only timed quizzes. How would your confi dence in your assessment change?

Formative assessments—including observations, interviews, performance tasks, and journaling—have become common practice in many classrooms, with a recognition that by using different ways to assess chil-dren, we gain a more comprehensive, accurate picture of what they know, what they do not know, and their misconceptions. These data are then used to design instruction accordingly (Wiliam 2011). Yet, in spite of this trend in other areas of education, timed, skill-based assessments continue to be the prevalent measure of basic mathematics facts achievement. As a result, many rich opportunities for assessing basic fact fl uency are lost. In this article, we share a variety of ways to forma-tively assess basic fact fl uency. We defi ne fl uency, raise some issues related to timed testing, and then share a collection of classroom-tested ideas for authentic fact fl uency assessment.

Have you had it with timed tests, which present a number of concerns and limitations?

Try a variety of alternative assessments from this sampling that allows teachers to accurately and appropriately measure children’s fact fl uency.

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Defining fluencyA variety of interpretations exist for what proce-dural fluency (in general) and basic fact fluency (specifically) mean. Fortunately, recent stan-dards, research, and reports provide a unified vision of what these terms signify. The Common Core State Standards for Mathematics (CCSSM) document describes procedural fluency as “skill in carrying out procedures flexibly, accurately, efficiently, and appropriately” (CCSSI 2010, p. 6). Likewise, Baroody (2006) describes basic fact fluency as “the efficient, appropriate, and flexible application of single-digit calculation skills and . . . an essential aspect of mathemati-cal proficiency” (p. 22). These definitions reflect what has been described for years in research and standards documents (e.g., NCTM 2000, 2006; NRC 2001) as well as CCSSM grade-level expectations related to basic facts (see table 1).

Notice that the CCSSM expectations use two key phrases; the first is to fluently add and sub-tract (or multiply and divide), and the second is to know from memory all sums (products) of two one-digit numbers. To assess basic fact fluency, all four tenets of fluency (flexibility, appropri-ate strategy use, efficiency, and accuracy) must

be addressed. Additionally, assessments must provide data on which facts students know from memory. Timed tests are commonly used to supply such data—but how effective are they in doing so?

Limitations and risks of timed mathematics testsTimed tests offer little insight about how flexible students are in their use of strategies or even which strategies a student selects. And evidence suggests that efficiency and accuracy may actu-ally be negatively influenced by timed testing. A study of nearly 300 first graders found a nega-tive correlation between timed testing and fact retrieval and number sense (Henry and Brown 2008). Children who were frequently exposed to timed testing demonstrated lower progress toward knowing facts from memory than their counterparts who had not experienced as many timed tests. In fact, growing evidence suggests that timed testing has a negative impact on students (Boaler 2012, Henry and Brown 2008, Ramirez et al. 2013). Surprisingly, the anxiety that many children experience over timed testing is unrelated to how well they do on the tests. Even high-achieving children share such concerns as, “I feel nervous. I know my facts, but this just scares me” (Boaler 2012). Math anxiety appears as early as first grade, and this anxiety does not correlate with reading achievement (Ramirez et al. 2013). In other words, children’s anxiety is specific to mathematics, not general academic work. And the struggling learner is not the only one who experiences anxiety: Ramirez and his colleagues found that children demonstrating a higher use of “working memory” (i.e., those who tended to use mathematical strategies that were more sophisticated) experienced the most negative impact on achievement as a result of math anxiety. Thus, it appears that some of our best mathematical thinkers are often those most negatively influenced by timed testing.

Fortunately, children can learn facts effec-tively without the use of timed testing. In a lon-gitudinal study of twenty second graders, Kling found that without any timed testing or other rote fact activities, by the end of the year, the children demonstrated automaticity with addi-tion facts (solved within 3 seconds) 95 percent of the time. Interestingly, the children performed strategies (e.g., making ten or near doubles) so

Past mathematics documents—as well as current standards, studies, and reports—offer a unified vision of what procedural fluency means.

CCSSM standards that address fluency and mastery with basic facts (italics added)

Kindergarten K.OA.A.5

Fluently add and subtract within 5

Grade 1 1.OA.C.6

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making 10; decomposing a number leading to a 10; using the relationship between addition and subtraction; and creating equivalent but easier or known sums

Grade 2 2.OA.B.2

Fluently add and subtract within 20 using mental strategies (refer to 1.OA.6). By the end of grade 2, know from memory all sums of 2 one-digit numbers

Grade 3 3.OA.C.7

Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 / 5 = 8) or properties of operations. By the end of grade 3, know from memory all products of 2 one-digit numbers.

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quickly that it was impossible to distinguish between highly efficient strategy application and “knowing from memory.” Since the begin-ning of first grade, fact practice for these chil-dren had involved (a) activities within textbook lessons (b) weekly fact games and (c) activities such as Quick Images with ten frames that were used to foster discussion around strategies (see Kling 2011, Bay-Williams and Kling in press). This research suggests that timed assessments and drill may not be necessary for children to achieve fact mastery.

If timed mathematics assessments have questionable value and potentially negative psychological, emotional, and educational impact, why are they still so frequently used? We commonly hear three reasons. First, fluency is interpreted as synonymous with speed. We have already addressed that fluency is more comprehensive than speed. Second, some feel that timed tests prepare children for high-stakes tests. The research shared here con-vincingly shows it may do the opposite. Third, timed tests are the only assessments widely available for assessing fluency of basic facts. As we seek to rectify this last concern, the remain-der of this article shares methods for more comprehensively and appropriately assessing students’ basic fact fluency.

Using formative assessment strategiesWith an eye on the aspects of fluency (accuracy, efficiency, flexibility, and appropriate strategy selection), we can use various assessment strat-egies to see what students know (and do not know) and determine what our next instruc-tional steps might be. All are approaches we have used with children in grades 1–4, and when used in combination with one another, these methods provide a comprehensive picture of a student’s level of fact mastery.

1. InterviewsInterviews provide the extraordinary opportu-nity to hear children explain what they know about a topic in a discussion format, during which teachers can ask follow-up and clarifying questions (Hodges, Rose, and Hicks 2012; Van de Walle et al. 2014). The insights gained from listening to a child can be invaluable for plan-ning individualized instruction or interven-

tions. And insights from a selection of students can inform instruction for the whole class. Con-sider which aspects of fluency to address using the questions posed in each of the sample inter-view protocols (see fig. 1). We see the interview as a quick way to get at all four categories of fluency (as well as to see if a student just knows

Below are protocols for student interviews, which are a way to quickly assess all four categories of fluency and see if a student just knows a fact. Insights from a selection of interviews can inform instruction for the whole class.

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Protocol A. Assess fluency

1. Write 4 × 5 on a card [point at card]. What does 4 × 5 mean?

2. What is the answer to 4 × 5?

3. How did you find the answer to 4 × 5? Could you find it another way?

4. If your friend was having trouble remembering this fact, what strategy might you suggest to him or her?

Protocol B. Assess flexibility and strategy selection

1. What is 8 + 5?

2. How can you use 8 + 2 to help you solve 8 + 5?

OR

1. How can you use 3 × 7 to solve 6 × 7?

Protocol C. Assess use of appropriate strategy (adapted from Henry and Brown 2008)

ProbesWhat is 7 + 8?How did you figure it out? [Ask regardless of how quickly or accurately they solve the fact.]

Codes R = Recall A = Automatic (within 3 seconds) M10 = Making 10 Strategy ND = Near Doubles Strategy D = Some other derived fact strategy CO = Counting on CA = Counting all MCA = Modeling and counting all


the fact). Accuracy is assessed as soon as the stu-dent responds, and efficiency is observed on the basis of how long it takes a student to solve the fact. Flexibility and appropriate strategy selec-tion are addressed by such follow-up prompts as, “How did you figure it out?” or “How could you use this strategy to solve this fact?” Codes, such as those suggested in Protocol C, can facili-tate recording during an interview (see fig. 1).

Interviews need not be one-on-one, sit-down events. They can be quick exchanges in the midst of other activities. For example, as stu-dents are lining up, ask, “Aaron, what is six times seven? How did you figure it out?” Furthermore, interviews have an added benefit of allow-ing students the opportunity to self-correct. For example, during interview assessments with thirty-eight beginning first graders, Kling

found that 54 percent of the time children self-corrected incorrect answers as they explained how they figured out the fact. Furthermore, the children’s self-reported strategies (i.e., “I counted on my fingers” or “I just knew it”) were consistent with what the interviewer was able to observe 97 percent of the time. This suggests the potential of interviews as highly reliable and informative assessment tools.

2. ObservationsObservation is a natural part of teaching, and recognizing which strategies students know can supply valuable insights to help support students as they learn new strategies and tackle unknown facts. To create organized and accu-rate records of observations, a list of students and facts can be attached to a clipboard (see

Codes can facilitate recording during an interview.

(a) Use an accuracy table to review students’ progression with addition facts.

Within 5 Foundational facts Within 10 Within 20

Name/facts 0, 1, 2 Combinations that make 10 Doubles

Nicholas

Kayla

Cynthia

Robbie

(b) A table can show the frequency of addition fact strategy use at a glance.

Name/strategies 1 more/ 2 more

Combinations that make 10 Making 10 Doubles Find 5s Applies commutativity

Nicholas + + + +

Kayla + + +

Cynthia + + +

Robbie + + + +

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table 2a); or a list can be tracked on the basis of which strategies students use (see table 2b). Equipped with these charts, you can observe as students engage in facts games, such as math-ematized versions of classic games of War, Go Fish, Concentration, Old Maid, and Memory (for other games to teach and review basic facts, see Forbringer and Fahsl 2010; Kamii and Anderson 2003; Van de Walle, Karp, and Bay-Williams 2013; Kling 2011; Bay-Williams and Kling, in press; Kling and Bay-Williams, in press).

A critical aspect of meaningful use of games is to ask students to tell their teammates both the answer and how they found it. As students turn over cards, observe to see and hear how efficient each student is as well as whether he or she chose an appropriate strategy or if they just knew. The teacher might observe, for example, that many students are more efficient at solv-ing 5 + 3 than they are at 3 + 8. These students may “just know” facts within ten but may apply strategies for the facts that have sums over ten. Such insights gained through observation can help the teacher select appropriate activities for continued learning and practice.

To enhance opportunities for assessing stu-dents during game play, consider having groups rotate through centers, stationing the teacher at one center and using probing questions, such as “How did you figure that out?” or “Are there any other ways you could figure it out?” One first-grade teacher had the following to say after using an observational checklist to formatively assess her students:

This is an important tool that provides a more comprehensive check of which specific strat-egies a student has successfully mastered toward developing fluency with their basic facts. CCSSM provide specific strategies that students are expected to understand and use, and the chart provides me the opportunity to

learn which strategies are being used effec-tively and where there are opportunities for further instruction and practice.

3. JournalingWriting provides an excellent opportunity to assess flexibility and understanding of strategy selection and application. Children at any grade level can find ways to incorporate pictures,

Various responses to a journal prompt illustrate the strategies that first graders used and reveal which children were able to appropriately select and explain an efficient strategy for the task.

If your friend did not know the answer to 4 + 5, how could he figure it out?

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As students turn over cards, observe to see and hear how efficient each student is as well as whether he or she chose an appro-priate strategy or just knew.


words, and numbers to communicate their strategies. For example, figure 2 shows a variety of first graders’ responses to the journal prompt, “If your friend did not know the answer to 4 + 5, how could he figure it out?” Carefully review the responses, considering what they illustrate about the strategies used by the children. In contrast to what can be learned from a child’s answer to 4 + 5 on a timed test, these samples offer rich opportunities to recognize which children can appropriately select and explain an efficient strategy for the task. This is impor-tant for deepening strategy understanding and also is reflected in the expectations of CCSSM

and the related, forthcoming assessments. For example, Smarter Balanced Assessment Con-sortium (SBAC) lists the following as “evidence required” for grade 3. Note the application of strategies inherent in these expectations. The student—

• multiplies and divides facts accurately ;• multiplies and divides facts using strategies,

such as the relationship between multiplica-tion and division or properties of operations; and

• uses multiplication and division facts (SBAC 2012) (emphasis added).

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of strategy selection and application.

Children at any

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This collection of prompts addresses the four components of fluency with basic facts. Writing about their strategies on a weekly basis engages students in self-reflection and monitoring, as well as emphasizes the importance of strategies in practicing basic facts.

Writing prompts for developing fluency with the basic facts

Appropriate strategy selection Flexibility

• Explain how to use the “count on” strategy for 3 + 9.

• What strategy did you use to solve 6 + 8?

• A friend is having trouble with some of his times 6 facts. What strategy might you teach him?

• Emily solved 6 + 8 by changing it in her mind to 4 + 10. What did she do? Is this a good strategy? Tell why or why not.

• How can you use 7 × 10 to find the answer to 7 × 9?

• Solve 6 × 7 using one strategy. Now try solving it using a different strategy.

• Emily solved 6 + 8 by changing it in her mind to 4 + 10. What did she do? Does this strategy always work?

Efficiency Accuracy

• What strategy did you use to solve 9 + 3?

• How can you use 7 × 7 to solve 7 × 8?• Which facts do you “just know”? For which

facts do you use a strategy?

• Crystal explains that 6 + 7 is 12. Is she correct? Explain how you know.

• What is the answer to 7 × 8? How do you know it is correct (how might you check it)?

Creative writing ideas that address several components

• Develop a “Face the facts” or “Ask Cougar” column (like Dear Abby) for the class. (Pick a fun name for the column that makes sense for the class, such as the school mascot.) Students send a letter about a tough fact. Rotate different students into the role of responder. The responder writes letters back, suggesting a strategy for the tough fact.

• Create a strategy rhyme (e.g., If times four is giving me trouble, I’ll remember to double and double).

• Make a facts survival guide. Children prepare pages illustrating with visuals (e.g., ten frames or arrays) of how find “tough” facts.

• Write a yearbook entry to some facts (e.g., Dear 8 × 7, I …)

(See McIntosh 1997 for many more ideas).


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e 3 Quizzes that focus on fluency are alternatives to timed tests.

(a) Quiz questions can be used to see if students “just know” foundational facts.

Solve these problems and tell how you solved them.

4 × 5 = ______ Check one: ______ I used this strategy: ________________________ ______ I just knew.10 × 6 = ______ Check one: ______ I used this strategy: ________________________ ______ I just knew.6 × 2 = ______ Check one: ______ I used this strategy: ________________________ ______ I just knew.5 × 3 = ______ Check one: ______ I used this strategy: ________________________ ______ I just knew.2 × 9 = ______ Check one: ______ I used this strategy: ________________________ ______ I just knew.3 × 10 = ______ Check one: ______ I used this strategy: ________________________ ______ I just knew.5 × 7 = ______ Check one: ______ I used this strategy: ________________________ ______ I just knew.8 × 10 = ______ Check one: ______ I used this strategy: ________________________ ______ I just knew.

(b) A quiz assesses if students recognize the commutativity of addition for one-more-than facts. Notice that these examples are shorter, not timed, and also focus on strategies. On completion, say to class, “Circle the row that was easier for you to solve. If they were both the same, write ‘same’.“

Solve these addition problems.

ROW A: 9 + 1 = 8+ 1 5 + 1 = 3 + 1 =

6+ 1

ROW B: 1 + 8 = 1 + 7 = 1+ 4

1+ 2 1 + 9 =

Meaningful writing tasks can be used across grade levels and operations. Table 3 presents a collection of writing prompts that address the four components of fluency. Having an oppor-tunity to write about strategies on a weekly basis engages students in self-reflection and self-monitoring as well as emphasizes the impor-tance of strategies in practicing basic facts.

4. QuizzesYou may be surprised to see this section, given the major concerns raised earlier related to timed tests, but quizzes can be used effectively to assess efficiency as well as strategy use. Ensure that students “just know” their foundational facts before moving on to derived facts. Foundational facts are so named because they can be used

to generate all the other facts using a strategy. Foundational facts in addition include one- and two-more-than, combinations that make ten, and doubles. For multiplication, they include × 1, × 2, × 5, and × 10. (See Kling and Bay-Williams, in press, for a discussion of foundational facts.) From these facts, we can derive all other facts. Quiz questions (see fig. 3a) can be used to see if students “just know” foundational facts.

Similarly, quizzes can be used to monitor facts that come more easily to students. For example, a quiz (see fig. 3b) assesses if students are recognizing the commutativity of addi-tion for one-more-than facts. Notice that these examples are shorter, not timed, and also focus on strategies. The following adaptations can enhance the effectiveness of facts quizzes:


• Choose one of the problems above and write about how you solved it.

• Tell which helper fact you used the most on this quiz.

• Circle facts you “just knew.” Highlight those for which you used a strategy.

• Circle facts you are sure about. Draw a square around facts that took you longer to solve.

Meaningful fact assessment for teachers and studentsWe recognize that using timed tests is a deeply rooted practice for measuring basic fact mas-tery. We hope that we have effectively made a case for why this practice must change and how to make such a change. As the NCTM Assessment Principle states, “Assessment should support the learning of important mathematics and furnish useful informa-tion to both teachers and students” (NCTM 2000, p. 11). Using the range of assessments described above accomplishes these goals, as they provide an opportunity for meaningful, targeted feedback to students that far exceeds the “right or wrong, fast or slow” feedback pro-vided by timed testing. In fact, these assess-ments infuse a fifth and critical category of assessment: self-assessment. Interviews, jour-nals, and quizzes on basic facts can and should encourage students to reflect on which facts and strategies they know well and which ones are tough for them. This self-assessment can be effectively followed up by having children identify and record strategies that could be used to efficiently determine the “tough” facts in the future. Over time, this self-assessment practice encourages children to instinctively apply effective strategies for challenging facts they encounter. As both teachers and students critique their growth with use of appropriate strategies, efficiency, flexibility, and accuracy, then true fluency with basic facts can become a reality for every child.

RefeRencesBaroody, Arthur J. 2006. “Why Children Have

Difficulties Mastering the Basic Number Com-binations and How to Help Them.” Teaching Children Mathematics 13 (August): 22–31.

Bay-Williams, Jennifer M., and Gina Kling. In press. “Enriching Addition and Subtraction

➺ reflect and discuss

“assessing basic Fact Fluency”Reflective teaching is a process of self-observation and self-evaluation. It means looking at your classroom practice, thinking about what you do and why you do it, and then evaluating whether it works. By collecting information about what goes on in our classrooms and then analyzing and evaluating this information, we identify and explore our own practices and underlying beliefs.

The following questions, related to “Assessing Basic Fact Fluency” by Gina Kling and Jennifer M. Bay-Williams, are suggested prompts to aid you in reflecting on the article and on how the authors’ ideas might benefit your own classroom practice. Consider the article independently and then discuss it with your colleagues.

Think of the basic facts assessments that you are currently using with respect to the following:

• Flexibility• Efficiency• Appropriate strategy use• Accuracy

1. With your current assessments, what percentage of emphasis might you assign to each of the four categories above? Is this balance what you would like it to be? If not, how might you alter your assessments to equitably address the four areas of fluency?

2. As you reflect on your students’ basic facts fluency, what would you like to know more about? Which of the assessment tools from the article might help you gain this knowledge? How might you use that assessment tool?

3. Discuss your reactions to the issue of timed tests. What might you do as a teacher or leader to avoid potential negative impacts of timed tests?

4. How might we help parents better understand fluency and help their children in the areas of flexibility and selecting appropriate strategies? How might you communicate the purpose of alternative assessment tools for basic facts with your students, parents, and school leadership?

Tell us how you used Reflect and Discuss as part of your professional development. The Editorial Panel appreciates the interest and values the views of those who take the time to send us their comments. Letters may be submitted to Teaching Children Mathematics at [email protected]. Please include Readers Exchange in the subject line. Because of space limitations, letters and rejoinders from authors beyond the 250-word limit are subject to abridgment. Letters are also edited for style and content.

Adopt more flexibility and variety in how you assess students’ basic fact fluency.

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Fact Mastery through Games.” Teaching Children Mathematics.

Boaler, Jo. 2012. “Timed Tests and the Development of Math Anxiety.” Education Week. Online July 3, 2012.

Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards .org/assets/CCSSI_Math%20Standards.pdf

Forbringer, Linda, and Allison J. Fahsl. 2010. “Differentiating Practice to Help Students Master Basic Facts.” In Responding to Diversity: Grades Pre-K–5, edited by Dorothy Y. White and Julie S. Spitzer, pp. 7–22. Reston, VA: National Council of Teachers of Mathematics.

Henry, Valerie J., and Richard S. Brown. 2008. “First-Grade Basic Facts: An Investigation into Teaching and Learning of an Accelerated, High-Demand Memorization Standard.” Journal for Research in Mathematics Education 39 (March): 153–83.

Hodges, Thomas E., Terry D. Rose, and April D. Hicks. 2012. “Interviews as RtI Tools.” Teaching Children Mathematics 19 (August): 30–36.

Kamii, Constance, and Catherine Anderson. 2003. “Multiplication Games: How We Made and Used Them.” Teaching Children Mathematics 10 (November): 135–41.

Kling, Gina. 2011. “Fluency with Basic Addition.” Teaching Children Mathematics 18 (September): 80–88.

Kling, Gina, and Jennifer M. Bay-Williams. In press. “Three Steps to Mastering Multiplication Facts.” Teaching Children Mathematics.

McIntosh, Margaret E. 1997. “500+ Writing Formats.” Mathematics Teaching in the Middle School 2 (March/April): 354–58.

National Council of Teachers of Mathematics (NCTM). 2000. Principles and Standards for School Mathematics. Reston, VA: NCTM.

———. 2006. Curriculum Focal Points for Pre-kindergarten through Grade 8: A Quest for Coherence. Reston, VA: NCTM.

National Research Council (NRC). 2001. Adding It Up: Helping Children Learn Mathematics. Edited by Jeremy Kilpatrick, Jane Swafford,

and Brad Findell. Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academies Press.

Ramirez, Gerardo, Elizabeth A. Gunderson, Susan C. Levine, and Sian L. Beilock. 2013. “Math Anxiety, Working Memory, and Math Achieve-ment in Early Elementary School.” Journal of Cognition and Development 14 (2): 187–202.

SBAC (Smarter Balanced Assessment Consortium). 2012. Smarter Balanced Assessments. http://www.smarterbalanced .org/smarter-balanced-assessments/#item

Van de Walle, John. A., Karen S. Karp, and Jennifer M. Bay-Williams. 2013. Elementary and Middle School Mathematics: Teaching Developmentally. Professional Development Edition. New York: Pearson.

Van de Walle, John A., Lou Ann H. Lovin, Karen S. Karp, and Jennifer M. Bay-Williams. 2014. Teaching Student Centered Mathematics: Grades K–2. 2nd edition. New York: Pearson.

Wiliam, Dylan. 2011. Embedded Formative Assess-ment. Bloomington, IN: Solution Tree Press.

Gina Kling, gina.garza [email protected], of Western Michigan Uni- versity and Jennifer M. Bay-Williams,

[email protected], of the University of Louisville, share enthusiasm for helping children develop fluency with their basic facts. The authors thank Meaghan Gorzenski of St. Monica School and Vicky Kudwa of St. Augustine Cathedral School in Kalamazoo, Michigan, and their first-grade students for their contributions to this article.

Readers may be interested in “Research Suggests That Timed Tests Cause Math Anxiety” (p. 469), in which Jo Boaler makes a case for alternative assessment strategies.

Download a free app for your smartphone. Then scan this code to access quizzes and observation checklists appended to this article at www.nctm.org/tcm060.

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What is FLUENCY? - · PDF filewhat is fluency? collaborate to craft a working definition of mathematical fluency. write your definition on a blank sheet to share with the whole