Broadening Our Defini;on for Student Success DeAnn Huinker - - PowerPoint PPT Presentation

Fluency is Not a Simple Idea: Broadening Our Defini;on for Student Success

DeAnn Huinker University of Wisconsin-Milwaukee

Wisconsin Mathema;cs Council Annual Mee;ng Thursday, May 4, 2017, 1:00-2:00 pm Session #179 (Bauer – Morehouse C) Focus: Grades K-5

Session Goal

“To have a conversa;on about fluency and unravel some of its complexi;es.” § What is the meaning of fluency in math? § What are some instruc;onal implica;ons related to fluency?

Fluency . . .

• Fluency in math means . . .
• Concerns I have about what fluency

seems to mean to others . . .

(e.g., parents, students, administrators, teachers) Stop

&

Jot

Study the student work

• samples. Assume this

is typical of the student.

• Is each student

fluent with two-digit addi;on?

• What observa;ons can you make about

each students’ level of fluency?

25 + 37 = ?

Student 1 Student 2 Student 3 Student 4

Na;onal Council of Teachers of Mathema;cs.(2000). Principles and Standards for School Mathema:cs. Reston, VA: Author.

Fluency Having and using efficient and accurate methods for compu;ng.

(NCTM, 2000, p. 32, 79, 84, 144)

Fluency: accuracy is important, but it isn’t enough.

Message #1

What fluency is not!

The Story of Ryan, Grade 4

I know all my mul;plica;on facts. I’m the fastest in my class!

Do I add or subtract? I hate word problems.

I’m good at adding, subtrac;ng, mul;plying, and dividing numbers!

What does it mean to be smart or good in mathema;cs? What would your students say?

Featherstone, H., Crespo, S., Jilk, L. M., Oslund, J. A., Parks, A. N., & Wood, M. B. (2011). Smarter together! Collabora:on and equity in the elementary math classroom. Reston, VA: Na;onal Council of Teachers of Mathema;cs..

It means to be smart and know what 6th graders

• know. It also means to do problems very fast.

You answer problems quickly. You do good on tests.

Grade 3 Students

I think it means that all the math problems just come to your mind no maier how big the problem is, even 50 x 50 = 2500.

Grade 3 Student

“It’s a widespread belief that to be good at math means to be fast at computa;on. But this belief may in fact do more harm as good.” “Overemphasizing fast fact recall at the expense of problem solving and conceptual experiences give students a distorted experience and they ooen turn away from mathema;cs for years, some;mes forever.”

• - Cathy Seeley, NCTM Past-President

Seeley, Cathy L. Faster isn't smarter: messages about math, teaching, and learning in the 21st century: a resource for teachers, leaders, policy makers, and families. Sausalito, CA: Math Solu;ons, 2009. (p. 93, 95)

Ashcrao, M. H. (2002). Math anxiety: Personal, educa;onal, and cogni;ve consequences. Current Direc:ons in Psychological Science, 11(5), 181-185 Ramirez, G., Gunderson, E. A., Levine, S. C., & Beilock, S. L. (2013). Math anxiety, working memory, and math achievement in early elementary

• school. Journal of Cogni:on and Development, 14(2), 187-202..

A rush to fluency undermines students’ confidence and interest in mathema:cs and is considered a cause of mathema:cs anxiety.

(Ashcrao 2002; Ramirez et al., 2013)

Steve Leinwand, American Ins;tutes for Research, Washington, D.C. Sue O’Connell, Quality Teacher Development, Millersville, MD

“Fluency is not instantaneous recall, fluency is not memory. Fluency requires understanding and strategies. And there is a maier of efficiency. Fluency efficiency, not fluency speed.”

• - Steve Leinwand

Fluency ≠ Speed

Message #2

Fluency ≠ Recall

Efficiency . . .

What does Steve mean by: “Fluency efficiency, not fluency speed.”

• -Susan Jo Russell, TERC

Russell, Susan Jo. "Developing computa;onal fluency with whole numbers." Teaching Children Mathema:cs 7, no. 3 (2000): 154-158. (p. 158)

Efficiency implies that the student does not

get bogged down in too many steps or lose track of the logic of the strategy. An efficient strategy is one that the student can carry out easily, keeping track of subproblems and making use of intermediate results to solve the problem.

6 x 12 = ?

Javier, Grade 5

(1:10 minutes)

hip://www.sci.sdsu.edu/CRMSE/sdsu-pdc/nickerson/imap/files/clips/Javier.mov

Efficiency ≠ speed Efficiency à strategies

Message #3 Efficiency involves using strategies based on number and opera;on rela;onships while keeping track of the quan;;es and groups.

Let’s work on our own “efficiency” in reasoning. . .

17 x 5= ?

Mental Math Solve in your head. No pencil or paper!

8 x ? = 336 6 x 98 = ? 24 x 25= ?

Focus: Efficiency

Consider how you used these understandings:

n Composing and decomposing n Place value ideas n Proper;es of the opera;ons n Rela;onships among the opera;ons

17 x 5= ? 8 x ? = 336 6 x 98 = ? 24 x 25= ?

24 x 25 = ?

I thought 24 x 100 = 2400, and 2400 ÷ 4 = 600.

Iden%ty & Inverse Ideas 25 = 4 x 25 ÷ 4 (24 x 100) ÷ 4 = 2400 ÷ 4 = 600

I thought 25 x 25 = 625 and then I subtracted 25. 625 – 25 = 600.

Distribu%ve Property: (25 – 1) x 25 = (25 x 25) – (1 x 25)

I figured that there are 4 twenty-fives in 100, and there are 6 fours in 24, so 6 x 100 = 600.

Associa%ve Property: (6 x 4) x 25) = 6 x (4 x 25)

“I would try to mul;ply in my head, but I can't do that.”

24 x 25 = ?

25 x 4 = 100, 6 x 100 = 600, 600 + 100 = 700. “Got lost!” Well, 10 x 25 = 250, 2(10 x 25) = 500, 500 x 4 = 2000. Trying to use the distribu%ve property.

• 1. Accuracy is important, but not enough.
• 2. Fluency ≠ speed (or recall).
• 3. Efficiency is using strategies based on number and
• pera;on rela;onships while keeping track of

quan;;es and groups (not geyng lost).

• 4. ...

M e s s a g e s

Mathema;cal Proficiency

and the Role of Procedural Fluency

The Strands of Mathema;cal Proficiency

Adap;ve Reasoning Strategic Competence Conceptual Understanding Produc;ve Disposi;on

Na;onal Research Council. (2001). Adding it Up: Helping Children Learn Mathema:cs. Washington, D.C.: Na;onal Academy Press.

Procedural Fluency

The Strands of Mathema;cal Proficiency

Adap;ve Reasoning Strategic Competence Conceptual Understanding Produc;ve Disposi;on Procedural Fluency

See math as sensible, useful, & doable. Pose, represent, and solve problems. Grasp concepts,

• pera;ons, & rela;ons.

Use logic to explain & jus;fy approaches.

Na;onal Research Council. (2001). Adding it Up: Helping Children Learn Mathema:cs. Washington, D.C.: Na;onal Academy Press.

Procedural fluency refers to:

• knowledge of procedures,
• knowledge of when and how to

use them appropriately, and

• skill in performing them flexibly,

accurately, and efficiently. (NRC, 2001, p. 121)

Na;onal Research Council. (2001). Adding it Up: Helping Children Learn Mathema:cs. Washington, D.C.: Na;onal Academy Press.

Procedural fluency refers to:

• knowledge of procedures,
• knowledge of when and how to

use them appropriately, and

• skill in performing them flexibly,

accurately, and efficiently. (NRC, 2001, p. 121)

Na;onal Research Council. (2001). Adding it Up: Helping Children Learn Mathema:cs. Washington, D.C.: Na;onal Academy Press.

• -Susan Jo Russell, TERC

Flexibility requires knowing more than one

approach to solving a par;cular kind of problem. Students need to be flexible to be able to choose an appropriate strategy for the problem at hand and be able to use another method to double-check the results.

Russell, Susan Jo. "Developing computa;onal fluency with whole numbers." Teaching Children Mathema:cs 7, no. 3 (2000): 154-158. (p. 158)

Flexibility à Good Choices

Message #4

Flexibility involves choosing a strategy appropriate for the numbers in the problem and for the opera;on.

Let’s work on our “flexibility” in compu;ng. . .

564 + 9 = ?

Mental Math Solve in your head. No pencil or paper!

2005 – 998 = ? 75 + 78 = ? 72 – 29 = ?

Focus: Flexibility

8 + 9 = ? 2002 – 998 = ? 25 + 37 = ? 72 – 29 = ?

Consider how you used these understandings:

n Composing and decomposing n Place value ideas n Proper;es of the opera;ons n Rela;onships among the opera;ons

72 – 29 = ?

Start at 29, add 1 to get to 30, add 40 to get to 70, add 2 to get to 72, so 1 + 40 + 2 = 43. (Add up in chunks) Subtract 72 – 30, that’s 42. Removed 1 too many, so I put it back on; answer is 43.

(Use a friendly number, then compensate)

I changed the problem to 73 - 30, that equals 43.

(Constant difference)

• 1. Accuracy is important, but not enough.
• 2. Fluency ≠ speed (or recall).
• 3. Efficiency is using strategies based on number and
• pera;on rela;onships while keeping track of

quan;;es and groups (not geyng lost).

• 4. Flexibility involves choosing a strategy appropriate

for the numbers in the problem and the opera;on.

M e s s a g e s

Revisit your thoughts about fluency, add addi;onal thoughts or notes. What adjustments did you make to your ini%al defini%on? Which ideas were affirmed or deepened?

Stop

&

Jot

Developing Fluency

Steve Leinwand, American Ins;tutes for Research, Washington, D.C. Sue O’Connell, Quality Teacher Development, Millersville, MD

Start at 1:13

“Too many ;mes what we‘re seeing is people interpre;ng fluency as the opposite of understanding or strategic understanding. Fluency is built on conceptual understanding and strategic understanding. They’re not two separate things. It’s not about fluency being in and of itself. It comes from that understanding and those strategies.

• - Sue O’Connell

Fluency is built on conceptual understanding.

Message #5

Principles to Ac;ons

Establish math goals to focus learning Implement tasks that promote reasoning & problem solving Use and connect mathema;cal representa;ons Facilitate meaningful mathema;cal discourse Pose purposeful ques;ons Build procedural fluency from conceptual understanding Support produc;ve struggle in learning mathema;cs Elicit & use evidence

• f student thinking

Effec;ve Mathema;cs Teaching Prac;ces

Establish math goals to focus learning Implement tasks that promote reasoning & problem solving Use and connect mathema;cal representa;ons Facilitate meaningful mathema;cal discourse Pose purposeful ques;ons Build procedural fluency from conceptual understanding Support produc;ve struggle in learning mathema;cs Elicit & use evidence

• f student thinking

Effec;ve Mathema;cs Teaching Prac;ces

What does it mean to be fluent?

Principles to Ac:ons (NCTM, 2014, p. 42)

Being fluent means that students are able to choose flexibly among methods and strategies to solve contextual and mathema;cal problems, they understand and are able to explain their approaches, and they are able to produce accurate answers efficiently.

What does it mean to be fluent?

Principles to Ac:ons (NCTM, 2014, p. 42)

• Choose flexibly among methods and strategies.
• Solve contextual and mathema;cal problems.
• Understand and can explain their approaches.
• Produce accurate answers efficiently.

2.NBT.9. Explain why addi;on and subtrac;on strategies work, using place value and the proper;es of opera;ons.

Favorite Standards…

Fluency involves understanding and explaining

• ne’s approaches.

Message #6

Favorite Standards…

3.OA.3. Use mul;plica;on and division within 100 to solve word problems in situa;ons involving equal groups, arrays, and measurement quan;;es, e.g., by using drawings and equa;ons with a symbol for the unknown number to represent the problem.

In Grades K-8, how many standards reference “real-world contexts” or “word problems”?

54 standards

Grade K: OA Grade 1: OA Grade 2: OA, MD Grade 3: OA, MD Grade 4: OA, NF, MD Grade 5: NF, MD, G Grade 6: RP, EE, NS, G Grade 7: RP, EE, NS, G Grade 8: EE, G

24% of K-8 standards

6 x 8 = ?

• Find the answer.
• Explain reasoning.
• Draw a picture.
• Write a story.

Na;onal Council of Teachers of Mathema;cs. (2014). Principles to ac:ons: Ensuring mathema:cal success for all. Reston, VA: Author.

“Being able to translate among all representa;ons, in both direc;ons, is an example of empowering students.”

Fluency includes being able to apply strategies to solve word problems.

Message #7

• 5. Fluency is built on conceptual

understanding.

• 6. Fluency involves understanding and explaining
• ne’s approaches.
• 7. Fluency includes being able to apply strategies

to solve word problems.

• 8. .....

M e s s a g e s

The Common Core and Fluency

In Grades K-8, how many standards reference using “strategies”?

26 standards

Grade K: CC Grade 1: OA, NBT Grade 2: OA, NBT Grade 3: OA, NBT Grade 4: NBT, NF Grade 5: NBT Grade 7: NS, EE

11% of K-8 standards

Standard 1.OA.6: “Basic Facts”

Add and subtract within 20, demonstra;ng fluency for addi;on and subtrac;on within 10. Use strategies such as coun;ng on; making ten; decomposing a number leading to a ten; using the rela;onship between addi;on and subtrac;on; and crea;ng equivalent but easier or known sums.

In Grades K-8, how many standards reference “proper;es of the opera;ons”?

28 standards

Grade 1: OA, NBT Grade 2: NBT Grade 3: OA, NBT Grade 4: NBT, NF Grade 5: NBT Grade 6: NS, EE Grade 7: NS, EE Grade 8: NS

12% of K-8 standards

Standard 3.OA.5: Basic Facts

Apply proper;es of opera;ons as strategies to mul;ply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commuta;ve property of mul;plica;on.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associa;ve property of mul;plica;on.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distribu;ve property.)

Read the excerpt from the Opera;ons and Algebraic Thinking (OA) Progressions Document. What are some salient ideas regarding fluency in the Standards?

The word fluent is used in the Standards to mean “fast and accurate.” However . . . this is not a maker of ins;lling facts divorced from their meanings, but rather as an outcome of a mul;-year process.

Fluency develops over ;me.

Message #8

Math Teaching Prac;ce:

Build Procedural Fluency from Conceptual Understanding

Build procedural fluency from conceptual understanding

Procedural Fluency:

• Builds on a founda;on of conceptual understanding.
• Develops over ;me (months, years), results in known

facts and generalized methods for solving problems.

• Enables students to flexibly choose among methods

to solve contextual and mathema;cal problems.

Baroody, 2006; Fuson & Beckmann, 2012/2013; Fuson, Kalchman, & Bransford, 2005; Russell, 2006

Principles to Ac:ons (NCTM, 2014, p. 47-48)

Recapping the

Fluency Messages

Key Messages

• 1. Accuracy is important, but it’s not enough.
• 2. Fluency ≠ speed (or recall).
• 3. Efficiency is using strategies based on number and
• pera;on rela;onships while keeping track of

quan;;es and groups (not geyng lost).

• 4. Flexibility involves choosing a strategy appropriate

for the numbers in the problem and the opera;on.

Key Messages

• 5. Fluency is built on conceptual understanding.
• 6. Fluency involves understanding and explaining
• ne’s approaches.
• 7. Fluency involves being able to use strategies to

solve word problems.

• 8. Fluency develops over ;me.

Closing Comments. . .

Grade 3 Student

It means you can solve math problems and have strategies to solve harder problems. And you always check your work before handing it in.

Grade 3 Student

It’s not about how fast you work, it means to know a lot of strategies and things that help you.

Maggie, Grade 2

Thank You!

• Dr. DeAnn Huinker

University of Wisconsin–Milwaukee huinker@uwm.edu

www.nctm.org/pta www.nctm.org/ptatoolkit @dh11235