Fall 2013

Putting the Mathematical

Practices Into Action

Welcome“Who’s in the Room”

Norms

A thoughtful and honest conversation amongst those of us who are engaged in our learning community today…

Session Outcome

• Know what it means to be mathematically proficient in the Standards for Mathematical Practice

• Examine and develop an in-depth understanding of the CCSS for Mathematical Practice

• Explore and practice strategies for implementing the CCSS for Mathematical Practice as a vehicle to develop CCSS content standards

Let’s Do

SomeMath!

Exploring The Mathematical Practices

Crossing The River

Exploring The Mathematical Practices

Thanks to Karen McPhersonBuncombe County, NC

What areThe Standards for

Mathematical Practice?

Putting the Practices into Action

Read “Introduction” pgs. XI-XV

“BLOCK PARTY”

•Pick a quote to record on the index card that describes a mathematically proficient student.

•Mingle around the room sharing your quote illustrating what mathematically proficient students do.

Mathematically proficient students:

• blend content knowledge with the ability to apply content knowledge to solve problems

• communicate math ideas

• justify solutions

• model math concepts

• reason to make sense of mathematics

“Putting the Practices into Action”

“Putting the Practices into Action”

PLC Study Guide p.148

“Reflection and Table Talk”

Exploring Standard 8

Look for and express regularity in repeated

reasoning.

Let’s Do

SomeMath!

Exploring Standard 8

Exploring Standard 8

Hexagon Trains

Train 1 Train 2 Train 3

Thanks to Karen McPhersonBuncombe County, NC

Understanding Standard 8

Why Focus on Repetition?

What Type of Repetition is Necessary?

Understanding Standard 8

When our students recognize and analyze what they are seeing repeatedly, they can

make generalizations which lead to algorithms or formulas that make the

tasks easier.

Understanding Standard 8

Looking for and Express Regularity in Repeated Reasoning

“”

Mathematically Proficient Students:•Notice Repetition.

•Discover Shortcuts and Generalizations.

Developing Standard 8

Suggestions for Looking for and Expressing Regularity in Repeated Reasoning

p. 130

Exploring Standard 7

Look for and make use of structure.

Let’s Do

Some Math!

Exploring Standard 7

Exploring Standard 7

Delivery Trucks

Thanks to Karen McPhersonBuncombe County, NC

Exploring Standard 7

Turn and Talk!

What is structure in mathematics?

Why focus on structure?

Developing Standard 7

Suggestions for Looking for and Making use of Structurep. 117-118

Exploring Standard 6

Attend to precision.

Let’s Do

Some Math!

Exploring Standard 6

Slope, Rate of Change, and Steepness:Do Students Understand These Concepts?

Developing Standard 6

Graph of f(x)

Understanding Standard 6

Why should we expect students to be precise?

How do we expect students to show precision?

Is precision always necessary? Explain.

Understanding Standard 6Precision in communicating mathematics includes…

•Thoroughly describing mathematical ideas

•Explaining how the problem was solved

•Justifying the solution within the context of the problem

•Provide examples as they construct mathematical arguments

Understanding Standard 6

1. What about vocabulary? 2. When and how should the

mathematics vocabulary of a lesson be introduced?

Developing Standard 6

Tips for Writing about Math

1.Explaining a Process (How they solved a problem or performed a complex calculation)

• Steps and Order are Key Elements

2. Justify an Answer (Reasoning of how they arrived at the answer)

• Solution and Data• Thinking that Defends Solution

Developing Standard 6

Attend to Precision

Mathematically Proficient Students•Calculate accurately

•Carefully perform math tasks

•Communicate Precisely

Developing Standard 6

Suggestions for Mathematical Precisionp. 102-103

Exploring Standard 5

Use appropriate tools strategically.

Let’s Do

Some Math!

Exploring Standard 5

Tony is buying a Car….

Calculate and explain which car will cost Tony the least to buy and use.

Exploring Standard 5

Freeclipart.com

Exploring Standard 5How would your students perform these operations? Should they choose a calculator, paper and pencil, or mental math?

Explain.

3.508 x 17.338

13.77 – 12.64

4.5 x 12

20% of 84

40% of 78

Understanding Standard 5

What tools are used when doing mathematics?

How are these tools used to do mathematics?

Are tools always appropriate when doing mathematics? Why?

Understanding Standard 5

Use Appropriate Tools Strategically

“Which tool?”

Mathematically Proficient Students:•Decide when to use tools and select appropriate tools.

•Use tools appropriately and accurately.

Developing Standard 5

Suggestions for Using Appropriate Tools Efficientlyp. 88-89

Exploring Standard 4

Model with mathematics.

Let’s Do

Some Math!

Exploring Standard 4

Shooting Free Throws, Probability, and the

Golden Ratio

Bing.com

Exploring Standard 4

Understanding Standard 4

What are Models?

Why Model?

Exploring Standard 4

Understanding Standard 4

Model with Mathematics“Representing Understanding”

Mathematically Proficient Students:•Model in a variety of methods

•Analyze Models

•Draw Conclusions

•Solve Problems

Developing Standard 4

Suggestions for Modeling Mathematicsp.

Exploring Standard 3

Construct viable arguments and critique the reasoning of

others.

Constructing Arguments…..•Select important data

•Provide solution

•Interpret solution in the context of the problem

•Explain your reasoning

•Use specific examples

•Cite logical reasoning to support ideas

•Defend the process or strategy used

Understanding Standard 3

Let’s Do

Some Math!

Exploring Standard 3

Corey The Camel

Critiquing Arguments…..•Listen to argument

•Evaluate thinking and reasoning

•Agree or disagree

•What should be kept? Why?

•What should be deleted? Why?

•What should be added to improve argument?

•Provide clear, specific reasoning

Understanding Standard 3

Construct an argument….

Perry has 2 cats, Claw and Callie. At the vet last year, Claw weighed 9 lbs. and Callie weighed 14 lbs. At this year’s vet visit, Claw weighed 12 lbs. and Callie weighed 17 lbs. Did Claw or Callie Grow more? Why?

Because……..

Understanding Standard 3

Critique an argument….

Understanding Standard 3

Understanding Standard 3Construct Viable Arguments

and Critique the Reasoning of Others

“Organizing Thoughts and Defending Reasoning”

Mathematically Proficient Students:• Construct viable arguments, both orally and in writing

• Clearly communicate their thinking to others

• Listen to and critique the reasoning of others

1. How does constructing arguments deepen understanding of math concepts?

2. How will you help students construct strong arguments?

3. What are the benefits of students listening to and critiquing each other’s arguments?

4. How will you help students effectively critique mathematical arguments?

Developing Standard 3

Reflection Questions

Student Justification Rubric p. 59

Developing Standard 3

Suggestions for Constructing and Critiquing Argumentsp. 57-58

Exploring Standard 2

Reason abstractly and quantitatively.

Let’s Do

Some Math!

Exploring Standard 2

Exploring Standard 2

Sorting Out Ideas about FunctionsHillen, A.F. & Malik, L (2013)Mathematics Teacher, 106(7), p. 526

Understanding Standard 2

Quantitative Reasoning

vs

Abstract Reasoning

Understanding Standard 2

Reason Abstractly and Quantitatively“Decontextualize and Contextualize”

Mathematically Proficient Students:• Represent quantities in a variety of ways

• Remove the problem context to solve the problem in an abstract way

• Refer back to the problem context to understand and evaluate the solution.

1. Are students able to represent quantitative relationships abstractly? Explain.

2. When creating equations to solve problems, do students struggle? How and why?

3. How does constructing word problems to match a given equation or graph strengthen students’ understanding?

Developing Standard 2

Reflection Questions

Developing Standard 2

Suggestions for Reasoning Abstractly and Quantitatively p. 40

Exploring Standard 1

Make Sense of Problems and Persevering in Solving Them.

Let’s Do

Some Math!

Exploring Standard 1

Exploring Standard 1

Exploring Standard 1Characteristics of Problem Solvers

•Determine and articulate what the problem is asking

•Find a starting point by understanding mathematical situations

•Identify relevant data for solving the problem

•Identify an appropriate way to solve the problem

•Connect problem situations to abstract representations of the problem in order to clarify the task

•Persist through the problem solving process arriving at a solution

•Identify or understand different ways to solve the problem

Exploring Standard 1

Positive Student Dispositions (p. 11)•Believe they can solve the problem

•Recognize confusion is part of the process

•Self-monitor

•Check the reasonableness of approaches

•Modify courses of action

•Display that persistence pays off

Developing Standard 1

Read, “How Do We Get There?”, pgs. 12-13

1.What can we do each day in our classroom to build persevering problem-solvers?

2.How do we encourage our students to seek multiple approaches to problem-solving?

3.How does selecting and sequencing student work foster positive attitudes toward mathematics?

QuestioningFocused

PurposefulGuided

Open-Ended

Bookmark guide

p. 17 “Questions to Guide Student Thinking”

p. 25 “Open-Ended Questions to Promote Problem Solving”

Developing Standard 1

1. Do I routinely provide opportunities for my students to share their solutions and processes with partners, groups, and the whole class?

2. Do I show my students that I not only value process and the correct answer, I also value “what”, and “how” they are thinking because I always ask “why”?

3. Do I pose problems that require perseverance? Do I use thoughtful questions to guide and encourage students as they struggle with problems?

4. Do I routinely think aloud to show students how to change course when needed during the problem solving process?

Developing Standard 1Reflection Questions

Understanding Standard 1Make Sense of Problems

and Persevere in Solving Them

Mathematically Proficient Students:•Understanding of the problem-solving process and how to navigate through it

•A repertoire of problem-solving strategies and the ability to select the appropriate strategy for a given problem

•The disposition to deal with confusion and persevere until a problem is solved.

Developing Standard 1

Suggestions for Developing Problem-Solving Skills pgs. 26-27

“Putting the Practices into Action”

“The Mathematical Practices explore and explain the habits of proficient mathematicians, and when we perform any task, particularly one as complex as math, our actions can not be simplified into compartments. Each standard has a unique focus, but each standard also intermingles with the others as we put it into practice. Rather than trying to compartmentalize these Practices, think about blending the Practices to empower your students to use math and to think mathematically.”

What questions do you have?

www.ncdpi.wikispaces.net

DPI Mathematics Section

Kitty RutherfordElementary Mathematics Consultant919-807-3841kitty.rutherford@dpi.nc.gov

Denise SchulzElementary Mathematics Consultant919-807-3839denise.schulz@dpi.nc.gov

Johannah MaynorSecondary Mathematics Consultant919-807-3842johannah.maynor@dpi.nc.gov

VacantSecondary Mathematics Consultant919-807-3934

Dr. Jennifer CurtisK – 12 Mathematics Section Chief919-807-3838Jennifer.curtis@dpi.nc.gov

Susan HartMathematics Program Assistant919-807-3846susan.hart@dpi.nc.gov