instructional shifts and the common core math practices · illustrating the standards for...

1 National Council of Supervisors of Mathematics

Illustrating the Standards for Mathematical Practice: Getting Started with the Practices

Instructional Shifts and the Common Core Math Practices

Ohio Middle Level Association

State Conference February 20, 2014

Jean C. Richardson Math Specialist K-8

Mayfield City School District [email protected]

440-995-7879

mailto:[email protected]



Goal of Presentation

We will engage in a conversation about the

importance of incorporating the eight

mathematical practices into our pedagogy with

the goal of developing mathematically proficient

students.



Turn and Talk

After every segment of the presentation, you will be given a few minutes to turn and talk with a person sitting near you.

The questions to discuss are on p. 2 and 3 of your packet. Be ready to turn back to these pages as the presentation continues.

p. 2-3



The Power of Imagination

http://www.youtube.com/watch?v=ywtLnd3xOVU

Ken Robinson, Ph.D.





Common Core State Standards for Mathematics

“In this changing world, those who understand and can do

mathematics will have significantly enhanced opportunities and

options for shaping their futures. Mathematical competence

opens doors to productive futures. A lack of mathematical

competence keeps those doors closed. All students should have

the opportunity and the support necessary to learn significant

mathematics with depth and understanding.”

NCTM (2000, p.50)



Shifts in Mathematics

Shift 1 Focus Teachers significantly narrow and deepen the scope of how time and energy is spent in the math classroom. They do so in order to focus deeply on only the concepts that are prioritized in the standards.

p. 4




Shift 2 Coherence Principals and teachers carefully connect the learning within and across grades so that students can build new understanding onto foundations built in previous years.

p. 4




Shift 3 Fluency Students are expected to have speed and accuracy with simple calculations; teachers structure class time and/or homework time for students to memorize, through repetition, core functions.

p. 4



Learning Progressions: Fluency (Automaticity)

Grade Level Standard I Can Statement

K K.OA.5 I can fluently add and subtract within 5.

1 1.OA.6 I can fluently add and subtract within 10.

2 2.OA.2 2.NBT.5

I can fluently add and subtract within 20 using mental strategies. Know from memory all sums of two one-digit numbers. I can fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

p. 5



Learning Progressions: Fluency (Automaticity)

Grade Level Standard I Can Statement

3 3.NBT.2 3.OA.7

I can fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. I can fluently multiply and divide within 100. Know from memory all products of two one-digit numbers.

4 4.NBT.4 I can fluently add and subtract multi-digit whole numbers using the standard algorithm.

5 5.NT.5 I can fluently multiply multi-digit whole numbers using the standard algorithm.

6 6.NS.2 I can fluently divide multi-digit numbers using the standard algorithm. p. 5




Shift 4 Deep Understanding

Students deeply understand and can operate easily within a math concept before moving on. They learn more than the trick to get the answer right. They learn the math.

p. 4




Shift 5 Application Students are expected to use math and choose the appropriate concept for application even when they are not prompted to do so.

p. 4




Shift 6 Dual Intensity Students are practicing and understanding. There is more than a balance between these two things in the classroom – both are occurring with intensity.

p. 4



Common Core State Standards

Mathematics Standards for Content



Common Core State Standards

Learning Goal: To examine the standards for Mathematical Practice Pre-Assessment p. 6-8



16

Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

p. 9-11



My Personal Learning Goal

p. 12



Standards for Mathematical Practice

“The Standards for Mathematical

Practice describe varieties of expertise

that mathematics educators at all

levels should seek to develop in their

students. These practices rest on

important “processes and

proficiencies” with longstanding

importance in mathematics

education.” (CCSS, 2010)



19

Underlying Frameworks

5 Process Standards

• Problem Solving

• Reasoning and Proof

• Communication

• Connections

• Representations



20

Underlying Frameworks

5 Proficiency Standards

• Conceptual Understanding

• Procedural Fluency

• Strategic Competence

• Adaptive Reasoning

• Productive Disposition



21

Mathematical Proficiency Conceptual Understanding • comprehension of mathematical concepts, operations, and relations

Procedural Fluency • skill in carrying out procedures flexibly, accurately, efficiently, and

appropriately

Strategic Competence • ability to formulate, represent, and solve mathematical problems

Adaptive Reasoning • capacity for logical thought, reflection, explanation, and justification

Productive Disposition • habitual inclination to see mathematics as sensible, useful, and

worthwhile, coupled with a belief in diligence and one’s own efficacy.



22

Six Components of Mathematics Classrooms

1. Creating an environment that offers all students an equal opportunity to learn

2. Focusing on a balance of conceptual understanding and procedural fluency

3. Ensuring active student engagement in the mathematical practices

4. Using technology to enhance understanding

5. Incorporating multiple assessments aligned with instructional goals and mathematical practices

6. Helping students recognize the power of sound reasoning and mathematical integrity



Turn and Talk

Individually review the Standards for Mathematical Practice revised in student language in your packet.

Then discuss the following question with a partner:

What implications might the Instructional Shifts and the Standards for Mathematical Practice have on the culture of your mathematics classroom?



24

NRC (2001). Adding It Up. Washington, D.C.: National

Academies Press.



25

Problem Solving and Precision

1.Make sense of problems and persevere in solving them.

6.Attend to precision.



26

Problem Solving

1.Make sense of problems and persevere in solving them. Mathematically proficient students:

understand the problem-solving process and how to navigate through the process from start to finish.

have a repertoire of strategies for solving problems and the ability to select a strategy that makes sense for a given problem.

have the disposition to deal with confusion and persevere until a problem is solved.



27

Problem Solving

Make sense of problems and persevere in solving them.

I can make sense of math problems and keep trying even when problems are challenging.



28

Problem-Based or Inquiry Approach

When students explore a problem and the mathematical

ideas are later connected to that experience.

It is through inquiry that

students are activating

their own knowledge and

trying to make new

knowledge (meaning).

This builds conceptual

understanding.



29

A Three-Part Format for Problem-Based Lessons



30

Two Machines, One Job

Ron's Recycle Shop started when Ron bought a used paper-shredding machine. Business was good, so Ron bought a new shredding machine. The old machine could shred a truckload of paper in 4 hours. The new machine could shred the same truckload in only 2 hours. How long will it take to shred a truckload of paper if Ron runs both shredders at the same time?

p. 13



31

Two Machines, One Job

Old Machine New Machine

1 T = 4H 1T = 2H

¼ T = 1H ½ T = 1H

½ + ¼ = ¾ T in 1 hour; therefore, ¼ T = 20 minutes

Therefore, with both shredders running at the same time, Ron could shred a truckload of paper in 1 hour and 20 minutes.

Old ¼ truckload I hour New ½ truckload 1 hour So, ¾ of a truckload is shredded in one hour.

Both If ¾ T = 60 min, then ¼ T= 20 min ¼ + ½ + ¼ = 1



Making Sense

= new problem



Problem Solving from NCTM’s Principles and Standards for School Mathematics

“Problem solving means engaging in a task for which the solution method is not known in advance. In order to find a solution, students must draw on their knowledge, and through this process, they will often develop new mathematical understandings.”

NCTM (2000). Principles and Standards for School Mathematics. Reston, VA: Author. (p. 52)


Illustrating the Standards for Mathematical Practice: Problem Solving Across the Grades

Problem Solving from NCTM’s Principles and Standards for School Mathematics

“Students should have frequent opportunities to formulate, grapple with and solve complex problems that require a significant amount of effort and should be encouraged to reflect on their thinking.”

NCTM (2000). Principles and Standards for School Mathematics. Reston, VA: Author. (p. 52)



Ignition Deep Practice Master Coaching Sweet Spot



36

The Sweet Spot “There is a place, right on the edge of your ability, where you learn best and fastest. It’s called the sweet spot. Here’s how to find it.” D. Coyle, 2012

Zone Sensations Percentage of Successful Attempts

Comfort Zone Ease, effortlessness. You’re working, but not reaching or struggling.

80 % or above

Sweet Spot Frustration, difficulty, alertness to errors. You’re fully engaged in an intense struggle – as if you’re stretching with all your might for a nearly unreachable goal, brushing it with your fingertips, then reaching again.

50-80%

Survival Zone Confusion, desperation. You’re overmatched: scrambling, thrashing, and guessing. You guess right sometimes, but it’s mostly luck.

Below 50%



37

Traditional vs. Rich Problems

Traditional Problem Rich Problem

What is 6 + 4? Ten children went to the movie. How many were girls? How many were boys? Explain your answer. Could there be other answers?

Molly has a quarter, 2 nickels, and a dime. How much money does she have?

Molly has 6 coins in her piggy bank. She has more than 85₵, but less than $1.10. What coins could she have? Explain your answers.

Three children shared a pizza. They each ate the same amount. What fraction did each child eat?

Three children are sharing a pizza. How might they share it? What fraction of the pizza could each child get? Justify your answers.



38

Give It a Try!

Focus on the Question

Dan Meyer

http://www.101qs.com/

p. 14




39

What’s the first question that comes to your mind?



40

What’s the first question that comes to your mind?



41

Open-Ended Questions

to Promote

Problem Solving

Before – During - After

p. 15



42

Precision

6.Attend to precision.

Mathematically proficient students:

calculate accurately and perform math tasks with precision.

communicate precisely.



43

Precision

Attend to precision.

I am accurate when I compute and I am specific when I talk about math ideas.



44

Give It a Try!

Translate the Symbol

Heads Up/Name That Category

Test Analysis



45

Give It a Try!

Translate the Symbol



46



Turn and Talk

Discuss the following question with a partner:

What opportunities do your students currently have to grapple with non-routine complex tasks and to reflect on their thinking and consolidate new mathematical ideas and problem solving solutions? Should a student’s ability to be precise in language and computation be calculated into a child’s grade?

p. 2-3



49

Reasoning and Explaining

2.Reason abstractly and quantitatively.

3.Construct viable arguments and critique the reasoning of others.



50

Reasoning

2.Reason abstractly and quantitatively.


Represent quantities in a variety of ways.

Remove the problem context to solve the problem in an abstract way (equation).

Refer back to the problem context, when needed, to

understand and evaluate the answer.



51

Reasoning

Reason abstractly and quantitatively.

I use numbers and symbols to describe math situations.



What generalization is suggested by these problems?



Adding 1 to a Factor

“The number that is not increased is the number that the answer goes up by.”

“I think that the factor you increase, it goes up by the other factor.”



Reasoning through Representation

Draw a picture for the original equation below; then change it just enough to match the two new equations. Make an array for the original equation below; then change it just enough to match the new equations. Write a story for the original equation below; then change it just enough to match the new equations. Example: Original equation 7 x 5 = 35 New equations 7 x 6 = 42 and 8 x 5 = 40



Frannie’s Story Context

There are 7 jewelry boxes and each box has 5 pieces of jewelry. There are 35 pieces of jewelry altogether.

Jewelry Boxes

7 x 5

Seven boxes with five pieces of jewelry in each box

35 pieces of jewelry

7 x 5 8 x 5 Eight Seven boxes with five pieces of jewelry in each box 35 pieces of jewelry + 5 pieces of jewelry 40 pieces of jewelry

Jewelry Boxes

7 x 5



Jewelry Boxes

7 x 5 7 x 6

six



+ 7 pieces of jewelry


Jewelry Boxes

Explain how the array changes from 7 x 5 to 8 x 5 and from 7 x 5 to 7 x 6.

Making Sense of Multiplication



Contextualize and Decontextualize

100 students and 5 chaperones went on the field trip. Each bus held 35 people. How many buses were needed?

Decontextualize: 105 ÷ 35 = n 105 ÷ 35 = 3 Recontextualize: 3 means 3 buses



Contextualize and Decontextualize

120 students and 5 chaperones went on the field trip. Each bus held 35 people. How many buses were needed?

Decontextualize: 120 ÷ 35 = 3.5714 buses Recontextualize: That means I need one more bus, so the

answer is 4 buses.



64

Give It a Try!

Number Webs

Pinch Cards

Reverse It

Match It

Question It

p. 16-19



65

Explaining

3.Construct viable arguments and critique the reasoning of others.


construct viable arguments, both orally and in writing.

listen to and critique the reasoning of others.



66

Explaining

Construct viable arguments and critique the reasoning of others.

I can justify my strategies and listen to see if other students’ ideas make sense.



67

Give It a Try!

Eliminate It

Which one does not belong with the others?

Tell or show why it does not belong.

3 7

5 8

p. 20



68

Give It a Try!

Eliminate It

Which one does not belong with the others?

Tell or show why it does not belong.

⅓ ¼

⅔ ⅖

p. 20



69

Give It a Try!

Agree or Disagree

9 is an even number.

3.5 feet is more than 42 inches.

15 is a prime number.

A square is a rectangle.

p. 20



Turn and Talk


What opportunities do your students currently have to reason about and explain their mathematical thinking, make claims, and construct viable arguments?

p. 2-3



71

Modeling and Using Tools

4.Model with mathematics.

5.Use appropriate tools strategically.



72

Modeling

4. Model with mathematics


Model math ideas and problems in varied ways.

Analyze models to draw conclusions and solve problems.



73

Modeling

Model with mathematics.

I can make models of math ideas.



An example of Modeling with Mathematics: Lunch Money

Lunches in our school cost $2 each. How much do 2 lunches cost? 3 lunches, 4 lunches . . . 10 lunches? More lunches? Create at least 2 models of this situation. You can choose a physical model, a table, a graph, and/or an equation. Your model should show number of lunches and cost of the lunches. You should be able to use your model to find the cost of a certain number of lunches.



Lunch Money: Student Work

One lunch costs $2.

Number of lunches Number of dollars

1 lunch 2 dollars

2 lunches 4 dollars

3 lunches

4 lunches

5 lunches



Lunch Money: Student Work

One lunch costs $2.



Other Models

Penny Jar: Start with 1 penny in the jar. Add 3 pennies each day.

Staircase Tower: Start with a tower of 1. Add 3 cubes for each new

tower.



Model 3.5

Stephanie used base-ten blocks. Maria shaded 10 x 10 grids. Luci showed 3.5 on a number line.



79

Give It a Try!

Model It

Jack and Jill shared a pizza. Jack ate one third of the pizza and Jill ate one half of the pizza. How much of the pizza was left?



80

Using Tools

5.Use appropriate tools strategically.


decide when to use tools and select appropriate tools.

use tools appropriately and accurately.

use mental math when appropriate



81

Using Tools

Use appropriate tools strategically.

I can decide which math tool to use and I know how to use it correctly.



82

Give It a Try!

In My Head?

734 x 82

63 x 4

930 ÷ 3

1/4 + 2/8



Turn and Talk


What opportunities do your students currently have to share their mathematical thinking by modeling and using tools?

p. 2-3



84

Seeing Structure and Generalizing

7.Look for and make use of structure.

8.Look for and express regularity in repeated reasoning.



85

Seeing Structure

7.Look for and make use of structure.


see the flexibility of numbers

recognize patterns and functions



86

Seeing Structure

Look for and make use of structure.

I can use what I know about numbers, patterns, and properties to help me find answers.



87

The Flexibility of Numbers

There were 1 and ½ cupcakes left on the plate and Liam and Molly decided they would eat them. How much might each person have eaten? Be ready to justify your answers.



Recognizing Patterns and Functions

Maya is using blocks to make a wall grow.



Student L



Student Z



91

Give It a Try!

Pattern Cover-Up

Patterns in the Hundred Chart or Multiplication Chart

Ratio Tables

p. 21-22



92

Give It a Try!

Pattern Cover-Up

9

8 9

8 9 7

10 8 9 7

9

4 9

4 9 19

1.5 4 9 19

p. 21



93

Generalizing

8.Look for and express regularity in repeated reasoning.


notice repetition.

discover shortcuts and generalizations.



94

Generalizing

Look for and express regularity in repeated reasoning.

I notice when things repeatedly happen and try to figure out rules to explain what is happening.



95

Give It a Try!

Organizing and Displaying Data to Discover Rules

Erica was decorating gingerbread men with 2 raisins for eyes. How many raisins will she need to make 6 gingerbread men? Tell how you know.

Alice jumps rope faster than anyone in her class. She can jump 8 times in 4 seconds. How long will it take her to jump 40 times? Justify your answer.



96

Give It a Try!

Organizing and Displaying Data to Discover Rules

1. Show 3 or 4 steps of a pattern.

2. Give students hands-on materials to explore and extend the pattern.

3. Have them tell you what comes next and why.

4. Have them record the data on a table.



Turn and Talk


What opportunities do your students currently have to share their mathematical thinking by seeing structure and generalizing?

p. 2-3



98

NRC (2001). Adding It Up. Washington, D.C.: National

Academies Press.



Next Steps and Resources

How to Start Implementing Mathematical Practice Standards

Posters

Return to the pre-assessment that you completed at the beginning of the day. With a different color pen/pencil, re-evaluate your levels of understanding and reflect upon what you have learned.

p. 24-27

p. 23

p. 6-8



End of Session Review

Review of all eight mathematical practices in chart form.

p. 29-36



ENTHUSIASM/LEARNING

LE

AR

NIN

G

A

Quite A

Bit

Some

Not Much

Nothing

Hated It Don’t like It Okay Liked It Loved It

ENTHUSIASM

p. 28



Bibliography Books:

The Talent Code by Daniel Coyle

The Little Book of Talent by Daniel Coyle

Teaching Student-Centered Mathematics by Van de Walle and Lovin

Principals and Standards for School Mathematics by NCTM

Putting the Practices into Action by O’Connell and SanGiovanni

Websites:

http://www.mathedleadership.org/

http://www.engageny.org/

http://www.marzanoresearch.com/site/

http://www.gatewaytomastery.org/

http://www.everydaymath.com/

http://investigations.terc.edu/


http://www.mathedleadership.org/

http://www.engageny.org/

http://www.marzanoresearch.com/site/

http://www.gatewaytomastery.org/

http://www.everydaymath.com/






Jean C. Richardson Math Specialist K-8

Mayfield City School District

[email protected]

440-995-7879

mailto:[email protected]

instructional shifts and the common core math practices · illustrating the standards for...

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