1 a presentation by lee v. stiff north carolina state university first annual title i mathematics...

1

a presentation by

Lee V. StiffNorth Carolina State University

First Annual Title I Mathematics Summit

Fulton County SchoolsAtlanta, GA

There is Nothing More Uncommon

than Common Core

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Common Core State Standards

Mathematical Practices

& Teacher

Behaviors

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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.

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4.Model with mathematics.

5.Use appropriate tools strategically.

6.Attend to precision.

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7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

NCTM Process Standards andthe CCSSM’s Mathematical Practices

NCTM Process Standards

Problem Solving

Reasoning and Proof

Communication

Connections

Representations

CCSSM’s Mathematical Practices

1.Make sense of problems and persevere in solving them.5.Use appropriate tools strategically.

2.Reason abstractly and quantitatively.3.Critique the reasoning of others.8.Look for and express regularity in repeated reasoning.

9.Construct viable arguments

6. Attend to precision.7.Look for and make use of structure.

4. Model with mathematics.

Common Core State Standards for Mathematics

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It Ain’t the Kidz!It Ain’t the Kidz!

It’s us who must It’s us who must

have a have a visionvision of of

high high quality quality

mathematics.mathematics.

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But, But,

what will that what will that

vision vision bebe??

Common Core GPS Mathematics




Problem Solving

Reasoning and Proof

Communication

Connections

Representations







Common Core State Standards for Mathematics

2008 EDSTAR, Raleigh-Durham, N.C.All rights reserved.

WHY DOESN’T THIS MAKE ANY SENSE?

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WHY DOESN’T THIS MAKE ANY SENSE?

So, the kidz ask:

© 2009 EDSTAR Analytics, Inc.

The Lenses of RigorThe Lenses of Rigor

Rigor: What Is It and Why Does It Matter?

Rigor: It Affects Student Performance in Mathematics.

Lessons built on low expectations, skill-building activities

vs.Lessons built on high expectations,

concept-building activities


Teacher beliefs and behaviors affect math performance.

Teacher ExpectationsQuality of Instruction

Rigor


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When Black and White children of comparable ability experience the same instruction, they do about

equally well, and this is true when the instruction is excellent in

quality and when it is not.

(Dreeben, R. (1987). Closing the divide: What teachers and

administrators can do to help Black students reach their potential. American Educator, 11(4), 28-35.)


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Overwhelming evidence suggests that we have greatly underestimated human ability

by holding expectations that are too low for too many children,

and by holding differential expectations where such

differentiation is not necessary.

(Weinstein, R. S. (2002). Reaching higher: The power of expectations in schooling. Cambridge, MA: Harvard University Press.)


According to Webster,

Rigor is strict precision

or exactness.

© 2009 EDSTAR, Inc.


Rigor is having theorems

that follow from axioms by means

of systematic reasoning.

According to mathematicians,


What is rigor What is rigor in in

school mathematics?school mathematics?

In schools,Rigor

is teaching and learning

that is active,

deep, and

engaging.


Active learning involves conversation Active learning involves conversation and hands-on, and hands-on,

minds-on activities.minds-on activities.

Questioning & Questioning & discovery discovery

learning goes on!learning goes on!

© 2009 EDSTAR, Inc.© 2009 EDSTAR, Inc.

Deep learning is focused, attention

given to details and explanations, maybe

project-oriented. Students are really concentrating on

the intricacies of a skill, concept, or

activity.

When learning is engaging, students make a real connection with the content.

There is a feeling that, while learning may be difficult, it is satisfying.


Rigorpromotes

Mathematical Practices.

Mathematical Practices revolve around

lessons that embrace mathematical rigor


Are You Ready for Rigor?

Attainability – are you prepared and equipped?

Sustainability – Is there a plan to maintain? Are there safety nets?

Are You Ready to Implement Mathematical Practices?



Are You Ready to Change Your Behavior?

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Stages of Change Model

maintenancecontemplation

preparation action

relapse

consistent behavior

pre-contemplation

smoking

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Pre-contemplation – Not yet

acknowledging that there is a problem

behavior that needs to be changed.

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Pre-contemplation – Not yet

acknowledging that there is a problem

behavior that needs to be changed.

People at this stage are:unaware,

under-aware, or

in denial!

“It’s the kids, not me!”

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Contemplation – Acknowledging that there is a problem

but not yet ready, or sure of, wanting to make a change.

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Contemplation – Acknowledging that there is a problem

but not yet ready or sure of wanting to make a change.

People at this stage: doubt that the long-term benefits associated with change outweigh the

short-term costs.

“I can retire soon!”

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Preparation – (Determination) Getting ready to

change.

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People at this stage: make a commitment to change. They seek steps or

information for modifying

their behavior.

“What resources are available to me?”

Preparation – Determination;

Getting ready to change.

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Action –

Actually changing behavior.

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Action –

Actually changing behavior.

People at this stage: engage

change behaviors; modify their

environment; seek support from others.

“It’s different, but I can do this!”

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Maintenance –

Maintaining the change in behavior.

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Maintenance –

Maintaining the change in behavior.

People at this stage: value the change behaviors to

avoid a relapse; know that “practice

makes perfect.”

“It’s hard to do, but it’s better!”

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Relapse –

Returning to older behaviors and

abandoning the new changes.

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Relapse –

Returning to older behaviors and

abandoning the new changes.

Relapses are expected. When they

happen, don’t abandon the

desired behaviors; learn from

your mistakes; renew your

commitment.

“I see what happened; let’s do this!”


What can teachers do to bring rigor

into the classroom?

Rubric for RigorActive Check

Includes elements of different concepts or from other disciplines

Employs hands-on and/or minds-on activities

Uses active questioning and verbal interactions that engages students

Creates opportunities to use problem-solving skills and/or discovery learning

Deep CheckReflects on problem-solving situations and skills when they are implemented

Makes connections to previous lessons or lays the foundation for future lessons

Maintains a sharp focus on the lesson objectives

Challenges students to analyze concepts and relationships, not just demonstrate what they know

Engaging CheckMakes connections between the lesson and real-life situations

or other areas of studyDemonstrates the benefit of applying known skills, concepts, and relationships to

new onesHelps students appreciate and seek challenging problem-solving situations

Conveys enthusiasm for the subject

TypicalLesson

RigorousLesson

Scoring: 1-High; 2-Medium; 3-Low


What do teachers frequently do when planning a lesson?

• IDENTIFY the worksheets and other resources they will use

• TALK about what the students cannot do!

Lesson Planning



Problem Solving

Reasoning and Proof

Communication

Connections

Representations








Teaching Mathematical

Practices

Teaching Mathematical Practices

Problem Solving

Understand the Problem

Devise a Plan

Carry Out the Plan

Look Back


Problem Solving


The Focus: Students take their time to comprehend the main idea/question; they think before coming up with a plan or solution.

Problem Solving

Student Actions


Problem Solving


The Focus: Students take their time to comprehend the main idea/question; they think before coming up with a plan or solution.

Problem Solving

Teacher Actions Student Actions

• Model thoughts and actions; question students on vocabulary and key ideas.• Model how to summarize question.• Use questioning skills to focus and guide students’ thinking.• Facilitate the reading of words, graphs, & symbols using strategies such as think-pair-share or small group reading.

• Read problem at least two times.• Explain the problem situation in your own words.• Restate the problem using half as many words.• Demonstrate an understanding of the vocabulary, graphs, & symbols.• Categorize the type of answer.• Identify the key concepts.• Use a graphic organizer.


Problem Solving

Devise a Plan

The Focus: Students determine how the question points to a plan; students make decisions about the steps they will take.

Problem Solving


Problem Solving

Devise a Plan

The Focus: Students determine how the question points to a plan; students make decisions about the steps they will take.

Problem Solving


• Use probing questions: “Have you seen a problem like this before?” “What tools (table, formula, compass, etc.) do you need?”• Discuss possible strategies; show/discuss alternate plans.• Propose a graphic organizer or diagram of problem situation.

• Choose/adapt a strategy/plan.• Identify key information (circling, underlining, highlighting).• Discuss possible steps with others.• Create equation or expression.• Identify a simpler case.• Implement ideas from class notes.• Identify a similar (known) problem.


Problem Solving

The Focus: Students implement the strategy by performing known skills and procedures and applying known concepts.

Problem Solving

Carry Out the Plan


Problem Solving

The Focus: Students implement the strategy by performing known skills and procedures and applying known concepts.

Problem Solving


• Have students check and recheck for understanding.• Examine the different methods used by students to expand the class’ understanding of the problem.• Evaluate the application of students’ plans.• Have students explain their thinking.

• Work the problem using the selected strategy.• Explain the steps in completing the problem.• Discuss the creation of your strategy with others.• Examine the strategies of your classmates.• Provide justifications for steps used in the solution.

Carry Out the Plan


Problem Solving

The Focus: Students should make connections, evaluate the problem solving process; develop critical thinking; and devise alternate solutions.

Problem Solving

Look Back


Problem Solving

The Focus: Students should make connections, evaluate the problem solving process; develop critical thinking; and devise alternate solutions.

Problem Solving


• Provide tools/strategies for checking students’ work.• Use clarification questions to help students make connections.• Ask students to justify their work.• Discuss multiple representations of the problem.• Require students to use proper math language to explain their work.• Provide time for students to reflect.

• Write a complete sentence that answers the question.• Compare/contrast other strategies for solving the problem.• Use examples, graphs, symbols, tables, written/oral explanations to justify your solution.• Demonstrate how you would check your answer.• Revise/edit your solution;

Look Back

Rubric for RigorActive Check

Includes elements of different concepts or from other disciplines

Employs hands-on and/or minds-on activities

Uses active questioning and verbal interactions that engages students

Creates opportunities to use problem-solving skills and/or discovery learning

Deep CheckReflects on problem-solving situations and skills when they are implemented

Makes connections to previous lessons or lays the foundation for future lessons

Maintains a sharp focus on the lesson objectives

Challenges students to analyze concepts and relationships, not just demonstrate what they know

Engaging CheckMakes connections between the lesson and real-life situations

or other areas of studyDemonstrates the benefit of applying known skills, concepts, and relationships to

new onesHelps students appreciate and seek challenging problem-solving situations

Conveys enthusiasm for the subject

TypicalLesson

RigorousLesson

Scoring: 1-High; 2-Medium; 3-Low



into the classroom?


Just remember:Rigor is a process-

not a problem.


This is what rigor looks like:1. Name the polygon.

3. Label the vertices using letters A-F.

2. Describe the polygon using the following terms: congruent, parallel, perpendicular, angle, measure, base, height, sides.

4. Describe the relationship between and .5. Identify congruent sides using the appropriate notation.

6. For each angle, provide an estimate, with justification, of its measure.

AB DEB

C

E

F

D

A


This is what rigor looks like:7. Is this a regular or irregular polygon? Write a descriptive paragraph to support your answer. Include diagrams.

8. Explain a method you would use to find the perimeter of the polygon.

9. Using a ruler, determine the perimeter to the nearest centimeter..

10. Describe a method to find the area. Label your steps in sequential order. Use pictures to describe your steps if you want.


This is what rigor looks like:11. Formulate an expression that represents the area of the polygon.

12. Implement your method to find the area of the polygon.

13. If the lengths of the sides were doubled, predict how the perimeter would be affected.14. If the lengths of the sides were doubled, predict how the area would be affected.15. If the measures of some angles increased, how would the lengths of the sides change? Justify your response.


This is what rigor looks like:16. Measure each angle and find the sum of the angle measures. Compare the sum of the angle measures to the sum of the angle measures in a triangle, a quadrilateral, and a pentagon. What pattern do you notice?

17. If the polygon were the base of a 3-dimensional figure, what type of figure could it be? Explain your answer.

18. If the polygon is the bottom of a hexagonal prism, what would its sides look like?


This is what rigor looks like:19. How many faces, vertices, and edges would the hexagonal prism have? 20. Explain how you could determine the volume of the hexagonal prism. Compare your method to a classmate’s. How are the two methods alike? How are the two methods different?

21. How many lines of symmetry can you draw in the polygon?

22. Name a line segment that shows a line of symmetry.

23. Use mathematical notation to identify parallel sides.


This is what rigor looks like:24. Draw the polygon in Quadrant I of a coordinate plane.

25. Identify the coordinate pairs of each vertex of the polygon.

26. If you translated the polygon 2 units to the right and 3 units down, what would the new coordinate pairs be for each vertex?

27. If you rotate the polygon 90°, in which quadrant would it be located?

28. Draw a 90°rotation.


This is what rigor looks like:29. Reflect the original polygon in Quadrant I over the x-axis. Identify the coordinate pairs of the image polygon.

30. What type of transformation would have occurred if the image of the original polygon in Quadrant I were in Quadrant 3? Illustrate your answer.

31. If the original polygon in Quadrant I were dilated by a scale factor of ½, what would the coordinate pairs of the new polygon be?

32. Draw a similar figure and write a proportion that shows their similarity.


Rigor is an activity: Use writing in math

to support rigor.

Writing in math



into the classroom?


Remember thatrigor is a process,

not a problem.

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Teaching Mathematical PracticesTeaching Mathematical Practices

1 2 3 4

??

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1 2 3 4

??Pos #

# SQs

1 2 3 4 n…

1 3 5 7 ?…


79

1 2 3 n

??

Find the nth term.

…


80

1 2 3 4

??Pos #

# s

1 2 3 4 n…

1 2 3 4 ?…

4 6 8 10# s ?…


81

4??Pos #

# s

1 2 3 4 n…

1 2 3 4 n…

4 6 8 10# s

1 2 3

TotalTotal

…

5 8 11 14 …

2n+2

3n+2


82

0

5

10

15

20

25

1 2 3 4 5 6 7

Pattern Position

Tota

l Num

ber o

f Sha

pes


Lesson Planningfor Mathematical Practices

• Study and analyze the Common Core GPS.• Identify resources aligned with Common Core. • Develop or identify diagnostic, formative, and summative assessments throughout the lesson cycle.• Develop or identify activities/lesson that are rigorous.• Develop or identify questions that are rigorous.• Address learning styles; present lessons in a variety of ways.

Create instructional strategies that will address: 1. common misconceptions, 2. errors, 3. differentiation of instruction, 4. student engagement, 5. reflection opportunities, 6. mathematical communication,7. vocabulary, and8. multiple representations of mathematical concepts.

Lesson Planningfor Mathematical Practices


Create classrooms where students are…Create classrooms where students are…• Talking about mathematicsTalking about mathematics

• Making connectionsMaking connections

• Solving problemsSolving problems

• ReasoningReasoning

Remember…Rigor Promotes

Mathematical Practices!

87

a presentation by

Lee V. StiffNorth Carolina State University

First Annual Title I Mathematics Summit

Fulton County SchoolsAtlanta, GA

There is Nothing More Uncommon

than Common Core

1 a presentation by lee v. stiff north carolina state university first annual title i mathematics...

Documents

common core slide

lenses of rigor slide

mathematical rigor

edstar analytics

school mathematics

repeated reasoning

systematic reasoning

use of structure