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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
2
Common Core State Standards
Mathematical Practices
& Teacher
Behaviors
3
Mathematical Practices
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
Mathematical Practices
4.Model with mathematics.
5.Use appropriate tools strategically.
6.Attend to precision.
5
Mathematical Practices
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
7
8
9
10
11
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.
12
But, But,
what will that what will that
vision vision bebe??
Common Core GPS Mathematics
Common Core GPS Mathematics
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
2008 EDSTAR, Raleigh-Durham, N.C.All rights reserved.
WHY DOESN’T THIS MAKE ANY SENSE?
16
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
© 2009 EDSTAR Analytics, Inc.
Teacher beliefs and behaviors affect math performance.
Teacher ExpectationsQuality of Instruction
Rigor
© 2009 EDSTAR Analytics, Inc.
22
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.)
© 2009 EDSTAR Analytics, Inc.
23
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.)
© 2009 EDSTAR Analytics, Inc.
According to Webster,
Rigor is strict precision
or exactness.
© 2009 EDSTAR, Inc.
© 2009 EDSTAR, Inc.
Rigor is having theorems
that follow from axioms by means
of systematic reasoning.
According to mathematicians,
© 2009 EDSTAR, Inc.
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.
© 2009 EDSTAR, Inc.
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.
© 2009 EDSTAR, Inc.
Rigorpromotes
Mathematical Practices.
Mathematical Practices revolve around
lessons that embrace mathematical rigor
© 2009 EDSTAR Analytics, Inc.
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?
Common Core GPS Mathematics
Common Core GPS Mathematics
Are You Ready to Change Your Behavior?
35
Stages of Change Model
maintenancecontemplation
preparation action
relapse
consistent behavior
pre-contemplation
smoking
36
Stages of Change Model
Pre-contemplation – Not yet
acknowledging that there is a problem
behavior that needs to be changed.
37
Stages of Change Model
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!”
38
Stages of Change Model
Contemplation – Acknowledging that there is a problem
but not yet ready, or sure of, wanting to make a change.
39
Stages of Change Model
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!”
40
Stages of Change Model
Preparation – (Determination) Getting ready to
change.
41
Stages of Change Model
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.
42
Stages of Change Model
Action –
Actually changing behavior.
43
Stages of Change Model
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!”
44
Stages of Change Model
Maintenance –
Maintaining the change in behavior.
45
Stages of Change Model
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!”
46
Stages of Change Model
Relapse –
Returning to older behaviors and
abandoning the new changes.
47
Stages of Change Model
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!”
© 2009 EDSTAR, Inc.
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
© 2009 EDSTAR, Inc.
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
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.
© 2009 EDSTAR, Inc.
Teaching Mathematical
Practices
Teaching Mathematical Practices
Problem Solving
Understand the Problem
Devise a Plan
Carry Out the Plan
Look Back
Teaching Mathematical Practices
Problem Solving
Understand the Problem
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
Teaching Mathematical Practices
Problem Solving
Understand the Problem
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.
Teaching Mathematical Practices
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
Teaching Mathematical Practices
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
Teacher Actions Student Actions
• 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.
Teaching Mathematical Practices
Problem Solving
The Focus: Students implement the strategy by performing known skills and procedures and applying known concepts.
Problem Solving
Carry Out the Plan
Teaching Mathematical Practices
Problem Solving
The Focus: Students implement the strategy by performing known skills and procedures and applying known concepts.
Problem Solving
Teacher Actions Student Actions
• 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
Teaching Mathematical Practices
Problem Solving
The Focus: Students should make connections, evaluate the problem solving process; develop critical thinking; and devise alternate solutions.
Problem Solving
Look Back
Teaching Mathematical Practices
Problem Solving
The Focus: Students should make connections, evaluate the problem solving process; develop critical thinking; and devise alternate solutions.
Problem Solving
Teacher Actions Student Actions
• 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
© 2009 EDSTAR, Inc.
What can teachers do to bring rigor
into the classroom?
© 2009 EDSTAR, Inc.
Just remember:Rigor is a process-
not a problem.
Teaching Mathematical Practices
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
Teaching Mathematical Practices
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.
Teaching Mathematical Practices
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.
Teaching Mathematical Practices
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?
Teaching Mathematical Practices
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.
Teaching Mathematical Practices
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.
Teaching Mathematical Practices
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.
© 2009 EDSTAR, Inc.
Rigor is an activity: Use writing in math
to support rigor.
Writing in math
© 2009 EDSTAR, Inc.
What can teachers do to bring rigor
into the classroom?
© 2009 EDSTAR, Inc.
Remember thatrigor is a process,
not a problem.
77
Teaching Mathematical PracticesTeaching Mathematical Practices
1 2 3 4
??
78
1 2 3 4
??Pos #
# SQs
1 2 3 4 n…
1 3 5 7 ?…
Teaching Mathematical PracticesTeaching Mathematical Practices
79
1 2 3 n
??
Find the nth term.
…
Teaching Mathematical PracticesTeaching Mathematical Practices
80
1 2 3 4
??Pos #
# s
1 2 3 4 n…
1 2 3 4 ?…
4 6 8 10# s ?…
Teaching Mathematical PracticesTeaching Mathematical Practices
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
Teaching Mathematical PracticesTeaching Mathematical Practices
82
0
5
10
15
20
25
1 2 3 4 5 6 7
Pattern Position
Tota
l Num
ber o
f Sha
pes
Teaching Mathematical PracticesTeaching Mathematical Practices
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
© 2009 EDSTAR, Inc.
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