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1 a presentation by Lee V. Stiff North Carolina State University First Annual Title I Mathematics Summit Fulton County Schools Atlanta, GA There is Nothing More Uncommon than Common Core

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2 Common Core State Standards Mathematical Practices & Teacher Behaviors

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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.

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4 Mathematical Practices 4.Model with mathematics. 5.Use appropriate tools strategically. 6.Attend to precision.

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5 Mathematical Practices 7.Look for and make use of structure. 8.Look for and express regularity in repeated reasoning.

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NCTM Process Standards and the CCSSMs Mathematical Practices NCTM Process Standards Problem Solving Reasoning and Proof Communication Connections Representations CCSSMs 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. 3.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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11 It Aint the Kidz! Its us who must have a vision of high quality mathematics.

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12 But, what will that vision be ?

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Common Core GPS Mathematics Common Core GPS Mathematics

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NCTM Process Standards and the CCSSMs Mathematical Practices NCTM Process Standards Problem Solving Reasoning and Proof Communication Connections Representations CCSSMs 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. 3.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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2008 EDSTAR, Raleigh-Durham, N.C. All rights reserved. © 2009 EDSTAR, Inc. WHY DOESNT THIS MAKE ANY SENSE?

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16 WHY DOESNT THIS MAKE ANY SENSE? So, the kidz ask:

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© 2009 EDSTAR Analytics, Inc. The Lenses of Rigor

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Rigor: What Is It and Why Does It Matter?

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Rigor: It Affects Student Performance in Mathematics.

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Lessons built on low expectations, skill-building activities vs. Lessons built on high expectations, concept-building activities © 2009 EDSTAR Analytics, Inc.

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Teacher beliefs and behaviors affect math performance. Teacher Expectations Quality of Instruction Rigor © 2009 EDSTAR Analytics, Inc.

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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), ) © 2009 EDSTAR Analytics, Inc.

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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.

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According to Webster, Rigor is strict precision or exactness. © 2009 EDSTAR, Inc.

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Rigor is having theorems that follow from axioms by means of systematic reasoning. According to mathematicians,

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© 2009 EDSTAR, Inc. What is rigor in school mathematics?

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© 2009 EDSTAR, Inc. In schools, Rigor is teaching and learning that is active, deep, and engaging.

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© 2009 EDSTAR, Inc. Active learning involves conversation and hands-on, minds-on activities. Questioning & discovery learning goes on!

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© 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.

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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.

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Rigor promotes Mathematical Practices.

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Mathematical Practices revolve around lessons that embrace mathematical rigor © 2009 EDSTAR Analytics, Inc.

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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?

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Common Core GPS Mathematics Common Core GPS Mathematics Are You Ready to Change Your Behavior?

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35 Stages of Change Model maintenance contemplation preparation action relapse consistent behavior pre- contemplation smoking

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36 Stages of Change Model Pre-contemplation – Not yet acknowledging that there is a problem behavior that needs to be changed.

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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! Its the kids, not me!

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38 Stages of Change Model Contemplation – Acknowledging that there is a problem but not yet ready, or sure of, wanting to make a change.

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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!

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40 Stages of Change Model Preparation – (Determination) Getting ready to change.

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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.

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42 Stages of Change Model Action – Actually changing behavior.

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43 Stages of Change Model Action – Actually changing behavior. People at this stage: engage change behaviors; modify their environment; seek support from others. Its different, but I can do this!

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44 Stages of Change Model Maintenance – Maintaining the change in behavior.

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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. Its hard to do, but its better!

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46 Stages of Change Model Relapse – Returning to older behaviors and abandoning the new changes.

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47 Stages of Change Model Relapse – Returning to older behaviors and abandoning the new changes. Relapses are expected. When they happen, dont abandon the desired behaviors; learn from your mistakes; renew your commitment. I see what happened; lets do this!

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© 2009 EDSTAR, Inc. What can teachers do to bring rigor into the classroom?

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Rubric for Rigor ActiveCheck 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 DeepCheck Reflects 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 EngagingCheck Makes connections between the lesson and real-life situations or other areas of study Demonstrates the benefit of applying known skills, concepts, and relationships to new ones Helps students appreciate and seek challenging problem-solving situations Conveys enthusiasm for the subject Typical Lesson Rigorous Lesson Scoring: 1-High; 2-Medium; 3-Low

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© 2009 EDSTAR, Inc. What do teachers frequently do when planning a lesson?

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IDENTIFY the worksheets and other resources they will use TALK about what the students cannot do! Lesson Planning

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NCTM Process Standards and the CCSSMs Mathematical Practices NCTM Process Standards Problem Solving Reasoning and Proof Communication Connections Representations CCSSMs 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. 3.Construct viable arguments 6. Attend to precision. 7.Look for and make use of structure. 4. Model with mathematics.

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© 2009 EDSTAR, Inc. Teaching Mathematical Practices

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Problem Solving Understand the Problem Devise a Plan Carry Out the Plan Look Back

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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

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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 ActionsStudent 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.

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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

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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 ActionsStudent 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.

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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

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Teaching Mathematical Practices Problem Solving The Focus: Students implement the strategy by performing known skills and procedures and applying known concepts. Problem Solving Teacher ActionsStudent 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

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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

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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 ActionsStudent 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

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Rubric for Rigor ActiveCheck 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 DeepCheck Reflects 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 EngagingCheck Makes connections between the lesson and real-life situations or other areas of study Demonstrates the benefit of applying known skills, concepts, and relationships to new ones Helps students appreciate and seek challenging problem-solving situations Conveys enthusiasm for the subject Typical Lesson Rigorous Lesson Scoring: 1-High; 2-Medium; 3-Low

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© 2009 EDSTAR, Inc. What can teachers do to bring rigor into the classroom?

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© 2009 EDSTAR, Inc. Just remember: Rigor is a process- not a problem.

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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. B C E F D A

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

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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.

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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?

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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 classmates. 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.

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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.

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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.

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© 2009 EDSTAR, Inc. Rigor is an activity: Use writing in math to support rigor.

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Writing in math

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© 2009 EDSTAR, Inc. What can teachers do to bring rigor into the classroom?

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© 2009 EDSTAR, Inc. Remember that rigor is a process, not a problem.

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77 Teaching Mathematical Practices ?

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? Pos # # SQs n… ?… Teaching Mathematical Practices

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n ? 2n-1 n + n-1 n 2 - (n-1) 2 Find the nth term. … Teaching Mathematical Practices

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? Pos # # s n… ?… # s?… Teaching Mathematical Practices

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81 4 ? Pos # # s n… n… # s Total … … 2n+2 3n+2 Teaching Mathematical Practices

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82 x f(x)=3n Teaching Mathematical Practices

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Lesson Planning for 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.

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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, and 8.multiple representations of mathematical concepts. Lesson Planning for Mathematical Practices

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© 2009 EDSTAR, Inc. Create classrooms where students are… Talking about mathematicsTalking about mathematics Making connectionsMaking connections Solving problemsSolving problems ReasoningReasoning

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Remember… Rigor Promotes Mathematical Practices!

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87 a presentation by Lee V. Stiff North Carolina State University First Annual Title I Mathematics Summit Fulton County Schools Atlanta, GA There is Nothing More Uncommon than Common Core

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