1. North Country Inservice HS Mathematics Common Core State Standards Mathematics Practice and Content Standards Day 1 Friday, October 19, 2012 Presenter: Elaine Watson, Ed.D.

2. Introductions Share What feeds your soul personally? What is your professional role? What feeds your soul professionally?

3. Volunteers for Breaks I need volunteers to remind me when we need breaks! Every 20 minutes, we need a 2-minute “movement break” to help our blood circulate to our brains. Every hour we need a 5-minute bathroom break.

4. Formative Assessment How familiar are you with the CCSSM?

5. Setting the Stage Dan Meyer’s TED Talk Math Class Needs a Makeover Go to link: watsonmath.com “North Country High School Math Inservice October 19, 2012”

6. CCSSM Equally Focuses on… Standards for Mathematical Practice Standards for Mathematical Content Same for All Grade Levels Specific to Grade Level

7. 8 Practice Standards Look at the handout SMP Lesson Alignment Template For an electronic copy to use later, go to watsonmath.com “North Country High School Math Inservice October 19, 2012”

8. Standards for Mathematical Practice Describe ways in which student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity

9. Standards for Mathematical Practice Provide a balanced combination of Procedure and Understanding Shift the focus to ensure mathematical understanding over computation skills

10. Standards for Mathematical Practice Students will be able to: 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.

11. Video of NYC High School Piloting the CCSS Watch first 5 minutes on Math See link in watsonmath.com

12. Standards for Mathematical Practice Some of the following slides on the Practice Standards have been adapted from slides presented in several online EdWeb Webinars in February through May 2012 discussing that focused on the Practice Standards by Sara Delano Moore, Ph.D.

13. The 8 Standards for Mathematical Practice can be divided into 4 Categories Overarching Habits of Mind of a Mathematical Thinker (# 1 and # 6) Reasoning and Explaining (# 2 and # 3) Modeling and Using Tools (# 4 and # 5) Seeing Structure and Generalizing (# 7 and # 8)

14. The 8 Standards for Mathematical Practice are fluidly connected to each other. One action that a student performs, either internally or externally, when solving a problem can take on characteristics from several of the 8 Practice Standards.

15. Overarching Habits of Mind of a Mathematical Thinker 1.Make sense of problems & persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments & critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. 6. Attend to precision. Look for & make use of structure. Look for & express regularity in repeated reasoning.

16. Start with Good Problems Characteristics Example from Illustrative Mathematics (F-BF.A.1.a, F-IF.B.4, F-IF.B.5 ) Context relevant to students Incorporates rich mathematics Entry points/solution pathways not readily apparent Mike likes to canoe. He can paddle 150 feet per minute. He is planning a river trip that will take him to a destination about 30,000 feet upstream (that is, against the current). The speed of the current will work against the speed that he can paddle. Let s be the speed of the current in feet per minute. Write an expression for r(s), the speed at which Mike is moving relative to the river bank, in terms of s. Mike wants to know how long it will take him to travel the 30,000 feet upstream. Write an expression for T(s), the time in minutes it will take, in terms of s. What is the vertical intercept of T? What does this point represent in terms of Mike’s canoe trip? At what value of s does the graph have a vertical asymptote? Explain why this makes sense in the situation. For what values of s does T(s) make sense in the context of the problem?

17. Make Sense of Problems (part I) Mathematically proficient students…  Explain the meaning of the problem to themselves  Look for entry points to the solution  Analyze givens, constraints, relationships, goals

18. Mike’s Canoe Trip – Explain the meaning of the problem – Entry points – Givens, constraints, relationships, goals Mike likes to canoe. He can paddle 150 feet per minute. He is planning a river trip that will take him to a destination about 30,000 feet upstream (that is, against the current). The speed of the current will work against the speed that he can paddle. Let s be the speed of the current in feet per minute. Write an expression for r(s), the speed at which Mike is moving relative to the river bank, in terms of s. Mike wants to know how long it will take him to travel the 30,000 feet upstream. Write an expression for T(s), the time in minutes it will take, in terms of s. What is the vertical intercept of T? What does this point represent in terms of Mike’s canoe trip? At what value of s does the graph have a vertical asymptote? Explain why this makes sense in the situation. For what values of s does T(s) make sense in the context of the problem?

19. Persevere in Solving Them Mathematically proficient students….  Plan a solution pathway  Consider analogous cases and alternate forms  Monitor progress and change course if necessary

20. Persevere in Solving Them “It's not that I'm so smart, it's just that I stay with problems longer.” - Albert Einstein

21. Mike’s Canoe Trip – Possible solution pathways/strategies – Consider analogous cases & alternate forms – Monitor progress and change course if needed Mike likes to canoe. He can paddle 150 feet per minute. He is planning a river trip that will take him to a destination about 30,000 feet upstream (that is, against the current). The speed of the current will work against the speed that he can paddle. Let s be the speed of the current in feet per minute. Write an expression for r(s), the speed at which Mike is moving relative to the river bank, in terms of s. Mike wants to know how long it will take him to travel the 30,000 feet upstream. Write an expression for T(s), the time in minutes it will take, in terms of s. What is the vertical intercept of T? What does this point represent in terms of Mike’s canoe trip? At what value of s does the graph have a vertical asymptote? Explain why this makes sense in the situation. For what values of s does T(s) make sense in the context of the problem?

22. Make Sense of Problems (part II) Mathematically proficient students…  Explain correspondence and search for trends  Check their answers using alternate methods  Continually ask themselves, “Does this make sense?”  Understand the approaches of others

23. What can teachers do? Select rich mathematical tasks – Connected to rigorous mathematics content Resources for rigorous mathematical tasks can be found on www.watsonmath,.com “North Country High School Math Inservice October 19, 2012” Illustrative Mathematics MARS Tasks Inside Mathematics 3 – Act Math Tasks Dan Meyer Andrew Stadel Others

24. What can teachers do? Ask good questions – Is that true every time? Explain how you know. – Have you found all the possibilities? How can you be sure? – Does anyone have the same answer but a different way to explain it? – Can you explain what you’ve done so far? What else is there to do?

25. What can teachers do? Communicate to students the final solution to a problem is less important than the skills they develop during the process of finding the solution. The skills developed in working through the process are long-lasting skills that will serve them in other areas of life.

26. Attend to Precision – In Vocabulary – In Mathematical Symbols – In Computation – In Measurement – In Communication

27. How is the teacher ensuring that students are making sense of problem and attending to precision? See Video: Discovering Properties of Quadrilaterals on Watsonmath.com

28. Challenges to Precision Vocabulary – Similar, adjacent Mathematical Symbols –=–= Computation and Measurement – Accurate computation – Estimating when appropriate – Appropriate units of measure Communication – Formulate explanations carefully – Make explicit use of definitions

29. Mike’s Canoe Trip – Vocabulary – Mathematical Symbols – Computation & Measurement – Communication Mike likes to canoe. He can paddle 150 feet per minute. He is planning a river trip that will take him to a destination about 30,000 feet upstream (that is, against the current). The speed of the current will work against the speed that he can paddle. Let s be the speed of the current in feet per minute. Write an expression for r(s), the speed at which Mike is moving relative to the river bank, in terms of s. Mike wants to know how long it will take him to travel the 30,000 feet upstream. Write an expression for T(s), the time in minutes it will take, in terms of s. What is the vertical intercept of T? What does this point represent in terms of Mike’s canoe trip? At what value of s does the graph have a vertical asymptote? Explain why this makes sense in the situation. For what values of s does T(s) make sense in the context of the problem?

30. Reasoning and Explaining Make sense of problems & persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments & critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for & make use of structure. Look for & express regularity in repeated reasoning.

31. 2. Reason abstractly & quantitatively Mathematics in and out of context Working with symbols as abstractions Quantitative reasoning requires number sense Using properties of operations and objects Considering the units involved Attending to the meaning of quantities, not just computation

32. Construct viable arguments… Understand and use assumptions, definitions, and prior results – Think about precision (MP6) Make conjectures and build logical progressions to support those conjectures – Not just two column proofs in high school Analyze situations by cases – Positive values of X and negative values of X – Two-digit numbers vs three-digit numbers Recognize & use counter-examples – Maximum area problem

33. How do we help children learn how to reason and explain? Provide rich problems where multiple pathways and solutions are possible Celebrate multiple pathways to the same answer – Monitor students as they work to choose approaches to share with the whole class Provide plenty of opportunities for students to talk to each other Recognize the difference between a viable argument and opinion Provide scaffolds for them…but not too many!

34. How do we help children learn how to reason and explain? Provide plenty of opportunities for students to talk to each other. Create a classroom culture in which all students feel safe to express their thinking Make sure students recognize the difference between a viable argument and an opinion Create a classroom culture where it’s safe to critique each other in a respectful way Provide scripts (sentence frames)for them to use such as those from Accountable Talk (see resources on watsonmath.com)

35. Teacher Moves in Group Discussion By scaffolding students' responses and contributions, teachers can quickly make a difference in the level of rigor and productivity in classroom talk. Teachers can bring everyone's attention to a key point By "marking" a student's contribution "that's an important point” By asking the student to repeat the remark—or restating it in their own words—and indicating why the point is important. From ACCOUNTABLE TALK® SOURCEBOOK: FOR CLASSROOM CONVERSATION THAT WORKS Version 3.1 By Sarah Michaels (Clark University), Mary Catherine O’Connor (Boston University), Megan Williams Hall (University of Pittsburgh), with Lauren B. Resnick (University of Pittsburgh)

36. Teacher Moves in Group Discussion If someone asks a thought-provoking question, the teacher might turn the question back to the group ”Good question, what do you think?” as a way to encourage students to push their own thinking. By citing facts and posing counterexamples, teachers can challenge students to elaborate or clarify their arguments "but what about...?” From ACCOUNTABLE TALK® SOURCEBOOK: FOR CLASSROOM CONVERSATION THAT WORKS Version 3.1 By Sarah Michaels (Clark University), Mary Catherine O’Connor (Boston University), Megan Williams Hall (University of Pittsburgh), with Lauren B. Resnick (University of Pittsburgh)

37. Teacher Moves in Group Discussion Teacher can model what desirable behaviors and habits of mind look like, ”Here’s what good problem solvers do when they're monitoring their own process." Teachers can focus the group's thinking by recapping or summarizing key points that have been brought up in a discussion. From ACCOUNTABLE TALK® SOURCEBOOK: FOR CLASSROOM CONVERSATION THAT WORKS Version 3.1 By Sarah Michaels (Clark University), Mary Catherine O’Connor (Boston University), Megan Williams Hall (University of Pittsburgh), with Lauren B. Resnick (University of Pittsburgh)

38. Teacher Moves That Support Accountability to the Learning Community Accountability to the learning community requires that students listen to one another, attending carefully so that they can use and build on one another's ideas. Students and teachers agree and disagree respectfully, challenging a claim, not the person who made it. To support this kind of accountability, teachers must establish a classroom environment where everyone can hear each other, and where everyone knows how important it is to hear and be heard. From ACCOUNTABLE TALK® SOURCEBOOK: FOR CLASSROOM CONVERSATION THAT WORKS Version 3.1 By Sarah Michaels (Clark University), Mary Catherine O’Connor (Boston University), Megan Williams Hall (University of Pittsburgh), with Lauren B. Resnick (University of Pittsburgh)

39. Teacher Moves That Support Accountability to the Learning Community Keeping the channels open: "Did everyone hear that?" Keeping everyone together: "Who can repeat...?" Linking contributions: "Who wants to add on...?" Verifying and clarifying: "So, are you saying...?”

40. Teacher Moves That Support Accountability to Accurate Knowledge Pressing for accuracy: "Where can we find that?" Building on prior knowledge: "How does this connect?"

41. Teacher Moves That Support Accountability to Rigorous Thinking Pressing for reasoning: "Why do you think that?” Expanding reasoning: "Take your time; say more."

42. Teacher Moves That Support Accountability to Rigorous Thinking Pressing for reasoning: "Why do you think that?” Expanding reasoning: "Take your time; say more."

43. Other Ideas For Getting Students Talking Pose a question and then say, “Turn and talk to your neighbor” Bring the discussion back to the whole group. Pose a question to the whole class and then draw names out of a “hat”

44. The Importance of Wait Time Increased wait time of at least 2.7, and preferably at least 3, seconds can have these effects on students: 1) The length of student responses increases between 300% and 700%. 2) More inferences are supported by evidence and logical argument. 3) The incidence of speculative thinking increases. 4) The number of questions asked by students increases.

45. The Importance of Wait Time Increased wait time of at least 2.7, and preferably at least 3, seconds can have these effects on students: 5) Student-student exchanges increase; teacher- centered “show and tell” behavior decreases. 6) Failures to respond decrease. 7) Disciplinary moves decrease.

46. The Importance of Wait Time Increased wait time of at least 2.7, and preferably at least 3, seconds can have these effects on students: 8) The variety of students participating voluntarily increases. Also the number of unsolicited, but appropriate contributions by students increases. 9) Student confidence, as reflected in fewer inflected responses, increases. 10) Achievement improves on written measures where the items are cognitively complex.

47. Teacher Moves in Group Discussion How can you show that your computation is correct? – Use a different tool or strategy – Compare your work with someone else How can you explain why your answer is best? – What possibilities did you consider? – What criteria did you use? – Why did you reject some options? – What made you choose this option? Embedding logic into your thinking – Does one part depend on another part? – Does changing one aspect of the problem change the result? – What are you sure about? What comes next?

48. …and critique the reasoning of others Compare two plausible arguments Distinguish correct from flawed reasoning – Explain/correct the flaw Ask useful questions to clarify and improve arguments

49. Modeling and Using Tools Make sense of problems & persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments & critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. Attend to precision. Look for & make use of structure. Look for & express regularity in repeated reasoning.

50. 4. Model with Mathematics Modeling is both a K - 12 Practice Standard and a 9 – 12 Content Standard.

53. Modeling Defined Modeling is the process of choosing and using appropriate mathematics and statistics to: analyze empirical situations understand them better improve decisions

54. Modeling Defined The word “modeling” in this context is used as a verb that describes the creative process of transforming a real situation into an abstract mathematical model.

55. Modeling Defined A model can be very simple, such as: writing total cost as a product of unit price and number bought, using a geometric shape to describe a physical object like a coin. Even such simple models involve making choices. It is up to us whether to model a coin as a three- dimensional cylinder, or whether a two- dimensional disk works well enough for our purposes.

56. Modeling Defined Other situations may need more elaborate models: modeling a delivery route, modeling a production schedule, modeling a comparison of loan amortizations Knowledge from other disciplines may be necessary Technology may or may not be needed

57. Modeling Defined Real-world situations are not organized and labeled for analysis. Formulating tractable models, representing such models, and analyzing them is a creative process. Like every such process, this depends on acquired expertise as well as creativity.

58. Modeling Defined There is no one correct model. The models students create increase in sophistication, elegance, and efficiency as they mature mathematically Modeling is best done as a collaborative activity

59. Modeling Defined The models devised depend on a number of factors: How precise an answer do we want or need? What aspects of the situation do we most need to understand, control, or optimize? What resources of time and tools do we have?

60. Model with Mathematics Apply mathematics to solve problems in the real world Make assumptions and approximations to simplify Identify important quantities and map relationships among them Interpret mathematical results in context and make adjustments to the model

61. Use appropriate tools strategically Consider available mathematical tools Know enough about tools to make informed decisions Recognize strengths and limitations of tools Identify and use relevant external tools, including technological resources

62. What are mathematical tools? Tools for doing mathematics – Paper and pencil – Calculators & Spreadsheets – Measuring Tools Tools for learning mathematics – Tools for doing mathematics – Physical Manipulatives – Technological tools

63. Using Manipulatives for Learning Mathematics A R C Concrete Representational Abstract

64. Seeing Structure and Generalizing  Make sense of problems & persevere in solving them.  Reason abstractly and quantitatively.  Construct viable arguments & critique the reasoning of others.  Model with mathematics.  Use appropriate tools strategically.  Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

65. Look for and make use of structure Patterns, patterns, patterns Properties of Operations – 3 + 7 = 7 + 3 – 7 x 8 = 7 x 5 + 7 x 3 Geometric Structure – Sorting geometric shapes – Reasoning about the attributes of shapes

66. Look for and make use of structure Look closely to discern a pattern or structure – In x 2 + 9x + 14, can see the 14 as 2 x 7 and the 9 as 2 + 7 – Can see complicated algebraic expressions as being composed of several objects: 5 – 3 (x – y) 2 is seen as 5 minus a positive number times a square, so its value can’t be more than 5 for any real numbers x and y

67. Look for and make use of structure Watch the Algebra video. (see watsonmath)watsonmath Use your SMP Lesson Plan Template Record the SMPs the teachers are using. How is structure used in these lessons?

68. Using Properties to See Structure Properties of Operations – Commutative Property of multiplication and addition – Distributive Property of multiplication over addition – Identity Property of multiplication and addition Properties of Equality – Transitive Property (if a=b and b=c, then a=c) Properties of Inequality – Exactly one of the following is true: a > b, a = b, a < b

69. Van Hiele Levels of Geometric Thinking shows how students increasing see more structure in shapes as they mature mathematically Level 0 (Pre-recognition) – Students do not yet see shapes clearly enough to compare with prototypes Level 1 (Visualization) – Students understand shapes by comparing to prototypes – Students do not see properties – Students make decisions based on perception, not reasoning Level 2 (Analysis) – Students see shapes as collections of properties – Students do not identify necessary and sufficient properties

70. Van Hiele Levels (cont) Level 3 (Abstraction) – Students see relationships among figures and properties – Students can create meaningful definitions and reason informally Level 4 (Deduction) – Students can construct proofs – Students understand necessary & sufficient conditions Level 5 (Rigor) – Students can understand non-Euclidean systems – Students can use indirect proof and formal deduction

71. Look for and express regularity in repeated reasoning Focus on computation here 1 ÷ 3 = Examining points on a line and slope – (1,2), m=3 (y-2)/(y-1) = 3 Attending to intermediate results

72. Look for and express regularity in repeated reasoning These practices are about seeing the underlying mathematical principles and generalizations. These practices have more subtlety, and are often hard to distinguish between each other.

74. Pyramid of Pennies Work through Dan Meyer’s Pyramid of Pennies Problem See link on watsonmath.comwatsonmath.com Use your SMP Lesson Planning Template and fill out what Math Practices you used when solving the problem.

76. Content Standard Activity Work with a partner and on your own laptop, go to Illustrative Mathematics on watsonmath.com Go to the HS Standards Check out the illustrations. Which SMPs would they reinforce? Which illustrations would you use in your classroom? Check out the modeling standards (those with a star) Why do you think these standards were chosen as modeling standards?

77. Last But Not Least… Formative Assessment Watch the two videos on watsonmath.com:watsonmath.com My Favorite No Daily Assessment with Tiered Exit Cards

78. Resources For resources used and/or discussed in this presentation, go to: www.watsonmath.com North Country High School Math Inservice October 19, 2012 Check out other resources available on watsonmath.com: archives of past posts resource links in the right hand column Contact Elaine at elaine.watson0729@gmail.comelaine.watson0729@gmail.com