1. Presented by Pitt County Schools Common Core Math Team

2. To examine the Mathematical Practices of Common Core, not the content (standards) of the Common Core

3. Opening Activity Make a list of things you have your students do, so that you know they understand the math concepts. On your own (3 minutes)

4. Old Boxes

5. Hong Kong / US Data Hong Kong had the highest scores in the most recent TIMSS. US students ranked near the bottom. Hong Kong students were taught 45% of objectives tested. Hong Kong students outperformed US students on US content that they were not taught. US students ‘covered’ 80% of TIMSS content. US students were outperformed by students not taught the same objectives.

6. Lessons Learned Mile wide and inch deep does not work. The task ahead is not so much about how many specific topics are taught; rather, it is more about ways of thinking. To change students’ ways of thinking, we must change how we teach.

7. The first focus is not the what, but the HOW Before we can change what we are teaching on a daily basis, we have to consider how we are teaching.

8. Common Core Attributes Focus and coherence Focus on key topics at each grade level Coherent progression across grade level Balance of concepts and skills Content standards require both conceptual understanding and procedural fluency Mathematical Practices Fosters reasoning and sense-making in mathematics College and career readiness Level is ambitious but achievable

9. Benefits from the Common Core Development of common assessments Policy and achievement comparisons across states and districts Development of curriculum, professional development and assessments through collaborative groups

10. 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 Standards for Mathematical Practices

11. Grouping of Math Practices Reasoning and Explaining 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others Modeling and Using Tools 4. Model with mathematics 5. Use appropriate tools strategically Seeing Structure and Generalizing 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Overarching Habits of Mind of a Productive Mathematical Thinker 1. Make sense of problems and persevere in solving them 6. Attend to precision Adapted from (McCallum, 2011)

12. The Standards for Mathematical Practice Take a moment to examine the first three words of each of the 8 mathematical practices… what do you notice? Mathematically Proficient Students…

13. Unpacking the Practice Standard 1. Make sense of problems and persevere in solving them. Individually read the practice standard ? – “I have a question about this.” * - “This makes sense to me.” With your group, discuss questions and highlight key ideas in this standard. Write unanswered questions on a post-it note for the facilitator to collect at the end of the session.

14. Students will need: Rich problems to consider. Time to reflect on their own thinking. Opportunities to dialogue with other students. A safe environment to share their solutions with other students.

15. Don’t forget: You can copy- paste this slide into other presentations, and move or resize the poll.

16. Why give students problems to solve? Answer: d. To learn mathematics. Answers are part of the process, they are not the product. The product is the student’s mathematical knowledge and know-how. The ‘correctness’ of answers is also part of the process. Yes, an important part. BUT…

17. Wrong Answers …are part of the process, too What was the student thinking? Was it an error of haste or a stubborn misconception?

18. Three Responses to a Math Problem 1. Answer getting 2. Making sense of the problem situation 3. Making sense of the mathematics you can learn from working on the problem

19. How Do I Help Students Make Sense and Persevere??? What do I do when a student “gets stuck” or says, “I don’t get it!”

20. 1. Make sense of problems and persevere in solving them. Students should be able to: Explain the meaning of the problem. May use concrete objects and/or pictorial representations. Come up with a strategy for solving the problem. Identify the connections between two different approaches to a problem. Determine whether or not the solution makes sense.

21. Unpacking the Practice Standard 2. Reason abstractly and quantitatively Individually read the practice standard ? – “I have a question about this.” * - “This makes sense to me.” With your group, discuss questions and highlight key ideas in this standard. Write unanswered questions on a post-it note for the facilitator to collect at the end of the session.

22. 2. Reason abstractly and quantitatively Students make sense of quantities and their relationships to problem situations. Use different forms of a number

23. 2. Reason abstractly and quantitatively. Students should be able to: De-contextualize – comprehend a given situation and represent it symbolically. Contextualize – consider the referents for the symbols they are working with. Understand the meaning of the quantities, not just how to compute them.

24. 3. Construct viable arguments and critique the reasoning of others Students understand and use stated assumptions, definitions, and previously established results in constructing arguments. I used the strategy that Jill shared yesterday to solve this problem.

25. The Standards for [Student] Mathematical Practice Consider the verbs that illustrate the student actions each practice. Examine Practice #3: Construct viable arguments and critique the reasoning of others. Highlight the verbs. Discuss with a partner: What jumps out at you?

26. Mathematical Practice #3: Construct viable arguments and critique the reasoning of others Mathematically proficient students: understand and use stated assumptions, definitions, and previously established results in constructing arguments. make conjectures and build a logical progression of statements to explore the truth of their conjectures. analyze situations by breaking them into cases, and can recognize and use counterexamples. justify their conclusions, communicate them to others, and respond to the arguments of others. reason inductively about data, making plausible arguments that take into account the context from which the data arose. compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is. construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. determine domains to which an argument applies. listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

27. The Standards for [Student] Mathematical Practice On a scale of 1 (low) to 6 (high), to what extent is your school promoting students’ proficiency in Practice 3? Evidence for your rating?

28. Math Discussion Help the children prepare by asking them to talk with their partners about what they want to share. Have them make a poster of the ideas or strategies they want to share and discuss. You may encourage them to walk around and look at other posters during a “gallery walk” during which they write comments or questions on sticky notes and place them on the posters. This engages children in reading and commenting on each other’s mathematics.

29. Math Discussion Math Discussion What do I do when a student gives wrong answers? Don’t try to fix the mathematics, work with the mathematician. The point is not to fix the mistakes in the children's work or to get everyone to agree with your answer, but to support the students’ development as mathematicians. Challenge them to think. As them to reflect on inconsistencies and answers that aren’t reasonable. Invite them to inquire further. “Wonder with them” about appearing patterns. (practice 7) Model the joy of mathematical inquiry.

30. Math Discussion Not just a whole- class share. There is no time for everyone to share. Many strategies will be similar. Powerful math discussion are structured to push the mathematical development of your class. Carefully select which pieces of work to use and the order of presentation.

31. As work is presented, focus the community discussion on the idea and push for generalization with questions like these: Do you agree that this strategy will always work? Why is this so? Could we prove it? When is it helpful? When not?

32. 3. Construct viable arguments and critique the reasoning of others. Students should be able to: Make conjectures. Use counterexamples in their arguments. Justify their conclusions and explain them to others. Listen and/or read other’s arguments and determine if they make sense. Ask questions to get clarification of an explanation.

33. 4. Model with mathematics Individually read Standard 4 ? – “I have a question about this.” * - “This makes sense to me.” With your group, discuss questions and highlight key ideas in this standard. Write unanswered questions on a post-it note for the facilitator to collect at the end of the session.

34. 4. Model with Mathematics Students should be able to: Apply the mathematics they know to solve everyday problems. Use equations, graphs, tables, diagrams, etc., to show the mathematical relationships in their model. Think about whether the model they have created makes sense and modify it if necessary.

35. Mathematical Modeling is … “the process of using mathematics to study a question from outside the field of mathematics. A mathematical model is a representation of a particular phenomenon using structures such as graphs, equations, or algorithms.” Abrams, J.P. (2001). Mathematical modeling: Teaching the open-ended application of mathematics. Retrieved from http://www.meaningfulmath.org/ http://www.meaningfulmath.org/

36. Unpacking the Practice Standard 5. Use appropriate tools strategically Individually read the practice standard ? – “I have a question about this.” * - “This makes sense to me.” With your group, discuss questions and highlight key ideas in this standard. Write unanswered questions on a post-it note for the facilitator to collect at the end of the session.

37. 5. Use appropriate tools strategically “I don’t use manipulatives because they can’t use them on the EOG!” Beyond the calculator! Students consider the available tools when solving a mathematical problem.

38. Not everyone uses good tools effectively… Phil Daro, NCCTM Leadership Seminar October 2010

39. 5. Use appropriate tools strategically. Students can: Consider which available tools they might use when solving a problem. Examples of tools: calculator, ruler, concrete objects Recognize the strengths and limitations of the tools they are using. Identify additional external resources, such as a website.

40. Unpacking the Practice Standard 6. Attend to precision Individually read the practice standard ? – “I have a question about this.” * - “This makes sense to me.” With your group, discuss questions and highlight key ideas in this standard. Write unanswered questions on a post-it note for the facilitator to collect at the end of the session.

41. 6. Attend to Precision Students calculate accurately and effectively, attending to the appropriate form of the number that is needed to answer a question. They understand when an estimation is acceptable (and how precise their estimation needs to be – hundreds, thousands or thousandths) and when an exact number is needed. Key terms and concepts are highlighted and explained in each lesson. Students revisit key terms and must provide explicit definitions or explanations of the terms as they progress.

42. 6. Attend to precision. Students should be able to: Communicate precisely to others. Use clear definitions in discussion. Explain the meaning of the symbols they choose. Specify units of measure and label axes. Calculate accurately and efficiently.

43. Unpacking the Practice Standard 7. Look for and make use of structure Individually read the practice standard ? – “I have a question about this.” * - “This makes sense to me.” With your group, discuss questions and highlight key ideas in this standard. Write unanswered questions on a post-it note for the facilitator to collect at the end of the session.

44. 7. Look for and make use of structure Students look closely to discern a pattern or structure. This boils down to understanding the form of a number and being able to be flexible with numbers and operations. Breaking a complex problem into simpler mathematics. For example, using the distributive property to figure out 7 x 8 if I don’t know my 7’s, but I know 5’s and 3’s – (7 x 5) + (7 x 3).

45. 7. Look for and make use of structure. Students should be able to: Look closely to identify a pattern or structure. Step back for an overview and shift perspective. See complicated things as single objects or as being composed of several objects.

46. 8. Look for and express regularity in repeated reasoning. Students should be able to: Notice if calculations are repeated, and look for both general methods and for short cuts. Maintain oversight of the process, while attending to the details.

47. Unpacking the Practice Standard 8. Look for and express regularity in repeated reasoning Individually read the practice standard ? – “I have a question about this.” * - “This makes sense to me.” With your group, discuss questions and highlight key ideas in this standard. Write unanswered questions on a post-it note for the facilitator to collect at the end of the session.

48. It’s not so much about “answer getting.” It’s attending to the process (and knowing that it doesn’t make sense to use other operations) that I’m using when problem solving. Focusing on this standard will decrease the number of students who choose the numbers out of a word problem and guess the operation to use without reasoning.

49. Question: So how do we help kids develop these behaviors? In your math class, who is doing the talking? Who is doing the math?

50. Quotes to think about! Covering does not translate to achievement The answer is part of the process not the product. Once you show them how, you cannot find out how they are thinking. When a child makes a mistake, they are showing a way of thinking that needs to be addressed to the whole class.

51. Mathematical Practices and Understanding Mathematics Students who understand a concept can: a. Use it to make sense of and explain quantitative situations (see “Model with Mathematics” in Practices) b. Incorporate it into their own arguments and use it to evaluate the arguments of others (see "Construct viable arguments and critique the reasoning of others” in Practices) c. Use it to find solutions to problems (see “Make sense of problems and persevere in solving them”) d. Make connections between it and related concepts

52. Conflicting Messages? What is the message that we send to students through our CURRENT assessment practices?

54. Reflection How do the standards for mathematical practices connect to what you know of past standards (e.g. NCTM & Adding it up) as well as what you know about best practices? What do you anticipate to be challenges in supporting students in developing proficiency in these practice standards? What actions will you take to address and overcome these challenges? How does addressing misconceptions and context help students engage in the mathematical practices? How has this activity increased your understanding of the instructional core?

55. Questions?