1. © 2013 UNIVERSITY OF PITTSBURGH Supporting Rigorous Mathematics Teaching and Learning Constructing an Argument and Critiquing the Reasoning of Others Tennessee Department of Education Middle School Mathematics Grade 6

2. Mathematical Understandings [In the TIMSS report the fact] that 89% of the U.S. lessons’ content received the lowest quality rating suggests a general lack of attention among teachers to the ideas students develop. Instead, U.S. lessons tended to focus on having students do things and remember what they have done. Little emphasis was placed on having students develop robust ideas that could be generalized. The emergence of conversations about goals of instruction – understandings we intend that students develop – is an important catalyst for changing the present situation. Thompson and Saldanha (2003). Fractions and Multiplicative Reasoning. In Kilpatrick et al. (Eds.), Research companion to the principles and standards for school mathematics, Reston: NCTM. P. 96. In this module, we will analyze student reasoning to determine attributes of student responses and then we will consider how teachers can scaffold student reasoning. 2

3. © 2013 UNIVERSITY OF PITTSBURGH Session Goals Participants will learn about: elements of Mathematical Practice Standard 3; students’ mathematical reasoning that is clear, faulty, or unclear; teachers’ questioning focused on mathematical reasoning; and strategies for supporting writing. 3

4. © 2013 UNIVERSITY OF PITTSBURGH Overview of Activities Participants will: analyze a video and discuss students’ mathematical reasoning that is clear, faulty, or unclear; analyze student work to differentiate between writing about process versus writing about mathematical reasoning; and study strategies for supporting writing. 4

5. © 2013 UNIVERSITY OF PITTSBURGH Making Sense of Mathematical Practice Standard 3 Study Mathematical Practice Standard 3: Construct a viable argument and critique the reasoning of others, and summarize the authors’ key messages. 5

6. Common Core State Standards: Mathematical Practice Standard 3 The Common Core State Standards recommend that students: construct viable arguments and critique the reasoning of others; 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; recognize and use counterexamples; justify 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; and 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. Modified from the Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO 6

7. NCTM Focal Points: Reasoning and sense making are of particular importance, but historically “reasoning” has been limited to very selected areas of the high school curriculum, and sense making is in many instances not present at all. However, an emphasis on student reasoning and sense making can help students organize their knowledge in ways that enhance the development of number sense, algebraic fluency, functional relationships, geometric reasoning, and statistical thinking. NCTM, 2008, Focus in High School Mathematics: Reasoning and Sense Making 7

8. The Relationship Between Talk and Understanding We come to an understanding in the course of communicating it. That is to say, we set out by offering an understanding and that understanding takes shape as we work on it to share it. And finally we may arrive cooperatively at a joint understanding as we talk or in some other way interact with someone else (p. 115). This view is supported by Chin and Osborne‘s (2008) study. They state that when students engage socially in talk activities about shared ideas or problems, students must be given ample opportunities for formulating their own ideas about science concepts, for inferring relationships between and among these concepts, and for combining them into an increasingly more complex network of theoretical propositions. For Hand (2008), the oral language component is heavily emphasized in the social negotiated processes in which students exchange, challenge, and debate arguments in order to reach a consensus. (Chen, Ying Chih, 2011 Examining the integration of talk and writing for student knowledge construction through argumentation.) 8

9. © 2013 UNIVERSITY OF PITTSBURGH Hiking Task Dia’Monique and Yanely picnicked together. Then Dia’Monique hiked 17 miles. Yanely hiked 14 miles in the opposite direction. What is their distance from each other? Draw a picture to show their distance from each other. 9

10. © 2013 UNIVERSITY OF PITTSBURGH Determining Student Understanding What will you need to see and hear to know that students understand the concepts of a lesson? Watch the video. Be prepared to say what students know or do not know. Cite evidence from the lesson. 10

11. © 2013 UNIVERSITY OF PITTSBURGH Context for the Lesson The teacher has been working on both the Standards for Mathematical Content and Mathematical Practice. She is interested in gaining a better understanding of ways of encouraging classroom talk. The students have been exposed to negative integers in the past. Teacher:Jessica Winkle Principal:Dr. Nancy Meador School:Madison Middle School School District:Metropolitan Nashville Public Schools, TN Grade:6 Date:March 7, 2013 11

12. The CCSS for Mathematics: Grade 6 The Number System 6.NS Apply and extend previous understandings of numbers to the system of rational numbers. 6.NS.C.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. 6.NS.C.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. 6.NS.C.6a Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite. Common Core State Standards, 2010, p. 43, NGA Center/CCSSO 12

13. The CCSS for Mathematics: Grade 6 The Number System 6.NS Apply and extend previous understandings of numbers to the system of rational numbers. 6.NS.C.6c Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. 6.NS.C.7c Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of -30 dollars, write |-30| = 30 to describe the size of the debt in dollars. Common Core State Standards, 2010, p. 43, NGA Center/CCSSO 13

14. © 2013 UNIVERSITY OF PITTSBURGH Essential UnderstandingCCSS Positive and Negative Numbers Can be Used to Represent Real- World Quantities Positive numbers represent values greater than 0 and negative numbers represent values less than 0. Many real-world situations can be modeled with both positive and negative values because it is possible to measure above and below a baseline value (often 0). 6.NS.C.5 Rational Numbers Can be Located on a Number Line Any rational number can be modeled using a point on the number line because the real number line extends infinitely in the positive and negative directions. The sign and the magnitude of the number determine the location of the point. 6.NS.C.6 Absolute Value is a Measure of a Number’s Distance From 0 The absolute value of a number is the number’s magnitude or distance from 0. If two rational numbers differ only by their signs, they have the same absolute value because they are the same distance from zero. 6.NS.C.7c Distance Between Two Values on a Number Line Can be Determined Using Arithmetic The distance between a positive and negative value on a number line is equal to the sum of their absolute values because they are located on opposite sides of zero. 6.NS.C.7c Essential Understandings 14

15. The CCSS for Mathematical Practice 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. Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO 15

16. © 2013 UNIVERSITY OF PITTSBURGH Determining Student Understanding (Small Group Work) What did students know and what is your evidence? Where in the lesson do you need additional information to know if students understood the mathematics or the model? Cite evidence from the lesson. 16

17. © 2013 UNIVERSITY OF PITTSBURGH Determining Student Understanding (Whole Group Discussion) In what ways did students make use of the third Standard for Mathematical Practice? Let’s step back now and identify ways in which student understanding shifted or changed during the lesson. Did student understanding evolve over the course of the lesson? If so, what ideas did you see changing over time? What do you think was causing the shifts? 17

18. Common Core State Standards: Mathematical Practice Standard 3 (Whole Group Discussion) How many of the MP3 elements did we observe and if not, what are we wondering about since this was just a short segment? The Common Core State Standards recommend that students: construct viable arguments and critique the reasoning of others; 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; recognize and use counterexamples; justify 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; and 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. Modified from the Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO 18

19. Imagine Publicly Marking Student Behavior In addition to stressing the importance of effort, the teachers were very clear about the particular ways of working in which students needed to engage. D. Cohen and Ball (2001) described ways of working that are needed for learning as learning practices. For example, the teachers would stop the students as they were working and talking to point out valuable ways in which they were working. (Boaler, (2001) How a Detracked Math Approach Promoted Respect, Responsibility and High Achievement.) 19

20. © 2013 UNIVERSITY OF PITTSBURGH Talk is NOT GOOD ENOUGH. Writing is NEEDED! 20

21. The Writing Process In the writing process, students begin to gather, formulate, and organize old and new knowledge, concepts, and strategies, to synthesize this information as a new structure that becomes a part of their own knowledge network. Nahrgang & Petersen, 1998 When writing, students feel empowered as learners because they learn to take charge of their learning by increasing their access to and control of their thoughts. Weissglass, Mumme, & Cronin, 1990 21

22. Talk Alone is NOT GOOD ENOUGH! Several researchers have reported that students tend to process information on a surface level when they only use talk as a learning tool in the context of science education. (Hogan, 1999; Kelly, Druker, & Chen, 1998; McNeill & Pimentel, 2010) After examining all classroom discussions without writing support, they concluded that persuasive interactions only occurred regularly in one teacher’s classroom. In the other two classes, the students rarely responded to their peers by using their claims, evidence, and reasoning. Most of the time, students were simply seeking the correct answers to respond to teachers’ or peers’ questions. Current research also suggests that students have a great deal of difficulty revising ideas through argumentative discourse alone. (Berland & Reiser, 2011; D. Kuhn, Black, Keselman, & Kaplan, 2000) Writing involves understanding the processes involved in producing and evaluating thoughts rather than the processes involved in translating thoughts into language. (Galbraith, Waes, and Torrance (2007, p. 3). (Chen, Ying Chih, 2011 Examining the integration of talk and writing for student knowledge construction through argumentation.) 22

23. The Importance of Writing Yore and Treagust (2006) note that writing plays an important role―to document ownership of these claims, to reveal patterns of events and arguments, and to connect and position claims within canonical science (p.296). That is, the writing undertaken as a critical role of the argumentative process requires students to build connections between the elements of the argument (question, claim, and evidence). When students write, they reflect on their thinking and come to a better understanding of what they know and what gaps remain in their knowledge (Rivard, 1994). (Chen, Ying Chih, 2011 Examining the integration of talk and writing for student knowledge construction through argumentation.) 23

24. Writing Assists Teachers, TOO Writing assists the teacher in thinking about the child as learner. It is a glimpse of the child’s reality, allowing the teacher to set up new situations for children to explain and build their mathematics understanding. Weissglass, Mumme, & Cronin, 1990 24

25. Comparing Temperatures Task Ms. Winkle gave the chart below to her students: This chart shows the temperatures in different cities on a cold day. Ms. Winkle asked Bill, “What is the difference between the highest temperature and the lowest temperature?” Bill did not know how to answer Ms. Winkle’s question. Find the highest and lowest temperatures in the chart. Show or explain how to find the difference between the highest and the lowest temperatures. If you used the thermometer below, be very complete in explaining how you used the thermometer to find your answer. What is the difference between the highest temperature and the lowest temperature? QUASAR Project QUASAR Cognitive Assessment Instrument CityTemperature Charleston, South Carolina 35  Amherst, Massachusetts -5  Madison, Wisconsin -10  Reading, Pennsylvania 0  Eugene, Oregon 20  Los Angeles, California 45  25

26. © 2013 UNIVERSITY OF PITTSBURGH Analyzing Student Work Analyze the student work. Sort the work into two groups—work that shows mathematical reasoning and work that does not show sound mathematical reasoning. What can be learned about student thinking in each of these groups, the group showing reasoning and the group that does not show sound reasoning? 26

27. © 2013 UNIVERSITY OF PITTSBURGH Student 1 27

28. © 2013 UNIVERSITY OF PITTSBURGH Student 2 28

29. © 2013 UNIVERSITY OF PITTSBURGH Student 3 29

30. © 2013 UNIVERSITY OF PITTSBURGH Student 4 30

31. © 2013 UNIVERSITY OF PITTSBURGH Student 5 31

32. © 2013 UNIVERSITY OF PITTSBURGH Student 6 32

33. © 2013 UNIVERSITY OF PITTSBURGH Student 7 33

34. © 2013 UNIVERSITY OF PITTSBURGH Essential Understanding Positive and Negative Numbers Can be Used to Represent Real-World Quantities Positive numbers represent values greater than 0 and negative numbers represent values less than 0. Many real-world situations can be modeled with both positive and negative values because it is possible to measure above and below a baseline value (often 0). Rational Numbers Can be Located on a Number Line Any rational number can be modeled using a point on the number line because the real number line extends infinitely in the positive and negative directions. The sign and the magnitude of the number determine the location of the point. Absolute Value is a Measure of a Number’s Distance From 0 The absolute value of a number is the number’s magnitude or distance from 0. If two rational numbers differ only by their signs, they have the same absolute value because they are the same distance from zero. Distance Between Two Values on a Number Line Can be Determined Using Arithmetic The distance between a positive and negative value on a number line is equal to the sum of their absolute values because they are located on opposite sides of zero. Essential Understandings 34

35. Common Core State Standards: Mathematical Practice Standard 3 The Common Core State Standards recommend that students: construct viable arguments and critique the reasoning of others; 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; recognize and use counterexamples; justify 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; and 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. Modified from the Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO 35

36. © 2013 UNIVERSITY OF PITTSBURGH Two Forms of Writing Consider the forms of writing below. What is the purpose of each form of writing? How do they differ from each other? Writing about your problem-solving process/steps when solving a problem Writing about the meaning of a mathematical concept/idea or relationships 36

37. A Balance: Writing About Process Versus Writing About Reasoning Students and groups who seemed preoccupied with “doing” typically did not do well compared with their peers. Beneficial considerations tended to be conceptual in nature, focusing on thinking about ways to think about the situations (e.g., relationships among “givens” or interpretations of “givens” or “goals” rather than ways to get from “givens” to “goals”). This conceptual versus procedural distinction was especially important during the early stages of solution attempts when students’ conceptual models were more unstable. Lesh & Zawojewski, 1983 37

38. © 2013 UNIVERSITY OF PITTSBURGH Strategies for Supporting Writing 38

39. © 2013 UNIVERSITY OF PITTSBURGH Strategies for Supporting Writing How might use of these processes or strategies assist students in writing about mathematics? Record your responses on the recording sheet in your participant handout. Reflect on the potential benefit of using strategies to support writing. 1.Make Time for the Think-Talk-Reflect-Write Process 2.The Use of Multiple Representations 3.Construct a Concept Web with Students 4.Co-Construct Criteria for Quality Math Work 5.Engage Students in Doing Quick Writes 6.Encourage Pattern Finding and Formulating and Testing Conjectures 39

40. 1. Make Time for the Think-Talk-Reflect- Write Process Think: Work privately to prepare a written response to one of the prompts, “What is division?” Talk: What is division? Keep a written record of the ideas shared. Reflect: Reflect privately. Consider the ideas raised. How do they connect with one another? Which ideas help you understand the concept better? Write: Write an explanation for the question, “What is division?” Think about what everyone in your group said, and then use words, pictures, and examples to explain what division means. Go ahead and write. Hunker & Lauglin, 1996 40

41. Pictures Written Symbols Manipulative Models Real-world Situations Oral & Written Language Modified from Van De Walle, 2004, p. 30 2. Encourage the Use of Multiple Representations of Mathematical Ideas 41

42. © 2013 UNIVERSITY OF PITTSBURGH 3. Construct a Concept Web with Students Analyze the concept web. Students developed the concept web with the teacher over the course of several months. How might developing and referencing a concept web help students when they are asked to write about mathematical ideas? 42

43. © 2013 UNIVERSITY OF PITTSBURGH 3. Construct a Concept Web with Students (continued) 43

44. © 2013 UNIVERSITY OF PITTSBURGH 4. Co-Construct Criteria for Quality Math Work: 4 th Grade Students worked to solve high-level tasks for several weeks. The teacher asked assessing and advancing questions daily. Throughout the week, the teacher pressed students to do quality work. After several days of work, the teacher showed the students a quality piece of work, told them that the work was “quality work,” and asked them to identify what the characteristics of the work were that made it quality work. Together, they generated this list of criteria. 44

45. © 2013 UNIVERSITY OF PITTSBURGH 45

46. © 2013 UNIVERSITY OF PITTSBURGH 5. Engage Students in Doing Quick Writes A Quick Write is a narrow prompt given to students after they have studied a concept and should have gained some understanding of the concept. Some types of Quick Writes might include: compare concepts; use a strategy or compare strategies; reflect on a misconception; and write about a generalization. 46

47. © 2013 UNIVERSITY OF PITTSBURGH Brainstorming Quick Writes What are some Quick Writes that you can ask students to respond to for your focus concept? 47

48. © 2013 UNIVERSITY OF PITTSBURGH 6. Encourage Pattern Finding and Formulating and Testing Conjectures How might students benefit from having their conjectures recorded? What message are you sending when you honor and record students’ conjectures? Why should we make it possible for students to investigate their conjectures? 48

49. Checking In: Construct Viable Arguments and Critique the Reasoning of Others How many of the MP3 elements did we observe and if not, what are we wondering about since this was just a short segment? The Common Core State Standards recommend that students: construct viable arguments and critique the reasoning of others; 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; recognize and use counterexamples; justify 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; and 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. Modified from the Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO 49