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 8

2. Mathematical Understandings [In the TIMSS report the fact] that 89% of the United States 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 Scuba Math Task Serena and Trevon are taking a scuba diving course while on vacation in Hawaii. Serena begins swimming toward the surface as Trevon begins his dive. The tables below represent their depth in feet with respect to time in seconds. At what time will Trevon and Serena be at the same depth? Show your work and explain your reasoning. Time (seconds) Serena’s Depth (feet) 0-90 10-85 20-80 30-75 40-70 Time (seconds) Trevon’s Depth (feet) 00 20-25 40-50 60-75 80-100 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 Standards for Mathematical Practice. She is interested in gaining a better understanding of ways of encouraging classroom talk. The students have worked with linear equations in the past, but this is their first exposure to systems of equations. Teacher:Heather Mainegra Principal:Dr. Kristi Jefferson School:Freedom Middle School School District:Franklin Special School District, TN Grade:8 Date:March 8, 2013 11

12. The CCSS for Mathematics: Grade 8 Expressions and Equations 8.EE Analyze and solve linear equations and pairs of simultaneous linear equations. 8.EE.C.7 Solve linear equations in one variable. 8.EE.C.8 Analyze and solve pairs of simultaneous linear equations. 8.EE.C.8a Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. 8.EE.C.8b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. 8.EE.C.8c Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Common Core State Standards, 2010, p. 54 - 55, NGA Center/CCSSO 12

13. The CCSS for Mathematics: Grade 8 Functions 8.F Use functions to model relationships between quantities. 8.F.B.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Common Core State Standards, 2010, p. 55, NGA Center/CCSSO 13

14. Essential UnderstandingCCSS Solutions Make the Equations True The solution(s) of a system two linear equations is the ordered pair or pairs (x, y) that satisfy both equations. 8.EE.C.8 Systems of Equations Can be Solved Graphically The line representing a linear equation consists of all of the ordered pairs (x, y) that satisfy the equation. So, the solution of a system of linear equations is represented graphically by the intersection of the lines representing the equations because the point(s) at the intersection satisfy both equations. 8.EE.C.8a Systems of Equations Can be Solved Algebraically Using Substitution Given a true equation in two variables, the equation formed by isolating either of the variables is also true. Therefore, so is the equation you get by substituting an expression equal to one of the variables into a second true equation. 8.EE.C.8b A System of Two Linear Equations Can Have Zero, One, or Infinitely Many Solutions Two distinct lines will intersect at one point if and only if they do not have the same slope. Therefore, a system of two linear equations representing distinct lines with different slopes has one solution. Parallel lines have no points in common. Therefore, a system of two linear equations representing distinct parallel lines has no solutions. Linear equations representing the same line have infinitely many points in common. Therefore, a system of two linear equations representing the same line has infinitely many solutions. 8.EE.C.8a 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. © 2013 UNIVERSITY OF PITTSBURGH Pick Your Programs Task In the past, cable television companies charged a flat rate for a cable TV package. Recently, some cable companies are starting to offer packages where you pay a flat rate for access plus a fee per channel. TV Party charges a $40.00 flat rate plus $1.50 per channel. Cable Club charges a flat rate of $20.00 plus $3.00 per channel. Your friend, Felicia, argues that since TV Party’s flat rate is two times as much as Cable Club and their price per channel is half of Cable Club’s price per channel, the cost will be the same for any number of channels. Explain why you agree or disagree with Felicia. Use tables, graphs, or equations to justify your response. 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. Essential Understandings Essential Understanding Solutions Make the Equations True The solution(s) of a system of two linear equations is the ordered pair or pairs (x, y) that satisfy both equations. Systems of Equations Can be Solved Graphically The line representing a linear equation consists of all of the ordered pairs (x, y) that satisfy the equation. So, the solution of a system of linear equations is represented graphically by the intersection of the lines representing the equations because the point(s) at the intersection satisfy both equations. Systems of Equations Can be Solved Algebraically Using Substitution Given a true equation in two variables, the equation formed by isolating either of the variables is also true. Therefore, so is the equation you get by substituting an expression equal to one of the variables into a second true equation. A System of Two Linear Equations Can Have Zero, One, or Infinitely Many Solutions Two distinct lines will intersect at one point if and only if they do not have the same slope. Therefore, a system of two linear equations representing distinct lines with different slopes has one solution. Parallel lines have no points in common. Therefore, a system of two linear equations representing distinct parallel lines has no solutions. Linear equations representing the same line have infinitely many points in common. Therefore, a system of two linear equations representing the same line has infinitely many solutions. 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 on page 27 of 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 PITTSBURGH45

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 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