Tasks, Tools, and Talk: A Framework for Enacting the CCSS Mathematical Practices

1. Tasks, Tools, and Talk:A Framework for Enacting the CCSS Mathematical Practices Peg Smith University of Pittsburgh North Carolina Council of Teachers of Mathematics Leadership Seminar October 24, 2012

2. Position Developing students’ capacity to engage in the mathematical practices specified in the Common Core State Standards will ONLY be accomplished by engaging students in solving challenging mathematical tasks, providing students with tools to support their thinking and reasoning, and orchestrating opportunities for students to talk about mathematics and make their thinking public. It is the combination of these three dimensions of classrooms, working in unison, that promote understanding.

3. Standards for Mathematical Practice Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and critique the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Common Core State Standards for Mathematics, 2010, pp.6-7

4. Overview Discuss the task, tools, and talk framework Review and discuss examples of tasks that support engagement in the mathematical practices Analyze and discuss a narrative case with respect to the task, tools, and talk Discuss the potential of the task, tools, and talk framework for supporting your work with teachers related to the CCSS.

5. Tasks, Tools, and TalkFramework the tasks or activities in which students engage should provide opportunities for them to “figure things out for themselves” (NCTM, 2009, p.11), and to justify and communicate the outcome of their investigation; tools(i.e., language, materials, and symbols) should be available to provide external support for learning (Hiebert, et al, 1997); and productive classroom talkshould make students’ thinking and reasoning public so that it can be refined and/or extended (Chapin, O’Conner, & Anderson, 2009).

6. Comparing Two Versions of a Task Compare the two versions of the Adding Odd Numbers Task and consider how they are the same and how they are different Consider the opportunities each task provides to engage in the Standards for Mathematical Practice

7. Comparing Two Versions of a Task Adding Odds - Version 1 MAKING CONJECTURES Complete the conjecture based on the pattern you observe in the specific cases. 29. Conjecture: The sum of any two odd numbers is ______? 1 + 1 = 2 7 + 11 = 18 1 + 3 = 4 13 + 19 = 32 3 + 5 = 8 201 + 305 = 506 30. Conjecture: The product of any two odd numbers is ____? 1 x 1 = 1 7 x 11 = 77 1 x 3 = 3 13 x 19 = 247 3 x 5 = 15 201 x 305 = 61,305 Adding Odds - Version 2 For problems 29 and 30, complete the conjecture based on the pattern you observe in the examples. Then explain why the conjecture is always true or show a case in which it is not true. MAKING CONJECTURES Complete the conjecture based on the pattern you observe in the specific cases. 29. Conjecture: The sum of any two odd numbers is ______? 1 + 1 = 2 7 + 11 = 18 1 + 3 = 4 13 + 19 = 32 3 + 5 = 8 201 + 305 = 506 30. Conjecture: The product of any two odd numbers is ____? 1 x 1 = 1 7 x 11 = 77 1 x 3 = 3 13 x 19 = 247 3 x 5 = 15 201 x 305 = 61,305

8. Comparing Two Versions of a Task Same Both ask students to complete a conjecture about odd numbers based on a set of finite examples that are provided Different V2 asks students to develop an argument that explains why the conjecture is always true (or not) V1 can be completed with limited effort; V2 requires considerable effort – students need to figure out WHY this conjecture holds up The number of ways to enter and solve the problem

9. Comparing Two Versions of a TaskOpportunities to Engage in the Mathematical Practices Version 1 MP 7 – look for and make use of structure Version 2 MP 7 – look for and make use of structure MP1 – Make sense of problems and persevere in solving them MP3 – Construct viable arguments and critique the reasoning of others MP5 – Use appropriate tools strategically

10. Characteristics of Tasks Aligned with SMP • High cognitive demand (Stein et. al, 1996; Boaler & Staples, 2008) • Significant content (i.e., they have the potential to leave behind important residue) (Hiebert et. al, 1997) • Require justification or explanation (Boaler & Staples, 2008) • Make connections between two or more representations (Lesh, Post & Behr, 1988) • Open-ended (Lotan, 2003; Borasi & Fonzi, 2002) • Multiple ways to enter the task and to show competence (Lotan, 2003)

11. Comparing Two Versions of a Task Compare the two versions of the Tiling a Patio Task and consider the extent to which each exemplifies the characteristics of tasks that align with the Standards for Mathematical Practice.

12. Tiling a Patio Alfredo Gomez is designing patios. Each patio has a rectangular garden area in the center. Alfredo uses black tiles to represent the soil of the garden. Around each garden, he designs a border of white tiles. The pictures shown below show the three smallest patios that he can design with black tiles for the garden and white tiles for the border. Patio 1 Patio 2 Patio 3

13. Tiling a Patio: Aligned with SMP? • High cognitive demand - no specified pathway to follow, requires students to explore relationships Significant content - equivalence, rate of change • Require justification or explanation - explain in d and e • Make connections between two or more representations - connect rule to visual; could also connect with tables and graphs • Open-ended - different descriptions and rules can be written and in different forms • Multiple ways to enter the task and to show competence (Lotan, 2003)--build patios, draw pictures, make tables, write equations, draw graphs

14. Mathematical Tasks:A Critical Starting Point for Instruction Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking. Stein, Smith, Henningsen, & Silver, 2000

15. Mathematical Tasks:A Critical Starting Point for Instruction The level and kind of thinking in which students engage determines what they will learn. Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997

16. Mathematical Tasks:A Critical Starting Point for Instruction If we want students to develop the capacity to think, reason, and problem solve then we need to start with high-level, cognitively complex tasks. Stein & Lane, 1996

17. Mathematical Tasks:A Critical Starting Point for Instruction There is no decision that teachers make that has a greater impact on students’ opportunities to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages students in studying mathematics. Lappan & Briars, 1995

18. Mathematical Tasks:A Critical Starting Point for Instruction If we want students to develop the capacity to think, reason, and problem solve then we need to startwith high-level, cognitively complex tasks. Stein & Lane, 1996

19. Tools Tools can be thought of as “amplifiers of human capacities” (Brunner, 1966, p.81). “Tools should help students do things more easily or help students do things they could not do alone” (Hiebert, et al, 1997, p.53).

20. Pictures Representations as Tools Manipulative Models Written Symbols Real-world Situations Oral Language Lesh, Post, and Behr, 1987

21. Tools • Adding Odds Task • Square tiles that can be used to build the rectangular model • Drawing of dots that can be group by two • Use of symbolic notation 2x is even; 2x + 1 odd

22. The Fencing Task Ms. Brown’s class will raise rabbits for their spring science fair. They have 24 feet of fencing with which to build a rectangular rabbit pen in which to keep the rabbits. • If Ms. Brown's students want their rabbits to have as much room as possible, how long would each of the sides of the pen be? • How long would each of the sides of the pen be if they had only 16 feet of fencing? • How would you go about determining the pen with the most room for any amount of fencing? Organize your work so that someone else who reads it will understand it. • Stein, Smith, Henningsen, & Silver, 2009, p. xvii

23. The Fencing Task What tools could you provide that would help students engage in this task? What difference do you think the tools would make?

24. Fencing Task Approaches • Build pens with physical materials • Draw pens on grid paper • Make a table of the dimensions of possible pens • Make a graph that shows the relationship between one linear dimension and the area • Set up an algebraic equation and solve

25. Fencing Task Approaches • Build pens with physical materials (linear and area pieces) • Draw pens on grid paper (grid paper) • Make a table of the dimensions of possible pens • Make a graph that shows the relationship between one linear dimension and the area (graph paper or graphing calculator) • Set up an algebraic equation and solve

26. Talk Students must talk, with one another as well as in response to the teacher. When the teacher talks most, the flow of ideas and knowledge is primarily from teacher to student. When students make public conjectures and reason with others about mathematics, ideas and knowledge are developed collaboratively, revealing mathematics as constructed by human beings within an intellectual community. NCTM, 1991, p.34

27. The Case of Darcy Dunn Read the Case of Darcy Dunn Consider the way tasks, tools, and talked supported students engagement in the Standards for Mathematical Practice.

28. The Case of Darcy Dunn Teacher selected a task that had the potential to engage students in SMP (e.g., 1, 2, 3, 4, 5, 7) Teacher provided students with tools they could use to explored the problem (tiles, grid paper, colored pencils, calculators) Teacher provided students with diagrams of the patios that helped them explain their reasoning to the class Teacher pressed for explanations and encouraged students to questions each other Teacher engaged the class in creating a mathematical model that was consistent with the verbal description given by a student and the diagram of the patio Teacher gave homework that required providing and justifying a conclusion to the question, “Can they all be right?”

29. The Case of Darcy Dunn • Evidence that students were engaged in the mathematical practices • MP1 – Students were able to connect verbal descriptions with the diagram and with the equation. • MP2 – Beth, Faith, and Devon were able to make sense of quantities and their relationships in problem situations (Tamika’s table that didn’t get shared yet is another example) • MP3 – Beth, Faith, and Devon justified their conclusions and communicated them to others; all students were asked to consider the equivalence of the 3 equations for homework and justify their conclusions) • MP4- The class was able to write algebraic equations for the situations described by Beth, Faith, and Devon. • MP7 – Beth, Faith, Devon, and others identified the underlying structure of the pattern that they used to generalize

30. Reflect In what ways might the Task, Tools, and Talk framework help you in your work with teachers?

31. THANK YOU!

32. Draw a Picture Every odd number (like 11 and 13) has one loner number. Add the two loner numbers and you will get an even number (24). Now add all together the loner numbers and the other two (now even) numbers.

33. Build a Model If I take the numbers 5 and 11 and organize the counters as shown, you can see the pattern. You can see that when you put the sets together (add the numbers), the two extra blocks will form a pair and the answer is always even. This is because any odd number will have an extra block and the two extra blocks for any set of two odd numbers will always form a pair.

34. Use Algebra If a and b are odd integers, then a and b can be written a = 2m + 1 and b = 2n + 1, where m and n are other integers. If a = 2m + 1 and b = 2n + 1, then a + b = 2m + 2n + 2. If a + b = 2m + 2n + 2, then a + b = 2(m + n + 1). If a + b = 2(m + n + 1), then a + b is an even integer.

35. Logical Argument An odd number = [an] even number + 1. e.g. 9 = 8 + 1 So when you add two odd numbers you are adding an even no. + an even no. + 1 + 1. So you get an even number. This is because it has already been proved that an even number + an even number = an even number. Therefore as an odd number = an even number + 1, if you add two of them together, you get an even number + 2, which is still an even number.

36. Tiling a Patio Using a Visual Model to Find a Pattern of Growth T = 2p + 6 T = 2(p + 2) + 2 T = 3(p + 2) - p

37. Tiling a Patio Making a Table to Find the Pattern of Growth

38. Tiling a Patio Using a Graph to Determine the Pattern of Growth

39. The Fencing TaskBuilding Pens

40. The Fencing TaskBuilding Pens

41. The Fencing TaskDiagrams on Grid Paper

42. The Fencing TaskUsing a Table

43. The Fencing TaskGraph of Length and Area

44. The Fencing TaskGraph of Length and Area

45. The Fencing TaskEquation and Graph P = 2l + 2w 24 = 2l + 2w 12 = l + w l = 12 - w A = l x w A = l(12 – l) A = 12l – l2

46. The Fencing TaskEquation and Calculus A = 12l – l2. This is a quadratic equation of a parabola that has a maximum. Finding the derivative of the equation, then setting that derivative equal to zero, will give us the l value for the maximum. A(l) = 12l – l2 A’(l) = 12 – 2l 12 – 2l = 0 l = 6 If l is 6, then the width is 12 – 6 or 6. Thus, the configuration with the maximum area is 6 x 6,

47. Task Analysis Guide