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1Peg Smith University of Pittsburgh Tasks, Tools, and Talk: A Framework for Enacting the CCSS Mathematical PracticesPeg SmithUniversity of PittsburghNorth Carolina Council of Teachers of Mathematics Leadership SeminarOctober 24, 2012
2PositionDeveloping 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.
3Standards for Mathematical Practice Make sense of problems and persevere in solving themReason abstractly and quantitativelyConstruct viable arguments and critique the reasoning of othersModel with mathematicsUse appropriate tools strategicallyAttend to precisionLook for and make use of structureLook for and express regularity in repeated reasoningCommon Core State Standards for Mathematics, 2010, pp.6-7
4Overview Discuss the task, tools, and talk framework Review and discuss examples of tasks that support engagement in the mathematical practicesAnalyze and discuss a narrative case with respect to the task, tools, and talkDiscuss the potential of the task, tools, and talk framework for supporting your work with teachers related to the CCSS.
5Tasks, Tools, and Talk Framework 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); andproductive classroom talk should make students’ thinking and reasoning public so that it can be refined and/or extended (Chapin, O’Conner, & Anderson, 2009).I am going to go through component of the framework and then we will explore each one in more detail.TASKSAll tasks are not created equal. They provide students with very different opportunities to think. For example, tasks that ask students to apply memorized procedures to routine problems require one kind of thinking. Tasks that ask students to engage with concepts and to make connections between ideas and representations ask for a very different kind of thinking.All students need the opportunity to engage in both types of tasks. Applying memorized procedures is important for developing fluency, but engaging in thinking and reasoning at a high level is necessary to develop problem-solving skills. And all students are capable of engaging in those tasks when they're carefully selected by the teacher.
6Comparing 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 differentConsider the opportunities each task provides to engage in the Standards for Mathematical Practice
7Comparing Two Versions of a Task Adding Odds - Version 1MAKING CONJECTURES Complete theconjecture based on the pattern you observein the specific cases.29. Conjecture: The sum of any two odd numbers is ______?1 + 1 = = 181 + 3 = = 323 + 5 = = 50630. Conjecture: The product of any two odd numbers is ____?1 x 1 = x 11 = 771 x 3 = 3 13 x 19 = 2473 x 5 = x 305 = 61,305Adding Odds - Version 2For 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 theconjecture based on the pattern you observein the specific cases.29. Conjecture: The sum of any two odd numbers is ______?1 + 1 = = 181 + 3 = = 323 + 5 = = 50630. Conjecture: The product of any two odd numbers is ____?1 x 1 = 1 7 x 11 = 771 x 3 = 3 13 x 19 = 2473 x 5 = x 305 = 61,305Take a few minutes and consider how they are the same and how they are different AND the opportunities to engage in the mathematical practices that are afforded by the task.
8Comparing Two Versions of a Task SameBoth ask students to complete a conjecture about odd numbers based on a set of finite examples that are providedDifferentV2 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 upThe number of ways to enter and solve the problemLet's take, for example, two versions of the Odd and Even task. In version one, we see that students are simply asked to fill in the blank after looking at a small number of examples. When they look at the sum of two odd numbers, they need to be able to say that the sum is in fact even. When they look at the product of two odd numbers, they simply need to say that the product is odd. There's nothing else that's required of the student.When you look at version two, however, a slight modification of the task, we see that students actually have to explain why it does or doesn't work the way it is. They have to come up with a reason why the sum of two odd numbers is even or provide an example that shows that that is not true.So, students really have to think about what they know and figure out how to frame an argument that would be convincing to somebody else. So in version one, very little is required of students. We would call that a low-level task. It's simply checking on a piece of knowledge that students should have acquired somewhere along the way. In version two, we see a very different kind of task where students have to make sense of the situation and justify the answer that they've come up with. This would be a high-level task.Now, all students should have the opportunity to engage in high-level tasks and they can as long as the task is at a level that's appropriate for the students and provides them with sufficient access to the problem.Both tasks ask students to make conjectures but Task A does not require students to think about why this is true. Hence there is no pressing of the student to figure out what this works the way it does. Furthermore, task A could leave students with the impression that finding a pattern and making a conjecture is sufficient to convince something that something is always true. (It isn’t!)
9Comparing Two Versions of a Task Opportunities to Engage in the Mathematical Practices MP 7 – look for and make use of structureVersion 2MP 7 – look for and make use of structureMP1 – Make sense of problems and persevere in solving themMP3 – Construct viable arguments and critique the reasoning of othersMP5 – Use appropriate tools strategicallyMP5 – we saw students using tiles and drawing pictures
10Characteristics 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)Here is a list of characteristics that might be helpful to you in identifying tasks that aligned with SMP – you have a copy of this list on the back of the mathematical practices handout.Lets consider version 2 of adding odd numbers task through this lens.
11Comparing 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.2 tasks are very similar in the potential to engage in the practices
12Tiling a PatioAlfredo 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 1Patio 2Patio 3
13Tiling a Patio: Aligned with SMP? High cognitive demand - no specified pathway to follow, requires students to explore relationships Significant content - equivalence, rate of changeRequire justification or explanation - explain in d and eMake connections between two or more representations - connect rule to visual; could also connect with tables and graphsOpen-ended - different descriptions and rules can be written and in different formsMultiple ways to enter the task and to show competence (Lotan, 2003)-- build patios, draw pictures, make tables, write equations, draw graphsBoth versions would be considered high level because there is no explicit pathway given to solving the task, so students do have to figure that out. They have to determine what's the underlying structure of the pattern and how are they going to use that to explain how you would find the number of tiles in any patio in the sequence. But in version one, students are asked first and only to create an equation that would be used to find the number of tiles in the patio, of any patio.But, in version two, the task begins by asking students to draw the next two patios in sequence and then to make as many observations as they can about the patios that they have in front of them. Now I’m going to argue that the first two questions in version two allow students a foothold in the problem.ENABLING PROMPLTS It allows them an opportunity to do something and by trying to get out all the things that they notice about the patios, by attending to how the patios change as you move from one to the next. The teacher then has access to students’ thinking and what they already know and understand about the situation and can ask additional questions based on what they’ve learned.
14Mathematical 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
15Mathematical 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
16Mathematical 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
17Mathematical 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
18Mathematical 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, 1996START is the operative word here. CDMT are a starting point but they provide no guarantee that students will actually engage in them as intended or that the desired mathematics will emerge from having done so.It is the discussion around such tasks that is both critically important and incredibly difficult.
19ToolsTools 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).Encourage the use of tools (i.e., language, materials, and symbols) to provide external support for learning (Hiebert, et al, 1997). Tools are used to represent mathematical ideas. They can be suggested by the teacher or invented by students.As Hiebert and his colleagues noted, “using tools enables students to develop deeper meanings of the mathematics that the tools are being used to examine. This is especially true as students explore relationships between the tools. So it seems that developing meaning for mathematical tools and for mathematics are all wrapped up together” (Hiebert et al., 1997, p.62).The five different representations of mathematical ideas shown in figure (Van de Walle, Karp, & Kay-Williams, 2013, p. 24), an expansion of the tools discussed by Hiebert et al. Research suggest that good problem solvers are able to move flexibly between and among different representational forms (pictures, written symbols, oral language real-world situations, manipulative models) and that strengthening these translation abilities facilitates learning (Lesh, Post, and Behr, 1987).
20Representations as Tools PicturesWritten SymbolsManipulative ModelsReal-worldSituationsOralLanguageLesh, Post, and Behr, 1987
21ToolsAdding Odds TaskSquare tiles that can be used to build the rectangular modelDrawing of dots that can be group by twoUse of symbolic notation2x is even; 2x + 1 oddThe tools you provide make a difference – they allow entry to students.
22The Fencing TaskMs. 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
23The Fencing TaskWhat tools could you provide that would help students engage in this task?What difference do you think the tools would make?
24Fencing Task Approaches Build pens with physical materialsDraw pens on grid paperMake a table of the dimensions of possible pensMake a graph that shows the relationship between one linear dimension and the areaSet up an algebraic equation and solve
25Fencing 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 pensMake 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
26TalkStudents 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
27The 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.
28The Case of Darcy DunnTeacher 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 classTeacher pressed for explanations and encouraged students to questions each otherTeacher engaged the class in creating a mathematical model that was consistent with the verbal description given by a student and the diagram of the patioTeacher gave homework that required providing and justifying a conclusion to the question, “Can they all be right?”The thing to consider is the way tasks, tools, and talk work to support students engagement and learning.
29The Case of Darcy DunnEvidence that students were engaged in the mathematical practicesMP1 – 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 generalizeThe thing to consider is the way tasks, tools, and talk work to support students engagement and learning.
30ReflectIn what ways might the Task, Tools, and Talk framework help you in your work with teachers?
32Draw a PictureEvery 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.
33Build a ModelIf 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.
34Use Algebra If a and b are odd integers, then a and b can be written If a and b are odd integers, then a and b can be writtena = 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.
35Logical 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 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.
36Tiling a Patio Using a Visual Model to Find a Pattern of Growth T = 2p + 6T = 2(p + 2) + 2Build the patios with two color tilesDraw Patios on Grid PaperT = 3(p + 2) - p
37Tiling a Patio Making a Table to Find the Pattern of Growth
38Using a Graph to Determine the Pattern of Growth Tiling a PatioUsing a Graph to Determine the Pattern of Growth
42The Fencing Task Using a Table LengthWidthPerimeterArea1112421020392748325735636The table shows that all the configurations have a perimeter of 24, but different areas. The area for the 6 x 6 pen is the largest; both before and after that, the areas are smaller than 36 ft2.
45The Fencing Task Equation and Graph P = 2l + 2w24 = 2l + 2w12 = l + wl = wA = l x wA = l(12 – l)A = 12l – l2The total perimeter of the pen has to be 24 ft, which can be thought of as two lengths plus two widths, or 2l + 2w. The length and width have to add up to 12; therefore, we can write the length as l and the width as 12 – l.The area of the pen is found by multiplying the length and the width (A = lw). In this case, the area can be expressed by the equation A = l(12 – l) or A = 12l – l2.This equation, when graphed, is the equation of a parabola. (See graph below.)The parabola graphed below has a maximum value at (6,36). Thus, a length of 6 and a width of 12 – 6 = 6 gives the greatest area of the pen.
46The Fencing Task Equation 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 – l2A’(l) = 12 – 2l12 – 2l = 0l = 6If l is 6, then the width is 12 – 6 or 6. Thus, theconfiguration with the maximum area is 6 x 6,an equation that models the area of the pen is A = 12l – l2. This is a quadratic equation of a parabola that has a maximum (we know this because the coefficient in front of l2 is negative). 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 – l2A’(l) = 12 – 2l12 – 2l = 0l = 6If l is 6, then the width is 12 – 6 or 6. Thus, the configuration with the maximum area is 6 x 6,