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© 2013 UNIVERSITY OF PITTSBURGH Mathematics Instruction: Planning, Teaching, and Reflecting Modifying Tasks to Increase the Cognitive Demand Tennessee Department of Education Elementary School Mathematics Grade 4

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Rationale There is wide agreement regarding the value of teachers attending to and basing their instructional decisions on the mathematical thinking of their students (Warfield, 2001). By engaging in an analysis of a lesson-planning process, teachers will have the opportunity to consider the ways in which the process can be used to help them plan and reflect, both individually and collectively, on instructional activities that are based on student thinking and understanding. 2

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© 2013 UNIVERSITY OF PITTSBURGH 3 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 By determining the cognitive demands of tasks and being cognizant of the features of tasks that make them high- level or low-level tasks, teachers will be prepared to select or modify tasks that create opportunities for students to engage with more tasks that are high-level tasks. Rationale

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© 2013 UNIVERSITY OF PITTSBURGH Session Goals Participants will: deepen understanding of the cognitive demand of a task; analyze a set of original and modified tasks to learn strategies for increasing the cognitive demand of a task; and recognize how increasing the cognitive demand of a task gives students access to the Common Core State Standards (CCSS) for Mathematical Practice. 4

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© 2013 UNIVERSITY OF PITTSBURGH Overview of Activities Participants will: discuss and compare the cognitive demand of mathematical tasks; identify strategies for modifying tasks; and modify tasks to increase the cognitive demand of the tasks. 5

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© 2013 UNIVERSITY OF PITTSBURGH 6 Mathematical Tasks: A Critical Starting Point for Instruction All tasks are not created equal−different tasks require different levels and kinds of student thinking. Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards- based mathematics instruction: A casebook for professional development, p. 3. New York: Teachers College Press.

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© 2013 UNIVERSITY OF PITTSBURGH 7 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, Olivier, & Human, 1997

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© 2013 UNIVERSITY OF PITTSBURGH 8 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

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© 2013 UNIVERSITY OF PITTSBURGH 9 Revisiting the Characteristics of Cognitively Demanding Tasks

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© 2013 UNIVERSITY OF PITTSBURGH Comparing the Cognitive Demand of Two Tasks Compare the two tasks. How are the tasks similar? How are the tasks different? 10

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© 2013 UNIVERSITY OF PITTSBURGH Task #1: A Place Value Task Identify the place value for each of the underlined digits. a.351 b.76 c.4,789 11

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© 2013 UNIVERSITY OF PITTSBURGH Task #2: What is Changing? Solve each equation = ___ = ___ = ___ = ___ = ___ = ___ When ten is added to each of the numbers above, how is the sum changing from one equation to the next? Sometimes the tens place changes and sometimes the hundreds place change when ten is added to the number. Why does this happen and when does it happen? Look at the number 2,399. Which numbers will change when ten is added to this number and why? 12

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© 2013 UNIVERSITY OF PITTSBURGH Linking to Research/Literature: The QUASAR Project Low-Level tasks –Memorization –Procedures without Connections High-Level tasks –Procedures with Connections –Doing Mathematics 13

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© 2013 UNIVERSITY OF PITTSBURGH The Mathematical Task Analysis Guide Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000) Implementing standards-based mathematics instruction: A casebook for professional development, p. 16. New York: Teachers College Press. 14

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TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning The Mathematical Tasks Framework Stein, Smith, Henningsen, & Silver, 2000 Linking to Research/Literature: The QUASAR Project 15

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TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning The Mathematical Tasks Framework Stein, Smith, Henningsen, & Silver, 2000 Linking to Research/Literature: The QUASAR Project Setting Goals Selecting Tasks Anticipating Student Responses Orchestrating Productive Discussion Monitoring students as they work Asking assessing and advancing questions Selecting solution paths Sequencing student responses Connecting student responses via Accountable Talk discussions 16

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© 2013 UNIVERSITY OF PITTSBURGH Identify Goals for Instruction and Select an Appropriate Task 17

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© 2013 UNIVERSITY OF PITTSBURGH The Structure and Routines of a Lesson The Explore Phase/Private Work Time Generate Solutions The Explore Phase/ Small Group Problem Solving 1.Generate and Compare Solutions 2.Assess and Advance Student Learning Share, Discuss, and Analyze Phase of the Lesson 1. Share and Model 2. Compare Solutions 3.Focus the Discussion on Key Mathematical Ideas 4. Engage in a Quick Write MONITOR: Teacher selects examples for the Share, Discuss, and Analyze Phase based on: Different solution paths to the same task Different representations Errors Misconceptions SHARE: Students explain their methods, repeat others’ ideas, put ideas into their own words, add on to ideas and ask for clarification. REPEAT THE CYCLE FOR EACH SOLUTION PATH COMPARE: Students discuss similarities and difference between solution paths. FOCUS: Discuss the meaning of mathematical ideas in each representation REFLECT: Engage students in a Quick Write or a discussion of the process. Set Up the Task Set Up of the Task 18

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The CCSS for Mathematics: Grade 4 Number and Operations – Fractions 4.NF Extend understanding of fraction equivalence and ordering. 4.NF.A.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 4.NF.A.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Common Core State Standards, 2010, p. 30, NGA Center/CCSSO 19

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The CCSS for Mathematics: Grade 4 Number and Operations – Fractions 4.NF Understand decimal notation for fractions, and compare decimal fractions. 4.NF.C.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/ NF.C.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. 4.NF.C.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. Common Core State Standards, 2010, p. 31, NGA Center/CCSSO 20

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Mathematical Practice Standards Related to the Task 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 21

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© 2013 UNIVERSITY OF PITTSBURGH Identify Goals: Solving the Task (Small Group Discussion) Revisit the Pizza Task. Solve the task. Discuss the possible solution paths to the task. 22

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© 2013 UNIVERSITY OF PITTSBURGH The Pizza Task

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© 2013 UNIVERSITY OF PITTSBURGH The Pizza Task (continued) Jolla’s Pizza Tim’s Pizza Juan’s Pizza Sarah’s Pizza Maria’s PizzaJake’s Pizza

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© 2013 UNIVERSITY OF PITTSBURGH Identify Goals Related to the Task (Whole Group Discussion) Does the task provide opportunities for students to access the Mathematical Content Standards and Practice Standards that we have identified for student learning? 25

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© 2013 UNIVERSITY OF PITTSBURGH 26 Giving it a Go: Modifying Textbook Tasks to Increase the Cognitive Demand of Tasks

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© 2013 UNIVERSITY OF PITTSBURGH 27 Your Turn to Modify Tasks Form groups of no more than three people. Discuss briefly important NEW mathematical concepts, processes, or relationships you will want students to uncover by the textbook page. Consult the CCSS. Determine the current demand of the task. Modify the textbook task by using one or more of the Textbook Modification Strategies. You will be posting your modified task for others to analyze and offer comments.

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© 2013 UNIVERSITY OF PITTSBURGH 28 Strategies for Modifying Textbook Tasks Increasing the cognitive demands of tasks: Ask students to create real-world stories for “naked number” problems. Include a prompt that asks students to represent the information another way (with a picture, in a table, a graph, an equation, with a context). Include a prompt that requires students to make a generalization. Use a task “out of sequence” before students have memorized a rule or have practiced a procedure that can be routinely applied. Eliminate components of the task that provide too much scaffolding.

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© 2013 UNIVERSITY OF PITTSBURGH 29 Strategies for Modifying Textbook Tasks (continued) Increasing the cognitive demands of tasks: Adapt a task so as to provide more opportunities for students to think and reason—let students figure things out for themselves. Create a prompt that asks students to write about the meaning of the mathematics concept. Add a prompt that asks students to make note of a pattern or to make a mathematical conjecture and to test their conjecture. Include a prompt that requires students to compare solution paths or mathematical relationships and write about the relationship between strategies or concepts. Select numbers carefully so students are more inclined to note relationships between quantities (e.g., two tables can be used to think about the solutions to the four, six, or eight tables).

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© 2013 UNIVERSITY OF PITTSBURGH 30 Gallery Walk Post the modified tasks. Circulate, analyzing the modified tasks. On a “Post-It” Note,” describe ways in which the tasks were modified and the benefit to students. If the task was not modified to increase the cognitive demand of the task, then ask a wondering about a way the task might be modified.

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© 2013 UNIVERSITY OF PITTSBURGH 31 The Cognitive Demand of Tasks Does the demand of the task matter? What are you now wondering about with respect to the task demands?

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The CCSS for Mathematics: Grade 4 Number and Operations – Fractions 4.NF Extend understanding of fraction equivalence and ordering. 4.NF.A.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 4.NF.A.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Common Core State Standards, 2010, p. 30, NGA Center/CCSSO 32

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The CCSS for Mathematics: Grade 4 Number and Operations – Fractions 4.NF Understand decimal notation for fractions, and compare decimal fractions. 4.NF.C.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/ NF.C.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. 4.NF.C.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. Common Core State Standards, 2010, p. 31, NGA Center/CCSSO 33

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© 2013 UNIVERSITY OF PITTSBURGH Accountable Talk Discussions Recall what you know about the Accountable Talk features and indicators. In order to recall what you know: Study the chart with the Accountable Talk moves. You are already familiar with the Accountable Talk moves that can be used to Ensure Purposeful, Coherent, and Productive Group Discussion. Study the Accountable Talk moves associated with creating accountability to: the learning community; knowledge; and rigorous thinking. 34

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© 2013 UNIVERSITY OF PITTSBURGH Accountable Talk Features and Indicators Accountability to the Learning Community Active participation in classroom talk. Listen attentively. Elaborate and build on each others’ ideas. Work to clarify or expand a proposition. Accountability to Knowledge Specific and accurate knowledge. Appropriate evidence for claims and arguments. Commitment to getting it right. Accountability to Rigorous Thinking Synthesize several sources of information. Construct explanations and test understanding of concepts. Formulate conjectures and hypotheses. Employ generally accepted standards of reasoning. Challenge the quality of evidence and reasoning. 35

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© 2013 UNIVERSITY OF PITTSBURGH Accountable Talk Moves FunctionExample To Ensure Purposeful, Coherent, and Productive Group Discussion MarkingDirect attention to the value and importance of a student’s contribution. That’s an important point. One factor tells use the number of groups and the other factor tells us how many items in the group. ChallengingRedirect a question back to the students or use students’ contributions as a source for further challenge or query. Let me challenge you: Is that always true? RevoicingAlign a student’s explanation with content or connect two or more contributions with the goal of advancing the discussion of the content. S: You said three groups of four. RecappingMake public in a concise, coherent form, the group’s achievement at creating a shared understanding of the phenomenon under discussion. Let me put these ideas all together. What have we discovered? To Support Accountability to Community Keeping the Channels Open Ensure that students can hear each other, and remind them that they must hear what others have said. Say that again and louder. Can someone repeat what was just said? Keeping Everyone Together Ensure that everyone not only heard, but also understood, what a speaker said. Can someone add on to what was said? Did everyone hear that? Linking Contributions Make explicit the relationship between a new contribution and what has gone before. Does anyone have a similar idea? Do you agree or disagree with what was said? Your idea sounds similar to his idea. Verifying and Clarifying Revoice a student’s contribution, thereby helping both speakers and listeners to engage more profitably in the conversation. So are you saying..? Can you say more? Who understood what was said? 36

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© 2013 UNIVERSITY OF PITTSBURGH To Support Accountability to Knowledge Pressing for Accuracy Hold students accountable for the accuracy, credibility, and clarity of their contributions. Why does that happen? Someone give me the term for that. Building on Prior Knowledge Tie a current contribution back to knowledge accumulated by the class at a previous time. What have we learned in the past that links with this? To Support Accountability to Rigorous Thinking Pressing for Reasoning Elicit evidence to establish what contribution a student’s utterance is intended to make within the group’s larger enterprise. Say why this works. What does this mean? Who can make a claim and then tell us what their claim means? Expanding Reasoning Open up extra time and space in the conversation for student reasoning. Does the idea work if I change the context? Use bigger numbers? Accountable Talk Moves (continued) 37

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© 2013 UNIVERSITY OF PITTSBURGH An Example: Accountable Talk Discussion 38

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© 2013 UNIVERSITY OF PITTSBURGH An Example: Accountable Talk Discussion 39

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© 2013 UNIVERSITY OF PITTSBURGH Reflecting: The Accountable Talk Discussion The observer has 2 minutes to share observations related to the lessons. The observations should be shared as “noticings.” Others in the group have 1 minute to share their “noticings.” 40

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© 2013 UNIVERSITY OF PITTSBURGH Step Back and Application to Our Work What have you learned today that you will apply when planning or teaching in your classroom? 41

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