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**Alabama College- and Career-Ready Standards for Mathematics**

Thinking Through A Lesson Alabama College- and Career-Ready Standards for Mathematics Quarterly Meeting #3

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**Thinking Through a Lesson: Successfully Implementing High-Level Tasks Session I:**

Outcomes: Discuss and examine the thought process of developing a lesson/unit that is standards-based. Select rich tasks that match student friendly outcomes and provide evidence of student learning.

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**The effectiveness of a lesson depends significantly on the **

Why Lesson Planning? The effectiveness of a lesson depends significantly on the care with which the lesson plan is prepared. (Brahier, 2000) SO I WANT TO ARGUE THAT THIS IS A HLP 3

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Why Lesson Planning? Good planning “shoulders much of the burden” of teaching by replacing “on the fly" decision making” during a lesson with careful investigation into the what and how of instruction before the lesson is taught. (Stigler & Hiebert, 1999, p.156) SO I WANT TO ARGUE THAT THIS IS A HLP 4

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**Why Lesson Planning? During the planning phase, teachers make**

decisions that affect instruction dramatically. They decide what to teach, how they are going to teach, how to organize the classroom, what routines to use, and how to adapt instruction for individuals. (Fennema & Franke, 1992, p. 156) SO I WANT TO ARGUE THAT THIS IS A HLP

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**An effective mathematical task is needed to challenge and engage students intellectually.**

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**Read an excerpt from the article:**

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**What causes high- level cognitive demand tasks to decline?**

Have participants discuss the challenges they experience in implementing cognitively challenging tasks that promote thinking, reasoning, and problem solving. At the end of the discussion, suggest that this article presents a framework for lesson planning that may help with these challenges

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**Alabama College and Career Standards for Mathematics Addressed**

2nd Grade: Reason with shapes and their attributes. 25. Partition a rectangle into rows and columns of same-size squares, and count to find the total number of them. [2-G2] 3rd Grade: Geometric measurement: understand concepts of area and relate area to multiplication and to addition. 20. Recognize area as an attribute of plane figures, and understand concepts of area measurement. [3-MD5] a. A square with side length 1 unit called ―a unit square,‖ is said to have ―one square unit‖ of area and can be used to measure area. [3-MD5a] b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. [3-MD5b] 21. Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). [3-MD6] 22. Relate area to the operations of multiplication and addition. [3-MD7] a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. [3-MD7a] b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real-world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning. [3-MD7b] c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning. [3-MD7c] d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into nonoverlapping rectangles and adding the areas of the nonoverlapping parts, applying this technique to solve real-world problems. [3-MD7d] Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures. 23. Solve real-world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters. [3-MD8] 4th Grade: Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. 21. Apply the area and perimeter formulas for rectangles in real-world and mathematical problems. [4-MD3] Example: Find the width of a rectangular room given the area of the flooring and the length by viewing the area formula as a multiplication equation with an unknown factor. 6th Grade: Solve real-world and mathematical problems involving area, surface area, and volume. 21. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. [6-G1] 7th Grade: Draw, construct, and describe geometrical figures and describe the relationships between them. 11. Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. [7-G1] Solve real-world and mathematical problems involving angle measure, area, surface area, and volume. 14. Know the formulas for the area and circumference of a circle, and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. [7-G4] Geometry Use coordinates to prove simple geometric theorems algebraically. (Include distance formula; relate to Pythagorean Theorem.) 34. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.* [G-GPE7] Find arc lengths and areas of sectors of circles. (Radian introduced only as unit of measure.) 29. Derive, using similarity, the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. [G-C5]

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**Comparing Two Mathematical Tasks**

Solve Two Tasks: Martha’s Carpeting Task The Fencing Task Participants work each task on their own first, and then share with partner at their table. Note various solution paths used by participants. Prepare to have them share solutions in the following order Martha’s carpeting Formula Diagram Fencing Task Trial and error Ordered set of diagrams Information organized in a table Algebraic equation in vertex form Graph of parabola showing vertex Calculus using derivative Chose participants to share out to the whole group. Record solutions on chart paper. In small groups, have them complete Similarities and Differences table. Share out responses in whole group. Then ask, “Do the differences between the Fencing Task and Martha’s Carpeting Task matter? Why or why not?” SAS Secondary Mathematics Teacher Leadership Academy, Year 1

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**Comparing Two Mathematical Tasks**

How are Martha’s Carpeting Task and the Fencing Task the same and how are they different? Participants work each task on their own first, and then share with partner at their table. Note various solution paths used by participants. Prepare to have them share solutions in the following order Martha’s carpeting Formula Diagram Fencing Task Trial and error Ordered set of diagrams Information organized in a table Algebraic equation in vertex form Graph of parabola showing vertex Calculus using derivative Chose participants to share out to the whole group. Record solutions on chart paper. In small groups, have them complete Similarities and Differences table. Share out responses in whole group. Then ask, “Do the differences between the Fencing Task and Martha’s Carpeting Task matter? Why or why not?”

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**Comparing Two Mathematical Tasks**

Do the differences between the Fencing Task and Martha’s Carpeting Task matter? Why or Why not? Then ask, “Do the differences between the Fencing Task and Martha’s Carpeting Task matter? Why or why not?” If there is time, you may want participants to generate similarities and differences before sharing the upcoming slide.

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**Similarities and Differences**

Both are “area” problems Both require prior knowledge of area Differences The amount of thinking and reasoning required The number of ways the problem can be solved Way in which the area formula is used The need to generalize The range of ways to enter the problem Way in which area formula is used: Martha’s Carpeting can be solved by knowing and using the area formula but this formula alone is not sufficient to solve the Fencing Task The need to generalize: Martha’s carpeting does not lead to a generalization but the Fencing Task does The range of ways to enter the problem: Martha’s Carpeting Task cannot be started by a student who does not know the formula for area; the Fencing Task can be started in other ways, such as sketches on graph paper. 13

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**Does Maintaining Cognitive Demand Matter? YES **

Have participants discuss the challenges they experience in implementing cognitively challenging tasks that promote thinking, reasoning, and problem solving. At the end of the discussion, suggest that this article presents a framework for lesson planning that may help with these challenges

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**Linking to Literature/ Research: The QUASAR Project**

The Mathematical Tasks Framework TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Assisted schools in economically disadvantaged communities to develop instructional programs that emphasize thinking, reasoning and problem solving in mathematics. Worked with lowest achieving middle schools in six urban sites. Studied the impact of high quality curricula and professional development upon student achievement. The Math Tasks Framework is designed to consider the evolution of tasks during a lesson. The fact that tasks take on lives of their own after being introduced into classroom settings has been noted by a variety of classroom researchers. As mathematical tasks are enacted in classroom settings, they become intertwined with the goals, intentions, actions, and interactions of teachers and students. We will now consider each phase of the math task framework. The first phase – Tasks as they appear in curricular or instructional materials. The Set-up Phase includes the teacher’s communication to students regarding what they are expected to do, how they are expected to do it, and with what resources. The teachers set-up of the task can be as brief as directing students’ attention to a task that appears on the blackboard and telling them to start working on it. Or it can be as long ad involved as discussing how students should work on the problem in small groups, working through a sample problem, and discussing the forms of solutions that will be acceptable. It is not unusual for a teacher to alter the cognitive demand of the task as she is setting if up for the class. In other words, she may, either purposefully or unwittingly, change the task from how it appeared in the curricular or instructional print material from which she originally took the idea. The Implementation Phase starts as soon as the students begin to work on the task and continues until the teacher and students turn their attention to a new mathematical task. During the implementation phase, both students and the teacher are viewed as important contributors to how the task is carried out. Although the students’ levels of cognitive engagement ultimately determines what is learned, the ways and extent to which the teacher supports students’ thinking and reasoning is a crucial ingredient in the ultimate fate of high-level tasks. For example, teachers can promote sense-making and deeper levels of understanding by consistently asking students to explain how they are thinking about the task. Or, conversely, they may cut off opportunities hot sense-making by hurrying students through the tasks, thereby not allowing the time to grapple with perplexing ideas. The ultimate reason for focusing on instructional tasks is to influence student learning. Research has demonstrated that the cognitive demands of mathematical instructional tasks are related to the level and kind of student learning. Within the QUASAR project, students who performed best on the QUASAR Cognitive Assessment Instrument were in classrooms in which tasks were more likely to be set up and implemented at high levels of cognitive demand. For these students, having the opportunity to work on challenging tasks in a supportive classroom environment translated into substantial learning gains on an instrument specially designed to measure student thinking, reasoning, problem solving, and communication. This suggests the importance of being mindful, both at the outset and during the various task phases, of the kinds of cognitive activity with which the students should be and actually are engaged in the classroom. Student Learning Stein, Smith, Henningsen, & Silver, 2000, p. 4 SAS Secondary Mathematics Teacher Leadership Academy, Year 1

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**Patterns of Set up, Implementation, and Student Learning**

Task Set Up Task Implementation Student Learning A. High High High B. Low Low Low Have participants read and discuss the slide. Ask, What do you notice? What are the implications for instruction? Evidence gathered across scores of middle school classrooms in four QUASAR middle schools has shown that students who performed the best on project-based measures of reasoning and problem solving were in classrooms in which tasks were more likely to be set up and implemented at high levels of cognitive demand. Results from QUASAR also show that students who had the lowest performance on project assessments were in classrooms where they had limited exposure to tasks that required thinking and reasoning (Stein & Lane, 1996). C. High Low Moderate Stein & Lane, 2012 17

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**Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands**

Routinizing problematic aspects of the task Shifting the emphasis from meaning, concepts, or understanding to the correctness or completeness of the answer Providing insufficient time to wrestle with the demanding aspects of the task or so much time that students drift into off- task behavior Engaging in high-level cognitive activities is prevented due to classroom management problems Selecting a task that is inappropriate for a given group of students Failing to hold students accountable for high-level products or processes (Stein, Grover & Henningsen, 2012) This slide identifies the classroom factors that researchers observed when high-level tasks declined during a lesson.

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**Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands**

Scaffolding of student thinking and reasoning Providing a means by which students can monitor their own progress Modeling of high-level performance by teacher or capable students Pressing for justifications, explanations, and/or meaning through questioning, comments, and/or feedback Selecting tasks that build on students’ prior knowledge Drawing frequent conceptual connections Providing sufficient time to explore (Stein, Grover & Henningsen, 2012) By contrast, this slide identifies the classroom factors that researchers observed when the cognitive demands of high-level tasks were maintained during a lesson.

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**Maintenance Decline Scaffolding of student thinking and reasoning**

Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands Maintenance Scaffolding of student thinking and reasoning Providing a means by which students can monitor their own progress Modeling of high-level performance by teacher or capable students Pressing for justifications, explanations, and/or meaning through questioning, comments, and/or feedback Selecting tasks that build on students’ prior knowledge Drawing frequent conceptual connections Providing sufficient time to explore Decline Routinizing problematic aspects of the task Shifting the emphasis from meaning, concepts, or understanding to the correctness or completeness of the answer Providing insufficient time to wrestle with the demanding aspects of the task or so much time that students drift into off-task behavior Engaging in high-level cognitive activities is prevented due to classroom management problems Selecting a task that is inappropriate for a given group of students Failing to hold students accountable for high-level products or processes 20

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“Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.” (Stein, Smith, Henningsen, & Silver, 2011) “The level and kind of thinking in which students engage determines what whey will learn.” (Hiebert et al., 2011)

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[Common Core State Standards for Mathematics] represent a significant departure from what mathematics is currently taught in most classrooms and how it is taught. Developing teachers’ capacity to enact these new standards in ways that support the intended student learning outcomes will require considerable changes in mathematics instruction in our nation’s classrooms. Such changes are likely to occur only through sustained and focused professional development opportunities for those who teach mathematics. Sztajn, Marrongelle, & Smith, 2011

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Job-Alike Reflection 3. How can we help teachers improve their capacity to plan (and enact) lessons that support students’ learning? 4. How would you conduct professional development with teachers to see how lesson planning addresses content rigor, task selection, student engagement and math practices, formative assessment, generating student work samples for assessment, and assisting struggling students? Have participants discuss the challenges they experience in implementing cognitively challenging tasks that promote thinking, reasoning, and problem solving. At the end of the discussion, suggest that this article presents a framework for lesson planning that may help with these challenges

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**Thinking Through a Lesson: Successfully Implementing High-Level Tasks Session 2**

Outcomes: Develop student friendly outcomes that reflect the rigor and depth of content. Select rich tasks that match student friendly outcomes and provide evidence of student learning. Model teacher collaboration on identifying resources for implementing CCRS Math Standards.

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**Have participants count off into triads.**

Read their section; Come back and work with groups to set up task.

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**Thinking Through a Lesson Protocol (TTLP) Planning Template**

Learning Goals (Residue) What understandings will students take away from this lesson? Evidence What will students say, do, produce, etc. that will provide evidence of their understandings? Task What is the main activity that students will be working on in this lesson? Instructional Support—Tools, Resources What tools or resources will students have to use in their work that will give them entry to, and help them reason through, the activity? Task Enactment What are the various ways that students might complete the activity? Instructional Support—Teacher What questions might you ask students that will support their exploration of the activity and bridge between what they did and what you want them to learn? To be clear on what students actually did, begin by asking a set of assessing questions such as: What did you do? How did you get that? What does this mean? Once you have a clearer sense of what the student understands, move on to appropriate set of questions below. Sharing and Discussing the Task Selecting and Sequencing Which solutions do you want to have shared during the lesson? In what order? Why? Connecting Responses What specific questions will you ask so that students make sense of the mathematical ideas that you want them to learn make connections among the different strategies/solutions that are presented

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**Thinking Through a Lesson Protocol Backwards Planning**

What will you see or hear that lets you know students are developing understanding of the concepts? What questions will you need to ask to build mathematical understanding? What mathematical concepts will be developed in the implementation of this task? What do you expect your students to do as they engage in the lesson? Share, Discuss, and Analyze Set Up Explore SAS Secondary Mathematics Teacher Leadership Academy, Year 1

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**Please do not spend time going to Internet to identify standards**

Please do not spend time going to Internet to identify standards. Participants can use the Math Course of Study to write objectives. See the Mathematical Tasks Facilitators’ Guides to address content questions.

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K-2 Sample Task Dan’s older brother Sam collects car stickers for his scrapbook. Sam decides to give Dan some of his stickers. Sam gives Dan 7 stickers. Sam now has 12 stickers. How many stickers did Sam have before giving some to Dan?

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K-2 Sample Task

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3-5 Sample Task Use the digits 3, 4, 5 and 6 to complete each number sentence. A digit can only be used once in each number sentence. + = 108 + = 99 + = 81 - = 31 + = 90 - = 18

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3-5 Sample Task

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Grades 6 – 8 Sample Task REPRESENTING AND INTERPRETING PROPORTIONAL RELATIONSHIPS: Computer Games

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**Grades 6 – 8 Sample Task Slippery Slopes: Saving Money**

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Grades 9 – 12 Sample Task

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Grades 9 – 12 Sample Task

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**Reflection What are some of the potential benefits of using the TTLP?**

With which components of the lesson planning protocol do you feel especially comfortable? Which components would you like to focus on more? How can the ideas and tools from this activity support you with that? Reflection Have participants discuss the challenges they experience in implementing cognitively challenging tasks that promote thinking, reasoning, and problem solving. At the end of the discussion, suggest that this article presents a framework for lesson planning that may help with these challenges

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Conclusion Not all tasks are created equal -- they should provide a variety of opportunities for students to learn mathematics. High level tasks are the most difficult to carry out in a consistent manner. Engagement in cognitively challenging mathematical tasks leads to the greatest learning gains for students. Professional development is needed to help teachers build the capacity to enact high level tasks in ways that maintain the rigor of the task. 39

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5. Are there any aspects of your own thinking and/or practice that our work today has caused you to consider or reconsider? Explain. 6. Are there any aspects of your students’ mathematical learning that our work today has caused you to consider or reconsider? Explain. 6a. What would you like more information about? Job-Alike Reflection 40

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Resources

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Insidemathematics.org

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Insidemathematics.org

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**map.mathshell.org.uk/materials**

Mathematics Assessment Project (MAP) map.mathshell.org.uk/materials 45

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**map.mathshell.org.uk/materials**

Mathematics Assessment Project (MAP) 20 ready-to-use Lesson Units for Formative Assessment for high school. cross referenced to CCSS content and practices standards. (Ultimately 20 per grade 7-12) Summative assessments, aimed at “College- and Career- Readiness,” presented in two forms: (1) a Task Collection with each task cross-referenced to the CCSS, and (2) a set of Prototype Test Forms showing how the tasks might be assembled into balanced assessments. Professional development modules map.mathshell.org.uk/materials

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**MAP Formative Assessment Lessons**

Assessment task, individual work (15 min) Teacher reviews work, creates questions to improve solutions (Whole group discussion) Partner or small group task to increase understanding, address misconceptions Debriefing discussion Revision of work on original assessment 47

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**The Illustrative Mathematics Project**

Hyperlinked CCSS Developing a complete set of tasks for each standard Range of difficulty Simple illustrations of single standards to complex tasks spanning many standards. Provide a process for submitting, discussing, reviewing, and publishing tasks. Launch Team: Phil Daro, William McCallum (chair), Jason Zimba illustrativemathematics.org

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illuminations.nctm.org/Lessons.aspx

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mathedleadership.org

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commoncoretools.wordpress.com

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**Collaborative Team Tools**

Available at nctm.org

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