Presentation Transcript

1. Study Group 3 - High School Math (Algebra 1 & 2, Geometry) Welcome Back! Let’s spend some quality time discussing what we learned from our Bridge to Practice exercises.
2. Let’s Go Over Bridge to Practice #2: Time to Reflect on Our Learning Part 1: For Algebra 1, Using the Bike and Truck Task: For Algebra 2, Using the Missing Function Task: For Geometry, Using the Building a New Playground Task: a. Choose the Content Standards from the relevant pages in your module 2 handout (or view the standards on the following slides for each subject area Alg 1: 6-9, Alg 2: 16-19, Geometry: 25-28) b. Choose the Practice Standards students will have the opportunity to use while solving this task and find evidence to support them.
3. For Algebra 1: Bike and Truck Task Distance from start of road (in feet) Time (in seconds) A bicycle traveling at a steady rate and a truck are moving along a road in the same direction. The graph below shows their positions as a function of time. Let B(t) represent the bicycle’s distance and K(t) represent the truck’s distance.
4. For Algebra 1 - Bike and Truck Task Label the graphs appropriately with B(t) and K(t). Explain how you made your decision. Describe the movement of the truck. Explain how you used the values of B(t) and K(t) to make decisions about your description. Which vehicle was first to reach 300 feet from the start of the road? How can you use the domain and/or range to determine which vehicle was the first to reach 300 feet? Explain your reasoning in words. Jack claims that the average rate of change for both the bicycle and the truck was the same in the first 17 seconds of travel. Explain why you agree or disagree with Jack and why.
5. Algebra 1 - Reflecting on Our Learning Which CCSS for Mathematical Content did we discuss? Which CCSS for Mathematical Practice did you use when solving the task?
6. The CCSS for Mathematical ContentAlgebra 1 TaskCCSS Conceptual Category – Algebra *Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard. Common Core State Standards, 2010, p. 65, NGA Center/CCSSO
7. The CCSS for Mathematical ContentAlgebra 1 Task CCSS Conceptual Category – Algebra Common Core State Standards, 2010, p. 65, NGA Center/CCSSO
8. The CCSS for Mathematical ContentAlgebra 1 TaskCCSS Conceptual Category – Algebra Common Core State Standards, 2010, p. 65, NGA Center/CCSSO
9. The CCSS for Mathematical ContentAlgebra 1 TaskCCSS Conceptual Category – Functions Common Core State Standards, 2010, p. 69, NGA Center/CCSSO
10. For Algebra 1 Task:What standards for mathematical practice made it possible for us to learn? Common Core State Standards for Mathematics, 2010 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.
11. Part 3 - Underlying Mathematical Ideas Related to the Lesson – For Algebra 1 (Essential Understandings) The language of change and rate of change (increasing, decreasing, constant, relative maximum or minimum) can be used to describe how two quantities vary together over a range of possible values.  A rate of change describes how one variable quantity changes with respect to another – in other words, a rate of change describes the covariation between two variables (NCTM, EU 2b). The average rate of change is the change in the dependent variable over a specified interval in the domain.  Linear functions are the only family of functions for which the average rate of change is the same on every interval in the domain.
12. Essential Understandings – Algebra 1 Cooney, T.J., Beckmann, S., & Lloyd, G.M., & Wilson, P.S. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12 (p. 8-10). Reston, VA: National Council of Teachers of Mathematics.
13. Essential Understandings – Algebra 1 Cooney, T.J., Beckmann, S., & Lloyd, G.M., & Wilson, P.S. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12 (p. 8-10). Reston, VA: National Council of Teachers of Mathematics.
14. For Algebra 2: Missing Function Task If h(x) = f(x) · g(x), what can you determine about g(x) from the given table and graph? Explain your reasoning.
15. Algebra 2 - Reflecting on Our Learning Which CCSS for Mathematical Content did we discuss? Which CCSS for Mathematical Practice did you use when solving the task?
16. The CCSS for Mathematical Content – Alg 2 TaskCCSS Conceptual Category – Number and Quantity Common Core State Standards, 2010, p. 60, NGA Center/CCSSO
17. The CCSS for Mathematical Content- Alg 2 Task CCSS Conceptual Category – Algebra Common Core State Standards, 2010, p. 64, NGA Center/CCSSO
18. The CCSS for Mathematical Content – Alg 2 TaskCCSS Conceptual Category – Algebra Common Core State Standards, 2010, p. 64, NGA Center/CCSSO
19. The CCSS for Mathematical Content – Alg 2 TaskCCSS Conceptual Category – Functions Common Core State Standards, 2010, p. 70, NGA Center/CCSSO
20. For Algebra 2 Task:What math practices made it possible for us to learn? Common Core State Standards for Mathematics, 2010 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.
21. Part 3 - Underlying Mathematical Ideas Related to the Lesson – For Algebra 2 (Essential Understandings) The product of two or more linear functions is a polynomial function.  The resulting function will have the same x-intercepts as the original functions because the original functions are factors of the polynomial. Two or more polynomial functions can be multiplied using the algebraic representations by applying the distributive property and combining like terms. Two or more polynomial functions can be added using their graphs or tables of values because given two functions f(x) and g(x) and a specific x-value, x1, the point (x1, f(x1)+g(x1)) will be on the graph of the sum f(x)+g(x). (This is true for subtraction and multiplication as well.)
22. For Geometry:Building a New Playground Task The City Planning Commission is considering building a new playground. They would like the playground to be equidistant from the two elementary schools, represented by points A and B in the coordinate grid that is shown.
23. For Geometry:Building a New Playground PART A Determine at least three possible locations for the park that are equidistant from points A and B. Explain how you know that all three possible locations are equidistant from the elementary schools. Make a conjecture about the location of all points that are equidistant from A and B. Prove this conjecture. PART B The City Planning Commission is planning to build a third elementary school located at (8, -6) on the coordinate grid. Determine a location for the park that is equidistant from all three schools. Explain how you know that all three schools are equidistant from the park. Describe a strategy for determining a point equidistant from any three points.
24. Geometry - Reflecting on Our Learning Which CCSS for Mathematical Content did we discuss? Which CCSS for Mathematical Practice did you use when solving the task?
25. The CCSS for Mathematical ContentCCSS Conceptual Category – Geometry Common Core State Standards, 2010, p. 76, NGA Center/CCSSO
26. The CCSS for Mathematical ContentCCSS Conceptual Category – Geometry Common Core State Standards, 2010, p. 76, NGA Center/CCSSO
27. The CCSS for Mathematical ContentCCSS Conceptual Category – Geometry Common Core State Standards, 2010, p. 77, NGA Center/CCSSO
28. The CCSS for Mathematical ContentCCSS Conceptual Category – Geometry Common Core State Standards, 2010, p. 78, NGA Center/CCSSO
29. For Geometry Task:Which Standards for Mathematical Practice made it possible for us to learn? Common Core State Standards for Mathematics, 2010 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.
30. Part 3 - Underlying Mathematical Ideas Related to the Lesson - For Geometry(Essential Understandings) Coordinate Geometry can be used to form and test conjectures about geometric properties of lines, angles and assorted polygons. Coordinate Geometry can be used to prove geometric theorems by replacing specific coordinates with variables, thereby showing that a relationship remains true regardless of the coordinates. The set of points that are equidistant from two points A and B lie on the perpendicular bisector of line segment AB, because every point on the perpendicular bisector can be used to construct two triangles that are congruent by reflection and/or Side-Angle-Side; corresponding parts of congruent triangles are congruent. It is sometimes necessary to prove both 'If A, then B' and 'If B, then A' in order to fully prove a theorem; this situation is referred to as an "if and only if" situation; notations for such situations include <=> and iff.
31. Part 2 - Research Connection: Findings by Tharp and GallimoreThis slide pertains to Alg 1, Alg 2, & Geometry Tharp & Gallimore, 1991 For teaching to have occurred - Teachers must “be aware of the students’ ever-changing relationships to the subject matter.” They [teachers] can assist because, while the learning process is alive and unfolding, they see and feel the student's progression through the zone, as well as the stumbles and errors that call for support. For the development of thinking skills—the [students’] ability to form, express, and exchange ideas in speech and writing—the critical form of assisting learners is dialogue -- the questioning and sharing of ideas and knowledge that happen in conversation.
32. End of review of Bridge to Practice #2 Now we will move into our new Study Group Module 3 which is divided into two parts: The impact of teacher implementation of a high level task on student learning Using assessing and advancing questions to support student learning

33. Supporting Rigorous Mathematics Teaching and Learning

    Part 1 Enacting Instructional Tasks: Maintaining the Demands of the Tasks Tennessee Department of Education High School Mathematics
34. Using the Assessment to Think About Instruction In order for students to perform well on the Constructed Response Assessments (CRAs), what are the implications for instruction? What kinds of instructional tasks will need to be used in the classroom? What will teaching and learning look like and sound like in the classroom?
35. Rationale Effective teaching requires being able to support students as they work on challenging tasks without taking over the process of thinking for them (NCTM, 2000). By analyzing the classroom actions and interactions of six teachers enacting the same high-level task, teachers will begin to identify classroom-based factors that are associated with supporting or inhibiting students’ high-level engagement during instruction.
36. Session Goals Participants will: learn about characteristics of the written task that impact students’ opportunities to think and reason about mathematics learn about the factors of implementation that contribute to the maintenance and decline of thinking and reasoning analyze student work to determine what students know and can do develop assessing and advancing questions based on student work (this will be part of the Bridge to Practice #3)
37. The Mathematical Tasks Framework TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning Stein M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development, p. 4. New York: Teachers College Press.
38. The Enactment of the Task(Private Think Time) Read the vignettes. Consider the following question: What are students learning in each classroom? Scenario A – Mrs. Fox Scenario B – Mr. Chambers Scenario C – Ms. Fagan Scenario D – Ms. Jackson Scenario E – Mr. Cooper Scenario F– Ms. Gorman
39. The Enactment of the Task(Small Group Discussion) Discuss the following questions and cite evidence from the cases: What are students learning in each classroom? What made it possible for them to learn?
40. The Enactment of the Task(Whole Group Discussion) What opportunities did students have to think and reasonin each of the classes?
41. TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning Research Findings: The Fate of Tasks
42. Linking to Research/Literature: The QUASAR Project How High-Level Tasks Can Evolve During a Lesson: Maintenance of high-level demands. Decline into procedures without connection to meaning. Decline into unsystematic and nonproductive exploration. Decline into no mathematical activity.
43. Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands Decline Problematic aspects of the task become routinized. Understanding shifts to correctness, completeness. Insufficient time to wrestle with the demanding aspects of the task. Classroom management problems. Inappropriate task for a given group of students. Accountability for high-level products or processes not expected.
44. Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands Maintenance Scaffolds of student thinking and reasoning provided. A means by which students can monitor their own progress is provided. High-level performance is modeled. A press for justifications, explanations through questioning and feedback. Tasks build on students’ prior knowledge. Frequent conceptual connections are made. Sufficient time to explore.
45. Linking to Research/Literature: The QUASAR Project Student Learning Task Set-Up Task Implementation High High High A. Low Low Low B. Moderate High Low C. Stein & Lane, 1996
46. 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
47. Supporting Rigorous Mathematics Teaching and Learning Part 2 Illuminating Student Thinking: Assessing and Advancing Questions Tennessee Department of Education High School Mathematics
48. Rationale Effective teaching requires being able to support students as they work on challenging tasks without taking over the process of thinking for them (NCTM, 2000). Asking questions that assess student understanding of mathematical ideas, strategies or representations provides teachers with insights into what students know and can do. The insights gained from these questions prepare teachers to then ask questions that advancestudent understanding of mathematical ideas, strategies or connections to representations.
49. The Structures and Routines of a Lesson MONITOR: Teacher selects examples for the Share, Discuss, and Analyze Phase based on: Different solution paths to the same task Different representations Errors Misconceptions Set Up of the Task The Explore Phase/Private Work Time Generate Solutions The Explore Phase/Small Group Problem Solving Generate and Compare Solutions Assess and Advance Student Learning 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: By engaging students in a quick write or a discussion of the process. 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
50. Small Groups based on subject area(Algebra 1, Algebra 2, or Geometry) Participants will: analyze given student work for their subject area to determine what the students knowand what they can do based only on the evidence from student work
51. Bike and Truck Task – Algebra 1 Distance from start of road (in feet) Time (in seconds) A bicycle traveling at a steady rate and a truck are moving along a road in the same direction. The graph below shows their positions as a function of time. Let B(t) represent the bicycle’s distance and K(t) represent the truck’s distance.
52. Bike and Truck Task - Algebra 1 Label the graphs appropriately with B(t) and K(t). Explain how you made your decision. Describe the movement of the truck. Explain how you used the values of B(t) and K(t) to make decisions about your description. Which vehicle was first to reach 300 feet from the start of the road? How can you use the domain and/or range to determine which vehicle was the first to reach 300 feet? Explain your reasoning in words. Jack claims that the average rate of change for both the bicycle and the truck was the same in the first 17 seconds of travel. Explain why you agree or disagree with Jack.
53. What Does Each Student Know?Algebra 1 Now we will focus on three pieces of student work. Individually examine the three pieces of student work A, B, and C for the Bike and Truck Task in your participant handout. What does each student know? Be prepared to share and justify your conclusions.
54. Response A - Algebra 1 54
55. Response B - Algebra 1 55
56. Response C - Algebra 1 56
57. Missing Function Task – Algebra 2 If h(x) = f(x) · g(x), what can you determine about g(x) from the given table and graph? Explain your reasoning.
58. What Does Each Student Know?Algebra 2 Now we will focus on three pieces of student work. Individually examine the three pieces of student work A, B, and C for theMissing Function Task in your Participant Handout. What does each student know? Be prepared to share and justify your conclusions.
59. Response A – Algebra 2 59
60. Response B – Algebra 2 60
61. Response C – Algebra 2 61
62. Building a New Playground Task-Geometry The City Planning Commission is considering building a new playground. They would like the playground to be equidistant from the two elementary schools, represented by points A and B in the coordinate grid that is shown.
63. Building a New Playground - Geometry PART A Determine at least three possible locations for the park that are equidistant from points A and B. Explain how you know that all three possible locations are equidistant from the elementary schools. Make a conjecture about the location of all points that are equidistant from A and B. Prove this conjecture. PART B The City Planning Commission is planning to build a third elementary school located at (8, -6) on the coordinate grid. Determine a location for the park that is equidistant from all three schools. Explain how you know that all three schools are equidistant from the park. Describe a strategy for determining a point equidistant from any three points.
64. What Does Each Student Know?Geometry Now we will focus on three pieces of student work. Individually examine the three pieces of student work A, B, and C for theBuilding a New PlaygroundTaskin your Participant Handout. What does each student know? Be prepared to share and justify your conclusions.
65. Response A - Geometry 65
66. Response B - Geometry 66
67. Response C - Geometry 67
68. Group D - Cannot Get Started Imagine that you are walking around the room, observing your students as they work on the task for either Algebra 1 or 2, or Geometry. Group D has little or nothing on their papers. Consider an assessing question and an advancing question for Group D. Be prepared to share and justify your conclusions. Reminder: You cannot TELL Group D how to start. What questions can you ask them?
69. Before Beginning Bridge to Practice #3: As you complete your next Bridge to Practice, reflect on the Content Standards and Essential Understandings as needed to help focus our discussion For Algebra 1, Using the Bike and Truck Task: F-IF.B.4; F-IF.B.5; F-IF.B.6 For Algebra 2, Using the Missing Function Task: A-APR.A.1; A-APR (cluster); F-BF.A.1b For Geometry, Using the Building a New Playground Task: G-GPE.B.4; G-GPE.B.5; G-GPE.B.6 a
70. Bridge to Practice #3 Part A: Use the list developed of what the students know and what they can do from the Student Work A-D to develop questions to be asked during the Explore Phase of the lesson Develop at least one assessing question for Students A-D for your subject area Develop at least one advancing question for Students A-D for your subject area
71. Bridge to Practice #3 Part B: Now that you have solved the task, examined some student work, and developed your assessing and advancing questions, facilitate this task with your students and record your assessing and advancing questions during the small group explore phase of the lesson. Note: Record could be audio or video using a device such as your phone, or have a colleague script your questions for you. Come prepared to share the questioning from your lesson.