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Coherence & Instructional Practice in Grades 6-8: Connecting Learning and Supporting Students

Explore the importance of coherence across grades in math education and learn how to adapt lessons for students below grade level. Discover the progressions of content and identify prerequisites for standards. Gain insights into the vertical coherence challenge and mapping the progressions. Join us for a productive and interactive learning experience!

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Coherence & Instructional Practice in Grades 6-8: Connecting Learning and Supporting Students

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  1. Across Grade Coherence and Instructional Practice in Grades 6-8 July 2016

  2. ACROSS GRADE COHERENCE IN GRADES 6-8Welcome Back!

  3. ACROSS GRADE COHERENCE IN GRADES 6-8Thank You for Your Feedback! +

  4. ACROSS GRADE COHERENCE IN GRADES 6-8Norms That Support Our Learning • Take responsibility for yourself as a learner • Honor timeframes (start, end, activity) • Be an active and hands-on learner • Use technology to enhance learning • Strive for equity of voice • Contribute to a learning environment in which it is “safe to not know”

  5. ACROSS GRADE COHERENCE IN GRADES 6-8This Week “Do the math” Connect to our practice

  6. ACROSS GRADE COHERENCE IN GRADES 6-8 Today • Morning: Across Grade Coherence in Grades 6-8 • Afternoon: Instructional Practice in Grades

  7. ACROSS GRADE COHERENCE IN GRADES 6-8Morning Objectives • Participants will be able to clearly define the across-grade part of the Coherence shift. • Participants will be able to describe the rationale for across grade coherence. • Participants will be able identify at least 3 prerequisites for a standard in their grade. • Participants will be able to adapt a lesson for students below grade level by adding just-in-time scaffolds based on progressions of content.

  8. ACROSS GRADE COHERENCE IN GRADES 6-8Morning Agenda • Across Grade Coherence • Vertical Coherence Challenge • Mapping the Progressions • Tools for Understanding the Progressions • Adapting Lessons for Students Below Grade Level

  9. ACROSS GRADE COHERENCE IN GRADES 6-8 Across Grade Coherence Lena paid $18.96 for 3 pounds of coffee. How would you teach students to graph the proportional relationship between the number of pounds of coffee and the total cost?   

  10. ACROSS GRADE COHERENCE IN GRADES 6-8 What is the Right Order? Grade 6 Use ratio and rate reasoning to solve real-world and mathematical problems… by making tables of equivalent ratios, finding missing values in tables, and plotting the pairs of values on the coordinate plane. Grade 8 Graph proportional relationships, interpreting the unit rate as the slope of the graph; compare different proportional relationships in different ways. Grade 7 Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

  11. ACROSS GRADE COHERENCE IN GRADES 6-8 Coherence is Key “A focused, coherent progression of mathematics learning, with an emphasis on proficiency with key topics, should become the norm in elementary and middle school mathematics curricula. Any approach that continually revisits topics year after year without closure is to be avoided. By the term focused, the Panel means that curriculum must include (and engage with adequate depth) the most important topics underlying success in school algebra. By the term coherent, the Panel means that the curriculum is marked by effective, logical progressions from earlier, less sophisticated topics into later, more sophisticated ones. Improvements like those suggested in this report promise immediate positive results with minimal additional cost.” - National Mathematics Advisory Panel

  12. ACROSS GRADE COHERENCE IN GRADES 6-8The Progressions

  13. Across Grade Coherence: Learning is carefully connected across grades so that students can build new understanding onto foundations built in previous years. 

  14. ACROSS GRADE COHERENCE IN GRADES 6-8Vertical Coherence Challenge • In your groups, you have 13 standards on pieces of paper. Most standards come from the Ratios & Proportions domain in grades 6-8. • The standards are not labeled! • Determine which standards are prerequisites for other standards. • Note: There is more than one vertical strand. • Bonus: Can you determine which standards belong in which grade?

  15. ACROSS GRADE COHERENCE IN GRADES 6-8 A Picture of Coherence E F-BF.1 D 7.RP.1 C 8.F.2 F 6.RP.2 B 5.NF.3 K 7.RP.2 J 4.MD.2 A 8.EE.6 H 7.G.1 L 6.RP.3a I 8.EE.8 G 5.G.2 M 7.RP.3

  16. ACROSS GRADE COHERENCE IN GRADES 6-8Progressions of Content • How does understanding the progression of content support our understanding of grade-level content? • How does understanding the progression of content help us support students below grade level?

  17. ACROSS GRADE COHERENCE IN GRADES 6-8Standards Mapping Protocol: • Identify 3 prerequisite standards – the standards do not have to be in 3 different grades • Identify the aspects of rigor for each prerequisite. • Discuss with a partner: • How does each prerequisite support the standard? • Why is it important to pay attention to the rigor of the prerequisite standard? The Standards: Grade 6 - 6.EE.B.7 Grade 7 - 7.NS.A.2 Grade 8 - 8.F.A.2

  18. ACROSS GRADE COHERENCE IN GRADES 6-86thGrade - 6.EE.B.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. 5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. 6.EE.B.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. 6.NS.A.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.

  19. ACROSS GRADE COHERENCE IN GRADES 6-87th Grade- 7.NS.A.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. 6.NS.A.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.  5.NF.B.3 Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. 5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

  20. ACROSS GRADE COHERENCE IN GRADES 6-88th Grade- 8.F.A.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. 8.F.A.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. 8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. 7.RP.A.2 Recognize and represent proportional relationships between quantities.

  21. Break

  22. ACROSS GRADE COHERENCE IN GRADES 6-8Understanding the Progressions Content Guides The Progressions Documents Wiring Diagram

  23. ACROSS GRADE COHERENCE IN GRADES K-2 Leveraging the Progressions How can we leverage progressions of content to give students access to grade-level content? 23 23

  24. ACROSS GRADE COHERENCE IN GRADES 6-8Adapting Lessons for Students Below Grade Level Protocol: • Review Lesson 1 and identify the targeted standard. • Identify the prerequisite standards from prior grades that support the targeted standard. • What is the aspect of rigor for each prerequisite? • Discuss with a partner: • How does each prerequisite support the standard? • How could you strategically use these prerequisite standards to support students who are not on grade level? • Annotate the lesson with specific supports. • With your table: • Each pair shares out the specific adaptations you and your partner made. Explain why you made these adaptations.

  25. ACROSS GRADE COHERENCE IN GRADES 6-8Adapting Lessons for Students Below Grade Level Protocol: time • 10 min: Individual work time • 15 min: Partner work • 10 min: Table share out • Goals for this activity: • Review Lesson 1 and identify the targeted standard(s). • What are the prerequisite standards from prior grades that support this standard(s)? • What aspects of rigor are highlighted in the prerequisite standards?

  26. Transition to Partner Time!

  27. ACROSS GRADE COHERENCE IN GRADES 6-8 Adapting Lessons for Students Below Grade Level • Goals for this activity: • How do these prerequisite standards support the grade-level standard(s)? • How could you strategically use these prerequisite standards to support students who are not on grade level? • Annotate the lesson with specific supports Protocol: time • 10 min: Individual work time • 15 min: Partner work • 10 min: Table share out

  28. Transition to Table Share!

  29. ACROSS GRADE COHERENCE IN GRADES 6-8Adapting Lessons for Students Below Grade Level Protocol: time • 10 min: Individual work time • 15 min: Partner work • 10 min: Table share out • Goals for this activity: • Each pair shares out the specific adaptations made and explainswhy these adaptation were made.

  30. ACROSS GRADE COHERENCE IN GRADES 6-8Adapting Lessons for Students Below Grade Level

  31. ACROSS GRADE COHERENCE IN GRADES 6-8Summary What is the shift of coherence? How does understanding the progression of content help us support students below grade level?

  32. ACROSS GRADE COHERENCE IN GRADES K-2 Lunch 12:00-1:00 Lunch 12:00 – 1:00 32 32

  33. INSTRUCTIONAL PRACTICE IN GRADES 6-8Today • Morning: Across Grade Coherence in Grades 6-8 • Afternoon: Instructional Practice in Grades 6-8

  34. INSTRUCTIONAL PRACTICE IN GRADES 6-8 Afternoon Objectives • Participants will be able to use the Instructional Practice Guide (IPG) as a lesson planning tool and a coaching tool. • Participants will be able to identify where, in lessons and videos, teachers engage in Core Actions.

  35. INSTRUCTIONAL PRACTICE IN GRADES 6-8 Afternoon Agenda • Intro to the Instructional Practice Guide (IPG) • Core Actions in Action! • Lesson Planning with the IPG • Connect to Practice

  36. ...effective teaching is the non-negotiable core that ensures that all students learn mathematics at high levels... - Principles to Actions: Ensuring Mathematical Success for All (NCTM) INSTRUCTIONAL PRACTICE IN GRADES K-2Instructional Practice ...effective teaching is the nonnegotiable core that ensures that all students learn mathematics at high levels... - Principles to Actions: Ensuring Mathematical Success for All (NCTM)

  37. INSTRUCTIONAL PRACTICE IN GRADES 6-8Instructional Practice Guide (IPG) The Instructional Practice Guide includes coaching and lesson planning tools to help teachers and those who support teachers to make the Key Shifts in instructional practice required by the Common Core State Standards (CCSS).

  38. INSTRUCTIONAL PRACTICE IN GRADES 6-8Core Actions • Ensure the work of the lesson reflects the Shifts required by the CCSS for Mathematics. • Employ instructional practices that allow all students to learn the content of the lesson. • Provide all students with opportunities to exhibit mathematical practices while engaging with the content of the lesson.

  39. INSTRUCTIONAL PRACTICE IN GRADES 6-8 Core Action 1 Ensure the work of the lesson reflects the Shifts required by the CCSS for Mathematics. Indicators • The lesson focuses on the depth of grade-level cluster(s), grade-level content standard(s) or part(s) thereof. • The lesson intentionally relates new concepts to students’ prior skills and knowledge. • The lesson intentionally targets the aspect(s) of rigor (conceptual understanding, procedural skill and fluency, application) called for by the standard(s) being addressed.

  40. INSTRUCTIONAL PRACTICE IN GRADES 6-8 Core Action 2 Employ instructional practices that allow all students to learn the content of the lesson. Indicators • The teacher makes the mathematics of the lesson explicit by using explanations, representations, and/or examples. • The teacher provides opportunities for students to work with and practice grade-level problems and exercises. • The teacher strengthens all students’ understanding of the content by sharing a variety of students’ representations and solution methods. • The teacher deliberately checks for understanding throughout the lesson and adapts the lesson according to student understanding. • The teacher summarizes the mathematics with references to student work and discussion in order to reinforce the focus of the lesson.

  41. INSTRUCTIONAL PRACTICE IN GRADES 6-8 Core Action 3 Provide all students with opportunities to exhibit mathematical practices while engaging with the content of the lesson. Indicators • The teacher poses high-quality questions and problems that prompt students to share their developing thinking about the content of the lesson. Students share their developing thinking about the content of the lesson. • The teacher encourages reasoning and problem solving by posing challenging problems that offer opportunities for productive struggle. Students persevere in solving problems in the face of initial difficulty. • The teacher establishes a classroom culture in which students explain their thinking. Students elaborate with a second sentence (spontaneously or prompted by the teacher or another student) to explain their thinking and connect it to their first sentence.

  42. INSTRUCTIONAL PRACTICE IN GRADES 6-8Core Action 3 – Indicators (cont’d) • The teacher creates the conditions for student conversations where students are encouraged to talk about each other’s thinking. Students talk about and ask questions about each other’s thinking, in order to clarify or improve their own mathematical understanding. • The teacher connects and develops students’ informal language to precise mathematical language appropriate to their grade. Students use precise mathematical language in their explanations and discussions. • The teacher establishes a classroom culture in which students choose and use appropriate tools when solving a problem. Students use appropriate tools strategically when solving a problem. • The teacher asks students to explain and justify work and provides feedback that helps students revise initial work. Student work includes revisions, especially revised explanations and justifications.

  43. INSTRUCTIONAL PRACTICE IN GRADES 6-8Deeper Dive With the IPG • Small Group Protocol • Read the indicators of the Core Action for your group (pp. 5-10). • Discuss the following with your small group: • How does this Core Action (including the indicators) support teachers and coaches in building understanding of CCSSM-aligned instruction? • What are the essential teacher practices that support the Indicators? • What resonates with you the most about this Core Action?

  44. INSTRUCTIONAL PRACTICE IN GRADES 6-8Deeper Dive With the IPG • Table Discussion Protocol • Turn and teach. • Discuss the following with your table group: • How does this tool support teachers and coaches in building understanding of CCSSM-aligned instruction? • What are essential teacher practices that support each Core Action? • Where do each of the Standards for Mathematical Practice show up in the IPG?

  45. INSTRUCTIONAL PRACTICE IN GRADES 6-8Deeper Dive With the IPG Whole Group Discussion Protocol How does this tool support teachers and coaches in building understanding of CCSSM-aligned instruction? Where do each of the Standards for Mathematical Practice show up in the IPG? What Core Actions are you most struck by and why?

  46. INSTRUCTIONAL PRACTICE IN GRADES 6-8 IPG Summary • Useful in both planning & coaching • Evidence for the indicators can come from lesson materials, teacher actions, student discussion and student work • When using as a coaching tool, not all indicators may be evident in a single class period • Not to be used as an evaluation instrument

  47. INSTRUCTIONAL PRACTICE IN GRADES 6-8 Core Actions in Action! What Core Actions are visible?

  48. Break

  49. INSTRUCTIONAL PRACTICE IN GRADES 6-8Lesson Planning with the IPG • How can we use the Core Actions and indicators? • Planning • Evaluating • Reflecting

  50. INSTRUCTIONAL PRACTICE IN GRADES 6-8Lesson Planning The Core Actions should be evident in planning and observable in instruction. • What parts of the lesson plan are vital to show evidence of Core Action 1? Annotate the lesson to show these. • What are some of the things you could do to ensure alignment with the indicators for Core Actions 2 and 3? What to Review: Grade 6, Module 1, Lesson 2 Grade 7, Module 1, Lesson 2 Grade 8, Module 1, Lesson 2

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