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Application LessonsAfter presenting content, how will I design and deliver lessons that help students generate and defend claims through knowledge application?6. Using Strategies That Appear in All Types of LessonsThroughout all types of lessons, what strategies will I use to help students continually integrate new knowledge with old knowledge and revise their understanding accordingly?Context7. Using Engagement StrategiesWhat engagement strategies will I use to help students pay attention, be energized, be intrigued, and be inspired?8. Implementing Rules and ProceduresWhat strategies will I use to help students understand and follow rules and procedures?9. Building RelationshipsWhat strategies will I use to help students feel welcome, accepted, and valued?10. Communicating High ExpectationsWhat strategies will I use to help typically reluctant students feel valued and comfortable interacting with their peers and me?

      Source: Marzano, 2017, pp. 6–7.

      Within the ten categories of teacher actions, we have organized sets of strategies in even more fine-grained categories, called elements. As teachers think about each design question, they can then consider specific elements within the design area.

       Forty-Three Elements

      The forty-three elements provide detailed guidance about the nature and purpose of a category of strategies. Table I.3 depicts the elements that correspond to each design area. For example, the design area of providing and communicating clear learning goals involves three elements.

      1. Providing scales and rubrics (element 1)

      2. Tracking student progress (element 2)

      3. Celebrating success (element 3)

      As a teacher considers how to provide and communicate clear learning goals that help students understand the progression of knowledge he or she expects them to master and where they are along that progression (design question 1), the teacher might think more specifically about providing scales and rubrics, tracking student progress, and celebrating success. These are the elements within the first design area.

      Finally, these forty-three elements encompass hundreds of specific instructional strategies, some of which we explore in this book in relation to the mathematics classroom. Table I.3 lists the forty-three separate elements in the New Art and Science of Teaching framework beneath their respective design areas.

The Need for Subject-Specific Models

      General frameworks like The New Art and Science of Teaching certainly have their place in a teacher’s understanding of effective instruction. However, a content-specific model of instruction can be a useful supplement to the more general framework in The New Art and Science of Teaching. The content-specific model should fit within the context of the general framework, but it should be based on content-specific research and should take into account the unique challenges of teaching a particular content area. For mathematics, such a content-specific model should address important aspects of mathematics and mathematics instruction, such as higher cognitive thinking, reasoning, and problem solving, and address the important concept areas of number sense, operations, measurement and data, and algebraic thinking. A content-specific model for mathematics should address these aspects in depth and relate back to the general framework of instruction. We designed this book to provide just such a model. Specifically, in the following chapters, we address the three overarching categories—(1) feedback, (2) content, and (3) context—with their corresponding ten categories of instruction and the embedded forty-three elements that feature specific strategies expressly for mathematics.

Table I.3: Elements Within the Ten Design Areas

Feedback	Content	Context
Providing and Communicating Clear Learning Goals1. Providing scales and rubrics2. Tracking student progress3. Celebrating successUsing Assessments4. Using informal assessments of the whole class5. Using formal assessments of individual students	Conducting Direct Instruction Lessons6. Chunking content7. Processing content8. Recording and representing contentConducting Practicing and Deepening Lessons9. Using structured practice sessions10. Examining similarities and differences11. Examining errors in reasoningConducting Knowledge Application Lessons12. Engaging students in cognitively complex tasks13. Providing resources and guidance14. Generating and defending claimsUsing Strategies That Appear in All Types of Lessons15. Previewing strategies16. Highlighting critical information17. Reviewing content18. Revising knowledge19. Reflecting on learning20. Assigning purposeful homework21. Elaborating on information22. Organizing students to interact	Using Engagement Strategies23. Noticing and reacting when students are not engaged24. Increasing response rates25. Using physical movement26. Maintaining a lively pace27. Demonstrating intensity and enthusiasm28. Presenting unusual information29. Using friendly controversy30. Using academic games31. Providing opportunities for students to talk about themselves32. Motivating and inspiring studentsImplementing Rules and Procedures33. Establishing rules and procedures34. Organizing the physical layout of the classroom35. Demonstrating withitness36. Acknowledging adherence to rules and procedures37. Acknowledging lack of adherence to rules and proceduresBuilding Relationships38. Using verbal and nonverbal behaviors that indicate affection for students39. Understanding students’ backgrounds and interests40. Displaying objectivity and controlCommunicating High Expectations41. Demonstrating value and respect for reluctant learners42. Asking in-depth questions of reluctant learners43. Probing incorrect answers with reluctant learners

      Source: Marzano, 2017, p. 8.

      Although this text predominantly provides suggestions to support lesson planning around mathematics instruction, we encourage readers to explore the foundational book The New Art and Science of Teaching (Marzano, 2017). In doing so, they will likely infuse their content areas and grade levels with additional strategies.

About This Book

      In chapters 1 through 10, we situate a mathematics-specific model within the broader context of The New Art and Science of Teaching framework. Part I, focused on feedback, begins with chapter 1, which describes how teachers can effectively articulate learning goals for mathematics content within scales and rubrics, create learning progressions (called proficiency scales), and use those scales to track students’ progress and celebrate their success. In chapter 2, we explain strategies for how to assess students’ current mathematics status using both informal and formal assessment.

      Part II addresses content. In chapters 3, 4, 5, and 6, we articulate instructional strategies for teaching the mathematics content that students need to learn. Chapter 3 focuses on conducting direct instruction lessons, chapter 4 on conducting practicing and deepening lessons, chapter 5 on conducting

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