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Common Core Mathematics in a PLC at Workâ„¢, Grades 6-8

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This teacher guide illustrates how to sustain successful implementation of the Common Core State Standards for mathematics, grades 6–8. Discover what students should learn and how they should learn it at each grade level. Comprehensive research-affirmed analysis tools and strategies will help you and your collaborative team develop and assess student demonstrations of deep conceptual understanding and procedural fluency.

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Chapter 1: Using High-Performing Collaborative Teams for Mathematics

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CHAPTER 1

Using High-Performing Collaborative Teams for Mathematics

The Common Core State Standards provide a consistent, clear understanding of what students are expected to learn, so that teachers and parents know what they need to do to help them learn. The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers. With American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy.

—NGA & CCSSO

Students arrive at middle school with many challenges, and grades 6–8 teachers are expected to ensure all students achieve proficiency in the rigorous Standards for mathematics content as well as the Standards for Mathematical Practice described in the CCSS. How can you successfully help your middle school students achieve these expectations?

One of the characteristics of high-performing and high-impact schools that are successfully closing achievement gaps is their focus on teacher collaboration as a key to improving instruction and reaching all students (Education Trust, 2005; Kersaint, 2007). A collaborative culture is one of the best ways for teachers to acquire both the instructional knowledge and skills required to meet this challenge, as well as the energy and support necessary to reach all students (Leithwood & Seashore Louis, 1998). Seeley (2009) characterizes this challenge by noting, “Alone we can accomplish great things. … But together, with creativity, wisdom, energy, and, most of all commitment, there is no end to what we might do” (pp. 225–226).

 

Chapter 2: Implementing the Common Core Standards for Mathematical Practice

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CHAPTER 2

Implementing the Common Core Standards for Mathematical Practice

The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.

—NGA & CCSSO

The CCSS for mathematics include standards for student proficiency in mathematical practice as well as content standards for developing student understanding. According to the CCSS, “The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students” (NGA & CCSSO, 2010a, p. 6). The ultimate goal is to equip your middle school students with expertise that will help them be successful in doing and using mathematics not only in grades 6–8 but in their high school, college, and career work, as well as their personal life. College instructors of entry-level college courses across disciplines rated the Standards for Mathematical Practice of higher value for students to master in order to succeed in their courses than any of the content standard domains. This was true for college instructors in each of the fields of mathematics, language, science, and social science (Conley, Drummond, de Gonzalez, Rooseboom, & Stout, 2011).

 

Chapter 3: Implementing the Common Core Mathematics Content in Your Curriculum

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CHAPTER 3

Implementing the Common Core Mathematics Content in Your Curriculum

The standards are meant to be a blueprint for math instruction that is more focused and coherent. The focus and coherence in this blueprint are largely in the way the standards progress from each other, coordinate with each other and most importantly cluster together into coherent bodies of knowledge. … Maintaining these progressions in the implementation of the standards will be important for helping all students learn mathematics at a higher level. … Fragmenting the Standards into individual standards, or individual bits of standards, erases all these relationships and produces a sum of parts that is decidedly less than the whole.

—Daro, McCallum, & Zimba

Chapter 2 addressed one of the two types of Common Core standards—the Standards for Mathematical Practice—and illustrated instructional practices that promote students’ proficiency in these practices. This chapter analyzes the second type of standard—the Standards for Mathematical Content. As you read this chapter, keep in mind that the Standards for Mathematical Practice and the Standards for Mathematical Content together form the Common Core State Standards. Mathematical proficiency is defined both by the content and skills that students need to know and be able to use and the mathematical habits of mind they have acquired.

 

Chapter 4: Implementing the Teaching-Assessing-Learning Cycle

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CHAPTER 4

Implementing the Teaching-Assessing-Learning Cycle

An assessment functions formatively to the extent that evidence about student achievement is elicited, interpreted, and used by teachers, learners, or their peers to make decisions about the next steps in instruction that are likely to be better, or better founded, than the decisions they would have made in absence of that evidence.

—Dylan Wiliam

The focus of this chapter is to illustrate the appropriate use of ongoing student assessment as part of an interactive, cyclical, and systemic collaborative team formative process on a unit-by-unit basis. You and your collaborative team can use this chapter as the engine that will drive your systematic development and support for the student attainment of the Common Core mathematics content expectations as described in chapters 2 and 3.

When led well, ongoing unit-by-unit mathematics assessments—whether in-class, during the lesson checks or end-of-unit assessment instruments like tests, quizzes, or projects—serve as a feedback bridge within the teaching-assessing-learning cycle. The cycle requires your team to identify core learning targets or standards for the unit, create cognitively demanding common mathematics tasks that reflect the learning targets, create in-class formative assessments of those targets, and design common assessment instruments to be used during and at the end of a unit of instruction.

 

Chapter 5: Implementing Required Response to Intervention

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CHAPTER 5

Implementing Required Response to Intervention

Ultimately there are two kinds of schools: learning enriched schools and learning impoverished schools. I have yet to see a school where the learning curves … of the adults were steep upward and those of the students were not. Teachers and students go hand in hand as learners … or they don’t go at all.

—Roland Barth

As the curriculum is written, the learning targets are set, and your assessments are in place, your instructional processes need to meet the needs of each student in the courses you teach. As you read the grades 6–8 Common Core mathematics for the first time, what went through your mind? Were you thinking about the students in your class, your school, or part of your district and wondering, “Will they be able to respond positively to the expected complexity in each grade level?” Did you reflect on how you would be able to develop the CCSS Mathematical Practices in each student? How will each student be able to succeed with rich and rigorous mathematical tasks? Are there different learning opportunities for different groups of students, depending on their mathematics ability or diversity? How can you generate equitable learning experiences so that each student is prepared to meet the demands of the Common Core mathematics as described in this book? The key to answering these questions is part of the essential work of your collaborative team. To create an equitable mathematics program, you and your colleagues must ensure current structures for teaching and learning will generate greater access and opportunity to learn for each student.

 

Epilogue: Your Mathematics Professional Development Model

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EPILOGUE

Your Mathematics Professional Development Model

Implementing the Common Core State Standards for mathematics presents you with both new challenges and new opportunities. The unprecedented adoption of a common set of mathematics standards by nearly every state provides the opportunity for U.S. educators to press the reset button on mathematics education (Larson, 2011). Collectively, you and your colleagues have the opportunity to rededicate yourselves to ensuring all students are provided with exemplary teaching and learning experiences, and you have access to the supports necessary to guarantee all students the opportunity to develop mathematical proficiency.

The CCSS college and career aspirations and vision for teaching, learning, and assessing students usher in an opportunity for unprecedented implementation of research-informed practices in your school or district’s mathematics program. In order to meet the expectations of the five fundamental paradigm shifts described in this book, you will want to assess your current practice and reality as a school against the roadmap to implementation described in figure E.1 (page 180).

 

Appendix A: Standards for Mathematical Practice

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APPENDIX A

Standards for Mathematical Practice

Source: NGA & CCSSO, 2010a, pp. 6–8. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. Used with permission.

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

 

Appendix B: Standards for Mathematical Content, Grade 6

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APPENDIX B

Standards for Mathematical Content, Grade 6

Source: NGA & CCSSO, 2010a, pp. 39–45. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. Used with permission.

In Grade 6, instructional time should focus on four critical areas: (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking.

(1)  Students use reasoning about multiplication and division to solve ratio and rate problems about quantities. By viewing equivalent ratios and rates as deriving from, and extending, pairs of rows (or columns) in the multiplication table, and by analyzing simple drawings that indicate the relative size of quantities, students connect their understanding of multiplication and division with ratios and rates. Thus students expand the scope of problems for which they can use multiplication and division to solve problems, and they connect ratios and fractions. Students solve a wide variety of problems involving ratios and rates.

 

Appendix C: Standards for Mathematical Content, Grade 7

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APPENDIX C

Standards for Mathematical Content, Grade 7

Source: NGA & CCSSO, 2010a, pp. 46–51. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. Used with permission.

In Grade 7, instructional time should focus on four critical areas: (1) developing understanding of and applying proportional relationships; (2) developing understanding of operations with rational numbers and working with expressions and linear equations; (3) solving problems involving scale drawings and informal geometric constructions, and working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume; and (4) drawing inferences about populations based on samples.

(1)  Students extend their understanding of ratios and develop understanding of proportionality to solve single- and multi-step problems. Students use their understanding of ratios and proportionality to solve a wide variety of percent problems, including those involving discounts, interest, taxes, tips, and percent increase or decrease. Students solve problems about scale drawings by relating corresponding lengths between the objects or by using the fact that relationships of lengths within an object are preserved in similar objects. Students graph proportional relationships and understand the unit rate informally as a measure of the steepness of the related line, called the slope. They distinguish proportional relationships from other relationships.

 

Appendix D: Standards for Mathematical Content, Grade 8

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APPENDIX D

Standards for Mathematical Content, Grade 8

Source: NGA & CCSSO, 2010a, pp. 52–56. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. Used with permission.

In Grade 8, instructional time should focus on three critical areas: (1) formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.

(1)  Students use linear equations and systems of linear equations to represent, analyze, and solve a variety of problems. Students recognize equations for proportions (y/x = m or y = mx) as special linear equations (y = mx + b), understanding that the constant of proportionality (m) is the slope, and the graphs are lines through the origin. They understand that the slope (m) of a line is a constant rate of change, so that if the input or x-coordinate changes by an amount A, the output or y-coordinate changes by the amount m·A. Students also use a linear equation to describe the association between two quantities in bivariate data (such as arm span vs. height for students in a classroom). At this grade, fitting the model, and assessing its fit to the data are done informally. Interpreting the model in the context of the data requires students to express a relationship between the two quantities in question and to interpret components of the relationship (such as slope and y-intercept) in terms of the situation.

 

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