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Common Core Mathematics in a PLC at Work: Grades K-2

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This teacher guide illustrates how to sustain successful implementation of the Common Core State Standards for mathematics, grades K–2. 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

Far too frequently, your mathematics professional development experience as a preK–2 teacher likely feels inadequate. Why? It could be because you receive little or no professional development time dedicated to teaching, assessing, and learning mathematics. Unless you are in the process of implementing a new mathematics curriculum, which may happen every six to eight years, the focus of most professional development time is in another major area of need—literacy.

To be certain, professional development in literacy for grades preK–2 is essential. After all, the evidence is clear that students who struggle to read in your class often struggle in mathematics as well. Skill in reading is necessary for success in mathematics (Gersten, Jordan, & Flojo, 2005; Jordan & Hanich, 2003). However, in order for you to transition to the Common Core State Standards for mathematics, you will need to shift the same amount of priority time to your professional development in mathematics (National Governors Association Center for Best Practices [NGA] & Council of Chief State School Officers [CCSSO], 2010).

 

Chapter 2 Implementing the Common Core Standards for Mathematical Practice

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CHAPTER 2

The Common Core State Standards for mathematics include standards for mathematics content as well as standards describing expectations for mathematical practice. As the Common Core State Standards (NGA & CCSSO, 2010) state, “The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students” (p. 6). The Standards for Mathematical Practice are presented in appendix B (page 157). Implementation of the Standards for Mathematical Practice addresses the second required paradigm shift: a shift to include within your daily lesson plans intentional strategies to teach mathematics in different ways—in ways that focus on the process of learning and developing deep student understanding of the mathematical content. Your goal will be to develop in students both conceptual understanding and procedural fluency of the CCSS content through the collaborative selection of high-cognitive-demand mathematical tasks with a focus on engagement with the Standards for Mathematical Practice.

 

Chapter 3 Implementing the Common Core Mathematics Content in Your Curriculum

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CHAPTER 3

Chapter 2 presented a strong argument for designing the transition to the CCSS through the window of teacher and student engagement in the Mathematical Practices (see appendix B, page 157). This makes a lot of sense as you consider the mathematics content of the CCSS. What’s the content? How does this mathematics differ from what you are now teaching or have previously taught? Are there particular standards that require additional focus? What about topics that appeared to be a struggle for your students last year or throughout your career? These “in my room with my kids” concerns are legitimate at every grade level.

This chapter provides a number of analysis tools for examining your classroom, school, or district implementation of the content domains and expectations of the Common Core State Standards. As you work collaboratively with colleagues, you will be able to address and become conversant with the paradigm shift less is more. The Common Core standards require you to shift to less (fewer standards) is more (opportunity to dig deeper with understanding) at each grade level in your school.

 

Chapter 4 Implementing the Teaching-Assessing-Learning Cycle

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CHAPTER 4

The vision of the Common Core State Standards for mathematics (NGA & CCSSO, 2010) is one that interprets the learning of important mathematics as consisting of both mathematics content and mathematical practices—a vision the National Council of Teachers of Mathematics (2010) shares in Making It Happen. Much of your implementation effort surrounding the CCSS will focus on making sure your instruction is aligned with the more coherent and focused CCSS content standards and that the Mathematical Practices are interpreted as both essential mathematics to be learned and as ways in which your students should engage in learning the mathematics content. These are crucial steps in implementing the CCSS. Ensuring that these interpretations are well understood and implemented in your classroom, however, is just the beginning work necessary to ensure student attainment of the CCSS.

This chapter examines the paradigm shift from the traditional use of summative assessment instruments (strictly to grade and evaluate student learning) to the use of collaboratively developed formative assessment processes to guide your instruction. When implemented effectively, your assessment practices are a critical instructional tool to improve teaching and student learning. Since the enactment of the No Child Left Behind Act of 2001 (NCLB, 2002), most of the assessment focus has been directed at preparing all students to perform well on state accountability tests. In many states, this led to tests and instruction, which essentially narrowed the curriculum to focus on lower-level procedural skills—skills that make up but one component of the more balanced vision of the CCSS, which states that “mathematical understanding and procedural skill are equally important” (NGA & CCSSO, 2010, p. 4). Consequently, past improvements on state accountability tests may not reflect actual improved student learning in the broader set of skills and concepts called for in the CCSS. Think about the process of assessment teachers in your school often use at your grade level. For student mathematics learning to improve according to the CCSS, you will need to shift your assessment work to using formative assessment as a collaboratively developed process for improving your instruction and student learning.

 

Chapter 5 Implementing Required Response to Intervention

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CHAPTER 5

All students are entitled to quality instruction within an equitable learning environment designed to meet their specific learning needs. You have a professional obligation to create and maintain an equitable learning environment and provide high-quality instruction. This chapter focuses on the final paradigm shift required to ensure successful CCSS implementation—the need to create directive response to intervention programs to support all students in meeting the expectations of the CCSS. When such programs are in place, then intervention serves the goal of equity.

The National Council of Teachers of Mathematics (2008) notes:

Excellence in mathematics education rests on equity—high expectations, respect, understanding, and strong support for all students. Policies, practices, attitudes, and beliefs related to mathematics teaching and learning must be assessed continually to ensure that all students have equal access to the resources with the greatest potential to promote learning. A culture of equity maximizes the learning potential of all students. (p. 1)

 

Epilogue Your Mathematics Professional Development Model

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EPILOGUE

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 that 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 in the early grades that leads to successful student preparation for the CCSS K–12 college- and career-ready mathematics standards.

The CCSS college and career aspirations and vision for teaching, learning, and the assessment for student learning 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 clearly define your reality as a school against the roadmap to implementation described in figure E.1 (page 152).

 

Appendix A Sample Standards for Mathematical Content, PreKindergarten

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APPENDIX A

The New York State Education Department and the Maryland State Department of Education have created their own set of preK content area standards in mathematics. Organized under “cognition and knowledge of the world,” the New York standards present the following preK mathematics benchmarks:

• Children will demonstrate an understanding of numbers, ways to represent numbers, relationships among numbers, and the number system.

• Children will understand the beginning principles of addition and subtraction.

• Children will demonstrate understanding of geometric and spatial relations.

• Children will understand directionality, order, and position.

• Children will sort, classify, and organize objects by size, number, attributes, and other properties.

• Children will demonstrate knowledge of measurement.

New York’s preK learning standards are available in the “New York State Pre-kindergarten Foundation for the Common Core” (New York State Education Department, n.d.; www.p12.nysed.gov/ciai/common_core_standards/pdfdocs/nyslsprek.pdf).

 

Appendix B Standards for Mathematical Practice

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APPENDIX B

Source: NGA & CCSSO, 2010, 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 C Standards for Mathematical Content, Kindergarten

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APPENDIX C

Source: NGA & CCSSO, 2010, pp. 9–12. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. Used with permission.

In Kindergarten, instructional time should focus on two critical areas: (1) representing, relating, and operating on whole numbers, initially with sets of objects; (2) describing shapes and space. More learning time in Kindergarten should be devoted to number than to other topics. (1) Students use numbers, including written numerals, to represent quantities and to solve quantitative problems, such as counting objects in a set; counting out a given number of objects; comparing sets or numerals; and modeling simple joining and separating situations with sets of objects, or eventually with equations such as 5 + 2 = 7 and 7 - 2 = 5. (Kindergarten students should see addition and subtraction equations, and student writing of equations in kindergarten is encouraged, but it is not required.) Students choose, combine, and apply effective strategies for answering quantitative questions, including quickly recognizing the cardinalities of small sets of objects, counting and producing sets of given sizes, counting the number of objects in combined sets, or counting the number of objects that remain in a set after some are taken away. (2) Students describe their physical world using geometric ideas (e.g., shape, orientation, spatial relations) and vocabulary. They identify, name, and describe basic two-dimensional shapes, such as squares, triangles, circles, rectangles, and hexagons, presented in a variety of ways (e.g., with different sizes and orientations), as well as three-dimensional shapes such as cubes, cones, cylinders, and spheres. They use basic shapes and spatial reasoning to model objects in their environment and to construct more complex shapes.

 

Appendix D Standards for Mathematical Content, Grade 1

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APPENDIX D

Source: NGA & CCSSO, 2010, pp. 13–16. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. Used with permission.

In Grade 1, instructional time should focus on four critical areas: (1) developing understanding of addition, subtraction, and strategies for addition and subtraction within 20; (2) developing understanding of whole number relationships and place value, including grouping in tens and ones; (3) developing understanding of linear measurement and measuring lengths as iterating length units; and (4) reasoning about attributes of, and composing and decomposing geometric shapes.

(1)  Students develop strategies for adding and subtracting whole numbers based on their prior work with small numbers. They use a variety of models, including discrete objects and length-based models (e.g., cubes connected to form lengths), to model add-to, take-from, put-together, take-apart, and compare situations to develop meaning for the operations of addition and subtraction, and to develop strategies to solve arithmetic problems with these operations. Students understand connections between counting and addition and subtraction (e.g., adding two is the same as counting on two). They use properties of addition to add whole numbers and to create and use increasingly sophisticated strategies based on these properties (e.g., “making tens”) to solve addition and subtraction problems within 20. By comparing a variety of solution strategies, children build their understanding of the relationship between addition and subtraction.

 

Appendix E Standards for Mathematical Content, Grade 2

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APPENDIX E

Source: NGA & CCSSO, 2010, pp. 13–16. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. Used with permission.

In Grade 2, instructional time should focus on four critical areas: (1) extending understanding of base-ten notation; (2) building fluency with addition and subtraction; (3) using standard units of measure; and (4) describing and analyzing shapes.

(1)  Students extend their understanding of the base-ten system. This includes ideas of counting in fives, tens, and multiples of hundreds, tens, and ones, as well as number relationships involving these units, including comparing. Students understand multi-digit numbers (up to 1000) written in base-ten notation, recognizing that the digits in each place represent amounts of thousands, hundreds, tens, or ones (e.g., 853 is 8 hundreds + 5 tens + 3 ones).

(2)  Students use their understanding of addition to develop fluency with addition and subtraction within 100. They solve problems within 1000 by applying their understanding of models for addition and subtraction, and they develop, discuss, and use efficient, accurate, and generalizable methods to compute sums and differences of whole numbers in base-ten notation, using their understanding of place value and the properties of operations. They select and accurately apply methods that are appropriate for the context and the numbers involved to mentally calculate sums and differences for numbers with only tens or only hundreds.

 

Appendix F Changes in Mathematics Standards, 1989–2010

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APPENDIX F

Helping students use their prior knowledge to enable them to recognize what is new and different in their learning is a key element of scaffolded instruction. Similarly, as you explore the CCSS for mathematics, it will be helpful to compare aspects of mathematics standards that have framed your previous instruction so that you can identify what is familiar, what is new and challenging, and what changes are required in the content delivered to your students. As you examine the CCSS mathematics standards for your grade level, you may find it helpful to refer to the standards that have formed the basis of your instruction recently. In all likelihood these standards are based on the landmark documents that have influenced mathematics instruction since 1989, when the National Council of Teachers of Mathematics published Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989). In 2000, NCTM updated these curriculum standards in their Principles and Standards for School Mathematics (PSSM), which has served as the blueprint for revised state standards throughout 2000–2010.

 

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