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**Common Core State Standards in Mathematics: ECE-5**

August 2012 Welcome!

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**Agenda Instructional Shifts in Mathematics**

Activating the Instructional Shifts Learning Trajectory Standards for Mathematical Practice Using classroom video to make connections DPS Tools/Resources Our agenda for today that is designed to help you answer the essential questions.

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**Instructional Shifts in Mathematics**

Focus: Narrow and deep emphasis based on standards. Coherence: Across grades and to major topics within grades. Rigor: Equal intensity of conceptual understanding, procedural skills and fluency, and application. Introduce the three shifts. Turn and Talk. Then provide additional insight into the three shifts using the next 3 slides. Turn and Talk: What do these shifts mean?

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FOCUS Focus: Teachers significantly narrow and deepen the scope of how time and energy is spent in the math classroom. They do so in order to focus deeply on the concepts that are emphasized in the standards so that students can engage in the mathematical practices, reach strong foundational knowledge and deep conceptual understanding, and transfer mathematical skills and understanding across concepts and grades.

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COHERENCE Coherence: Teachers connect the learning within and across grades, so that, for example, fractions or multiplication develop across grade levels and students can build new understanding onto foundations built in previous years. Teachers can then begin to count on deep conceptual understanding of core content and build on it. Each standard is not a new events, but an extension of previous learning. An example of linking within grade is in third grade where the concepts of multiplication and area are linked.

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**Conceptual Understanding**

RIGOR Procedural Fluency Conceptual Understanding Application Rigor: Equal intensity of conceptual understanding, procedural skills and fluency, and application. Conceptual understanding: Teachers teach more than “how to get the answer” and support students’ ability to access concepts from a number of perspectives so that students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of core math concepts by solving short conceptual problems, applying math in new situations, and speaking about their understanding. Procedural skill and fluency: Students are expected to have speed and accuracy in calculation. Fluency supports students in being able to understand and manipulate more complex concepts. To be fluent is to flow: Fluent isn’t halting, stumbling or reversing oneself. It means more or less the same as when someone is said to be fluent in a foreign language. Application: Students are expected to use math and choose the appropriate concept for application even when they are not prompted to do so. Teachers provide opportunities at all grade levels for students to apply math concepts in “real world” situations.

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**CCSS Domain Progression**

Recall that domains are groups of related standards (“big buckets”). The K-5 standards provide students with a solid foundation in whole numbers, fractions and decimals The 6-8 standards describe robust learning in geometry, algebra, and probability and statistics Modeled after the focus of standards from high-performing nations, the standards for grades 7 and 8 include significant algebra and geometry content Students who have completed 7th grade and mastered the content and skills will be prepared for algebra, in 8th grade or after

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**Instructional Shift: Focus**

K: Counting and Cardinality, Joining (addition) and Separating (subtraction); Place Value 1: Addition and Subtraction, Place Value, Measurement 2: Addition and Subtraction (with application to measurement), Place Value 3: Multiplication and Division, Fractions as Number; Area and Measurement 4: Four Operations, Place Value, Fraction Equivalence (including decimals) and Operations 5: Place Value (including decimals), Operations with Fractions, Volume These are the priorities in support of rich instruction and expectations of fluency and conceptual understanding. Turn and Talk: What do you notice? How is this different than our current practice?

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**Development of Fluency**

A key aspect of fluency is that it is NOT something that happens all at once in a single grade but requires attention to student understanding along the way. It is important to ensure that sufficient practice and extra support are provided at each grade to allow all students to meet the standards that call explicitly for fluency. Fluency is not meant to come at the expense of understanding but is an outcome of a progression of learning and sufficient thoughtful practice. It is important to provide the conceptual building blocks that develop understanding in tandem with skill along the way to fluency; the roots of this conceptual understanding often extend one or more grades earlier in the standards than the grade when fluency is finally expected.

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**Mathematics Inequalities**

Turn and Talk: Fluency with Basic Facts ≠ Memorized Basic Facts Knowing from Memory ≠ Memorization Fluency is called out in the CCSS as one component of Rigor. Fluency requires that students develop automaticity and accuracy. These two mathematics inequalities give us an opportunity to think about how fluency is developed. Show the first and have folks turn and talk and then the second one. No need to report out.

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**Activate Knowledge about Shifts**

Tomas and Sarah collect baseball cards. Sarah has 18 more cards than Tomas. She has 200 cards. How many cards does Tomas have? A pencil costs 59¢ and a sticker costs 20¢ less. How much do a pencil and a sticker cost together? Let’s do math and explore a couple of problems that exemplify the instructional shifts called for in K-5 mathematics. Individual think time. How are these problems the same as or different from current practice? How do these problems reflect the instructional shifts (focus, coherence, and rigor)?

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**Connecting: Practice Standards with Content Standards**

Standards for Mathematical Practice: Ways students should engage with content as they grow in mathematical maturity and expertise Standards for Mathematical Content: Balanced combination of procedure and understanding Content standards which set an expectation of understanding are “points of intersection” between the Standards for Mathematical Content AND the 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. The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations in the Content Standards that begin with the word “understand” are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In this respect, those content standards which set an expectation of understanding are potential “points of intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve student achievement in mathematics.

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**Rationale for Trajectory: Addition and Subtraction**

Important starting point for professional development in the new standards Major area of emphasis at multiple grade levels Provides a framework for analyzing Development of representational models/tools Key contexts such as joining, separating, and comparing Multiple strategies (e.g., place-value strategies and properties of operations) Since addition and subtraction fluency is important in K-5, we’ll begin with studying the progression of addition and subtraction across the grades. We want to enter the study of the standards through a learning trajectory so that we can develop a deeper understanding of student learning of addition and subtraction across the grades. This will help us to support students in building concepts and making connections as they develop fluency and apply their understanding.

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**Organization of Math Standards**

Again, we want to remind ourselves of the organization of the standards and the naming conventions. Standards define what students should understand and be able to do. Clusters are groups of related standards. Note that standards from different clusters may sometimes be closely related, because mathematics is a connected subject. Domains are larger groups of related standards. Standards from different domains may sometimes be closely related. A Cluster Heading provides a description of how the cluster of standards relate to a bigger idea. Note: When we read an individual standard, it’s important to refer back to the Cluster Heading under which it is located.

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**Studying the Learning Trajectory**

Read the standards for each grade level (begin with K) and discuss what students need to know and be able to do in that grade. As you shift to the next grade level, analyze how the demands of the standards change. Consider changes in: Content (e.g., number system and magnitude of numbers) Representations (e.g., use of number sentence, manipulatives, graphs, diagrams, charts, tables) Processes (e.g., problem solving, reasoning, mental math, communication) Document your findings in the table provided. We’ll begin at the bottom of the page in the grade level where the topic begins. We want to understand the foundation students will have as they move up the grades. Go through directions with participants. They should work with a partner but both record so they have a record. Note: Magnitude of numbers (e.g., up to 10, up to 20) Note: Teachers will turnkey this Trajectory in August on the Green day for Standards and will have all of the instructions and materials to lead this. For now, we ask that they engage deeply with the content. Note to us: It’s not about getting the “right answer” rather it’s about the collegial conversations.

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**Learning Trajectory Discussion**

What are the key learnings from this kind of analysis? What evidence of the instructional shifts can you identify from your study of the learning trajectory? What are the key ideas that support the Big Idea from the Learning Trajectory? ES: key ideas to “pull out”: composition and decomposition of numbers, properties of operations, and inverse relationship between addition and subtraction ES: Where do standard algorithms come in? (+/-) ES: What properties are we talking about? (+/-)

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**Implications of Learning Trajectory**

Therefore, it’s about . . . Understanding alignment and its implications Collaborative processes K-12 perspective on alignment Collegial conversations The purpose of the learning trajectory is to deepen our understanding of the standards AND to: It is about understanding alignment and its implications for teaching and learning. It is about engaging in collaborative processes and constructing meaning using those processes. It is not only about specific grade-level content. It is about developing a K-12 perspective on alignment. It is about collegial conversations focused on the standards (mathematical content and mathematical practices), instruction, and students.

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**Activate Knowledge about Shifts**

Luke wrote a note to his mother. It took him 68 seconds to find a pen, 42 seconds to find paper, and 41 seconds to write it. Did it take Luke more than 3 minutes to complete the note? Explain. What content has to be understood to solve this problem? Where on our learning trajectories do you see the content necessary to solve this problem?

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**Standards for Mathematical Practice**

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. Again, recall that we worked with the Standards for Mathematical Practice this year but now, based on our trajectory, we want to look for close connections of the Standards for Mathematical Practice with the Standards for Mathematical Content.

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**SMP: 1 and 7 Carefully read these Standards for Mathematical Practice**

Share the verbs. We will chart these. How do these verbs show evidence of the instructional shifts in math? The Standards for Mathematical Practice exemplify the ways in which students engage deeply with the content in the Standards for Mathematical Content. Some of you may be familiar with this activity from last summer’s teacher leader professional development or from the turnkey sessions back at your school. We will begin today’s work in a similar way but we will apply it to the content standards in a few minutes. Read each of the two Standards for Mathematical Practice and focus on the verbs. Before turn and talk: Full group popcorn share-out of verbs. Turn & Talk

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Learning from Video The videos are not used to critique the teaching we will see. Teaching is far too complex for a thoughtful judgment to be made from a video segment that allows us no more than a brief glimpse into the classroom. We want to look at video as a way to apply our learning to classroom practice. Let’s remind ourselves that the teachers and students in the video we will see have given us a gift of allowing us to learn from them. We will only step into their classroom for a few minutes and witness only a brief part of a lesson.

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**Common Core Classrooms in Action**

How might second grade students solve the following problems? Before we look at the video, we want to give you a few minutes to engage in the problem that has been posed to the students. Think privately and then share with a shoulder partner.

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**Common Core Classrooms in Action: Take 1**

As you watch the video, consider: How is conceptual understanding of addition demonstrated by the students in the video? How do the students use place value to solve the addition problems? How does this represent the Instructional Shifts? How are students developing their expertise in the Standards for Mathematical Practice? We’ve chosen a video to share of a classroom where the topic of instruction is on addition and subtraction is. We do not suggest that this video is an exemplar; rather, it is a video that will allow us to think deeply about the learning in the classroom. Share the two focus questions and orient teachers to the template in their notebooks to use. What you just talked about were the Instructional Shifts, Focus, Coherence, Rigor. Make connections to the trajectory. Instructional Shifts- Focus, Coherence, and Rigor Focus-deep emphasis based on standards Coherence-Across grades and to major topics within a grade Rigor-Conceptual Understanding, Fluency, and Application with equal intensity

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**Common Core Classrooms in Action: Take 2**

As you watch the video a second time, consider: How does the teacher’s questioning reveal student understanding? How are these teacher moves reflected in the Framework for Effective Teaching? Focus on I-2 to I- 4 (pp 9-13) both teacher and student behaviors. after discussing the first question, ask participants to take out the Framework to answer the second question and look at teacher moves and student behaviors.

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Instructional Tasks Provide students with necessary skills to engage successfully in complex tasks or non-routine situations Provide students with opportunities that prepare them for the rigor of the new standards Provide teachers with strategies to support students’ engagement and perseverance in complex tasks Purpose of Instructional Tasks: Provide students with necessary skills (practices, content, and productive dispositions) to engage successfully in complex tasks/problems. Designed to motivate students to use knowledge in routine or non-routine situations Provide students with opportunities that will prepare them for the higher expectations of the new assessments Provide teachers with a protocol to support students in learning how to engage in complex tasks and to persevere in solving and successfully completing complex tasks

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Instructional Tasks Incorporated in a current unit of study (will replace a lesson, an activity, or an assessment) Provide opportunity to apply the Standards to specific tasks with explicit reference to the CCSS for Mathematics Explicit connection to the Framework for Effective Teaching—New this year! Includes content/language objective--New this year! There will be four instructional tasks this year. New features this year will be explicit connections to the Framework and a content/language objective for each task.

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**Instructional Task: Do the Math!**

Individually, do the math. Individually, consider: What mathematical content and practices can be learned from the task? What are all the ways the task can be solved? What misconceptions might students have? What errors might students make? What resources/tools might you want to have available? Share at table. This is incorporated in the PCK Planning Module for Instructional Tasks. Distribute task (we will all do the same task at each grade band so that we can have a common discussion). Use private think time to do the math and consider the questions. We’ll cue you when to share at table.

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**Instructional Tasks: Connections**

How will the Instructional Task support the Instructional Shifts? What opportunities does the Task provide for students to develop expertise in the Standards for Mathematical Practice? Which Standard(s)? How? In looking at the Expectations and Indicators in the Framework for Effective Teaching, how are these enacted in the Task? Take a few minutes to individually and privately respond to these three questions. Distribute Teacher Packet and ask for partners to share response and refer to Standards for Mathematical Practice and Framework connections in the Teacher Packet for Instructional Task.

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**Components of DPS Content/Language Objective**

Content: What will students learn based on the lesson? Targeted Domain: What domain (speaking, listening, reading, writing) will be targeted in lesson? Language Function: How will students use the language in the lesson? Language Form: What grammatical structures and what academic vocabulary will be used? Differentiated Language Supports: What supports will my students need to understand the content? We are training this out in this form and in this color-coded fashion: Red=function and domain; green=language form; black bold=content; blue= supports; a, b, c=differentiated supports However, teachers are free to reword components in “student-friendly” language (and are not “required” to use the color coding, unless they choose to); the important point, is to ensure that all four components are clearly evident in the lesson. The flexible groups & supports are also not limited to the three groups stated. Teachers must plan for the groups they have. If they do not have levels 1-2 students, they do not have to plan for that group, for example. Note: Inclusion of the supports is for planning purposes. They should be visible in the lesson but supports don’t necessarily need to posted in the room with the objective.

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**Content/Language Objective**

Students will add fractions to make the sum of one whole and explain their thinking, orally and in writing, using academic language (e.g., equal, equation, equivalent fractions) and supports such as fraction bars, drawings, or other classroom manipulatives. ES. The new tasks and revised tasks will have a content/language objective. What parts/components do you notice in the objective?

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**DPS Tools/Resources to Support Instructional Shifts**

Instructional Tasks with PCK Modules Time Frame documents Provide a guide to ensure that sufficient time is devoted to the topics of major emphasis Essential Learning Goal (ELG) documents Describe the key learnings for students based on the topics of major emphasis in the CCSS We want to look at the time frame documents and essential learning goal documents and consider how they will support the instructional shifts in mathematics. Tomorrow we’ll look at a new instructional task and again consider how the task will support the instructional shifts in mathematics. Select one grade level or course to study the Time Frame and ELGs for that grade/course. On the ELG documents, you will notice the inclusion of M, A, S Not all of the content in a given grade is emphasized equally in the standards. Some clusters require greater emphasis than the others based on the depth of the ideas, the time that they take to master, and/or their importance to future mathematics or the demands of college and career readiness. In addition, an intense focus on the most critical material at each grade allow depth in learning, which is carried out through the Standards for Mathematical Practice. Thus: Major Clusters—areas of intensive focus, where students need fluent understanding and application of the core concepts Supporting Clusters—areas that support and strengthen area of major emphasis Additional Clusters—areas that expose students to other mathematical topics. Note: Some supporting and additional clusters move to major clusters in later years. Study the Time Frame and ELG documents. How are these documents the same as/different from the current Time Frame and ELG documents? How are the Instructional Shifts represented in the documents?

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Reflection on Session Quick Write: How will this learning impact your work this year? Turn and Talk: Share your responses with a partner at your table. Keep in mind: is a transition and learning year to prepare us for full implementation of CCSS in

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