© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Rigor Breakdown A Three Part Series

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Rigor Breakdown 2

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Rigor Breakdown Part 1: Conceptual Understanding Grades 3-5

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Session Objectives 4 Understand the conceptual understanding component of rigor called for in the Standards, as defined by guiding documents Examine various activities that promote conceptual understanding in A Story of Units Compare and contrast conceptual understanding activities and analyze the impact and advantages of each Highlight Standards for Mathematical Practice in the conceptual understanding activities in A Story of Units Recognize the balance and intensity of all three components of rigor in A Story of Units

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Conceptual Understanding Defined by the Instructional Shifts 5 “Teachers teach more than ‘how to get the answer’ and instead 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 applying them to new situations as well as writing and speaking about their understanding.”

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Conceptual Understanding Defined by the Publishers’ Criteria 6 “The word ‘understand’ is used in the Standards to set explicit expectations for conceptual understanding…” (page 5) “Materials amply feature high-quality conceptual problems and questions that can serve as fertile conversation-starters in a classroom if students are unable to answer them...” (pages 9-10)

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Conceptual Understanding Defined by the Publishers’ Criteria 7 “…This includes brief conceptual problems with low computational difficulty (e.g., ‘Find a number greater than 1/5 and less than 1/4.’); brief conceptual questions (e.g., ‘If the divisor does not change and the dividend increases, what happens to the quotient?’); and problems that involve identifying correspondences across different mathematical representation of quantitative relationships.” (page 10)

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units AGENDA 8

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Video Clip – Concrete and Pictorial Models 9 Reflections: Compare and contrast the example with how you develop this concept in your classroom today. Analyze the impacts and advantages of developing conceptual understanding using concrete and pictorial models. Identify Standards of Mathematical Practice.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Video Clip: Fractions as Division – Concrete to Pictorial 10 This video is posted in the video library on EngageNY:

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Video Clip – Concrete and Pictorial Models 11 Reflections: Compare and contrast the example with how you develop this concept in your classroom today. Analyze the impacts and advantages of developing conceptual understanding using concrete and pictorial models. Identify Standards of Mathematical Practice.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Mathematical Practices 12 “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.” (CCSSM, page 8)

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Concrete and Pictorial Models – Key Points 13 Concrete materials give students an experiential understanding of concepts. Pictorial representations offer greater flexibility than concrete models, challenging student understanding at a deeper level while maintaining their connection to the contextual situation. Without the concrete or pictorial models, operations become disconnected from meaning, rendering students unable to judge when and where they apply.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units AGENDA 14

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Lesson Engagement – Conceptual Questioning 15 Reflections: Where did you notice conceptual questioning taking place? What Mathematical Practices did you notice? Compare and contrast Conceptual Questioning with Concrete and Pictorial Models analyzing the impacts and advantages of each.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Video Clip – Conceptual Questioning 16 Reflections: Identify conceptual questioning in the example. Compare and contrast the impacts and advantages of conceptual questioning with the example of concrete and pictorial models.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Video Clip: 10 Cakes Shared Among 4 17 This video is posted in the video library on EngageNY:

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Video Clip – Conceptual Questioning 18 Reflections: Identify conceptual questioning in the example. Compare and contrast the impacts and advantages of conceptual questioning with the example of concrete and pictorial models.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Conceptual Questioning – Key Points 19 Goes beyond getting the right answer Goes beyond Yes/No questions Encourages recognition of subtleties and exposes current level of student understanding “Can you think of a case where that would not work?” “Someone else says the answer is this. Can you prove that they are right/wrong?” “When we get a like unit for these two fractions, will the like unit be bigger or smaller than the units we have?” “Can you think of a number between 1/4 and 1/5?”

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units AGENDA 20

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Writing and Speaking about Understanding 21 From the Shifts: “Deep Understanding: …. Students demonstrate deep conceptual understanding of core math concepts by applying them to new situations as well as writing and speaking about their understanding.”

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Video Clip – Writing and Speaking about Understanding 22 Reflections: How does the speaking about understanding evident in the video compare with what occurs in your classroom/school/district today?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Video Clip – Writing and Speaking about Understanding 23 Word Problem 2/5 of the washing machines in a store were sold at a total price of $3840. If 18 washing machines remained in the store. What was the cost of one washing machine?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Video Clip: Washing Machine Problem 24 This video is posted in the video library on EngageNY:

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Video Clip – Writing and Speaking about Understanding 25 Reflections: How does the speaking about understanding evident in the video compare with what occurs in your classroom/school/district today?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Lesson Engagement – Writing and Speaking About Understanding 26 Reflections: Look for evidence of the “writing and speaking about their understanding” requirement as you complete the worksheet.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Writing and Speaking About Understanding Key Points 27 Speaking about understanding can occur among students debating a problem or with a teacher questioning students individually or as a group. Writing about understanding can occur at the board, on worksheets, on homework, or in student journals. Both speaking and writing are valuable ways to consolidate learning and reveal students’ current level of understanding.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Conceptual Understanding – Three Examples 28 Concrete and pictorial models Conceptual questioning Writing and speaking about understanding

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Key Points 29 Conceptual understanding can be promoted in a variety of ways including use of concrete and pictorial models, conceptual questioning, as well as writing and speaking about understanding. A Story of Units provides frequent, rich opportunities for students to develop conceptual understanding. These opportunities for conceptual development are often also opportunities to nurture the Standards for Mathematical Practice.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Next Steps 30 How can you increase students’ deep understanding of the concepts you will be covering when you return to your schools? What can you share with your colleagues about implementing the conceptual understanding component of rigor?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units A Call for Equal Intensity and Balance 31 The Instructional Shifts: “Students are practicing and understanding. There is more than a balance between these two things in the classroom – both are occurring with intensity. Teachers create opportunities for students to participate in ‘drills’ and make use of those skills through extended application of math concepts...”

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units A Call for Equal Intensity and Balance 32 The Publishers’ Criteria: “To help students meet the expectations of the Standards, educators will need to pursue, with equal intensity, three aspects of rigor in the major work of each grade: conceptual understanding, procedural skill and fluency, and applications.” (page 5) “Materials and tools reflect the balances in the Standards…” (page 9)

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Intensity and Balance in A Story of Units 33

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Intensity and Balance in A Story of Units 34