© 2012 common core, inc. all rights reserved. commoncore.org nys common core mathematics curriculum...

© 2012 Common Core, Inc. All rights reserved. commoncore.org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units

Rigor BreakdownA Three Part Series


N YS C O M M O N CO R E M AT H E M AT I C S C U R R I C U LU M A Story of Units

Rigor Breakdown

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NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units

Rigor BreakdownPart 1: Conceptual Understanding

Grades 3-5



Session Objectives

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• 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



Conceptual Understanding Defined by the Instructional Shifts

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“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.”



Conceptual Understanding Defined by the Publishers’ Criteria

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“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)



Conceptual Understanding Defined by the Publishers’ Criteria

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“…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)



AGENDA

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Conceptual Understanding – Concrete and Pictorial Materials

Conceptual Understanding – Conceptual QuestioningConceptual Understanding – Writing and Speaking

about Understanding



Video Clip – Concrete and Pictorial Models

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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.



Video Clip: Fractions as Division – Concrete to Pictorial

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This video is posted in the video library on EngageNY:

http://engageny.org/video-library




Video Clip – Concrete and Pictorial Models

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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.



Mathematical Practices

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“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)



Concrete and Pictorial Models – Key Points

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• 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.



AGENDA

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about Understanding



Lesson Engagement – Conceptual Questioning

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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.



Video Clip – Conceptual Questioning

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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.



Video Clip: 10 Cakes Shared Among 4

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Video Clip – Conceptual Questioning

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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.



Conceptual Questioning – Key Points

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• 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?”



AGENDA

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about Understanding



Writing and Speaking about Understanding

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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.”



Video Clip – Writing and Speaking about Understanding

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Reflections:• How does the speaking about

understanding evident in the video compare with what occurs in your classroom/school/district today?




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Word Problem2/5 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?



Video Clip: Washing Machine Problem

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Reflections:• How does the speaking about

understanding evident in the video compare with what occurs in your classroom/school/district today?



Lesson Engagement – Writing and Speaking About Understanding

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Reflections:• Look for evidence of the “writing and

speaking about their understanding” requirement as you complete the worksheet.



Writing and Speaking About UnderstandingKey Points

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• 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.



Conceptual Understanding – Three Examples

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• Concrete and pictorial models• Conceptual questioning• Writing and speaking about understanding



Key Points

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• 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.



Next Steps

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• 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?



A Call for Equal Intensity and Balance

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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...”



A Call for Equal Intensity and Balance

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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)



Intensity and Balance in A Story of Units

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Intensity and Balance in A Story of Units

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© 2012 common core, inc. all rights reserved. commoncore.org nys common core mathematics curriculum...

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