Connecticut Core Standards for Mathematics

Systems of Professional LearningModule 4 Grades K–5: Focus on Designing Learning

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Focus on Designing LearningBy the end of this session you will have:

Strengthened your working relationships with peer Core Standards CoachesExplored the structure of the Standards and the Progressions documents on which the Standards were developedCreated effective learning targets for use in planning lessonsDescribed a process and strategies for conducting informal needs assessment in order to identify student needs

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Focus on Designing Learning (cont'd)

By the end of this session you will have:Identified elements that need to be considered when designing and implementing CCS-Math lessonsDesigned and/or modified CCS-Math lessons in order to meet the needs of all studentsMade plans for next steps in your CCS-Math implementation

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Morning SessionWelcome and Introductions

Sharing Implementation Experiences

Creating Learning Targets

Identifying an Entry Point into the Mathematics

Afternoon SessionDesigning CCS-Math Lessons

Reflecting on the Lesson Design

Supporting Teachers

Post-Assessment, Session Evaluation, & Wrap Up

Today’s Agenda

Introductory Activity:Pre-Assessment – CCS-Math

Please complete the Pre-Assessment

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Page 4

Sharing Implementation Experiences

Section 1Page 6

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In Module 3 you:Articulated a common understanding of UDL.Identified the importance of incorporating UDL Principles into lessons. Described the alignment of instructional practices and learning expectations of the CCS-Math.Planned for implementing UDL strategies within classroom lessons.

Module 3 Review

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UDL is a set of three principles developed for providing all individuals equal opportunities to learn.

Universal Design for Learning (UDL)

CAST (2014)

Principle 1: Provide Multiple Means of RepresentationPrinciple 2: Provide Multiple Means of Action and ExpressionPrinciple 3: Provide Multiple Means of Engagement

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Universal Design for Learning Principles include guidance on:

Providing flexibility; Reducing barriers; Providing appropriate accommodations, supports, and challenges; andSetting expectations for all students.

AND…..

Key Points About UDL

CAST (2014)

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Key Points for Getting Started with UDLUDL can support teachers’ implementation of the CCS-Math Standards.The strategies that have been discussed for implementing the CCS-Math Standards overlap with the strategies that can be used to meet the UDL Principles.Think about, plan for, and implement the UDL Principles strategically. Begin with those that will have the greatest impact on YOUR students.

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In Module 3 you:Developed a shared understanding of the four attributes of the formative assessment process. Reflected on the role of students in the formative assessment process.Explored strategies for supporting teachers as they make changes to their classroom practices. Made plans for next steps in your CCS-Math implementation.

Module 3 Review

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What are your goals for assessment?

Assessment OF Learning

Assessment FOR Learning

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Formative Assessment: Four Attributes

(SBAC)

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Key Strategies Focusing on Students’ Role in the Formative Assessment Process:

Activating students as learning resources for one another

Activating students as owners of their own learning

Wiliam, D. (2011)

Sharing Implementation Experiences: CSS-Math

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Sharing Implementation Experiences

1. Each participant will discuss with their table group one positive highlight, one question that they have as coaches, and one ongoing need for teachers from their personal implementation of the CCS-Math thus far.

2. Each table group will then determine two positive highlights, two questions, and two ongoing needs that they will present to the larger group.

3. Participants will record notes and “New Ideas” generated from the discussion.

Guide Pages

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Positive Highlights

Questions

Ongoing Needs

Pages 6-7

Creating Learning Targets

Section 2

Page 9

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Key Ideas from Module 1

The CCS-Math embodies a core shift in teaching and learning.

Three Shifts: Focus, Coherence, and RigorStandards for Mathematical PracticeProblem TypesAccessibilityKey Classroom PracticesFormative Assessment Process

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Key Idea for Module 4

In order to design learning experiences that align with the CCS-Math, teachers must develop a deep understanding of the structure of the Standards.

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The Grain-size of Mathematics

“Mathematics does not break down into lesson-sized pieces.”

- Phil Daro

Daro, P. (2013)

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Understanding the Structure

“…individual statements of what students are expected to understand and be able to do are embedded within domain headings and cluster headings designed to convey the structure of the subject.”“The Standards” refers to all elements of the design – the wording of domain headings, cluster headings, and individual statements; the text of the grade level introductions and high school category descriptions; the placement of the standards for mathematical practice at each grade level.”

Daro, McCallum, & Zimba (2012)

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Fewer topics allow for focusing on the major work for each grade.

Focus

http://www.corestandards.org/

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Provides the critical areas for the grade level.

Describes key ideas that students should learn in relation to each of the critical areas.

http://www.corestandards.org/

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Domains

Cluster Headings

http://www.corestandards.org/

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http://www.corestandards.org/

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Understanding the Structure

“…the Standards were crafted to reward study on multiple levels: from close inspection of details, to a coherent grasp of the whole.”“Specific phrases in specific standards are worth study and can carry important meaning; yet this meaning is also importantly shaped by the cluster heading in which the standard is found. At higher levels, domain headings give structure to the subject matter of the discipline, and the practices’ yearly refrain communicates the varieties of expertise which study of the discipline develops in an educated person.”

Daro, McCallum, & Zimba (2012)

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The Grain-size of Mathematics

“Mathematics does not break down into lesson-sized pieces.”

- Phil Daro

“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, P. (2013)

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A Progression of Understanding

Why is paying attention to the structure important?

Why not just focus on the individual standard statements?

How were the major work, critical areas, cluster headings, and individual standard statements determined?

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Exploring the Content Standards

Part 2

Examine your assigned domain across the K-5 grade band and record five general observations about the progression and two observations about the relationship between the Content and Practice Standards.

2.NBT.1Understand that three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g.

706 equals 7 hundreds, 0 tens, and 6 ones.

K.NBT.1

Compose and decompose

numbers from 11to19 into ten

ones and some further ones

e.g. by using objects,

drawings, and record each

composition or decomposition

by a drawing or equation.

1.NBT.2a

Understand that numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

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A Progression of Understanding

Access the CCS-Math Progressions here: http://ime.math.arizona.edu/progressions/.Click on the first bulleted item under the heading: Draft Front Matter.Within the Draft Front Matter, scroll down to page 9 which begins the section titled: Organization of the Common Core State Standards for Mathematics.

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A Progression of Understanding

What information do the Progressions documents provide?

Why is this information important?

How can this information be useful to teachers?

Guide Pages14-15

Page 11

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Let’s Take A Break…

…Be back in 10 minutes

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Designing Meaningful Learning

How do we use the structure of the Standards and the information in the Progressions documents to determine the learning targets of day-to-day lessons if the Standards themselves are not broken down into lesson-sized pieces?Why do we need to focus on learning targets, why not just focus on the individual standards themselves?

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Formative Assessment: Four Attributes

Learning targets: Focus on what students will learn, not what they will do.

Success Criteria: Observable and measurable behaviors to be reached.

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Focus on the Major Work: Grade 3 Example

Grade 3 Introduction http://www.corestandards.org/Math/Content/3/introduction/

Mathematics Grade 3In Grade 3, instructional time should focus on four critical areas: (1) developing understanding of multiplication and division and strategies for multiplication and division within100; (2) developing understanding of fractions, especially unit fractions (fractions with numerator 1); (3) developing understanding of the structure of rectangular arrays and of area; and (4) describing and analyzing two-dimensional shapes.

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Focus on the Major Work:Grade 3 Example

Grade 3 Critical Area #2developing understanding of fractions, especially unit fractions (fractions with numerator 1)

Grade 3 Introduction http://www.corestandards.org/Math/Content/3/introduction/

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Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative to the size of the whole. Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.

Grade 3 Introduction http://www.corestandards.org/Math/Content/3/introduction/

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(1) Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size. Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors. By comparing a variety of solution strategies, students learn the relationship between multiplication and division.

Grade 3 Introduction http://www.corestandards.org/Math/Content/3/introduction/

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3) Students recognize area as an attribute of two-dimensional regions. They measure the area of a shape by finding the total number of same-size units of area required to cover the shape without gaps or overlaps, a square with sides of unit length being the standard unit for measuring area. Students understand that rectangular arrays can be decomposed into identical rows or into identical columns. By decomposing rectangles into rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to determine the area of a rectangle.

Grade 3 Introduction http://www.corestandards.org/Math/Content/3/introduction/

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2) Students develop an understanding of fractions, beginning with unit fractions. Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative to the size of the whole. For example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is divided into 5 equal parts. Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.

Grade 3 Introduction http://www.corestandards.org/Math/Content/3/introduction/

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4) Students describe, analyze, and compare properties of two-dimensional shapes. They compare and classify shapes by their sides and angles, and connect these with definitions of shapes. Students also relate their fraction work to geometry by expressing the area of part of a shape as a unit fraction of the whole.

Grade 3 Introduction http://www.corestandards.org/Math/Content/3/introduction/

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Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative to the size of the whole. Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.

Grade 3 Introduction: Key Ideas of Critical Area #2 http://www.corestandards.org/Math/Content/3/introduction/

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Students view fractions in general as being built out of unit fractions, Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole.

Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative to the size of the whole. Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.

Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative to the size of the whole.

Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative to the size of the whole. Students are able to use fractions to represent numbers equal to, less than, and greater than one.

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Part 1: Critical Areas and Key Ideas

With your group, do the following: 1. Choose a grade level and critical area for which

you will focus on designing learning throughout the remainder of this Module.

2. Identify the key ideas that support the critical area.

3. As you work, complete Part 1 of the Learning Target Planning Template

Page 12

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Making Connections to Clusters and the Individual Standards

Students view fractions in general as being built out of unit fractions.

“Mathematics does not break down into lesson-sized pieces.”

- Phil Daro

Daro, P. (2013)

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Making Connections to Clusters and the Individual Standards

“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, P. (2013)

Critical AreaKey Idea Key Idea Key Idea Key IdeaLesson(s)

Lesson(s)Lesson(s)

Lesson(s)

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Making Connections to Clusters and the Individual Standards

Students view fractions in general as being built out of unit fractions.

Cluster: Develop understanding of fractions as numbers.Associated Standard

3.NF.A.1: Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

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Part 2: Connections to Clusters and Individual StandardsWith your group, do the following: 1. Explore the domain Progressions document that aligns

with your identified critical area. 2. Determine which key idea your group will use as the

focus for your first lesson(s) within this critical area. 3. Identify the cluster headings and individual

standard(s) that support your selected key idea. 4. Complete Part 2 of the Learning Target Planning

Template.Page 13

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Creating Learning TargetsStudents view fractions in general as being built out of unit fractions.

Cluster: Develop understanding of fractions as numbers.Associated Standard

3.NF.A.1: Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

Access the CCS-Math Progressions here: http://ime.math.arizona.edu/progressions/.

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Creating Lesson Level Learning Targets

If I want my students to be able to understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

Students will understand that fraction notation consists of the numerator and the denominator.Students will _______________________________.Students will _______________________________.Students will _______________________________.

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Part 3: Creating Learning TargetsWith your group, do the following:

1. Create the lesson level learning targets for the general entry point lessons into the mathematics of the identified critical area.

2. Place your critical area, key idea, standard(s), and learning targets on chart paper.

Additional resource: http://commoncoretools.me/wp-content/uploads/2014/02/Elaborations.pdf

Page 14

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Time to Share

With your group, review the work of others who worked at your same grade level. Using sticky notes, provide your peers with questions to consider, suggestions on additional learning targets, and any general feedback that would be helpful in developing out their lesson design.

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Three Key Points to Communicate

The major work for each grade level provides the critical areas and supporting key ideas that students should explore at each grade level.Every lesson and resource should be designed and considered for the impact it will make on students’ understanding those key ideas. The progression of the mathematics is an important part of the design process.

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Bon Appétit

Identifying an Entry Point into the Mathematics

Section 3Page 16

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…you find that your students have gaps in their content knowledge?

What happens when….

ScaffoldingOpen Questions

Parallel Tasks

Examples of Strategies for Differentiating Cognitively Rigorous Tasks

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Determining Prior Knowledge

Determining the prior knowledge: Use the learning target information that you created in Section 2 and the information from the Progressions documents to identify what students should already know and be able to do prior to the work at the current grade level.Be sure to identify prior knowledge based on both the Content and the Practice Standards.

Guide Page

X

Page 16

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Determining Prior Knowledge3.NF.A.1: Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

Prior Knowledge2.G.3: Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

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Determining Prior KnowledgeWhat will be assessed:

2.G.3 – Can students partition shapes into two, three, or four equal shares?2.G.3 – Do students recognize that equal shares of identical wholes need not have the same shape?MP4 – Can students draw a picture of circles and rectangles and partition the shape in their drawing into halves, thirds, and or fourths?MP6 – Do students use the words halves, thirds, and fourths? Can students label their models with the correct fraction notation?

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Determining Prior Knowledge

Determining the prior knowledge:Use the learning target information that you created in Section 2 and the information from the Progressions documents to identify what students should already know and be able to do prior to the work at the current grade level.Be sure to identify prior knowledge based on both the Content and the Practice standards.

Guide Page

X

Page 16

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Determining Prior Knowledge

Was there anything interesting or surprising about what you found?

How did you determine where students should be with their development of the Practice Standards?

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Assessing Prior Knowledge

(SBAC)

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Assessing Prior KnowledgeDecisions to be made:

How will you elicit evidence?

Formally: pre-tests, diagnostic assessment

Informally: questions, tasks, student work

When will you elicit evidence?

Prior to planning the upcoming lessons

Fractions Example: Activities built into Geometry lessons in the unit that precedes the work with fractions.

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Assessing Prior Knowledge

With your group, determine how you will assess students’ prior knowledge.

Be prepared to talk about your assessment decisions.

Page 17

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Addressing Gaps in Prior Knowledge

(SBAC)

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…you find that your students have gaps in their content knowledge?

What happens when….

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Addressing Gaps in Prior Knowledge“The natural distribution of prior knowledge in classrooms should not prompt abandoning instruction in grade level content, but should prompt explicit attention to connecting grade level content to content from previous learning.”“Much unfinished learning from earlier grades can be managed best inside grade level work when the progressions are used to understand student thinking.”

Daro, McCallum, & Zimba (2012)

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Addressing Gaps in Prior Knowledge

With your group, brainstorm ideas for addressing gaps in prior knowledge that you will work into your lesson plan in Section 4. Mix, Mingle, and Share

When prompted, find two other people with whom you will share your ideas. When time is called, return to your table group and share any new ideas gained from your one-on-one discussions.

Page 18

Designing CCS-Math Lessons

Section 4Page 20

What are the ingredients?

Effective CCS-Math Lessons

Example: Formative assessment practices are used throughout the lesson.

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Sample CCS-Math Lesson Design Template:

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Page 21

Use the template to design a lesson that will address:

The learning targets you identified in Section 2.

The needs of all students through the pre-assessment data you created in Section 3.

Designing a CCS-Math Lesson

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Illustrative Mathematics – http://www.illustrativemathematics.orgAchieve the Core – http://achievethecore.orgSmarter Balanced – http://smarterbalanced.orgMathematics Assessment Project – http://map.mathshell.org/materials/index.php

Resources for Planning Lessons

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Page 24

Now that you have used it –How will the regular use of a template like this affect classroom instruction in your setting?Is there anything you would change about the template?

Reflect on the use of the Sample CCS-Math Lesson Design Template

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Page 24

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Let’s Take A Break…

…Be back in 10 minutes

Reflecting on the Lesson Design

Section 5Page 26

Use the following tools to evaluate the lesson you created/modified:

EQuIP rubric (introduced in Module 1)UDL Principles (introduced in Module 3)

Record your observations and modifications on page 27.

Evaluating the Lesson Design

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Page 27

What were the strengths of your lesson design?

What areas of your lesson design were deficient?

Which of the evaluation tools did you find most helpful in finding these deficiencies?

Questions for Reflection

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Supporting Teachers

Section 6Page 29

What do you want to know?Create the questions you want to answer around teachers’ readiness, prior knowledge, and understanding of the structure of the Standards and how to design learning that is aligned to the CCS-Math.

How will you find out?Conduct observations? Attend grade level meetings? Talk one-on-one?

What will you do with your findings?

Determining an Entry Point

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Pages 29-30

Closing Activities

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Page 32

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By the end of this session you will have:Strengthened your working relationships with peer Core Standards CoachesExplored the structure of the Standards and the Progressions on which the Standards were developedCreated effective learning targets for use in planning lessonsDescribed a process and strategies for conducting informal needs assessment in order to identify student needs

Focus on Designing Learning

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By the end of this session you will have:Identified elements that need to be considered when designing and implementing CCS-Math lessonsDesigned and/or modified CCS-Math lessons in order to meet the needs of all studentsMade plans for next steps in your CCS-Math implementation

Focus on Designing Learning (cont'd)

Post Assessment and Session EvaluationWhere Are You Now?

Assessing Your Learning.

Please complete an online Session Evaluation. Your feedback is very important to us! The survey is located here:

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http://surveys.pcgus.com/s3/CT-Math-Module-4-K-5

Some Key Resources

achievethecore.orgamericaachieves.org

ctcorestandards.orgengageny.org

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Thanks, and see you next time!