Focus, Coherence, and Rigor May 2013 Common Core Training for Administrators Middle Grades Mathematics Division of Academics, Accountability, and School

Focus, Coherence, and Rigor

May 2013

Common Core Training for Administrators

Middle Grades Mathematics

Division of Academics, Accountability, and School Improvement

Mathematical Shifts of the Common Core State Standards:

Focus, Coherence and Rigor

AGENDA

Purpose and Vision of CCSSMImplementation TimelineSix Shifts in MathematicsDesign and OrganizationInstructional Implications: Classroom Look-forsExpectations of Student PerformanceCCSSM Resources: WebsitesReflections / Questions and Answers

Community Norms

We are all learners todayWe are respectful of each otherWe welcome questionsWe share discussion timeWe turn off all electronic devices__________________

Why Common Standards?

• Previously, every state had its own set of academic standards and different expectations of student performance.

Consistency

• Common standards can help create more equal access to an excellent education.Equity

• Students need the knowledge and skills that will prepare them for college and career in our global economy. Opportunity

• These new standards are clear and coherent in order to help students, parents, and teachers understand what is expected.

Clarity

College and Career Readiness: Anchor for the Common Core

• The Common Core State Standards were back-mapped from the anchor of college and career readiness because governors and state school chiefs realized there was a significant gap between high school expectations for students and what students are expected to do in college/career.

– Among high school graduates, about only half are academically prepared for postsecondary education.

Remediation rates and costs are staggering• As much as 40% of all students entering 4-year colleges need remediation in one or more courses• As much as 63% in 2-year colleges

Degree attainment rates are disappointing•Fewer than 42% of adults aged 25-34 hold college degrees

Source: The College Completion Agenda 2010 Progress Report, The College Board

College Remediation and Graduation Rates

Are We Mathematically Ready for College and Careers?

The Common Core State Standards provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need to do to help them. The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers. With American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy.

Common Core State Standards Mission

F – Full Implementation of CCSSML – Full implementation of content area literacy standards including: text complexity,

quality and range in all grades (K-12)B – Blended instruction of CCSS with NGSSS; last year of NGSSS assessed on FCAT 2.0

(Grades 3-8); 4th quarter will focus on NGSSS/CCSSM grade level content gaps

Florida’s Common Core State Standards Implementation TimelineM-DCPS

Year / Grade level K 1 2 3 – 8 9 – 12

2011-2012 F L L L L L

2012-2013 F L F L L L L

2013-2014 CCSS fully implemented

F L F L F L B L B L

2014-2015 CCSS fully implemented

and assessed

F L F L F L F L F L

F L

F L

2010-2011

FCAT 2.0

2011-2012

FCAT 2.0

2012-2013

FCAT 2.0

2013-2014FCAT 2.0

2014-2015

PARCC

2015-2016

PARCC

2016-2017

PARCC

2017-2018

PARCC

2018-2019

PARCC

K 1 2 3 4 5 6

K 1 2 3 4 5 6 7

K 1 2 3 (4th 9 weeks Common

Core Lockdown)

4 5 6 7 8

1 2 3 4 (4th 9 weeks Common

Core Lockdown)

5 6 7 8 9


Core Lockdown)

6 7 8 9 10


Core Lockdown)

7 8 9 10 11


Core Lockdown)

8 9 10 11 12


Core Lockdown)

9 10 11 12

NGSSS

NGSSS

NGSSS

NGSSS

NGSSS

CCSSM

CCSSM

CCSSM

CCSSM

CCSSM

CCSSM

CCSSM

CCSSM

CCSSM

NGSSS

FOCUS deeply on what is emphasized in the StandardsCOHERENCE: Think across grades, and link to major topics within grades

RIGOR: Require- Fluency

Dual Intensity

Deep Understanding

Model/Apply

Mathematical Shifts

Shift 1: Focus

Teachers use the power of the eraser and 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 only the concepts that are prioritized in the standards so that students reach strong foundational knowledge and deep conceptual understanding and are able to transfer mathematical skills and understanding across concepts and grades.

Students are able to transfer mathematical skills and understanding across concepts and grades.–

Spend more time on Fewer Concepts

Achievethecore.org

What the Student Does… What the Teacher Does…• Spend more time on fewer

concepts• Extract content from the

curriculum

• Focus instructional time on priority concepts

• Give students the gift of time

Mathematics Shift 1: Focus

Spend more time on Fewer Concepts

http://www.fldoe.org/schools/ccc.asp

- Ministry of Education, Singapore

K-8 Priorities in MathPriorities in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding

K–2 Addition and subtraction, measurement using whole number quantities

3–5 Multiplication and division of whole numbers and fractions

6 Ratios and proportional reasoning; early expressions and equations

7 Ratios and proportional reasoning; arithmetic of rational numbers

8 Linear algebra

Achievethecore.org

Shift 2: Coherence

Principals and teachers carefully connect the learning within and across grades so that students can build new understanding onto foundations built in previous years.

Teachers can begin to count on deep conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning. A student’s understanding of learning progressions can help them recognize if they are on track.

Keep Building on learning year after year

Achievethecore.org

What the Student Does… What the Teacher Does…• Build on knowledge from year to

year, in a coherent learning progression

• Connect the threads of critical areas across grade levels

• Connect to the way content was taught the year before and will be taught the following years

• Focus on priority progressions

Mathematics Shift 2: Coherence

Keep Building on learning year after year


Shift 3: Fluency

Teachers help students to study algorithms as “general procedures” so they can gain insights to the structure of mathematics (e.g. organization, patterns, predictability).

Students are able to apply a variety of appropriate procedures flexibly as they solve problems.

Students are expected to have speed and accuracy with simple calculations and procedures so that they are more able to understand and manipulate more complex concepts.

Spend time Practicing

Achievethecore.org (First Component of RigorRigor)

What the Student Does… What the Teacher Does…

• Spend time practicing, with intensity, skills (in high volume)

• Push students to know basic skills at a greater level of fluency

• Focus on the listed fluencies by grade level

• Uses high quality problem sets, in high volume

Mathematics Shift 3: Fluency

Spend time Practicing


Grade Required FluencyK Add/subtract within 5

1 Add/subtract within 10

2Add/subtract within 20

Add/subtract within 100 (pencil and paper)

3Multiply/divide within 100

Add/subtract within 1000

4 Add/subtract within 1,000,000

5 Multi-digit multiplication

6Multi-digit division

Multi-digit decimal operations

7 Solve px + q = r, p(x + q) = r

8 Solve simple 22 systems by inspection

K-8 Key Fluencies

Achievethecore.org

Shift 4: Deep Conceptual 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.

Understand Math, Do Math, and Prove it

Teachers teach more than “how to get the answer;” they 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.

Achievethecore.org (Second Component of RigorRigor)


• Show mastery of material at a meaningful level

• Articulate mathematical reasoning

• Demonstrate deep conceptual understanding of priority concepts

• Explain and justify their thinking

• Create opportunities for students to understand the “answer” from a variety of access points

• Ensure that students understand WHY they are doing what they’re doing-ASK PROBING QUESTIONS

• Guide student thinking instead of telling the next step

• Continuously self reflect and build knowledge of concepts being taught

Mathematics Shift 4: Deep Understanding

Understand Math, Do Math, and Prove ithttp://www.fldoe.org/schools/ccc.asp

Shift 5: Applications (Modeling)

Students are expected to use math and choose the appropriate concept for application even when they are not prompted to do so.

Apply math in Real World situations

Teachers provide opportunities to apply math concepts in “real world” situations. Teachers in content areas outside of math ensure that students are using math to make meaning of and access content.

(Third Component of RigorRigor)


• Utilize math in other content areas and situations, as relevant

• Choose the right math concept to solve a problem when not necessarily prompted to do so

• Apply math including areas/ courses where it is not directly required (i.e. in science)

• Provide students with real world experiences and opportunities to apply what they have learned

Mathematics Shift 5: Application (Modeling)

Apply math in Real World situations


Shift 6: Dual Intensity

There is a balance between practice and understanding; 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.

Think fast and Solve problems

Achievethecore.org (Fourth Component of RigorRigor)

What the Student Does… What the Teacher Does…• Practice math skills with an

intensity that results in fluency

• Practice math concepts with an intensity that forces application in novel situations

• Find the balance between conceptual understanding and practice within different periods or different units

• Be ambitious in demands for fluency and practice, as well as the range of application

Mathematics Shift 6: Dual Intensity

Think fast and Solve problems


Design Design and and

OrganizatioOrganizationn

Design and Organization

Standards for Mathematical Content K-8 standards presented by grade level Organized into domains that progress over several

grades Grade introductions give 2–4 focal points at each

grade level

Standards for Mathematical Practice Carry across all grade levels Describe habits of mind of a mathematically expert

student

Mathematical Practices

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics

5. Use appropriate tools strategically

6. Attend to precision

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoning

Learning ExperiencesIt matters It matters howhow students learn students learn

• Learning mathematics is more than just learning concepts and skills. Equally important are the cognitive and metacognitive process skills. These processes are learned through carefully constructed learning experiences.

• For example, to encourage students to be inquisitive and have a deeper understanding of mathematics, the learning experiences must include carefully structured opportunities where students discover mathematical relationships and principles on their own.

Reasoning and

Explaining

Seeing Structure

and Generalizin

g

Overarching Habits of Mind of a Productive Mathematical Thinker



Modeling and

Using Tools





1. Make sense of problems and persevere in solving them6. Attend to precision

Mathematically proficient students can…

explain the meaning of the problem and look for entry points to its solution

monitor and evaluate their progress and change course if necessary

use a variety of strategies to solve problems


MP 1: Make sense of problems and persevere in solving them.

Gather Information

Make a plan

Anticipate possible solutions

Continuously evaluate progress

Check results

Question sense of solutions


use mathematical vocabulary to communicate reasoning and formulate precise explanations

calculate accurately and efficiently and specify units of measure and labels within the context of the situation

MP 6: Attend to precision


have the ability to contextualize and decontextualize problems involving quantitative relationships:

decontextualize - to abstract a given situation and represent it symbolically and manipulate the representing symbols

contextualize - to pause as needed during the manipulation process in order to probe into the referents for the symbols involved.

Reasoning and Explaining

MP 2: Reason abstractly and quantitatively.


make a mathematical statement (conjecture) and justify it

listen, compare, and critique conjectures and statements

MP 3: Construct viable arguments and critique the reasoning of others


apply mathematics to solve problems that arise in everyday life

reflect and make revisions to improve their model as necessary

map mathematical relationships using tools such as diagrams, two-way tables, graphs, flowcharts and formulas.

Modeling and Using Tools

MP 4: Model with Mathematics.


consider the available tools when solving a problem (i.e. ruler, calculator, protractor, manipulatives, software)

use technological tools to explore and deepen their understanding of concepts

MP 5: Use appropriate tools strategically- In early grades, this might be as

simple as writing an addition equation to describe a situation.

- In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community.

- In early grades, this might be as simple as writing an addition equation to describe a situation.

- In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community.


look closely to determine possible patterns and structure (properties) within a problem

analyze a complex problem by breaking it down into smaller parts

Seeing Structure and Generalizing

MP 7: Look for and make use of structure


notice repeating calculations and look for efficient methods/ representations to solve a problem

generalize the process to create a shortcut which may lead to developing rules or creating a formula

MP 8: Look for and express regularity in repeated reasoning

GROUP ACTIVITY

Each group will receive:•An envelope containing 8 small cards and a handout of the listed mathematical practices.

Instructions:•Match each of the 8 small cards with its mathematical practice (using the handout of the listed mathematical practices).

Reasoning and

Explaining

Seeing Structure

and Generalizin

g




Modeling and

Using Tools





1. Make sense of problems and persevere in solving them6. Attend to precision

Content Content Standards and Standards and ProgressionsProgressions

DomainsDomains are larger groups/categories of standards that progress across grades

ClustersClusters are groups of related standards Content standards Content standards define what students should

understand and be able to do

ClusterStandard

Domain

New Florida Coding for CCSSMNew Florida Coding for CCSSM

MACC.2.OA.1.1

MathCommon

Core

Grade Level

Domain

Cluster

MACC.7.EE.1.2

Standard

Note: In the state of Florida, Note: In the state of Florida, clustersclusters will be will be numberednumbered.

Florida Coding Scheme for Common Core State Standards

MACC.8.EE.3.7• Identify the cluster• Identify the grade level• Identify the standard• Identify the domain• Find the standard in your CCSSM binder

Common CoreProgressions

• The Standards are designed around coherent coherent progressionsprogressions from grade to grade.

• Teachers carefully connect the learning across gradesacross grades so that students can build new understanding onto foundations built in previous years.

• Each standard is not a new event, but an Each standard is not a new event, but an extension to previous learning.extension to previous learning.

Think across grades and link to major topics within gradesThink across grades and link to major topics within grades

6

3Understand that shapes in different categories may share attributes.

6Find the area of right triangles, other triangles, special quadrilaterals, and polygons.

Geometry

Progression- Activity

In your groups, use the geometry progression handout to identify the grade level corresponding to each bullet.

HS

6

3Understand that shapes in different categories may share attributes.

6Find the area of right triangles, other triangles, special quadrilaterals, and polygons.

KCorrectly name shapes regardless of their orientations and overall size.

1Distinguish between defining attributes versus non defining attributes .‐

2Recognize and draw shapes having special attributes.

4Classify two-dimensional figures based on the presence or absence of parallel and perpendicular lines.

5Understand that attributes belonging to a category of two dimensional figures also belong to all ‐subcategories of that category.

7Solve real world and mathematical problems involving area, volume and surface area.‐

8Know the formulas for the volumes of cones, cylinders, and spheres.

Geometry

Common CoreProgressions

Grades 3-5Grades 3-5

Elementary grades work with part to whole relationships in fraction form.

Grade 6Grade 6

In 6th grade, the concept of ratio is introduced, and students look at tables of equivalent ratios and their representations on a graph.

Grade 7Grade 7

In 7th grade, students focus more on the use of multiple representations to solve proportion problems, including ratio tables, graphs, and equations, with a special focus on the constant of proportionality or unit rate.

Grade 8Grade 8

Finally in 8th grade, students extend the proportional reasoning developed in 6th and 7th grade to examine linear relationships with multiple representations.

Ratio and Proportionality Grade Level Progression

PARCC Sample Items vs. FCAT 2.0 Sample Items

• Next generation assessment system

• Technology-based• Assesses at a conceptually DEEP level

The Standards & The Assessment

• Define what students should understand and be able to do in their study of mathematics

• These standards are “focused” and “coherent” (i.e., conceptually DEEP)

PARCC Assessments

WHAT

PROBLEMS WORTH DOINGMulti-step problems, conceptual questions, applications, and substantial procedures will be common, as in an excellent classroom.

BETTER STANDARDS DEMAND BETTER QUESTIONS

Instead of reusing existing items, PARCC will develop custom items to the Standards.

FOCUSPARCC assessments will focus strongly on where the Standards focus. Students will have more time to master concepts at a deeper level.

HOW

DRAG & DROP

FILL-IN RESPONSES

COMPARISONS

RADIO BUTTONS / MC

CHECK BOXES

WRITTEN RESPONSES

Transformative Formats

FOCUS

PARCC assessments will focus strongly on where the Standards focus. Students will have more time to master concepts at a deeper level.PROBLEMS WORTH DOING

Multi-step problems, conceptual questions, applications, and substantial procedures will be common, as in an excellent classroom.BETTER STANDARDS DEMAND BETTER

QUESTIONS

Instead of reusing existing items, PARCC will develop custom items to the Standards.

Overview of PARCC Mathematics Task Types

Task Type Description of Task Type

I. Tasks assessing concepts, skills and procedures

• Balance of conceptual understanding, fluency, and application• Can involve any or all mathematical practice standards• Machine scorable including innovative, computer-based formats• Will appear on the End of Year and Performance Based Assessment components

II. Tasks assessing expressing mathematical reasoning

• Each task calls for written arguments / justifications, critique of reasoning, or precision in mathematical statements (MP.3, 6).

• Can involve other mathematical practice standards• May include a mix of machine scored and hand scored responses• Included on the Performance Based Assessment component

III. Tasks assessing modeling / applications

• Each task calls for modeling/application in a real-world context or scenario (MP.4)

• Can involve other mathematical practice standards• May include a mix of machine scored and hand scored responses• Included on the Performance Based Assessment component

Design of PARCC Math Summative Assessments

• Performance Based Assessment (PBA)– Type I items (Machine-scorable)– Type II items (Mathematical Reasoning/Hand-Scored –

scoring rubrics are drafted)– Type III items (Mathematical Modeling/Hand-Scored and/or

Machine-scored - scoring rubrics are drafted)

• End-of-Year Assessment (EOY)– Type I items only (All Machine-scorable)

Grade 6 Example

Numbering / Ordering Numbers / Absolute Value

FCAT 2.0 – Grade 6

PARCC – Grade 6Part a

PARCC – Grade 6Part b

PARCC – Grade 6Part c

PARCC – Grade 6Part d

http://commoncore.dadeschools.net/

Office of Academics and Transformation

Division of Academics, Accountability, & School Improvement

Questions/Concerns:Department of Mathematics and Science

Middle Grades Mathematics1501 N.E. 2nd Avenue, Suite 326

Miami, FL 33132Office: 305-995-1939

Fax: 305-995-1991

Documents

Focus, Coherence, and Rigor May 2013 Common Core Training for Administrators Middle Grades Mathematics Division of Academics, Accountability, and School