focus, coherence, and rigor

Focus, Coherence, and Rigor

Mathematical Shifts of the Common Core State

Standards

May 2013

Common Core Training for Administrators

High School Mathematics

Division of Academics, Accountability, and School Improvement

Common Core State Standards for Mathematics:

Focus, Coherence and Rigor

AGENDAPurpose and Vision of CCSSMShifts in MathematicsDesign and OrganizationExpectations of Student PerformanceInstructional Implications: Engaging in Mathematical Practices’ Look-fors

CCSSM Resources: Websites

Reflections / 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__________________

How do you know?

Good Mathematics is NOT how many answers you know…

but how you behave when you don’t know.

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

Purpose and Vision of the

CCSSM

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

“Teach Less, Learn More…”

Ministry of EducationSingapore

http://www.google.com/url?sa=i&rct=j&q=teach+less+learn+more&source=images&cd=&cad=rja&docid=AEzkth732sIhzM&tbnid=oUsz3XSTYYKGhM:&ved=0CAUQjRw&url=http://www3.moe.edu.sg/bluesky/tllm.htm&ei=-GBfUf6hNIPO8QTdk4HoCA&bvm=bv.44770516,d.eWU&psig=AFQjCNGQg9ZdlmM8LTh_Ah0h_fAQv8CuAQ&ust=1365291638128930

Fluency, Deep Understanding, Application, Intensity

Mathematical Shifts of the Common Core State

StandardsFocus, Coherence, and Rigor

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

RIGOR: Requires 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 ConceptsAchievethecore.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

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

Pair-share activity:

Describe what is and what is not FOCUS

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 yearAchievethecore.org


• Build on knowledge from year to year, in a coherent learning progression

• Connect the threads of math focus areas across grade levels

• Connect to the way content was taught the year before and the years after

• Focus on priority progressions

Mathematics Shift 2: Coherence

Keep Building on learning year after yearhttp://www.fldoe.org/schools/ccc.asp


Describe what is and what is not COHERENCE

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

(First Component of Rigor)Achievethecore.org

Grade Required FluencyK Add/subtract within 5

1 Add/subtract within 10

2Add/subtract within 20Add/subtract within 100 (pencil and paper)

3Multiply/divide within 100Add/subtract within 1000

4 Add/subtract within 1,000,000

5 Multi-digit multiplication

6Multi-digit divisionMulti-digit decimal operations

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

8 Solve simple 22 systems by inspection

Key Fluencies

Achievethecore.org

Fluency in High SchoolAlgebra I Fluency Recommendations

Analytic geometry of lines Add, subtract, and multiply polynomials. Transforming expressions and “chunking” (seeing parts of an expression as a

single object)

Geometry Fluency Recommendations

Triangle congruency and similarity Use coordinates to establish geometric results Use construction tools

Algebra II Fluency Recommendations

Divide polynomials with remainders by inspection in simple cases. Rewrite expressions Translate between recursive definitions and closed forms


• Spend time practicing and applying skills

• 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

Mathematics Shift 3: Fluency

Spend time Practicing http://www.fldoe.org/schools/ccc.asp


Describe what is and what is not FLUENCY

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.

(Second Component of Rigor)Achievethecore.org


• Show mastery of material at a deep level

• Articulate mathematical reasoning

• Demonstrate deep conceptual understanding of priority concepts

• 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 it



Describe what is and what is not DEEP CONCEPTUAL UNDERSTANDING

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 Rigor)Achievethecore.org


• Apply 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 where its 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

Apply math in Real World situations



Describe what is and what is not APPLICATIONS /MODELING

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

(Fourth Component of Rigor)Achievethecore.org


• Practice math skills with an intensity that results in fluency

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

• Find the dual intensity between 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 problemshttp://www.fldoe.org/schools/ccc.asp


Describe what is and what is not DUAL INTENSITY

Design and

Organization

Standards for Mathematical Practice Carry across all grade levels Connect with content standards in each grade Describe habits of mind of a mathematically expert student

K - 8 grade-by-grade standards organized by domains that progress over several grades.

9 – 12 high school standards organized by conceptual categories

Design and Organization

Standards for Mathematical Content

Standards for Mathematical

Practice

Standards for Mathematical Practices

“The Standards for Mathematical Practice are unique in that they describe how teachers need to teach to ensure their students become mathematically proficient. We were purposeful in calling them standards because then they won’t be ignored.”

~ Bill McCallum

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

Reasoning and

Explaining

Seeing Structure

and Generalizin

g

Overarching Habits of Mind of a Productive Mathematical Thinker

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

Modeling and

Using Tools

4. Model with Mathematics

5. Use appropriate tools strategically

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoning

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

Mathematically proficient students can…

explain the meaning of the problem

monitor and evaluate their progress “Does this make sense?”

use a variety of strategies to solve problems

Overarching Habits of Mind of a Productive Mathematical Thinker

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 clear definitions and mathematical vocabulary to communicate reasoning

carefully specify units of measure and labels to clarify the correspondence with quantities in a problem

MP 6: Attend to precision

manipulatives pictures symbols


have the ability to contextualize and decontextualize (navigate between the concrete and the abstract).

understand and explain the computation methods they use.

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 on their attempt to solve problems and make revisions to improve their model as necessary

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


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

analyze patterns and apply them in appropriate mathematical context

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

evaluate the reasonableness of their results throughout the problem solving process.

MP 8: Look for and express regularity in repeated reasoning

Mathematical Practices Indicators

a second look . . .

A second look . . .

Match each given set of student indicators to a Mathematical PracticeMatch each given set of teacher indicators to a Mathematical PracticeFind another group so you can complete a set of Mathematical Practices Post your conclusions on chart paper for sharing out

#. Mathematical

Practice . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

.

Content Standards and Progressions

Expect students to practice applying mathematical ways of thinking to real world issues and challenges

Require students to develop a depth of understanding and ability to apply mathematics to novel situations, as college students and employees regularly are called to do

The High School Mathematics Standards

Emphasize mathematical modeling, the use of mathematics and statistics to analyze empirical situations, understand them better, and improve decisions

Identify the mathematics that all students should study in order to be college and career ready

We need to shift our focus from H. S. completion to College and Career Readiness for All students.

Format of the High School Standards

The high school standards are organized around five conceptual categories:

Number and QuantityAlgebraFunctionsGeometryStatistics and Probability

Modeling is a sixth category that is embedded within all other conceptual categories.

Conceptual categories portray a coherent view of high school mathematics; a student’s work with functions, for example crosses a number of traditional course boundaries, potentially up through and including calculus.


Content categories: overarching ideas that describe strands of content in high school

Domains/Clusters: groups of standards that describe coherent aspects of the content category

Standards: define what students should know and be able to do at each grade level

Notations within the Common Core State Standards for Mathematics

Standards indicated as (+) are beyond the college and career readiness level but are necessary for advanced mathematics courses, such as calculus, discrete mathematics, and advanced statistics.

Standards with a (+) may still be found in courses expected for all students.

Modeling Standards are indicated with a


Domain

Cluster

Conceptual Category Overview


Conceptual Category: Number and Quantity Overview

Standards

Cluster

Domain

Beyond the college and career readiness level

New Florida Coding for CCSSM

Mathematics

CommonCore

Grade Level

Domain- Subdomain

Cluster

MACC.912.N-CN.3.9

Standard

Common CoreProgressions

Common CoreProgressions

DefinitionDescribes a topic across a number of grade levels based on conceptual development and the logical structure of mathematics.

Progression

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

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

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

Think across grades and link to major topics within grades

Learning Progressions by Domain

Mathematics Common Core State Standards

Major Flows

Elementary to High School

Operations & Algebraic Thinking

Expressions & Equations

Algebra

Number: Base Ten

Number & Quantity

Number:Fractions

Geometry

Geometry

Geometry

Measurement & Data

Statistics &Probability

Statistics &Probability

Functions Functions

The NumberSystem

K - 5 6 - 8 9 - 12

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.

Prove theorems about parallelograms.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 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

PARCC Sample Itemsvs.

EOC Sample Items

Expectations of Student Performance

• 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 Priorities

1. Determine whether students are college and career ready or on track

2. Connect to the Common Core State Standards

3. Measure the full range of student performance, including that of high- and low-achieving students

4. Provide educators data throughout the year to inform instruction

5. Create innovative 21st century, technology-based assessments

6. Be affordable and sustainable

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.

Transition to Common Core Assessments PARCC

2012 – 2013(NGSSS)

2013 – 2014(NGSSS)

2014 – 2015(CCSSM)

End-Of-Course Algebra I Geometry

Algebra I Geometry

PARCC

HS Math EOCs 3 subjects TBDField Test(sampled schools)

HS Math EOCs 3 subjects TBDBaseline

Florida Department of Education/ARM

PARCC has two required assessment components that make up a student’s overall score: the performance-based assessment (PBA) component and the end-of-year (EOY) assessment component.

High School – Algebra Example

Functions

EOC – Algebra

PARCC – Algebra

Part a

PARCC – Algebra

Part b

PARCC – Algebra

Part c

Discussion

Looking at what the Common Core assessments require of students, discuss the similarities and differences between FCAT/EOC and PARCC Format Depth and Rigor Level on Webb’s Depth of Knowledge

Analyze how verbs describe the new expectations for students

Instructional Implications

Engaging in Mathematical Practices Look-fors

~ Francis “Skip” Fennell

InsideMathematics.org

A Common Core video

http://www.insidemathematics.org/index.php/exemplary-lessons-integrating-practice-standards

Questions to Consider:

How can teachers help students learn to apply math and think about problem solving outside of the math classroom?

What does the emphasis on "depth" look like in practice?

How can we make adapting to Common Core a reflection point in our practice? Will the new standards change the way we teach?

CCSSM Resources

Website

http://commoncore.dadeschools.net/



Reflections

How can administration guide and support teachers in the effective implementation of the mathematical instructional shifts?

How will the implementation of Mathematical Practices shape future classroom instruction?

Office of Academics and Transformation

Division of Academics, Accountability, & School Improvement

Questions/Concerns:Department of Mathematics and Science

High School Mathematics1501 N.E. 2nd Avenue, Suite 326

Miami, Fl 33132Office: 305-995-1939

Fax: 305-995-1991

http://www.dadeschools.net/

focus, coherence, and rigor

Documents

math module

speaker notes

math classroom

common core training

standardscontent standards

f lf l66purpose

school improvement oelcs

implementation of ccssml