focus, coherence, and rigor
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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. Focus, Coherence, and Rigor. Common Core State Standards for Mathematics: - PowerPoint PPT PresentationTRANSCRIPT
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
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
What the Student Does… What the Teacher Does…
• 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
Pair-share activity:
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
What the Student Does… What the Teacher Does…
• 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
Pair-share activity:
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
What the Student Does… What the Teacher Does…
• 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
http://www.fldoe.org/schools/ccc.asp
Pair-share activity:
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
What the Student Does… What the Teacher Does…
• 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
http://www.fldoe.org/schools/ccc.asp
Pair-share activity:
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
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 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
Pair-share activity:
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
Mathematically proficient students can…
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
Mathematically proficient students can…
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.
Mathematically proficient students can…
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
Mathematically proficient students can…
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.
Mathematically proficient students can…
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
Mathematically proficient students can…
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
Mathematically proficient students can…
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.
Format of the High School Standards
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
Format of the High School Standards
Domain
Cluster
Conceptual Category Overview
Format of the High School Standards
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
~ Francis “Skip” Fennell
InsideMathematics.org
A Common Core video
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