California Common Core Standards for Mathematics
Overview
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General Content Overview
◦ Focus and Coherence
◦ Similarities
◦ Shifts
◦ Structure/Organization
◦ Activity
Objectives
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Mathematical Practices Overview
◦ Purpose
◦Structure
◦8 Practices
◦Standards Alignment
◦Activity
Common Core Standards Overview:Toward Greater Focus and Coherence
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Avoid the problem of “mile wide and an inch deep”
Recognize that “fewer standards” are no substitute for focused standards
Aim for clarity and specificity
© 2011 California County Superintendents Educational Services Association • Mathematics General Overview
Topics and performances are logical over time
Based on learning progressions research on how students learn
Reflect hierarchical nature of the content
Evolve from particulars to deeper structures
Coherence Design
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Define what students should understand and be able to do in their study of mathematics
◦ Is the ability to justify appropriate to student’s math maturity
◦ Understanding and procedural skill are equally important and can be assessed using tasks of sufficient richness
Are internationally benchmarked
◦ Reflect rigor, focus and coherence of standards in top-performing countries
Common Core Standards
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Do:
◦Set grade-level standards K-8
◦Identify standards for Algebra 1
◦Provide conceptual cluster standards in high school
◦Provide clear signposts along the way toward the goal of college and career readiness for all students
Common Core Standards
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Do not:
◦Define intervention methods or materials
◦Define the full range of supports for English learners, students with special needs and students who are well above or below grade level expectations
◦Dictate curriculum or teaching methods
Common Core Standards
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Common Core Standards for Mathematics: Two Types
Mathematical Content (different at each grade level)
Mathematical Practice (recurring throughout the grades)
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Mathematical Content
Common Core State Standards for CA DOMAINS
California Standards • Grades K-7 STRANDS
K-5•Counting and Cardinality (K only)•Operations and Algebraic Thinking•Number and Operations in Base 10•Number and Operations-Fractions•Measurement and Data
6-8•Ratio and Proportional Relationships (grade 6-7)•The Number System•Expressions and Equations•Functions (Grade 8)•Geometry•Statistics and probability
• Number Sense
• Algebra and Functions
• Measurement and Geometry
• Statistics, Data Analysis and Probability
• Mathematical Reasoning
Mathematical Content Comparison
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Comparison Example
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Domains K-8
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K 1 2 3 4 5 6 7 8
Counting & Cardinality x
Operations & Algebraic Thinking
x x x x x X
Number & Base Ten x x x x X x
Numbers & Operations-Fractions
x x X
Ratios & Proportional Relationships
x x
The Number system x x x
Expressions & Equation x x X
Functions x
Geometry x x x x x x x x X
Measurement & Data x x x x x X
Statistics & Probability x x x
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Grade Shifts: Examples
Developed by SCFIRD
Concept 1997 Standards CCCS
Compose simple shapes to form larger shapes (e.g., 2 triangles to form a rectangle)
Grade2
K
Introduction to ProbabilityGrade
3Grade
7
Introduction of fractions as numbers
Grade2
Grade3
Add and subtract simple fractions
Grade3
Grade4
Introduction of integers Grade 4
Grade6
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Teaching Algebra and Functions
https://www.teachingchannel.org/videos/algebra-lesson-planning
Example of Common Core Math in an Intermediate Classroom
Overview page◦ Lists domains, clusters and mathematical practices
Domains◦ Larger groups of related standards. Standards from
different domains may be closely related.
Standards-by grade level◦ Defines what students should understand and be able to do
Clusters◦ Groups of related standards. Standards from different
clusters may be closely related
Mathematical ContentGrade K-8 Standards
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Overview Page
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Reading the Standards
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Standards-by grade level: Defines what students should understand and be able to do
Clusters: Groups of related standards. Standards from different clusters may be closely related
Domains: Larger groups of related standards. Standards from different domains may be closely related.
Domain
StandardCLUSTER
Operations and Algebraic Thinking 5.OA
Write and interpret numerical expressions.1. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.2. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.Analyze patterns and relationships.3. Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.Number and Operations in Base Ten 5.
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Label the Domain, Standards, and Clusters for Grade 5 Common Core Mathematics.
Circle, underline or highlight key academic vocabulary within the standards and compare to DOK chart.
Go to U Drive, Curriculum, Common Core State Standards, Common Core Unpacked Math, k-12_math_crosswalk pgs. 77-96.
Using the CCSS Math and the crosswalk, choose a standard and compare similarities and differences.
Be prepared to discuss.
Mathematical Content Activity
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Mathematical Practice
Common Core Standards: Mathematical Proficiency
Standards for Mathematical Practice
◦Describe habits of mind of a mathematically expert student
◦Relate to mathematical proficiency as defined by the California Framework
◦Consistent throughout grade levels
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Problem Solving
Computational & Procedural
Skills
DOING MATH
Conceptual Understanding
“WHERE” THE MATHEMATICSWORK
“HOW” THE
MATHEMATICSWORK
“WHY” THE
MATHEMATICSWORK
Mathematical Proficiency as defined by the California Framework
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Standards for Mathematical Practice…
“ …describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high schools years.”
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Structuring the Standards for Mathematical Practice
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1. Make sense of problems and persevere in solving them.
6. Attend to precision
2. Reason abstractly and quantitatively3. Construct viable arguments and critique the reasoning of other
Reasoning and explaining
4. Model with mathematics
5. Use appropriate tools strategies
Modeling and using tools
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
Seeing structure and generalizing
Standards for Mathematical Practice
Mathematically Proficient Students:
1. Make sense of problems and
persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and
critique the reasoning
4. Model with mathematics
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5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for an express regularity in repeated
reasoning
Standards for Mathematical PracticeMathematically proficient students:
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Example of Mathematical Practices in the 5th Grade Classroom: Proportions and Ratios
http://www.insidemathematics.org/index.php/exemplary-lessons-integrating-practice-standards
Inside Mathematics Video
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Activity #1◦ Label the Mathematical Practices that align with
specific standards.◦ Choose a standard and its Mathematical
Practice(s)◦ Go to U Drive, Curriculum, Common Core State
Standards, Common Core Unpacked Math, Find your grade level
◦ Brain storm how you can integrate the Mathematical Practice with the Common Core Standard you chose.
◦ Be prepared to share.
Mathematical Practices Aligned to Common Core Standards
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Activity #2◦ Choose a standard and its Mathematical
Practice(s)◦ Go to U Drive, Curriculum, Common Core State
Standards, Common Core Unpacked Math, Find CCSSMathTasks_Grade5
◦ Review a performance task and brainstorm how you might be able to integrate one into a lesson.
◦ Be prepared to share.
Mathematical Practices Aligned to Common Core Standards
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SBAC Math Testhttp://www.smarterbalanced.org/practice-test/
SCROLL DOWN AND CLICK ON PRACTICE TEST