Mathematics
Standards and Model Curriculum
Targeted Professional
Development Meeting
Presenter Name
Date
Targeted Professional Development
Meetings
Goal:
To provide opportunities for Ohio educators
to develop an understanding of the revised
standards and model curricula in all four
content areas: English language arts,
mathematics, science and social studies
Overview
• A Look Inside the CCSSM
– K- 8
– High School
• Digging Deeper
• Model Curriculum
• Progressions
• Resources
• What Should Districts Be Doing Now?
Change always comes bearing gifts.
–
~Price Pritchett
– Continuity gives us roots;
Change gives us branches,
letting us stretch and grow and
reach new heights.
~ Pauline R. Kezer
CCSS Principles
• Focus
– Identifies key ideas, understandings and skills for
each grade or course
– Stresses deep learning, which means applying
concepts and skills within the same grade or course
• Coherence
– Articulates a progression of topics across grades and
connects to other topics
– Vertical growth that reflects the nature of the
discipline
CCSS 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
Reading Literacy Standards
Grades 6-8
What does literacy look like in the
mathematics classroom?
• Learning to read mathematical text
• Communicating using correct mathematical terminology
• Reading, discussing and applying the mathematics
found in literature
• Researching mathematics topics or related problems
• Reading appropriate text providing explanations for
mathematical concepts, reasoning or procedures
• Applying readings as citing for mathematical reasoning
• Listening and critiquing peer explanations
• Justifying orally and in writing mathematical reasoning
• Representing and interpreting data
Format of K-8 Standards
Grade Level
Domain
Standard
Cluster
Grade Level Introduction
Cross-cutting
themes
Critical Area of
Focus
Grade Level Overview
Grade 4 Overview
Operations and Algebraic Thinking
Use the four operations with whole numbers to solve problems.
Gain familiarity with factors and multiples.
Generate and analyze patterns.
Number and Operations in Base Ten
Generalize place value understanding for multi-digit whole
numbers.
Use place value understanding and properties of operations to
perform multi-digit arithmetic.
Number and Operations—Fractions
Extend understanding of fraction equivalence and ordering.
Build fractions from unit fractions by applying and extending
previous understandings of operations on whole numbers.
Understand decimal notation for fractions, and compare decimal
fractions.
Measurement and Data
Solve problems involving measurement and conversion of
measurements from a larger unit to a smaller unit.
Represent and interpret data.
Geometric measurement: understand concepts of angle and
measure angles.
Geometry
Draw and identify lines and angles, and classify shapes by
properties of their lines and angles.
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
Change of Emphasis K- Grade 5
K-2
• Greater development of how numbers work
• Data analysis is just a tool for working with
numbers and shapes
Grades 3-5
• Fractions then decimals
• Multiplication with inverse division
• Operation strategies and relationships
developed BEFORE algorithm procedures
Change of Emphasis Grades 6-8
• Beginning of Data Analysis and Probability
• Introduction of Integers, Coordinate Graphing
• Focus on Linear Algebra: numerically,
graphically and symbolically
• Completion of Operations with fractions and
decimals
CCSS for High School
Mathematics
• Organized in “Conceptual Categories”
–
–
–
–
–
–
Number and Quantity
Algebra
Functions
Modeling
Geometry
Statistics and Probability
• Conceptual categories are not courses
• Additional mathematics for advanced
courses indicated by (+)
• Standards with connections to modeling
indicated by (★)
Format of High School Standards
Domain
Cluster
Standard
Advanced
Conceptual Category
Introduction
Conceptual Category Overview
Domain
Cluster
HS CCSS: Changing Content
Emphases
• Number and Quantity
– Number systems, attention to units
• Modeling
– Threaded throughout the standards
• Geometry
– Proof for all, based on transformations
• Algebra and Functions
– Organized by mathematical practices
• Statistics and Probability
– Inference for all, based on simulation
High School Mathematical Pathways
• Two main pathways:
Typical
in U.S.
– Traditional: Two algebra courses and a geometry course,
with statistics and probability in each
– Integrated: Three courses, each of which includes
algebra, geometry, statistics, and probability
• Both pathways:
–
–
–
–
Typical
outside U.S.
Complete the Common Core in the third year
Include the same “critical areas”
Require rethinking high school mathematics
Prepare students for a menu of fourth-year courses
Two Main Pathways
Pathway Overview
Course Overview: Critical Areas (units)
Course Detail by Unit (critical area)
Digging Deeper into the CCSS
Standards for Mathematical Practice
Mathematical ‘Habits of Mind’
Activity 1:
Standards for Mathematical Practice
• Read the assigned Standard for
Mathematical Practice
• Think – Write – Pair – Share
– What is the meaning of the
practice?
– How will the practice look at my
grade level?
• Group Sharing
Activity 2:
K-8 Critical Areas of Focus
HS Critical Areas
• Read a K-8 grade level’s
Critical Areas of Focus or
HS Critical Area
– What are the concepts?
– What are the skills and
procedures?
– What relationships are
students to make?
Concepts, Skills and Procedures
Concepts
• Big ideas
• Understandings or meanings
• Strategies
• Relationships
Understanding concepts underlies the development and
usage of skills and procedures and leads to connections
and transfer.
Skills and Procedures
• Rules
• Routines
• Algorithms
Skills and procedures evolve from the understanding and
usage of concepts.
Concepts, Skills and Procedures
Grade 4 Number and Operations in Base Ten
Generalize place value understanding for multi-digit whole
numbers.
• Recognize that in a multi-digit whole number, a digit
in one place represents ten times what it represents
in the place to its right. For example, recognize that
700  70 = 10 by applying concepts of place value
and division.
• Read and write multi-digit whole numbers using
base-ten numerals, number names, and expanded
form. Compare two multi-digit numbers based on
meanings of the digits in each place, using >, =,
and < symbols to record the results of comparisons.
• Use place value understanding to round multi-digit
whole numbers to any place.
Activity 2
Critical Areas
• Read the grade level Critical Areas of Focus or
HS Critical Areas
What are the concepts?
What are the procedures and skills?
What relationships are students to make?
• Look at the domains, clusters and standards for
the same grade(s) or High School Course
How do the Critical Areas inform their instruction?
Model Curriculum
Model Curriculum
Model Curriculum
Instructional Strategies
Instructional Resources and
Tools
Common Misconceptions
• Progressions
– Describe a sequence of increasing
sophistication in understanding and
skill within an area of study
• Three types of progressions
– Learning progressions
– Standards progressions
– Task progressions
Learning Progression for
Single-Digit Addition
From Adding It Up: Helping Children Learn Mathematics, NRC, 2001.
Learning Progressions Document for
CCSSM
http://ime.math.arizona.edu/progressions/
• Narratives
• Typical learning progression of a topic
• Children's cognitive development
• The logical structure of mathematics
• Math Common Core Writing Team with
Bill McCallum as Creator/Lead Author
CCSS Domain Progression
K
1
2
3
4
5
6
7
8
HS
Counting &
Cardinality
Number and Operations in Base Ten
Number and Operations –
Fractions
Ratios and Proportional
Relationships
The Number System
Expressions and Equations
Number &
Quantity
Algebra
Operations and Algebraic Thinking
Functions
Geometry
Measurement and Data
Functions
Geometry
Statistics and Probability
Statistics &
Probability
Standards Progression:
Number and Operations in Base Ten
Use Place Value Understanding
Grade 1
Grade 2
Grade 3
Use place value understanding
and properties of operations to
add and subtract.
4. Add within 100, including adding a
two-digit number and a one-digit
number, and adding a two-digit
number and a multiple of 10, using
concrete models or drawings and
strategies based on place value,
properties of operations, and/or the
relationship between addition and
subtraction; relate the strategy to a
written method and explain the
reasoning used.
Understand that in adding two-digit
numbers, one adds tens and tens,
ones and ones; and sometimes it is
necessary to compose a ten.
5. Given a two-digit number, mentally
find 10 more or 10 less than the
number, without having to count;
explain the reasoning used.
6. Subtract multiples of 10 in the
range 10-90 from multiples of 10 in
the range 10-90 (positive or zero
differences), using concrete models
or drawings and strategies based on
place value, properties of operations,
Use place value understanding
and properties of operations to
add and subtract.
5. Fluently add and subtract within
100 using strategies based on place
value, properties of operations,
and/or the relationship between
addition and subtraction.
6. Add up to four two-digit numbers
using strategies based on place
value and properties of operations.
7. Add and subtract within 1000,
using concrete models or drawings
and strategies based on place value,
properties of operations, and/or the
relationship between addition and
subtraction; relate the strategy to a
written method. Understand that in
adding or subtracting three digit
numbers, one adds or subtracts
hundreds and hundreds, tens and
tens, ones and ones; and sometimes
it is necessary to compose or
decompose tens or hundreds.
8. Mentally add 10 or 100 to a given
number 100–900, and mentally
subtract 10 or 100 from a given
number 100–900.
Use place value understanding
and properties of operations to
perform multi-digit arithmetic.
1. Use place value understanding to
round whole numbers to the nearest
10 or 100.
2. Fluently add and subtract within
1000 using strategies and algorithms
based on place value, properties of
operations, and/or the relationship
between addition and subtraction.
3. Multiply one-digit whole numbers
by multiples of 10 in the range 10–90
(e.g., 9 × 80, 5 × 60) using strategies
based on place value and properties
of operations.
Flows Leading to Algebra
Activity 3:
The Standards Progressions
• Get a partner
• K-8 Choose a Standards Progression
HS Choose the same Conceptual Category
in both Pathways
–
–
–
–
Read over the Progression/Conceptual Category
What’s New?
What’s the Same?
What’s Missing?
• Share with another pair within K-8 or HS
Task Progression
• A rich
mathematical
task can be
reframed or
resized to serve
different
mathematical
goals
CCSS Support Materials
• Mathematics Common Core State
Standards and Model Curriculum
– K-8 Comparative Analysis
– Standards Progressions View
– K-8 Critical Areas of Focus
– Crosswalks: Cluster to Benchmark
Comparison
– What should districts be doing?
– FAQ
Grade Level Comparative Analysis
Content that is new to Grade 8
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The Number System Know that
there are numbers that are not
rational, and approximate them by
rational numbers. (8.NS.1-2)
Functions Define, evaluate, and
compare functions. (8.F.1-3)
Functions Use functions to model
relationships between quantities.
(8.F.4-5)
Geometry Understand congruence
and similarity using physical
models, transparencies, or
geometry software.[initial
introduction] (8.G.1-2)
Geometry Understand and apply
the Pythagorean Theorem. [initial
introduction] (8.G.6-8)
Statistics and Probability
Investigate patterns of association
in bivariate data. (8.SP.4)
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Content that is still included at Grade 8, but
may be modified or at a greater depth
Expressions and Equations Work with
radicals and integer exponents.
(8.EE.1-4)
Expressions and Equations Understand
the connections between proportional
relationships, lines, and linear equations.
[derive y=mx] (8.EE.5-6)
Expressions and Equations Analyze and
solve linear equations and pairs of
simultaneous linear equations.
(8.EE.7-8)
Geometry Understand congruence and
similarity using physical models,
transparencies, or geometry software.
(8.G.3-5)
Geometry Solve real-world and
mathematical problems involving volume
of cylinders, cones, and spheres. (8.G.9)
Statistics and Probability Draw informal
comparative inferences about two
populations. (7.SP.3-4)
Statistics and Probability Investigate
patterns of association in bivariate data.
(8.SP.1-3)
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Content that is no longer a focus at
Grade 8
Number, Number Sense and
Operations Ratio, proportion percent
problems (See Grade 7.RP)
Measurement Order and conversion
of units of measure (See Grade 6.G)
Measurement Rates (See Grade
7.RP)
Geometry Geometric figures on
coordinate plane (See Grades 6-7.G)
Geometry Nets (See 6.G.4)
Patterns, Functions and Algebra
Algebraic expressions
(See Grades 6-7.EE)
Patterns, Functions and Algebra
Grade 8 learning is limited to linear
equations
Patterns, Functions and Algebra
Quadratic equations (See HS)
Data Analysis Graphical
representation analysis
(See Grade 6.SP)
Data Analysis Measures of center
and spread; sampling
(See Grade 7.SP)
Probability (See Grade 7.SP)
CCSS Support Materials Future Development
• Pod Casts
– Common Core State Standards – 101
– Ohio’s CCSS Model Curriculum – 102
– Standards for Mathematical Practice and the Critical
Areas of Focus – 103
• Resource Alignment Tool
• Eye of Integration
External Resources for CCSSM
• CCSSO
– www.ccsso.org/
• Achieve
– www.achieve.org
• NCTM
– www.Nctm.org
• Center for K-12 Assessment & Performance
Management at ETS
– www.k12center.org
• YouTube Video Vignettes explaining the CCSS
– http://www.Youtube.com/user/TheHuntInstitute#P/a
Resources for H.S. Improvement
• NCTM’s high school reports
– Focus on Reasoning and Sense Making
• Use the Common Core State Standards
– Identify A2E content for all students
• Use Pathways and Standards Progressions
– Reduce redundancy and incoherence
– Use previous mathematics in service of new ideas
• Ohio’s Model Curriculum
– Adopted in March 2011
What Should Districts Do Now?
• Deepen your understanding of the CCSSM in
Professional Learning Communities through:
–
–
–
–
–
the Standards for Mathematical Practice
the Critical Areas
the Model Curriculum
the Standards Progressions
the Comparative Analysis
• Begin focusing instruction around:
– the Mathematical Practices
– The Critical Areas
• Develop support structures for reaching all students
– Use previous mathematics in service of new ideas
– Provide all students access to the regular curriculum; RtI
1 CCSS, 2010, p. 5
2 PARCC – Draft Content Framework - 2011
ODE Mathematics Consultants
• Brian Roget brian.roget@ode.state.oh.us
• Anita Jones anita.jones@ode.state.oh.us
• Ann Carlson
ann.carlson@ode.state.oh.us
• Yelena Palayeva
yelena.palayeva@ode.state.oh.us