Common Core State Standards Math Workgroup Training

Math
Common Core State Standards

Dr. Marci Shepard
Orting School District
CCSS Math Workgroup
April 2012

Includes information from OSPI, ESDs, NCTM, Ohio Department of Education and other sources

Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012

Did you miss previous sessions?

http://www.orting.wednet.edu/education/components/layout/default.php?sectionid=374&

Common Core State Standards
The Big Ideas in MATH

Office of Superintendent of Public Instruction
Randy I. Dorn, State Superintendent

Focusing on the Foundation…
Washington’s Implementation Timeline & Activities
2010-11 2011-12 2012-13 2013-14 2014-15

Phase 1: CCSS Exploration

Phase 2: Build Awareness & Begin
Building Statewide Capacity

Phase 3: Build Statewide Capacity
and Classroom Transitions

Phase 4: Statewide Application and
Assessment

Ongoing: Statewide Coordination
and Collaboration to Support
Implementation

January 2012 CCSS Webinar Series Part 2: Mathematics 4

http://www.youtube.com/watch?v=dnjbwJdcPjE&list=UUF0pa3nE3aZAfBMT8pqM5PA&index=5&feature=plcp

Orting School District * Teaching, Learning and Assessment * 2012

Content Progressions and Major Shifts

Major Shifts
Focus Coherence Application
• Fewer big ideas --- • Articulated progressions of topics • Being able to apply
learn more and performances that are concepts and skills to
• Learning of concepts developmental and connected to new situations
is emphasized other progressions
January 2012 CCSS Webinar Series Part 2: Mathematics 6

Structural Comparison:
WA Standards vs. CCSS Mathematics
WA Mathematics Standards Common Core State Standards

Grades K-8, high school standards
presented through six mathematical
Presentation of Grade K-8, high school standards presented domains including specially noted
Standards in traditional and integrated pathways. STEM standards - denoted by (+)
symbols.

Grade-level standards are broken into
Grade-level standards are broken into core
clusters of learning under several
Organization content areas, additional key content, and
domains and all have Standards for
mathematical processes.
Mathematical Practice.

Standards are accompanied by explanatory Standards have occasional examples
Examples
comments and examples. in italics.

Kindergarten | Grade 1 | Grade 2 | Grade 3 | Grade 4 | Grade 5 | Grade 6 | Grade 7 |
Grade 8
Transition for Algebra I | Transition for Geometry | Integrated Math I | Integrated Math II

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

Organization of the Standards


CCSS Design and Organization


Format of K-8 Standards Grade Level

Domain

Standard

Cluster


Cross-cutting
Grade Level Introduction themes

Critical Area of
Focus


Grade Level Overview
Grade 4 Overview Mathematical Practices
Operations and Algebraic Thinking 1. Make sense of problems and
Use the four operations with whole numbers to solve problems. persevere in solving them
Gain familiarity with factors and multiples. 2. Reason abstractly and
Generate and analyze patterns. quantitatively
Number and Operations in Base Ten 3. Construct viable arguments and
Generalize place value understanding for multi-digit whole numbers. critique the reasoning of others
Use place value understanding and properties of operations to
perform multi-digit arithmetic. 4. Model with mathematics
Number and Operations—Fractions 5. Use appropriate tools
Extend understanding of fraction equivalence and ordering. strategically
Build fractions from unit fractions by applying and extending previous 6. Attend to precision
understandings of operations on whole numbers.
Understand decimal notation for fractions, and compare decimal 7. Look for and make use of
fractions. structure
Measurement and Data 8. Look for and express regularity
Solve problems involving measurement and conversion of in repeated reasoning
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.


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


Format of Standards


Conceptual Category Introduction


Conceptual Category Overview

Domain

Cluster


High School Mathematical Pathways
Typical
• Two main pathways: 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
Typical
outside U.S.
• Both pathways:
– 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)


Understanding the Math Common Core State Standards

QUESTIONS 1-4


Content


Critical Areas in Mathematics
Priorities in Support of Rich Instruction and
Grade
Expectations of Fluency and Conceptual Understanding

Addition and subtraction, measurement using
K–2
whole number quantities
Multiplication and division of whole numbers
3–5
and fractions
Ratios and proportional reasoning; early
6
expressions and equations
Ratios and proportional reasoning; arithmetic
7
of rational numbers
8 Linear algebra


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?


Critical Areas of Focus


Digging into the Standards….
Focusing on the Domain

• Read using a highlighter to identify language
someone might have difficulty with
• Develop parent friendly language and/or
examples for 2nd column of template



QUESTION 5


Progressions


• 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
Ratios and Proportional
Number and Operations in Base Ten
Relationships Number &
Number and Operations – Quantity
The Number System
Fractions

Expressions and Equations Algebra
Operations and Algebraic Thinking
Functions Functions

Geometry Geometry

Statistics &
Measurement and Data Statistics and Probability
Probability


Standards Progression:
Number and Operations in Base Ten


Use Place Value Understanding
Grade 1 Grade 2 Grade 3
Use place value understanding and Use place value understanding and Use place value understanding and
properties of operations to add and properties of operations to add and properties of operations to perform
subtract. subtract. multi-digit arithmetic.
4. Add within 100, including adding a 5. Fluently add and subtract within 100 1. Use place value understanding to
two-digit number and a one-digit using strategies based on place value, round whole numbers to the nearest 10
number, and adding a two-digit number properties of operations, and/or the or 100.
and a multiple of 10, using concrete relationship between addition and 2. Fluently add and subtract within 1000
models or drawings and strategies based subtraction. using strategies and algorithms based on
on place value, properties of operations, 6. Add up to four two-digit numbers place value, properties of operations,
and/or the relationship between using strategies based on place value and/or the relationship between
addition and subtraction; relate the and properties of operations. addition and subtraction.
strategy to a written method and explain 7. Add and subtract within 1000, using 3. Multiply one-digit whole numbers by
the reasoning used. concrete models or drawings and multiples of 10 in the range 10–90 (e.g.,
Understand that in adding two-digit strategies based on place value, 9 × 80, 5 × 60) using strategies based on
numbers, one adds tens and tens, ones properties of operations, and/or the place value and properties of
and ones; and sometimes it is necessary relationship between addition and operations.
to compose a ten. subtraction; relate the strategy to a
5. Given a two-digit number, mentally written method. Understand that in
find 10 more or 10 less than the number, adding or subtracting three digit
without having to count; explain the numbers, one adds or subtracts
reasoning used. hundreds and hundreds, tens and tens,
6. Subtract multiples of 10 in the range ones and ones; and sometimes it is
10-90 from multiples of 10 in the range necessary to compose or decompose
10-90 (positive or zero differences), tens or hundreds.
using concrete models or drawings and 8. Mentally add 10 or 100 to a given
strategies based on place value, number 100–900, and mentally subtract
properties of operations, and/or the 10 or 100 from a given number 100–
relationship between addition and 900.

High School Pathways
• The CCSSM Model Pathways
– Two models that organize the CCSSM into coherent, rigorous
courses
– NOT required. The two sequences are examples, not
mandates

• Pathway A: Consists of two algebra courses and a geometry
course, with some data, probability and statistics infused
throughout each (traditional)
• Pathway B: Typically seen internationally that consists of a
sequence of 3 courses each of which treats aspects of
algebra, geometry and data, probability, and statistics.


Flows Leading to Algebra


(later in presentation)

TASK PROGRESSION



QUESTIONS 6-14


Practices


8 CCSSM Mathematical Practices
Standards for Mathematical Practice

– Make sense of problems and persevere in solving them
– Reason abstractly and quantitatively
– Construct viable arguments and critique the reasoning of
others
– Model with mathematics
– Use appropriate tools strategically
– Attend to precision
– Look for and make use of structure
– Look for and express regularity in repeated reasoning


Standards for Mathematical Practices

Graphic



Take a moment to examine the first three words
of each of the 8 mathematical practices… what
do you notice?

Mathematically Proficient Students…



• Consider the verbs that illustrate the student
actions each practice.
• For example, examine Practice #3: Construct
viable arguments and critique the reasoning
of others.
Highlight the verbs.
Discuss with a partner: What jumps out at you?


Mathematical Practice #3: Construct viable
arguments and critique the reasoning of others
Mathematically proficient students understand and use stated assumptions,
definitions, and previously established results in constructing arguments. They make
conjectures and build a logical progression of statements to explore the truth of their
conjectures. They are able to analyze situations by breaking them into cases, and can
recognize and use counterexamples. They justify their conclusions, communicate them
to others, and respond to the arguments of others. They reason inductively about
data, making plausible arguments that take into account the context from which the
data arose. Mathematically proficient students are also able to compare the
effectiveness of two plausible arguments, distinguish correct logic or reasoning from
that which is flawed, and if there is a flaw in an argument-explain what it is.
Elementary students can construct arguments using concrete referents such as
objects, drawings, diagrams, and actions. Such arguments can make sense and be
correct, even though they are not generalized or made formal until later grades. Later
students learn to determine domains to which an argument applies. Students at all
grades can listen or read the arguments of others, decide whether they make sense,
and ask useful questions to clarify or improve the arguments.


Mathematical Practice #3: Construct viable
arguments and critique the reasoning of others
Mathematically proficient students:
• understand and use stated assumptions, definitions, and previously established results in
constructing arguments.
• make conjectures and build a logical progression of statements to explore the truth of their
conjectures.
• analyze situations by breaking them into cases, and can recognize and use counterexamples.
• justify their conclusions, communicate them to others, and respond to the arguments of others.
• reason inductively about data, making plausible arguments that take into account the context
from which the data arose.
• compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning
from that which is flawed, and-if there is a flaw in an argument-explain what it is.
• construct arguments using concrete referents such as objects, drawings, diagrams, and actions.
Such arguments can make sense and be correct, even though they are not generalized or made
formal until later grades.
• determine domains to which an argument applies.
• listen or read the arguments of others, decide whether they make sense, and ask useful
questions to clarify or improve the arguments.

Observations
• What do you notice?

• What will students be doing differently?

• What will teachers be doing differently?


The Standards for [Student]
Mathematical Practice

On a scale of 1 (low) to 6 (high),
to what extent is your school/our district
promoting students’ proficiency in Practice 3?
Evidence for your rating?


The Standards for [Student]
• SMP1: Explain and make conjectures…
• SMP2: Make sense of…
• SMP3: Understand and use…
• SMP4: Apply and interpret…
• SMP5: Consider and detect…
• SMP6: Communicate precisely to others…
• SMP7: Discern and recognize…
• SMP8: Notice and pay attention to…


…describe the thinking processes, habits of mind and
dispositions that students need to develop a
deep, flexible, and enduring understanding of
mathematics; in this sense they are also a means to an
end

SP1. Make sense of problems
“….they [students] analyze
givens, constraints, relationships and goals. ….they
monitor and evaluate
their progress and change course if necessary. …. and
they continually ask themselves “Does this make sense?”

AND….
describe mathematical content students need to learn

SP1. Make sense of problems
“……. students can explain correspondences between
equations, verbal descriptions, tables, and graphs or
draw diagrams of important features and
relationships, graph data, and search for regularity or
trends.”


Buttons Task
Gita plays with her grandmother’s collection of black & white buttons.
She arranges them in patterns. Her first 3 patterns are shown below.

Pattern #1 Pattern #2 Pattern #3 Pattern #4

1. Draw pattern 4 next to pattern 3.

2. How many white buttons does Gita need for Pattern 5 and Pattern 6?
Explain how you figured this out.

3. How many buttons in all does Gita need to make Pattern 11? Explain
how you figured this out.

4. Gita thinks she needs 69 buttons in all to make Pattern 24. How do
you know that she is not correct? How many buttons does she need to
make Pattern 24? Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012

Buttons Task
1. Individually complete parts 1 - 3.
2. Then work with a partner to compare your
work and complete part 4. (Look for as many
ways to solve parts 3 and 4 as possible.)


Buttons Task
Gita plays with her grandmother’s collection of black & white buttons.
She arranges them in patterns. Her first 3 patterns are shown below.

Pattern #1 Pattern #2 Pattern #3 Pattern #4

1. Draw pattern 4 next to pattern 3. 15 buttons and 18
buttons

2. How many white buttons does Gita need for Pattern 5 and Pattern 6?
Explain how you figured this out.
34 buttons
3. How many buttons in all does Gita need to make Pattern 11? Explain
how you figured this out.
73 buttons
4. Gita thinks she needs 69 buttons in all to make Pattern 24. How do
you know that she is not correct?
How many buttons does she need to make Pattern 24?

Buttons Task


Buttons Task
Which mathematical practices are needed complete
the task?
Indicate the primary practice.
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.

Standards for [Student]
“Not all tasks are created equal, and different
tasks will provoke different levels and kinds
of student thinking.”
Stein, Smith, Henningsen, & Silver, 2000

“The level and kind of thinking in which
students engage determines what they
will learn.”
Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997


The Nature of Tasks Used in the
Classroom …
…Will Impact Student Learning!


But, WHAT TEACHERS DO
with the tasks matters too!

The Mathematical Tasks Framework

Stein, Grover & Henningsen (1996)
Smith & Stein (1998)
Stein, Smith, Henningsen & Silver (2000)

http://www.insidemathematics.org/index.php/classroom-videovisits/public-lessons-numerical-
patterning/218-numerical-patterninglesson-planning?phpMyAdmin=NqJS1x3gaJqDM-1-
LXtX3WJ4e8


Learner A


Learner B


Buttons Task Revisited
What might a teacher get out of using the same
math task two days in a row, rather than
switching to a different task(s)?
– Address common misconceptions
– Support students in moving from less to
more sophisticated solutions


Buttons Task Revisited
Which of the Standards of Mathematical Practice did the
students engage in when they revisited the task?
Indicate the primary practice.
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.


But, WHAT TEACHERS DO
with the tasks matters too!
The Mathematical Tasks Framework

Stein, Grover & Henningsen (1996)
Smith & Stein (1998)
Stein, Smith, Henningsen & Silver (2000)


The 8 Standards for Mathematical Practice –
place an emphasis on student demonstrations
of learning…
Equity begins with an understanding of how
the selection of tasks, the assessment of tasks,
the student learning environment creates great
inequity in our schools…


To what extent do all students in your
class, school or our district have the
opportunity to engage in tasks that
promote attainment of the mathematical
practices on a regular basis?
Please rate on a scale of
1 (low) to 6 (high).


Content and Practices


Cognitive Complexity


Depth of Knowledge Levels


Sorting Activity
• Categorize tasks into level 1, 2, 3, or 4 using
Cognitive Complexity Levels. Record your
responses on the provided worksheet.
• Share results and come to consensus at your
table. One person will record results on the
“master” copy.
• Share results and review criteria groups used
for low and high levels.


Sorting questions to ponder…
• How did you determine between levels 2 & 3?
• Does a task presented as a word problem
always have a high level of cognitive
complexity?
• If a task requires an explanation, does it have
a high level of cognitive complexity?


Changing the Cognitive Complexity Level
• Pick out a task that was placed in level 1 or 2.
Determine how you would modify your task to
be a level 3 task.
• Share task out with whole group.


Cognitive Complexity and
Mathematical Practices

Which levels of cognitive complexity allow
students to develop the mathematical
practices?


Task Progression

• A rich
mathematical
task can be
reframed or
resized to serve
different
mathematical
goals


Are there various levels of Cognitive
Complexity in your instructional materials?
• Review several types of problems/tasks found
in your instructional materials.
• What level of cognitive complexity are these
tasks?
– Level 1 (recall)
– Level 2 (skill/concept)
– Level 3 (strategic thinking)
– Level 4 (extended thinking)


Are there various levels of Cognitive
Complexity in your instructional materials?
Share the types of problems/tasks you found.
• What are the prevalent levels of complexity in
your instructional materials?
• How will this impact meeting the standards
for mathematical practice?


Gas Mileage Problem
• With scaffolding
• Without scaffolding


Who’s Doing the Work?
TEDtalk: Dan Meyer Video

http://www.youtube.com/watch?v=BlvKWEvKSi8

Video Debrief
• How much is too much support; how much is
too little?
• How does scaffolding interfere/promote
standards for mathematical practice?

• Compare/contrast Gas Mileage activities


Appendix A

Questions 15-22


Tables


Question 23


Transition Plans


Three-Year Transition Plan for Common Core State Standards for Mathematics
by Grade Level


State Resources for Transition

Grade-level transition documents describe:

– What standards to continue
– What standards to remove
– What standards to move to


OSD RESOURCES
MATH COMMON CORE STATE STANDARDS


OSD Teaching, Learning and Assessment Website: Common Core State Standards:
http://www.orting.wednet.edu/education/components/scrapbook/default.php?sectiondetailid=3910&

Common Core State Standards for Math: http://www.k12.wa.us/CoreStandards/Mathematics/pubdocs/CCSSI_MathStandards.pdf
Designing High School Mathematics Courses (Appendix A): http://www.corestandards.org/assets/CCSSI_Mathematics_Appendix_A.pdf

Illustrative Math: http://illustrativemathematics.org/


Mathematical Practices by Grade Level: http://www.azed.gov/standards-practices/files/2011/10/2010mathglossary.pdf

3-Year Transition Plan: http://www.k12.wa.us/CoreStandards/pubdocs/Three-YearDomainImplementation.pdf


1 CCSS, 2010, p. 5 Dr. Marci Shepard  Orting School District  Teaching, Learning & Assessment  2012
2 PARCC – Draft Content Framework - 2011

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