STRATEGY MEETING

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Math Tools for Unpacking Addressing the West
Virginia
Next Generation Math Standards
Elementary School Version
West Virginia RESAs 3 and 7Charleston and
Morgantown, WV
April, 2013
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Essential Workshop Questions

1. What is the relationship between the Common Core
   Standards an the Next Generation Math Standards,
   and why were they developed?
2. How are the Next Generation Math Standards
   organized?
3. What are the Six Instructional Shifts and the
   Eight Mathematical Practices What are their role
   in the Next Generation Standards?
4. What processes are useful for unpacking the
   standards?
5. What are the implications of the Standards on the
   way we approach the teaching and learning of
   mathematics?

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VKR Math Vocabulary Activity

• Assess your Vocabulary Knowledge Rating (VKR) of
  personal knowledge of these important workshop
  words.
• Consider each word and check the appropriate
  column. Check 4 column, if you could explain and
  teach others. Check 3 column if you know the
  term well, but would not want to teach others.
  Check 2 column if you have heard of the term.
  Check 1 column if the word is new to you.

CCSS WV Institute 4 3 2 1
VKR
Standards for Mathematical Practices
Content Standards
etc.
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VKR Math Vocabulary Activity
1 2 3 4
Common Core StandardsNext Generation
StandardsStandardClusterObjectiveTeaching
StrategyStudent Engagement ActivityFive Stages
of TL MathSix Instructional Shifts Eight
Mathematical Practices
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Whats the connection between the Common Core
Standards and the Next Generation Standards, and
why were these standards developed?
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What are the Common Core Standards?
The Common Core Standards are a product of a U.S.
education initiative that seeks to bring diverse
state curricula into alignment with each other by
following the principles of standards-based
education reform. The initiative is sponsored by
the National Governors Association (NGA) and the
Council of Chief State School Officers (CCSSO).
At this time, 45 U.S. States are committed to
implementing the Common Core Standards.
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What are the Next Generation Standards?
The Next Generation Standards are West Virginias
education standards. These standards parallel
the Common Core Standards, and contain
modifications that meet the specific needs of
West Virginia. The Next Generation Standards
represent the next logical step in the
progression of the statewide movement called
EducateWV Enhancing Learning. For Now. For the
Future.
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Why were the new Standards developed?

• The Next Generation Standards were developed
  to 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. be robust and relevant to
  the real world reflect the knowledge and
  skills that our young people need for success
  in college and careers

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Why were the new Standards developed?

• The Next Generation Standards were developed
  to be sure that American students are fully
  prepared for success in the global economy
• help teachers zero in on the most important
  knowledge and skills
• establish shared goals among students,
  parents, and teachers

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Why were the new Standards developed?

• The Next Generation Standards were developed
  to help states and districts assess the
  effectiveness of schools and classrooms and
  give all students an equal opportunity for
  high achievement
• help solve the problem of discrepancies
  between States test results and
  International test results
• replace the discrepant array of curriculums
  that existed across the country

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How are the Next Generation Standards organized?
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Common Core and Next Generation Organization
Terminology
Common Core Standards (CCS)
Next Generation Standards (NGS)
Domain (CCS only)
Standards (CCS and NGS)
Cluster (CCS and NGS)
Objective (NGS only)
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Common Core Organization/Terminology
In the Common Core Standards, the terms domain,
standard, and cluster have the following
meanings. domain used for the broad math
strand or category name standard
more specific math category name (next
level beyond domain) cluster group of
specific learning objectives that
connect with the standard
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Common Core Standards for MathExample of how
they are organized

• Grade 5 Standard Operations and Algebraic
  Thinking
• Write and interpret numerical
  expressionsM.5.OA.1 Use parentheses, brackets
  or braces in numerical expressions, and evaluate
  expressions with these symbols.
• M.5.OA.2 Write simple expressions that record
  calculations with numbers.

Domain
Standard
Cluster
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Next Generation Organization/Terminology
In the Next Generation Standards, the terms
standard, cluster, and objective have the
following meanings. standard used for the
broad math strand or category name
(replaces the CC word domain) cluster more
specific math category name (next
level beyond standard, replaces the CC
word standard) objectives specific things
that students should learn and be able
to do (listed in each cluster)
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How are the Next GenerationMath Standards
organized?

• Grade 5 Standard Operations and Algebraic
  Thinking
• Write and interpret numerical
  expressionsM.5.OA.1 Use parentheses, brackets
  or braces in numerical expressions, and evaluate
  expressions with these symbols.
• M.5.OA.2 Write simple expressions that record
  calculations with numbers.

Standard
Cluster
Objectives
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The Next Generation Math Standards for Grades K-5

• The next five slides show the standards (broad
  math categories or strands) for grades K-5. Note
  the similarities and differences among the grade
  levels.

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The Next Generation Math Standards for Grades K-5

• Kindergarten Standards
• Counting and CardinalityQuestions and Algebraic
  ThinkingNumbers and Operations in Base
  TenMeasurement and DataGeometry

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The Next Generation Math Standards for Grades K-5

• First Grade Standards
• Operations and Algebraic ThinkingNumbers and
  Operations in Base TenMeasurement and
  DataGeometry

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The Next Generation Math Standards for Grades K-5

• Second Grade Standards
• Operations and Algebraic ThinkingNumbers and
  Operations in Base TenMeasurement and
  DataGeometry

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The Next Generation Math Standards for Grades K-5

• Third Grade Standards
• Operations and Algebraic ThinkingNumbers and
  Operations in Base TenNumbers and Operations
  with FractionsMeasurement and DataGeometry

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The Next Generation Math Standards for Grades K-5

• Fourth Grade Standards
• Operations and Algebraic ThinkingNumbers and
  Operations in Base TenNumbers and Operations
  with FractionsMeasurement and DataGeometry

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The Next Generation Math Standards for Grades K-5

• Fifth Grade Standards
• Operations and Algebraic ThinkingNumbers and
  Operations in Base TenNumbers and Operations
  with FractionsMeasurement and DataGeometry

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The Next Generation Math Standards for Grades K-5

• The next five slides show the breakdown of the
  common Operations and Algebraic Thinking
  (Questions and Algebraic Thinking for
  Kindergarten) standard for grades K-5. Each
  slide shows the clusters for the standard, and
  the number of objectives associated with each
  cluster.

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The Next Generation Math Standards for Grades K-5

• Take note of the standard, cluster, and number of
  objectives for each cluster. Work with a partner
  from your grade level, and see if you can guess
  what the objectives are for your grade-level
  clusters.

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How are the Next Generation Math Standards
organized?

• Kindergarten Standard and Cluster
• Questions and Algebraic Thinking Understand
  addition as putting together and adding to, and
  understand subtraction as taking apart and taking
  from (5 objectives)

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How are the Next Generation Math Standards
organized?

• First Grade Standard and Cluster
• Operations and Algebraic ThinkingRepresent and
  Solve Problems Involving Addition and
  Subtraction- (2 objectives) Understand and
  Apply Properties of Operations and the
  Relationship between Addition and
  Subtraction- (2 objectives) Add and Subtract
  within 20- (2 objectives) Work with Addition
  and Subtraction Equations- (2 objectives)

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How are the Next Generation Math Standards
organized?

• Second Grade Standard and Cluster
• Operations and Algebraic Thinking Represent and
  Solve Problems Involving Addition and
  Subtraction- (1 objective) Add and Subtract
  within 20- (1 objective) Work with Equal
  Groups of Objects to Gain Foundations for
  Multiplication- (2 objectives)

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How are the Next Generation Math Standards
organized?

• Third Grade Standard and Cluster
• Operations and Algebraic Thinking Represent and
  Solve Problems Involving Multiplication and
  Division- (4 objectives) Understand
  Properties of Multiplication and the Relationship
  between Multiplication and Division- (2
  objectives) Multiply and Divide within 100- (1
  objective) Solve Problems Involving the Four
  Operations and Identify and Explain Patterns
  in Arithmetic- (2 objectives)

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How are the Next Generation Math Standards
organized?

• Fourth Grade Standard and Cluster
• Operations and Algebraic Thinking Use the Four
  Operations with Whole Numbers to Solve Problems-
  (3 objectives) Gain Familiarity with Factors
  and Multiples- (1 objective) Generate and
  Analyze- (1 objective)

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How are the Next Generation Math Standards
organized?

• Fifth Grade Standard and Cluster
• Operations and Algebraic Thinking Write and
  Interpret Numerical Expressions- (2
  objectives) Analyze Patterns and Relationships-
  (1 objective)

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The Next Generation Math Standards for Grades K-5

• After guessing what the objectives are for each
  cluster, work in grade-level teams and read the
  objectives for each cluster identified in this
  activity. For each objective, work together to
  create a math problem that captures the essence
  of the objective. The standard, clusters,
  objectives and sample problems will be share with
  the entire group to provide a K-5 vertical view
  of the teaching and learning progressions
  associated with the K-5 math program.

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Six Instructional Shifts Associated with West
Virginias Next Generation Math Standards
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Six Instructional Shifts in Math
Focus
Coherence
Fluency
Understanding
Applications
Dual Intensity
New Points of Emphasis for Teaching the Next
Generation Standards
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Instructional Shifts

• Instructional Shifts within the common core are
  needed for students to attain the standards.

Kelly L. Watts, RESA 3
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6 Shifts in Mathematics

• Focus
• Coherence
• Fluency
• Deep Understanding
• Applications
• Dual Intensity

Kelly L. Watts, RESA 3
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Focus

• In reference to the TIMMS study, there is power
  of the eraser and a gift of time. The Core is
  asking us to prioritize student and teacher time,
  to excise out much of what is currently being
  taught so that we can put an end to the mile
  wide, inch deep phenomenon that is American Math
  education and create opportunities for students
  to dive deeply into the central and critical math
  concepts. We are asking teachers to focus their
  time and energy so that the students are able to
  do the same.

Kelly L. Watts, RESA 3
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Focus

• Teachers
• Make conscious decisions about what to excise
  from the curriculum and what to focus on
• Pay more attention to high leverage content and
  invest the appropriate time for all students to
  learn before moving onto the next topic
• Think about how the concepts connect to one
  another
• Build knowledge, fluency, and understanding of
  why and how we do certain math concepts.

• Students
• Spend more time thinking and working on fewer
  concepts
• Being able to understand concepts as well as
  processes. (algorithms)

Kelly L. Watts, RESA 3
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Coherence

• We need to ask ourselves
• How does the work Im doing affect work at the
  next grade level?
• Coherence is about the scope and sequence of
  those priority standards across grade bands.
• How does multiplication get addressed across
  grades 3-5?
• How do linear equations get handled between 8 and
  9?
• What must students know when they arrive, what
  will they know when they leave a certain grade
  level?

Kelly L. Watts, RESA 3
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Coherence

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

• Teachers
• Connect the threads of math focus areas across
  grade levels
• Think deeply about what youre focusing on and
  the ways in which those focus areas connect to
  the way it was taught the year before and the
  years after

Kelly L. Watts, RESA 3
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Fluency

• Fluency is the quick mathematical content what
  you should quickly know. It should be recalled
  very quickly. It allows students to get to
  application much faster and get to deeper
  understanding. We need to create contests in our
  schools around these fluencies. This can be a
  fun project. Deeper understanding is a result of
  fluency. Students are able to articulate their
  mathematical reasoning, they are able to access
  their answers through a couple of different
  vantage points its not just getting the answer
  but knowing why. Students and teachers need to
  have a very deep understanding of the priority
  math concepts in order to manipulate them,
  articulate them, and come at them from different
  directions.

Kelly L. Watts, RESA 3
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Fluency

• Students
• Spend time practicing, with intensity, skills (in
  high volume)

• Teacher
• Push students to know basic skills at a greater
  level of fluency
• Focus on the listed fluencies by grade level
• Create high quality worksheets, problem sets, in
  high volume

Kelly L. Watts, RESA 3
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Deep Understanding

• The Common Core is built on the assumption that
  only through deep conceptual understanding can
  students build their math skills over time and
  arrive at college and career readiness by the
  time they leave high school. The assumption here
  is that students who have deep conceptual
  understanding can
• Find answers through a number of different
  routes
• Articulate their mathematical reasoning
• Be fluent in the necessary baseline functions in
  math, so that they are able to spend their
  thinking and processing time unpacking
  mathematical facts and make meaning out of them.
  
• Rely on their teachers deep conceptual
  understanding and intimacy with the math concepts

Kelly L. Watts, RESA 3
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Deep Understanding

• Students
• Show, through numerous ways, mastery of material
  at a deep level
• Use mathematical practices to demonstrate
  understanding of different material and concepts

• Teacher
• Ask yourself what mastery/proficiency really
  looks like and means
• Plan for progressions of levels of understanding
• Spend the time to gain the depth of the
  understanding
• Become flexible and comfortable in own depth of
  content knowledge

Kelly L. Watts, RESA 3
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Applications

• The Common Core demands that all students engage
  in real world application of math concepts.
  Through applications, teachers teach and measure
  students ability to determine which math is
  appropriate and how their reasoning should be
  used to solve complex problems. In college and
  career, students will need to solve math problems
  on a regular basis without being prompted to do
  so.

Kelly L. Watts, RESA 3
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Applications

• Students
• 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

• Teachers
• Apply math in areas where its not directly
  required (i.e. science)
• Provide students with real world experiences and
  opportunities to apply what they have learned

Kelly L. Watts, RESA 3
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Dual Intensity

• This is an end to the false dichotomy of the
  math wars. It is really about dual intensity
  the need to be able to practice and do the
  application. Both things are critical.

Kelly L. Watts, RESA 3
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Dual Intensity

• Students
• Practice math skills with a intensity that
  results in fluency
• Practice math concepts with an intensity that
  forces application in novel situations

• Teacher
• Find the dual intensity between understanding and
  practice within different periods or different
  units
• Be ambitious in demands for fluency and
  practices, as well as the range of application

Kelly L. Watts, RESA 3
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The Next Generation Math Standards for Grades K-5

• The next six slides show the six instructional
  shifts and short instructional scenarios that
  each connect with one of the shifts. Read each
  scenario and determine the instructional shift
  that it represents.

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Six Instructional Shifts in Math
Fluency
Mrs. Johnson, a fifth-grade teacher, delivered
two informational lessons on the concept of
parentheses, brackets, braces, and numeric
expressions. After two days of paper/pencil
practice, she decided to teach her students the
550 Game (demonstrated in the Corwin/Silver
Strong workshop) and to let them compete in
pairs. Her goal was to help her 5th graders to
sharpen their proficiency with numeric
expressions and math symbols, and to mentally
process numbers faster.
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Six Instructional Shifts in Math
Focus
In planning a unit on Place Value, Mrs. Smith
used the Five Stages planning tool (demonstrated
in the Corwin/Silver Strong workshop) to ensure
that she would design lessons and student
engagement activities that would help her
students to develop a strong knowledge base,
understanding of concepts, proficiency of skills,
and the ability to solve a variety of related
problems.
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Six Instructional Shifts in Math
Dual Intensity
Principal Joe visited several math classes and
noticed that the lessons all emphasized
procedures, skills, and practice. Joe met with
the teachers and complimented them on their
thorough approach to skill development. Joe also
encouraged them to work together and to devise a
plan to show students how those math skills are
used in the real world. The goal would be to
continue to strengthen students skills, and to
teach students how to use those skills in problem
solving.
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Six Instructional Shifts in Math
Coherence
Several math teachers and administrators from
participated in a joint exercise where they
investigated several Next Generation math
objectives from grades levels K-5. The
participants developed sample math problems that
aligned with the K-5 objectives and shared their
work with each other, so they could all
understand how the curriculum pieces fit
together.
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Six Instructional Shifts in Math
Understanding
Prior to learning the rules associated with
operations on fractions and mixed numbers,
students participated in the Fraction Paper
Cutting Activity (demonstrated in the
Corwin/Silver Strong workshop). The
student-centered activity allowed students to cut
paper, form fraction pieces, and use their paper
pieces to model and investigate a variety of
fraction problems.
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Six Instructional Shifts in Math
Applications
Mr. Williams noticed that his fourth-grade
science curriculum presented a number of
opportunities to integrate math into several
science lessons, and vice versa. Mr. Williams
decided to create a simple correlation of science
concepts with math concepts that featured common
math concepts and skills, so they can be taught
together.
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The Six Instructional Shifts

• Can you remember the Six Instructional Shifts?
  The Great Coverup Strategy, shown on the next
  slide, will challenge you to see how many shifts
  you can recall and recite.

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Six Instructional Shifts
Focus
Coherence
Fluency
Understanding
Application
Dual Intensity
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Standards for the Eight Mathematical Practices
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Making a case . . .
Work individually and investigate the result of
adding two even whole numbers. Is the sum always,
sometimes, or never even? Create a sensible rule
for adding two even whole numbers and the
expected result. Explain why your rule
works. Continue to work individually and
investigate the result of adding two odd whole
numbers. Is the sum always, sometimes, or never
odd? Create a sensible rule for adding two odd
whole numbers and the expected result. Explain
why your rule works.
Share your findings, rules, and explanations with
a learning partner. Will your rules always work?
Be sure to critique your partners argument.
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Making a case . . .
In the preceding activity, participants had
opportunities to think about math, investigate
math, draw conclusions, communicate their
findings to other participants, and critique each
others thinking. This kind of math engagement
satisfies one of the 8 Mathematical Practices
shown below.
Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others
The Eight Mathematical Practice are shown on the
next slide.
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The 8 Mathematical Practices
Building insights about meaning, and learning how
to communicate those insights

• 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 tool strategically.
• 6. Attend to precision.
• 7. Look for and make use of structure.
• 8. Look for and express regularity in repeated
  reasoning.

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Eight Mathematical Practices Applied to a Real
Standard
Review the list of Eight Mathematical Practices.
How can they be applied to the standard and
objectives below?

• Grade 5 Standard Operations and Algebraic
  Thinking
• Cluster Write and interpret numerical
  expressionsM.5.OA.1 Use parentheses, brackets or
  braces in numerical expressions, and evaluate
  expressions with these symbols.
• M.5.OA.2 Write simple expressions that record
  calculations with numbers.

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Unpacking the Standards
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Unpacking the Standards

• Many organization templates and tools exist and
  can be used to unpack math standards. One such
  tool is the Five Stages Unpacking Tool for Math
  Standards. This tool is aligned with the Five
  Stages of Teaching and Learning Mathematics. The
  next three slides provide an explanation of the
  Five Stages of Teaching and Learning Math.

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Try this . . .
1. Write the numerical expression for the sum of
the interior angles of a polygon with n sides.
(n 2)180
2. Explain why this formula works.
3. Use the formula to calculate the sum of the
interior angles of an octagon.
(8 2)180 6(180) 600 480 1080 degrees
4. Knowing that 3 interior angles of home plate
are right angles, find the measures of the other
two.
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Try this . . .
4. Knowing that 3 interior angles of home plate
are right angles, find the measures of the other
two.
(n 2)180
(5 2)180
(3)180
540
540 270 270
270 2 135o
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Try this . . .
1. Write the numerical expression for the sum of
the interior angles of a polygon with n sides.
(n 2)180
Knowledge
2. Explain why this formula works.
Understanding
3. Use the formula to calculate the sum of the
interior angles of an octagon.
Proficiency of Skills
(8 2)180 1080 degrees
4. Knowing that 3 interior angles of home plate
are right angles, find the measures of the other
two.
Applications
Each angle 135 degrees
Retention
5. Now that you know how to solve this kind of
problem, what will help you to remember how to
solve the
problem for future applications?
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The Five Stages of Teaching and Learning
Mathematics

• Success or failure associated with solving an
  arbitrary math problem comes down to five
  questions. 1. Did the student know the math
  vocabulary, terms, formulas, and number facts
  associated with the problem?2. Did the student
  understand the math concepts, hidden questions,
  and math connections in the problem?3. Was the
  student fluent with respect to the math
  procedures and skills needed to solve the
  problem?4. Was the student able to apply the
  knowledge, understanding, and skills in relation
  to the real-world context of the problem?5. Was
  the student able to retain or remember important
  math facts, skills, and concepts needed to solve
  the problem?

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The Five Stages of Teaching and Learning
Mathematics

• The Five Stages of Teaching and Learning
  Mathematics is a helpful framework for planning,
  teaching, and assessing a math lesson or unit.
• The Five Stages of Teaching and Learning
  Mathematics can also serve as a model for
  unpacking a math standard.

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The Five Stages of Teaching and Learning Math
Knowledge
Understanding
Proficiency of Skills
Applications
Retention
Great Considerations for Unpacking a Math Standard
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The Five Stages of Teaching and Learning
Mathematics

• The next three slides provide an example of how
  the Five Stages of Teaching and Learning Math can
  be used to unpack a math objective. A sample
  objective is shown below.
• Grade 4 M.4.NF4 Apply and extend previous
  understandings of multiplication to multiply a
  fraction by a whole number.

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Unpacking Grade4 M.4.NF4

• Grade 4 M.4.NF4 Apply and extend previous
  understandings of multiplication to multiply a
  fraction by a whole number.

Knowledge
Teaching Strategies
product- answer to multiplication
problem The fractional equivalent to a whole
number n is n/1. 1 times any number is the
number itself. 0 times any number is zero n x
a/b na/b how to simplify an improper fraction
Mental Math Strings that feature these
facts The Great Cover Up Convergence
Mastery Proceduralizing
Understanding
Teaching Strategies
For any fraction a/b, a is the number of
times that 1/b occurs If n gt1, then n x a/b is
greater than a/b. The concept of n x a/b
expresses the idea of bringing the amount a/b to
the table n times. improper fraction and proper
fraction equivalencies
The hands-on/multiplication component of the
Fraction Paper Cutting Activity
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Unpacking Grade4 M.4.NF4

• Grade 4 M.4.NF4 Apply and extend previous
  understandings of multiplication to multiply a
  fraction by a whole number.

Proficiency of Skills
Teaching Strategies
Multiply any whole number n times any of
the common fractions a/b where b 1, 2, 3, 4, 5,
6, 8, 10, and 12. Simplify problems of the
type n x a/b n x a/b m and n x a/b
c/b
Mental Math Strings that feature these
facts The Great Cover Up Algebra War Games
(modified) Timed Challenges (for fractions)
Convergence Mastery
Applications
Teaching Strategies
Work with Tangram pieces Solve problems
involving fractional pieces of Hersheys
chocolate bars Solve two-step word problems
Solve problems involving fractional parts of time
and money
Task Rotation applied to problem solving
Graduated Difficulty Modeling and
Experimentation
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Unpacking Grade4 M.4.NF4

• Grade 4 M.4.NF4 Apply and extend previous
  understandings of multiplication to multiply a
  fraction by a whole number.

Retention
Teaching Strategies
General Math Facts Measurement
Equivalencies Properties of Fractions
Patterns
Review math facts using Timed Challenges
Incorporate measurement equivalencies in fraction
problems Create patterns based on whole numbers
x fractions
8 Math Practices that apply
1. Make sense of problems and persevere in
solving them. (All problems and experiences)2.
Reason abstractly and quantitatively. (Fraction
Paper Cutting Activity)3. Construct viable
arguments and critique the reasoning of others
(Is nxa/b always gt a/b?)4. Model with
mathematics. (Fraction Paper Cutting Activity,
Tangrams, Candy bars)5. Use appropriate tool
strategically.6. Attend to precision. (Computing
exact answers, not estimates)7. Look for and make
use of structure.8. Look for and express
regularity in repeated reasoning. (n x a/b always
equals na/b.)
75
The Five Stages of Teaching and Learning
Mathematics

• The next two slides provide a sample objective
  for grades K-5. Work with a grade level partner.
  Unpack the objective using the Five Stages
  Unpacking Tool. Make connections between the
  Eight Mathematical Practices and the things that
  students will learn and experience as they learn
  the math associated with the objective.

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Unpacking the Common Core Math Standards
The Five Stages of Teaching and Learning
Mathematics
Grade K Solve addition and subtraction word
problems, by adding and subtracting within 10, by
using objects or drawings Grade 1 Apply
properties of operations as strategies to add and
subtract within 20 Grade 2 Use addition and
subtraction within 100 to solve one and two-step
word problems
Knowledge
Understanding
Proficiency of Skills
Applications
Retention
Work with a partner, choose a standard, and
unpack the standard using the Five Stages tool.
77
Unpacking the Next Generation Math Standards
The Five Stages of Teaching and Learning
Mathematics
Grade 3 Fluently multiply and divide within 100,
using strategies such as the relationship between
multiplication and division Grade 4 Solve
multi-step word problems, posed with whole
numbers, using the four operations Grade 5 Use
parentheses, brackets, or braces in numerical
expressions, and evaluate expressions with these
symbols
Knowledge
Understanding
Proficiency of Skills
Applications
Retention
Work with a partner, choose a standard, and
unpack the standard using the Five Stages tool.
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Instructional Considerations
The 3- 4- 5- Math Instructional Model
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The 3- 4- 5- Math Instructional Model
RVD
3
Repetition, Variation of Context, Depth of Study
4
The Four Learning Styles and Task Rotation
5
The Five Stages of Teaching and Learning Math
80
Teaching math associated with the Next Generation
Standards Mathematics

• The next slides provide important information
  about The RVD Instructional Model, The Four
  Learning Styles of students, and The Five
  Stages of Teaching and Learning Math
• Each of these have important roles in the
  teaching and learning of mathematics.

81
R - V - D
RVD provides teachers with three important ideas
that can be applied to the teaching and learning
process. Repetition reminds us that practice is
an essential tool for developing fluency and
proficiency with math skills and procedures.
Variation reminds us that students need to
experience math in more than one context.
Different instructional and application contexts
give students opportunities to make important
connections and deepen their understanding of
math. Depth reminds us that students need to
learn and experience all aspects of a math
concept and not superficially engage in exercises
that only scratch the surface.
82
Introduction to the Four Learning Styles
Interpersonal Learner
Mastery Learner
Understanding Learner
Self-Expressive Learner
83
Introduction to the Four Learning Styles
Mastery Learners
Want to learn practical information and
procedures
Like math problems that are algorithmic
Approach problem solving in a step by step
manner
Experience difficulty when math becomes abstract
Are not comfortable with non-routine problems
Want a math teacher who models new skills,
allows time for practice, and builds in feedback
and coaching sessions
84
Introduction to the Four Learning Styles
Interpersonal Learners
Want to learn math through dialogue and
collaboration
Like math problems that focus on real world
applications
Approach problem solving as an open discussion
among a community of problem solvers
Experience difficulty when instruction focuses
on independent seatwork
Want a math teacher who pays attention to their
successes and struggles in math
Want a math teacher who pays attention to their
successes and struggles in math
Want a math teacher who pays attention to their
successes and struggles in math
85
Introduction to the Four Learning Styles
Understanding Learners
Want to understand why the math they learn works
Like math problems that ask them to explain or
prove
Approach problem solving by looking for
patterns and identifying hidden questions
Experience difficulty when there is a focus on
the social environment of the classroom
Want a math teacher who challenges them to
think and who lets them explain their thinking
Want a math teacher who pays attention to their
successes and struggles in math
Want a math teacher who pays attention to their
successes and struggles in math
86
Introduction to the Four Learning Styles
Self-Expressive Learners
Want to use their imagination to explore math
Like math problems that are non-routine
Approach problem solving by visualizing the
problem, generating possible solutions and
explaining alternatives
Experience difficulty when instruction focuses
on drill and practice and rote problem solving
Want a math teacher who invites imagination and
creative problem solving into the math classroom
Want a math teacher who pays attention to their
successes and struggles in math
Want a math teacher who pays attention to their
successes and struggles in math
87
The Four Learning Styles
Research shows that student learn in different
ways. The Four Learning Styles provide the basis
for a teaching and learning framework that
addresses the different ways students learn. By
providing rich learning experiences that reflect
the different learning styles, teachers can lead
more students to success in math. The Task
Rotation Teaching Strategy provides four tasks,
one for each type of learner. Students who study
math through the contexts of different learning
styles will increase their levels of success in
math.
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The Five Stages of Teaching and Learning Math
Knowledge
Understanding
Proficiency of Skills
Applications
Retention
Great Considerations for Planning, Teaching, and
Assessing a Math Lesson
89
The Five Stages of Teaching and Learning
Mathematics

• Success or failure associated with solving a math
  problem comes down to five questions. 1. Did
  the student know the math terms, formulas, and
  number facts associated with the problem?2. Did
  the student understand the math concepts, hidden
  questions, and math connections in the
  problem?3. Was the student fluent with respect
  to the math procedures and skills needed to solve
  the problem?4. Was the student able to apply
  the knowledge, understanding, and skills in the
  context of the problem?5. Was the student able
  to retain or remember important math facts,
  skills, and concepts needed to solve the problem.

90
Cooperative Planning Activity

• Work together and talk about how you will use the
  information and strategies, featured in this
  workshop, to improve math instruction and
  achievement in your classroom(s).

91
Workshop Reflections
Specific facts and ideas that I learned today
Things I learned that will really help me in my
classroom
Why the things I learned will help my students to
learn math
Creative modifications and extentions to the
things I learned today