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Permission is granted to the Alabama Department of Education for dissemination and use in any whole or part in any form within the Alabama Department
of Education region.
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Foundations of the Standards
Exploring Grade Level Intent
Exploring the Content Standards’ Structure
Exploring Standards for Mathematical Practice
Exploring Mathematical Understanding
Exploring the Expectations of Understanding
Exploring Two Standards
Exploring Vertical Connections
Determining Implications and Next Steps
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1. To understand the foundation of the
Alabama College-and Career-Ready
Standards
2. To explore the critical focus areas by grade
level
3. To explore the grade level standards
4. To explore mathematical understanding
5. To reflect on implications to your practice
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Cannot/should not be rushed–a marathon, not a
race.
Your LEAs teacher leaders are needed.
Our focus–to learn HOW to explore these
standards.
We aren’t exploring all standards today. You will
be given a process that can be duplicated in your
school.
We won’t be aligning today–because alignment
cannot be done effectively without careful
exploration.
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1. Copy of the 2010 Alabama Course
of Study: Mathematics
2. The Explorations Guide
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6.
Highlighters
Pen or pencil
Tables for group work
Timer/timekeeper
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InformationGiving
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Attentive listening
Open mindset to
receive new ideas and
information
Note-taking
Group Work &
Recording
 Open mindset
 Professional conversations
 Careful note-taking (for taking
back)
 Deep thinking
 Record questions–to be
addressed later
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Now … for some
background
information
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Joint initiative of:
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Supported by:
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The College-and Career-Ready Standards are
not intended to be new names for old ways of
doing business. They are a call to take the next
step. It is time to recognize that standards are
not just promises to our children, but promises
we intend to keep.
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What Is College and Career Readiness?
College readiness means students are prepared for
credit-bearing academic courses at 2-year or 4-year
postsecondary schools.
Career readiness means students are prepared to
succeed in workforce training programs in careers
that:
1) Offer competitive, livable salaries above the
poverty line
2) Offer opportunities for career advancement
3) Are in a growing or sustainable industry
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College & Career Focus
Consistent
Mobility
Student Ownership
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How can you ensure that the work
in your classroom is preparing
students to be ready for essential
workforce and college
responsibilities?
Alabama
Added
Content
Alabama
Added
Content
Grades K- 12
Grades K-8
Grades 9-12
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Identifies key ideas, understandings and skills
for each grade or course
Stresses deep learning (addresses mile-wide,
inch-deep issue)
Connects topics and standards within grade
or course
Requires applying concepts and skills within
same grade or course
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FOCUS: Increased Clarity and
Specificity
“It is important to recognize that “fewer
standards” are no substitute for
focused standards. Achieving “fewer
standards” would be easy to do by
resorting to broad, general statements.
Instead, these Standards aim for clarity
and specificity.” CCSS page 3.
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Provides the opportunity to make
connections between mathematical ideas
Occurs both within a grade and across
grades
Is necessary because mathematics
instruction is not just a checklist of topics
to cover, but a set of interrelated, powerful
ideas
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LEARNING PROGRESSIONS
Learning Trajectories – sometimes called
learning progressions – are sequences of
learning experiences hypothesized and designed to
build a deep and increasingly sophisticated
understanding of core concepts and practices within
various disciplines. The trajectories are based
on empirical evidence of how students’
understanding actually develops in response
to instruction and where it might break
down.
Daro, Mosher, & Corcoran, 2011
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K
1
2
3
4
5
6
7
8
HS
Counting and
Cardinality
Number and Operations in Base Ten
Number and Operations –
Fractions
Ratios and Proportional
Relationships
The Number System
Number and
Quantity
Expressions and Equations
Algebra
Operations and Algebraic Thinking
Functions
Geometry
Measurement and Data
Functions
Geometry
Statistics and Probability
Statistics and
Probability
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Know what to expect about students’ preparation.
Manage more readily the range of preparation of
students in your class.
Know what teachers in the next grade expect of
your students.
Identify clusters of related concepts at grade level.
Provide clarity about the student thinking and
discourse to focus on conceptual development.
Engage in rich uses of classroom assessment.
District curricula, assessments, and
instruction (the exterior and interior)
Administrator and teacher awareness of new
standards, grade-specific training in the critical areas
of focus, structure of the standards, verbs and
vocabulary, student practice standards, etc…
Alabama College- and Career-Ready Standards
(2010 Alabama Course of Study: Mathematics)
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Exploring the
Standards:
Grade 3
Mathematics
Activity
#1
Important descriptions at the beginning of
each grade level.
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Provide the intent of the mathematics at each grade.
Provide 3-4 critical focus areas for the grade level .
Provide a sense of …
◦ The sophistication for mathematical
understanding at the grade level.
◦ The learning progressions for the grade.
◦ Extensions from prior standards.
◦ What’s important at the grade level.
Activity
#1
Grade 3
Narrative
Turn to page 27
in your ACOS
for Grade 3.
Activity
#1
Grade 3
Activity
#2
Number &
Quantity
Statistics &
Probability
Algebra
Geometry
Functions
Modeling
Mathematical
Practice
Standards
Mathematical
Content
Standards
Activity
#2
Standards for Mathematical
Practice
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Carry across all grade levels
Describe habits of mind of a
mathematically expert student
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8.
ACOS –
pages 6-8
Make sense of problems and persevere in solving
them
Reason abstractly and quantitatively
Construct viable arguments & 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
Grade 3 Overview
Activity
#2
Standards for
Mathematical Practice
are provided in detail
on pages 6-8 of the
ACOS.
The Practices are also
listed on the Grade 3
Overview.
Activity
#2
Refer to the
ACOS
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K-8 standards presented by grade level
Organized into domains that progress over several
grades
Grades K-8 introductions give 2 to 4 focal points at
each grade level
High school standards presented by conceptual
theme (Number & Quantity, Algebra, Functions,
Modeling, Geometry, Statistics & Probability)
Activity
#2
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Content standards define what students should
understand and be able to do
Clusters are groups of related standards
 Domains are larger groups that progress across
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grades
Domain
Cluster
Statement
Standards
Content Standard Identifiers
Cluster
Activity
#2
“…grade placements for specific topics
have been made on the basis of state
and international comparisons and the
collective experience and collective
professional judgment of educators,
researchers and mathematicians.”
(2010 Alabama Course of Study, p.1)
Activity
#2
Grade 3
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Activity
#3
“The Standards for Mathematical
Practice describe varieties of
expertise that mathematics
educators at all levels should seek
to develop in their students.”
(2010 Alabama Course of Study, p. 6)
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Underlying Frameworks
Activity
#3
National Council of Teachers of Mathematics
5 PROCESS Standards
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Problem Solving
Reasoning and Proof
Communication
Connections
Representations
NCTM (2000M). Principles and
Standards for School Mathematics.
Reston, VA: Author.
Underlying Frameworks
Activity
#3
National Research Council
Strands of Mathematical Proficiency
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Conceptual Understanding
Procedural Fluency
Strategic Competence
Adaptive Reasoning
Productive Disposition
NRC (2001). Adding It Up.
Washington, D.C.: National
Academies Press.
Activity
#3
1. Make sense of problems and persevere in
solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments & 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
Refer to pages 6-8
in the ACOS
When presented with a problem,
I can make a plan, carry out my
plan, and evaluate its success.
BEFORE…
EXPLAIN the problem to
myself.
• Have I solved a problem like
this before?
ORGANIZE information.
• What is the question I need
to answer?
• What is given?
• What is not given?
• What tools will I use?
• What prior knowledge do I
have to help me?
#1
AFTER…
CHECK
DURING…
• Is my answer correct?
• How do my representations
PERSEVERE
connect to my algorithms?
MONITOR my work EVALUATE
CHANGE my plan if • What worked?
it isn’t working out
ASK myself, “Does this
make sense? “
• What didn’t work?
• What other strategies
were used?
• How was my solution
similar to or different from
my classmates?
I can use reasoning habits
to help me contextualize and
decontextualize problems.
CONTEXTUALIZE
I can take numbers and put them in
a real-world context.
For example, if given
3 x 2.5 = 7.5,
I can create a context:
I walked 2.5 miles per day for 3
days. I walked a total of 7.5
miles.
#2
DECONTEXTUALIZE
I can take numbers out of
context and work mathematically
with them.
For example, if given
I walked 2.5 miles per day for
3 days, How far did I walk?
I can write and solve
3 x 2.5 = 7.5.
I can make conjectures and
critique the mathematical
thinking of others.
I can construct, justify, and
communicate arguments by…
• considering context
• using examples and nonexamples
• using objects, drawings,
diagrams and actions
#3
I can critique the reasoning
of others by…
• listening
• comparing arguments
• identifying flawed logic
• asking questions to
clarify or improve
arguments
#4
I can recognize math in
everyday life and use math I
know to solve everyday
problems.
I can…
• make assumptions and
estimate to make
complex problems
easier.
• identify important
quantities and use tools
to show their relationships.
•evaluate my answer and
make changes if needed.
concrete
models
symbols
Represent
Math
oral
language
pictures
real-world
situations
#5
I know when to use
certain tools to help me
explore and deepen my
math understanding.
I have a math toolbox.
I know HOW to use math
tools.
I know WHEN to use math
tools.
I can reason: “Did the tool I
used give me an answer
that makes sense?”
#6
I can use precision when
solving problems and
communicating my ideas.
Problem Solving
I can calculate accurately.
I can calculate efficiently.
My answer matches what
the problem asked me to
do–estimate or find an
exact answer.
Communicating
I can SPEAK, READ, WRITE,
and LISTEN mathematically.
I can correctly use…
• math symbols.
• math vocabulary.
• units of measure.
I can see and understand how
numbers and spaces are organized
and put together as parts and
wholes.
SHAPES
NUMBERS
For example:
• Base 10 structure
• Operations and properties
•Terms, coefficients, exponents
For example:
• Dimension
• Location
• Attributes
• Transformation
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I can notice when calculations are
repeated. Then, I can find more
efficient methods and short cuts.
Patterns:
1/9 = 0.1111….
2/9 = 0.2222…
3/9 = 0.3333…
4/9 = 0.4444….
5/9 = 0.5555….
I notice the
pattern which leads
to an efficient
shortcut!!!
Activity
#3
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These practices rest on important “processes and proficiencies” with
longstanding importance in mathematics education. The first of these
are the NCTM process standards of problem solving, reasoning and
proof, communication, representation, and connections.
Lets look at mathematical practices in action in a classroom.
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Video Clip
What Mathematical Practices did you see the students use?
Activity
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Grade 3
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Activity
#4
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Domains are common learning progressions that
can progress across grade levels.
Domains do not dictate curriculum or teaching
methods.
Topics within domains are not meant to be taught
in the order presented.
Teachers must present the standards in a manner
that is consistent with decisions that are made in
collaboration with their K-12 mathematics team.
Activity
#4
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Mathematical language may be different than everyday
language and other disciplinary area language.
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Questions may arise about the meaning of the
mathematical language used. This is a good opportunity
for discussions and sense making in the ACOS.
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Questions about mathematical language can be
answered by investigating the progression of the
concepts in the standards throughout other grades.
Activity
#4
Grade 3
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Domain
Clusters
Standards
Operations & Algebraic
Thinking
Number and Operations in
Base Ten
4
9
1
3
Number and Operations-Fractions
Measurement and
Data
1
3
4
8
Geometry
TOTAL
1
2
25 Total
Standards
Activity
#5
The Alabama College- and Career-Ready
Standards for mathematics provide a major
focus on UNDERSTANDING.
Questions to think about …
What is meant by understanding?
How do we see it in students?
How do we teach it?
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Activity
#5
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Activity
#6
Interpretation
From Kindergarten through
to Grade 12, there is a
strong emphasis and
specificity on ways that
students will be expected
to show their
understanding.
Explanation
Application
Mathematics Procedural Skills
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Students who understand
a concept can:
Activity
#6
explain … interpret …apply
For example, they can …
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use it to make sense of and explain quantitative
situations (see "Model with Mathematics" in Practices)
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incorporate it into their own arguments and use it to
evaluate the arguments of others (see " Construct viable
arguments and critique the reasoning of others" in Practices)
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bring it to bear on the solutions to problems (see
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make connections between it and related concepts
"Make sense of problems and persevere in solving them")
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Activity
#6
Grade 3
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Activity
#7
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Lens #1: Student-Friendly Language
Lens #2: Key Vocabulary
Lens #3: Mathematical Practices
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Activity
#7
Explaining the intended learning in student-friendly
terms at the outset of a lesson is the critical first step
in helping students know where they are
going...Students cannot assess their own learning or
set goals to work toward without a clear vision of the
intended learning. When they do try to assess their
own achievement without understanding the learning
targets they have been working toward, their
conclusions are vague and unhelpful.
-Stiggins, Arter, Chappuis & Chappuis,
2004, pp. 58-59
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Activity
#7
Why identify key vocabulary in the
standards for instruction?
◦ To clarify the teacher’s
understanding
◦ To activate prior vocabulary in
context
◦ To make connections to the
prior learning and experiences
of students
◦ To observe how vocabulary is
developed in the learning
progressions of the standards
What implications does
the vocabulary of the
standards hold for
teacher professional
development?
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Activity
#7
“…those content standards which set an expectation of
understanding are potential ‘points of intersection’
between the Standards for Mathematical Content and
the Standards for Mathematical Practice.”
“…attend to the need to connect the mathematical
practices to mathematical content in mathematics
instruction.”
(2010 Alabama Course of Study: Mathematics, page 9)
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Activity
#7
Grade 3
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Activity
#8
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All Standards in
mathematics have a
connection to early and
subsequent concepts
and skills.
Current Standard
The flow of those
connections is
documented by how a
student develops the
concepts.
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Activity
#8
(from Phil Daro, one of three lead writers on
the Common Core Standards for Mathematics)
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Properties of operations: their role in arithmetic and
algebra
Mental math and algebra vs. algorithms (Inspection)
Units and unitizing
Operations and the problems they solve
Quantities
Variables
Functions
Modeling (As a
sequence across grades)
Number
Operations
Expressions
Equations
(As a sequence across grades)
Modeling
Practices
Activity
#8
Fractions Progression
K-2
Equal
Partitioning
Understanding
that arithmetic
of fractions
draws upon
four prior
progressions
that informed
the CCSS
Unitizing in
Base 10 and in
Measurement
Number Line
in Quantity
and
Measurement
Properties of
Operations
3-5
6-8
Rates, proportional
and Linear
Relationships
Rational numbers
Fractions
Rational
Expressions
Activity
#8
Gr. 3. Develop an understanding of fractions as numbers.
Gr. 4. Extend understanding of fraction equivalence and
ordering.
Gr. 4. Build fractions from unit fractions by applying and
extending previous understandings of operations on whole
numbers.
Gr. 4. Understand decimal notation for fractions, and compare
decimal fractions.
Gr. 5. Use equivalent fractions as a strategy to add and subtract
fractions.
Gr. 5. Apply and extend previous understandings of
multiplication and division to multiply and divide fractions.
Gr. 6. Apply and extend previous understandings of
multiplication and division to divide by fractions.
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Functions and Equation Progression
K-2
3-6
Activity
#8
7-12
Quantity and
Measurement
Operations and
Algebraic
Thinking
Ratio and
Proportional
Relationships
Functions
Expressions
and Equations
Modeling Practices
Modeling
(with
Functions)
Activity
#8
Grade 3
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Activity
#9
We’ve been exploring the
standards–now, what do we do?
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Activity
#9
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Activity
#10
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Alabama College- and CareerReady Initiative
http://alex.state.al.us/ccrs
Send questions or comments to
Cindy Freeman, [email protected]
Shelia Patterson, [email protected]
Please complete the exit ticket as instructed.
3. Write three significant things that you
learned today.
2. Write about two areas that are still
confusing to you.
1. Write one immediate step you will take to
implement the 2010 Alabama Course of
Study: Mathematics when you return to your
LEA.
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