Transcript Slide 1

Mathematical Shifts of the
Common Core State Standards
Focus, Coherence, and Rigor
May 2013
Common Core Training for Administrators
Middle Grades Mathematics
Division of Academics, Accountability, and School Improvement
Mathematical Shifts of the Common Core State Standards:
Focus, Coherence and Rigor
AGENDA
Purpose and Vision of CCSSM
Implementation Timeline
Six Shifts in Mathematics
Design and Organization
Instructional Implications: Classroom Look-fors
Expectations of Student Performance
CCSSM Resources: Websites
Reflections / Questions and Answers
Community Norms
We are all learners today
We are respectful of each other
We welcome questions
We share discussion time
We turn off all electronic devices
__________________
Why Common Standards?
Consistency
• Previously, every state had its own set of academic
standards and different expectations of student
performance.
Equity
• Common standards can help create more equal access to
an excellent education.
Opportunity
• Students need the knowledge and skills that will prepare
them for college and career in our global economy.
Clarity
• These new standards are clear and coherent in order to
help students, parents, and teachers understand what is
expected.
College and Career Readiness:
Anchor for the Common Core
• The Common Core State Standards were back-mapped
from the anchor of college and career readiness
because governors and state school chiefs realized
there was a significant gap between high school
expectations for students and what students are
expected to do in college/career.
– Among high school graduates, about only half are
academically prepared for postsecondary education.
College Remediation and Graduation Rates
Remediation rates and costs are staggering
• As much as 40% of all students entering 4-year colleges
need remediation in one or more courses
• As much as 63% in 2-year colleges
100%
Degree attainment rates are disappointing
80%
• Fewer than 42% of adults aged 25-34 hold college
degrees
60%
40%
20%
0%
Enter High School Graduate from
High School
Enroll in College
Persist to
Bachelor's Degree
Sophomore year
within 6 yrs
Source: The College Completion Agenda 2010 Progress Report, The College Board
Are We Mathematically Ready
for College and Careers?
Common Core State Standards Mission
The Common Core State Standards 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. The standards are
designed to be robust and relevant to the real world,
reflecting the knowledge and skills that our young
people need for success in college and careers. With
American students fully prepared for the future, our
communities will be best positioned to compete
successfully in the global economy.
Florida’s Common Core State Standards
M-DCPS
Implementation Timeline
Year / Grade level
K
1
2
3–8
9 – 12
2011-2012
F L
F LL
L
L
L
2012-2013
F L
F L
F LL
L
L
2013-2014
F L
F L
F L
B L
B L
F L
F L
F L
F L
F L
CCSS fully implemented
2014-2015
CCSS fully implemented
and assessed
F – Full Implementation of CCSSM
L – Full implementation of content area literacy standards including: text complexity, quality
and range in all grades (K-12)
B – Blended instruction of CCSS with NGSSS; last year of NGSSS assessed on FCAT 2.0
(Grades 3-8); 4th quarter will focus on NGSSS/CCSSM grade level content gaps
20102011
20112012
20122013
FCAT 2.0
FCAT 2.0
FCAT 2.0
K
NGSSS
1
2013-2014
FCAT 2.0
20142015
PARCC
20152016
PARCC
20162017
PARCC
20172018
PARCC
20182019
PARCC
K
1
2
3
4
5
6
K
1
2
3
4
5
6
7
1
2
4
5
6
7
8
7
8
9
8
9
10
10
11
10
11
12
11
12
CCSSM
2
3
3
4
NGSSS
3
4
5
4
NGSSS
5
NGSSS
6
NGSSS
CCSSM
CCSSM
(4th
9 weeks Common
Core Lockdown)
4
5
6
6
7
(4th 9 weeks Common
Core Lockdown)
NGSSS
2
3
CCSSM
5
CCSSM
(4th 9 weeks Common
Core Lockdown)
5
6
7
8
7
8
9
9
10
(4th 9 weeks Common
Core Lockdown)
7
8
(4th 9 weeks Common
Core Lockdown)
9
CCSSM
(4th 9 weeks Common
Core Lockdown)
6
CCSSM
CCSSM
CCSSM
Mathematical Shifts
FOCUS deeply on what is emphasized in the
Standards
COHERENCE: Think across grades, and link to
major topics within grades
RIGOR: Require-
Fluency
Deep Understanding
Model/Apply
Dual Intensity
Shift 1: Focus
Teachers use the power of the eraser and significantly
narrow and deepen the scope of how time and energy is
spent in the math classroom. They do so in order to
focus deeply on only the concepts that are prioritized in
the standards so that students reach strong foundational
knowledge and deep conceptual understanding and are
able to transfer mathematical skills and understanding
across concepts and grades.
Students are able to transfer mathematical skills
and understanding across
concepts and grades.
–
Spend more time on Fewer Concepts
Achievethecore.org
Mathematics Shift 1: Focus
What the Student Does…
• Spend more time on fewer
concepts
What the Teacher Does…
• Extract content from the
curriculum
• Focus instructional time on
priority concepts
• Give students the gift of time
Spend more time on Fewer Concepts
http://www.fldoe.org/schools/ccc.asp
- Ministry of Education, Singapore
K-8 Priorities in Math
Priorities in Support of Rich Instruction and Expectations of
Fluency and Conceptual Understanding
K–2
Addition and subtraction, measurement using whole
number quantities
3–5
Multiplication and division of whole numbers and
fractions
6
Ratios and proportional reasoning; early
expressions and equations
7
Ratios and proportional reasoning; arithmetic of
rational numbers
8
Linear algebra
Achievethecore.org
Shift 2: Coherence
Principals and teachers carefully connect the learning
within and across grades so that students can build new
understanding onto foundations built in previous years.
Teachers can begin to count on deep conceptual
understanding of core content and build on it. Each
standard is not a new event, but an extension of previous
learning.
A student’s understanding of learning progressions can
help them recognize if they are on track.
Keep Building on learning year after year
Achievethecore.org
Mathematics Shift 2: Coherence
What the Student Does…
• Build on knowledge from year to
year, in a coherent learning
progression
What the Teacher Does…
• Connect the threads of critical
areas across grade levels
• Connect to the way content was
taught the year before and will be
taught the following years
• Focus on priority progressions
Keep Building on learning year after year
http://www.fldoe.org/schools/ccc.asp
Shift 3: Fluency
Teachers help students to study algorithms as “general
procedures” so they can gain insights to the structure of
mathematics (e.g. organization, patterns, predictability).
Students are expected to have speed and accuracy with
simple calculations and procedures so that they are more
able to understand and manipulate more complex
concepts.
Students are able to apply a variety of appropriate
procedures flexibly as they solve problems.
Spend time Practicing
Achievethecore.org
(First Component of Rigor)
Mathematics Shift 3: Fluency
What the Student Does…
• Spend time practicing, with
intensity, skills (in high volume)
What the Teacher Does…
• Push students to know basic skills
at a greater level of fluency
• Focus on the listed fluencies by
grade level
• Uses high quality problem sets, in
high volume
Spend time Practicing
http://www.fldoe.org/schools/ccc.asp
K-8 Key Fluencies
Grade
K
Add/subtract within 5
1
Add/subtract within 10
2
3
Add/subtract within 20
Add/subtract within 100 (pencil and paper)
Multiply/divide within 100
Add/subtract within 1000
4
Add/subtract within 1,000,000
5
Multi-digit multiplication
6
Achievethecore.org
Required Fluency
Multi-digit division
Multi-digit decimal operations
7
Solve px + q = r, p(x + q) = r
8
Solve simple 22 systems by inspection
Shift 4: Deep Conceptual Understanding
Teachers teach more than “how to get the answer;” they
support students’ ability to access concepts from a
number of perspectives so that students are able to see
math as more than a set of mnemonics or discrete
procedures.
Students demonstrate deep conceptual understanding of
core math concepts by applying them to new situations as
well as writing and speaking about their understanding.
Understand Math, Do Math, and Prove it
Achievethecore.org
(Second Component of Rigor)
Mathematics Shift 4: Deep Understanding
What the Student Does…
• Show mastery of material at a
meaningful level
• Articulate mathematical
reasoning
• Demonstrate deep conceptual
understanding of priority
concepts
• Explain and justify their thinking
What the Teacher Does…
• Create opportunities for students
to understand the “answer” from
a variety of access points
• Ensure that students understand
WHY they are doing what they’re
doing-ASK PROBING QUESTIONS
• Guide student thinking instead of
telling the next step
• Continuously self reflect and
build knowledge of concepts
being taught
Understand Math, Do Math, and Prove it
http://www.fldoe.org/schools/ccc.asp
Shift 5: Applications (Modeling)
Teachers provide opportunities to apply math concepts
in “real world” situations. Teachers in content areas
outside of math ensure that students are using math to
make meaning of and access content.
Students are expected to use math and choose the
appropriate concept for application even when they are
not prompted to do so.
Apply math in Real World situations
(Third Component of Rigor)
Mathematics Shift 5: Application (Modeling)
What the Student Does…
• Utilize math in other content
areas and situations, as relevant
What the Teacher Does…
• Apply math including areas/
courses where it is not directly
required (i.e. in science)
• Choose the right math concept to
solve a problem when not
• Provide students with real world
necessarily prompted to do so
experiences and opportunities
to apply what they have learned
Apply math in Real World situations
http://www.fldoe.org/schools/ccc.asp
Shift 6: Dual Intensity
There is a balance between practice and understanding;
both are occurring with intensity. Teachers create
opportunities for students to participate in “drills” and
make use of those skills through extended application of
math concepts.
Think fast and Solve problems
Achievethecore.org
(Fourth Component of Rigor)
Mathematics Shift 6: Dual Intensity
What the Student Does…
• Practice math skills with an
intensity that results in fluency
What the Teacher Does…
• Find the balance between
conceptual understanding and
practice within different periods
or different units
• Practice math concepts with an
intensity that forces application in
novel situations
• Be ambitious in demands for
fluency and practice, as well as
the range of application
Think fast and Solve problems
http://www.fldoe.org/schools/ccc.asp
Design
and
Organization
Design and Organization
Standards for Mathematical Content
 K-8 standards presented by grade level
 Organized into domains that progress over several
grades
 Grade introductions give 2–4 focal points at each
grade level
Standards for Mathematical Practice
 Carry across all grade levels
 Describe habits of mind of a mathematically expert
student
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
Learning Experiences
It matters how students learn
•
Learning mathematics is more than just learning
concepts and skills. Equally important are the
cognitive and metacognitive process skills. These
processes are learned through carefully constructed
learning experiences.
•
For example, to encourage students to be inquisitive
and have a deeper understanding of mathematics,
the learning experiences must include carefully
structured opportunities where students discover
mathematical relationships and principles on their
own.
Overarching Habits of Mind of a
Productive Mathematical Thinker
1. Make sense of problems and persevere in solving them
6. Attend to precision
Reasoning
and
Explaining
2. Reason abstractly
and quantitatively
3. Construct viable
arguments and
critique the
reasoning of others
Modeling
and
Using Tools
Seeing
Structure and
Generalizing
4. Model with
mathematics
7. Look for and make
use of structure
5. Use appropriate
tools strategically
8. Look for and
express regularity
in repeated
reasoning
Overarching Habits of Mind of a Productive Mathematical Thinker
MP 1: Make sense of problems
and persevere in solving them.
Mathematically proficient
students can…
MP 6: Attend to precision
Mathematically proficient
students can…
use mathematical
vocabulary to
communicate reasoning
and formulate precise
explanations
explain the meaning of the
problem and look for entry
points to its solution
monitor and evaluate their
progress and change course if
necessary
calculate accurately and
efficiently and specify
units of measure and
labels within the context of
the situation
use a variety of strategies to
solve problems
Gather
Information
Make a
plan
Anticipate
possible
solutions
Continuously
evaluate progress
Check
results
Question
sense of
solutions
Reasoning and Explaining
MP 2: Reason abstractly and
quantitatively.
Mathematically proficient
students can…
have the ability to contextualize and
decontextualize problems involving
quantitative relationships:


decontextualize - to abstract a given
situation and represent it
symbolically and manipulate the
representing symbols
contextualize - to pause as needed
during the manipulation process in
order to probe into the referents for
the symbols involved.
MP 3: Construct viable
arguments and critique the
reasoning of others
Mathematically proficient
students can…
make a mathematical
statement (conjecture)
and justify it
listen, compare, and
critique conjectures and
statements
Modeling and Using Tools
- In early grades, this might be as
Mathematically proficient
Mathematically proficient
simple
as
writing
an
addition
students can…
students can…
apply mathematicsto
to solve
consider
the available
equation
describe
a
situation.
problems that arise in
tools when solving a
MP 4: Model with
Mathematics.
everyday life
reflect and make revisions to
improve their model as
necessary
MP 5: Use appropriate tools
strategically
problem (i.e. ruler,
calculator, protractor,
manipulatives, software)
- In middle grades, a student might
use technological tools to
explore and deepen their
apply
proportional
reasoning
to
map mathematical
understanding of concepts
relationships using tools
plan
a school
such as diagrams,
two-way event or analyze a
tables, graphs, flowcharts
problem
in the community.
and formulas.
Seeing Structure and Generalizing
MP 7: Look for and make use
of structure
Mathematically proficient
students can…
look closely to determine
possible patterns and
structure (properties)
within a problem
analyze a complex
problem by breaking it
down into smaller parts
MP 8: Look for and express
regularity in repeated
reasoning
Mathematically proficient
students can…
notice repeating
calculations and look for
efficient methods/
representations to solve a
problem
generalize the process to
create a shortcut which
may lead to developing
rules or creating a formula
GROUP ACTIVITY
Each group will receive:
• An envelope containing 8 small cards
and a handout of the listed
mathematical practices.
Instructions:
• Match each of the 8 small cards with
its mathematical practice (using the
handout of the listed mathematical
practices).
Overarching Habits of Mind of a
Productive Mathematical Thinker
1. Make sense of problems and persevere in solving them
6. Attend to precision
Reasoning
and
Explaining
2. Reason abstractly
and quantitatively
3. Construct viable
arguments and
critique the
reasoning of others
Modeling
and
Using Tools
Seeing
Structure and
Generalizing
4. Model with
mathematics
7. Look for and make
use of structure
5. Use appropriate
tools strategically
8. Look for and
express regularity
in repeated
reasoning
Content Standards
and
Progressions
 Domains are larger groups/categories of standards
that progress across grades
 Clusters are groups of related standards
 Content standards define what students should
understand and be able to do
Domain
Standard
Cluster
New Florida Coding for CCSSM
Note: In the state of Florida, clusters will be numbered.
MACC.7.EE.1.2
MACC.2.OA.1.1
Common
Math
Core
Grade
Level
Domain
Standard
Cluster
Florida Coding Scheme for Common
Core State Standards
MACC.8.EE.3.7
•
•
•
•
•
Identify the cluster
Identify the grade level
Identify the standard
Identify the domain
Find the standard in your CCSSM binder
Common Core
Progressions
Think across grades and link to major topics within grades
• The Standards are designed around coherent
progressions from grade to grade.
• Teachers carefully connect the learning
across grades so that students can build new
understanding onto foundations built in
previous years.
• Each standard is not a new event, but an
extension to previous learning.
G
e
o
m
e
t
r
y
3
Understand that shapes in different categories may share attributes.
6
Find the area of right triangles, other triangles, special quadrilaterals, and polygons.
6
Progression- Activity
In your groups, use the
geometry progression handout
to identify the grade level
corresponding to each bullet.
G
e
o
m
e
t
r
y
K
1
Correctly name shapes regardless of their orientations and overall size.
Distinguish between defining attributes versus non‐defining attributes .
Recognize and draw shapes having special attributes.
2
3
4
5
Understand that shapes in different categories may share attributes.
Classify two-dimensional figures based on the presence or absence of parallel and perpendicular
lines.
Understand that attributes belonging to a category of two‐dimensional figures also belong to all
subcategories of that category.
6
Find the area of right triangles, other triangles, special quadrilaterals, and polygons.
6
Solve real‐world and mathematical problems involving area, volume and surface area.
7
Know the formulas for the volumes of cones, cylinders, and spheres.
8
Prove theorems about parallelograms.
HS
Common Core
Progressions
Elementary
grades work
with part to
whole
relationships in
fraction form.
In 6th grade, the
concept of ratio
is introduced,
and students
look at tables of
equivalent ratios
and their
representations
on a graph.
Grades 3-5
In 7th grade, students
focus more on the
use of multiple
representations to
solve proportion
problems, including
ratio tables, graphs,
and equations, with a
special focus on the
constant of
proportionality or
unit rate.
Finally in 8th
grade, students
extend the
proportional
reasoning
developed in 6th
and 7th grade to
examine linear
relationships
with multiple
representations.
Grade 6
Grade 7
Grade 8
Ratio and Proportionality Grade Level Progression
PARCC Sample Items vs. FCAT 2.0 Sample Items
The Standards & The Assessment
• Define what students should
understand and be able to do
in their study of mathematics
• These standards are “focused”
and “coherent” (i.e.,
conceptually DEEP)
• Next generation
assessment system
• Technology-based
• Assesses at a
conceptually DEEP level
PARCC Assessments
Transformative Formats
FOCUS
PARCC assessments will focus
strongly on where the Standards
focus. Students will have more
time to master concepts at a
deeper level.
RADIO BUTTONS / MC
DRAG & DROP
FOCUS
PARCC assessments will focus strongly on
where the Standards focus. Students will
CHECK BOXES
have
more
time
to
master
concepts
at a
PROBLEMS WORTH DOING
PROBLEMS
WORTH
DOING
deeper
level. conceptual
Multi-step problems,
questions, applications, and
substantial procedures will be
common, as in an excellent
classroom.
Multi-step problems, conceptual
questions, applications, and substantial
FILL-IN RESPONSES
procedures will be common, as in an
BETTER
STANDARDS
DEMAND
BETTER
STANDARDS
DEMAND BETTER
excellent
classroom.
BETTER QUESTIONS
Instead of reusing existingQUESTIONS
items,
PARCC will develop custom items
to the Standards.
WRITTEN
RESPONSES
Instead of reusing existing items,
PARCC
will develop custom items to the
Standards.
COMPARISONS
Overview of PARCC Mathematics Task
Types
Task Type
Description of Task Type
I. Tasks assessing
concepts, skills
and procedures
•
•
•
•
Balance of conceptual understanding, fluency, and application
Can involve any or all mathematical practice standards
Machine scorable including innovative, computer-based formats
Will appear on the End of Year and Performance Based Assessment components
II. Tasks assessing
expressing
mathematical
reasoning
•
•
•
•
Each task calls for written arguments / justifications, critique of reasoning, or
precision in mathematical statements (MP.3, 6).
Can involve other mathematical practice standards
May include a mix of machine scored and hand scored responses
Included on the Performance Based Assessment component
III. Tasks assessing
modeling /
applications
•
•
•
•
Each task calls for modeling/application in a real-world context or scenario (MP.4)
Can involve other mathematical practice standards
May include a mix of machine scored and hand scored responses
Included on the Performance Based Assessment component
Design of PARCC Math Summative
Assessments
• Performance Based Assessment (PBA)
– Type I items (Machine-scorable)
– Type II items (Mathematical Reasoning/Hand-Scored –
scoring rubrics are drafted)
– Type III items (Mathematical Modeling/Hand-Scored and/or
Machine-scored - scoring rubrics are drafted)
• End-of-Year Assessment (EOY)
– Type I items only (All Machine-scorable)
Grade 6 Example
Numbering / Ordering Numbers / Absolute Value
FCAT 2.0 – Grade 6
PARCC – Grade 6
PARCC – Grade 6
PARCC – Grade 6
PARCC – Grade 6
http://commoncore.dadeschools.net/
How will the implementation
of the Common Core
Mathematical Practices
shape future classroom
instruction?
Office of Academics and Transformation
Division of Academics, Accountability, & School Improvement
Questions/Concerns:
Department of Mathematics and Science
Middle Grades Mathematics
1501 N.E. 2nd Avenue, Suite 326
Miami, FL 33132
Office: 305-995-1939
Fax: 305-995-1991