Download Intent of the Common Core

Making Sense of
The CT Mathematics Standards
(Common Core State Standards)
GRADES K – 2
ATOMIC CONFERENCE
NOVEMBER 29, 2011
Kathy St. Onge
Ann Spinelli
M a r y S a n t i l l i M a rc i a Fe rre i ra
Intent of the Common Core
 Same goals for all students



Coherence
Focus
Clarity, rigor and specificity
 Opportunities for broadening the discussion
about the teaching and learning of
mathematics
45 states have adopted (as of December 2011)
CCSS Assessment Projects
 SBAC
SMARTER Balanced Assessment Consortium

-
30 states
http://www.k12.wa.us/smarter/
 PARCC (Partnership for the Assessment of Readiness for College and
Careers)
 25 states
-
http://www.achieve.org/PARCC
“These Standards are not intended
to be new names for old ways of
doing business.”
CCSSM, p. 5
Organization of the CCSS
 Standards for Mathematical Practice
 Math Content Standards
 Domains
 Clusters
 Standards
Connecting the Practices to the Content
 Math content standards describe what
students should understand and be able to do.
 Math practices describe ways in which
students should interact with mathematics.
 Curricula, assessment and professional
development should be focused on
connecting the mathematical practices and
the content standards.
(CCSS p. 8)
Standards for Mathematical Practices
 8 Mathematical Practices
 Related to the NCTM Process Standards
(2000) and the Strands of Mathematical
Proficiency (Adding It Up, 2001)
Standards for Mathematical Practice
The standards for mathematical practices are located in
the front of the mathematics standards and within the
“nature of mathematics” section at each grade level.
The standards for mathematical practice illustrate the
connection between 21st century skills and mathematical
content and instruction.
The standards for mathematical practices should be
considered when creating curricula, assessments, and
professional development for teachers, and administrators.
Standards for
Mathematical Practice
“…describe the varieties of
expertise that mathematics
educators at all levels should
seek to develop in their
students.”
Mathematically Proficient Students Will…
Adapted from Inside Mathematics
Standards for Mathematical Practice
1. Make sense of problems and persevere in
solving them
6. Attend to precision
Overarching habits of mind of a productive
mathematical thinker.
2. Reason abstractly and
quantitatively
Reasoning and explaining
3. Construct viable arguments
and critique the reasoning of
others
4. Model with mathematics
5. Use appropriate tools
strategically
Modeling and using tools
7. Look for and make use of
structure.
8. Look for and express regularity
in repeated reasoning
Seeing structure and generalizing
1. Make Sense of Problems and Persevere in
Solving Them
 Engage in problem solving on a regular basis
 Foster a “productive disposition” - build success early on
 Involve students in sharing solutions, methods, and
reasoning
 Frame the class environment to encourage student
interaction and conversation – math discourse
 Allow students to “struggle” with the mathematical tasks
– avoid rescuing too soon to diminish the cognitive load
 Emphasize equivalent representations of a given situation
or mathematical relationship
2. Reason Abstractly and Quantitatively
Mathematical
Problem
Decontextualize
Represent as symbols, abstraction
Refer back to the situation
Contextualize
4
x x x x
5
2 =?
 Teach concepts in context – symbols have meaning
 Base instruction on making sense and select practice that
involves the application of concepts being learned
 Emphasize reasoning as opposed to only learning
procedures
 Allow students to develop a representation of
mathematical problems on a regular basis
3. Construct Viable Arguments and Critique the
Reasoning of Others
 Encourage interaction and conversation on a regular





basis
Use problem-based activities – rich tasks
Practice the language of “argument,” conjecture, and
discourse while students are engaged in mathematical
tasks
Facilitate student discourse – “talk moves” *
Encourage taking risks, defending solutions
Have students present solutions and
ideas on a regular basis
*Classroom Discussions: Using Math Talk to Help Students Learn, 2nd edition, Grades K-6
by Suzanne Chapin, Catherine O’Connor and Nancy Anderson, Math Solutions, 2009.
4. Model with Mathematics
Problems in
everyday life…
 Use physical objects, drawings and physical
gestures to represent math situations
 Encourage student verbal descriptions
 Encourage representing the same situation in
different ways
 Guide students to see similarities in different
ways to represent the same situations
5. Use Appropriate Tools Strategically
 Provide mathematical tools in the classroom
 Ensure that students know how to use the
appropriate tools effectively
 Discuss criteria to help make a decision as to
when to use a mathematical tool
 Encourage students use their rationale for using
a tool in their explanation of their solution
6. Attend to Precision
 Make mathematical tools available in the classroom
 Display and provide instruction on mathematical




vocabulary – interactive word wall
Hold students accountable for using vocabulary in
discussion and written explanations
Embed instruction about math symbols (7, +, =, >,)
Discuss answers in terms of the context of the
problems to give students experience with the idea of
a “reasonable” answer
Review processes for computational skills; include
error analysis and feedback to develop accuracy and
proficiency
7. Look for and Make Use of Structure
 Encourage students to always look for patterns to help




develop conceptual understanding
Provide opportunities for students to generalize
Use mental math to practice patterns in our number
system
Provide opportunities to work on tasks that generate
data that can be used to develop a generalization
Foster a class environment that values and encourages
student reasoning as opposed to teacher “telling”
8. Look for and Express Regularity
in Repeated Reasoning
 Encourage students to always look for patterns or an
opportunity to generalize about computational skills
 Use mental math to practice patterns in our number
system that can be used to develop more efficient
computation methods
 Incorporate lessons and activities that use pattern or
structure to help develop conceptual understanding
 Foster a class environment that values and encourages
student reasoning as opposed to teacher “telling” what
to notice or how to do a skill
The Leadership and Learning Center Seminar- “Digging Deeper into the Common Core State Standards”
Incorporating the Practice Standards…
 Examine
the math problems.
 Think
about the Mathematical Practices that
students would engage in when solving the
problems.
 Share
with someone next to you your reasoning.
Summary
All Standards for Mathematical Practice
will not be demonstrated with every
math exercise given, but multiple
standards should be evident in every
mathematics lesson.
Common Core State Standards
K-12 Mathematics Learning Progressions
Kindergarten
1
2
3
4
5
6
7
8
Counting
and
Cardinality
HS
Number
and
Quantity
Number and Operations in Base Ten
Number and Operations:
Fractions
Operations and Algebraic Thinking
The Number System
Ratios and Proportional
Relationships (6 and 7)
Expressions and
Equations
Functions
Geometry
Measurement and Data
Algebra
Functions
Geometry
Geometry
Statistics and Probability
Statistics
and
Probability
http://education.ohio.gov/GD/Templates/Pages/ODE/ODEDetail.aspx?page=3&TopicRelationID=1704&ContentID=83475&Content=102764
Priorities in Mathematics
Grade
K–2
3–5
6
7
8
Priorities in Support of Rich Instruction and Expectations of
Fluency and Conceptual Understanding
Addition and subtraction, measurement using whole
number quantities
Multiplication and division of whole numbers and
fractions
Ratios and proportional reasoning; early expressions
and equations
Ratios and proportional reasoning; arithmetic of
rational numbers
Linear algebra
http://commoncoretools.wordpress.com/
Key Fluencies
Grade
Required Fluency
K
Add/subtract within 5
1
Add/subtract within 10
Add/subtract within 20
2
3
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
Multi-digit division
Multi-digit decimal operations
7
Solve px + q = r, p(x + q) = r
8
Solve simple 22 systems by
inspection
http://commoncoretools.wordpress.com/
Content Standards
K-2
Grade Level Overview
Critical Areas of Focus
A description of the key areas where instruction & learning time should be focused.
Mathematics | Kindergarten
In Kindergarten, instructional time should focus on two critical areas: (1) representing, relating,
and operating on whole numbers, initially with sets of objects; (2) describing shapes and space.
More learning time in Kindergarten should be devoted to number than to other topics.
(1) Students use numbers, including written numerals, to represent quantities and to solve quantitative
problems, such as counting objects in a set; counting out a given number of objects; comparing sets or
numerals; and modeling simple joining and separating situations with sets of objects, or eventually with
equations such as 5 + 2 = 7 and 7 – 2 = 5. (Kindergarten students should see addition and subtraction
equations, and student writing of equations in kindergarten is encouraged, but it is not required.) Students
choose, combine, and apply effective strategies for answering quantitative questions, including quickly
recognizing the cardinalities of small sets of objects, counting and producing sets of given sizes, counting the
number of objects in combined sets, or counting the number of objects that remain in a set after some are
taken away.
(2) Students describe their physical world using geometric ideas (e.g., shape, orientation, spatial relations) and
vocabulary. They identify, name, and describe basic two-dimensional shapes, such as squares, triangles,
circles, rectangles, and hexagons, presented in a variety of ways (e.g., with different sizes and orientations),
as well as three-dimensional shapes such as cubes, cones, cylinders, and spheres. They use basic shapes
and spatial reasoning to model objects in their environment and to construct more complex shapes.
Format of Pre-K-8 Standards
Domain
Standard
2.NBT.1 (code)
Cluster
Process Used to Develop
Framework for District
Curriculum Work
Grade Level Band Teams
•
•
•
•
K–2
3–5
6–8
High School
A Frame for District Curriculum Work
Prioritized
Standards
Vertical
Alignment
Named Units &
Assigned
Standards
Developed a
Suggested
Sequence of
Instruction
Created
Sample
Assessment
Items
Developed Unit
Pacing Plan
ALL Standards are Important
KINDERGARTEN
SECOND GRADE
Counting and Cardinality
Number and Operations in Base
Ten
Know number names and the
count sequence.
1. Count to 100 by ones and by
tens.
2. Count forward beginning from
a given number within the known
sequence (instead of having to
begin at 1).
3. Write numbers from 0 to 20.
Represent a number of objects
with a written numeral 0-20
(with 0 representing a count of
no objects).
Understand place value.
1. Understand that the three
digits of a three-digit number
represent amounts of hundreds,
tens, and ones; e.g., 706 equals 7
hundreds, 0 tens, and 6 ones.
2. Count within 1000; skip-count
by 5s, 10s, and 100s.
3. Read and write numbers to 1000
using base-ten numerals, number
names, and expanded form.
Priority
Standards
Supporting
Standards
Units
of
Study
Critical Areas of
Focus:
Key areas where
instruction &
learning time should
be focused.
Kindergarten Units
 Counting and Matching Numerals 0-5 with Comparing
 Counting and Matching Numerals 6- 10 with Comparing
 Counting and Matching Numerals 11-20
 Teen Numbers (11-19) & Counting to 100
 Fluency with Addition & Subtraction within 5
 Exploring Addition & Subtraction within 10
 Identify & Describe 2D & 3D Shapes
 Compare, Analyze and Compose 2D & 3D Shapes
 Measurement by Direct Comparison
Suggested
Pacing
Timeframe
•
•
•
•
Mapped out year using school calendars
Developmentally appropriate
Critical Areas of Focus for grade level
Time to Process & Practice
FIRST GRADE
Suggested Unit Sequence
1
Fluency with Addition & Subtraction within 10
Pacing
5 weeks
2
Exploring Addition & Subtraction within 20
4 weeks
3
Counting & Place Value
5 weeks
4
Exploring Addition and Subtraction within 100
5 weeks
5
Defining Attributes of 2D & 3D Shapes
2 weeks
6
Partitioning Circles & Rectangles
2 weeks
7
Measuring Length with Non-Standard Units
2 weeks
8
Time to the Hour and Half-Hour
2 weeks
Unit Planning Organizer
Development of Unit Planning Organizer (in process)
 Mathematical Practices
 Domain & Standards Overview
 Priority & Supporting CCSS
 Explanations & Examples
 Concepts Students Need to Know
 Skills Students Need to Be Able to Do
 Bloom’s Taxonomy Levels
 Unit Assessment Items
Transition Guide
Transition Guide: Displaced Grade Level Concepts
Assessment
Grade Level
Band Teams
Assessment
Items Based on
CCSS
Unit Planning Organizer
Items for use during
instruction when
appropriate
Kindergarten Assessment Items
Unit 1 - Counting and Matching Numerals 0 – 5 with Comparing
Test Mode: Administer one on one
Rote Count
Teacher: Count out loud starting at 1 and count as high as you can.
Record highest number student accurately counts to.
Ex: Child counts from 1-15 accurately, then skips 16. Stop student
and record last correct number stated.
Kindergarten Assessment Items
Unit 2 - Counting and Matching Numerals 6 – 10 with Comparing
Match Numerals
Preparation: In advance, teacher puts out groups of objects (ex: counters,
unifix cubes or bears) and numeral cards 0-10. Objects should be arranged in
groups of 3, 5, 8 and 10.
Teacher: Give students the shuffled set of numeral cards.
Count each group. Put the matching numeral card next to each set.
Observe and record ( or - ) if student correctly matches all four sets.
Kindergarten Assessment Items
Unit 4 - Fluency with Addition and Subtraction within 5
There are 5 apples in a bowl.
Some apples are red. Some apples are green.
•How many of each color apple could be in the bowl?
___ red apples
___ green apples
•Find a different answer.
___ red apples ___ green apples
Grade 1 - Assessment Items
Unit 4 - Exploring Addition and Subtraction within 100
Constructed Response
Write a number sentence and solve the problem. Use manipulatives (base-ten
blocks, hundreds chart, number lines) or a drawing to show how to solve this
problem.
Mrs. Jones needs 42 cupcakes for the class picnic.
She has 32 cupcakes.
How many more cupcakes does she need to buy?
This is how Joe found the answer to 29 + 30 + 1
29 + 30 + 1 = 30 + 30 = 60
What did Joe do to solve the problem?
Grade 2 - Assessment Items
Unit 2 - Place Value to 1,000
Multiple Choice
Circle all the statements that are equal to this number.
823
a) 8 hundreds and 23 tens
c) 7 hundreds, 12 tens and 3 ones
e) 8 hundreds and 23 ones
What is another way to show 729?
700 + 2 + 90
700 + 20 + 9
70 + 200 + 9
7 + 20 + 900
b) 823 ones
d) 82 tens and 3 ones
f) 7 hundreds and 23 tens
Grade 2 - Assessment Items
Unit 3 - Fluency with Addition and Subtraction within 100
Constructed Response
Solve the problem.
54
- 29
Show or explain how to find the answer two different ways.
Write an equation for this problem. Solve the equation to
find the answer.
The teacher is 70 inches tall.
The student is 47 inches tall.
How much taller is the teacher than the student?
QUESTIONS