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Mathematics Rigor with Common Core State Standards
DPI AIG Team SessionNCAGT Conference
March, 2012
Kitty Rutherford [email protected]
Robin Barbour [email protected]
www.corestandards.org
Critical Areas
Critical AreaCritical Area
Focal PointsFocal Points
04/20/23 • page 4
Mathematical Mathematical PracticesPractices
Grade Level
Overview
Grade Grade LevelLevelDomainDomain Standards
ClusterCluster
High School Themes
• Number and Quantity
• Algebra
• Functions
• Modeling
• Geometry
• Statistics and Probability
High School Standards Notation
Perform operations on matrices and use matrices in applications. 6. (+) Use matrices to represent and manipulate data, e.g., to represent
payoffs of incidence relationship in a network.
11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y =g(x intersect are the solutions of the equations f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★
K-8 DomainsDomains K 1 2 3 4 5 6 7 8
Counting and Cardinality
Operations and Algebraic Thinking
Number and Operations in Base Ten
Measurement and Data
Geometry
Number and Operations - Fractions
Ratios and Proportional Relationships
The Number System
Expressions and Equations
Statistics and Probability
Functions
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the
reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated
reasoning.
Standards for Mathematical Practices
Mathematical practices describe the habits of mind of mathematically proficient students…
• Who is doing the talking?
• Who is doing the thinking?
• Who is doing the math?
04/20/23 • page 12
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04/20/23 • page 14
Make sense of problems and preserve in solving them.
When planning, ask
“What task can I give that will build student
understanding?”rather than
“How can I explain clearly so they will understand?”
Grayson Wheatley, NCCTM, 2002
04/20/23 • page 15
61
77
84
41
57
69
15 1613
54
24
17
0
10
20
30
40
50
60
70
80
90
100
Australia Czech Republic Hong Kong J apan Netherlands United States
Using ProceduresMaking Connections
31
1618
20 19
59
8
52
4648
37
00
10
20
30
40
50
60
70
80
90
100
Australia Czech Republic Hong Kong J apan Netherlands United States
Using Procedures
Making Connections
Types of Math Problems Presented
How Teachers ImplementedMaking Connections Math
Problems
Lesson ComparisonUnited States and Japan
The emphasis on skill acquisition is evident in the steps most common in U.S. classrooms
The emphasis on understanding is evident in the steps of a typical Japanese lesson
•Teacher instructs students in concept or skill
•Teacher solves example problems with class
•Students practice on their own while teacher assists individual students
•Teacher poses a thought provoking problem
•Students and teachers explore the problem
•Various students present ideas or solutions to the class
•Teacher summarizes the class solutions
•Students solve similar problems
17
What do you see?
40 4
10 2
30 4
18
Predict some additional data.
40 4
10 2
30 4
19
How close were you?
40 4
10 2
30 4
20 3
20
All the numbers – so?
45 4
25 3
15 2
40 4
10 2
30 4
20 3
21
Where are you?
Roller Coaster 45 4
Ferris Wheel 25 3
Bumper Cars 15 2
Rocket Ride 40 4
Merry-go-Round 10 2
Water Slide 30 4
Fun House 20 3
22
Fill in the blanks.
Ride ??? ???Roller Coaster 45 4
Ferris Wheel 25 3
Bumper Cars 15 2
Rocket Ride 40 4
Merry-go-Round 10 2
Water Slide 30 4
Fun House 20 3
23
The Amusement Park
Ride Time Tickets
Roller Coaster 45 4
Ferris Wheel 25 3
Bumper Cars 15 2
Rocket Ride 40 4
Merry-go-Round 10 2
Water Slide 30 4
Fun House 20 3
24
The Amusement Park
The 4th and 2nd graders in your school are going on a trip to the Amusement Park. Each 4th grader is going to be a buddy to a 2nd grader.
Your buddy for the trip has never been to an amusement park before. Your buddy want to go on as many different rides as possible. However, there may not be enough time to go on every ride and you may not have enough tickets to go on every ride.
25
The bus will drop you off at 10:00 a.m. and pick you up at 1:00 p.m. Each student will get 20 tickets for rides.
Use the information in the chart to write a letter to your buddy and create a plan for a fun day at the amusement park for you and your buddy.
26
The Amusement Park
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the
reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated
reasoning.
Standards for Mathematical Practices
Year Standards To Be TaughtStandards To Be
Assessed
2011 – 2012 2003 NCSCOS 2003 NCSCOS
2012 – 2013 CCSS CCSS (NC)
2013 – 2014 CCSS CCSS (NC)
2014 – 2015 CCSS CCSS (SBAC)
Common Core State Standards Adopted June, 2010
Mathematics Claims
The Smarter Balanced Assessment Consortium has released a document outlining four claims about what mathematically proficient students can do.
The claims are a synthesis of the Standards for Mathematical Practice, and form the guiding principles to be used in creating assessments.
Mathematics Claim 1 & 2
• Students can explain and apply mathematical concepts and carry out mathematical procedures with precision and fluency.
• Students can frame and solve a range of complex problems in pure and applied mathematics.
Mathematics Claim 3 & 4 • Students can clearly and precisely
construct viable arguments to support their own reasoning and to critique the reasoning of others.
• Students can analyze complex, real-world scenarios and can use mathematical models to interpret and solve problems.
http://www.k12.wa.us/smarter/
RIGOR
ConceptualUnderstanding
Application
Skills and Procedures
Rigor through Standards
Rigor though Standards
6th Grade Critical Area:
Students use the meaning of fractions, the meanings of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for dividing fractions make sense. Students use these operations to solve problems.
• Skipping material will create gaps in learning which jeopardizes foundational content needed to maximize the likelihood of success in High School Mathematics
Content Acceleration
• Compacting 3 years of content into 2 is supported by research; 2 years into 1 is considered too challenging
• Considering high school courses is essential when making middle school recommendations
6th Grade 7th Grade 8th Grade
100 %100% 7th grade ??% 8th grade
??% 8th grade Math 1 Standards
Option 1:
Option 2:
6th Grade 7th Grade 8th Grade
100 % 6th grade Remaining 7th gradeRemaining 8th grade
Math 1 Standards??% 7th grade ??% 8th grade
High School Courses in Middle SchoolGetting Students Ready
04/20/23 • page 43
•Integer Exponents (8.EE.1)
•Multiplication and Division with Scientific Notation (8.EE.4)
•Solving Systems by Substitution (8.EE.8)
•Volume of Pyramids, Cones and Spheres (7.G.6, 8.G.9)
•Surface Area of Pyramids (6.G.4, 7.G.6)
At A GlanceInstructional Implications High School
04/20/23 • page 44
•Angles (7.G.5, 8.G.5)
•Using Pythagorean Theorem in 3-D Figures (8.G.7)
•Mean Absolute Deviation (6.SP.5c)
•Two-way Tables (8.SP.4)
•Qualitative Graphs (8.F.5)
•Graphing Proportional Relationships (7.RP.2a, b, c, d, 8.EE.5)
At A GlanceInstructional Implications Math One
www.ncdpi.wikispaces.net
04/20/2345
www.ncdpi.wikispaces.net
04/20/2346
Grade Band Pages
04/20/2347
Crosswalk Documents
04/20/23 • page 48
The crosswalks reflect a comparison between the Common Core State Standards and the North Carolina Standard Course of Study. They inform educators about how the current standards align with the CCSS Standards.
Mathematics Crosswalk
CAUTION!!
04/20/23 • page 50
CONTENT APPEARING TO BE THE SAME MAY ACTUALLY BE
DIFFERENT!!
The CCSS Requires CLOSE Reading!!!
Unpacking Documents
04/20/23 • page 51
The purpose of the Unpacking Documents is to increase student achievement by ensuring educators understand the new standards.
The “unpacking” of the standards done in these documents is an effort to answer a simple question “What does this standard mean that a student must know and be able to do?” and to ensure the description is helpful, specific and comprehensive for educators.
Unpacking – At a Glance
04/20/23 • page 52
04/20/23 • page 53
Unpacking – Standards for Mathematical Practice
Unpacked Content
04/20/23 • page 54
Common Core GlossaryTable 1. Common multiplication and division situations
K-5 Units
Students learn mathematics by exploring mathematically-rich tasks and sharing strategies, ideas, and approaches with one another. (practices)
K Adding and Subtraction
1st Exploring Two-Digit Numbers
2nd Two- & Three-Digit Addition & Subtraction
3rd Unit on Area and Perimeter
4th Fractions
5th Fractions04/20/23 • page 58
Format of the Lessons
The phases of the lesson:
• Engage - Brief opening activity
• Explore - Mathematically-rich task
• Explain - Discussion of task and concepts
• Elaborate - Follow-up activity
• Evaluate - description of formative and summative assessments
04/20/23 • page 59
Additional Wiki Information
Year Who Can Apply Nomination
Deadline
Application
Deadline
2012Elementary Teachers
Grades K - 6April 1, 2012 May 1, 2012
2013Secondary Teachers
Grades 7 - 12April 1, 2013 May 1, 2013
www.paemst.org
Presidential Awards for Excellence in
Mathematics and Science Teaching
Elementary Mathematics Add-on Licensure
• 18-hour Graduate program (6 courses)• Participating Universities
– East Carolina University– Appalachian State University– NC State University– UNC Chapel Hill– UNC Charlotte– UNC Greensboro– UNC Wilmington
• Dr. Sid Rachlin ([email protected])
Webinars
Archived Webinars:- November 17th: CCSS and Math I Standards- January 10th: K-12 Getting Started:
Organization Tools and Instructional Planning Model
- February 9th: Making Mathematics Accessible (K-12)
- March 8th: K – 2 Assessment and Calendar Time
http://illustrativemathematics.org/ http://commoncoretools.wordpress.com K, Counting and Cardinality;K–2, Operations and Algebraic ThinkingK, Counting and Cardinality; K–5, Operations and Algebraic Thinking K–3, Categorical Data; Grades 2–5, Measurement Data* (data part of the Measurement and Data Progression)3-5, Number and Operations – Fractions 6-7, Ratio and Proportional Relationships6-8, Progression for Statistics and Probability6–8, Expressions and Equations
ADDITIONAL RESOURCES:
QUESTIONS
COMMENTS
Kitty [email protected]
Robin [email protected]
Contact Information
Website: www.ncdpi.wikispaces.net