Post on 19-Feb-2016
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Please sit with a mixture of grade levels.
Conditions for Learning
Choose to be present Choose to be engaged Be an active listener Learning is a process not an event
Take 30 seconds each to share one important condition that supports your learning.
Objectives
Grounding in the College & Career Readiness Standards and the implications for you and your students.
The College & Career Readiness Standards (CCRS) are divided into the:
Standards for Mathematical Content which present a balanced combination of concept development, procedural fluency and application
Standards for Mathematical Practice which rely on processes and proficiencies.
College & Career Readiness StandardsThe Shifts
The Background of the College & Career Readiness Standards
Initiated by the National Governors Association (NGA) and Council of Chief State School Officers (CCSSO) with the following design principles: Result in College and Career Readiness Based on solid research and practice evidence fewer, higher, and clearer
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CCR Standards
What is your comfort and knowledge regarding the defined shifts of math standards?
Read the article “Making the Shifts.” Please use the following code to annotate as you read Circle your agreements Box your ponderings Underline for more information
Debrief the Article
At your table: Do a whip around and share one
coding you feel most strongly about. If your first choice has been shared, be ready to share an alternate.
CCR Standards Require Three Shifts in Mathematics
1. Focus: Focus strongly where the Standards focus.2. Coherence: Think across grades and link to major
topics within grades. 3. Rigor: In major topics, pursue conceptual
understanding, procedural skill and fluency, and application.
Focus on the Major Work of the Grade
Two levels of focus: - What’s in/What’s out - The shape of the content that is in
Mathematics topics
intended at each grade by
at least two-thirds of A+
countries
Mathematics topics intended at each grade by at least two-thirds of 21 U.S. states
1 Schmidt, Houang, & Cogan, “A Coherent Curriculum: The Case of Mathematics.” (2002).
The shape of math in A+ countries
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Traditional U.S. Approach
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K 12
Number and Operations
Measurement and Geometry
Algebra and Functions
Statistics and Probability
Focusing Attention Within Number and Operations
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Operations and Algebraic Thinking
Expressions and Equations
Algebra
→ →
Number and Operations—Base Ten →
The Number System
→
Number and Operations—Fractions
→
K 1 2 3 4 5 6 7 8 High School
Shift #1: Focus Strongly Where the Standards Focus
Significantly narrow the scope of content and deepen how time and energy is spent in the math classroom.
Focus deeply on what is emphasized in the standards, so that students gain strong foundations.
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Key Areas of Focus in Mathematics
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Grade Focus Areas in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding
K–2 Addition and subtraction - concepts, skills, and problem solving and place value
3–5 Multiplication and division of whole numbers and fractions – concepts, skills, and problem solving
6 Ratios and proportional reasoning; early expressions and equations
7 Ratios and proportional reasoning; arithmetic of rational numbers
8 Linear algebra and linear functions
Engaging with the shift: What do you think belongs in the major work of each grade?
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Grade Which two of the following represent areas of major focus for the indicated grade?K Compare numbers Use tally marks Understand meaning of addition and
subtraction
1 Add and subtract within 20 Measure lengths indirectly and by iterating length units Create and extend patterns and sequences
2 Work with equal groups of objects to gain foundations for multiplication Understand place value Identify line of symmetry in two
dimensional figures
3 Multiply and divide within 100 Identify the measures of central tendency and distribution
Develop understanding of fractions as numbers
4 Examine transformations on the coordinate plane
Generalize place value understanding for multi-digit whole numbers
Extend understanding of fraction equivalence and ordering
5 Understand and calculate probability of single events Understand the place value system
Apply and extend previous understandings of multiplication and division to multiply and divide fractions
6 Understand ratio concepts and use ratio reasoning to solve problems Identify and utilize rules of divisibility Apply and extend previous understandings
of arithmetic to algebraic expressions
7Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers
Use properties of operations to generate equivalent expressions
Generate the prime factorization of numbers to solve problems
8 Standard form of a linear equation Define, evaluate, and compare functions Understand and apply the Pythagorean Theorem
Alg.1 Quadratic inequalities Linear and quadratic functions Creating equations to model situations
Alg.2 Exponential and logarithmic functions Polar coordinates Using functions to model situations
Activity Debrief
Was there anything that surprised or shocked you?
PARCC Content Model Framework
Major standards K-2: 85% of instructional time 3-5: 75% of instructional time
Supporting standards Additional standards
Group Discussion
Shift #1: Focus strongly where the Standards focus. In your groups, discuss ways to respond to the
following question, “Why focus? There’s so much math that students could be learning, why limit them to just a few things?”
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What’s in/What’s outIt is because of this level of focus that teachers will have the time to go deeper with the math that is most important. Compared to the typical state standards of the past, the College & Career Readiness Standards for math have fewer standards which are manageable and it is clear what is expected of the teachers and students at each grade level.
PARCC Example
BREAK – 10 MINUTES
Coherence Across and Within Grades
It’s about math making sense. The power and elegance of math comes out through carefully laid progressions and connections within grades.
Shift #2: Coherence: Think Across Grades, and Link to Major Topics Within Grades Carefully connect the learning within and across
grades so that students can build new understanding on foundations built in previous years.
Begin to count on solid conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning.
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Alignment in Context: Neighboring Grades and Progressions
One of several staircases to algebra designed in the OA domain.
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Looking for Coherence Across Grades
Coherence is an important design element of the standards.
“The Standards are not so much built from topics as they are woven out of progressions.”
Structure is the Standards, Publishers’ Criteria for Mathematics, Appendix
Coherence Card Activity Activity: Place the standards of each color under the
appropriate grade (K-8).
Determine a “theme” for each color. No grade has two of the same color card. Some “themes” that have only a few cards might represent
consecutive grades and some may not. Read each card in its entirety to help determine placement. Do not check your Standards until you and your colleagues
agree on the final product. Discuss horizontal and vertical observations with your
partners.
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Check your answers
Use the key to check your answers.
What surprised you? What are your thoughts?
Group DiscussionShift #2: Coherence: Think across grades, link to major topics within grades In your groups, discuss what coherence
in the math curriculum means to you. Be sure to address both elements—coherence within the grade and coherence across grades. Cite specific examples.
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Rigor: Illustrations of Conceptual Understanding, Fluency, and Application
Here rigor does not mean “hard problems.”It’s a balance of three fundamental components that result in deep mathematical understanding.There must be variety in what students are asked to produce.
Shift #3: Rigor: In Major Topics, Pursue Conceptual Understanding, Procedural Skill and Fluency, and Application
The CCRS-Math require a balance of: Solid conceptual understanding Procedural skill and fluency Application of skills in problem solving
situations Pursuit of all three requires equal intensity in
time, activities, and resources.
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Solid Conceptual Understanding
Teach more than “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives
Students are able to see math as more than a set of mnemonics or discrete procedures
Conceptual understanding supports the other aspects of rigor (fluency and application)
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Fluency
The standards require speed and accuracy in calculation.
Teachers structure class time and/or homework time for students to practice core functions such as single-digit multiplication so that they are more able to understand and manipulate more complex concepts
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Grade Standard Required FluencyK K.OA.5 Add/subtract within 5
1 1.OA.6 Add/subtract within 10
2 2.OA.22.NBT.5
Add/subtract within 20 (know single-digit sums from memory)Add/subtract within 100
3 3.OA.73.NBT.2
Multiply/divide within 100 (know single-digit products from memory)Add/subtract within 1000
4 4.NBT.4 Add/subtract within 1,000,000
5 5.NBT.5 Multi-digit multiplication
6 6.NS.2,3 Multi-digit divisionMulti-digit decimal operations
Required Fluencies in K-6
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Application Students can use appropriate concepts and
procedures for application even when not prompted to do so.
Teachers provide opportunities at all grade levels for students to apply math concepts in “real world” situations, recognizing this means different things in K-5, 6-8, and HS.
Teachers in content areas outside of math, particularly science, ensure that students are using grade-level-appropriate math to make meaning of and access science content.
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Coherence Card Activity: Rigor Focus
Activity: Identify the type of rigor within each theme for your grade band.
More than one strand of rigor may be assigned to each theme
Underline the word(s) that indicate the strand of rigor identified
Group DiscussionShift #3: Rigor: Expect fluency, deep understanding, and application In your groups, discuss why the
balance of the three strands of rigor are important to building student understanding of mathematics and what is balance.
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Frequently Asked Questions How can we assess fluency other
than giving a timed test? Is it really possible to assess
conceptual understanding? What does it look like?
Are the Standards for Math all about application and meaningful tasks?