Transcript of Introduction to the K-8 Publishers’ Criteria for the Common Core State Standards for Mathematics...
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- Introduction to the K-8 Publishers Criteria for the Common Core
State Standards for Mathematics October 23, 2012
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- 2 Agenda Brief overview of the shifts required by the CCSS-M
Walkthrough/discussion of Publishers Criteria Opportunity for
questions
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- Focus strongly where the standards focus Coherence: Think
across grades and link to major topics within grades Rigor: In
major topics, pursue conceptual understanding, procedural skill and
fluency, and application with equal intensity 3 The Three
Shifts
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- 4 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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- Focus Move away from "mile wide, inch deep" curricula
identified in TIMSS. Learn from international comparisons. Teach
less, learn more. Less topic coverage can be associated with higher
scores on those topics covered because students have more time to
master the content that is taught. Ginsburg et al., 2005 5
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- 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 The shape of math in A+
countries 1 Schmidt, Houang, & Cogan, A Coherent Curriculum:
The Case of Mathematics. (2002). 6
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- 7 K 12 Number and Operations Measurement and Geometry Algebra
and Functions Statistics and Probability Traditional U.S.
Approach
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- 8 Focusing Attention Within Number and Operations Operations
and Algebraic Thinking Expressions and Equations Algebra Number and
Operations Base Ten The Number System Number and Operations
Fractions K12345678High School
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- 9 Grade Focus Areas in Support of Rich Instruction and
Expectations of Fluency and Conceptual Understanding K2 Addition
and subtraction - concepts, skills, and problem solving and place
value 35 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 Key Areas of Focus in Mathematics
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- 10 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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- 11 Coherence: Think Across Grades Example: Fractions The
coherence and sequential nature of mathematics dictate the
foundational skills that are necessary for the learning of algebra.
The most important foundational skill not presently developed
appears to be proficiency with fractions (including decimals,
percents, and negative fractions). The teaching of fractions must
be acknowledged as critically important and improved before an
increase in student achievement in algebra can be expected. Final
Report of the National Mathematics Advisory Panel (2008, p.
18)
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- 4.NF.4. Apply and extend previous understandings of
multiplication to multiply a fraction by a whole number. 5.NF.4.
Apply and extend previous understandings of multiplication to
multiply a fraction or whole number by a fraction. 5.NF.7. Apply
and extend previous understandings of division to divide unit
fractions by whole numbers and whole numbers by unit fractions.
6.NS. Apply and extend previous understandings of multiplication
and division to divide fractions by fractions. 6.NS.1. Interpret
and compute quotients of fractions, and solve word problems
involving division of fractions by fractions, e.g., by using visual
fraction models and equations to represent the problem. Grade 4
Grade 5 Grade 6 CCSS 12 Informing Grades 1-6 Mathematics Standards
Development: What Can Be Learned from High-Performing Hong Kong,
Singapore, and Korea? American Institutes for Research (2009, p.
13)
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- 13 Coherence: Link to Major Topics Within Grades Example: Data
Representation Standard 3.MD.3
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- 14 Example: Geometric Measurement 3.MD, third cluster
Coherence: Link to Major Topics Within Grades
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- 15 The CCSSM require a balance of: Solid conceptual
understanding Procedural skill and fluency Application of skills in
problem solving situations Pursuit of all threes requires equal
intensity in time, activities, and resources. Shift #3: Rigor: In
Major Topics, Pursue Conceptual Understanding, Procedural Skill and
Fluency, and Application
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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 19
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- 20 Required Fluencies in K-6 GradeStandardRequired Fluency
KK.OA.5Add/subtract within 5 11.OA.6Add/subtract within 10 2 2.OA.2
2.NBT.5 Add/subtract within 20 (know single-digit sums from memory)
Add/subtract within 100 3 3.OA.7 3.NBT.2 Multiply/divide within 100
(know single-digit products from memory) Add/subtract within 1000
44.NBT.4Add/subtract within 1,000,000 55.NBT.5Multi-digit
multiplication 66.NS.2,3 Multi-digit division Multi-digit decimal
operations
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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. 21
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- Overview of the K-8 Publishers Criteria for Mathematics 22
Available on www.corestandards.org/resources
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- Using the Criteria 23 As guidance for publishers Informing
purchases and adoptions, and/or Working with previously purchased
materials Reviewing teacher-developed materials and guiding their
development As a tool for professional development
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- Supporting Innovation in Materials and Tools 24 CCSS presents a
historic opportunity Different forms of materials can meet the
criteria Workbooks Targeted interventions Multi-year programs
Digital materials offer substantial promise Focus and coherence can
be greatly enhanced through dynamic navigation
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- 25 These standards are not intended to be new names for old
ways of doing business. They are a call to take the next step.
CCSSM, page 5
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- Some Old Ways of Doing Business (1 of 2) A different topic
every day Every topic treated as equally important Elementary
students dipping into advanced topics at the expense of mastering
fundamentals Infinitesimal advance in each grade; endless review
Incoherence and illogic bizarre associations, or lacking a thread
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- Some Old Ways of Doing Business (2 of 2) Lack of rigor Reliance
on rote learning at expense of concepts Aversion to repetitious
practice Severe restriction to stereotyped problems lending
themselves to mnemonics or tricks Lack of quality applied problems
and real-world contexts Lack of variety in what students produce
E.g., overwhelmingly only answers are produced, not arguments,
diagrams, models, etc. 27 From....To. 856 = ___ hundreds, ___ tens,
___ ones1 hundredth = ___ tenths x2 10x + 21 = 0 c(c 1) = c
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- 1/2 + 1/3 + 1/4 = ? 3/4 + 1/3 = ? The Criteria do not replace
the Standards 28
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- Supporting special populations 29 All students must have the
opportunity to learn and meet the same high standards if they are
to access the knowledge and skills necessary in their post-school
lives. The Standards should be read as allowing for the widest
possible range of students to participate fully from the outset,
along with appropriate accommodations to ensure maximum
participation of students with special education needs. (CCSSM, p.
4) As stated in the Standards, an over-arching criterion for
materials and tools is that they provide supports for special
populations such as students with disabilities, English language
learners, and gifted students.
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- #1. Focus on Major Work In any single grade, students and
teachers using the materials as designed spend the large majority
of their time, approximately three-quarters,* on the major work of
each grade. Major work should especially predominate at the
beginning of the year. Especially careful treatment of the clusters
leading to algebra (and their interconnections) * Approx. 2/3 in
grade 7 30
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- 31 Progress to Algebra in Grades K-8
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- #2. Focus in Early Grades Materials do not assess any of the
following topics before the grade level indicated. Topic Grade
Introduced Probability, including chance, likely outcomes,
probability models.7 Statistical distributions, including center,
variation, clumping, outliers, mean, median, mode, range,
quartiles, and statistical association or trends, including two-way
tables, bivariate measurement data, scatter plots, trend line, line
of best fit, correlation. 6 Similarity, congruence, or geometric
transformations.8 Symmetry of shapes, including line/reflection
symmetry, rotational symmetry. 4 32 Additionally, materials do not
assess pattern problems in K-5 that do not support the focus on
arithmetic, such as find the next one problems.
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- Supporting content does not detract from focus, but rather
enhances focus and coherence simultaneously by engaging students in
the major work of the grade. 33 #3. Focus and Coherence through
Supporting Work
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- #4. Rigor and Balance Materials and tools reflect the balances
in the Standards and help students meet the Standards rigorous
expectations, by: a)Developing students conceptual understanding of
key mathematical concepts, where called for in specific content
standards or cluster headings b)Giving attention throughout the
year to individual standards that set an expectation of fluency.
c)Allowing teachers and students using the materials as designed to
spend sufficient time working with engaging applications, without
losing focus on the major work of each grade. 34
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- 1)The three aspects of rigor are not always separate in
materials. 2)Nor are the three aspects of rigor always together in
materials. Digital and online materials with no fixed lesson flow
or pacing plan are not designed for superficial browsing but rather
instantiate the Rigor and Balance criterion and promote depth and
mastery. 35 Additional Aspects of the Rigor and Balance
Criterion
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- 36 #5. Consistent Progressions Materials are consistent with
the progressions in the Standards, by (all of the following):
a)Basing content progressions on the grade-by-grade progressions in
the Standards. b)Giving all students extensive work with
grade-level problems. c)Relating grade-level concepts explicitly to
prior knowledge from earlier grades.
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- 37 #6. Coherent Connections Materials foster coherence through
connections at a single grade, where appropriate and where required
by the Standards, by (all of the following): a)Including learning
objectives that are visibly shaped by CCSSM cluster headings, with
meaningful consequences for the associated problems and activities.
b)Including problems and activities that serve to connect two or
more clusters in a domain, or two or more domains in a grade, in
cases where these connections are natural and important.
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- 38 #7. Practice-Content Connections Materials meaningfully
connect content standards and practice standards. Designers of
curricula, assessments, and professional development should all
attend to the need to connect the mathematical practices to
mathematical content in mathematics instruction. (CCSSM, p.
8.)
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- 39 #7. Practice-Content Connections What does it look like for
materials to meaningfully connect content and practice standards?
(1 of 2) Over the course of any given year of instruction, each
mathematical practice standard is meaningfully present and
well-grounded in the content standards. Materials are accompanied
by an analysis, aimed at evaluators, of how the authors have
approached each practice standard in relation to content within
each applicable grade or grade band.
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- 40 #7. Practice-Content Connections What does it look like for
materials to meaningfully connect content and practice standards?
(2 of 2) Materials do not treat the practice standards as static
across grades or grade bands, but instead tailor the connections to
the content of the grade and to grade-level-appropriate student
thinking. Materials also include teacher-directed materials that
explain the role of the practice standards in the classroom and in
students mathematical development.
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- 41 #8. Focus and Coherence via Practice Standards Materials
promote focus and coherence by connecting practice standards with
content that is emphasized in the Standards. Content and practice
standards are not connected mechanistically or randomly, but
instead support focus and coherence. For example: Materials connect
looking for and making use of structure (MP.7) with structural
themes emphasized in the standards such as properties of
operations, place value decompositions of numbers, numerators and
denominators of fractions, numerical and algebraic expressions,
etc.
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- 42 #8. Focus and Coherence via Practice Standards Materials
connect looking for and expressing regularity in repeated reasoning
(MP.8) with major topics by using regularity in repetitive
reasoning as a tool with which to explore major topics. In K-5,
shed light on, e.g., the 10 x 10 addition table, the 10 x 10
multiplication table, the properties of operations, the
relationship between addition and subtraction or multiplication and
division, and the place value system; in 6-8, materials shed light
on proportional relationships and linear functions; in high school,
materials shed light on formal algebra as well as functions,
particularly recursive definitions of functions.
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- 43 #9. Careful attention to Each Practice Standard Materials
attend to the full meaning of each practice standard. A few
examples (1 of 3) MP.1 does not say, Solve problems. Or Make sense
of problems. Or Make sense of problems and solve them. It says Make
sense of problems and persevere in solving them. Thus, students
using the materials as designed build their perseverance in
grade-level-appropriate ways by occasionally solving problems that
require them to persevere to a solution beyond the point when they
would like to give up.
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- 44 #9. Careful attention to Each Practice Standard Materials
attend to the full meaning of each practice standard. A few
examples (2 of 3) MP.5 does not say, Use tools. Or Use appropriate
tools. It says Use appropriate tools strategically. Thus, materials
include problems that reward students strategic decisions about how
to use tools, or about whether to use them at all.
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- 45 #9. Careful attention to Each Practice Standard Materials
attend to the full meaning of each practice standard. A few
examples (3 of 3) MP.8 does not say, Extend patterns. Or Engage in
repetitive reasoning. It says Look for and express regularity in
repeated reasoning. Thus, it is not enough for students to extend
patterns or perform repeated calculations. Those repeated
calculations must lead to an insight (e.g., When I add a multiple
of 3 to another multiple of 3, then I get a multiple of 3.).
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- 46 #10. Emphasis on Mathematical Reasoning Materials support
the Standards emphasis on mathematical reasoning, by (all of the
following): a)Prompting students to construct viable arguments and
critique the arguments of others concerning key grade-level
mathematics that is detailed in the content standards (cf. MP.3).
b)Engaging students in problem solving as a form of argument.
c)Explicitly attending to the specialized language of
mathematics.
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- Indicators of Quality (1 of 2) Problems are worth doing Variety
in what students produce Variety in the pacing and grain size of
content coverage Separate teacher materials that support and reward
teacher study Use of manipulatives follows best practices Materials
are carefully reviewed (freedom from mathematical errors,
grade-level appropriateness, freedom from bias, freedom from
construct-irrelevant language complexity) 47
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- Indicators of Quality (2 of 2) Visual design isnt distracting,
chaotic, aimed at adult purchasers serves only to support young
students in engaging thoughtfully with the subject Support for
English language learners is thoughtful and helps those learners to
meet the same standards as all other students (For paper-based
materials.) A textbook that is focused is short. For example, by
design Japanese textbooks have less than one page per lesson.
Elementary textbooks should be less than 200 pages, middle and
secondary less than 500 pages 48
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- K-8 Publishers Criteria for Mathematics: Some Next Steps for
Publishers 49 For additional resources for educators, go to
achievethecore.org. Publishers designing new materials and tools
can use the criteria to shape these projects. Publishers currently
modifying their materials and tools can use the criteria to
determine the direction of the changes needed. Publishers with
materials and tools already aligned can revisit their thinking on
what alignment looks like. Publishers can develop innovative
materials and tools specifically aimed at addressing identified
weaknesses of widespread textbooks or programs. 1. 2. 3. 4.
Publishers can make fulfillment of the letter and spirit of the
criteria a selling point. 5.
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- 50 Questions?