### Document

```Introduction to the K-8 Publishers’
Criteria for the Common Core
State Standards for Mathematics
October 23, 2012
Agenda
•
•
•
Brief overview of the shifts required by the CCSS-M
Walkthrough/discussion of Publishers’ Criteria
Opportunity for questions
2
The Three Shifts
• Focus strongly where the
standards focus
• Coherence: Think across
• Rigor: In major topics, pursue
conceptual understanding,
procedural skill and fluency,
and application with equal
intensity
3
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.
4
Focus
•
Move away from "mile wide, inch deep" curricula
identified in TIMSS.
•
•
Learn from international comparisons.
“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
The shape of math in A+ countries
Mathematics
topics
intended at
at least twothirds of A+
countries
Mathematics
topics
intended at
at least twothirds of 21
U.S. states
1 Schmidt,
Houang, & Cogan, “A Coherent Curriculum: The Case of Mathematics.” (2002).
6
K
12
Number and
Operations
Measurement
and Geometry
Algebra and
Functions
Statistics and
Probability
7
Focusing Attention Within Number and
Operations
Operations and Algebraic
Thinking
Expressions
→ and
Equations
Number and Operations—
Base Ten
→
K
1
2
3
4
Algebra
The Number
System
Number and
Operations—
Fractions
→
→
→
5
6
7
8
High School
8
Key Areas of Focus in Mathematics
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
9
Shift #2: Coherence: Think Across Grades, and
•
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.
10
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)
11
CCSS
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.
Standards Development: What Can Be
Learned from High-Performing Hong
Kong, Singapore, and Korea? American
Institutes for Research (2009, p. 13)
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.
12
Example: Data Representation
Standard
3.MD.3
13
Example: Geometric Measurement
3.MD, third
cluster
14
Shift #3: Rigor: In Major Topics, Pursue Conceptual Understanding,
Procedural Skill and Fluency, and Application
•
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.
15
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)
16
17
18
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 singledigit multiplication so that they are more able to
understand and manipulate more complex concepts
19
Required Fluencies in K-6
Standard
Required Fluency
K
K.OA.5
1
1.OA.6
2
2.OA.2
2.NBT.5
Add/subtract within 20 (know single-digit sums from
memory)
3
3.OA.7
3.NBT.2
Multiply/divide within 100 (know single-digit products
from memory)
4
4.NBT.4
5
5.NBT.5
Multi-digit multiplication
6
6.NS.2,3
Multi-digit division
Multi-digit decimal operations
20
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-levelappropriate math to make meaning of and access science
content.
21
Overview of the K-8 Publishers’ Criteria for
Mathematics
Available on
www.corestandards.org/resources
22
Using the Criteria
• 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
23
Supporting Innovation in Materials and Tools
• 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
24
“These standards are
not intended to be
new names for old
ways of doing
call to take the next
step.”
CCSSM, page 5
25
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
• Incoherence and illogic – bizarre associations, or
26
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
From....
To….
856 = ___ hundreds, ___ tens, ___ ones
x2 – 10x + 21 = 0
1 hundredth = ___ tenths
¾ c(c –1) = c
• 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
The Criteria do not replace the Standards
3/4 + 1/3 = ?
1/2 + 1/3 + 1/4 = ?
28
Supporting special populations
“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.
29
#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 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
Progress to Algebra in Grades K-8
31
Materials do not assess any of the following topics
Topic
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
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.
32
#3. Focus and Coherence through Supporting
Work
Supporting content does not detract from focus,
but rather enhances focus and coherence
simultaneously by engaging students in the major
33
#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
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
34
Additional Aspects of the Rigor and Balance
Criterion
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
#5. Consistent Progressions
Materials are consistent with the progressions in the
Standards, by (all of the following):
progressions in the Standards.
b) Giving all students extensive work with grade-level
problems.
c) Relating grade-level concepts explicitly to prior
36
#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
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.
37
#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.)
38
#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
39
#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
student thinking.
•
Materials also include teacher-directed materials that
explain the role of the practice standards in the classroom
and in students’ mathematical development.
40
#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.
41
#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.
42
#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
occasionally solving problems that require them to
persevere to a solution beyond the point when they would
like to give up.
43
#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’
whether to use them at all.
44
#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.”).
45
#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.
46
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
Indicators of Quality (2 of 2)
• Visual design isn’t 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
K-8 Publishers’ Criteria for Mathematics: Some Next
Steps for Publishers
1.
Publishers designing new materials and tools can use the criteria to shape
these projects.
2.
Publishers currently modifying their materials and tools can use the criteria
to determine the direction of the changes needed.
3.
Publishers with materials and tools ‘already aligned’ can revisit their
thinking on what alignment looks like.
4.
Publishers can develop innovative materials and tools specifically aimed at