Designing Common Core State Standards Systemic Mathematics Curriculum: Part 2

Designing
Common
Core
State
Standards

Systemic
Mathema4cs
Curriculum

Part
2

Presented
By
Janet
Hale



www.CurriculumMapping101.com

Architects Contractors
design. build.
LEARN TEACH

Aspen Grove Mentality

Systemic Design - Interdependent
1 Grove…1 Root System

Phase 1 Design / Phase 2 Design
Vertical Alignment Horizontal Alignment
Design units that represent Design units of study that
K-12 learning continuum integrate learning among
(e.g., Geometry, Measurement/Data) strands in one grade level
by domains (e.g., intradisciplinary,
with cross-connections program-based,
interdisciplinary)

Designing Systemic K-12
CCSS Math Collaborative Maps

How long will it take for
the K-12 Task Force to
complete Phase 1?

Part 1: Laying a Solid Foundation

•  Unit Names

•  Enduring Understandings/
Essential Questions

•  Standards for
Mathematical Practice

•  Vocabulary

Part 2
•  Breaking Apart (Unpacking)
Standards
(Explicit Process, Implicit Influences)

•  Systemic Content / Skills Development
(Process: Format … Collaborative Agreement on Tight and Loose)

•  PreK-12 Vertical Reviews
(Internal Alignment – Content/Skills &
External Alignment to Standards; if using, EUs/EQs)

Implicit Influences
•  Breaking Apart (Unpacking) Standards
Teachers as designers will spend time explicitly studying
the codes, but need to first consider design influences.

•  Math CCSS - The 3 Key Shifts

•  Depth of Knowledge

•  When to incorporate the use of
state-provided Crosswalk and
Gap Analysis documents.

CCSS Mathematics – 3 Key Shifts
(www.achievethecore.org)

1.  FOCUS
Focus Strongly Where the Standards Focus
(narrow scope of content to allow in-depth learning; no “but we have so much to cover”;
“inch wide, mile deep” mindset to ensure time necessary for students to have time to
explore, test, and reach conceptual understanding)

2. COHERENCE
Think across grade levels (systemic design)
(each new standard is not a “new event” … each new standard is an extension of previous
distinct or linked learning)
Link learning among domains within one grade level (leverage)
(conceptual relationships across and among standards to aid in conceptual understanding
and reasoning)

3. RIGOR
Equitable, balanced curriculum
(learning/teaching):
–Conceptual Understanding
–Procedural Skills and Fluencies
–Application of Math Process using
real-world/authentic problems/tasks
(within/across disciplines)

1.  FOCUS

2.  COHERENCE

Presenta)on
Slide
from
CCSS
for
Mathema-cs:
Key
Shi4s
-‐Sandra
Alber),
Student
Achievement
Partners

3.
RIGOR
Grade
7

(Content
lis-ng
in
an
Essen-al
Map
unit)

--Conceptual Understanding Algebraic
Representa)ons:

--Procedural Skills and Fluencies
--Application of Math Process

Equa)on
Fluency
Involving
4
Opera)ons

Mul)-‐Step
Word
Problems
(Posi)ve/Nega)ve

Ra)onal
Numbers,
Inequali)es,
Complex

www.achievethecore.org

Frac)ons)

CCSS
Fluency
≠
Rote
Memoriza4on

CCSS
Fluency

=

Speed
and
Accuracy

using
self-‐selected
strategies

High
School
Fluencies:
Algebra,
Func4ons,

Geometry,
Sta4s4cs
&
Probability,
and
Modeling

1.  Use
mul)media.

2.  Encourage

student

intui)on.

3.  Ask
the
shortest

ques)on
you

can.

4.  Let
students

build
the

problem.

5.  Be
less
helpful.

hMp://www.ted.com/talks/lang/en/dan_meyer_math_curriculum_makeover.html

Cognitive Complexity
New
BLOOM’S

DOK

The
Three-‐Story
Intellect

New
Bloom’s

Duplicate Identify Paraphrase
Count List Recite Input Understanding/
Bena Kallick Define Memorize Locate R/U Remembering
CCSS
Presentation Describe Name Reproduce

The
Three-‐Story
Intellect

Compare Distinguish Analyze
New Contrast Explain Synthesize Process
Bloom’s Classify Discriminate Reason A/A
Analyzing/
Applying

Infer Sequence Interpret
CCSS

The
Three-‐Story
Intellect

Evaluate Predict Hypothesize
Generate Speculate Forecast Output Creating/
Imagine If/then Create E/C Evaluating
Judge Apply Speculate
Compare Distinguish Analyze
New Contrast Explain Synthesize Process Analyzing/
Bloom’s Classify Discriminate Reason A/A Applying

Infer Sequence Interpret
CCSS

New
BLOOM’S

R/U A/A E/C
Input Process Output

DOK

Norman
Webb’s

Depths
of
Knowledge

DOK
Model
(1997)
created
to
analyze
the
cogni)ve
expecta)on

demanded
by
standards,
curricular
ac4vi4es,
and
assessment
tasks.

redesign.rcu.msstate.edu

Several
things
are
involved,

including
the
content,
the
ac4vity

and/or
thinking
processes,
and
the

complexity
of
both
the
content
and

ac4vity/thinking
processes.

-‐-‐Debbie
Baughman,

The
Standards
Company

DOK

Four
Levels

Level
1
Recall/Reproduc4on

Recall
facts,
informa)on,
procedures,

basic
concept
founda)ons

(minor
comprehension
involved
at
this
level,

no
depth,
no
complexity)

Level
2
Skill/Concept

Apply/process
facts,
informa)on,
procedures,

conceptual
understanding
involving
at
least
two

steps
that
require
reasoning

(a
need
to
interpret
material
and
make
simple
decisions
about
how
to

approach
a
problem,
but
does
not
yet
have
a
deep
complexity)

DOK

Four
Levels

Level
3
Strategic
Thinking

Requires
deeper
reasoning,
developing
a

plan
or
sequence
of
steps
to
complete
a
task;

more
than
one
possible
solu)on/answer

(deal
with
abstrac-ons
and
open-‐ended
conclusions
and
able
to

support
one’s
reasoning;
wrestle
with
complex
concepts,
tasks,
material)

Level
4
Extended
Thinking

Process
mul)ple
condi)ons
and
solu)ons
for
the

problem;
extend
thinking
by
comple)ng
much
deeper

and
complex
tasks

(according
to
Webb,
higher-‐level
thinking
is
absolutely
central;

interac-on
with
concepts,
tasks,
material
is
in-‐depth
and
purposeful)

CAUTION!

Bloom’s
Verbs
cannot
be
applied
with
the
same
mindset

for
what
students
must
cogni0vely
do
when

applying
Webb’s
Depth
Of
Knowledge
(DOK)
to

student
learning,
teaching,
and
assessment
items/tasks.

This
cau4on
inﬂuences
wri4ng
skills…

Measurable
Verb
+
Descriptor

DOK
1
–
Describe
shape-‐paMern
term/number-‐paMern
rule

using
real-‐world
examples
(e.g.,
Pretend
you
are
walking
outside.

Draw
and
explain
a
natural
or
man-‐made
paJern’s
term.)

DOK
2
–
Describe
number/shape
paMerns
that
follow

determined
term/rule
and
jus)fy
reasoning
(e.g.,
Look
at
the

bowling
pins
paJern.
What
will
the
next
two
rows
look
like
in
this

paJern?
Explain
the
increase
using
textual,
visual,
and
number

representa-ons.
Without
drawing,
what
would
be
the
number
of
pins
in

the
15th
row?
Explain
your
reasoning.
)

New
BLOOM’S

R/U A/A E/C
Input Process Output

1 2 3 4
Recall/ Skill/ Strategic Extended
Reproduction Concept Thinking Thinking

DOK
PARCC Smarter Balanced
www.smarterbalanced.org/wordpress/wp-content/uploads/2012/03/
www.parcconline.org/parcc-content-frameworks DRAFTMathItemSpecsShowcase2.pdf

walk

C ross

Gap
A
nalys
is

walk
Most
Beneficial
Use
of

C ross Crosswalk
Documents

for
Curriculum
Design

Verifies
what
task-‐force
members

discover
acer
first

studying
and
analyzing

ver)cal/horizontal
Math
CCSS.

nalysis

G ap
A

Most
Beneﬁcial
Use
of
Gap
Analysis
Documents
for
Curriculum
Design

Indicates
to
task-‐force
members
who
may
be
called
upon
to
be

resident
experts
for
fellow
task-‐force
members
who

may
need
some
or
a
lot
of
aid
in
designing
unfamiliar
learning
(content
–
skills).

walk

ss
Best
to
incorporate
use
of

C ro Crosswalk
and
Gap
Analysis

documents
AFTER
task-‐force

teachers
have
systemically

studied
and
discussed

the
Math
CCSS
Domain(s).

Gap
A
nalys
www.LiveBinders.com

Search
for:
Curriculum
21
is

Arizona
http://www.azed.gov/standards-practices/
mathematics-standards/

Ohio
http://education.ohio.gov/GD/Templates/Pages/ODE/
ODEDetail.aspxPage=3&TopicRelationID=1704&Con
tent=123507 (Transitional Tools)

Math CCSS Courses
UNIT NAME SIGNIFIERS: ____________
Math (K-8)
GEOMETRY (K-8) = Suggested
Starting Point K-8
DATA: MEASUREMENT/DATA (K-5)
DATA: STATISTICS/PROBABILITY (6-8)
NUMBER/QUANTITATIVE: COUNTING/CARDINALITY (K)
NUMBER/ALGEBRAIC: NUMBER BASE 10/OPERATIONS (K-5)
NUMBER: NUMBER SYSTEM/EXPRESSIONS/EQUATIONS (6-8)
QUANTITATIVE: RATIOS/PROPORTIONAL RELATIONSHIPS (6-8)
Coordinate Algebra (9) (Integrated Pathway)
EXPRESSIONS/EQUATIONS
LINEAR FUNCTIONS Determine
Desired
EXPOTENTIAL FUNCTIONS Pathway
DATA ANALYSIS
COORDINATE PLANE Math CCSS
INEQUALITIES Appendix A

Analytic Geometry (10) Advanced Algebra (11)
(Above examples based on work in Muscogee CSD, Columbus, GA)

Math CCSS Courses
Progression of Systemic
UNIT NAMES
once task-force teachers began studying the architectural
codes (clusters/standards) while considering influences and
beginning to determine potential content listings.

GEOMETRY (Grades K-2)

GEOMETRY/MEASUREMENT (Grades 3-5)

GEOMETRY (Grade 6)
Phase 2 … GEOMETRY: AREA AND VOLUME

GEOMETRY (Grade 7)
Phase 2 … GEOMETRY: GEOMETRY

GEOMETRY (Grade 8)
Phase 2 … GEOMETRY: GEOMETRIC APPLICATIONS
OF EXPONENTS
(Above examples based on work in Muscogee CSD, Columbus, GA)

Content = What Students Must KNOW
Noun/Noun Phrase: Descriptor
Think … Table of “Contents”
3-Dimensional Shapes
: Needs A Descriptor

3-Dimensional Shapes: Sphere, Cone, Cylinder

3-Dimensional Shapes: Sphere

Time: Analog Face (Hour Hand, Minute Hand), Digital Face (Hour Digits,
Colon, Minute Digits) / O-clock, Thirty

Time: Nearest Minute, Elapsed

Time: Comparison (Second, Minute, Hour, Day, Week, Months, Year,
Decade, Century, Millennium) / Conversion Of Units (Within Single
System)

Distance/Weight/Mass/Capacity/Time/Money: Conversion Of Unlike Units
- Same System (Smaller to Larger/Larger to Smaller) / 2-Step & Multi-
Step Word Problems

Noun/Noun Phrase: Descriptor
Length/Height/Width: ½ Inch, Inch, Foot, Yard / Centimeter,
Meter

Fractions/Equal Shares: 2 Shares, 3 Shares, 4 Shares
(Circle, Rectangle)

Quadratic Polynomials: Fundamental Theorem of Algebra

Measures Of Central Tendency: Mean, Median, Mode,
Range, Outlier

Whenever possible (in any element field),
use numeral instead of number word for 0-9.
It is easier on the eye/mind to
locate information quickly.

Use of Intra-Alignment Coding for Internal Alignment

The
legers’
job
is
to
align
Content
to
Skills

Skills = What students must cognitively be able to
DO in relationship to the intra-aligned KNOWing
There are technically three parts to a quality written skill
statement:

1.  Measurable Verb (can “see” the thinking)
2.  Target (how learning is measured - assessments)
3.  Descriptor (provides clarity/detail related to the intra-
aligned Content)

Do not begin skill statements with:
The student will…

A skill statement starts
with a capital letter,
but needs no period
at end of statement

1.  Measurable Verb
Not Measurable Verbs = Demonstrate,
Understand, Know, Show, Use
Use appropriate tools to measure and record to nearest abbreviated
unit (½ in., 1 in., 1 ft. / 1 cm, 1 m, 1 yd)

☺Measure manipulatively and record in writing to nearest unit nearest
abbreviated unit (½ in., 1 in., 1 ft. / 1 cm, 1 m, 1 yd) using appropriate
tools

Use number line to order up to 5 rational numbers (based on place
value of each number)

☺Order up to 5 rational numbers (based on place value of each number)
using number line

If using Apply… Apply to what?
Apply in writing self-selected strategies to
solve real-world 2-step word problems and
justify reasoning

2.
Target

The
measurement
mode(s)
…
In
other
words,
“how”

learning
is
formally
assessed.

-‐-‐in
wri4ng,
orally

/
-‐-‐manipula4vely,
visually

Note:
in
wri4ng
=
not
orally

•  (Gr.
3)
Iden)fy
in
wri)ng
value
of
given
digit
based
on
base-‐ten

placement
(e.g.,
3,491
…
What
does
the
digit
3
represent
in
3,491?
Student

writes:
The
digit
3
represents
3
thousands.)

•  (Gr.
K
–
Qtr
1.)
Correlate
manipula)vely,
visually,
and
orally

concept
of
“1
real/illustrated
object
in
pictograph
represents

1
unit
in
bar
graph”
using
labeled
graph
paper/adult
support

•  (FUNCTIONS)
Iden)fy
in
wri)ng
center
and

radius
of
circle
by
comple)ng
the
square

2.
Target

Important
Note:
Some4mes
a
measureable
verb
does

not
need
a
target.

Obvious:
Recite…

Write…

Mathema4cs/Science:
Solve…
Find…
Calculate…

Construct…
and
other
math-‐speciﬁc
cogni4ons

are
considered
to
be
in
wri4ng
unless
otherwise

noted.

Ba.
Solve
for
unknown
to
make
equa)on
true

(e.g.,
53
+
□
=
73
…
53
+
20
=
73)

Universal
Verbs:
Such
as
Iden4fy…
Compare
and

contrast…
do
need
a
target/targets

Classify Differentiate Justify
Compare Discriminate Organize
Communicate Distinguish Predict
Contrast Estimate Prioritize
Convert Evaluate Prove
Correlate Explain Rank
Criticize Generalize Recognize
Critique Identify Sequence
Define Infer Summarize
Describe Interpret Transfer
Determine Invent

*This is a sample list and therefore not comprehensive.

Design
Note
…
Use
of
parentheses
in
skill
statements

Reduce
complex
frac)on
(frac)on
over
frac)on)
by
mul)plying

by
common
denominator
(e.g.,
see
complex
frac-on
example)

Describe
orally
and
in
wri)ng
par))oned
shares
using

6
terms
(halves,
half
of,
thirds,
third
of,
quarters,
quarter
of)

(e.g.,
Ali
par--ons
a
circle
into
2
shares.
She
writes:
The
circle
has
2
equal

shares
or
2
halves.)

(e.g.,
_____________
)
=
for
example

(i.e.,
______________)
=
that
is

(______________)

=
that
is

3.
Descriptors
add
clarity/details
to
the
aligned

content
lis4ng,
not
duplicate
it.

Not
Yet
Quality

Content

C.
Addi)on/Subtrac)on:
1-‐Step/2-‐Step
Word
Problems

Skills

C.
Solve
word
problems

If
you
were
the
Assessment
Fairy,

could
you
design
a
quality

assessment
to
measure
the

required
skill
learning?

Math
CCSS
Glossary,
Page
88

Quality

Content

C.
Addi)on/Subtrac)on:
1-‐Step/2-‐Step
Word
Problems

Skills

Ca.
Solve
5
types
(adding
to,
taking
from,
puing
together,
taking
apart,

comparing
(more
/
less))
of
1-‐step
word
problems
(compose/
decompose/
regroup
with
minuend
up
to
100,
sum
up
to
100)
using

self-‐selected
strategies
and
jus)fy
reasoning:

-‐-‐Adding
to
(Result
Unknown)
(e.g.,
Twenty
bunnies
sat
on
the
grass.
Thirty-‐
three
more
bunnies
hopped
there.
How
many
bunnies
are
on
the
grass
now?)

-‐-‐Adding
to
(Change
Unknown)
(e.g.,
Twelve
bunnies
were
si]ng
on
the
grass.

Some
more
bunnies
hopped
there.
Then
there
were
thirty-‐seven.
How
many

bunnies
hopped
over
to
the
twelve?)

-‐-‐Adding
to
(Start
Unknown)
(e.g.,
Some
bunnies
were
si]ng...)

Could
the
skill
statement

s4ll
be
improved
design-‐wise?

Decided
to
create
an
agachment
associated
with
the
Skill
/
Table
1,
but
“more
teacher
friendly.”

Addition and Subtraction Word Problem Types
Add To (addition) – Involve physical action, Take From (subtraction) – Involve physical
change in quantity over a period of time action, change in quantity over a period of
time

Result Change Start Result Change Start
Unknown Unknown Unknown Unknown Unknown Unknown

Lela had four dolls. Lela had four dolls. Lela had some dolls. Luke had ten Luke had ten Luke had some
She bought two She bought some She bought two more cookies. He ate cookies. He ate cookies. He ate
more. How many more dolls. Now, she dolls. Now she has six three cookies. How some of the cookies. three cookies. Now
dolls does she have has six dolls. How dolls. How many dolls many cookies does Now he has seven he has seven
now? many dolls did Lela did Lela have before Luke have left? cookies. How many cookies left. How
buy? she bought some cookies did Luke many cookies did
more? _____ + 2 = 6 eat? Luke have to start
4 + 2 = ______ 4 + _____ = 6 10 - 3 = ______ 10 - _____ = 7 with? _____ - 3 = 7
4 + 2 = ______
4 + 2 = ______
Put Together/Take Apart – A visual relationship 10 - 3 = ______

representation between whole and its two separate Compare – A visual comparison of two
parts unconnected sets

Total Addend Both
Addends Difference Bigger Smaller
Unknown Unknown
Unknown Unknown Unknown Unknown

Zach has two brothers. Cam has one more
Luke had ten cookies. Luke has five cookies Zach has two brothers.
Cam has three brother than Zach.
Luke had ten cookies. He ate some of the and two plates. How Cam has one more
brothers. Cam has how Cam has three
He ate three cookies. cookies. Now he has many different brother than Zach.
many more brothers brothers. How many
How many cookies seven cookies. How combinations of How many brothers
than Zach? brothers does Zach
does Luke have left? many cookies did Luke cookies can he make? does Cam have?
have?
eat?
3 – 2 = _____ _____ + 1 = 3
10 - 3 = ______ 10 - _____ = 7 5=5+0 5=0+5 2 + 1 = _____
10 - 3 = _____ 5=1+4 5=4+1
2 + _____ = 3 3 – 1 = _____
5=2+3 5=3+2

Pendergast
Elementary
School
District

Phoenix,
AZ

Content
G.
Scale
Drawings:

Square,
Right
Triangle,
Rectangle
/
Unique
Figures

Skills
G.
Reduce
larger
image
using
graph
paper
to
reproduce
larger
image
as
a
similar

figure
(e.g.,
see
Edwin's
Reduc-on
example)

G.
Enlarge
smaller
image
using
graph
paper
to
reproduce
smaller
image
as
a
similar

figure

G.
Iden)fy
in
wri)ng
scale
key
in
scale
drawings
(e.g.,
city
maps,
architect
blueprints,

landscape
plans,
engineering
blueprint
/
e.g.,
see
Scale
Keys
example)

G.
Correlate
in
wri)ng
scale
key
to
drawing
representa)on(s)
(e.g.,
Michael
is

looking
at
a
scale
drawing.
The
scale
key
reads:

1"
=
1
Foot

Michael
writes:
In
this

scale
drawing,
1
inch
represents
1
foot
of
the
actual
boat.)

G.
Reduce
larger
image
(2-‐dimensional
quadrilaterals)
based
on
given
scale
using

graph
paper
(e.g.,
see
Campbelle's
Parallelogram
example)

G.
Enlarge
smaller
image
(2-‐dimensional
quadrilaterals)
based
on
given
scale
using

graph
paper
(e.g.,
see
Harper's
Rectangles
example)

Ga.
Determine
dimensions
of
figures
when
given
scale
and
iden)fy
impact
of
scale

on
actual
length
(1-‐dimension),
area
(2-‐dimension)
and
jus)fy
reasoning
(e.g.,
see

Coach
Clegg's
Problems
example)

A bit of “fine tuning” is
oftentimes needed when
first writing skill statements.
Sometimes difficult for task-force
members to differentiate between
a SKILL/STRATEGY or ACTIVITY.
A skill is focused on what students
must cognitively be able to do.

An activity is focused on
providing practice concerning
a particular skill’s or skills’
development or reinforcement.
No: Practice orally addition facts
No: Review steps to solve …
No: Walk around room to measure…

Wearing
the
right
design
“gear”
…
Dive
on
in!

(Even
though
it
may
feel
a
ligle
unnerving
at
ﬁrst…)

While
breaking
apart
standards
and

determining
content,
and
especially
skills,

task-‐force
members
are
always
ques4oning

and
inquiring
using
on-‐line
resources
as

well
as
each
another’s
exper4se.

Researchers

www.LiveBinders.com
Search for: Janet Hale > Select Author > Click Search

www.azed.gov/standards-practices/mathematics-standards/

http://www.azed.gov/standards-practices/mathematics-standards/

www.illustrativemathematics.org

Seven-Step Review Process
 1. Collecting the Data
 2. First Read-Through
 3. Small Group Review
 4. Large Group
Comparisons
 5. Determine Immediate
Revision Points
 6. Determine Points
Requiring Research and
Planning
 7. Plan for Next Review
Heidi
Hayes
Jacobs,
Mapping
The
Big
Picture
(1997)

Seven-Step Review Process
 1. Collecting the Data
 2. First Read-Through
 3. Small Group Review
 4. Large Group
Comparisons
 5. Determine Immediate
Revision Points
 6. Determine Points
Requiring Research and
Planning
 7. Plan for Next Review

1.
Collec4ng
the
Data

2.
First
Read-‐Through

3.
Small
Group
Review

4.
Large
Group

Comparisons

5.
Determine
Immediate
Revision
Points

Modify
exis4ng
and/or

add
new
skill
statements
,

examples,
and
agachments.

6.
Determine
Points
Requiring

Research
and
Development
(?)

7.
Plan
For
Next
Review

Janet
Hale

www.CurriculumMapping101.com

teachtucson@aol.com

520-‐241-‐8797

Designing Common Core State Standards Systemic Mathematics Curriculum: Part 2

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