The PARCC Institute – High School Math:
Preparing for PARCC!
NJ’s Next Generation Standardized
Assessment System
FEA/PSA - Judith T. Brendel - Spring 2014
Welcome…
2
Desired outcomes…
•understand the overall substance, design and technical
aspects of next generation PARCC assessments
•gain access and practice with PARCC tasks and related
resources
•better enable educators to effectively lead the transition to
CCSS and PARCC assessments.
3
What should teaching and learning in
the 21st century look like?”
“…revolutionary transformation rather than
evolutionary tinkering.” US DOE, NETP 2010
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Education for a
Contemporary Age
Preparing a Globally Competent
and Innovative Workforce
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A Changing World…
A growing need for a paradigm shift in the way we teach and
learn…
•
A Nation at Risk (1983) http://www2.ed.gov/pubs/NatAtRisk/findings.html
http://www.ed.gov/pubs/NatAtRisk/index.html
•
SCANS Report and America 2000
http://wdr.doleta.gov/SCANS/whatwork/whatwork.pdf
(What Work Requires of Schools)
“The best time to plant a tree is twenty years ago; the second best time
is today.” (Ancient Chinese Proverb)
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A Changing Student… (recommended resources)
•
Digital Natives, Digital Immigrants – Parts I and II – Do
They Really Think Differently? (Prensky, 2001)
http://www.marcprensky.com/writing/default.asp
•
Marc Prensky’s Essential 21st Century Skills
http://www.marcprensky.com/writing/PrenskyEssential_21stCenturySkills.pdf
“Different kinds of experiences lead to different brain structures.”
Dr. Bruce D. Perry, Baylor College of Medicine
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A Changing Student… (recommended resources)
•Authentic
Intellectual Work and Standardized Tests:
Conflict or Coexistence (2001)
http://ccsr.uchicago.edu/content/publications.php?pub_id=38#
•How
People Learn: Brain, Mind, Experience and School
(1999, 2003) http://www.nap.edu/openbook.php?record_id=6160
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Listen to our students!
“I am a 21st Century Learner!”
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How? Fostering essential skills…
•
•
•
•
•
Communication
Collaborative
problem solving
Critical thinking
Creativity
Innovation
•
•
•
•
•
Global
competence
Content
knowledge
Entrepreneurship
Adaptability
Interpersonal
… through innovative pedagogy and
ubiquitous technology.
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What conditions will
foster these skills?
“I do not teach my
students; I simply create
the conditions for their
learning.”
(Albert Einstein)
2006 Innovative Teachers Forum.
Retrieved May 10, 2010 on : http://www.edlabgroup.org/tl/resources/InnovativeTeachers2006.pdf
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21st learning…
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…what should it look like?
21st learning…
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http://www.pascack.k12.nj.us/cms/lib5/NJ01000238/Centricity/Domain/87
/Writing%20Equations%202.swf
14
Common standards,…
…common (online) assessments, and common
instruction that promote the essential skills.
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Reflection…
•What trends will continue to shape the world in
which our students will live and work?
•What skills will our students need to succeed in
their future?
•What type of learning environments would best
support to development of these skills?
…what are your thoughts?
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Let’s take a (quick) break!
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Let’s get to work…
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Let’s take our… (page-3)
PARCC Pre-Assessment!
Item 9: grade-7 level math w/new
CCSS (What’s different?)
ADD H.S. EXAMPLE
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A (Quick) Walk through the PARCC!
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Race to the Top Assessment Grants:
•
$350 million of Race-to-the-Top Fund to consortia of states to design and
develop common K-12 assessment systems aligned to CCSS
•
In September 2010, the U.S. Department of Education awarded grants to:
•
Partnership for Assessment of Readiness for College and Careers
(PARCC) – 18 States, & DC
•
Smarter Balanced Assessment Consortium (SBAC)
•
Winning consortia - four years - to develop assessments systems,
•
States participating will administer new assessments statewide by 2014-2015
•
ACHIEVE was selected through a competitive bid process, by PARCC to be
the Project Manager.
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Race to the Top Assessment Grants:
•
New Jersey is PARCC Governing Board State:
•
Will administer the assessment system statewide 2014-15 school year.
•
Will field test assessment items spring 2014.
•
Governing Board meets quarterly. The chief state school officer for each
of the governing states sits on the PARCC Governing Board. This to make
major policy, operational, design and financial decisions regarding PARCC.
•
Executive Committee meets weekly. (Governing Board Members from 6
states)
•
ACHIEVE won competitive bid, selected by PARCC to be the Project
Manager.
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Why New Common Assessments?
Current Assessment Systems…
•
Too many tests, often with disconnected purposes
(e.g., instructional improvement vs. accountability vs.
college admissions)
•
Not challenging enough to measure CCR
•
Do not measure the full range of college- and careerready knowledge and skills (such as research, analysis,
critical thinking, and collaboration)
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Why New Common Assessments?
Current Assessment Systems…
•
Fail to generate information for educators and
students quickly enough or at all
•
Are widely inconsistent across states making
results impossible to compare
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Why New Common Assessments?
Next-Generation Common Assessment Systems will…
•
Measure students’ mastery of Common Core State
Standards
•
Provide a common measure of college and career
readiness
•
Include a range of item types that allow for the
assessment of higher-order skills
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Why New Common Assessments?
Next-Generation Common Assessment Systems will…
•
Leverage new technologies in assessment and reporting
to get actionable student data to educators and parents
in real time
•
Mitigate challenges associated with student mobility by
ensuring students will have the same expectations
wherever they live
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PARCC Assessment Consortium
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K-2 formative
assessment
being
developed,
aligned to the
PARCC system
K-2
Timely student achievement
data showing students,
parents and educators
whether ALL students are
on-track to college and
career readiness
3-8
College
readiness score
to identify who
is ready for
college-level
coursework
High
School
Targeted
interventions &
supports:
•12th-grade bridge
courses
• PD for educators
SUCCESS IN
FIRST-YEAR,
CREDITBEARING,
POSTSECONDARY
COURSEWORK
ONGOING STUDENT
SUPPORTS/INTERVENTIONS
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What does it mean? “Students are on-track or ready for
college and careers” (Math)
Solve problems
involving the major
content for their grade
level with connections to
practices
Solve problems
involving the additional
and supporting content
for their grade level with
connections to practices
Use the modeling
practice to solve real
world problems
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Express mathematical
reasoning by
constructing
mathematical arguments
and critiques
Demonstrate fluency in
areas set forth in the
Standards for Content in
grades 3-6
Assessment Transition Timeline
Assessment Transition Timeline
“Transitional Assessments”
Spring 2012
NJ ASK
Aligned to
NJCCCS
Spring 2013
Spring 2014
NJ ASK
NJ ASK
Aligned to the
CCSS and
Aligned to the
NJCCCS
CCSS
SY 2014-15
Full
administration
of PARCC
assessments
(Except Gr. 6-8
Math)
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The PARCC Commitment…
PARCC is designed to promote quality
instruction aligned to the CC Standards,
so the assessment is worthy of preparation
rather than a distraction from good work.
TEACHING to the TEST may be a GOOD THING!
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Understanding the
Common Core State
Standards …is to
understand PARCC
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PVRSD Math TF – Work Session 1 | November 18, 2010
Why Common Core State Standards?
"The standards are designed to be robust
and relevant to the real world, reflecting the
knowledge and skills that our young people
need for success in college and careers,"
which will place American students in a
position in which they can compete in a
global economy.
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Why Common Core State Standards?
•
Before Common Core State Standards we had
standards, but rarely did we have standards-based
instruction.
 Long
lists of broad, vague statements
 Mysterious assessments
 Coverage mentality
 Focused on teacher behaviors – “the inputs”
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Principles of the CCSS…
•
•
•
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FEWER - CLEARER - HIGHER
Aligned to requirements for College and
Career Readiness
Based on evidence
Honest about time
The CCSS Difference: Grade 8 Mathematics
Before: NJCCCS (2004)
1. Understand and apply the Pythagorean Theorem.
After: CCSS (2010)
1. Explain a proof of the Pythagorean Theorem and its converse.
1.
Apply the Pythagorean Theorem to determine unknown side
lengths in right triangles in real-world and mathematical problems
in two and three dimensions.
1.
Apply the Pythagorean Theorem to find the distance between two
points in a coordinate system.
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a2 + b2
= c2
5
5
25
a=4
c=?
4
16
a
4
c
b
3
9
3
b=3
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Implications of the CCSS on
WHAT and HOW to teach…
What are the Critical Shifts to
Consider?
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COMMON CORE SHIFTS in ASSESSMENTS
Shift 1: Focus Priority standards = focus of the assessments.
Other standards deemphasized.
Shift 2: Coherence Assessments will reflect the progression of
content and concepts as depicted in the standards across grade levels.
Shift 3: Fluency It will be assumed students possess required
fluencies through grade 8; as such, students will not be allowed to use
calculators in grades 3-5.
Students will be allowed to use four-function calculators with a square
root key or scientific calculators in grade 6 and scientific calculators in
grades 7-8.
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CC SHIFTS in ASSESSMENTS continued ….
Shift 4: Deep Understanding
Each standard will be assessed from multiple perspectives, while
not veering from the primary target of measurement for the
. standard.
Shift 5: Application
Shift 6: Dual Intensity
Students will be expected to know grade-level mathematical content
with fluency and to know which mathematical concepts to employ
to solve real-world mathematics problems.
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PARCC sample
prototypes with Task
Type
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PVRSD Math TF – Work Session 1 | November 18, 2010
TASK TYPES – PARCC
I – Computation, Skills, Process
(x – 2)2 + 6 = 75
2x2 - 5x – 6 = 0
II – Mathematical Reasoning
xxx
xxxx
III – Modeling, Combined Standards, Applications
Compare the 3 sets of data (graph, chart, and
equation) and defend the argument that ….
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TASK TYPE-I
Tasks assessing concepts, skills and procedures.
- include a balance of conceptual understanding, fluency, and
application. These tasks can involve any or all mathematical
practice standards.
- will be machine scorable and will include innovative, computerbased formats.
-will appear on the End-of-Year and Performance Based
Assessment components and generate evidence for measuring major,
additional, and supporting content with connections to the
mathematical practices as indicated in the PARCC Model Content
Frameworks for Mathematics.*
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December 2, 2013
Mathematics online Sample Item/Prototypes:
PARCC TYPE-I TASKS (1, 2, or 4 point questions)
HS – FunctionsS
HS – Seeing Structure in Quadratic Equation
HS – Picture Frame (Algebra-I/Math 2)
http://www.parcconline.org/sites/parcc/files/HS-Alg1Math2PictureFrame.pdf
HS – Myla’s Swimming Poole (Algebra-I)
HS – Green Tea Observation Study (Algebra-2/Math 3)
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December 2, 2013
TYPE II: Tasks assessing expressing mathematical reasoning.
- tasks call for written arguments/justifications, critique of
reasoning, or precision in mathematical statements (MP. 3, 6).
These tasks can also involve other mathematical practice standards.
- tasks may include a mix of innovative, machine scored and
hand scored responses.
- tasks will be included on the Performance Based Assessment
component and generate evidence for measuring mathematical
reasoning with connections to content.
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December 2, 2013
Mathematics online Sample Item/Prototypes:
PARCC TYPE-II TASKS (4 point questions)
•HS – Graphs of Functions (Algebra-II/Math III) Nov. 2013
HS.C.6.2, A.REI.D, MP.3, 7 with 4-point scoring rubric
http://www.parcconline.org/sites/parcc/files/HighSchoolAlg2Math3-GraphsofFunctions.pdf
•HS – Michelle’s Conjectures (Algebra-II/Math III)
http://www.parcconline.org/sites/parcc/files/PARCC_SampleItems_Mathematics_HSAlgIMathIIMi
chelleConjecture_081913_Final_0.pdf
•HS – Geometric Construction Connection (Geometry/Math III)
PARCC_SampleItems_Mathematics_HSGeoMathIIIGeometricConnection_081913_Final_0.pdf
http://www.parcconline.org/sites/parcc/files
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Dec. 2, 2013
TYPE III: Tasks assessing modeling/applications.
(3 or 6 point questions)
- tasks call for modeling/application in a real-world context or
scenario (MP.4) and can also involve other mathematical practice
standards.
- tasks may include a mix of innovative, machine scored and
hand scored responses.
- tasks will be included on the Performance Based Assessment
component and generate evidence for measuring mathematical
modeling/application with connections to content.
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December 2, 2013
TYPE-I PARCC TASKS
include a balance of conceptual understanding, fluency, and
application, and can involve any or all Math Practice Standards.
will be machine scorable and will include innovative, computerbased formats.
will appear on the End of Year and Performance Based Assessment
components and generate evidence for measuring major, additional,
and supporting content with connections to the mathematical
practices as indicated in the
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PARCC Model Content Frameworks for Mathematics.*
SA MS/HS– PD Session 3| April 4., 2011
Functions http://www.parcconline.org/samples/mathematics/high-schoolfunctions
Given f(x) as a graphed parabola and g(x) as table of values
1.Compare y-intercepts
DIFFICULTY?
2.Compare f(3) and g(3)
3.Compare maximum values of each w/in range -5 ≤ x ≤ 5
4.Compare
f (5)- f (2)
g(5)- g(2)
5- 2
and
5- 2
STANDARDS?
A type I PARCC task
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J. Brendel 12.3.2013
Seeing Structure in a Quadratic Equations
http://www.parcconline.org/samples/mathematics/high-school-seeingstructure-quadratic-equation
(3x - 2)2 = 6x – 4
STANDARDS?
a Type-I PARCC task
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J. Brendel 12.3.2013
Seeing Structure in Equations
http://www.parcconline.org/samples/mathematics/high-school-seeing-structure-equation
24 + 10x – x2 = p – (x – 5)2 Solve for p
STANDARDS?
a Type I PARCC task
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J. Brendel 12.3.2013
PICTURE FRAME
http://www.parcconline.org/sites/parcc/files/HS-Alg1Math2PictureFrame.pdf
STANDARDS?
a Type I PARCC task
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J. Brendel 12.3.2013
GREEN TEA TYPE
http://www.parcconline.org/sites/parcc/files/PARCC_SampleItems_Mathematics_HS
AlgIIMathIIIGreenTeaStudy_081913_Final_0.pdf
STANDARDS?
a Type I PARCC task
53
J. Brendel 12.3.2013
TYPE-II PARCC TASKS
call for written arguments/justifications, critique of reasoning, or
precision in mathematical statements (MP. 3, 6). These tasks can
also involve other mathematical practice standards.
may include a mix of innovative, machine scored and hand scored
responses.
will be included on the Performance Based Assessment
component and generate evidence for measuring mathematical
reasoning with connections to content.
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GRAPH of FUNCTIONS
(posted November 2013)
http://www.parcconline.org/sites/parcc/files/HighSchoolAlg2Math3GraphsofFunctions.pdf
STANDARDS?
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a Type-II PARCC Task
GEOMETRY CONSTRUCTION CONNECTION
http://www.parcconline.org/sites/parcc/files/PARCC_SampleItems_Mathematics_HS
GeoMathIIIGeometricConnection_081913_Final_0.pdf
STANDARDS?
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a Type-II PARCC Task
MICHELLE’S CONJECTURES
(Aug.19, 2013)
http://www.parcconline.org/sites/parcc/files/PARCC_SampleItems_Mathematics_HSAlgIMat
hIIMichelleConjecture_081913_Final_0.pdf
POINTS ?
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a Type-II PARCC Task
TYPE-III PARCC TASKS
call for modeling/application in a real-world context or scenario
(MP.4) and can also involve other mathematical practice standards.
may include a mix of innovative, machine scored and hand scored
responses.
will be included on the Performance Based Assessment component
and generate evidence for measuring mathematical
modeling/application with connections to content.
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MINI GOLF PRICES
(November, 2013)
http://www.parcconline.org/sites/parcc/files/HSAlg1Math2MiniGolfPrices.pdf
POINTS ?
STANDARDS?
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a Type-III PARCC Task
BRETT’S RACE
(November, 2013)
http://www.parcconline.org/sites/parcc/files/HSMathematicsSampleItemBR.pdf
STANDARDS?
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a Type-III PARCC Task
POPCORN INVENTORY
(NOVEMBER 2013)
http://www.parcconline.org/sites/parcc/files/HSAlg1Math1PopcornInventory.pdf
POINTS ?
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a Type-III PARCC Task
TEMPERATURE CHANGES
http://www.parcconline.org/sites/parcc/files/PARCC_SampleItems_Mathematics_HS
AlgIIMathIIITemperatureChange_081913_Final_0.pdf
POINTS ?
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a Type-III PARCC Task
Task
Type
1
1
1
PARCC PROTOTYPE SAMPLE TASKS
As of 12/5/2013
Algebra-I/Math I
Geometry/Math II
Algebra-II / Math III
Seeing Structure in
Equations
(1 point)
Picture Frame
(2 points)
Myla’s Swimming Pool
(2 points)
2
Michele’s Conjectures
3
Popcorn Inventory
(6 points)
3
Brett’s Race
(3 points)
Golf Prices
(6 points)
3
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Seeing Structure in
Functions
Quadratic Equation
(1 point)
Green Tea
(2 points)
Geometry
Construction
(3 points)
Graph function
(4 points)
Temperature
Changes
(3 points)
PARCC RELEASED TASKS include elsewhere? Use this slide to ‘hold’ sites
file://localhost/Users/jbrendel/Desktop/PARCC%20HS%20Functions.h
tml (November/December 2013)
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https://www.teachingchannel.org/videos/laws-of-sines-cosineslesson# 14-min lesson “fly quadcopters” math/science/engineering
… a 4-month project … Ms. Brookins and Mr. James GRADE-11
How/if .. .include here?
https://www.teachingchannel.org/videos/statistical-analysis-lesson
Statistical analysis to rank baseball players GRADES 11-12
CCSS Standards: Math SID. 1, 2 and 3
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SA MS/HS– PD Session 3| April 4., 2011
What you SHOULD NOT see !
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y
m = y1 - y
6-2 4
= =
x1 - x2 3- 3 0
(3,6)
N
Z
0
slope =
4
67
Slope =
(3,2)
>
4
0
x
<
BRAINSTORM in GROUPS
What should teachers NOT be doing?
68
SA MS/HS– PD Session 3| April 4., 2011
READING CCSS and Mathematics
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Not Text-Dependent in ELA
•In “Casey at the Bat,” Casey strikes out.
Describe a time when you failed at
something.
•In “Letter from a Birmingham Jail,” Dr.
King discusses nonviolent protest.
Discuss, in writing, a time when you
wanted to fight against something that
you felt was unfair.
•In “The Gettysburg Address” Lincoln
says the nation is dedicated to the
proposition that all men are created
equal. Why is equality an important
70value to promote?
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Text-Dependent in ELA
What makes Casey’s experiences at bat
humorous?
What can you infer from King’s letter
about the letter that he received?
“The Gettysburg Address” mentions the
year 1776. According to Lincoln’s
speech, why is this year significant to
the events described in the speech?
Text-Dependent in mathematics
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December 2, 2013 author: J. Brendel Material not to be duplicated.
Regular Practice with Complex Text and
its Academic Language
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Regular Practice with Complex Text and
its Academic Language
Cramer’s Rule for solving systems of equations uses determinants
to solve for each variable. D is the determinant of the coefficient
matrix. Dy is the determinant formed by replacing the coefficients of
y in D with the constant terms.
Algebra-II
Sample academic language in a basic statistics task: measure of
central tendency, skewed, mean, median, mode, outlier, bimodal,
quartiles, box-and-whisker plot, percentile, random sample, …
73
New to Pre-Algebra, Algebra I or II
Regular Practice with Complex Text and
its Academic Language (“Adult Language”)
Example: Phone company ‘A’ charges an initiation fee plus a charge
per minute. The cost of using 300 minutes (plus initiation fee) is $25
and the cost of 700 minutes (plus initiation fee) is $33.
•Write an equation to represent the cost.
•What is the initiation fee?
•What is the cost per minute?
•Is this company’s offer better than the new phone company ‘B’s?
•Explain or compare with charts and graphs.
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Pre-Algebra, Algebra
Regular Practice with Complex Text and
its Academic Language
Basic constructions w/triangles using protractors, compasses or online
with simulation tools: Students construct, do proofs and solve
problems involving:
Medians, vertex angle, balancing point, centroid
Altitudes, acute triangles, obtuse triangles, right triangles, orthocenter,
interior, exterior or ….
Perpendicular Bisectors, concurrent lines, point of concurrency,
circumcenter of a triangle
Angle Bisectors, incenter, equidistant, ….
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Geometry
Adjusting Math Language to the CCSS
• Old habit to eliminate: Defining equality as “same as.”
The problem: This is mathematically incorrect and leads to
misconceptions.
New habit to adopt: Defining equality as “same value as.”
• Old habit to eliminate: “Addition makes things get bigger.”
The problem: When negative numbers are introduced, the old habit
has to be debugged.
New habit to adopt: Addition is about combining.
• Old habit to eliminate: “Subtraction makes things get smaller.”
The problem: As with addition, negative numbers make this wrong.
New habit to adopt: Subtraction is about difference.
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Regular Practice With Complex Text and its
Academic Language: Why?
77
•
Gap between complexity of college and high school
texts is huge.
•
What students can read, in terms of complexity is
greatest predictor of success in college (ACT study).
•
Too many students are reading at too low a level.
What are the Features of Complex Text?
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•
•
•
•
•
Subtle and/or frequent transitions
•
Complex sentences
Multiple and/or subtle themes and purposes
Density of information
Unfamiliar settings, topics or events
Lack of repetition, overlap or similarity in words
and sentences
What are the Features of Complex Text? (cont.)
79
•
•
Uncommon vocabulary
•
•
Longer paragraphs
Lack of words, sentences or paragraphs that review
or pull things together for the student
Any text structure which is less narrative and/or
mixes structures
It all boils down to…
Texts Worth Reading…and
Questions Worth Answering!
80
Let’s take a quick break!
81
Understanding the
Common Core Math
Standards …is to
understand PARCC
Math!
82
PVRSD Math TF – Work Session 1 | November 18, 2010
The Common Core Standards: Classroom Video Illustration
High School School
83
The Standards are NOT new names
for old ways of doing things…
84
NOT …
•
A different topic every day
•
Every topic/concept 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
85
85
Dan Meyer:
Math Class Needs a Makeover
86
From....
To….
856 = ___ hundreds, ___ tens, ___ ones
x2 + 10x + 21 = 0
Lack of rigor
87
1 hundredth = ___ tenths
¾ c(c –1) = c
• Reliance on rote learning
at expense of concepts
• Lack of quality applied
problems and real-world
contexts
• Aversion to repetitious
practice
• Lack of variety in what
students produce
• Severe restriction to
stereotyped problems
lending themselves to
mnemonics or tricks
NOT
• E.g., overwhelmingly
only answers are
produced, not arguments,
diagrams, models, etc.
Three (3) Significant 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
88
• application with equal intensity.
Activity No. 3: Shifts Happen - Mathematics
What is this
Opportunities
shift?
Why this shift?
Focus: 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.
89
Challenges
Three (3) Significant 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
90
• application with equal intensity.
FOCUS continued
70% or more of the assessment will measure the major work in
grades 3-8
Focus allows for a variety of problem types (task types) to get at
a concept in multiple ways.
Students will have more time to master concepts at a deeper
level.
91
The shape of math in A+ countries…
Mathematics topics intended at
each grade by at least two-thirds
of A+ countries
1 Schmidt,
Houang, &
Cogan, “A
Coherent
Curriculum: The
Case of
Mathematics.”
(2002).
92
Mathematics topics intended
at each grade by at least
two-thirds of 21 U.S. states
Traditional U.S. Approach…
K
12
Number and
Operations
Measurement
and Geometry
Algebra and
Functions
Statistics and
93
Probability
93
Focusing attention within Number and Operations
Operations and Algebraic Thinking 
Number and Operations—Base
Ten

94
1
2
3
4
Algebra

The Number
System
Number and
Operations—
Fractions
K
Expressions and

Equations

5
6
7
8
High School
Priorities in Mathematics (CCSS)
Grade
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
7
95
Focus Areas in Support of Rich Instruction and
Expectations of Fluency and Conceptual Understanding
8
Ratios and proportional reasoning; early expressions and
equations
Ratios and proportional reasoning; arithmetic of rational
numbers
Linear algebra, linear functions
95
Priorities in Mathematics (CCSS)- MS/HS
ALG. - 1
96
Focus Areas in Support of Rich Instruction and
Expectations of Fluency and Conceptual Understanding
UNIT-1
Relationships Between Quantities and Reasoning with Equations
UNIT-2
Linear Relationships
UNIT-3
Expressions and Equations
UNIT-4
Quadratic Functions and Modeling
UNIT-5
Functions and Descriptive Statistics
96
Priorities in Mathematics (CCSS) - HS
GEOMETRY
97
Focus Areas in Support of Rich Instruction and
Expectations of Fluency and Conceptual Understanding
UNIT-1
Congruence, Proof, and Constructions
UNIT-2
Similarity, Proof, and Trigonometry
UNIT-3
Extending to Three Dimensions
UNIT-4
Connecting Algebra and Geometry Through Coordinates
UNIT-5
Circles With and Without Coordinates
UNIT-6
Applications of Probability
97
Priorities in Mathematics (CCSS) – HS
ALG. - 2
Focus Areas in Support of Rich Instruction and
Expectations of Fluency and Conceptual Understanding
UNIT-1
Polynomial, Rational, and Radical Relationships
UNIT-2
Trigonometric Functions
UNIT-3
Modeling with Functions
UNIT-4
Inferences and Conclusions from Data
98
98
Three (3) Significant 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
99
• application with equal intensity.
99
COHERENCE
Integrative tasks draw on multiple standards to ensure
students are making important connections.
The Standards are not treated as a checklist.
100
COHERENCE
Need to include h.s. task(s) that incorporate
multiple standards to show connections.
101
Coherence and NEW multiple-choice questions.
Circle all that are true.
Given: All squares
A.are parallelograms and rectangles
B.have 4 acute angles
25 5
C.have only 2 diagonals
3
5
D.Area
= ½ base x height
21 7
E.Sum of consecutive angles = 180˚
F.Perimeter
= 2 x Side length
G.Area
> area
with the same perimeter.
102
*Activity 4: Work with your group (p. 15. Ex. 8)
What skills and understandings are assessed here?
Do they come from only one standard? One domain?
A.
subtraction w/decimals
B.
addition then subtraction
C.
write % as a decimal; multiply
D.
read carefully, add and subtract money
E.
know how to find an average
F.
know to ÷ or x; compute accurately
G.
know to add lowest, then highest; understand concept,
recognize > and < symbols
103
Coherence: Link to major topics within grades
Example: HIGH SCHOOL performance task; Algebra-I unit 1
A large truck has two fuel tanks, each with a capacity of 150
gallons. One of the tanks is half full, and the other is empty. Fuel is
pumped into the tanks until both tanks are full. The pump delivers
fuel at a constant rate of 5¾ gallons per minute.
a.Write an equation for the total number of gallons of fuel ….
a.How much fuel is in the tanks after the pump … 8 minutes?
b.Graph the equation … on the coordinate plane … label ….
104
Algebra-I Unit-1
104
Three (3) Significant 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
105
• application with equal intensity.
RIGOR
Need to include h.s. task(s) that incorporate
multiple standards to show connections.
106
See the Rigor – Early Elementary Grades
Ray’s father has 129 plants in his garden.
Billy’s father has 230 plants in his garden.
Ray says his father has more plants because 9 is bigger
than 2.
Is Ray correct? Yes ___ No ___
Explain how you know.
107
Guess the grade level!
See the Rigor in Grade-3
Fractions on a Number Line
http://www.illustrativemathematics.org/standards/hs
108
See the Rigor in Middle Grades
Understanding of Fractions
Write a number that is greater than
1
5
and less than
Hint: Find equivalent fractions for
denominators of 40 or 100.
1
5
and
1
4
1
4
with
Why wasn’t it helpful to use 20 as a denominator? (JB)
109
See Rigor in Grade 6: Ratios
PARCC - Grade-6 Slider
Ruler: type-I
http://www.parcconline.org/
samples/mathematics/grade6-slider-ruler
110
ONLINE
http://www.parcconline.org/samples/item-taskprototypes
Grade-3 Fractions on a Number Line: type- I
http://www.parcconline.org/samples/mathematics/grade-3mathematics-number-line
Grade-4 Fraction model: type-II
http://www.parcconline.org/sites/parcc/files/PARCC_SampleItems_
Mathematics_G4FractionModel_081913_Final.pdf
Grade-6 Slider Ruler: type-I
http://www.parcconline.org/samples/mathematics/grade-6-sliderruler
111
See the Rigor in High School Math
112
Linking to major topics – High School
Part A
115
Part B
Describe
Write
Compare
116
REQUIRED FLUENCIES
117
SA MS/HS– PD Session 3| April 4., 2011
Required Fluencies in K-6
Grade
Standard
K
K.OA.5
Add/subtract within 5
1
1.OA.6
Add/subtract within 10
2.OA.2
Add/subtract within 20 (know single-digit sums from memory)
2.NBT.5
Add/subtract within 100
Multiply/divide within 100 (know single-digit products from
memory)
2
3
118
3.OA.7
3.NBT.2
Required Fluency
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 division
Multi-digit decimal operations
118
HS EXPECTATIONS
ALGEBRA-I: Unit 2 – LINEAR RELATIONSHIPS
Find approximate solutions to
linear equations by making a table of values, using
technology to graph and successive approximations.
• F.IF.7/9 supporting Graph functions by hand (in
simple cases) and with technology ….
• A.REI.5, A.REI.6 additional Solve systems of linear
equations in two variables …
• A.REI.10, 11 major
119
Dec. 2, 2013 J.Brendel
Coherence: Link to Major Topics Across Grades
6. Apply properties
One of several staircases to
algebra designed in the
OA domain.
•
5.
expressions
3. multiply … divide
1. add … subtract
120
120
Coherence: Link to Major Topics Across Grades
One of several staircases to algebra designed in the OA domain.
ALGEBRA- II
ALGEBRA-I
121
121
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.
122
122
Fluency
FLUENCY and new type of MC question
123
HS Teachers outside of math
use grade-level-appropriate math
125
Part B
126
Part C
127
Part D
128
Standards of Mathematical Practices
1.
Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3. Construct
Constructviable
viablearguments
argumentsand
andcritique
critiquethe
thereasoning
reasoningofof
others.
others
4.
Modelwith
withmathematics.
mathematics.
Model
5.
Use appropriate tools strategically.
6.
Attend
Attend to
to precision.
7.
precision
Look
for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
129
Analysis of the PARCC
Mathematics Assessment:
Design Shifts
(Activity No. 4)
130
Shift No. 1…Focus where Standards Focus
Advance: PARCC assessments will focus strongly
where the Standards focus
(70% or more on the major work in grades 3-8).
 Focus allows for a variety of problem types to get at
concept in multiple ways.
 Students will have more time to master concepts at a
deeper level.
131
Shift No. 2…Coherence across/within grades
Advance: The assessment design is informed
by multi-grade progressions in the Standards
and the Model Content Frameworks.
 Key beginnings are stressed:
 e.g., ratio concepts in grade 6,
 e.g., fluency with the multiplication table in
grade 3
 Become key endpoints and takeaway skills
132
Shift No. 2…Coherence across/within grades
Advance: Integrative tasks draw on multiple
standards to ensure students are making
important connections.
 The Standards are not treated as a
checklist.
133
Shift No. 3…Rigor
Advance: PARCC assessments will reach the
rigor in the Standards through innovations in
technology and item design.
134
SAMPLE ALGEBRA-1
UNIT-I ASSESSMENT
and High School Reference Sheet
(separate packet)
• Solution Guide
• With notes to teachers
• Sample responses and scoring
135
December 2, 2013
The Mathematics Common Core Toolbox
Mathematics Sample Item/Prototypes:
PARCC online at the Dana Center
•Grade 4 (Deer in the Park) p. 23
•Grade 6 (Gasoline Consumption)
•High School (Isabella’s Credit Card) *p. 24-25)
•High School (Popcorn Inventory) new Nov. 2013
http://www.parcconline.org/sites/parcc/files HSAlg1Math1PopcornInventory.pdf
Activity: grade-2 (p. 26-27) *What makes it unique? ….
136
Performance Based Assessment (PBA)
• Type I items (machine-scorable)
• Type II items (Mathematical Reasoning/hand-scored)
• Type III items (Mathematical Modeling/hand scored
and/or machine-scored
End-of-Year Assessment (EOA)
• Type I items only (All machine-scorable)
137
Sample Online Performance Tasks
Mathematics Sample online Item/Prototypes:
Smarter Balanced: link on next slide
•Grade 3 - Lemonade Sale
•Grade 4 - Family Trip to the Zoo
•Grade 6 - Cereal Box (vol., surf.area, min s.a.)
•Grade 7 - Food Basket (calories, grams, protein)
•Grade 8 - Heartbeats (scatter plot ….)
•Grade 11- Speeding Ticket (graph, what is fair? ….)
•Grade 2 hardcopy-Trip to Zoo (gr. 4 modified; J. Brendel
Barron’s)
138
(page 27-28)
EXPERIENCE an ASSESSMENT
Activity 6
https://sbacpt.tds.airast.org/student/login.aspx?c=SB
AC_PT
• Sample online Smarter Balanced Assessments.
Select from grades 3, 4, 5, 6, 7, 8 or 11
• Select Examples or Performance Tasks
•
J.Brendel 11/2013
139
CIRCLES
http://sampleitems.smarterbalanced.org/itempreview/ModernShell.asp
x?config=SBAC%5CContent%5CCircle1.json
MATH II - WALLS
http://sampleitems.smarterbalanced.org/itempreview/ModernShell.asp
x?config=SBAC%5CContent%5CRoomWall.json
J. Brendel 11/2013
140
The Specs, Accountability,
and Resources:
What We Know Now!
141
PARCC Assessment Consortium
142
Dynamic Learning Maps (DLM) Alternate
Assessment Consortium
143
Estimated Time on Task: Grade 9-10
Performance Based Assessment
ELA/Literacy
Math
TOTAL
Grade
9-10
Alg I/
Math I
Geo/
Math II
144
Estimated
Time on
Task
(minutes)
Literacy
Analysis
Research
Narrative
Session
1
Session
2
80
85
50
50
50
315
Estimated Time on Task: Grade 9-10
End-of-Year (EOY) Assessment
ELA/Literacy
Math
TOTAL
Grade
9-10
Alg I/ Math I
Geo/ Math II
Estimated
Time on
Task
(minutes)
Session
1
Session
2
Session
1
Session
2
70
70
65
65
Grade 9-10 Summative Total:
145
270
9 Hours, 45 minutes
Estimated Time on Task: Grade 11
Performance Based Assessment
ELA/Literacy
Math
TOTAL
Grade
11
Alg II/
Math III
146
Estimated
Time on
Task
(minutes)
Literacy
Analysis
Research
Narrativ
e
Session
1
Session
2
80
85
50
65
65
345
Estimated Time on Task: Grade 11
End-of-Year (EOY) Assessment
ELA/Literacy
Math
TOTAL
Grade
11
Alg II/
Math III
Estimated
Time on
Task
(minutes)
Session
1
Session
2
Session
1
Session
2
70
70
55
55
250
Grade 11 Summative Total:
147
9 Hours, 55 minutes
PARCC HIGH LEVEL Blueprints – Mathematics
Assessment Items
EOY
Type 1
EOY TOTAL
PBA/MYA
Type 1
Type 2
P/M totals
Type 1
Type 2
Type 3
PBA/MYA TOTALS
Points Alg.1
Geom Alg2
1 pt.
21
19
19
2 pt.
11
12
12
4 pt.
3
3
3
35
34
34
1 pt.
10
10
10
2 pt.
3 pt.
2
2
2
4 pt.
2
2
2
3 pt.
2
2
2
6 pt.
2
2
3
10
10
10
4
4
5
4
4
5
18
18
20
End-of-Year
machine scored
~ 1/3 total
points = PBA
http://www.parcconline.org/sites/parcc/files/PARCC%20High%20Level%20Bluep
rints%20-%20Mathematics%20043013.pdf Grades 3-8, High School
148
PARCC High Level Blueprints Mathematics
Math Item Counts Per Form Integrated Math I, II, II
Assessment Item
EOY
Type 1
Points
Math-I
Math-II
Math-III
1 pt.
2 pt.
4 pt.
19
12
3
34
19
12
3
34
19
14
2
35
1 pt.
2 pt.
3 pt.
4 pt.
3 pt
6 pt.
10
2
2
2
2
10
10
2
2
2
2
10
10
2
3
2
3
10
4
4
18
4
4
18
5
5
20
EOY TOTAL
PBA/MYA
Type 1
Type 2
PBA/MYA
TOTALS
Type 1
Type 2
Type 3
PBA/MYA
TOTALS
149
End-of-Year
machine
scored
~ 1/3 total
points =
PBA
What Else…
 20 Day Window…for both the PBA and EOY
summative assessments (performance based and end-of-year)
 Summative Assessments Retest Opportunities
 Grades 3-8 - 1 retest opportunity/year
 HS - up to 3 retest opps/year for each assessment
 Computer-based (w/paper version available in rare
circumstances), contingency back-up
 PBA delay? No…
150
What Else…
Accessibility Features for ALL students
• Audio Amplification
• Blank Paper (provided by test administrator)
• Eliminate Answer Choices
• Flag Items to Review
• General Administration Directions (provided by test administrator)
• General Administration Directions Read Aloud and
Repeated (provided by test administrator)
151
What Else…
Accessibility Features for ALL students continued
• Highlight tool
• Headphones
• Magnification/Enlargement Device
• Notepad
• Pop-Up Glossary
• Redirect Student to Test (provided by test administrator)
• Spell Checker
• Writing Tool
152
What Else…
Accommodations / Modifications for Special Education Students
(from the 55-page PARCC Accessibility manual)
Accessibility Features Identified in Advance
• Answer Masking
• Background/Font Color (Color Contrast)
• General Masking
• Line Reader Tool
• Text-to-Speech for the Mathematics Assessments
http://parcconline.org/parcc-accessibility-accommodations-and-fairness
153
Modifications Accommodations NOT permitted
Modifications involve changes in the conditions under which a student takes an
assessment that result in unacceptable changes to the test itself, or what the test
measures (e.g., reducing or changing expectations for students) and are
therefore not permitted on PARCC assessments.
Examples of modifications that will result in invalidated results on PARCC
include:
•Requiring a student to be assessed on less content matter than other students
because he has been taught less material;
•Reducing the scope of assessments so a student needs to complete only a limited
number of problems or items;
154
•
Modifying the complexity of assessments to make them easier (e.g.,
deleting half of the response choices on a multiple-choice test so that a
student selects from two options instead of four);
•
Giving hints, clues, or other coaching that directs the student to correct
responses on assignments and tests;
•
Adults defining vocabulary on the test or explaining test items;
•
Allowing the student to complete an assessment of English language arts
in a language other than English; and
•
Using dictionaries that provide definitions (rather than an acceptable
word-to-word dual-language dictionary).
155
Accountability:
What We Know Now!
156
156
Results Types…
 Proficiency (PLDs)
 On-Track-to-College and
Career Readiness (CCRDs)
 Growth Data
Performance
Level
Descriptors
Performance Level
(PLDs)
157
Description
Level 5*
Distinguished Command
Level 4*
Strong Command
Level 3
Moderate Command
Level 2
Partial Command
Level 1
Minimal Command
*Achieved CCR Performance Threshold
What does “distinguished” (Level 5) mean?
“Students performing at this level demonstrate a distinguished command
of the knowledge, skills, and practices embodied by the Common Core
State Standards for English language arts/literacy assessed at grade 11.
They are academically well prepared to engage successfully in entry-level,
credit-bearing courses in College English Composition, Literature, and
technical courses requiring college-level reading and writing.
Students performing at this level are exempt from having to take and pass
placement tests in two- and four-year public institutions of higher
education designed to determine whether they are academically prepared
for such courses without need for remediation.”
158
What does it mean to be CCR? (Level-4)
“Students who earn a PARCC College- and Career-Ready
Determination by performing at Level 4 in mathematics
and enroll in College Algebra, Introductory College
Statistics, and technical courses requiring an equivalent
level of mathematics have approximately a 0.75
probability of earning college credit by attaining at least a
grade of C or its equivalent in those courses.”
159
What does it mean to be CCR? (cont.)
Will be used as a benchmark against which the CCR cut score on
the PARCC assessments will be validated through empirical
research.
•WHY - a “C” is the minimum grade needed to earn credit for a
course
•WHY - a reasonably high standard but not unattainable.
•WHY - similar to the criteria used by ACT,
•WHY - other important CCR skills will not be measured by PARCC
assessments.
160
What are the cut scores for each level?
As/PARCC, “the vendor selected through the RFP will
also work with state leaders, educators and eternal
experts to determine the cut scores to establish the five
PARCC Performance levels for each grade-level
assessment.
161
RFP for “Operational Assessment”
Vendor proposals are due by Dec. 11, 2013.
Timelines and Resources:
What We Know Now!
162
162
PARCC Timeline…
•
Jan – Aug 2013
2013-2014
2014-2015
163
•
Release of 2nd set of prototype assessment and instructional tasks
(spring 2013)
Release of online professional learning modules (spring 2013) (update:
expected June 2014)
Item tryouts (spring 2013) – see next slide
•
•
•
•
Full-scale pilot/field testing (spring 2014)
Partnership Resource Center launches (spring 2013)
Optional performance tasks for K-2 available (February 2014)
College readiness tools available (spring 2014)
•
•
•
Diagnostic assessments release (September 2014)
Full operational administration of PARCC assessments (spring 2015)
Setting of achievement levels, including college-ready performance
levels (late spring 2015 (post-administration)
•
PARCC Timeline…
*Over the next year, PARCC will develop five online training
modules for teachers, school leaders, and school site testing
coordinators – each with a unique purpose.
PARCC will produce modules with the following foci:
1. PARCC Common Assessments Overview
2. Introduction to the PARCC Mid-Year Assessment
3. Introduction to the PARCC Diagnostic Assessment
4. Introduction to the PARCC Speaking/Listening Assessment
5. PARCC Accessibility System
*It is anticipated that modules will be available in June 2014.
164
PARCC Timeline…
NJ Spotlight (Nov. 20, 2013)
For the first time since 1989, New Jersey will next year suspend its
requirement that high school graduates pass a state test in language
arts and math to receive their diplomas. (No HSPA) But that doesn't
mean there will be fewer tests, just that they'll be used for different
purposes.
For instance, current eighth, ninth, and 10th graders will be tested
in language arts and math now in three separate tests, and while
passing the exams will not be required for graduation, the scores
will be included in a student’s permanent transcript. (2014-15
school year)
165
NJDOE Model Curricula
(handouts Algebra-I, Geometry, Algebra-II)
PARCC Model Content
Frameworks
166
INITIAL QUESTIONS to CONSIDER
when choosing or developing a
Quality Math Assessment.
See handout page-31
JB modified from:
http://www.state.nj.us/education/AchieveNJ/teacher/
SGOGuidebook.pdf
167
GENERAL TEST PREP:
What does this look like?
See handout pages 32-33
Know the test in terms of the following:
• 10 Considerations for Success
•
168
www.achievethecore.org
169
Links to Model Math Curriculum Sites
Sample Assessments by grade:
http://www.achievethecore.corg/
Common Core and Special Education Students
http://www.ode.state.or.us/search/page/?id=3741
Common Core Practice Test
https://sbacot.tds.airast.org/student/login.aspz?c=SBAC.PT
http://sbac.portal.airast.org/practice-test/
Common Core Resources to use with students
http://www.illustrativemathematics.org
Dana Center Resources http://www.ccsstoolbox.org/
http://ccsstoolbox.agilemind.com/pdf/DanaCenter_YAG_HS.pdf
170
COMPANIES are developing new programs to assist districts:
ACUITY: a comprehensive K-12 assessment program
Tom Moellering from McGraw Hills as/DA/District Administration
(30 minute video/conversation)
•pre-built assessments (diagnostic) – or item banks to create your own;
•can be delivered online or in print; scan in software …
•automatically scores and results to teacher quickly
•teacher can see individual/class(es)/school/district results
•teacher can assign specific tasks
•instruction resources in general (including Performance Tasks)
December 2013 J.Brendel
171
COMPANIES are developing new programs to assist districts
as they transition to technology-based assessments:
Pearson’s online Assessment Roadmap 2014
www.Commoncore.pearsoned.com/index.cfm?locator=PS11Uz
Step-1: Conducting a Needs Analysis
Step-2: Developing a Transition Strategy
Step-3: Ensuring Interoperability
Step-4: Communicating Proactively
Step-5: Anticipating Ongoing Change
www.PearsonAssessments.com/NextGenRoadmap
Bryan Bleil (VP, online & tech), Ellen Stain Seymore
172
J. Brendel 11/2013
TEXTBOOK PUBLISHERS are
beginning to develop
new programs to assist districts as they transition to technology-based
assessments:
Many now have all resources online including assessment options.
Sample:
http://www.connected.mcgraw-hill.com
• Teacher username and password
• Select Subject: [Algebra-I CC Standards National]
• Select Standards: [National] [Assess] [Algebra-I]
[Create a New Test]
173
J. Brendel 11/2013
Two Activities
Self-Evaluation, Action Planning
and Next Steps
Response Priority Comments
Roadblocks (pages 28-29)
Action Planning/Next Steps (page 30)
174
Questions?
Concerns (silly question)?
Discussion/Debrief
175
SINCERE THANKS…
For your participation, collaboration and
dedicated efforts!!!
Judy
jbrendel@pascack.k12.nj.us
Brian
bchinni@tmieducation.com
176