Transcript Common Core State Standards for Mathematics 9

Pippen Consulting
Randy and Sue Pippen
2011-12
[email protected]
You have three playing cards lying face up,
side by side. A five is just to the right of a
two, a five is just to the left of a two, a spade
is just to the left of a club, and a spade is just
to the right of a spade. What are two
possibilities for the three cards?
Be ready to discuss your thinking!


Find a shoulder partner that is not in your
school or district – move if you have to.
Introduce yourselves to each other:
◦ Name, position, what you hope to learn today.

On a signal, tell the group what your partner
told you.
Turn to partner and discuss
1.
2.
3.
Does it look different at elementary, middle
and high school?
Is this design effective? What is our
evidence that it is? What is our evidence
that it is not?
How long have we used this model?

Signal your familiarity with the new Illinois
State Standards for Mathematics (Common
Core State Standards) by showing a signal of
1 to 5 with 1 being the lowest.
IDEA
M
O
C
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- understand that the Common Core Math State Standards are the
new Illinois State Math Standards and will be the basis for the Math
State Assessments for grades 9-12;
-learn for evaluation purposes that the new Common Core Math
State Standards involve content and practice standards - what
mathematics is to be taught and assessed, and what instructional
practices are expected to be used for grades 9-12;
-examine how grades 9-12 math instruction and assessment must
change in order to teach and assess for understanding, making
sense, and what to monitor through evaluation; and
-analyze the differences between the grades 9-12 scope and
sequence of the old Illinois Learning Standards
• Relate the New Common Core State Standards to
the Illinois Standards and the upcoming change
in State testing.
• Relate the new Mathematics Practice Standards to
the way instruction should look with the CCSSM.
• Familiarize administrators with the instructional
changes required for students to learn with
depth, understanding and making sense of the
mathematics.
• Relate the differences in the old Illinois Math
Standards and the new Illinois Math Standards
(CCSSM).
• Develop a plan to update staff on the key
components of the Content and Practice
Standards and how they will be assessed.
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Fewer, higher, more focused
Benchmarked Internationally
Equal emphasis of understanding and skills
Much more specific than old Illinois Learning
Standards
Emphasis on number early on, learning
trajectories develop through the grades
Highly visual and connected with multiple
representations of functions:
graphs/verbal/symbolic/numeric
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Emphasis on arithmetic and number patterns translating
to algebra
Congruence and similarity based on transformations
Resurgence of constructions, but in a variety of ways
Algebra 1, Geometry, and Algebra 2 for all students
Modeling, modeling, modeling or “What’s it good for?”
Precalculus only for students who will take calculus
Not all students should take calculus – STEM standards
(+)
A variety of fourth year courses
No longer push for more students in the 8th grade taking
high school algebra
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Currently sending too many underprepared
students to algebra at the 8th grade
Program may not be equivalent to high
school due to time constraints of middle
school, may not have a secondary-mathcertified teacher
There cannot be any skipping in CCSSM
There are other ways to accelerate (p. 81
Appendix A)
Not all students need calculus, therefore do
not need to accelerate at all.

Understand the connections between proportional
relationships, lines, and linear equations.
5. Graph proportional relationships, interpreting the unit rate as
the slope of the graph. Compare two different proportional
relationships represented in different ways. For example,
compare a distance-time graph to a distance-time equation to
determine which of two moving objects has greater speed.
6. Use similar triangles to explain why the slope m is the same
between any two distinct points on a non-vertical line in the
coordinate plane; derive the equation y = mx for a line through
the origin and the equation y = mx + b for a line intercepting the
vertical axis at b.

Analyze and solve linear equations and pairs of
simultaneous linear equations.
7. Solve linear equations in one variable.
a. Give examples of linear equations in one variable with one
solution, infinitely many solutions, or no solutions. Show which of
these possibilities is the case by successively transforming the
given equation into simpler forms, until an equivalent equation of
the form x = a, a = a, or a = b results (where a and b are
different numbers).
b. Solve linear equations with rational number coefficients,
including equations whose solutions require expanding
expressions using the distributive property and collecting like
terms.
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No ISAT or PSAE after 2013-2014.
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May be pilot items in ISAT in 2012-2014.
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Some areas tested by current state tests will no longer
be tested in new design.
NCLB has not been reauthorized nor made any
adjustments for CCSS. Many states are refusing to
continue with NCLB.
A waiver is to be available to states who meet the
criteria.to be released in September
English Language Arts and Mathematics, Grades 3 - 11
25%
50%
75%
90%
PARTNERSHIP RESOURCE CENTER: Digital library of released items, formative assessments,
model curriculum frameworks, curriculum resources, student and educator tutorials and
practice tests, scoring training modules, and professional development materials
Focused
ASSESSMENT 1
• ELA
• Math
Summative
assessment for
accountability
Focused
ASSESSMENT 2
• ELA
• Math
Required, but
not used tor
accountability
Focused
ASSESSMENT 3
• ELA
• Math
Focused
ASSESSMENT4
• Speaking
• Listening
END OF YEAR
COMPREHENSIVE
ASSESSMENT
Governing Board States
Participating States
1.
2.
3.
4.
5.
Create high-quality assessments
Build a pathway to college and career
readiness for all students
Support educators in the classroom
Develop 21st century, technology-based
assessments
Advance accountability at all levels
1. Determine whether students are college- and career-ready
or on track
2. Assess the full range of the Common Core Standards,
including standards that are difficult to measure
3. Measure the full range of student performance, including
the performance high and low performing students
4. Provide data during the academic year to inform
instruction, interventions and professional development
5. Provide data for accountability, including measures of
growth
6. Incorporate innovative approaches throughout the system
18

Summative Assessment Components:
◦ Performance-Based Assessment (PBA) administered as close to the end
of the school year as possible. The ELA/literacy PBA will focus on
writing effectively when analyzing text. The mathematics PBA will focus
on applying skills, concepts, and understandings to solve multi-step
problems requiring abstract reasoning, precision, perseverance, and
strategic use of tools.
◦ End-of-Year Assessment (EOY) administered after approx. 90% of the
school year. The ELA/literacy EOY will focus on reading comprehension
The math EOY will be comprised of innovative, machine-scorable items

Formative Assessment Components:
◦ Early Assessment designed to be an indicator of student knowledge and
skills so that instruction, supports and professional development can be
tailored to meet student needs
◦ Mid-Year Assessment comprised of performance-based items and
tasks, with an emphasis on hard-to-measure standards. After study,
individual states may consider including as a summative component
The PARCC assessments will allow us to make important
claims about students’ knowledge and skills.
 In English Language Arts/Literacy, whether students:
◦ Can Read and Comprehend Complex Literary and Informational Text
◦ Can Write Effectively When Analyzing Text
◦ Have attained overall proficiency in ELA/literacy

In Mathematics, whether students:
◦ Have mastered knowledge and skills in highlighted domains (e.g.
domain of highest importance for a particular grade level – number/
fractions in grade 4; proportional reasoning and ratios in grade 6)
◦ Have attained overall proficiency in mathematics
Flexible
Early Assessment
• Early indicator of
student
knowledge and
skills to inform
instruction,
supports, and PD
Summative
assessment for
accountability
Mid-Year
Assessment
• Performancebased
• Emphasis on hard
to measure
standards
• Potentially
summative
Formative
assessment
Performance-Based
Assessment (PBA)
• Extended tasks
• Applications of
concepts and skills
ELA/Literacy
• Speaking
• Listening
End-of-Year
Assessment
• Innovative,
computer-based
items
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
Targeted
interventions &
supports:
th
•12 -grade
bridge courses
• PD for
educators
High School
SUCCESS IN
FIRST-YEAR,
CREDIT-BEARING,
POSTSECONDARY
COURSEWORK
ONGOING STUDENT SUPPORTS/INTERVENTIONS
22
INSTRUCTIONAL TOOLS TO
SUPPORT IMPLEMENTATION
PROFESSIONAL DEVELOPMENT
MODULES
K-12
Educator
TIMELY STUDENT
ACHIEVEMENT DATA
EDUCATOR-LED TRAINING TO
SUPPORT “PEER-TO-PEER”
TRAINING
23
PARCC’s assessment will be computer-based and
leverage technology in a range of ways to:
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Item Development
◦ Develop innovative tasks that engage students in the
assessment process
Administration
◦ Reduce paperwork, increase security, reduce
shipping/receiving & storage
◦ Increase access to and provision of accommodations for SWDs
and ELLs
Scoring
◦ Make scoring more efficient by combining human and
automated approaches
Reporting
◦ Produce timely reports of students performance throughout the
year to inform instructional, interventions, and professional
24
development

PARCC assessments will be purposefully
designed to generate valid, reliable and timely
data, including measures of growth, for various
accountability uses including:
◦ School and district effectiveness
◦ Educator effectiveness
◦ Student placement into college-credit bearing courses
◦ Comparisons with other state and international
benchmarks

PARCC assessments will be designed for other
accountability uses as states deem appropriate
25
Oct. 2010
Sept. 2011
Launch and
design phase
begins
Development
phase begins
Sept. 2012
Sept. 2013
Sept. 2014
Summer 2015
First year field
testing and
related research
and data
collection
begins
Second year
field testing
begins and
related research
and data
collection
continues
Full
administration
of PARCC
assessments
begins
Set
achievement
levels,
including
college-ready
performance
levels
26
Technical Challenges
• Developing an
interoperable
technology platform
Challenges

• Transitioning to a
computer-based
assessment system
• Developing and
implementing
automated scoring
systems and
processes
• Identifying effective,
innovative item
types
Policy Challenges
Implementation


Estimating costs
over time, including
long-term
budgetary planning
Transitioning to the
new assessments at
the classroom level
Ensuring long-term
sustainability

Student supports
and interventions

Accountability
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High school
course
requirements
College
admissions/
placement
Perceptions
about what these
assessments can
do
27
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Cost effectiveness in a difficult economy
The three summative through-course
assessments could dictate the scope and
sequence of the curriculum limiting local
flexibility (not federal government right)
The potential that the required three
through-course assessments would
disrupt the instructional program on, and
in preparation for, testing days
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Intended to ensure results will be reported in categories
consistent with the CCSS.
Separate scores in ELA for reading and writing as well as
an overall score indicating on track to college and career
readiness.
Separate score in a “highlighted domain” that reflects the
CCSS’s emphasis at each grade level (e.g., fractions in
grade 4, rations and proportional relationships at grade
6), as well as an overall math score indicating on track to
college readiness.
Measures student growth over a full academic year or
course
Provides data during the academic year to inform
instruction, interventions and professional development
activities.
Accessible to all students including disabled and ELL
Must be approved by the US Department of Education
Grade or
HS Category
K
1
2
3
4
5
6
7
8
HS-NQ
HS-A
HS-F
HS-M
HS-G
HS-SP
Highlighted
Domains
CC
OA
NBT
OA
NF
NF
RP. EE
RP, NS
EE, G
RN
SSE, REI
IF, BF
No separate score
CO, GPE
ID
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Listen to directions
See what it looks like
Stand up and try it
Old Illinois Learning
Standards
Number
NCTM Standards
Number Sense
Common Core State
Standards
Number and Quantity
Measurement
Measurement
Modeling
Algebra
Algebra
Algebra
Functions
Modeling
Geometry
Probability and
Statistics
Geometry
Geometry
Probability and
Statistics
Modeling
Probability and
Statistics
Modeling
Old Illinois Learning
Standards
Solve Problems
NCTM Standards
Problem Solving
Common Core State Standards
Model with Mathematics
Make sense of problems and persevere
in solving them
Look for and express regularity in
repeated reasoning
Look for and make use of structure
Working on Teams
Use appropriate tools strategically
Using Technology
Communicating
Communication
Construct viable arguments and critique
the reasoning of others
Making Connections
Connections
Look for and express regularity in
repeated reasoning
Attend to precision (language)
Representation
Attend to precision
Reasoning and Proof
Reason abstractly and quantitatively
Construct viable arguments and critique
the reasoning of others
Kindergarten
1
2
3
4
5
6
7
8
HS
Counting and
Cardinality
Number and Operations in Base Ten
Number and Operations - Fractions
Number
and
Quantity
Ratios and
Proportionality
The Number System
Expressions and Equations
Algebra
Operations and Algebraic Thinking
Functions
Geometry
Measurement and Data
Functions
Geometry
Geometry
Statistics and Probability
Statistics
and
Probability
EARLY
ELEMENTARY
LATE
ELEMENTARY
MIDDLE/JUNIOR
HIGH SCHOOL
EARLY HIGH
SCHOOL
LATE HIGH
SCHOOL
6.A.1a Identify
whole numbers
and compare them
using the symbols
<, >, or = and the
words “less than”,
“greater than”, or
“equal to”,
applying counting,
grouping and
place value
concepts.
6.A.2 Compare
and order whole
numbers, fractions
and decimals
using concrete
materials,
drawings and
mathematical
symbols.
6.A.3 Represent
fractions,
decimals, percentages,
exponents and
scientific notation
in equivalent
forms.
6.A.4 Identify and
apply the
associative,
commutative,
distributive and
identity properties
of real numbers,
including special
numbers such as
pi and square
roots.
6.A.5 Perform
addition,
subtraction and
multiplication of
complex numbers
and graph the
results in the
complex plane.
6.A.1b Identify
and model
fractions using
concrete materials
and pictorial
representations.
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N.Q.1 Use units as a way to understand
problems and to guide the solution of multistep problems; choose and interpret units
consistently in formulas; choose and interpret
the scale and the origin in graphs and data
displays.
N.Q.2 Define appropriate quantities for the
purpose of descriptive modeling.
N.Q.3 Choose a level of accuracy appropriate
to limitations on measurement when
reporting quantities.
1a. How do you solve 3x + 1 = -14 ?
1b. Why did you do it the way you did?

Switch roles
2a. How do you graph y = ½ x -3?
2b. Why did you do it the way you did?
EARLY
ELEMENTARY
LATE
ELEMENTARY
MIDDLE/JUNIOR
HIGH SCHOOL
EARLY HIGH
SCHOOL
LATE HIGH
SCHOOL
8.D.1 Find the
unknown numbers
in whole-number
addition,
subtraction,
multiplication and
division situations.
8.D.2 Solve linear
equations involving
whole numbers.
8.D.3a Solve
problems using
numeric, graphic or
symbolic
representations of
variables,
expressions,
equations and
inequalities.
8.D.4 Formulate
and solve linear
and quadratic
equations and
linear inequalities
algebraically and
investigate
nonlinear
inequalities using
graphs, tables,
calculators and
computers.
8.D.5 Formulate
and solve nonlinear
equations and
systems including
problems involving
inverse variation
and exponential
and logarithmic
growth and decay.
8.D.3b Propose
and solve problems
using proportions,
formulas and linear
functions.
8.D.3c Apply
properties of
powers, perfect
squares and
square roots.

A.REI.1 Explain each step in solving a simple
equation as following from the equality of
numbers asserted at the previous step,
starting from the assumption that the original
equation has a solution. Construct a viable
argument to justify a solution method.
EARLY
ELEMENTARY
LATE
ELEMENTARY
MIDDLE/JUNIOR
HIGH SCHOOL
EARLY HIGH
SCHOOL
LATE HIGH
SCHOOL
9.B.1a Identify
and describe
characteristics,
similarities and
differences of
geometric shapes.
9.B.2 Compare
geometric figures
and determine
their properties
including parallel,
perpendicular,
similar, congruent
and line symmetry.
9.B.3 Identify,
describe, classify
and compare twoand threedimensional
geometric figures
and models
according to their
properties.
9.B.4 Recognize
and apply
relationships within
and among
geometric figures.
9.B.5 Construct
and use two- and
three-dimensional
models of objects
that have practical
applications (e.g.,
blueprints, topographical maps,
scale models).
9.B.1b Sort,
classify and
compare familiar
shapes.
9.B.1c Identify
lines of symmetry
in simple figures
and construct
symmetrical
figures using
various concrete
materials.

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G.CO.1 Know precise definitions of angle, circle, perpendicular line,
parallel line, and line segment, based on the undefined notions of
point, line, distance along a line, and distance around a circular arc.
G.CO.2 Represent transformations in the plane using, e.g.,
transparencies and geometry software; describe transformations as
functions that take points in the plane as inputs and give other points
as outputs. Compare transformations that preserve distance and angle
to those that do not (e.g., translation versus horizontal stretch).
G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular
polygon, describe the rotations and reflections that carry it onto itself.
G.CO.4 Develop definitions of rotations, reflections, and translations in
terms of angles, circles, perpendicular lines, parallel lines, and line
segments.
G.CO.5 Given a geometric figure and a rotation, reflection, or
translation, draw the transformed figure using, e.g., graph paper,
tracing paper, or geometry software. Specify a sequence of
transformations that will carry a given figure onto another.
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There is a train. It leaves a station an hour later than a plane
flying overhead, flying in the opposite direction.
The number of the train is a 3-digit number whose tens digit
is 3 more than its units digit.
The conductor of the train is half as old as the train was when
the conductor was a third as old, just a third as old.
The conductor’s niece and nephew are on the train. They
head toward the club car at the back of the train to buy mixed
nuts; some of the nuts are $1.79 a pound and some are $2.25
a pound.
They have quarters, dimes and nickels in their pockets to pay
for the nuts.
The niece starts first and walks at 2 miles per hour and the
nephew starts later and walks at 3 miles per hour.
How long will it take them to get to the back of the train if
they walk together?
•
If you know the width of a lawn
mower in inches, how can you find
how many square yards of lawn it
cuts in running a certain number of
feet?
▫ Problems Without Figures
▫ Gillan, 1909

Traditional Path or Integrated Path

Same fifteen units – distributed by course

Illinois will have to choose one or the other to
determine testing

Challenges: Materials for either path

Texts: May say they are aligned, probably not
Algebra I

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Unit 1 – Relationships
Between Quantities and
Reasoning with Equations
Unit 2 – Linear and
Exponential Relationships
Unit 3 – Descriptive
Statistics
Unit 4 - Expressions and
Equations
Unit 5 – Quadratic
Functions and Modeling
Mathematics I

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
Unit 1 – Relationships
Between Quantities
Unit 2 – Linear and
Exponential Relationships
Unit 3 – Reasoning with
Equations
Unit 4 – Descriptive Statistics
Unit 5 – Congruence, Proof
and Constructions
Unit 6 – Connecting Algebra
and Geometry through
Coordinates
Geometry
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
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
Mathematics II

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Unit 1 – Extending the
Number System
Unit 2 - Quadratic Functions
and Modeling
Unit 3 – Expressions and
Equations
Unit 4 – Applications of
Probability
Unit 5 – Similarity, Right
Triangle Trigonometry and
Proof
Unit 6 – Circles With and
Without Coordinates
Algebra II
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
Unit 1 – Polynomial,
Rational and Radical
Relationships
Unit 2 – Trigonometric
Functions
Unit 3 – Modeling with
Functions
Unit 4 – Inferences and
Conclusions from Data
Mathematics III




Unit 1 – Inferences and
Conclusions from Data
Unit 2 – Polynomial,
Rational and Radical
Relationships
Unit 3 – Trigonometry
of (+)General Triangles
and Trigonometric
Functions
Unit 4 – Mathematical
Modeling

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

More algebra at the eighth grade means a
different algebra in high school, more
technology for both
Geometry must be built upon grade school
transformations – most books are not written
that way
More Probability and Stats in all high school
courses
Advanced Algebra has less content but more
depth than previous courses, more
technology



Turn to your shoulder partner and talk about
what you see regarding the new and old ILS –
specifically, talk about implications for
instruction
Signal to start, signal to stop (about 2
minutes).
Whole Group Sharing



Listen to directions
See what it looks like
Stand up and try it


What is Mathematics Proficiency?
Two sources: Strands of Proficiency from
Adding It Up and Mathematical Practice
Standards (CCSSM)
Strands of Mathematical Proficiency
Conceptual
Understanding
Strategic
Competence
Adaptive
Reasoning
Productive
Disposition
Procedural
Fluency
NRC (2001). Adding It Up. Washington, D.C.:
National Academies Press.
52
Strands of Mathematical Proficiency
• Conceptual Understanding – comprehension of
mathematical concepts, operations, and relations
• Procedural Fluency – skill in carrying out procedures
flexibly, accurately, efficiently, and appropriately
• Strategic Competence – ability to formulate,
represent, and solve mathematical problems
• Adaptive Reasoning – capacity for logical thought,
reflection, explanation, and justification
• Productive Disposition – habitual inclination to see
mathematics as sensible, useful, and worthwhile,
coupled with a belief in diligence and one’s own
efficacy.
53
1.
2.
In pairs, review the Standards
for Mathematical Practice.
Take the standards two at a
time, one for each of you, then
share what you read. Return
to whole group to discuss.
Then back to pairs, repeat.
When finished with all eight,
discuss a new insight you had
into the practices.
◦ Make sense of problems and persevere in solving them
◦ Reason abstractly and quantitatively
◦ Construct viable arguments and critique the reasoning
of others
◦ Model with Mathematics
◦ Use appropriate tools strategically
◦ Attend to precision
◦ Look for and make use of structure
◦ Look for and express regularity in repeated reasoning


Are we there yet?
What will it take?


Brainstorming
Handout – What
Should I look for in a
Math Classroom?
LESS









Lecturing
Students passive
Value on student
silence
Worksheet/seatwork
“Coverage”
Competition
Rote memorization
Tracking/pullouts
Reliance on outside
tests
MORE

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
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

Experiential/hands-on
Active Learning
Student conversations
Higher order thinking
Deeper study of fewer
topics
Choice for students
Student responsibility
Help within classroom
Heterogeneous grouping
Teacher’s evaluation of
learning



It is not something
you do to others
Maximum
motivation occurs
when the person
believes he has
autonomy, mastery
and purpose
Control leads to
compliance,
autonomy leads to
engagement



Mastery is the desire
to get better and
better at something
that matters
Choice plays into
autonomy – turn
homework into
“homelearning”
“Now-that” rewards
instead of “if-then”
rewards, nontangible are best




1. They can extinguish
intrinsic motivation
2. They can diminish
performance.
3. They can crush
creativity
4. They can crowd
out good behavior



5. They can
encourage cheating,
shortcuts and
unethical behavior
6. They can become
addictive
7. They can foster
short-term thinking
 From Drive, Daniel Pink
 Praise
effort and strategy, not
intelligence
 Make praise specific, not general
 Praise in private, one-on-one
 Offer praise only when there is a
good reason for it
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“A curriculum is more than a collection of
activities; it must be coherent, focused on
important mathematics, and well articulated
across the grades.” NCTM Principles and
Standards for School Mathematics 2000
The curriculum is not the textbook!
NCTM Focal Points – a good elementary resource
Common Written Curriculum – Clear Objectives
Common Core State Standards
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“Assessment should support the learning of
important mathematics and furnish useful
information to both teachers and students.”
NCTM Principles and Standards, 2000.
Aligned to Objectives and Could be Arranged
by Objectives
Common Major Assessments
Frequent Informal Assessments with
Immediate Feedback
Feedback for Guiding Instruction and Goal
Setting
64
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Effective Professional development:
◦ Develops teachers’ knowledge of math content,
students and how they learn mathematics, effective
instructional and assessment practices
◦ Models examples of high-quality mathematics
teaching and learning
◦ Allows teachers to reflect on their practice and
student learning in their classroom
◦ Allows teachers to collaborate and share experience
with colleagues
◦ Connects to a comprehensive long-term plan that
includes student achievement
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Discussion
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Video
Discussion: What is the teacher doing, what
are the students doing?
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Handout – During the Observation
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Discussion
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Seating people at tables
◦ If each table can seat 8 people with three on a side
and one at each end.
◦ When tables are pushed together end to end,
people can sit on each side and only at each end.
◦ How many people can be seated at 2 tables end to
end?
 3 tables, end to end
 5 tables, end to end
 n tables, end to end
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Emphasis on the
mathematical meaning
Having students
constructing their meaning
Making connections
between mathematics and
other subject matter areas
Building on student
meanings and student
understandings
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Having students solve problems without
prior or concurrent skill development.
Allowing students to explore and develop
their own algorithms
Having students learn skill development
through problem solving, conjecturing and
verifying.
Drill on isolated skills can hinder making
sense of them later.
“The joy of the task is its own reward.”
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Students taught procedures tend to resist
new ideas and appeared to apply procedures
without understanding. (Kieran, 1984)
“Initial rote learning of a concept can create
interference to later meaningful learning”
(Pesek and Kirshner, 2000)
 Based on an article in Educational Leadership,
Video
Who is doing the work?
What is the engagement level of the students?
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Hands-on experiences enable students to
construct their own meanings.
Teachers must be knowledgeable in the
use of concrete materials.
Using the same material to teach different
ideas help shorten the time it takes to see
connections between mathematical ideas.
Do not limit to demonstrations.
Students must see the two-way
relationship between the concrete
materials and the notation used to
represent it.
2x - 4 = 8
Add 4 to each side and remove zero pairs.
Arrange the tiles into two equal groups on both
sides of the mat.
Answer?
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Changes the content, methods, and skill
requirements
Enables more high-level questions.
Actively involves students through asking
questions, conjecturing and exploring – lots of
exploring with discussion about what is
happening and why
Positive effects on graphing ability, conceptual
understanding of graphs, and relating graphs to
other representations.
Students using graphing calculators are more
flexible with strategies, have greater
perseverance, and trying to understand concepts.
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Teach through tasks instead of
“telling”
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Employ a variety of student thinking
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Recognize and value different
methods
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May include manipulatives, but most
of all relies on thinking and recording
thinking
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Make mathematics problematic – you have not
already taught them how
Connect with where students are – varied
levels of entry
Leave behind something of mathematical
value – mathematical learning
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Select tasks with goals in mind
Share essential information
Establish classroom culture
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Ideas and methods are valued
Students choose and share their methods
Mistakes are learning sites for everyone
Correctness resides in the mathematical argument
Mathematical tasks are a set of problems or
a single complex problem the purpose of
which is to focus students’ attention on a
particular mathematical idea.
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Tasks form the basis for students’ opportunities to
learn what mathematics is and how one does it;
Tasks influence learners by directing their attention to
particular aspects of content and by specifying ways to
process information;
The level and kind of thinking required by
mathematical instructional tasks influences what
students learn; and
Differences in the level and kind of thinking of tasks
used by different teachers, schools, and districts, is a
major source of inequity in students’ opportunities to
learn mathematics.
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The King asks Archimedes if his crown is made from
pure gold.
He knows that the crown is either pure gold or it may
have some silver in it.
Archimedes figures out that the volume of the crown
is 125 cm3 and that its mass is 1.8 kilograms.
He also knows that 1 kilogram of gold has a volume
of about 50 cm3 and 1 kilogram of silver has a
volume of about 100 cm3.
1. Is the crown pure gold? Explain how you know.
2. If the crown is not pure gold, then how much silver
is in it?
 Show all your work.
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A professional development resource
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Released in April from NCTM
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Aligns well with the CCSSM Mathematical
Practices
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Choose the task
Work it out and anticipate student methods
Conduct a classroom discussion to clarify the
task, but not direct the students to a solution or
method, close reading
Monitor the work and identify which groups are
using which methods or new methods
Select and record which groups will present
Sequence the presentations for maximum
discussion
Connect the ideas with a whole-class discussion
 From the 5 Practices book
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Adapt classroom problems – choose from the
end of the unit before teaching the unit –
make it an application problem.
Consult the Internet – see sources at the end
of the PowerPoint
Focus on the math you want them to learn
Talk to
a
partner
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To get better participation in classroom
conversations, move between three formats:
◦ Whole-class discussion – before a task, after a task
◦ Small-group discussion – time limit, specific
directions on what they are to do/discuss/produce
◦ Partner talk – short time limit to get more thinking
when the whole-class discussion stalls out, specific
directions on what they are to discuss (30 seconds)
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Five productive talk moves
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Revoicing (teacher)
Repeating (student)
Reasoning - Agree/disagree and why (student)
Adding on (student)
Wait time (teacher)
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Revoicing: “So you’re saying it’s an odd
number?”
Repeating: “Can you repeat what he just said in
your own words?”
Reasoning: “Miranda, do you agree or disagree
with what Paul just said?”
Adding on: “Would someone like to add
something more to this?”
Wait time: Wait beyond the time for a few
students to raise their hands. Wait for the
reluctant participants to think and offer an
explanation. (10 seconds or more)
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Five steps to implementing classroom talk
◦ Set the classroom climate, respectful and
supportive
◦ Focus the talk on the mathematics
◦ Provide for equitable participation
◦ Explain your expectations for the new forms of talk
and why talk in math is important
◦ Try only one challenging new thing at a time
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Identify talk moves in the video as the teacher
launches a lesson on linear equations.
http://www.insidemathematics.org/index.php/cl
assroom-video-visits/public-lessonscomparing-linear-functions/269-comparinglinear-functions-problem-2-parta?phpMyAdmin=NqJS1x3gaJqDM-18LXtX3WJ4e8
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Discussion
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Second Video - from book: 6.2
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Is the teacher always the one talking?
Do students present solutions?
Do students work together?
Do students converse about mathematics
with each other or with the teacher?
Are students building their own meaning or is
the teacher dispensing it?
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All learning, except for simple rote memorization,
requires the learner to actively construct meaning
Students’ prior understanding of and thoughts about a
topic or concept before instruction exert a tremendous
influence on what they learn during instruction
The teacher’s primary goal is to generate a change in
the learner’s cognitive structure or way of viewing and
organizing the world
Because learning is a process of active construction by
the learner, the teacher cannot do the work of learning
Learning in cooperation with others is an important
source of motivation, support, modeling and coaching
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Number off by 7s
Go to the numbered poster with a marker.
Write implications for instructional leaders
according to the topic at the top of the
poster.
At signal, move to next poster and repeat.
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Summary Discussion and Reflection
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Required of the academy
Your plan should be how to disseminate the
information you learned about today.
It must be submitted to Donna at the St. Clair
ROE to be entered into the system for you.
• Relate the New Common Core State Standards to
the Illinois Standards and the upcoming change
in State testing.
• Relate the new Mathematics Practice Standards to
the way instruction should look with the CCSSM.
• Familiarize administrators with the instructional
changes required for students to learn with
depth, understanding and making sense of the
mathematics.
• Relate the differences in the old Illinois Math
Standards and the new Illinois Math Standards
(CCSSM).
• Develop a plan to update staff on the key
components of the Content and Practice
Standards and how they will be assessed.
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What was most valuable to you today?
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Contact info: [email protected]
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If you want a copy of this PowerPoint:
http://dl.dropbox.com/u/26625625/2011%2
01169%20AA.ppt
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www.nctm.org
Illuminations
www.insidemathematics.org
www.nctm.org Navigations Books and Focus
Books
Coming: illustrativemathematics.org
Coming: www.mathedleadership.org – Great
Tasks and More (NCSM website)
www.mathedleadership.org - Common Core State
Standards (CCSS) Mathematics Curriculum Materials
Analysis Project
Challenge problems in texts
Enrichment activities – maybe
Word problems not taught yet
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Five Practices for Orchestrating Productive
Mathematics Discussions, Smith and Stein, NCTM,
2011.
Classroom Discussions, Using Math Talk to Help
Students Learn, Chapin, Math Solutions, 2009.
Handbook of Research on Improving Student
Achievement, Third Edition, Gordon Cawelti, Editor,
Educational Research Service, 2004.
Common Core Standards, NGA, CCSSO, 2010
Annenberg Media Videos
Drive, The Surprising Truth about What Motivates
Us, Daniel Pink, 2009.
Conferences
NCSM Annual Meeting, Philadelphia, April 2012
NCTM Annual Meeting, Philadelphia, April 2012