Principles to Actions:
Ensuring Mathematical Success For All
NCDPI Mathematics Department
Kitty Rutherford and Jennifer Curtis
Welcome
“Who’s in the Room”
A 25-year History of Standards-Based
Mathematics Education Reform
Standards Have Contributed
to Higher Achievement
• The percent of 4th graders scoring proficient
or above on NAEP rose from 13% in 1990 to
42% in 2013.
• The percent of 8th graders scoring proficient
or above on NAEP rose from 15% in 1990 to
36% in 2013.
• Between 1990 and 2012, the mean SATMath score increased from 501 to 514 and
the mean ACT-Math score increased from
19.9 to 21.0.
Trend in fourth-and-eigth grade NAEP
Mathematics Average Scores
http://nces.ed.gov/nationsreportcard/subject/publications/main2013/pdf/2014451.pdf
North Carolina NAEP Trends
in Mathematics
Grade
Source
1990
2013
Change
4
NC
223
254
Up 31
4
US
227
250
Up 23
8
NC
250
286
Up 36
8
US
262
284
Up 22
NAEP Scale Score
1990 –First year NAEP reported NC Scores
2013 – Latest NC NAEP Test Data
http://nces.ed.gov/nationsreportcard/subject/publications/main2013/pdf/2014451.pdf
NC EOG/EOC Percent
Solid or Superior Command (CCR)
Grade
2012-2013
2013-2014
3
46.8
48.2
4
47.6
47.1
5
47.7
50.3
6
38.9
39.6
7
38.5
39.0
8
34.2
34.6
Math I
42.6
46.9
Common Core State Standards
http://www.ncpublicschools.org/accountability/reporting/
Although We Have Made
Progress, Challenges Remain
• The average mathematics NAEP score for 17-yearolds has been essentially flat since 1973.
• Among 34 countries participating in the 2012
Programme for International Student Assessment
(PISA) of 15-year-olds, the U.S. ranked 26th in
mathematics.
• While many countries have increased their mean
scores on the PISA assessments between 2003 and
2012, the U.S. mean score declined.
• Significant learning differentials remain.
Brainstorm
What are the essential
conditions required to
make sure
mathematics works
for all students?
Curriculum
Tools and
Technology
Access and
Equity
Teaching and
Learning
Assessment
Effective
Mathematics
Programs
Professionalism
Principles to Actions:
Ensuring Mathematical Success for All
The primary purpose of
Principles to Actions is to fill
the gap between the adoption
of rigorous standards and the
enactment of practices,
policies, programs, and
actions required for
successful implementation of
those standards.
NCTM. (2014). Principles to Actions: Ensuring
Mathematical Success for All. Reston, VA: NCTM.
Principles to Actions:
Ensuring Mathematical Success for All
The overarching message is
that effective teaching is
the non-negotiable core
necessary to ensure that all
students learn mathematics.
The six guiding principles
constitute the foundation of
PtA that describe highquality mathematics
education.
NCTM. (2014). Principles to Actions: Ensuring
Mathematical Success for All. Reston, VA: NCTM.
Teacher Beliefs
“Teachers’ beliefs
influence the decisions
they make about the
manner in which they
teach mathematics.”
Principles to Actions pg. 10
Beliefs About Teaching and
Learning Mathematics
Students’ beliefs influence their perception
of what it means to learn mathematics and
how they feel toward the subject.
Principles to Actions pg. 10
High-Quality Standards are Necessary,
But Insufficient, for Effective Teaching
and Learning
Teaching mathematics requires
specialized expertise and professional
knowledge that includes not only knowing
mathematics but knowing it in ways that
will make it useful for the work of
teaching.
Ball and Forzani 2010
Teaching and Learning
Curriculum
Tools and
Technology
Access and
Equity
Teaching and
Learning
Assessment
Effective
Mathematics
Programs
Professionalism
Teaching and Learning
An excellent mathematics program
requires effective teaching that engages
students in meaningful learning through
individual and collaborative experiences
that promote their ability to make sense
of mathematical ideas and reason
mathematically.
Principles to Actions pg. 7
5 Interrelated Strands Constitute
Mathematical Proficiency
National Research Council, 2001
Obstacles to Implementing
High-Leverage Instructional Practices
Dominant cultural beliefs about the
teaching and learning of mathematics
continue to be obstacles to consistent
implementation of effective teaching and
learning in mathematics classrooms.
Eight High-Leverage
Instructional Practices
1. Establish mathematics goals to focus learning
2. Implement tasks that promote reasoning and
problem solving
3. Use and connect mathematical representations
4. Facilitate meaningful mathematical discourse
5. Pose purposeful questions
6. Build procedural fluency from conceptual
understanding
7. Support productive struggle in learning mathematics
8. Elicit and use evidence of student thinking
Not to be confused with…
What do you notice?
Overview of the
Eight Mathematics
Teaching Practices
1. Establish mathematics goals to
focus learning.
Effective teaching of mathematics
establishes clear goals for the
mathematics that students are learning,
situates goals within learning
progressions, and uses goals to guide
instructional decisions.
Principles to Actions pg. 12
What did you notice about the dialog?
Math coach intentionally shifts the
conversation to a discussion of the
mathematical ideas and learning that will
be the focus of instruction
Principles to Action – pg. 16
2. Implement Tasks That Promote
Reasoning and Problem Solving
Effective teaching of mathematics
engages students in solving and
discussing tasks that promote
mathematical reasoning and problem
solving and that allow for multiple entry
points and varied solution strategies.
Principles to Actions pg. 17
High or Low
Cognitive Demanding Task?
Cognitive Demand
Jigsaw/Sort
1.
2.
Everyone grab a puzzle piece/ticket from your table.
Move to the designated spot in the room for your color.
– Green/Orange: Memorization
– Yellow/Pink: Procedures without Connections
– Blue/Blue: Procedures with Connections
– Red/Purple: Doing Mathematics
3. Read page 18 and summarize the description associated with
your cognitive demand task type. Come to a shared
understanding of the demand task and be prepared to share
back at your table.
4. At your table, use the contents of the envelope to sort the tasks
by cognitive demand.
Table Talk
What are the attributes of a
mathematically strong task?
Math Tasks
There is no decision that teachers make
that has a greater impact on students’
opportunities to learn and on their
perception about what mathematics is
than the selection or creation of the tasks
with which the teacher engages students
in shaping mathematics.
Look on page 21
NCDPI – Task
http://commoncoretasks.ncdpi.wikispaces.net/
Principles to Action - page 24
Which Math Practices would students
be engaged in?
3. Use and connect mathematical
representations
Effective teaching of mathematics
engages students in making
connections among mathematical
representations to deepen
understanding of mathematics concepts
and procedures and as tools for
problem solving.
Principles to Actions pg. 24
Let’s Do Some Math!
The bowl holds 5 L of water. If we
use a scoop that holds 1/6 of a
liter, how many scoops will we
need in order to fill the entire
bowl?
NCDPI Unpacking Document, Grade 5, pg. 44
Use and connect mathematical
representations.
Table Talk
• Read the discussion on pages 24-26
• On chart paper, solve the problem using
each representation.
• Share something with your
group that resonates with you.
• Now, refer to the example on
page 27.
Which Math Practices would students
be engaged in?
4. Facilitate meaningful
mathematical discourse
Effective teaching of mathematics
facilitates discourse among students in
order to build shared understanding of
mathematical ideas by analyzing and
comparing student approaches and
arguments.
Principles to Actions pg. 29
Let’s Do Some Math
One bag of cat food will
feed 3 cats for 40 days. If I
buy 2 more cats, how long
will one bag of cat food
last?
Becoming a Problem Solving Genius, Zaccaro, pg. 5
3 cats x 40 days = 120 servings
per
bag
Number of Cats
Equation
Number of Days
120 ÷ 1 = 120
120
120 ÷ 2 = 60
60
120 ÷ 3 = 40
40
120 ÷ 4 = 30
30
120 ÷ 5 = 24
24
Table Talk
During the cat food task, how did you use
discourse to build shared understanding?
Which Math Practices would students
be engaged in?
5. Pose purposeful questions
Effective teaching of mathematics uses
purposeful questions to assess and
advance student reasoning and sense
making about important mathematical
ideas and relationships.
Principles to Actions pg. 35
Types of Questions-Four Corners
Gathering
Information
Probing
Thinking
Making the
mathematics
visible
Encouraging
reflection and
justification
Types of Questions
• In your table group, sort the question
descriptions and examples.
• Create at least 1 additional question
that fits each description.
• Are these types of questions important
in the classroom?
Principles to Actions pg. 36-37
Funneling vs Focusing
• Read pg 37, last two paragraphs
• Review Figure 16 on pg 39-40
• Using chart paper, illustrate funneling
vs focusing questioning patterns.
• What are some barriers that might
prevent teachers from moving from
funneling to focusing questions?
Principles to Actions pg. 41
6. Build procedural fluency from
conceptual understanding
Effective teaching of mathematics builds
fluency with procedures on a
foundation of conceptual
understanding so that students, over
time, become skillful in using
procedures flexibly as they solve
contextual and mathematical problems.
Principles to Actions pg. 42
Form two lines…
• How does computational fluency relate to
conceptual understanding?
• How do we move from conceptual
understanding to computational fluency?
• Where do we use computational fluency in
mathematics?
• Why are algorithms necessary?
How could Anna’s reasoning help
David understand
his
mistake?
Principles to Actions pg. 43
How Are these Methods Interrelated?
Principles to Actions pg. 45
http://maccss.ncdpi.wikispaces.net/Elementary+Webinars
Principles to Action - page 47
Which Math Practices would
students be engaged in?
7. Support productive struggle in
learning mathematics
Effective teaching of mathematics
consistently provides students,
individually and collectively, with
opportunities and supports to engage in
productive struggle as they grapple with
mathematical ideas and relationships.
Principles to Actions pg. 48
Shopping Trip Task
Joseph went to the mall with his friends to
spend the money that he had received for
his birthday. When he got home, he had
$24 remaining. He had spend 3/5 of his
birthday money at the mall on video games
and food. How much money did spend?
How
much money had he received for
hisbirthday?
Principles to Actions pg. 51
Classroombased indicators of
success
Expectations for
students
Teacher actions to
support students
Most tasks that
promote reasoning and
problem solving take
time to solve, and
frustration may occur,
but perseverance in the
face of initial difficulty
is important.
Use tasks that promote
reasoning and problem
solving; explicitly
encourage students to
persevere; find ways to
support students without
removing all the
challenges in a task.
Students are engaged
in the tasks and do not
give up. The teacher
supports students
when they are “stuck”
but does so in a way
that keeps the thinking
and reasoning at a high
level.
Correct solutions are
import- ant, but so is
being able to explain
and discuss how one
thought about and
solved particular tasks.
Ask students to explain
and justify how they
solved a task. Value the
quality of the explanation
as much as the final
solution.
Students explain how
they solved a task and
provide mathematical
justifications for their
reasoning.
Everyone has a
responsibility and an
obligation to make
sense of mathematics
by asking questions of
peers and the teacher
when he or she does
not understand.
Give students the
opportuni- ty to discuss
and determine the
validity and appropriateness of strategies and
solutions.
Students question and
critique the reasoning
of their peers and
reflect on their own
understanding.
Diagrams, sketches,
and hands-on materials
are im- portant tools to
use in making sense of
tasks.
Communicating about
one’s thinking during a
task makes it possible
for others to help that
person make progress
on the task.
Give students access to Students are able to
tools that will support
use tools to solve tasks
their thinking processes. that they can- not solve
without them.
Ask students to explain
their thinking and pose
questions that are
based on students’
reasoning, rather than
on the way that the
teacher is think- ing
about the task.
Students explain their
thinking about a task to
their peers and the
teacher. The teacher
asks probing questions
based on the students’
thinking.
Principles to Actions pg. 49, 51
Struggling to Learn
Carol Dweck, Psychologist
Growth Mind-set Research
The Teaching Channel
How does having a growth mindset relate to embracing and supporting student
struggle?
Principles to Actions pg. 53
Which Math Practices
would students be engaged
in?
Emergentmath.com
8. Elicit and use evidence
of student thinking
Effective teaching of mathematics uses
evidence of student thinking to assess
progress toward mathematical
understanding and to adjust instruction
continually in ways that support and
extend learning.
Principles to Actions pg. 53
“My Favorite No: Learning From Mistakes”
During the video;
•
Identify strategies the teacher uses to access, support, and extend student thinking.
•
How do these strategies allow for immediate re-teaching?
•
What student behaviors were associated with these instructional strategies?
Which Math Practices would
students be engaged in?
Sort the Beliefs
Unproductive Beliefs
Check your arrangement on Principles to Actions pg. 11
Beliefs About Teaching and Learning
Mathematics
Principles to Action – pg. 11
Essential Elements of Effective
Mathematics Programs
Curriculum
Tools and
Technology
Access and
Equity
Teaching and
Learning
Assessment
Effective
Mathematics
Programs
Professionalism
Principles to Actions
What action are you taking?
• Your role:
– Leaders and policymakers pgs 110-112
– Principals, coaches, specialists, other school
leaders pgs 112-114
– Teachers pgs 114-117
• Choose at least one action that you plan to
implement as a result of today’s session.
• Turn and share your plan with a shoulder
partner.
What questions do
you have?
2015 NCCTM Spring
Leadership Conference
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DPI Mathematics Section
Kitty Rutherford
Elementary Mathematics Consultant
919-807-3841
kitty.rutherford@dpi.nc.gov
Denise Schulz
Elementary Mathematics Consultant
919-807-3839
denise.schulz@dpi.nc.gov
Lisa Ashe
Secondary Mathematics Consultant
919-807-3909
lisa.ashe@dpi.nc.gov
Vacant
Secondary Mathematics Consultant
919-807-3842
@dpi.nc.gov
Dr. Jennifer Curtis
K – 12 Mathematics Section Chief
919-807-3838
jennifer.curtis@dpi.nc.gov
Susan Hart
Mathematics Program Assistant
919-807-3846
susan.hart@dpi.nc.gov
For all you do for our students!