Good morning…
• Find a seat and create a name
plate with the cardstock.
• Complete the Teacher
Beliefs Survey that can be
found on the table.
Principles to Actions:
Ensuring Mathematical
Success For All
NC DPI Mathematics Section
Fall 2015
Lisa Ashe lisa.ashe@dpi.nc.gov
Joseph Reaper joseph.reaper@dpi.nc.gov
Welcome
“Who’s in the Room”
Session Goals
To explore the effective teaching
practices that contribute to higher student
achievement using Principles to Actions.
To explore the links between the
Mathematics Teaching Practices and the
Standards for Mathematical Practice.
To examine the classroom “look-fors” that
promote student learning.
Norms
• Assume positive intentions
• Disagree with ideas, not
people
• Maintain Professionalism
• Participate Actively
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 (National Assessment of
Educational Progress) 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 SAT-Math
score increased from 501 to 514 and the mean
ACT-Math score increased from 19.9 to 21.0.
Trend in fourth-and-eighth 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
Average NAEP Scores for
NC 8th Graders
http://nces.ed.gov/nationsreportcard/subject/publications/stt2013/pdf/2014465NC8.pdf
NC EOG/EOC Percent
Solid or Superior Command (CCR)
Grade
2012-2013
2013-2014 2014-2015
3
46.8
48.2
48.8
4
47.6
47.1
48.5
5
47.7
50.3
51.3
6
38.9
39.6
41.0
7
38.5
39.0
40.0
8
34.2
34.6
35.8
Math I
42.6
46.9
48.5
http://www.ncpublicschools.org/accountability/reporting/
Although We Have Made Progress,
Challenges Remain
• The average mathematics NAEP score for 17year-olds 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.
Program for International Student
Assessment (PISA)
International and National Trends in Student
Performance
600
Scale Score
575
550
United States
525
OECD
500
Hong Kong
475
450
2000
2003
2006
2009
Year
2012
2015
Brainstorm
Administrators
& Leadership
Family
Students
Beliefs
about
Teaching
and
Learning
Teachers
Media
Community
Beliefs About Teaching and
Learning Mathematics
“Teachers’ beliefs
influence the decisions
they make about the
manner in which they
teach mathematics.”
“Students’ beliefs
influence their perception
of what it means to learn
mathematics and how
they feel toward the
subject.”
Examine the comic strip.
What do you see?
Principles to Actions pg. 10-11
Our Current Realities
• Too much focus is on learning procedures without any connection to
meaning, understanding, or the applications that require these
procedures.
• Too many students are limited by the lower expectations and narrow
curricula of remedial tracks from which few ever emerge.
• Too many teachers have limited access to the instructional materials,
tools, and technology that they need.
• Too much weight is placed on results from assessments – particularly
large-scale, high-stakes assessments – that emphasize skills and fact
recall and fail to give sufficient attention to problem solving and
reasoning.
• Too many teachers of mathematics remain professionally isolated,
without the benefit of collaborative structures and coaching, and with
inadequate opportunities for professional development related to
mathematics teaching and learning.
Principles to Actions pg. 3
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.
Teaching and
Learning
Guiding
Principles
for
School
Mathematics
Access and Equity
Curriculum
Tools and Technology
Assessment
Professionalism
Overarching Message
“Effective teaching is the non-negotiable
core that ensures that all students learn
mathematics at high levels.”
Principles to Actions (NCTM, 2014, p. 4)
Teaching and Learning Principle
“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 (NCTM, 2014, p. 7)
Why Focus on Teaching?
Student learning of mathematics
“depends fundamentally on what happens
inside the classroom as teachers and
learners interact over the curriculum.”
(Ball & Forzani, 2011, p. 17)
Ball, D. L, & Forzani, F. M. (2011). Building a common core for learning to teach, and
connecting professional learning to practice. American Educator, 35(2), 17-21.
Effective
Mathematics
Teaching Practices
Effective Mathematics Teaching 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.
Effective Mathematics Teaching 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.
Obstacles to Implementing
High-Leverage Instructional Practices
Many parents and
educators believe that
students should be taught
as they were taught,
through memorizing facts,
formulas, and procedures
and then practicing skills
over and over again.
Principles to Actions (NCTM, 2014, p. 9)
Let’s Do Some Math!
The Calling Plans Task
Long-distance company A charges a base rate of $5.00
per month plus 4 cents for each minute that you are on
the phone. Long-distance company B charges a base rate
of only $2.00 per month but charges you 10 cents for
every minute used.
How much time per month would you have to talk on the
phone before subscribing to company A would save you
money?
Create a phone plan, Company C, that costs the same as
Companies A and B at 50 minutes, but has a lower
monthly fee than either Company A or B.
The Calling Plans Task (MS)
The Context of Video Clip 1
Prior to the lesson:
• Students solved the Calling Plans Task – Part 1.
• The tables, graphs and equations they produced in
response to that task were posted in the classroom.
Video Clip 1 begins immediately after Mrs. Brovey
explained that students would be working on the
Calling Plans Task – Part 2 and read the problem to
students. Students first worked individually and
subsequently worked in small groups.
The Calling Plans Video
As you watch the video, make note of what the
teacher does to support student learning and
engagement as they work on the task.
Work with a partner. One of you will focus on the
teacher and the other will focus on the students.
o What is the teacher doing/saying?
o What are the students doing/saying?
Math
Teaching
Practice
1
Establish mathematics
goals to focus learning.
Formulating clear, explicit learning goals sets
the stage for everything else.
(Hiebert, Morris, Berk, & Janssen, 2007, p. 57)
Establishing mathematics goals to
focus learning.
As you read pgs. 12-14 and Table on pg. 16,
use the following symbols in your reading.
Symbol
Start
20
minute
Timer:
Meaning
✓
Confirms your thinking.
✕
Contradicts your thinking.
?
Raises questions you would like to discuss
with others.
*
Important…answers a question.
!
New, interesting or surprising
End
Table Talk
• Each person in the group share one
interesting thing that you read. No
responses at this time. You may want to
record or take notes to respond later.
• After everyone has shared, each person in
the group share a response to someone
else’s comment.
• Once everyone has responded, then you
may openly discuss.
Establish
mathematics
goals goals
Establish
mathematics
to focus learning
to focus learning
Learning Goals should:
• Clearly state what it is students are to learn
and understand about mathematics as the
result of instruction.
• Be situated within learning progressions.
• Frame the decisions that teachers make
during a lesson.
Daro, Mosher, & Corcoran, 2011; Hattie, 2009;
Hiebert, Morris, Berk, & Jensen., 2007; Wiliam, 2011
Reflecting on Calling
Plans Task
What learning goals can you create
for the task?
When complete, share with your
table. Each table needs to be
prepared to share 1 goal.
Mrs. Brovey’s
Mathematics Learning Goals
Students will understand that:
1. the point of intersection is a solution to each
equation (Companies A, B, and C);
2. the rate of change (cost per minute) determines
the steepness of the line;
3. if the y-intercept (monthly base rate) is lowered
then the rate of change (cost per minute) must
increase in order for the new equation to
intersect the other two at the same point.
Connections to the CCSS Content Standards
Define, Evaluate and Compare Functions
8.F
Uses Functions to Model Relationships Between Quantities
4. Construct a function to model a linear relationship between two
quantities. Determine the rate of change and initial value of the
function from a description of a relationship or from two (x, y)
values, including reading these from a table or from a graph.
Interpret the rate of change and initial value of a linear function in
terms of the situation it models, and in terms of its graph or a table
of values.
5. Describe qualitatively the functional relationship between two
quantities by analyzing a graph (e.g., where the function is
increasing or decreasing, linear or nonlinear). Sketch a graph that
exhibits the qualitative features of a function that has been
described verbally.
National Governors Association Center for Best Practices & Council of Chief State School Officers
(2010). Common core state standards for mathematics. Washington, DC: Authors
.
Mr. Torchon’s
Mathematics Learning Goals
Students will understand that:
1. The point of intersection is a solution to each
equation (Plans A and B);
2. A system of equations may be solved with a
table or a graph;
3. When the graph of one function is below the
graph of another function, the function with the
lower graph has y-values that are less than the
other function.
Connections to the CCSS Content
Standards
Reasoning with Equations and Inequalities
A-REI
Solve Systems of Equations
6. Solve systems of linear equations exactly and approximately
(e.g., with graphs), focusing on pairs of linear equations in two
variables.
Represent and solve equations and inequalities graphically
11. Explain why the x-coordinates of the points where the graphs of
the equations y = f(x) and y = g(x) intersect are the solutions of
★
the equation f(x) = g(x); find the solutions approximately.
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core
state standards for mathematics. Washington, DC: Authors.
Connections to the CCSS Content
Standards
Interpreting Functions
F-IF
Interpret functions that arise in applications in terms of the context
4. For a function that models a relationship between two quantities,
interpret key features of graphs and tables in terms of the quantities, and
sketch graphs showing key features given a verbal description of the
relationship. ★
5. Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. ★
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core
state standards for mathematics. Washington, DC: Authors.
Teaching Practices Jigsaw
1. Everyone grab a flip-flop cut out and
note-taking handout from your table.
2. Expert Groups: Move to find others
in the room with the same cut out.
3. Read your assigned practice. Come
to a shared understanding of the
teaching practice and be prepared to
share back at your table.
Start
10
minutes
End
Teaching Practice Reading
2.
3.
4.
5.
6.
7.
8.
Implement Tasks – Page 17
Use and Connect Representations – Page 24
Facilitate discourse – Page 29
Pose questions – Page 35
Start
Build procedural fluency – Page 42
10
minutes
Support productive struggle – Page 48
Elicit and use evidence – Page 53
End
Teaching Practices Share
1. Share Group: Now create a
group that has at least one of
every flip flop type.
2. Share important points about
each teaching practice.
3. Use the graphic organizer to take
notes on each practice.
Start
15
minutes
End
Math
Teaching
Practice
2
Implement tasks that
promote reasoning and
problem solving.
Student learning is greatest in classrooms where the
tasks consistently encourage high-level student
thinking and reasoning and least in classrooms where
the tasks are routinely procedural in nature.
(Boaler & Staples, 2008; Stein & Lane, 1996)
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.
Implement tasks that promote
reasoning and problem solving
Mathematical tasks should:
• Allow students to explore mathematical
ideas or use procedures in ways that are
connected to understanding concepts.
• Build on students’ current understanding
and experiences.
• Have multiple entry points.
• Allow for varied solution strategies.
Boaler & Staples, 2008; Hiebert et al., 1997;
Stein, Smith, Henningsen, & Silver, 2009
High or Low
Cognitive Demanding Task?
Low Cognitive Demand Tasks
High Cognitive Demand Tasks
Memorization
Procedures with
Connections
Procedures
without
Connections
Doing
Mathematics
Stein, Smith, Henningsen, & Silver (2000)
Cognitive Demand Sort
1. Individually study the Cognitive
Demand chart at your tables.
2. Grab a “tool” cut out from your
table
3. Find your group.
4. Using the chart as a guide, sort
the tasks in the envelope labeled
Task Sort by cognitive demand.
Start
10
minutes
End
Task
Implementation Student Learning
High
High
Low
Low
High
Low
High
Low
Moderate
Reflecting on Calling
Plans Task
Reflecting on your experience with the task
or the video, in what ways can the
implementation of the task allow for
multiple entry points and engage students
in reasoning and problem solving?
The Calling Plans 1 Task
The Context of Video Clip
• Students are working on the Calling Plans Task.
• Mr. Torchon is initially working with groups of
students, asking questions to help them focus on
their thinking and to progress toward a solution.
• The tables, graphs (and, in at least one case an
equation) students produce in response to the
task are then posted in the classroom and a whole
class discuss ensues.
Lens for Watching the HS Video
First Viewing
As you watch the video, make note of what the
teacher does to support student learning and
engagement as they work on the task.
In particular, identify any of the Effective
Mathematics Teaching Practices that you notice Mr.
Torchon using.
Be prepared to give examples and to cite line
numbers from the transcript to support your claims.
Math
Teaching
Practice
3
Use and connect
mathematical
representations.
Because of the abstract nature of
mathematics, people have access to
mathematical ideas only through the
representations of those ideas.
(National Research Council, 2001, p. 94)
Use and connect mathematical
representations
Different Representations should:
• Be introduced, discussed, and connected.
• Be used to focus students’ attention on the
structure of mathematical ideas by examining
essential features.
• Support students’ ability to justify and explain
their reasoning.
Lesh, Post, & Behr, 1987; Marshall, Superfine, & Canty, 2010;
Tripathi, 2008; Webb, Boswinkel, & Dekker, 2008
Important Mathematical Connections between
and within different types of representations
Visual
Physical
Contextual
Symbolic
Verbal
Principles to Actions (NCTM, 2014, p. 25)
(Adapted from Lesh, Post, & Behr, 1987)
Lens for Watching the Video
Second Viewing
As you watch the video this time, pay attention to the
ways in which representations are used and connected.
Specifically:
•
Does the task allow students to use multiple
representations?
•
What representations are students using?
•
In what ways are students making connections among the
mathematical representations?
•
In what ways are students contextualizing mathematical
ideas in real-world situations?
What mathematical
representations were
students working within the
lesson?
Physical
How did Mr. Torchon
support students in making
connections between and
within different types of
Contextual
representations?
Visual
Symbolic
Verbal
Math
Teaching
Practice
4
Facilitate meaningful
mathematical discourse.
Discussions that focus on cognitively challenging
mathematical tasks, namely those that promote
thinking, reasoning, and problem solving, are a
primary mechanism for promoting conceptual
understanding of mathematics.
(Hatano & Inagaki, 1991; Michaels, O’Connor, & Resnick, 2008)
Facilitate meaningful
mathematical discourse
Mathematical Discourse should:
• Build on and honor students’ thinking.
• Let students share ideas, clarify understandings,
and develop convincing arguments.
• Engage students in analyzing and comparing
student approaches.
• Advance the math learning of the whole class.
Carpenter, Franke, & Levi, 2003;
Fuson & Sherin, 2014; Smith & Stein, 2011
1. Anticipating
2. Monitoring
3. Selecting
4. Sequencing
5 Practices for
Orchestrating
Productive
Mathematics
Discussions
5. Connecting
(Smith & Stein, 2011)
Structuring Mathematical Discourse...
Given the Calling Plans Task, using the MS goal or the
HS goal, how would you structure the student
presentations to connect mathematical ideas and
encourage discourse?
• Selecting specific student representations and
strategies for discussion and analysis.
• Sequencing the various student approaches for
analysis and comparison.
• Connecting student approaches to key math ideas
and relationships.
Math
Teaching
Practice
5
Pose purposeful questions.
Teachers’ questions are crucial in helping students
make connections and learn important mathematics
and science concepts.
(Weiss & Pasley, 2004)
Pose purposeful questions
Effective Questions should:
• Reveal students’ current understandings.
• Encourage students to explain, elaborate, or
clarify their thinking.
• Make the targeted mathematical ideas more
visible and accessible for student examination
and discussion.
Boaler & Brodie, 2004; Chapin & O’Connor, 2007;
Herbel-Eisenmann & Breyfogle, 2005
The Calling Plans Task – Part 2
The Context of Video Clip 2
Following individual and small group work, Mrs. Brovey pulls
the class together for a whole group discussion. Several
different equations that satisfy the conditions of the problem
are offered by students. Jake, a student in the class then
proposed a theory that every time the rate increases by 1 cent
the base rate decreases by 50 cents. Mrs. Brovey records the
four possible phone plans for Company C (shown below) on
the board and ask the class what patterns they see.
C = .14m
C = .13m + $ .50
C = .12m + $1.00
C = .11m + $1.50
Video Clip 2 focuses on the discussion between teacher and
students regarding the patterns they notice.
Lens for Watching the MS
Video Clip 2
As you watch the second video of the MS this time, pay
attention to the questions the teacher asks. Specifically:
• What do the questions reveal about students’ current
understandings?
• To what extent do the questions encourage students
to explain, elaborate, or clarify their thinking?
• To what extent to the questions make mathematics
more visible and accessible for student examination
and discussion?
• How are the questions in this clip similar to or
different from the questions asked in video clip 1?
NCTM’s Core Set of
Effective Mathematics Teaching
Practices
“Although the important work of teaching is not
limited to the eight Mathematics Teaching
Practices, this core set of research-informed
practices is offered as a framework for
strengthening the teaching and learning of
mathematics.”
Principles to Actions (NCTM, 2014, p. 57)
Math Goals
What might be the math learning goals?
Tasks &
Representations
What representations might students use in
reasoning through and solving the problem?
Discourse &
Questions
How might we question students and structure
class discourse to advance student learning?
Fluency from
Understanding
How might we develop student understanding
to build toward aspects of procedural fluency?
Struggle &
Evidence
How might we check in on student thinking
and struggles and use it to inform instruction?
Reflections
and Next Steps
Start Small, Build Momentum,
and Persevere
The process of creating a new cultural norm
characterized by professional collaboration,
openness of practice, and continual learning
and improvement can begin with a single
team of grade-level or subject-based
mathematics teachers making the
commitment to collaborate on a single
lesson plan.
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
– Group Discussion about next steps
Principles to Actions
As you reflect on this framework for the
teaching and learning of mathematics, identify
1-2 mathematics teaching practices that want
to begin strengthening in your own instruction.
Working with a partner, develop a list of
actions to begin the next steps of your journey
toward ensuring mathematical success for all
of your students.
Food for thought….
① We are being asked to teach in distinctly
different ways from how we were taught.
② The traditional curriculum was designed to
meet societal needs that no longer exist.
③ It is unreasonable to ask a professional to
change more than 10 percent a year, but it is
unprofessional to change by much less than
10 percent a year.
④ If you don’t feel inadequate, you’re probably
not doing the job.
http://steveleinwand.com/wp-content/uploads/2014/08/FourPostulatesforChange.pdf
NCCTM Fall Leadership Conference
November 4th
Koury Convention Center
Featuring:
Daniel Brahier, lead author for NCTM’s Principles to Actions
Diane Briars, President of NCTM
Jon Wray, Howard County Public Schools
45th Annual State Math Conference
Principles to Actions in Action
November 5th and 6th
Koury Convention Center
Greensboro, NC
Anyone here being honored as their district’s NCCTM
Outstanding Elementary Mathematics Educator?
2013 PAEMST
Math State Winner
What questions do
you have?
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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
919-807-3842
denise.schulz@dpi.nc.gov
Lisa Ashe
Secondary Mathematics Consultant
919-807-3909
lisa.ashe@dpi.nc.gov
Joseph Reaper
Secondary Mathematics Consultant
919-807-3691
joseph.reaper@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!