1. Good morning…
Please complete the Survey that can be found on your table.
Principles to Actions, p. 10

2. Principles to Actions: Ensuring Mathematical Success For All
Kitty Rutherford & Denise Schulz
NC DPI Mathematics Section
Spring 2016

3. Welcome “Who’s in the Room?”

4. Norms Listen as an Ally Value Differences Maintain Professionalism
Participate Actively
Thumbs up if you agree with these norms. Are there other norms we need to add so that we have the best possible learning experience for all?

5. maccss.ncdpi.wikispaces.net

6. A 25-year History of Standards-Based Mathematics Education Reform

7. 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 SAT-Math score increased from 501 to 514 and the mean ACT-Math score increased from 19.9 to 21.0.

8. Trend in fourth-and-eighth grade NAEP Mathematics Average Scores

9. North Carolina NAEP Trends in Mathematics
Grade
Source
1990
2013
Change 4 NC
223
254
Up 31 US
227
250
Up 23 8
286
Up 36
262
284
Up 22
NAEP Scale Score
1990 –First year NAEP reported NC Scores
2013 – Latest NC NAEP Test Data

10. NC EOG/EOC Percent Solid or Superior Command (CCR)
Grade 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

11. Although We Have Made Progress, Challenges Remain
The average mathematics NAEP score for 17- year-olds has been essentially flat since 1973.
Among 34 countries participating in the 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 and 2012, the U.S. mean score declined.
Significant learning differentials remain.

12. Brainstorm Community Students Media Teachers Family
Beliefs about Teaching and Learning
Community
Students
Media
Administration and Leadership
Teachers
Family
What do you notice about the beliefs of the six groups? Is it positive? Negative? What are some beliefs that are part of each

13. 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?
Pass out books
Principles to Actions, p. 10
Principles to Actions pg

14. Kyle’s Beliefs Nobody likes math!
Make my appointment for 3 o’clock - that’s when we have math
We have it at the end if the day so if we don’t get to it, it is okay!
Teachers’ beliefs influence the decisions that they make about the manner in which they teach mathematics.
Students’ beliefs influence their perception of what it means to learn mathematics and their dispositions towards the subject.
Teachers belief influence the decisions that they make about the manner in which they teach mathematics. Student’s beliefs influence their perception of what it means to learn mathematics and their dispositions towards the subject.

15. 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

16. 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.

17. 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 high-quality mathematics education.”
NCTM. (2014). Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: NCTM.

18. Effective Mathematics Programs
Teaching and Learning
Access and Equity
Curriculum
Tools and Technology
Assessment
Professionalism
Although such teaching and learning form the nonnegotiable core of successful mathematics programs, they are part of a system of essential elements of excellent mathematics programs. Consistent implementation of effective teaching and learning of mathematics, as previously described in the eight Mathematics Teaching Practices, are possible only when school mathematics programs have in place—
a commitment to access and equity;
a powerful curriculum;
appropriate tools and technology;
meaningful and aligned assessment; and
a culture of professionalism.

19. 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

20. 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

21. 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.”
Principles to Actions pg. 9

22. Mathematics Teaching Practices
Establish mathematics goals to focus learning
Implement tasks that promote reasoning and problem solving
Use and connect mathematical representations
Facilitate meaningful mathematical discourse
Pose purposeful questions
Build procedural fluency from conceptual understanding
Support productive struggle in learning mathematics
Elicit and use evidence of student thinking

23. Mathematics Teaching Practices
Establish mathematics goals to focus learning.
Implement tasks that promote reasoning and problem solving.
Use and connect mathematical representations.
Facilitate meaningful mathematical discourse.
Pose purposeful questions.
Build procedural fluency from conceptual understanding.
Support productive struggle in learning mathematics.
Elicit and use evidence of student thinking.

24. Not to be confused with…

25. What do you notice?

26. 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

27. Role Play Choose a puzzle piece from the center of the table.
Find your group members.
In your group, role play the scenario on pgs
What do you notice about the dialog?
Use puzzle pieces/tickets to break into group of 6.
3 read and 3 observe

28. What did you notice about the dialog?
“The math coach intentionally shifts the conversation to a discussion of the mathematical ideas and learning that will be the focus of instruction.”
Principles to Actions pg. 14

29. Principles to Action – pg. 16
Now, let’s think about the standards for mathematical practice.
What do you notice?
Principles to Action – pg. 16

31. 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

32. High or Low Cognitive Demanding Task?

34. Cognitive Demand Sort
Read page 18 and summarize the description associated with each cognitive demand task type.
Memorization
Procedures without Connections
Procedures with Connections
Doing Mathematics
Come to a shared understanding of the demand task.
Then, use the contents of the envelope to sort the tasks by cognitive demand.

35. What are the attributes of a mathematically strong task?
Table Talk
What are the attributes of a mathematically strong task?

36. Task Implementation Student Learning
High
Low
High
Low
Moderate

37. 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.”

38. Look on page 21
NCDPI – Task

39. Principles to Action - page 24

41. Teaching Practices Jigsaw
Select a ticket from your table.
Green-Use and connect mathematical representations (pg. 24)
Red-Facilitate meaningful mathematical discourse (pg. 29)
Purple-Pose purposeful questions (pg. 35)
Yellow-Build procedural fluency from conceptual understanding (pg. 42)
Pink-Support productive struggle in learning mathematics (pg. 48)
Blue-Elicit and use evidence of student thinking (pg. 53)
Make a chart

42. Read your assigned teaching practice.
Find the others in the room with the same color ticket.
Come to a shared understanding of the teaching practice.
Create a chart with:
Discussion
Illustration
Teacher and student actions
Be prepared to share with your table during a gallery walk.

43. Gallery Walk
With your table group, take a walk to each practice poster.
If you are the expert on that practice, explain to your group what the practice is about, pointing out key ideas.
Take your book with you so you can make notes!

44. Tasks & Representations Fluency from Understanding
What might be the math learning goals?
Math Goals
What representations might students use in reasoning through and solving the problem?
Tasks & Representations
How might we question students and structure class discourse to advance student learning?
Discourse & Questions
How might we develop student understanding to build toward aspects of procedural fluency?
Fluency from Understanding
How might we check in on student thinking and struggles and use it to inform instruction?
Struggle & Evidence

45. Let’s Do Some Math!
The third grade class is responsible for setting up the chairs for the spring band concert. In preparation, the class needs to determine the total number of chairs that will be needed and ask the school’s engineer to retrieve that many chairs from the central storage area. The class needs to set up 7 rows of chairs with 20 chairs in each row, leaving space for a center aisle. How many chairs does the school’s engineer need to retrieve from the central storage area?
Principles to Actions pg. 27

46. What might be the math learning goals?

47. Establish mathematics goals to focus learning.
What might be the math learning goals?
Math Goals
Establish mathematics goals to focus learning.
Math
Teaching
Practice 1
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

48. Establish mathematics goals to focus learning.
Teaching
Practice 1
Formulating clear, explicit learning goals sets the stage for everything else.
(Hiebert, Morris, Berk, & Janssen, 2007, p. 57)

49. Mr. Harris’s Math Goals
Students will recognize the structure of multiplication as equal groups within and among different representations, focusing on identifying the number of equal groups and the size of each group within collections or arrays.
How did your goals differ?
Student Friendly Version:
We are learning to represent and solve word problems and explain how different representations match the story situation and the math operations.

50. Alignment to the Standards
Standard 3.OA.3. Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
Standard 3.NBT. 3. Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 x 80, 5 x 60) using strategies based on place value and properties of operations.

51. Table Talk Think back to the Band Concert Task.
Review the student work samples.
With the mathematical goal in mind, determine which students should present a solution, and in what order the solutions should be presented.
What questions should be asked to connect solutions?
Student Work:
Connections:
Refer to example on pg 27 showing students moving through representations

52. The Case of Mr. Harris and the Band Concert Task
Read the Case of Mr. Harris and study the strategies used by his students.
Make note of what Mr. Harris did before or during instruction to support his students’ developing understanding of multiplication.
Talk with a neighbor about the “Teaching Practices” Mr. Harris is using and how they support students’ progress in their learning.

53. Stations: Group Discussion Questions
Using your station number card, travel to the appropriate station for group discussion.
Your group can have a large group discussion, or several smaller group discussions.
You will transition to your next station when you hear the music

54. Tasks & Representations
What representations might students use in reasoning through and solving the problem?
Tasks & Representations
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

55. Implement tasks that promote reasoning and problem solving.
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)

56. Tasks & Representations
What representations might students use in reasoning through and solving the problem?
Tasks & 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

57. Use and connect mathematical representations.
Teaching
Practice 3
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)

58. Use and connect mathematical representations.
Illustrate, show, or work with mathematical ideas using diagrams, pictures, number lines, graphs, and other math drawings.
Use concrete objects to show, study, act upon, or manipulate mathematical ideas (e.g., cubes, counters, tiles, paper strips).
Record or work with mathematical ideas using numerals, variables, tables, and other symbols.
Solve the problem
How did you solve the problem? Find the type of representation and gather
Compare your work with your group.
Find someone from another group and compare how you each solved it
What do you notice about how most people solved it?
Situate mathematical ideas in everyday, real-world, imaginary, or mathematical situations and contexts.
Use language to interpret, discuss, define, or describe mathematical ideas, bridging informal and formal mathematical language.

59. Important Mathematical Connections between and within different types of representations
Contextual
Physical
Visual
Symbolic
Verbal
Principles to Actions (NCTM, 2014, p. 25)
(Adapted from Lesh, Post, & Behr, 1987)

60. Mathematical Discourse should: Build on and honor students’ thinking.
How might we question students and structure class discourse to advance student learning?
Discourse & Questions
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

61. Facilitate meaningful mathematical discourse
Teaching
Practice 4
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)

62. “What students learn is intertwined with how they learn it
“What students learn is intertwined with how they learn it. And the stage is set for the how of learning by the nature of classroom-based interactions between and among teacher and students.”
(Smith & Stein, 2011)
Teachers need to develop a range of ways of interacting with and engaging students as they work on tasks and share their thinking with other students.

63. 5 Practices for Orchestrating Productive Mathematics Discussions
Anticipating
Monitoring
Selecting
Sequencing
Connecting
Group jigsaw
Reference page 30

64. Structuring Mathematical Discourse
During the whole class discussion of the task, Mr. Harris was strategic in:
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.

65. Effective Questions should: Reveal students’ current understandings.
How might we question students and structure class discourse to advance student learning?
Discourse & 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

66. Pose purposeful questions.
Math
Teaching
Practice 5
Teachers’ questions are crucial in helping students make connections and learn important mathematics and science concepts.
(Weiss & Pasley, 2004)

67. Fluency from Understanding
How might we develop student understanding to build toward aspects of procedural fluency?
Fluency from Understanding
Procedural Fluency should:
Build on a foundation of conceptual understanding.
Over time (months, years), result in known facts and generalized methods for solving problems.
Enable students to flexibly choose among methods to solve contextual and mathematical problems.
Baroody, 2006; Fuson & Beckmann, 2012/2013; Fuson, Kalchman, & Bransford, 2005; Russell, 2006

68. Build procedural fluency from conceptual understanding.
Math
Teaching
Practice 6
Build procedural fluency from conceptual understanding.
A rush to fluency undermines students’ confidence and interest in mathematics and is considered a cause of mathematics anxiety.
(Ashcraft 2002; Ramirez Gunderson, Levine, & Beilock, 2013)

69. “Fluency builds from initial exploration and discussion of number concepts to using informal reasoning strategies based on meanings and properties of the operations to the eventual use of general methods as tools in solving problems.”
Principles to Actions (NCTM, 2014, p. 42)

71. Productive Struggle should:
How might we check in on student thinking and struggles and use it to inform instruction?
Struggle & Evidence
Productive Struggle should:
Be considered essential to learning mathematics with understanding.
Develop students’ capacity to persevere in the face of challenge.
Help students realize that they are capable of doing well in mathematics with effort.
Black, Trzesniewski, & Dweck, 2007; Dweck, 2008; Hiebert & Grouws, 2007; Kapur, 2010; Warshauer, 2011

72. Support productive struggle in learning mathematics.
Teaching
Practice 7
Support productive struggle in learning mathematics.
The struggle we have in mind comes from solving problems that are within reach and grappling with key mathematical ideas that are comprehendible but not yet well formed.
(Hiebert, Carpenter, Fennema, Fuson, Human, Murray, Olivier, & Wearne, 1996)

73. 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 spent 3/5 of his birthday money at the mall on video games and food. How much money did he spend? How much money had he received for his birthday?
Principles to Actions pg. 51

74. How Would You Approach This with Students?

75. Teacher A:
Tells students to draw a rectangle and shows them how to divide it into fifths to represent what Joseph had spent and what he had left.
Then, guides students step by step until they have labeled each one-fifth as worth $12.
Finally, she tells the students to use the information in the diagram to figure out the answers to the questions.

76. Teacher B:
Asks students to write down two things that they know about the problem and one thing that they wish they knew because it would help them make progress in solving the problem.
Initiates a discussion in which several ideas are offered for what to do next.
Encourages students to consider the various ideas that have been shared as they continue working on the task.

77. How do these approaches differ? What have students learned?

78. Teacher A wants students to be successful in answering, and begins to direct the work.
Teacher B resists the temptation to step in, but instead supports students in considering what they know and what they need to figure out.
Students in these classrooms have very different opportunities to learn.

79. “Teachers greatly influence how students perceive and approach struggle in the mathematics classroom. Even young students can learn to value struggle as an expected and natural part of learning.”
Principles to Actions pg. 50

80. Fixed vs. Growth Mindset
Fixed: those who believe intelligence is an innate trait; believe that learning should come naturally
Growth: those who believe intelligence can be developed through effort; likely to persevere through struggle because they see challenging work as an opportunity to learn and grow
Principles to Actions pg. 50

81. What is the central message in this video about productive struggle and student learning?
What is the central message about productive struggle and student learning?

82. Additional Resource on Mindset

83. Provide a window into students’ thinking.
How might we check in on student thinking and struggles and use it to inform instruction?
Struggle & Evidence
Evidence should:
Provide a window into students’ thinking.
Help the teacher determine the extent to which students are reaching the math learning goals.
Be used to make instructional decisions during the lesson and to prepare for subsequent lessons.
Chamberlin, 2005; Jacobs, Lamb, & Philipp, 2010;
Sleep & Boerst, 2010; van Es, 2010’ Wiliam, 2007

84. Elicit and use evidence of student thinking.
Math
Teaching
Practice 8
Elicit and use evidence
of student thinking.
Teachers using assessment for learning continually look for ways in which they can generate evidence of student learning, and they use this evidence to adapt their instruction to better meet their students’ learning needs.
(Leahy, Lyon, Thompson, & Wiliam, 2005, p. 23)

85. Preparation of each lesson needs to include intentional and systematic plans to elicit evidence that will provide “a constant stream of information about how student learning is evolving toward the desired goal.”
Principles to Actions pg. 53

86. “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?
1. Give each student an index card and ask them to individually answer the following problem. Create an array for 4 x 12 and then show the math sentences that could be used to solve it.
2. Each student responds on an index card.
3. Quickly collect the cards and sort by Yes (correct) and No (incorrect) answers. Don't let the students see how others did but announce each yes and no. Select an incorrect response that can promote a strong discussion about the problem.
4. Copy the problem on the board or on another card under the document camera.
5. Ask the children to find what is right about what this person did. Discuss.
6. Ask the children what did this person do incorrectly? Discuss. Both discussions should form a review of the meaning of multiplication, as well as give you further information on who may struggle with this problem

87. Community Students Media Teachers Family Administration and Leadership
Beliefs about Teaching and Learning
Community
Students
Media
Administration and Leadership
Teachers
Family

88. Unproductive Beliefs Sort the Beliefs
Check your arrangement on Principles to Actions pg. 11

89. Beliefs About Teaching and Learning Mathematics
These beliefs should not be viewed as good or bad. Beliefs should be understood as unproductive when they hinder the implementation of effective instructional practice or limit student access to important mathematics content and practices.
Principles to Action – pg. 11

90. Essential Elements of Effective Mathematics Programs
Teaching and Learning
Access and Equity
Curriculum
Tools and Technology
Assessment
Professionalism

91. 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.

92. Principles to Actions What action are you taking? Your role:
Leaders and policymakers pgs
Principals, coaches, specialists, other school leaders pgs
Teachers pgs
Group Discussion with your district to talk about next steps

93. For Next Time (February)
Think about the needs of your school/district.
Create a plan of action for implementing Teaching Practices.
Be prepared to share your plan and outcome with this group.

94. Next Steps As A Group
What would you like to see evolve in our follow-up session?

96. 2014 PAEMST Math State Finalists
Elementary
Kayonna Pitchford
Heather Landreth Meredith Stanley

97. What questions do you have?

98. Follow Us! NC Mathematics www.facebook.com/NorthCarolinaMathematics

99. DPI Mathematics Section
Kitty Rutherford
Elementary Mathematics Consultant
Denise Schulz
Lisa Ashe
Secondary Mathematics Consultant
Joseph Reaper
Dr. Jennifer Curtis
K – 12 Mathematics Section Chief
Susan Hart
Mathematics Program Assistant 99

100. For all you do for our students!