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Principles to Actions: Ensuring Mathematical Success For All Kitty Rutherford & Denise Schulz NC DPI Mathematics Section Fall 2015

Welcome “Who’s in the Room?”

Norms Listen as an Ally Value Differences Maintain Professionalism Participate Actively

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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 The percent of 8th graders scoring proficient or above on NAEP rose from 15% in 1990 to 36% in 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

North Carolina NAEP Trends in Mathematics GradeSource Change 4NC223254Up 31 4US227250Up 23 8NC250286Up 36 8US262284Up 22 NAEP Scale Score 1990 –First year NAEP reported NC Scores 2013 – Latest NC NAEP Test Data

NC EOG/EOC Percent Solid or Superior Command (CCR) Grade Math I

Although We Have Made Progress, Challenges Remain The average mathematics NAEP score for 17- year-olds has been essentially flat since 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

Principles to Actions pg 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.” Examine the comic strip. What do you see? “Teachers’ beliefs influence the decisions they make about the manner in which they teach mathematics.”

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.

“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.” Principles to Actions: Ensuring Mathematical Success for All NCTM. (2014). Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: NCTM.

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 “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

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

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

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.

Not to be confused with…

What do you notice?

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

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?

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

Principles to Action – pg. 16

Reflect On Your Role Reflecting on this practice, what would be your role in the implementation of this practice? –Teacher –Administrator –Coach –District Level

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? High Low

Principles to Actions pg. 18

Cognitive Demand Sort 1.Read page 18 and summarize the description associated with each cognitive demand task type. –Memorization –Procedures without Connections –Procedures with Connections –Doing Mathematics 2.Come to a shared understanding of the demand task. 3.Then, use the contents of the envelope to sort the tasks by cognitive demand.

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

TaskImplementation Student Learning

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.” Lappan & Briars, 1995

Look on page 21 NCDPI – Task

Principles to Action - page 24

Reflect On Your Role Reflecting on this practice, what would be your role in the implementation of this practice? –Teacher –Administrator –Coach –District Level

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)

1.Read your assigned teaching practice. 2.Find the others in the room with the same color ticket. 3.Come to a shared understanding of the teaching practice. 4.Create a chart with: –Discussion –Illustration –Teacher and student actions 5.Be prepared to share with your table during a gallery walk.

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!

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

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

What might be the math learning goals?

Math Teachin g Practic e 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)

Math Teachin g Practic e 1 Establish mathematics goals 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

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. 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.

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.

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:

MR. HARRIS 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.

Group Questions As a group, answer the questions regarding Mr. Harris and the teaching practices. Be prepared to share with the whole group.

What were the math expectations for student learning? In what ways did these math goals focus the teacher’s interactions with students throughout the lesson? Consider Case Lines 4-9, 21-24, Questions

Implement tasks that promote reasoning and problem solving. Math Teachin g Practic e 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)

Boaler & Staples, 2008; Hiebert et al., 1997; Stein, Smith, Henningsen, & Silver, 2009 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.

Questions In what ways did the implementation of the task allow for multiple entry points and engage students in reasoning and problem solving? Consider Case Lines &

Use and connect mathematical representations. 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)

Lesh, Post, & Behr, 1987; Marshall, Superfine, & Canty, 2010; Tripathi, 2008; Webb, Boswinkel, & Dekker, 2008 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.

Use and connect mathematical representations. Illustrate, show, or work with mathematical ideas using diagrams, pictures, number lines, graphs, and other math drawings. Record or work with mathematical ideas using numerals, variables, tables, and other symbols. Use language to interpret, discuss, define, or describe mathematical ideas, bridging informal and formal mathematical language. Situate mathematical ideas in everyday, real-world, imaginary, or mathematical situations and contexts. Use concrete objects to show, study, act upon, or manipulate mathematical ideas (e.g., cubes, counters, tiles, paper strips).

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

What mathematical representations were students working with in the lesson? How did Mr. Harris support students in making connections between and within different types of representations? Contextual Physical Visual Symboli c Verbal

JasmineKenneth Consider Lines In what ways did comparing representations strengthen the understanding of these students?

Molly Consider Lines How did comparing representations benefit Molly?

Facilitate meaningful mathematical discourse. Math Teachin g Practic e 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)

Carpenter, Franke, & Levi, 2003; Fuson & Sherin, 2014; Smith & Stein, 2011 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.

“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)

5 Practices for Orchestrating Productive Mathematics Discussions 1.Anticipating 2.Monitoring 3.Selecting 4.Sequencing 5.Connecting

Questions How did Mr. Harris structure the whole class discussion (lines 52-57) to advance student learning toward the intended math learning goals?

Jasmine Kenneth Teresa Consider Lines Why did Mr. Harris select and sequence the work of these three students and how did that support student learning?

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.

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

Boaler & Brodie, 2004; Chapin & O’Connor, 2007; Herbel-Eisenmann & Breyfogle, 2005 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.

Questions In what ways did Mr. Harris’ questioning on lines assess and advance student learning about important mathematical ideas and relationships?

Purposeful Questions Lines “How does your drawing show 7 rows?” “How does your drawing show that there are 20 chairs in each row? “How many twenties are you adding, and why?” “Why are you adding all those twenties? Math Learning Goal Students will recognize the structure of multiplication as equal groups within and among different representations— identify the number of equal groups and the size of each group within collections or arrays. Math Learning Goal Students will recognize the structure of multiplication as equal groups within and among different representations— identify the number of equal groups and the size of each group within collections or arrays.

Reflect On Your Role Reflecting on this practice, what would be your role in the implementation of this practice? –Teacher –Administrator –Coach –District Level

Build procedural fluency from conceptual understanding. Math Teachin g Practic e 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)

Baroody, 2006; Fuson & Beckmann, 2012/2013; Fuson, Kalchman, & Bransford, 2005; Russell, 2006 Build procedural fluency from conceptual 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.

Questions In what ways did this lesson develop a foundation of conceptual understanding for building toward procedural fluency in multiplying with multiples of ten?

What foundational understandings were students developing at each of these points in the lesson that are critical for moving toward procedural fluency? Questions Lines 59-69:Discussion of skip counting. Lines 70-76:Wrote the multiplication equation. Lines 78-81:Asked students to compare Tyrell and Ananda’s work.

TyrellAnanda Discuss ways to use this student work to develop the understanding that 14 tens = 140 and to meaningfully to build toward fluency in working with multiples of ten.

TyrellAnanda Discuss ways to use this student work to develop informal ideas of the distributive property—how numbers can be decomposed, combined meaningfully in parts, and then recomposed to find the total.

Principles to Actions (NCTM, 2014, p. 42) “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.”

Support productive struggle in learning mathematics. Math Teachin g Practic e 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)

Black, Trzesniewski, & Dweck, 2007; Dweck, 2008; Hiebert & Grouws, 2007; Kapur, 2010; Warshauer, 2011 Support productive struggle in learning mathematics. 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.

Questions How did Mr. Harris support productive struggle among his students, individually and collectively, as they grappled with important mathematical ideas and relationships? At which points in the lesson might Mr. Harris have consciously restrained himself from “taking over” the thinking of his students?

“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

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

What is the central message about productive struggle and student learning?

Additional Resource on Mindset how-to-normalize-mistake-making-and-struggle-in-class/

Elicit and use evidence of student thinking. Math Teachin g Practic e 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)

Chamberlin, 2005; Jacobs, Lamb, & Philipp, 2010; Sleep & Boerst, 2010; van Es, 2010’ Wiliam, 2007 Elicit and use evidence of student thinking. 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.

Questions Identify specific places during the lesson in which Mr. Harris elicited evidence of student learning. Discuss how he used or might use that evidence to adjust his instruction to support and extend student learning.

Examples of Eliciting and Using Evidence Throughout the lesson, Mr. Harris was eliciting and using evidence of student thinking. Lines 33-36:Purposeful questioning as students worked individually. Lines 43-51: Observations of student pairs discussing and comparing their representations. Lines 59-74: Whole class discussion. Lines 78-80:Asked students to respond in writing.

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

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? “My Favorite No: Learning From Mistakes”

Reflect On Your Role Reflecting on this practice, what would be your role in the implementation of this practice? –Teacher –Administrator –Coach –District Level

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

Principles to Action – pg. 11 Beliefs About Teaching and Learning Mathematics

Essential Elements of Effective Mathematics Programs

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 –Principals, coaches, specialists, other school leaders pgs –Teachers pgs –Group Discussion about next steps

For Next Time (December 1) 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.

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

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

45 th Annual State Math Conference Principles to Actions in Action November 5 th and 6 th Koury Convention Center Greensboro, NC Anyone here being honored as their district’s NCCTM Outstanding Elementary Mathematics Educator?

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

What questions do you have?

Follow Us! NC Mathematics

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

For all you do for our students!