Good morning Please complete the Survey that can be found on your table. Principles to Actions, p. 10 Principles to Actions: Ensuring Mathematical Success For All
Kitty Rutherford & Denise Schulz NC DPI Mathematics Section Spring 2016 Welcome Whos in the Room? 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? maccss.ncdpi.wikispaces.net 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 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
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 NCNAEP Test Data 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 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 inmathematics. While many countries have increased their meanscores on the PISA assessments between and 2012, the U.S. mean score declined. Significant learning differentials remain. Brainstorm Community Students Media Teachers Family
Beliefs about Teaching and Learning Community Students Media Administration and Leadership Teachers Family What doyou notice about the beliefs of the six groups? Is it positive? Negative? What are some beliefs that are part of each 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 Actionspg Kyles Beliefs Nobody likes math!
Make my appointment for 3 oclock- thats when we have math We have it at the end if the day so if we dont 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.Students beliefs influence their perception of what it means to learn mathematics and their dispositions towards the subject. 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 Actionspg. 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. 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. 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. High-Quality Standards are Necessary,But Insufficient, for Effective Teachingand 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 Actionspg. 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 Actionspg. 9 Mathematics Teaching Practices
Establish mathematics goals to focus learning Implement tasks that promote reasoning and problemsolving Use and connect mathematical representations Facilitate meaningful mathematical discourse Pose purposeful questions Build procedural fluency from conceptualunderstanding Support productive struggle in learning mathematics Elicit and use evidence of student thinking Mathematics Teaching Practices
Establish mathematics goals to focus learning. Implement tasks that promote reasoning andproblem solving. Use and connect mathematical representations. Facilitate meaningful mathematical discourse. Pose purposeful questions. Build procedural fluency from conceptualunderstanding. Support productive struggle in learning mathematics. 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 Actionspg. 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? Use puzzle pieces/tickets to break into group of 6. 3 read and 3 observe 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 Actionspg. 14 Principles to Action pg. 16
Now, lets think about the standards for mathematical practice. What do you notice? Principles to Action pg. 16 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 Actionspg. 17 High or Low Cognitive Demanding Task? 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. What are the attributes of a mathematically strong task?
Table Talk What are the attributes of a mathematically strong task? Task Implementation Student Learning
High Low High Low Moderate 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 Principles to Action - page 24 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 conceptualunderstanding (pg. 42) Pink-Support productive struggle in learning mathematics(pg. 48) Blue-Elicit and use evidence of student thinking (pg. 53) Make a chart 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. 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! 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 thinkingand struggles and use it to inform instruction? Struggle & Evidence Lets 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 schools 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 schools engineer need to retrieve from the central storage area? Principles to Actionspg. 27 What might be the math learning goals? Establish mathematics goals to focus learning.
What might be the math learning goals? Math Goals Establish mathematicsgoals 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 Establish mathematics goals to focus learning.
Teaching Practice 1 Formulating clear, explicit learninggoals sets the stage for everythingelse. (Hiebert, Morris, Berk, & Janssen, 2007, p. 57) Mr. Harriss 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. 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 1090 (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: Refer to example on pg 27 showing students moving through representations 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. 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 Tasks & Representations
What representations might students use in reasoning through and solving the problem? Tasks & Representations Mathematical tasks should: Allow students to explore mathematical ideasor use procedures in ways that are connectedto 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 Implement tasks that promote reasoning and problem solving.
Math Teaching Practice 2 Implement tasks that promote reasoning and problem solving. Student learning is greatest inclassrooms where the tasksconsistently encourage high-levelstudent thinking and reasoning andleast in classrooms where the tasksare routinely procedural in nature. (Boaler & Staples, 2008; Stein & Lane, 1996) 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 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) 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. 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) 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, clarifyunderstandings, and develop convincingarguments. Engage students in analyzing and comparingstudent approaches. Advance the math learning of the whole class. Carpenter, Franke, & Levi, 2003; Fuson & Sherin, 2014; Smith & Stein, 2011 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, OConnor, & Resnick, 2008) 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. 5 Practices for Orchestrating Productive Mathematics Discussions
Anticipating Monitoring Selecting Sequencing Connecting Group jigsaw Reference page 30 Structuring Mathematical Discourse
During the whole class discussionof the task, Mr. Harris was strategic in: Selecting specific student representationsand strategies for discussion and analysis. Sequencing the various student approachesfor analysis and comparison. Connecting student approaches to key math ideas and relationships. 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, orclarify their thinking. Make the targeted mathematical ideas morevisible and accessible for student examinationand discussion. Boaler & Brodie, 2004; Chapin & OConnor, 2007;Herbel-Eisenmann & Breyfogle, 2005 Pose purposeful questions.
Math Teaching Practice 5 Teachers questions are crucial inhelping students make connectionsand learn important mathematicsand science concepts. (Weiss & Pasley, 2004) 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 conceptualunderstanding. Over time (months, years), result in knownfacts and generalized methods for solvingproblems. Enable students to flexibly choose amongmethods to solve contextual andmathematical problems. Baroody, 2006; Fuson & Beckmann, 2012/2013;Fuson, Kalchman, & Bransford, 2005; Russell, 2006 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) 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) Productive Struggle should:
How might we check in on student thinkingand struggles and use it to inform instruction? Struggle & Evidence Productive Struggle should: Be considered essential to learningmathematics with understanding. Develop students capacity to persevere inthe face of challenge. Help students realize that they are capableof doing well in mathematics with effort. Black, Trzesniewski, & Dweck, 2007; Dweck, 2008;Hiebert & Grouws, 2007; Kapur, 2010; Warshauer, 2011 Support productive struggle in learning mathematics.
Teaching Practice 7 Support productive strugglein learning mathematics. The struggle we have in mind comes fromsolving problems that are within reachand grappling with key mathematicalideas that are comprehendible but not yetwell formed. (Hiebert, Carpenter, Fennema, Fuson, Human, Murray, Olivier, &Wearne, 1996) 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 Actionspg. 51 How Would You Approach This with Students? 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. 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. How do these approaches differ? What have students learned? 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. 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 Actionspg. 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 Actionspg. 50 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? Additional Resource on Mindset
Provide a window into students thinking.
How might we check in on student thinkingand struggles and use it to inform instruction? Struggle & Evidence Evidence should: Provide a window into students thinking. Help the teacher determine the extent towhich students are reaching the math learninggoals. Be used to make instructional decisionsduring the lesson and to prepare forsubsequent lessons. Chamberlin, 2005; Jacobs, Lamb, & Philipp, 2010; Sleep & Boerst, 2010; van Es, 2010Wiliam, 2007 Elicit and use evidence of student thinking.
Math Teaching Practice 8 Elicit and use evidence of student thinking. Teachers using assessment for learningcontinually look for ways in which theycan generate evidence of studentlearning, and they use this evidence toadapt their instruction to better meettheir students learning needs. (Leahy, Lyon, Thompson, & Wiliam, 2005, p. 23) 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 Actionspg. 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? 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 Community Students Media Teachers Family Administration and Leadership
Beliefs about Teaching and Learning Community Students Media Administration and Leadership Teachers Family Unproductive Beliefs Sort the Beliefs
Check your arrangement on Principles to Actionspg. 11 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 Essential Elements of Effective Mathematics Programs
Teaching and Learning Access and Equity Curriculum Tools and Technology Assessment Professionalism 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 with yourdistrict to talk about next steps 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. Next Steps As A Group What would you like to see evolve in our follow-up session? 2014 PAEMST Math State Finalists
Elementary Kayonna Pitchford Heather Landreth Meredith Stanley https://www.paemst.org/ What questions do you have? Follow Us! NC Mathematics www.facebook.com/NorthCarolinaMathematics
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 For all you do for our students!