Mathematical Representations Pathways to Teacher Leadership in Mathematics Project

Learning Intention We are learning to understand the importance of using and connecting mathematical representations for student learning as a high-leverage teaching practice. Success Criteria We will be successful when we become skilled in developing our students’ representational competence.

Background

Principles to Actions: Ensuring Mathematical Success for All Session Description: Quality teaching ensures success for all students. In its new document, NCTM frames a core set of highly effective teaching practices for mathematics, advancing our professional knowledge of representations, struggle, fluency, evidence, and more. We will examine this framework and help you prepare for your next professional actions.

New major NCTM Publication!! Released April 9, 2014

Filling the gap between standards and learning The primary purpose of Principles to Actions is to fill the gap between the development and adoption of CCSSM and other standards and the enactment of practices, policies, programs, and actions required for their widespread and successful implementation. CCSSM helps focus and clarify common outcomes. ... [it] does not tell teachers, coaches, administrators, parents, or policymakers what to do at the classroom, school, or district level or how to begin making essential changes to implement these standards. (NCTM, 2014, p. 11)

Principles to Actions NCTM undertook a major initiative to define and describe the principles and actions, including specific teaching practices, that are essential for a high-quality mathematics education for all students. Offers guidance to teachers, mathematics coaches, administrators, parents, and policymakers. ----- Meeting Notes (4/30/14 17:37) ----- notes etc

Principles to Actions Writing Team Steven Leinwand American Institutes for Research, D.C. Daniel J. Brahier Bowling Green State University, Ohio DeAnn Huinker University of Wisconsin–Milwaukee Robert Q. Berry III, University of Virginia, Charlottesville, VA Frederick L. Dillon, Strongsville City Schools, Ohio Matthew R. Larson Lincoln Public Schools, Nebraska Miriam A. Leiva University of North Carolina at Charlotte W. Gary Martin, Auburn University, Auburn, Alabama Margaret S. Smith, University of Pittsburgh, Pennsylvania

Guiding Principles

Principles describe features of excellent mathematics programs. Unproductive Beliefs Productive Beliefs Defined. Examined for unproductive and productive beliefs, Linked to effective practices. Illustrated with examples. Effective Practices Illustrations and Examples

Guiding Principles for School Mathematics Teaching and Learning Access and Equity Curriculum Tools and Technology Assessment Professionalism Guiding Principles for School Mathematics The Council first defined a set of Principles that “describe features of high-quality mathematics education” in Principles and Standards for School Mathematics in 2000. Now in Principles to Actions it articulates and builds on an updated set of six Guiding Principles that reflect more than a decade of experience and new research evidence about excellent mathematics programs, as well as significant obstacles and unproductive beliefs that continue to compromise progress. Candy Jar-more ..... spells out Guiding Principles and actions that must be taken in each of the following: Teaching and Learning, Access and Equity, Curriculum, Tools and Technology, Assessment, and Professionalism.

Guiding Principle: 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.

Overarching Message Effective teaching is the non-negotiable core that ensures that all students learn mathematics at high levels. Principles to Actions gives all stakeholders the ideas, the research, and the actions they need to dramatically improve mathematics education. It outlines the productive practices all teachers should adopt to improve their students’ mathematics learning, and it describes practical steps that math specialists and coaches, administrators, policymakers, and parents can take to support a high quality mathematics education.

Defining High-leverage Teaching Practices for Mathematics

Effective 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) Action: To work together as a profession toward implementation of a common set of high-leverage practices that underlie effective teaching ...... “those practices at the heart of the work of teaching that are most likely to affect student learning.” (Ball & Forzani, 2010, p. 45) Deborah Ball and Francesca Forzani Ball, Deborah Loewenberg, and Francesca M. Forzani. "Building a Common Core for Learning to Teach: And Connecting Professional Learning to Practice." American Educator 35, no. 2 (2011): 17-21. Ball, Deborah Loewenberg, and Francesca M. Forzani. "Teaching skillful teaching." Educational Leadership 68, no. 4 (2010): 40-45.

Mathematics Teaching Practices Establish math goals to focus learning Implement tasks that promote reasoning & problem solving Use and connect mathematical representations Facilitate meaningful mathematical discourse Pose purposeful questions Support productive struggle in learning mathematics Build procedural fluency from conceptual understanding Elicit & use evidence of student thinking Mathematics Teaching Practices These 8 Mathematics Teaching Practices represent as a core set of high-leverage practices and essential teaching skills necessary to promote deep learning of mathematics. The eight Mathematics Teaching Practices (see fig. 2) describe essential components of every lesson that are necessary to promote deep learning of mathematics (NCTM 2014). Effective teaching of mathematics begins with teachers clarifying and understanding the mathematics that students need to learn in order to establish clear goals for the selection of tasks that promote reasoning and problem solving while developing conceptual understanding and procedural fluency. Classrooms should be rich in mathematical discourse with students using and making connections among mathematical representations as they compare and analyze varied solution strategies. This discourse is carefully facilitated by the teacher with purposeful questioning. Teachers also acknowledge the value of productive struggle in learning mathematics and support students in developing a disposition to persevere in solving problems. Classroom teaching and learning interactions need to be guided by evidence of student thinking in order to assess and advance student reasoning and sense making about important mathematical ideas and relationships.

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

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. Read: PtA, p. 24-26 Identify one specific passage (e.g., sentence) to share with your group that was particularly salient for you. Take turns to share and discuss the passages and reasons for selecting these points.

#3. Use and connect representations Contextual Physical Visual Symbolic Verbal Effective teaching of mathematics engages students in making connections among representations to deepen understanding of concepts and procedures and as tools for problem solving. Marshall, Anne Marie, Alison Castro Superfine, and Reality S. Canty. “Star Students Make Connections.” Teaching Children Mathematics 17, no. 1 (2010): 39–47. To develop students’ representational competence: (1) Discuss the explicit connections among representations; (2) Alternate directionality in making connections; and (3) Encourage purposeful selection of representations. (Marshall, Superfine, & Canty, 2010) Effective mathematics teaching includes a strong focus on developing students’ “representational competence” through making important connections among contextual, visual, verbal, physical, and symbolic representational forms. Lesh, Post, & Behr, 1987; Marshall, Superfine, & Canty, 2010; Tripathi, 2008; Webb, Boswinkel, & Dekker, 2008

Unpacking the Readings on Mathematical Representations Pick a task card for your group. Study and discuss the prompt within your group. Prepare a chart to present key points to the whole group. (You can be creative!) Present key ideas and issues to the whole group.

Compare and contrast the PtA (NCTM, 2014) model to Lesh, Post, & Behr (1987) and to Marshall, Superfine, and Canty (2010). Explain and give examples of “translations” and “transformations.” Comment on the notion of “translation (dis)abilities” in regards to your students or your curriculum. Identify and give examples of the characteristics or components of representational competence. Then provide a critique, with examples, of your own students’ level of representational competence. Distinguish external embodiments from internal conceptualizations, speculate on the value of this distinction, and comment on how Lesh, Post, and Behr (1987) define student understanding.

Revisiting . . . . Learning Intention We are learning to understand the importance of using and connecting mathematical representations for student learning as a high-leverage teaching practice. Success Criteria We will be successful when we become skilled in developing our students’ representational competence.

Disclaimer Pathways to Teacher Leadership in Mathematics Project University of Wisconsin-Milwaukee, 2014-2017   This material was developed for the Pathways to Teacher Leadership in Mathematics project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Any other use of this work—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors—without prior written permission is prohibited. This project was supported through a grant from the Wisconsin ESEA Title II Improving Teacher Quality Program.