“A Story of Functions” Understanding the Structure of the Curriculum 7/17/2014 Erin Wheeler- Staff Development Specialist, Erie 2 BOCES Textbook What do you think when you hear the word “textbook?” What has your relationship with textbooks been like over the years. A Different Vision of “Textbook” Few U.S. textbooks paint mathematics as a dynamic, unfolding tale. They instead prioritize teaching procedures and employ a spiraling approach, in which topics are partially taught and then returned to— sometimes years later—with the unrealistic expectation that students will somehow connect the dots. But teaching procedures as skills without a rich context is ineffective. Students can too easily forget procedures and will fail if they do not have deeper, more concrete knowledge from which they can draw. “How to Implement a Story of Units” page 4 Teaching Mathematics Dr. Wu “What is Different About the Common Core Math Standards?” http://math.berkeley.edu/~wu/CommonCoreVI.pdf June 20, 2011 A Different Vision of “Textbook” “The textbook is the material on which Chinese teachers spend most of their time and devote most of their efforts to “study intensively.” They study it constantly throughout the school year when they teach it. (Knowing and Teaching Elementary Mathematics, Ma (1999, pg. 132) A Different Vision of “Textbook” First of all, they work for an understanding of “what it is.” They study how it interprets and illustrates the ideas in the (Standards), why the authors structured the book in a certain way, what the connections among the contents are… (Knowing and Teaching Elementary Mathematics, Ma (1999, pg. 132) Coherence at the High School Level A Trilogy of Mathematics 9-12 6-8 PK-5 A Story of Units A Story of Ratios A Story of Functions http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/ Textbooks that brings the world of mathematics research into the classroom. What is the goal of my math class? To be able to solve a certain type of problems by the end of today. To use the problems we are solving today to learn mathematics. https://www.youtube.com/watch?v=B6UQcwzyE1U (Phil Daro- Rationale of the Math Standards 5 min) A Misconception to Clear Up http://commoncore.org/ http://www.corestandards.org/about-the-standards/ Curriculum Map and Overview Story of Functions Curriculum Maphttps://www.engageny.org/r esource/grades-9-12mathematics-curriculummap Interactive Curriculum Maphttp://commoncore.org/ma ps/math/ Algebra 1 and Geometry Test Guides https://www.engageny.org/r esource/regents-examsmathematics Going Beyond the Map http://parcconline.org/sites/parcc/files/PARCCMCFMathematicsNovember2012V3_FINAL_0.pdf Going Beyond the Map NYS COMMON CORE MATHEMATICS CURRICULUM Key Points for Algebra 1 • Module 1 plants the seeds of the work of the year, connecting the work to real world contexts, and also provides a deep study of algebraic equivalence, the structure of expressions, and reasoning of solving equations. • Module 2 continues to connect the work of the year to real world contexts from a data perspective, motivating the intense study of exponential and quadratic functions to come. • Module 3 provides a deep study of exponential functions, introduces function notation. • Module 4 provides a deep study of quadratic functions. • Module 5 serves to synthesize the year by providing situations of linear, exponential, and quadratic forms, where the student is required to recognize the model to be used. © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM Key Themes of Geometry Module 1: Congruence, Proof, and Constructions • Module is anchored by the definition of congruence • Emphasis is placed on extending the meaning and use of vocabulary in constructions • There is an explicit recall and application of facts learned over the last few years in unknown angle problems and proofs • Triangle congruence criteria are indicators that a rigid motion exists that maps one triangle to another; each criterion can be proven to be true with the use of rigid motions. © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM Key Themes of Geometry Module 2: Similarity, Proof, and Trigonometry • Just as rigid motions are used to define congruence, so dilations are added to define similarity. • A close look at the way dilations are constructed yields information about how dilations behave. • Right triangle similarity is built on the concepts of withintriangle ratios and between-triangle ratios leading to an understanding of trigonometric ratios. © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios Going Beyond the Map Creating the Curriculum Map A Story of Units Module Structure Module Overview Topic A Topic B Topic C L1 L2 L3 L4 L5 L6 L7 L8 20 © 2012 Common Core, Inc. All rights reserved. commoncore.org Overview Narrative Explains the sequence outlined on the title page. Helps you see where we’ve been and where we’re going. Progressions Documents “…explain why standards are sequenced the way they are, point out cognitive difficulties and pedagogical solutions, and give more detail on particularly knotty areas of the mathematics. This would be useful in teacher preparation and professional development, organizing curriculum, and writing textbooks.” http://ime.math.arizona.edu/progressions/ This seems strange… http://ime.math.arizona. edu/progressions/ A Story of Units Module Structure Module Overview Topic A Topic B Topic C L1 L2 L3 L4 L5 L6 L7 L8 Simple Complex 25 © 2012 Common Core, Inc. All rights reserved. commoncore.org Making the Connections 9 + 6 = 15 Abstract Pictorial Concrete Universal Design for Learning Multiple Means of Representation Multiple Means of Action and Expression Multiple Means of Engagement “How to Implement a Story of Units” Differentiating Instruction pages 14-20 https://www.engageny.org/resour ce/pre-kindergarten-grade-5mathematics-curriculum-map-andguiding-documents NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Types of Lessons 1. Problem Set Students and teachers work through examples and complete exercises to develop or reinforce a concept or procedure. 2. Socratic Teacher leads students in a conversation to develop a specific concept or proof. 3. Exploration Independent or small group work on a challenging problem followed by debrief to clarify, expand or develop math knowledge 4. Modeling Students and teacher practice part of the modeling cycle with problems that are ill-defined and have a real world context. © 2012 Common Core, Inc. All rights reserved. commoncore.org Topic Opener- Narrative Math Publisher’s Criteria Rigor: in major topics, pursue with equal intensity • conceptual understanding • procedural skill and fluency • applications http://achievethecore.org/content/upload/Math_Publishers_Criteria_HS_Spring_2013_FINAL.pdf Fluency Fluency is designed to promote automaticity by engaging students in practice in ways that get their adrenaline flowing. Automaticity is critical so that students avoid using up too many of their attention resources with lower-level skills when they are addressing higher-level problems. Set a timer and stop when your fluency time is up! Fluency follows instruction! NYS COMMON CORE MATHEMATICS CURRICULUM What’s In a Lesson? Teacher Materials Lessons • Student Outcomes and Lesson Notes (in select lessons) • Classwork • General directions and guidance, including timing guidance • Bulleted discussion points with expected student responses • Student classwork with solutions (boxed) • Exit Ticket with Solutions • Problem Set with Solutions Student Materials • Classwork • Problem Set © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios Examples- Developing the Concept Closing- Debrief the patterns observed Exit Ticket • Follows the closing after the student have had an opportunity to share and hear classmates observations. • Provides a daily gauge of student understanding. Problem Set • Practice applying the day’s learning. • Consider a time frame vs. a task frame. Assessments Evidence of the Standards Evidence of the Standards Thank you for attending or watching!