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CCSSI FOR MATHEMATICS “STANDARDS OF PRACTICE” Collegial Conversations HIGH SCHOOL Today’s Goal To explore the Standards for Content and Practice for Mathematics Begin to consider how these new CCSS Standards are likely to impact your classroom practices What are the Common Core State Standards? Aligned with college and work expectations Focused and coherent Included rigorous content and application of knowledge through high-order skills Build upon strengths and lessons of current state standards Internationally benchmarked so that all students are prepared to succeed in our global economy and society Research and evidence based State led- coordinated by NGA Center and CCSSO Focus • Key ideas, understandings, and skills are identified • Deep learning of concepts is emphasized – That is, time is spent on a topic and on learning it well. This counters the “mile wide, inch deep” criticism leveled at most current U.S. standards. Benefits for States and Districts • • • • Allows collaborative professional development based on best practices Allows development of common assessments and other tools Enables comparison of policies and achievement across states and districts Creates potential for collaborative groups to get more economical mileage for: – Curriculum development, assessment, and professional development Common Core Development • Initially 48 states and three territories signed on • As of November 29, 2010, 42 states have officially adopted • Final Standards released June 2, 2010, at www.corestandards.org • Adoption required for Race to the Top funds Michigan’s Implementation Timeline • Held October and November of 2010 rollouts • District curricula and assessments that provide a K-12 progression for meeting the MMC requirements will require minimal adjustments to meet CCSS • Curriculum and assessment alignment in SY10-11 • Implementation SY11-12 • New assessment 2014-15 (Smarter Balanced Assessment Consortium or SBAC – replaces MEAP and MME) Background Responsibilities of States in the Consortium Each State that is a member of the Consortium in 2014– 2015 also agrees to do the following: Adopt common achievement standards no later than the 2014–2015 school year, Fully implement the Consortium summative assessment in grades 3–8 and high school for both mathematics and English language arts no later than the 2014–2015 school year, Adhere to the governance requirements, Agree to support the decisions of the Consortium, Agree to follow agreed-upon timelines, Be willing to participate in the decision-making process and, if a Governing State, final decisions, and Identify and implement a plan to address barriers in State law, statute, regulation, or policy to implementing the proposed assessment system and address any such barriers prior to full implementation of the summative assessment components of the system. Technology Approach SBAC Item Bank • Partitioned into a secure item bank for summative assessments and a non-secure bank for the interim/benchmark assessments: • • • • Traditional selected-response items Constructed-response items Curriculum-embedded performance events Technology-enhanced items (modeled after assessments in use by the U.S. military, the architecture licensure exam, and NAEP) HOW TO READ THE STANDARDS Domains are large groups of related standards. Standards from different domains may sometimes be closely related. Look for the name with the code number on it for a Domain. Common Core Format Clusters are groups of related standards. Standards from different clusters may sometimes be closely related, because mathematics is a connected subject. • Clusters appear inside domains. Standards define what students should be Common Core Format able to understand and be able to do – part of a cluster. They are content statements. An example content statement is: “Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2)”. •Progressions of increasing complexity from grade to grade Common Core - Clusters • May appear in multiple grade levels in the K-8 Common Core. There is increasing development as the grade levels progress • What students should know and be able to do at each grade level • Reflect both mathematical understandings and skills, which are equally important Common Core Format K-8 High School Grade Conceptual Category Domain Domain Cluster Cluster Standards (There are no preK Common Core Standards) Standards K – 5 DOMAINS Domains Grade Levels Counting and Cardinality K only Operations and Algebraic Thinking 1-5 Number and Operations in Base Ten 1-5 Number and Operations Fractions 3-5 Measurement and Data 1-5 Geometry 1-5 MIDDLE GRADES DOMAINS Domains Grade Levels Ratio and Proportional Relationships 6-7 The Number System 6-8 Expressions and Equations 6-8 Functions 8 Geometry 6-8 Statistics and Probability 6-8 Fractions, Grades 3–6 3. Develop an understanding of fractions as numbers. 4. Extend understanding of fraction equivalence and ordering. 4. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 4. Understand decimal notation for fractions, and compare decimal fractions. 5. Use equivalent fractions as a strategy to add and subtract fractions. 5. Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 6. Apply and extend previous understandings of multiplication and division to divide fractions by fractions. Statistics and Probability, Grade 6 Develop understanding of statistical variability • • • Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages. Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. Algebra, Grade 8 Graded ramp up to Algebra in Grade 8 • Properties of operations, similarity, ratio and proportional relationships, rational number system. Focus on linear equations and functions in Grade 8 • Expressions and Equations – Work with radicals and integer exponents. – Understand the connections between proportional relationships, lines, and linear equations. – Analyze and solve linear equations and pairs of simultaneous linear equations. • Functions – Define, evaluate, and compare functions. – Use functions to model relationships between quantities. High School Conceptual themes in high school • • • • • • Number and Quantity Algebra Functions Modeling Geometry Statistics and Probability College and career readiness threshold • (+) standards indicate material beyond the threshold; can be in courses required for all students. Format of High School Domain Cluster Standard Format of High School Standards Regular Standard Modeling STEM Common Core - Domain • Overarching “big ideas” that connect topics across the grades • Descriptions of the mathematical content to be learned, elaborated through clusters and standards Common Core - Clusters • May appear in multiple grade levels with increasing developmental standards as the grade levels progress • Indicate WHAT students should know and be able to do at each grade level • Reflect both mathematical understandings and skills, which are equally important Common Core - Standards • Content statements • Progressions of increasing complexity from grade to grade – In high school, this may occur over the course of one year or through several years HS Pathways 1.) Traditional (US) – 2 Algebra, Geometry and Data, probability and statistics included in each course 2.) International (integrated) three courses including number , algebra, geometry, probability and statistics each year 3.) Compacted version of traditional – grade 7/8 and algebra completed by end of 8th grade 4.) Compacted integrated model, allowing students to reach Calculus or other college level courses High School Pathways • The CCSS Model Pathways are NOT required. The two sequences are examples, not mandates • Two models that organize the CCSS into coherent, rigorous courses • Four years of mathematics: – One course in each of the first two years – Followed by two options for year 3 and a variety of relevant courses for year 4 • Course descriptions – Define what is covered in a course – Are not prescriptions for the curriculum or pedagogy High School Pathways • Four years of mathematics: – One course in each of the first two years – Followed by two options for year three and a variety of relevant courses for year four • Course descriptions – Define what is covered in a course – Are not prescriptions for the curriculum or pedagogy High School Pathways • Pathway A: Consists of two algebra courses and a geometry course, with some data, probability and statistics infused throughout each (traditional) • Pathway B: Typically seen internationally that consists of a sequence of 3 courses each of which treats aspects of algebra, geometry and data, probability, and statistics. Interrelationships • Algebra and Functions – Expressions can define functions – Determining the output of a function can involve evaluating an expression • Algebra and Geometry – Algebraically describing geometric shapes – Proving geometric theorems algebraically Numbers and Quantity • Extend the Real Numbers to include work with rational exponents and study of the properties of rational and irrational numbers • Use quantities and quantitative reasoning to solve problems. Numbers and Quantity Required for higher math and/or STEM • Compute with and use the Complex Numbers, use the Complex Number plane to represent numbers and operations • Represent and use vectors • Compute with matrices • Use vector and matrices in modeling Algebra and Functions • Two separate conceptual categories • Algebra category contains most of the typical “symbol manipulation” standards • Functions category is more conceptual • The two categories are interrelated Algebra • Creating, reading, and manipulating expressions – Understanding the structure of expressions – Includes operating with polynomials and simplifying rational expressions • Solving equations and inequalities – Symbolically and graphically Algebra Required for higher math and/or STEM • Expand a binomial using the Binomial Theorem • Represent a system of linear equations as a matrix equation • Find the inverse if it exists and use it to solve a system of equations Functions • Understanding, interpreting, and building functions – Includes multiple representations • Emphasis is on linear and exponential models • Extends trigonometric functions to functions defined in the unit circle and modeling periodic phenomena Functions Required for higher math and/or STEM • Graph rational functions and identify zeros and asymptotes • Compose functions • Prove the addition and subtraction formulas for trigonometric functions and use them to solve problems Functions Required for higher math and/or STEM • Inverse functions – Verify functions are inverses by composition – Find inverse values from a graph or table – Create an invertible function by restricting the domain – Use the inverse relationship between exponents and logarithms and in trigonometric functions High School - Modeling • Linking mathematics and statistics to everyday life, work, etc. • Process of choosing and using appropriate mathematics and statistics • Examples: pg 72 Modeling Modeling has no specific domains, clusters or standards. Modeling is included in the other conceptual categories and marked with a asterisk. Modeling Modeling links classroom mathematics and statistics to everyday life, work, and decision-making. Technology is valuable in modeling. A model can be very simple, such as writing total cost as a product of unit price and number bought, or using a geometric shape to describe a physical object. Modeling • Planning a table tennis tournament for 7 players at a club with 4 tables, where each player plays against each other player. • Analyzing stopping distance for a car. • Modeling savings account balance, bacterial colony growth, or investment growth. Geometry, High School Middle school foundations • • Hands-on experience with transformations. Low tech (transparencies) or high tech (dynamic geometry software). High school rigor and applications • • Properties of rotations, reflections, translations, and dilations are assumed, proofs start from there. Connections with algebra and modeling Geometry • Understanding congruence • Using similarity, right triangles, and trigonometry to solve problems Congruence, similarity, and symmetry are approached through geometric transformations Geometry • Circles • Expressing geometric properties with equations – Includes proving theorems and describing conic sections algebraically • Geometric measurement and dimension • Modeling with geometry Geometry Required for higher math and/or STEM • Non-right triangle trigonometry • Derive equations of hyperbolas and ellipses given foci and directrices • Give an informal argument using Cavalieri’s Principal for the formulas for the volume of solid figures Statistics and Probability • Analyze single a two variable data • Understand the role of randomization in experiments • Make decisions, use inference and justify conclusions from statistical studies • Use the rules of probability Key Advances Focus and coherence • • Focus on key topics at each grade level. Coherent progressions across grade levels. Balance of concepts and skills • Content standards require both conceptual understanding and procedural fluency. Mathematical practices • Foster reasoning and sense-making in mathematics. College and career readiness • Level is ambitious but achievable. Design and Organization Mathematical Practice – expertise students should acquire: (Processes & proficiencies) • NCTM five process standards: • • • • • Problem solving Reasoning and Proof Communication Connections Representations NCTM Process Standards and the CCSS Mathematical Practice Standards NCTM Process Standards CCSS Mathematical Practices Problem Solving Make sense of problems and persevere in solving them. Use appropriate tools strategically Reasoning and Proof Reason abstractly and quantitatively. Critique the reasoning of others. Look for and express regularity in repeated reasoning Communication Construct viable arguments Connections Attend to precision. Look for and make use of structure Representations Model with mathematics. Design and Organization • Mathematical proficiency (National Research Council’s report Adding It Up) – Adaptive reasoning – Strategic competence – Conceptual understanding (comprehension of mathematical concepts, operations, relations) – Procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently, and appropriately) – Productive disposition (ability to see mathematics as sensible, useful, and worthwhile Mathematics/Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Mathematics/Standards for Mathematical Practice “The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education.” CCSS, 2010 Standards for Mathematical Practice • Carry across all grade levels • Describe habits of a mathematically expert student Standards for Mathematical Content • • • • K-8 presented by grade level Organized into domains that progress over several grades Grade introductions give 2-4 focal points at each grade level High school standards presented by conceptual theme (Number & Quantity, Algebra, Functions, Modeling, Geometry, Statistics & Probability Standards of Mathematical Practice 1. Choose a partner at your table and “Pair Share” the Standards of Practice between you and your partner. 2. When you and your partner feel you understand generally each of the standards, discuss the following question: What implications might the standards of practice have on your classroom? Transition from Current State Standards & Assessments to New Common Core Standards • • • • • Develop Awareness Needs Assessment/Gap Analysis Planning Capacity Building Job-embedded Professional Development Transition Planning Next Steps: • Alignment of CCSS with curriculum • Gap analysis (content and skills that vary from the MEAP and MME) • What instructional practices will facilitate the transition? • What new assessment strategies will be needed? • Professional development needs? Transition Planning • Gather in teams from your schools and discuss – What are your immediate needs as a classroom teacher being asked to implement the CCSS? – What professional development is needed? – What initial gaps come to mind and how do you address these gaps? – As a school team, look at the eight Standards for Mathematical Practice. What do they look like? Sound like? What will students need in order to implement them? What will teachers need? What are the implications for assessment and grading? Select a recorder, time keeper and someone to report out for your group. Questions? Please contact: PUT YOUR INFORMATION HERE! Have a great day!