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Overview © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 1 Objectives General Overview ◦ Focus and Coherence Structure ◦ Organization ◦ Mathematical Proficiency ◦ Grade 8 Options Similarities Shifts Next Steps © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 2 Avoid the problem of “mile wide and an inch deep” Recognize that “fewer standards” are no substitute for focused standards Aim for clarity and specificity © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 3 Topics and performances are logical over time Based on learning progressions research on how students learn Reflect hierarchical nature of the content Evolve from particulars to deeper structures © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 4 Define what students should understand and be able to do in their study of mathematics ◦ Is the ability to justify appropriate to student’s math maturity ◦ Understanding and procedural skill are equally important and can be assessed using tasks of sufficient richness Are internationally benchmarked ◦ Reflect rigor, focus and coherence of standards in top-performing countries © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 5 Do: ◦ Set grade-level standards K-8 ◦ Identify standards for Algebra 1 ◦ Provide conceptual cluster standards in high school ◦ Provide clear signposts along the way toward the goal of college and career readiness for all students © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 6 Do not: ◦ Define intervention methods or materials ◦ Define the full range of supports for English learners, students with special needs and students who are well above or below grade level expectations ◦ Dictate curriculum or teaching methods © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 7 Mathematical Practice (recurring throughout the grades) Mathematical Content (different at each grade level) © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 8 Standards for Mathematical Practice ◦ Describe habits of mind of a mathematically expert student ◦ Relate to mathematical proficiency as defined by the California Framework © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 9 “WHERE” THE MATHEMATICS WORK Problem Solving Computational & Procedural Skills DOING MATH Conceptual Understanding “HOW” THE MATHEMATICS WORK “WHY” THE MATHEMATICS WORK © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 10 “ …describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high schools years.” © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 11 1. Make sense of problems and persevere in solving them …start by explaining to themselves the meaning of a problem and looking for entry points to its solution 2. Reason abstractly and quantitatively …make sense of quantities and their relationships to problem situations 3. Construct viable arguments and critique the reasoning of others …understand and use stated assumptions, definitions, and previously established results in constructing arguments 4. Model with mathematics …can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 12 Standards for Mathematical Practice Mathematically proficient students: 5. Use appropriate tools strategically …consider the available tools when solving a mathematical problem 6. Attend to precision …calculate accurately and efficiently 7. Look for and make use of structure …look closely to discern a pattern or structure 8. Look for and express regularity in repeated reasoning …notice if calculations are repeated, and look for both general methods and for shortcuts © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 13 Balanced combination of procedure and understanding “Understand” expectations connect practice to content. ◦ Lack of understanding prevents students from engaging in the mathematical practices ◦ Weighted toward central and generative concepts that most merit the time, resources, innovative energies and focus Build in complexity and provide more clarity for expected performance © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 14 Grade One ◦ Understand Place Value ◦ The two digits of a two-digit number represent amounts of tens and ones ◦ 10 can be thought of as a bundle of ten ones – called a “ten.” ◦ The numbers from 11 to 19 are composed of a ten and one, two, etc. ◦ The numbers 10, 20, 30, … refer to one, two, three, …tens and zero ones © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 15 Overview page ◦ Lists domains, clusters and mathematical practices Standards-by grade level ◦ Defines what students should understand and be able to do Clusters ◦ Groups of related standards. Standards from different clusters may be closely related Domains ◦ Larger groups of related standards. Standards from different domains may be closely related. Additional standard language or whole standards ◦ Bolded and underlined ◦ Added to maintain rigor of California expectations © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 16 © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 17 © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 18 © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 19 California Comparison Common Core State Standards for CA DOMAINS California Standards • Grades K-7 STRANDS K-5 •Counting and Cardinality (K only) •Operations and Algebraic Thinking •Number and Operations in Base 10 •Number and Operations-Fractions •Measurement and Data • Number Sense • Algebra and Functions • Measurement and Geometry • 6-8 •Ratio and Proportional Relationships (grade 6-7) •The Number System •Expressions and Equations •Functions (Grade 8) •Geometry •Statistics and probability Statistics, Data Analysis and Probability • Mathematical Reasoning © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 20 © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 21 © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 22 Develop Conceptual Understandings Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. (K.OA.2) Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. (2NBT.7) © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 23 Emphasis on Fluency Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g. knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of grade 3, know from memory all products of two one-digit numbers. (3.OA.7) Fluently multiply multi-digit whole numbers using the standard algorithm. (5.NBT.5) © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 24 A Strong Focus on Fractions Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. (3.NF.2.a) Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g. by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5+ 1/2 = 3/7, by observing that 3/7 < 1/2. (5.NF.2) © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 25 Fraction Concepts Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. ( 3.NF.3d) Discuss how you might compare pairs of fractions using a visual fraction model. For discussion purposes, use the following two fraction pairs: 7/9 and 4/9 (same denominator) 4/9 and 4/7 (same numerator) © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 26 Fraction Concepts Source: www.mathisfun.com/numbers/fraction-number-line.html © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 27 Fraction Concepts Source: www.mathisfun.com/numbers/fraction-number-line.html © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 28 Goal for 8th grade students is Algebra 1 Not all students have the necessary prerequisite skills for Algebra 1 Two sets of standards for grade 8 ◦ Each set will prepare students for college and career ◦ Standards for Algebra 1 Taken from 8th grade Common Core, high school Algebra content cluster and CA Algebra standards ◦ 8th grade Common Core Goal of grade 8 Common Core is to finalize preparation for students in high school K-7 standards as augmented prepare students for either set of standards © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 29 © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 30 Arranged by conceptual cluster (NOT by course): • Number and Quantity • Modeling • Algebra • Geometry • Functions • Statistics and Probability Same K-8 structure of domain, cluster and standard © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 31 Specify the math that all students should study to be college and career ready Identify additional math standards that students should learn in order to take advanced courses such as calculus, advanced statistics, or discrete mathematics. These are indicated by (+). Include the addition of two courses from California: ◦ Calculus ◦ Advanced Placement Statistics and Probability Development of suggested course descriptions will be done by CDE as part of their long-range implementation plan ◦ Traditional vs. Integrated © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 32 High School Example-Geometry Content Cluster © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 33 Modeling Cluster ◦ Not a collection of topics but viewed in relation to other standards ◦ A Standard of Mathematical Practice ◦ Specific modeling standards appear throughout the high school standards and are indicated by a star symbol (★) © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 34 Grade California Standard Kindergarten Use concrete objects to determine the answers to addition and subtraction problems (for two numbers that are each less than 10). Common Core Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. First Count, read, and write whole numbers to 100. Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral. Third Memorize to automaticity the multiplication table for numbers between 1 and 10. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division and the properties of operations. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 35 Grade Fifth Sixth Seventh California Standard Understand the concept of multiplication and division of fractions. Common Core Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (A unit fraction is one with a numerator of 1 and the denominator is a positive integer) Interpret and use ratios in different Understand the concept of a ratio and use contexts (e.g., batting averages, miles ratio language to describe a ratio relationship per hour) to show the relative sizes of between two quantities. two quantities, using appropriate notations ( a/b, a to b, a:b ). Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g., three is less than a number, half as large as area A). Use variables to represent quantities in realworld and mathematical problems and construct simple equations and inequalities to solve problems about the quantities. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 36 Grade/C ourse Seventh Algebra California Standard Construct and read drawings and models made to scale. Common Core Solve problems involving scale drawings of geometric figures, including actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Algebra 1: Algebra Content Cluster: Solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Geometry Geometry: Geometry Content Cluster: Use trigonometric functions to solve Use trigonometric ratios and the for an unknown length of a side of Pythagorean Theorem to solve right a right triangle, given an angle and triangles in applied problems. a length of a side. © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 37 Grade Shifts: Examples Concept 1997 Standards CCCS Compose simple shapes to form larger shapes (e.g., 2 triangles to form a rectangle) Grade 2 K Introduction to Probability Grade 3 Grade 7 Introduction of fractions as numbers Grade 2 Grade 3 Add and subtract simple fractions Grade 3 Grade 4 Introduction of integers Grade 4 Grade 6 Developed by SCFIRD © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 38 California’s Additional 15% Based on the following central questions: What K-12 CA Mathematics standards were not reflected in the CCS document? Which (of those) standards would substantively enhance and improve the CCS? Which would maintain the rigor of California’s standards? © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 39 Examples of Additional 15%: Added standards to develop ideas not included in CCS ◦ Grade 2-Operations and Algebraic Thinking ◦ Grade 5-Operations and Algebraic Thinking ◦ High School Geometry-Geometric Measurement and Dimension © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 40 Examples of Additional 15%: Added language to existing standard ◦ Grade 2-Measurement and Data ◦ Grade 4-Geometry © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 41 Examples of Additional 15%: Added a substantial section to an existing cluster ◦ Grade 6-The Number System © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 42 Examples of Additional 15%: Added two courses from California Standards: ◦ Calculus ◦ Advanced Placement Probability and Statistics © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 43 Stay the Course! ◦ More similarities than differences in the standards ◦ Implement a truly balanced math program as this will support the mathematical practices ◦ Continue to use quality assessments to inform and drive effective instruction ◦ Provide opportunities for teachers to collaborate and plan © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 44 Websites ◦ Common Core Standards: www.corestandards.org ◦ California Common Core Standards: Visit the California Department of Education’s Common Core State Standards Web page at: http://www.cde.ca.gov/be/st/cc/index.asp • • • • The standards Frequently asked questions Informational flyers Additional resources © 2011 California County Superintendents Educational Services Association • Mathematics General Overview 45