Common Core State Standards for Mathematics Core Curriculum: Mathematics Department Guy Barmoha, Miriam Sandbrand, Duke Chinn Central Area Assistant Principal’s Meeting Jan. 10, 2012

 Each state had its own set of academic standards, meaning public education students in each state were learning at different levels  All students had to be prepared to compete with not only their American peers in the next state, but with students from around the world Past Standards Initiatives

Common Core State Standards Initiative  A state‐led effort to create the next generation of standards for K-12 Mathematics and for K-12 English Language Arts and 6-12 Literacy in Social Studies/History, Science and Technical Subjects  A common set of K-12 standards to ensure that all students, no matter where they live, are prepared for success in college and work  Internationally benchmarked to ensure that our students are college and career ready in a 21 st century, globally competitive society  45 states and D.C. have adopted the CCSS

Year Grade K Fully Implement CCSS Text Complexity Literacy CCSS Literacy Standards in History/Social Studies, Science, and Technical Subjects Fully Implement CCSS Fully Implement CCSS Text Complexity Literacy CCSS Literacy Standards in History/Social Studies, Science, and Technical Subjects Fully Implement CCSS Fully Implement CCSS Implement Blended NGSSS and CCSS Implement Blended NGSS and CCSS Fully Implement and Assess CCSS Fully Implement and Assess CCSS ~ fully implement CCSS; assess FCAT ~ fully implement CCSS; assess PARCC

Key Advances in Mathematics 5 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

Organization of Common Core State Standards for Mathematics 6 Grade-Level Standards – K-8 grade-by-grade standards organized by domain – 9-12 high school standards organized by conceptual categories Standards for Mathematical Practice – Describe mathematical “habits of mind” – Connect with content standards in each grade

7 The K- 8 standards:  The K-5 standards provide students with a solid foundation in whole numbers, addition, subtraction, multiplication, division, fractions and decimals  The 6-8 standards describe robust learning in geometry, algebra, and probability and statistics  Modeled after the focus of standards from high-performing nations, the standards for grades 7 and 8 include significant algebra and geometry content  Students who have completed 7 th grade and mastered the content and skills will be prepared for algebra, in 8 th grade or after Overview of K-8 Mathematics Standards

8 Each grade includes an overview of cross- cutting themes and critical areas of study

9 Format of K-8 Mathematics Standards Domains: overarching ideas that connect topics across the grades Clusters: illustrate progression of increasing complexity from grade to grade Standards: define what students should know and be able to do at each grade level

NGSSS v.s. CCSS Content Standards Number and Operations in Base Ten 5.NBT Perform operations with multi-digit whole numbers and with decimals to hundredths. Number and Operations—Fractions 5.NF Apply and extend previous understandings of multiplication and division to multiply and divide fractions. Grade 6: BIG IDEA 1: Develop an understanding of and fluency with multiplication and division of fractions and decimals. NGSSS CCSS The Number System 6.NS Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

NGSSS v.s. CCSS Content Standards Data Analysis MA.6.S.6.1 Determine the measures of central tendency (mean, median, and mode) and variability (range) for a given set of data. MA.6.S.6.2 Select and analyze the measures of central tendency or variability to represent, describe, analyze and/or summarize a data set for the purposes of answering questions appropriately. NGSSS

NGSSS v.s. CCSS Content Standards 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. Statistics and Probability, Grade 6 CCSS

Content Standards Crosswalks

Overview of High School Mathematics Standards 14 The high school mathematics standards: – Call on students to practice applying mathematical ways of thinking to real world issues and challenges – Require students to develop a depth of understanding and ability to apply mathematics to novel situations, as college students and employees regularly are called to do – Emphasize mathematical modeling, the use of mathematics and statistics to analyze empirical situations, understand them better, and improve decisions – Identify the mathematics that all students should study in order to be college and career ready

Format of High School Mathematics Standards 15 – Content/Conceptual categories: overarching ideas that describe strands of content in high school – Domains/Clusters: groups of standards that describe coherent aspects of the content category – Standards: define what students should know and be able to do at each grade level – High school standards are organized around five conceptual categories: Number and Quantity, Algebra, Functions, Geometry, and Statistics and Probability – Modeling standards are distributed under the five major headings and are indicated with a (  ) symbol – Standards indicated as (+) are beyond the college and career readiness level but are necessary for advanced mathematics courses, such as calculus, discrete mathematics, and advanced statistics. Standards with a (+) may still be found in courses expected for all students

16 Format of High School Mathematics Standards Each content category includes an overview of the content found within it

Model Mathematics Pathways: – Developed by a panel of experts convened by Achieve, including many of the standards writers and reviewers – Organize the content of the standards into coherent and rigorous courses – Illustrate possible approaches—models, not mandates or prescriptions for organization, curriculum or pedagogy – Require completion of the Common Core in three years, allowing for specialization in the fourth year – Prepare students for a menu of courses in higher-level mathematics Model Course Pathways for Mathematics 17

Model Course Pathways for Mathematics 18

19 Model Course Pathways for Mathematics Pathway A Traditional in U.S. Geometry Algebra I Courses in higher level mathematics: Precalculus, Calculus (upon completion of Precalculus), Advanced Statistics, Discrete Mathematics, Advanced Quantitative Reasoning, or other courses to be designed at a later date, such as additional career technical courses. Pathway B International Integrated approach (typical outside of U.S.). Mathematics II Mathematics I Algebra II Mathematics III Model Course Pathways for Mathematics

Algebra: Reasoning with Equations and Inequalities (A-REI.1-12) Understand solving equations as a process of reasoning and explain the reasoning Solve equations and inequalities in one variable Solve systems of equations Represent and solve equations and inequalities graphically 8.EE.7-8 Analyze and solve linear equations and pairs of simultaneous linear equations. 7.EE.3-4 Solve real-life and mathematical problems using numerical and algebraic expressions and equations. 6.EE.5-8 Reason about and solve one-variable equations and inequalities. 5.OA.1-2 Write and interpret numerical expressions. 4.OA.1-3 Use the four operations with whole numbers to solve problems. 3.OA.1-4 Represent and solve problems involving multiplication and division. 2.OA.1 Represent and solve problems involving addition and subtraction. 1.OA.7-8 Work with addition and subtraction equations. K.OA.1-5 Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Standards for Mathematical Practice 21 Eight 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 understanding 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

2. Reason Abstractly and Quantitatively Reasoning abstractly and quantitatively often involves making sense of mathematics in real- world contexts. Word problems can provide examples of mathematics in real-world contexts. This is especially useful when the contexts are meaningful to the students.

Consider the following problems: Jessica has 7 key chains. Calvin has 8 key chains. How many key chains do they have all together? Jessica has 7 key chains. Alex has 15 key chains. How many more key chains does Alex have than Jessica? 2. Reason Abstractly and Quantitatively

Consider the following problems: Jessica has 7 key chains. Calvin has 8 key chains. How many key chains do they have all together? Jessica has 7 key chains. Alex has 15 key chains. How many more key chains does Alex have than Jessica? Key words seem helpful 2. Reason Abstractly and Quantitatively

Consider the following problems: Jessica has 7 key chains. Calvin has 8 key chains. How many key chains do they have all together? Jessica has 7 key chains. Alex has 15 key chains. How many more key chains does Alex have than Jessica? Key words seem helpful, or are they…. 2. Reason Abstractly and Quantitatively

Now consider this problem: Jessica has 7 key chains. How many more key chains does she need to have 15 key chains all together? 2. Reason Abstractly and Quantitatively

Now consider this problem: Jessica has 7 key chains. How many more key chains does she need to have 15 key chains all together? How would a child who has been conditioned to use key words solve it? 2. Reason Abstractly and Quantitatively

Now consider this problem: Jessica has 7 key chains. How many more key chains does she need to have 15 key chains all together? How would a child who has been conditioned to use key words solve it? How might a child reason abstractly and quantitatively to solve this problem? 2. Reason Abstractly and Quantitatively

Now consider this problem: Jessica has 7 key chains. How many more key chains does she need to have 15 key chains all together? 7 + __ = Reason Abstractly and Quantitatively

Now consider this problem: Jessica has 7 key chains. How many more key chains does she need to have 15 key chains all together? 7 + __ = 15(think = 10 and = 15 so = 15) Jessica needs to get 8 more key chains. 2. Reason Abstractly and Quantitatively

3. Construct viable arguments and critique t h e u n d e r s t a n d i n g o f o t h e r s Stay, Change, Flip

3. Construct viable arguments and critique t h e u n d e r s t a n d i n g o f o t h e r s

6. Attend to Precision Name this 2-dimensional figure

6. Attend to Precision

This statement is true… …always. …sometimes. …never. 6. Attend to Precision This statement is true… …always. …sometimes. …never.

Standards for Mathematical Practice 37 Eight 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 understanding 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

Standards for Mathematical Practice 38 Teachers need content knowledge for teaching mathematics to know the tasks to provide, the questions to ask, and how to assess for understanding. Math Talk needs to be supported in the classroom.

PARCC Timeline Development begins SY First year pilot/field testing and related research and data collection SY Second year pilot/field testing and related research and data collection SY Full admin. of PARCC assessment s Launch and design phase Summer 2015 Set achievement levels, including college-ready performance levels

40 PARCC: High-Quality Assessments End-of-Year Assessment Innovative, computer-based items Performance-Based Assessment (PBA) Extended tasks Applications of concepts and skills Summative assessment for accountability Formative assessment Early Assessment Early indicator of student knowledge and skills to inform instruction, supports, and PD E/LA/Literac y Speaking Listening Flexible Mid-Year Assessment Performance-based Emphasis on hard to measure standards Potentially summative

PARCC: Model Content Frameworks Higher Expectations: Conceptual Understanding, Fluency, and Application The standards are a rigorous set of expectations. According to these standards, it is not enough for students to… learn procedures by rote understand the concepts without being able to apply them to solve problems learn the important procedures of mathematics without attaining skill and fluency in them

Conceptual Understanding There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from. The student who can explain the rule understands the mathematics, and may have a better chance to succeed at a less familiar task such as expanding (a + b + c)(x + y). Conceptual understanding will be assessed using both short tasks and performance-based tasks as part of PARCC’s commitment to measure the full range of the standards.

Procedural Skill and Fluency Fluency means quickly and accurately. A key aspect of fluency in this sense is that it is not something that happens all at once in a single grade but requires attention to student understanding along the way. It is important to ensure that sufficient practice and extra support are provided at each grade to allow all students to meet the standards that call explicitly for fluency.

Grade Level Fluency

Application Application: an expectation that students will “apply the mathematics they know to problems arising in everyday life, society and the workplace. Furthermore, many individual content standards refer explicitly to real-world problems. The ability to apply mathematics will be assessed as part of PARCC’s commitment to measure the full range of the standards.

Application

More Information

Elementary Math Dept. CCSS Resources  Kindergarten IFC  Kindergarten and 1 st Grade ON CORE  Kindergarten Supplemental Assessments  K Friendly benchmarks  Mathematical Question Cards  Elementary Math Wiki Elementary Math Wiki

Secondary Math Dept. CCSS Resources  Crosswalks  Trainings  Secondary Math Wiki Secondary Math Wiki  Mathematical Question Cards