CCCSS Mathematics Presentation

Overview

© 2011 California County Superintendents Educational Services Association • Mathematics General Overview

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Objectives 

General Overview

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◦ Focus and Coherence

Structure ◦ Organization

◦ Mathematical Proficiency

◦ Grade 8 Options 

Similarities

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Shifts

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Next Steps

© 2011 California County Superintendents Educational Services Association • Mathematics General Overview

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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

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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

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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

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Are internationally benchmarked ◦ Reflect rigor, focus and coherence of standards in top-performing countries

© 2011 California County Superintendents Educational Services Association • Mathematics General Overview

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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

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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

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Mathematical Practice (recurring throughout the grades) Mathematical Content (different at each grade level)

© 2011 California County Superintendents Educational Services Association • Mathematics General Overview

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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

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“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

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“ …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

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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

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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

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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

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Build in complexity and provide more clarity for expected performance © 2011 California County Superintendents Educational Services Association • Mathematics General Overview

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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

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Overview page ◦ Lists domains, clusters and mathematical practices

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Standards-by grade level ◦ Defines what students should understand and be able to do

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Clusters ◦ Groups of related standards. Standards from different clusters may be closely related

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Domains ◦ Larger groups of related standards. Standards from different domains may be closely related.

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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

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© 2011 California County Superintendents Educational Services Association • Mathematics General Overview

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© 2011 California County Superintendents Educational Services Association • Mathematics General Overview

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© 2011 California County Superintendents Educational Services Association • Mathematics General Overview

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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

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Number Sense

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Algebra and Functions

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Measurement and Geometry

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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

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Mathematical Reasoning

© 2011 California County Superintendents Educational Services Association • Mathematics General Overview

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© 2011 California County Superintendents Educational Services Association • Mathematics General Overview

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© 2011 California County Superintendents Educational Services Association • Mathematics General Overview

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Develop Conceptual Understandings

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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

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Emphasis on Fluency 

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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

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A Strong Focus on Fractions 

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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

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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