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

Common Core State Standards for Mathematics

Core Curriculum: Mathematics DepartmentGuy Barmoha, Miriam Sandbrand, Duke Chinn

Central AreaAssistant Principal’s MeetingJan. 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 21st century, globally competitive society

45 states and D.C. have adopted the CCSS

YearGrade

K 1 2 3-5 6-12

2011-12 Fully Implement

CCSS

Text Complexity Text Complexity Text Complexity Literacy

CCSS Literacy Standards in History/Social Studies, Science, and

Technical Subjects


CCSS

Fully Implement CCSS

Text Complexity Text Complexity Literacy

CCSS Literacy Standards in History/Social Studies, Science, and

Technical Subjects


CCSS



Implement Blended NGSSS

and CCSS

Implement Blended NGSS

and CCSS

2014-15 Fully Implement and

Assess CCSS

Fully Implement and Assess CCSS



Fully Implement and Assess

CCSS

2013-14 ~ fully implement CCSS; assess FCAT 2.02014-15 ~ 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 decimalsThe 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 contentStudents who have completed 7th grade and mastered the content and skills will be prepared for algebra, in 8th grade or after

Overview of K-8 Mathematics Standards

8

Overview of K-8 Mathematics Standards

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. CCSSContent Standards

Number and Operations in Base Ten 5.NBTPerform operations with multi-digit whole numbers and with decimals to hundredths.

Number and Operations—Fractions 5.NFApply and extend previous understandings of multiplication anddivision 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.NSApply and extend previous understandings of multiplication anddivision to divide fractions by fractions.


Data AnalysisMA.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


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


18

19


Pathway ATraditional 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 BInternational Integrated approach (typical

outside of U.S.)

.

Mathematics II

Mathematics I

Algebra II Mathematics III


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?





Key words seem helpful





Key words seem helpful, or are they….


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 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?




7 + __ = 15




7 + __ = 15 (think 7 + 3 = 10 and 10 + 5 = 15 so 7 + 8 = 15)

Jessica needs to get 8 more key chains.


3.

Construct viable

arguments

and critique

the

understanding

of

others

€

4

15÷2

3 Stay, Change, Flip

€

= 415•3

2

3.

Construct viable

arguments

and critique

the

understanding

of

others

€

4

15÷2

3

€

=

4

152

3

€

=

4

15•3

22

3•3

2

€

=

4

15•3

21

€

= 415•3

2

6. Attend to Precision

Name this 2-dimensional figure

Name this 2-dimensional figure



€

x = xThis statement is true…

…always.…sometimes.…never.


€

x =−xThis statement is true…

…always.…sometimes.…never.


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


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 TimelinePARCC Timeline

2011-12

Development begins

SY 2012-13

First year pilot/field

testing and related

research and data collection

SY 2013-14

Second year pilot/field

testing and related

research and data collection

SY 2014-15

Full admin. of PARCC

assessments

2010-11

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-BasedAssessment (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/Literacy

•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 InformationMore Information

47

www.corestandards.orgwww.corestandards.org

www.PARCConline.orgwww.PARCConline.org

http://www.corestandards.org/




http://www.PARCConline.org/




Elementary Math Dept. CCSS Resources

Kindergarten IFC

Kindergarten and 1st Grade ON CORE

Kindergarten Supplemental Assessments

K Friendly benchmarks

Mathematical Question Cards

Elementary Math Wiki

Secondary Math Dept. CCSS Resources

Crosswalks

Trainings

Secondary Math Wiki

Mathematical Question Cards

Documents

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