Mathematics
© 2011 California County Superintendents Educational Services Association • Mathematics Teacher Overview
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Objectives

General Overview

◦ Focus and
Coherence
◦ Mathematical
Proficiency
Structure
◦ Organization
◦ Grade 8 Options

Similarities

Shifts

Next Steps
© 2011 California County Superintendents Educational Services Association • Mathematics Teacher 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 Teacher 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 Teacher 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

Are internationally benchmarked
◦ Reflect rigor, focus and coherence of standards in
top-performing countries
© 2011 California County Superintendents Educational Services Association • Mathematics Teacher 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 Teacher 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 Teacher 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 Teacher 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 Teacher 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 Teacher 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 Teacher 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 Teacher 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 Teacher Overview
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
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
Locate the Mathematical Practices
With a partner compare the eight
mathematical practices to the three
components of a balanced math program as
defined by the California Framework
Which practices align best to Conceptual
Understanding? Computation and
Procedures? Problem Solving?
Be ready to share out with the entire group.
© 2011 California County Superintendents Educational Services Association • Mathematics Teacher 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

Build in complexity and provide more clarity for
expected performance
© 2011 California County Superintendents Educational Services Association • Mathematics Teacher 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 Teacher Overview
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
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 Teacher Overview
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© 2011 California County Superintendents Educational Services Association • Mathematics Teacher Overview
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© 2011 California County Superintendents Educational Services
Association • Mathematics Teacher Overview
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© 2011 California County Superintendents Educational Services Association • Mathematics Teacher Overview
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



Locate one grade level in your standards handout
Find
◦ Introduction
◦ Domains
◦ Clusters
◦ Standards
At your table, pick a grade level and have each person
or partner group choose a domain and read. Share out
reactions to language, content and structure.
Share out with the large group.
© 2011 California County Superintendents Educational Services Association • Mathematics Teacher 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
•
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 Teacher Overview
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© 2011 California County Superintendents Educational Services Association •
Mathematics Teacher Overview
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© 2011 California County Superintendents Educational Services Association • Mathematics Teacher Overview
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
Choose a domain that covers at least two
grade levels and read the standards at each
grade level.
◦ What do you notice?
◦ What big ideas are repeated

Be prepared to share key findings with the
group.
© 2011 California County Superintendents Educational Services Association • Mathematics Teacher Overview
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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 Teacher Overview
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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 Teacher Overview
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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 Teacher 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 <, 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 Teacher Overview
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Fraction Concepts
Source: www.mathisfun.com/numbers/fraction-number-line.html
© 2011 California County Superintendents Educational Services Association • Mathematics Teacher Overview
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Fraction Concepts
Source: www.mathisfun.com/numbers/fraction-number-line.html
© 2011 California County Superintendents Educational Services Association • Mathematics Teacher Overview
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


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 Teacher Overview
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© 2011 California County Superintendents Educational Services Association • Mathematics Teacher Overview
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Locate the 8th grade and Algebra 1 standards in your
standards handout.
Identify the domains in each set of standards
◦ Which are the same? Different?
Read the ends of the Algebra 1 standards?
◦ Where did these standards come from?
◦ Why are they all underlined and bolded?
What implications does this choice for grade 8
mathematics have on your school/district?
© 2011 California County Superintendents Educational Services Association • Mathematics Teacher Overview
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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 Teacher Overview
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


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 Teacher Overview
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High School Example-Geometry Content Cluster
© 2011 California County Superintendents Educational Services
Association • Mathematics Teacher Overview
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
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 Teacher Overview
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

Turn to the High School Conceptual
Clusters
Choose one of the clusters and identify the
(+) and (★) standards
◦ What do you notice about them?

Discuss:
◦ How does this organization of standards work in
the context of your existing high school
structure?
◦ What implications do these standards have on
your instruction?
© 2011 California County Superintendents Educational Services Association • Mathematics Teacher Overview
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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 Teacher Overview
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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 Teacher Overview
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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 Teacher Overview
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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 Teacher Overview
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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 Teacher Overview
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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 Teacher Overview
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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 Teacher Overview
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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 Teacher Overview
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Examples of Additional 15%:

Added two courses from California
Standards:
◦ Calculus
◦ Advanced Placement Probability and Statistics
© 2011 California County Superintendents Educational Services Association • Mathematics Teacher Overview
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
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 Teacher Overview
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
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 Teacher Overview
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