Next Generation
Mathematics Standards for
Content and Practices
Grades 6-12
st
21
Presented to the
Century
After Schools Program
Lou Maynus, Mathematics Coordinator
WVDE
CCSS Math Content
PreK
K
1
2
3
4
5
6
7
8
HS
Mathematical Practices
Counting &
Cardinality
Number & Operations in Base Ten
Number & Operations Fractions
Operations & Algebraic Thinking
Ratios &
Proportional
Relationships
The Number System
Number
&
Quantity
Expressions & Equations Algebra
Functions
Geometry
Measurement & Data
Statistics & Probability
2
Practice Standards
1. Make sense of problems & persevere in solving
them.
2. Reason abstractly & quantitatively.
3. Construct viable arguments & critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for & make use of structure.
8. Look for & express regularity in repeated
reasoning.
3
Content & Practices
• Understand is used in the CCSS to mean that
students can explain the concept with
mathematical reasoning, including
– Justifying why a particular mathematical
statement is true,
– Explaining where a mathematical rule comes
from
– Giving concrete illustrations, and
– Providing mathematical representations &
example applications.
4
Content & Practices
Standards that begin with
“understand” are good
opportunities to connect the
Practices to the Content!
5
Content & Practices
• “A lack of understanding (of the content)
effectively prevents a student from
engaging in the Mathematical Practices.”
• “Conceptual understanding is not an
option – it is an expectation!”
Skip Fennell, former NCTM president
6
Standards for Mathematical Practice
• Carry across all grade levels
• Describe habits of mind of a mathematically
expert student
• Describe varieties of expertise that should
be developed in students
• Rely upon longstanding important processes
and proficiencies
• Articulate the behaviors of mathematically
competent students
7
Practice Standards
• A big part of the intent of the Math
Practices requires that the Content be
taught through the Practices. That way,
the connections are real – integrated
rather than interspersed.
• The Standards for Mathematical Practice
are what students who understand
mathematics “do” with the content.
8
17
Great Tasks
Matter!
18
Answer Getting vs Learning Math
• US:
– How can I teach my kids to get the
answer to this problem?
• Japan:
– How can I use this problem to teach
mathematics they don’t already know?
19
Tasks Matter
• “There is no decision that teachers make
that has a greater impact on students’
opportunities to learn and on their
perceptions about what mathematics is than
the selection or creation of the tasks.”
Lappan & Briars, 1995
• “If we want students to develop the capacity
to think, reason, and problem solve then we
need to start with high-level, cognitively
complex tasks.”
Stein & Lane, 1996
20
Tasks Matter
• “Not all tasks are created equal, and different
tasks will provoke different levels and kinds of
student thinking.”
Stein, Smith, Henningsen, & Silber, 2000
• “The level and kind of thinking in which
students engage determines what they will
learn.”
Heibert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997
• “The problems we encounter in the real world
rarely tell us exactly what question we need to
answer and they almost never tell us where to
begin.”
Thinkmath.edc.org
21
Tasks Matter
• “Mathematical understanding and
procedural skill are equally important,
and both are assessable using
mathematical tasks of sufficient
richness.”
Common Core Standards, 2010
• CCSS ask you to connect the Content
Standards and the Practice Standards.
This happens in the context of good
problems.
22
Dan Meyer:
Math Class Needs a Makeover
TED Talk
http://www.ted.com/talks/dan_meyer
_math_curriculum_makeover.html
23
What is a Great Task?
A Great Task:
1. Is open-ended
2. Does not have a solution path that is
immediately obvious (or implied)
3. Requires examination & perseverance –
challenges students
4. Requires students to think and not just
rely on memorized procedures
24
What is a Great Task?
A Great Task:
5. Begs for discussion – offering rich
discourse on the math involved
6. Requires students to connect
mathematical skill, understanding and
reason
7. Requires students to interpret and
communicate results
25
Why use a Great Task?
• The primary use of great tasks is as a tool for
teachers to evaluate the depth of student
understanding, to catch misconceptions, and as a
foundation for classroom discourse designed to
extend students’ thinking.
• Great tasks allow students to demonstrate their
thinking and give their teachers insight into
student understandings and misconceptions.
• Great tasks provide opportunities to evaluate
reasoning and examine sense making.
NCSM Great Tasks for Mathematics , 2011
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Tips to Help Students Persevere
• Pose problem, make sure students
understand the language, no hints at the
solution
• Focus students on the problem situation, not
the question/answer game. Hide question and
ask them to formulate questions
• Require students to start by explaining to
themselves the meaning of the problem and
looking for entry points
• Occasionally pose problems that ask only for
the analysis and not for a numeric answer to
its solution
27
North Shore Swim Club
Annual Breakfast
In Summary
• Promote the standards for mathematical
practice.
• Choice rich tasks that children will want to
solve and integrate the content with the
practices.