Transcript K-8 Mathematics Standards

Empowering Learners
through the Common
Core State Standards
in Grades K-2
Juli K. Dixon, Ph.D.
University of Central Florida
[email protected]
A student was asked to solve
48 + 25 and the student did
this…
40 + 20 = 60
A student was asked to solve
48 + 25 and the student did
this…
40 + 20 = 60
8 + 2 = 10
A student was asked to solve
48 + 25 and the student did
this…
40 + 20 = 60
8 + 2 = 10
60 + 10 = 70
A student was asked to solve
48 + 25 and the student did
this…
40 + 20 = 60
8 + 2 = 10
60 + 10 = 70
70 + 3 = 73
Perspective…
We know we want students to
explain and justify mathematics in
these ways, but how do we get
them here?
Perspective…
We know we want students to
explain and justify mathematics in
these ways, but how do we get
them here?
How is this related to the Common
Core State Standards (CCSS)?
Background of the CCSSM
• Published by the National Governor’s
Association and the Council of Chief State
School Officers in June 2010
• Result of collaboration among 48 states
• Provides a focused curriculum with an
emphasis on teaching for depth
• Consists of Content Standards and Standards
for Mathematical Practice
Background of the CCSSM
45 States + DC have adopted the Common Core State Standards
Minnesota adopted the CCSS in ELA/literacy only
Background of the CCSSM
“… standards must address the problem of a
curriculum that is ‘a mile wide and an inch
deep.’ These Standards are a substantial
answer to that challenge” (CCSS, 2010, p. 3).
Background of the CCSSM
“… standards must address the problem of a
curriculum that is ‘a mile wide and an inch
deep.’ These Standards are a substantial
answer to that challenge” (CCSS, 2010, p. 3).
So what do these standards look like anyway?
Content Standards Wordle
for Grades K-2
Content Standards
• Define expectations for students at each grade
level
• Use concepts from earlier grades
• Emphasize need to justify mathematical moves
• Indicate understanding and skill are equally
important
• Include expectations that students demonstrate
understanding of procedures
Content Standards
• Critical Areas – describe areas of focus
• Domains – group related clusters
• Clusters – group related standards
• Standards – define what students should know
and be able to do
Content Standards
Measurement and Data
K.MD
Describe and compare measurable attributes.
1.Describe measurable attributes of objects, such as length or weight.
Describe several measurable attributes of a single object.
2.Directly compare two objects with a measurable attribute in common, to
see which object has “more of”/“less of” the attribute, and describe the
difference. For example, directly compare the heights of two children and
describe one child as taller/shorter.
Classify objects and count the number of objects in each category.
3.Classify objects into given categories; count the numbers of objects in each
category and sort the categories by count.
Content Standards
Domain
Measurement and Data
K.MD
Cluster
Describe and compare measurable attributes.
Standard
1.Describe measurable attributes of objects, such as length or weight.
Describe several measurable attributes of a single object.
Standard
2.Directly compare two objects with a measurable attribute in common, to
see which object has “more of”/“less of” the attribute, and describe the
difference. For example, directly compare the heights of two children and
describe one child as taller/shorter.
Cluster
Classify objects and count the number of objects in each category.
Standard
3.Classify objects into given categories; count the numbers of objects in each
category and sort the categories by count.
Perspective…
“One hallmark of mathematical
understanding is the ability to justify, in a
way appropriate to the student’s
mathematical maturity, why a particular
mathematical statement is true or where
a mathematical rule comes from” (CCSS,
2010, p. 4).
Perspective…
“The Standards for Mathematical Practice
describe varieties of expertise that
mathematics educators at all levels should
seek to develop in their students”
(CCSS, 2010, p. 6)
Making Sense of the
Mathematical Practices
The Standards for Mathematical Practice
are based on:
• The National Council of Teachers of
Mathematics’ (NCTM) Principles and
Standards for School Mathematics
(NCTM, 2000), and
• The National Research Council’s (NRC)
Adding It Up (NRC, 2001).
Making Sense of the
Mathematical Practices
NCTM Process Standards:
• Problem Solving
• Reasoning and Proof
• Communication
• Representation
• Connections
Making Sense of the
Mathematical Practices
NRC Strands of Mathematical Proficiency:
• Adaptive Reasoning
• Strategic Competence
• Conceptual Understanding
• Procedural Fluency
• Productive Disposition
Making Sense of the
Mathematical Practices
NRC Strands of Mathematical Proficiency:
• Adaptive Reasoning
• Strategic Competence
• Conceptual Understanding
• Procedural Fluency
• Productive Disposition
Standards of Mathematical
Practice Wordle
Making Sense of the
Mathematical Practices
The 8 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 reasoning
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
Perspective…
The following represents a
recommendation from the Center for
the Study of Mathematics Curriculum
(CSMC, 2010)
Perspective…
Lead with Mathematical Practices
1Implement CCSS beginning with mathematical
practices,
2Revise current materials and assessments to
connect to practices, and
3Develop an observational scheme for principals
that supports developing mathematical practices.
(CSMC, 2010)
Content Standards
Domain
Counting and Cardinality
K.CC
Cluster
Compare Numbers.
Standard
6.
Identify whether the number of objects in one group is greater than,
less than, or equal to the number of objects in another group, eg., by using
matching and counting strategies.
Solve this…
Perspective…
What do you think Kindergarten
children will do?
Perspective…
What do you think Kindergarten
children will do?
Consider how the student is allowed to
struggle through a problem in this
kindergarten video.
Perspective…
Are you observing this sort of
productive struggle in classrooms?
Is it important?
Perspective…
What does this have to do with the
Common Core State Standards for
Mathematics (CCSSM)?
With which practices were the
grade K students engaged?
The 8 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 reasoning
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
With which practices were the
grade K students engaged?
The 8 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 reasoning
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
Perspective…
In an effort to simplify students’ learning
pathways and minimize barriers (stigler, et. al.,
1999), teachers often provide students with
efficient procedures too early.
When we do this – we minimize students’
opportunities to engage in these practices.
Reason
abstractly
and
2
quantitatively
Reason
abstractly
and
2
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.
Reason
abstractly
and
2
quantitatively
Consider the following problems:
Jessica has 8 key chains. Calvin has 9 key
chains. How many key chains do they have all
together?
Jessica has 8 key chains. Alex has 15 key
chains. How many more key chains does Alex
have than Jessica?
Reason
abstractly
and
2
quantitatively
Consider the following problems:
Jessica has 8 key chains. Calvin has 9 key
chains. How many key chains do they have all
together?
Jessica has 8 key chains. Alex has 15 key
chains. How many more key chains does Alex
have than Jessica?
Key words seem helpful
Reason
abstractly
and
2
quantitatively
Consider the following problems:
Jessica has 8 key chains. Calvin has 9 key
chains. How many key chains do they have all
together?
Jessica has 8 key chains. Alex has 15 key
chains. How many more key chains does Alex
have than Jessica?
Key words seem helpful, or are they….
Reason
abstractly
and
2
quantitatively
Now consider this problem:
Jessica has 8 key chains. How many more key
chains does she need to have 13 key chains
all together?
Reason
abstractly
and
2
quantitatively
Now consider this problem:
Jessica has 8 key chains. How many more key
chains does she need to have 13 key chains
all together?
How would a child who has been conditioned
to use key words solve it?
Reason
abstractly
and
2
quantitatively
Now consider this problem:
Jessica has 8 key chains. How many more key
chains does she need to have 13 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 these problems?
Reason
abstractly
and
2
quantitatively
Consider this problem:
Jessica has 8 key chains. Calvin has 9 key
chains. How many key chains do they have all
together?
I know that 8 + 8 = 16, so…
Reason
abstractly
and
2
quantitatively
Consider this problem:
Jessica has 8 key chains. Alex has 15 key
chains. How many more key chains does Alex
have than Jessica?
I know that 8 + 8 = 16, so…
Reason
abstractly
and
2
quantitatively
Now consider this problem:
Jessica has 8 key chains. How many more key
chains does she need to have 13 key chains
all together?
8 + __ = 13
(How might making a ten help?)
Reason
abstractly
and
2
quantitatively
What happens when this child gets to 2nd
grade?
Empowering Young
Learners
Consider this Kindergarten class.
With which Standard(s) for Mathematical
Practice are they engaged?
Empowering Young
Learners
Consider this Kindergarten class.
What did you notice?
With which practices were the
grade K students engaged?
The 8 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 reasoning
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
With which practices were the
grade K students engaged?
The 8 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 reasoning
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
Use appropriate tools
5
strategically




This practice supports hands-on learning
Tools must include technology
Tools also include non-technological tools
such as manipulatives, number lines, and
paper and pencil
Mathematically proficient students know
which tool to use for a given task.
With which practices were the
grade K students engaged?
The 8 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 reasoning
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
Construct viable arguments
3 and critique the reasoning
of others
What does this look like in a grade K2 class?
How can we be intentional about
providing opportunities for students
to engage in this practice?
Empowering Young
Learners
Consider this:
2 0 3
- 6 8
Empowering Young
Learners
Consider this:
2 0 3
- 6 8
Was the language you used in talking
through the solution “precise”?
Making Sense of the
Mathematical Practices
The 8 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 reasoning
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
Empowering Young
Learners
Consider this:
2 0 3
- 6 8
I heard a student say this…
6 Attend to precision
Consider this:
2 0 3
- 6 8
Does our math talk sound more like
this?
http://video.google.com/videoplay?docid=7106559846794044495
Making Sense of the
Mathematical Practices
The 8 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 reasoning
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
How do we support the
transition to the Common Core?



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.
Social norms need to be established in
classroom and professional development
settings to address misconceptions in
respectful ways.
Empowering Learners
through the Common
Core State Standards
in Grades K-2
Juli K. Dixon, Ph.D.
University of Central Florida
[email protected]