Common Core Standards, K – 12
Kentucky Core Academic
Standards
Mathematics
1
What do we mean by:
Understanding Mathematics?
Jake is a Genius
• Each person get 4 post-it notes and silently read pages 3, 4,
& 5, and the bottom of page 8.
– On the 1st post-it write down a couple of the most important ideas
in the Introduction (pg 3),
– On the 2nd post-it write down a couple of the most important
ideas about Understanding Mathematics (pg 4),
– On the 3rd post-it write down a couple of the most important
ideas about How to read the grade level standards and what the
standards do and do not mean (pg 5),
– On the 4th post-it write down a couple of the most important
ideas about Connecting the Standards for Mathematical Practice
to the Standards for Mathematical Content (pg 8).
2
What do we mean by:
Understanding Mathematics?
• When everyone has finished reading & jotting down their ideas, share your
ideas from each post-it by having each person read one idea for each topic.
Take turns letting a different person be the first to share their ideas for each
of the topics:
– Introduction
– Understanding Mathematics
– How to read the grade level standards & what the standards mean
– Connecting the Standards for Mathematical Practice to the Standards for
Mathematical Content
How did this process clarify your thinking about what was read?
3
Standards for
Mathematical Practice
The standards for mathematical Practice are important “processes and proficiencies”
and describe varieties of expertise that mathematics educators at all levels should
seek to develop in their students. See page 6. They are based on ideas important to
mathematics education from –
the NCTM process
the strands of mathematical proficiency specified in the
standards of:
National Research Council’s report Adding It Up:
 problem solving,
 adaptive reasoning,
 reasoning and proof,  strategic competence,
 communication,
 conceptual understanding,
(comprehension of mathematical concepts, operations and
 representation, and
relations),
 connections.
 procedural fluency, and
(skill in carrying out procedures flexibly, accurately, efficiently
and appropriately),

productive disposition.
(habitual inclination to see mathematics as sensible, useful,
and worthwhile, coupled with a belief in diligence and one’s
4
Standards for
Mathematical Practice – pgs 6-8
• Silently read the mathematical practice
assigned to your group and highlight your
MIPs.
• On chart paper, your group will represent your
mathematical practice in words and pictures.
• Be prepared to share out.
5
Standards for
Mathematical Practice – pgs 6-8
• Cut out your foldable and write one
mathematical practice on each front flap.
• As each group presents their mathematical
practice, use your foldable to record ideas
about the meaning of each practice.
foldable
6
7
Middle School Landscape
• Using the topics assigned to your group,
identify where the topic starts and stops in the
CCSS using the Cluster & Domain Progression
handout.
• Discuss with your group any differences you
notice between the progression of your topic
in the CCSS and the KY POS.
• Be prepared to share your findings.
8
High School Landscape
• Use the suggested CCSS Pathway to compare
the content of this course to the table of
contents of your assigned text.
• Discuss with your group any content
differences you notice between the two.
• Be prepared to share your findings.
9
10
Supplement
Practices
In what ways do the
differences
between the old
and new
arrangement of
content topics
impact any two of
these areas?
Interventions
Content
Knowledge
Assessments
11
Analyzing Cognitive Demand
• All mathematical tasks are not created equal.
• As we solve two problems dealing with area,
we will look for similarities and differences
between the two.
• Afterwards, we will sort new and different
problems into low and high level tasks.
12
Why instructional tasks
are important?
Comparing Two Mathematical Tasks:
1. Martha’s Carpeting Task
2. The Fencing Task
13
Martha’s Carpeting Task: Solve on your own,
then discuss with your table group.
Martha was recarpeting her bedroom, which
was 15 feet long and 10 feet wide. How many
square feet of carpeting will she need to
purchase?
14
Martha’s Carpeting Task
Solution Strategies:
Using the Area Formula
A=lxw
A = 15 x 10
A = 150 square feet
Drawing a Picture
10
15
15
The Fencing Task:
Solve on your own then discuss
with your table group.
Ms. Brown’s class will raise rabbits for
Their spring science fair. They have
24 feet of fencing with which to build a
rectangular rabbit pen to keep the rabbits.
– If Ms. Brown’s students want their rabbits to have as much
room as possible, how long would each of the sides of the
pen be?
– How long would each of the sides of the pen be if they had
only 16 feet of fencing?
– How would you go about determining the pen with the most
room for any amount of fencing? Organize your work so that
someone else who reads it will understand it.
16
The Fencing Task
Solution Strategies:
Diagrams on
Grid Paper
17
The Fencing Task
Solution Strategies: Using a Table
Length
1
2
Width
11
10
Perimeter
24
24
Area
11
20
3
4
5
9
8
7
24
24
24
27
32
35
6
6
24
36
7
5
24
35
18
The Fencing Task Solution Strategies:
Graph of Length and Area
40
Max
35
30
Area
25
20
15
10
5
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Length
19
Comparing Two Mathematical Tasks
•How are Martha’s Carpeting
Task and the Fencing Task
the same and how are they
different?
20
Similarities and Differences
Similarities
• Both are “area”
problems
• Both require prior
knowledge of area
Differences
• The amount of thinking and
reasoning required
• The number of ways the
problem can be solved
• Way in which the area
formula is used
• The need to generalize
• The range of ways to enter
the problem
21
• Would both the Carpet & Fencing tasks
likely be assigned by both a teacher who
accepted the student claim and a
teacher who explored the claim?
• Which task, Carpet or Fencing, would help
the student who made the claim to
actually understand why her claim is
22
Mathematical Tasks:
A Critical Starting Point for Instruction
Not all tasks are created equal, and
different tasks will provoke different levels
and kinds of student thinking.
Stein, Smith, Henningsen, & Silver, 2000
23
Mathematical Tasks:
The level and kind of thinking in which
students engage determines what they
will learn.
Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997
24
Learning
Targets
Tasks,
Problems,
Assignments
25
Mathematical Tasks:
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
with which the teacher engages students in
studying mathematics.
Lappan & Briars, 1995
26
Mathematical Tasks:
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
27
Task Sort
• Being able to sort tasks according to cognitive
demand allows teachers to better match tasks
with learning targets.
• As you sort through these tasks:
– Separate them by “cognitive demand”
– Map all applicable standards/ practices to these
tasks
28
Demonstrating
Competency
Does this always look
the same?
When does a teacher
need to consider what
this looks like?
How does this impact
teaching and learning,
assessment, and
interventions?
29
Demonstration
of Learning
Teaching and
Learning
Assessments
Standards/ Targets
30
Fortune Cookie
• Each team receives an envelope.
• One person draws a question, and makes one
statement about the topic, then passes it on.
• The next person adds their statement or
responds to the previous statement.
• When everyone has responded to the first
statement, another person draws from the
envelope and repeats the process.
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