Practicing the Standards for
Mathematical Practice (SMP)
Susie W. Håkansson
California Mathematics Project
Overview of Session
The CDE Professional Learning Module Series
K-12 SMP Module
Outline of K-12 SMP Module
Inside the K-12 SMP Module
Conclusion
Professional Learning Module Series
Overview of the Common
Core State Standards for
California Educators
CALIFORNIA DEPARTMENT OF EDUCATION
Tom Torlakson, State Superintendent of Public Instruction
The Common Core State Standards
Professional Learning Series
• Overview of the Common Core State Standards for
California Educators
• Mathematics: Kindergarten through Grade Eight
Learning Progressions
• Mathematics: Kindergarten through Grade Twelve
Standards for Mathematical Practice
• English Language Arts: Informational Text—Reading
4 | California Department of Education
Professional Learning Module Series
Mathematics: Kindergarten
through Grade 12
Standards for Mathematical
Practice (K-12 SMP)
CALIFORNIA DEPARTMENT OF EDUCATION
Tom Torlakson, State Superintendent of Public Instruction
K-12 SMP Module:
CMP Development Team
Susie W. Hakansson
Kyndall Brown
Patrick Callahan
Carol Cronk
Kathlan Latimer
Joanne Rossi Becker
Sheri Willebrand
Tsai-Tsai O-Lee
Diane Kinch
Ginny Wu
Joan Easterday
Patricia Dickenson
Welcome and Overview
Module Overview: Goals and Organization
Thinking about Thinking
Background of the CCSS for Mathematics
Pre-Assessment
Goal of Module
Throughout the module, the overarching
goals of the K-12 SMP module is two-fold:
Deepen the teachers’ understanding of
the SMP
Support the learning of all students
1. Make sense of problems and persevere in
solving them
6. Attend to precision
OVERARCHING HABITS OF MIND
CCSS Mathematical Practices
REASONING AND EXPLAINING
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
MODELING AND USING TOOLS
4. Model with mathematics
5. Use appropriate tools strategically
SEEING STRUCTURE AND GENERALIZING
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
Overview of Units
Unit 1: Teaching and Learning the Standards for
Mathematical Practice
Unit 2: Overarching Habits of Mind (MP1 and MP6)
Unit 3: Reasoning and Explaining (MP2 and MP3)
Unit 4: Modeling and Using Tools (MP4 and MP5)
Unit 5: Seeing Structure and Generalizing (MP7 and MP8)
Unit 6: Summary and Next Steps
Unit 1: Teaching and
Learning the SMP
Observing Students
Two Types of Mathematics Standards
Interaction of the Practice Standards and Content
Standards
Meeting the Needs of All Students
Looking Ahead
Students at Work
You will see a video of a 2nd Grade EL class
engaged in mathematics. Here are some
questions to think about as you watch the
video:
What do you see the students doing?
In what ways are the students engaged?
Describe your observations of students’
thinking, problem solving and interactions.
Pre Self-Assessment Survey
Take a few minutes to complete the Pre SelfAssessment Survey.
This will give you an indication of your current
comfort level and confidence in supporting all
students to be successful in the eight SMP.
We will ask users to complete this Self-Assessment
Survey after they have completed the module.
We expect huge increases.
Points of Intersection
Expectations that begin with
the word ‘understand’ are
often especially good
opportunities to connect the
practices to the content.
- CCSS for Mathematics, p.8
Grain Size
It is not expected that a single
problem would engage students
with all eight practice standards,
even a lesson may be too small to
accomplish that.
- Phil Daro, CCSS for Mathematics lead writer
Unit 2: Overarching Habits
of Mind (MP1 and MP6)
Unpacking MP1 and MP6
Sense Making and Mindsets
Student Self-Efficacy and Perseverance
Attending to Precision
Summary and Reflection
Students Attending to Precision
A student is asked if 5 = 5 is true. The
student says no.
Students Attending to Precision
A student is asked if the statement
3 + 5 = 3 + 5 is true. The student
says it is false.
Students Attending to Precision
A student is asked if 8 = 3 + 5 and 3 + 5 = 8
are the same number sentences. Her
answer is based on whether it has to
make sense to kids or if it has to be what
the student perceives the teacher wants.
Reflection
There is something off in the first two of
these students’ understanding of the equal
sign. Why do they say the statements are not
true?
The third student bases her answer on
whether or not it has to make sense or be
what the teacher wants to hear. Why do you
think she says this?
Unit 3: Reasoning and
Explaining (MP2 and MP3)
Unpacking MP2 and MP3
Beginning to Reason: Definitions and
Conjectures
Explaining and Justifying
Identifying Flaws in Reasoning
Making Arguments More Viable
Summary
Beginning to Reason
It is important in mathematics to give clear and logical
explanations.
Please take a few minutes to write down your ideas
What is the definition of an even number?
Explain why the sum of two even numbers is always
even.
Beginning to Reason
Over 300 students from grade five through high
school geometry responded to these prompts.
Consider the clarity and logic of the following
student generated definitions. Does the
student’s grade level impact the logic and/or
clarity?
th
5
Grade Students’
Responses
What is the definition of an even number?
A number that ends in 2, 4, 6, 8, 0. If you count by 2s,
you will eventually make it to that number. It can be
split in half equally.
To find out which ones are even and which ones are
odd, I think this one … 2, 4, 6, 8. Every other one is
going to be even.
th
5
Grade Student’s
Response
What is the definition of an even number?
4 is an even number because if you divide it by 2, it
will have an equal set of numbers (see diagram
below).
4
2
1
2
1
1
1
Middle School Students’
Responses
What is the definition of an even number?
The definition of an even number is like, they are
multiplying by 2, because it goes to 2, 4, 6, 8 and etc.
That’s what I like about math.
You skip the odd number and that’s the even number
A number that can be evenly divided by half
High School Students’
Responses
What is the definition of an even number
Usually not a prime number except two, and is not odd.
An even number, such as two, can always be divisible and
stays a composite number. The pattern of all numbers is
odd, even, odd, even, etc. So every other number is even. If
a number ends in an even number (2, 4, 6, 8 and technically
0) that number is automatically even.
Because when you go on a date, you go with another
person and there’s two of you. And if you go on a double
date, there’s four of you. But if there is a third wheel,
because their date didn’t show up that’s just awkward and
odd … no pun intended.
Reflection
Discuss what argument strategies (e.g.,
example, counter-examples, non-examples)
are evident in the student definitions. What
do you think should be included in a
definition to be considered complete and
clear?
Unit 4: Modeling and Using
Tools (MP4 and MP5)
Unpacking MP4 and MP5
Introduction to Modeling with Mathematics
(MP4)
Modeling with Mathematics at the Different
Grade Spans
Introduction to Using Appropriate Tools
Strategically (MP5)
Use of Tools at Different Grade Spans
Summary and Reflection
PISA Rubric
Three Levels
Level one problems are those that are
directly translatable from a context. An
example is a simple word problem from
which students can formulate an equation.
PISA Rubric
Three Levels
Level two problems are those where a model
can be modified to satisfy changed
conditions. Such problems allow students to
study patterns and relationship between
quantities, and represent these patterns and
relationships using words, numbers, symbols
and pictures. These can be problems that
have multiple solution strategies, but usually
have only one correct solution.
PISA Rubric
Three Levels
Level three problems have no predetermined
solution. Such problems require students to
make assumptions about the context,
develop a strategy to solve the problem,
check their answers, present results, and
possibly revise their solution strategies and
begin the process over again.
Modeling
Different Grade Spans
Grades K-2:
Connie has 13 marbles. Juan has 5 marbles. How many
more marbles doe Connie have than Juan?
Ellen walks 3 miles an hour. How many hours will it
take her to walk 15 miles?
The giraffe in the zoo is 3 times as tall as the
kangaroo. The kangaroo is 6 feet tall. How tall is the
giraffe?
These are examples of Level 1 modeling tasks.
Modeling
Different Grade Spans
Grades 3-5:
Lamar wants a new toy truck that sells for $25. Lamar
has $3 dollars now. Create a plan that would help
Lamar buy his truck three weeks from today.
This is an example of a Level 3 modeling task.
Modeling
Different Grade Spans
Grades 6-8:
The physicist Enrico Fermi enjoyed posing questions like
the ones below to his students and colleagues.
What is the weight of all the trash produced in your
house in a year?
What do you think is the volume of gasoline your car
uses in a year? How does this compare to the volume
of liquid (water, soda, coffee, etc.) you drink in a
year?
These are examples of Level 3 modeling tasks.
Modeling
Different Grade Spans
Grades 9-12:
Suppose a building has 5 floors (1-5), occupied by offices.
The ground floor (0) is not used for business purposes. Each
floor has 80 people working on it, and there are 4 elevators
available. Each elevator can hold 10 people at one time.
The elevators take 3 seconds to travel between floors and
average 22 seconds on each floor when someone enters
and exits. If all of the people arrive at work at about the
same time and enter the elevator on the ground floor, how
should the elevators be used to get the people to their
offices as quickly as possible?
This is an example of a Level 3 modeling task.
Modeling
Different Grade Spans
Grades 9-12:
Note: To make this problem more open-ended, allow
students to decide how many people work on each floor,
the times when people arrive in the morning, and how long
the elevator takes to travel between floors, as well as how
long it remains on each floor.
Unit 5: Seeing Structure and
Generalizing (MP7 and MP8)
Unpacking MP7 and MP8
Structure, Repeated Reasoning, and
Generalization
Making Sense of a Growing Pattern
Geometry Examples of Structure and
Generalization
Performance Tasks and Student Work
Summary and Reflection
Consecutive Sums
Some numbers can be written as a sum of
consecutive positive integers. For example:
6 = 1+2+3
15 = 4+5+6
= 1+2+3+4+5
Which numbers have this property? Explain.
Let’s look at what might be expected of
students at each grade span when working
on this problem.
Grades K-2
Write the first 10 numbers as a sum of other
numbers. Which of these sums contain only
consecutive numbers?
Grades 3-5
Some numbers can be written as a sum of
consecutive positive integers. For example:
6 = 1+2+3
15 = 4+5+6
= 1+2+3+4+5
In small groups, find all numbers from 1-100 that
can be written as a consecutive sum. Look for
patterns as you work. Conjecture which numbers
can and which cannot be expressed as a
consecutive sum. How can some of the sums be
used to find others?
Grades 6-8
Some numbers can be written as a sum of
consecutive positive integers. For example:
6 = 1+2+3
15 = 4+5+6
= 1+2+3+4+5
Which numbers have this property? Explain.
Sally made this conjecture. “Powers of two
cannot be expressed as a consecutive sum.”
Agree or disagree and explain your reasoning.
Grades 9-12
Some numbers can be written as a sum of
consecutive positive integers. For example:
6 = 1+2+3
15 = 4+5+6
= 1+2+3+4+5
Exactly which numbers have this property?
When investigating this problem, Joe made the
following conjecture: “A number with an odd factor
can be written as a consecutive sum and the odd
factor will be the same as the number of terms.”
Agree or disagree with this statement and explain
your reasoning.
Reflection
For each of the grade spans, how will
understanding the structure of the number
system support students as they work on this
problem? How will your students use
repeated reasoning?
Unit 6: Summary and Next
Steps
Summary
Take Action: Planning the Next Steps
Post Self-Assessment Survey
This is the same survey that you completed at the
beginning of this session.
You have only seen a glimpse of this module. Even
with this limited introduction, we hope that you have
gained some understanding of the SMP.
End of Module
Post-Assessment
Glossary
Resources
Acknowledgements
Module Evaluation
Going Online
Go to http://www.myboe.org
Select California’s Common Core State Standards
(right side of page)
Then select CCSS Professional Learning Modules
On the top right is the module list.
GSDMC Student Awards
You are invited to the TODOS San Diego Area
Student Awards beginning at 5:00 PM this
afternoon in the Starboard Room. Students,
their families, and nominating teachers will be
recognized. Refreshments will be served.
Support for this annual heartwarming event is
being provided by TODOS, GSDMC, and Texas
Instruments.
You can recommend your students (K-12) for
next year’s awards!
Speaker Evaluation
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Strongly
Agree
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Text your message to this Phone Number: 37607
“ 31165
___ ___ ___
Speaker was wellprepared and
knowledgeable Speaker was
engaging and an
effective presenter
Example: “ 31165
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Other comments,
Session matched suggestions, or
feedback
title and
description in
program book
545 Great session! ”
Thanks!
For more information, contact :
Susie W. Håkansson
shakans@ucla.edu
www.myboe.org