Balanced Math
Framework
August 15, 2013
Getting to Know You...
Math Style
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Grab a bingo card from the middle of
your table
Circulate the room searching for
teachers who can "sign" a box on
your bingo card (one signature per
box please)
Math Workshop
Readers and
Writers
=
Workshop
Framework
Framework
Math
Workshop
Math Review is.............
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Time to reinforce a previously taught
concept
Formative and based on daily student
understanding
Work that is de-briefed and discussed
Used to guide instruction
3 to 6 review problems (based on grade
level)
An opportunity to circulate and observe
Math Review is ...............
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Not time to teach a new concept or trick the
students
Not pre-printed or planned by yearlong or
unit objectives
Not work completed without discussion
Not used as a grade or graded by others
Not more than six problems
Not busy work
Math Review and Mental
Math
Problem Solving happens daily in the
classroom.
Conceptual Understanding
This is where you teach your
curriculum. You will use Math
Expressions, Glencoe and DMI
experiences as a resource.
Standards for Mathematical Practice
Mathematically Proficient Students...
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 repeating reasoning.
The Standards for Mathematical
Practice
Take a moment to examine
the first three words of each
of the 8 mathematical
practices... what do you
notice?
Mathematically Proficient Students...
The Standards for [Student]
Mathematical Practice
What are the verbs that illustrate the
student actions of each mathematical
practice?
Mathematical Practice #3: Construct viable
arguments and critique the reasoning of others
Mathematically proficient students:
• understand and use stated assumptions, definitions, and previously established results in constructing
arguments.
• make conjectures and build a logical progression of statements to explore the truth of their conjectures.
• analyze situations by breaking them into cases, and can recognize and use counterexamples.
• justify their conclusions, communicate them to others, and respond to the arguments of others.
• reason inductively about data, making plausible arguments that take into account the context from which the
data arose.
• compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which
is flawed, and-if there is a flaw in an argument-explain what it is.
• construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such
arguments can make sense and be correct, even though they are not generalized or made formal until
In the SJSD curriculum...
Standards for Mathematical Practice
Buttons Task
Standards for [Student]
Mathematical Practices
• "Not all tasks are created equal, and different
tasks will provoke different levels and kinds
of student thinking."
~ Stein, Smith, Henningsen, & Silver, 2000
• "The level and kind of thinking in which
students engage determines what they will
learn."
~ Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human
1997
Comparing Two Mathematical Tasks
Martha was re-carpeting her bedroom which
was 15 feet long and 10 feet wide. How many
square feet of carpeting will she need to
purchase?
~ Stein, Smith, Henningsen, & Silver, 2000, p. 1
Comparing Two Mathematical Tasks
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 in which to keep the rabbits.
1. 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?
2. How long would each of the sides of the pen be if they
had only 16 feet of fencing?
3. 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 read it will understand it.
~ Stein, Smith, Henningsen, & Silver, 2000, p.2
Comparing Two Mathematical Tasks
Discuss:
How are Martha's Carpeting Task and the
Fencing Task the same and how are they
different?
Comparing Two Mathematical Tasks
Lower-Level Tasks
Higher-Level Tasks
Reflection
My definition of a good teacher has changed
from "one who explains things so well that
students understand" to "one who gets
students to explain things so well that they
can be understood."
(Steven C. Reinhart, "Never say anything a kid can say!"
Mathematics Teaching in the Middle School 5, 8 [2000]:
478)
Richard Schaar
What I learned in school may be growing
increasingly obsolete today, but how I
learned to learn is what helps me keep up
with the world around me. I have the study
of mathematics to thank for that.
Rigor and Relevance
Rigor & Relevance
Framework
Relevance makes RIGOR possible, but only
when trusting and respectful relationships
among students, teachers, and staff are
embedded in instruction. Relationships
nurture both rigor and relevance.
Rigor is...
Article:
Tips for Using Rigor,
Relevance and
Relationships.
Rigor is...
Work that requires students to work at high
levels of Bloom's Taxonomy combined with
application to the real world.
3 Misconceptions of Rigor
•MORE – does not mean more
rigorous.
•DIFFICULT – increased difficulty
does not mean increased rigor.
•RIGID – “all assignments are due
by… no exception.”
RIGOR
Relevance
Why do I need to know this?
Misconceptions of
Relevance
•COOL – relevance doesn’t
exclusively mean what the
students do for “fun”
•EXCLUSIVE – relevance
without rigor does not ensure
success.
Relevance
Application Model
1. Knowledge in one discipline
2. Application within discipline
3. Application across disciplines
4. Application to real-world predictable situations
5. Application to real-world unpredictable situations
Putting it all together
Activity
Rigor and Relevance Card Sort
Six Questions All Students
Must Be Able to Answer
When seeking rigor, relevance, and relationships, all
students should be able to answer the following
questions:
1. What is the purpose of this lesson?
2. Why is this important to learn?
3. In what ways are you challenged to think in this
lesson?
4. How will you apply, assess, or communicate what
you've learned?
5. Do you know how good your work is and how you can
improve it?
6. Do you feel respected by other students in this class?
Mastery of Math Facts
After students have reached conceptual understanding, the following
fluencies are required by the CAS:
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K K.OA.5
Add/subtract within 5
1 1.OA.6
Add/subtract within 10
2 2.OA.2
Add/subtract within 20 (know single digit
products from memory)
2.NBT.5 Add/subtract within 100
3 3.0A.7 Multiply/divide within 100 (know single-digit products
from memory).
3.NBT.2 Add/subtract within 1000
4 4.NBT.4 Add/subtract within 1,000,000
5 5.NBT-5
Multi-digit multiplication
6 6.NS.2,3 Multi-digit division Multi-digit decimal operations
Common Formative
Assessment
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Kindergarten and First Grade Mathematics
Interviews
Math Fact Fluency - Reflex
Conference Notes - anecdotal records and Math
Reasoning Inventory
Performance Tasks
Mathematics Predictive Exams
Math Review and Mental Math
For Session 1:
Please read
Casebook pages 13-28
Cases 3, 4, 5
Lunch
Developing Mathematical
Ideas
August 15 & 16, 2013
DMI is about developing YOUR
mathematical understanding
“If our goal is to create mathematically
powerful children then we must also
create mathematically powerful teachers.”
--Lance Menster
What is DMI?
Developing Mathematical Ideas (DMI) is a
professional development curriculum
presented through a series of seminars.
The premise of the DMI materials
is that the art of teaching involves
helping students move from where
they are into the content to be
learned.
DMI Premises
DMI seminars bring together teachers from
kindergarten through middle grades to:
• learn mathematics content
• learn to recognize key mathematical ideas
with which their students are grappling
• learn how core mathematical ideas develop
across the grades
• learn how to continue learning about children
and mathematics
DMI is a Process
• This year we are working through the first
module: Building a System of Tens
• Today and tomorrow we are working in the
first 3 sessions
Session 1: Analyzing Addition Strategies
Session 2: Place Value and Multiplication
Session 3: The Mathematics of Algorithms
Session One:
Building a System of Tens
Student's Addition and Subtraction Strategies
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Mathematical Goals for Session One
I can use multiple strategies relying on the base
ten structure and properties of operation to add
and subtract multi-digit computations.
I can use a logical visual or physical
representation, such as a number line, base ten
blocks, arrays, etc. to explain why my strategy
works.
I can express the same amount in different ways
using powers of 10. For example, I can
decompose numbers using the powers of 10
(100 is 100 ones, or ten tens, or one ten and 90
ones, etc.).
Mental Math
57 + 24
Mental Math
83-56
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Mathematical Goals for Session One
I can use multiple strategies relying on the base
ten structure and properties of operation to add
and subtract multi-digit computations.
I can use a logical visual or physical
representation, such as a number line, base ten
blocks, arrays, etc. to explain why my strategy
works.
I can express the same amount in different ways
using powers of 10. For example, I can
decompose numbers using the powers of 10
(100 is 100 ones, or ten tens, or one ten and 90
ones, etc.).
Second Grade Strategies
40 - 26
Seventh Grade Strategies
123 - 76
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Mathematical Goals for Session One
I can use multiple strategies relying on the base
ten structure and properties of operation to add
and subtract multi-digit computations.
I can use a logical visual or physical
representation, such as a number line, base ten
blocks, arrays, etc. to explain why my strategy
works.
I can express the same amount in different ways
using powers of 10. For example, I can
decompose numbers using the powers of 10
(100 is 100 ones, or ten tens, or one ten and 90
ones, etc.).
Break
Chapter 1 Case Discussion
In your group, examine Focus
Questions 3, 4, and 5. Use any
manipulatives or chart paper you
need to work through these
questions.
Small Group Discussion
Whole Group
Discussion
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Mathematical Goals for Session One
I can use multiple strategies relying on the base
ten structure and properties of operation to add
and subtract multi-digit computations.
I can use a logical visual or physical
representation, such as a number line, base ten
blocks, arrays, etc. to explain why my strategy
works.
I can express the same amount in different ways
using powers of 10. For example, I can
decompose numbers using the powers of 10
(100 is 100 ones, or ten tens, or one ten and 90
ones, etc.).
Math Activity:
Close to 100 Game
The object of the game is to create two 2-digit
numbers whose sum is as close to 100 as
possible. Each game has five rounds. At the
end of five rounds the player with the lowest
total score wins.
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Mathematical Goals for Session One
I can use multiple strategies relying on the base
ten structure and properties of operation to add
and subtract multi-digit computations.
I can use a logical visual or physical
representation, such as a number line, base ten
blocks, arrays, etc. to explain why my strategy
works.
I can express the same amount in different ways
using powers of 10. For example, I can
decompose numbers using the powers of 10
(100 is 100 ones, or ten tens, or one ten and 90
ones, etc.).
For Session 2, please be sure to read:
Case studies 6, 7, & 10
Exit cards
1. What mathematical ideas did this session
highlight for you?
2. What was this session like for you as a
learner?
3. What burning questions do you have about
this session?
Session Two:
Building a System of Tens
The Base Ten Structure of Numbers
August 16, 2013
Mathematical Goal for Session Two
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The value of a number is determined by
multiplying the value of each digit by the
value of the place that it occupies and then
summing. For whole numbers, the value of
the place farthest to the right is 1; the value
of every other place is 10 times the value of
the place to its right.
Math Activity
Small Group: Representing
Multiplication
Whole-group Discussion:
Sharing Representations
Mathematical Goal for Session
Two
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The value of a number is determined by
multiplying the value of each digit by the
value of the place that it occupies and then
summing. For whole numbers, the value of
the place farthest to the right is 1; the value
of every other place is 10 times the value of
the place to its right.
DVD: Interview with Three
Students
Mathematical Goal for Session
Two
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The value of a number is determined by
multiplying the value of each digit by the
value of the place that it occupies and then
summing. For whole numbers, the value of
the place farthest to the right is 1; the value
of every other place is 10 times the value of
the place to its right.
Break
Case Discussion
Think about:
1. What is right about the student's thinking?
2. Where has the student's thinking gone awry?
Small-Group: Ideas about the
Number System
Whole-Group: Number
Lines
Mathematical Goal for Session
Two
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The value of a number is determined by
multiplying the value of each digit by the
value of the place that it occupies and then
summing. For whole numbers, the value of
the place farthest to the right is 1; the value
of every other place is 10 times the value of
the place to its right.
During your working lunch please be
sure to read Case 14
(pages 65 - 70 Casebook).
Lunch
Session Three:
Building a System of Tens
Making Sense of Addition and Subtraction Algorithms
Mathematical Goals for
Sessions Three
• Extend students’ knowledge of place value
(ones, tens, hundreds) to solving addition
and subtraction problems efficiently.
• Understand how place value underlies the
traditional algorithms for addition and
subtraction.
Whole Group: Addition and
Subtraction Strategies
Investigating addition strategies:
• Creating Verbal Descriptions
• Visual Representations
• Story Context
Small-Group: Addition and
Subtraction Strategies
Creating Subtraction Posters
Gallery Walk: addition and
subtraction strategies
Mathematical Goals for
Sessions Three
• Extend students’ knowledge of place
value
(ones, tens, hundreds) to solving addition
and subtraction problems efficiently.
• Understand how place value underlies the
traditional algorithms for addition and
subtraction.
Break!
DVD: Addition and Subtraction
Mathematical Goals for
Sessions Three
• Extend students’ knowledge of place value
(ones, tens, hundreds) to solving addition
and subtraction problems efficiently.
• Understand how place value underlies the
traditional algorithms for addition and
subtraction.
Small Group Discussion: Addition
and Subtraction Algorithms
Case 14
Focus Questions 3 and 4
Whole Group Discussion:
Addition and Subtraction
Algorithms
• What is the same about the two strategies in
case 14?
• What is different about the two strategies?
• What are the mathematical
principles underlying each of
the strategies the students
use?
Mathematical Goals for
Sessions Three
• Extend students’ knowledge of place value
(ones, tens, hundreds) to solving addition
and subtraction problems efficiently.
• Understand how place value underlies the
traditional algorithms for addition and
subtraction.
Task #1 - Read and discuss the article
"Orchestrating Discussions"
Task #2 - Do the Writing Assignment: A math
interview
Exit Cards..............
• What was important or significant to you in
the mathematics discussed at this session?
• What mathematics are you still wondering
about from this session?
• What do you want to tell us about
how the seminar is working for you?