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Balanced Math Framework
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Transcript of Balanced Math Framework
Balanced Math Framework
August 15, 2013
Getting to Know You...Math Style
• 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)
Readers and Math Writers =
Workshop Workshop
Framework Framework
Math Workshop
Math Review is.............• 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 common misconceptions or understandings
Math Review is ...............
• 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 othersMathematically 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 later grades.
• determine domains to which an argument applies.
• listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
In the SJSD curriculum...
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
ReflectionMy 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 Model1. 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
AnswerWhen 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 canimprove it?
6. Do you feel respected by other students in this class?
Mastery of Math FactsAfter students have reached conceptual understanding, the
following fluencies are required by the CAS:• 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
• 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 13-28Cases 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 materialsis that the art of teaching involves helping students move from where they are into the content to be learned.
DMI PremisesDMI 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 sessionsSession 1: Analyzing Addition StrategiesSession 2: Place Value and Multiplication
Session 3: The Mathematics of Algorithms
Session One:Building a System of
TensStudent's Addition and Subtraction Strategies
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
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
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
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.
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
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?
Exit cards
Session Two:Building a System of
TensThe Base Ten Structure of Numbers
August 16, 2013
Mathematical Goal for Session Two• 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• 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• 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 DiscussionThink 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• 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
TensMaking 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 14Focus 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 mathematicalprinciples 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?