math with meaning - fordham university€¦ · geometry – disequilibrium whether in geometry or...
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Math with Meaning
A Language Based
Approach to Math
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Goals for the day
Revisit Number Sense Foundations
Revisit Model Drawing Foundations
Connect Language/Speaking/Listening to CCS
Focus on Vocabulary Building Blocks
Build Language Around Manipulatives
Context for Vocabulary – Geometry
Use Mathematical Discourse
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Activity for Purposeful Practice
I am…you are
Two questions
Who/What am I?
Who/What are you?
Make a circle
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Common Core Framework
Conceptual Understanding: Comprehension of mathematical concepts, operations, and relations Procedural Fluency: Skill in carrying out procedures flexibly, accurately, efficiently, and appropriately Strategic Competence: Ability to formulate, represent, and solve mathematical problems Adaptive Reasoning: Capacity for logical thought, reflection, explanation, and justification Productive Disposition: To see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.
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Costas’ Method
Intellectual Functioning
Level 1 - focus on Gathering and recalling
– Level 2- focus on Processing and making sense of gathered information
– Level 3 - focus on Applying and evaluating information
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The Eight Mathematical Practices
1. Make sense of problems & persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments & critique others reasoning.
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.
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The Octa-Habits
1. Get it and stick with it!
2. Think about it and number it!
3. Defend and critique the thinking.
4. Play with manipulatives and models.
5. Use the tools.
6. Sweat the details.
7. Bag it and Tag it!
8. Find and use the patterns.
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Learning Begins with Questioning
What is the difference between
good questions and great questions?
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Language Frames
Listen and Record
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Questions by Practice 1. Get it…Stick with it!
How would you describe this problem in your own words?
How would you describe what you are trying to find?
What do you notice about…?
Describe the relationship between the quantities
Describe what you have already tried?
Talk me through the steps you have used at this point.
What steps are you most confident of?
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Questions by Practice 2. Think about it…Number it!
What do the numbers in the problem represent?
What is the relationship of the quantities?
How is ____ related to ____ ?
What does ____ (symbol, diagram, etc) mean to you?
What properties might we use to find the solution?
How did you decide in this task that you needed to use___?
Could we use another operation/property to solve this?
Why or why not?
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Questions by Practice 3. Defend…Critique the thinking!
What mathematical evidence would support your solution?
How can we be sure that …?
How could you prove that …?
Will it still work if …?
How did you test whether your approach worked?
How did you decide on the unknown?
Did you try a method that did not work? Why didn’t it?
Could you demonstrate a counter-example?
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Questions by Practice 4. Play with Models & Manipulatives
What number model could you construct to represent this?
What are some ways to represent the quantities?
What’s an equation/expression that matches the diagram?
Can you use a number line, chart, table, etc?
What are some ways to visually represent …?
Is there a real world situation that is similar to this?
What formula connects with this situation?
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Questions by Practice 5. Use the Tools
What mathematical tools could we use to visualize this?
What information do you have?
What do you know that is not stated in the text?
Are there different terms, measures, symbols needed?
Would it be helpful to use a protractor, ruler, compass?
Can manipulatives, diagrams, number-lines or graphs help?
What would a reasonable estimate be for a solution? Why?
What can a calculator do or not do here?
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Questions by Practice 6. Sweat the Details
What are mathematical terms that apply here
How do you know that your solution was reasonable?
How could you demonstrate that your solution solves this?
Is there a more efficient strategy?
How are you showing the meaning of the quantities?
What symbols/notations are important to use here?
What language/properties can you use to explain …?
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Questions by Practice 7. Bag it and Tag it!
What observations do you make about …?
What parts of the problem can you eliminate?
Can you simplify the problem?
What patterns do you find in …?
What have we learned before that we can use here?
Is this similar to other problems?
How does this relate to …?
How does this relate to other mathematical concepts?
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Questions by Practice 8. Look for patterns!
Will the same strategy work in other situations?
Is this always true, sometimes true, or never true?
How would we prove that …?
What do you notice here that you have noticed before?
What would happen if …?
Is there a mathematical rule for …?
What predictions/generalizations can this pattern support?
Is this consistent with other properties of mathematics?
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Teaching with the Brain in Mind
85% of students
ARE NOT
Auditory Learners
Then why do we do it?
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Teaching with the Brain in Mind
MODALITY BITS PER SECOND
AUDITORY 300,000 bps
KINESTHETIC 10,000,000 bps
VISUAL 100,000,000 bps
Sousa’s Brain Based Research on Learning
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Foundations of Number Sense
Concrete-Pictorial-Abstract approach
(Jerome Bruner)
Relational understanding
(Richard Skemp)
Multiple Models & Varied Practice/Experiences
(Zoltan Dienes)
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Foundations of Number Sense C-P-A
Instructional strategies build understanding
through activities that move students in a
sequence from the Concrete (manipulatives) to
the Pictorial (visual models) and finally to the
Abstract (symbolic) level. - Bruner-
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Counting Change
A student wanted to buy a $1 soda at a vending machine, but did not have the correct change. What is the greatest amount of change he could have and still not be able to make the exact change for the purchase?
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Counting Change
Reasoning and Proof
Even wrong answers have reasons behind them
Communication
Being precise means that we understand that
coins and change are two different things. The
problem is not asking about coins, but change.
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Foundations of Number Sense Perceptual Variability
Conceptual learning is maximized when
children are exposed to a concept through a
variety of physical contexts. Using a variety of
materials and experiences leads to an
understanding of mathematical abstraction.
Short, varied practice on a regular basis.
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Perceptual Variability
Base 10 blocks
Place value disks
Proportional everyday materials
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Perceptual Variability
What would a
progressing
sequence of
difficulty look like
with number
bracelets?
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Foundations of Number Sense Relational vs Instrumental
Know the difference between Relational
Understanding (knowing what and why) and
Instrumental Understanding (procedural what
and how).
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The Veddic Algorithm vs. Model
600
30
40
2
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How vs. Why
Can you explain “How” it works?
Can you explain “Why” it works?
Instrumental
Relational
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Model Drawing
1. Read and Reflect on the Problem
2. Rewrite the question in answer form
3. Determine who/what the problem is about
4. Draw the Unit Bar
5. Adjust, modify and place Question Mark
6. Calculate
7. Answer the question
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Model Drawing
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Model Drawing Sequence
TeHan has 3 balls and Holly has 2 balls. How many balls do they have altogether?
3 balls 2 balls
?
3 balls 2 balls
?
3 balls 2 balls
?
3+2=
3 balls 2 balls
? 3 2
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Part Whole Practice Problem
If of a number is 12, what is the number?
23
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Additive Comparison
Sean collected some clams at Nana June’s
Beach. Elliot collected 15 more clams than
Sean. If they collect 105 clams altogether, how
many clams did Sean collect?
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Additive Comparison
Rain is 8 years older than Jade and 2
years younger than Ramel. The total of
their ages is 63 years. How old is Ramel?
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From Model to Equation
1.
3
87
8
40
2.
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Multiplicative Comparison
Xavier had 4 times as many stamps in his collection as Shannon did. If they had 345 stamps altogether, how many more stamps did Xavier have than Shannon?
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Modeling Comparisons
A has 4 units. B is 2 more than A.
A
B
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Modeling Comparisons
A
B
A is 3 times B.
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Modeling Comparisons
B is 6 units and A is of B
A
B
1
2
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Modeling Comparisons
A is as big as B and C is twice as big as B.
A B C
2
3
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Fractions
Mary found of the flowers in her garden
are red, of the remainder are yellow and the
rest are pink. There are 18 pink flowers. How
many flowers are there altogether?
2 5
1 4
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Vocabulary Panels
Three components
WORD
DEFINITION
GRAPHIC OR EXAMPLE
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Vocabulary Panels
MEDIAN
The number that is exactly in the middle of an ordered set of values
2 3 3 3 4 7 7
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Student Work
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Student Work
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Student Work
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Vocabulary Panels
In your grade level teams
Generate Target Vocabulary
Generate Support Vocabulary
Generate a Word Panel
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Word Wall = Vocabulary Panels + Word Files
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Cuisenaire Rods
Comparisons
Matching
Patterns
Part Wholes
Place Value
Commutative/Associative
Fractions & Applying Properties
Factors
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Cuisenaire Rods & Language Frames
Comparison
Which rod is smaller?
Which rod is larger?
Are the rods the same?
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Cuisenaire Rods & Language Frames
Matching
How many smaller rods are needed to match
the length of the larger rod?
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Cuisenaire Rods & Language Frames
Sequencing
What comes next?
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Cuisenaire Rods & Language Frames
Part Wholes
If = 1 then ____ + ____ = ____ Explain.
If = 1 then ____ + ____ = ____ Explain.
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Cuisenaire Rods & Language Frames
Part Wholes
If = 1 then how many ways can I name
light green? Say it, and write it.
If =1 then how many ways can I name
purple? Say it, and write it.
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Cuisenaire Rods & Language Frames
Applying Operations
+ = or
If = 1 then 2 + 3 = 5 and …
What if =1 then what
changes? Why? How? Say it and write it.
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Geometry in Two Dimensions
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Geometry–Two Dimensional Scope
All Polygons
Group of Quadrilaterals
Group of Triangles
Group of Regular Polygons
Non-Polygons
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Geometric Language
What is the language of polygons?
What is the language of these shapes?
What is different?
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If this was a rubber stamp how many
different shapes could I stamp? Explain.
How many of each does this solid have?
What are the properties of each shape?
How do these individual stamp faces come
together to form this shape?
Geometry From Two to Three
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Geometry – Attending to Precision
What:
…Is the vocabulary of shapes used?
…Is the vocabulary of 2 dimensional shapes?
…Is the vocabulary of 3 dimensional shapes?
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Geometry - 300,000 bps
Without looking in the bag;
Reach in and feel the shape
Provide one unique description of the shape
Pass the bag to the next person
They will add their own unique description
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Geometry - 300,000 bps
Without looking in the bag;
Reach in and feel the solid
Provide one unique description of the solid
Pass the bag to the next person
They will add their own unique description
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Geometry – Stretching Language
Criteria for reference
Shape of faces
Number of faces
Variety of faces
Variety of nets
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Geometry – Find a Solid
Find a solid;
Identify any 2 Dimensional components
Which of these are edged?
Which of these are curved?
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Geometry – Disequilibrium
How do we use 2 Dimensional learning
to build 3 Dimensional concepts?
What elements do we need to add to
our dimensional framework?
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Geometry – Disequilibrium
Whether in Geometry or other content
areas, build on what the students know,
and have them explore new content on
their terms.
How would you apply this to what you are
teaching now to generate new discussions?
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Mathematical Discourse
AKA
•Number Talks
•Math Talks
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Research – 8 + 17
Above Average
Students
30% used known
facts
61% used number
sense
9% used a counting
on strategy
Below Average Students
6% used known facts
0% used number
sense
72% used a counting
on strategy
22% used a counting
all strategy
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Properties
CCSS K-8 math standards mention properties 66 times.
Properties are ways that numbers and operations
interact with one another.
Traditionally, these have been taught to students by
learning the definitions – this is changing!
Students need to be USING the properties in order to
appreciate them.
Using properties in math allows for flexible thinking,
which is linked to proficiency.
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Number Talk General Goals
Increased discourse among students
Greater conceptual learning
Greater problem solving skills
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Number Talk Progressive Goals
Goal: Improve computational fluency by
encouraging students to:
Use any strategy that makes sense to them
Then to be able to use more efficient strategies
that they have learned from their peers
And to finally progress to “just knowing it” or
using most efficient strategies for computation
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Achieving Mathematical Discourse
1. Increase opportunities for students to question, explore, and challenge each other’s thinking
2. Discuss relationships between numbers and operations
3. Discuss many different strategies for solving different problems
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Number Talks
High Trust Community
Inclusive/Value Based Class Discussion
Teacher-Neutral/Respectful/Encouraging Role
Role of Mental Math
Purposeful Problems
05 to 15 minutes
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How many unit squares?
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Number Talks - 12 x 15