this webinar was pre‐recorded on wednesday, …...to [email protected]. narrators: megan kidwell and...
TRANSCRIPT
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Copyright © 2016 Mathematically Connected Communities (MC2) Project. All rights reserved
Welcome!
This webinar was pre‐recorded on Wednesday, January 27, 2016.
Please submit any questions or comments to [email protected].
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Narrators: Megan Kidwell and Lisa MatthewsDevelopers: MC2 K-3Team
From Counting to Counting OnFostering the Development of Early Addition & Subtraction
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Les Steffe, UGA, 1980s
Identifies stages of counting that children progress through as they come to understand the operations of addition and subtraction
Related to Piaget’s theory of cognitive development
Stages of Early Counting: Research
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Stages of Early Counting: Making Sense with Concrete Collections
EmergentStudent is unable to accurately count a collection
Perceptual Student can count visible collections, but not collections that are covered up
Figurative Student can count covered collections, but always “counts everything” (always starts counting from one)
Counting On /Counting Back Student is able to start with one collection and count on or back
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Stages of Early Counting: Making Sense with Concrete Collections
EmergentStudent is unable to accurately count a collection
Perceptual Student can count visible collections, but not collections that are covered up
Figurative Student can count covered collections, but always “counts everything” (always starts counting from one)
Counting On /Counting Back Student is able to start with one collection and count on or back
Advancing to e
ach new stage
involves a cogn
itive reorganization
regarding how collect
ions work
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Features of Early CountingNumber sequences (verbal)
One-to-one correspondence
Keeping track
One total for two distinct collections
Cardinality: The answer to “How many?” can be a single number word (“5”)
Progresses from concrete to abstract:visible collections → concealed collections
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Number sequences (verbal)
One-to-one correspondence
Keeping track
One total for two distinct collections
Cardinality: The answer to “How many?” can be a single number word (“5”)
Progresses from concrete to abstract:visible collections → concealed collections
Features of Early CountingKnows number sequences (verbal)
Uses one-to-one correspondence
Monitors items as they are counted
Can give the total for two distinct collections
Cardinality: The answer to “How many?” can be a single number word (“5”)
Perceptual Counting: Students are counting things they can see
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Number sequences (verbal)
One-to-one correspondence
Keeping track
One total for two distinct collections
Cardinality: The answer to “How many?” can be a single number word (“5”)
Progresses from concrete to abstract:visible collections → concealed collections
Features of Early CountingKnows number sequences (verbal)
Uses one-to-one correspondence
Monitors items as they are counted
Can give the total for two distinct collections
Cardinality: The answer to “How many?” can be a single number word (“5”)
Perceptual Counting: Students are counting things they can seeEmergent→
Perceptual cogn
itive reorganization:
attends to Certain Feature
s when
determining quantity
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Number sequences (verbal)
One-to-one correspondence
Keeping track
One total for two distinct collections
Cardinality: The answer to “How many?” can be a single number word (“5”)
Progresses from concrete to abstract:visible collections → concealed collections
Features of Early Counting
Figurative Counting:
Students are counting
things that are covered up
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Number sequences (verbal)
One-to-one correspondence
Keeping track
One total for two distinct collections
Cardinality: The answer to “How many?” can be a single number word (“5”)
Progresses from concrete to abstract:visible collections → concealed collections
Features of Early Counting
Figurative Counting:
Students are counting
things that are covered upPerceptual→Figurativ
e cognitive reorganization:
does not have
to be able to s
ee
objects in order
to count them
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Features of Early Counting
Counting on / counting backStudent understands that, rather than counting everything in the first collection, they can refer to the entire first collection with a single number Student counts on to add, counts back to subtract
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Features of Early Counting
Counting on / counting backStudent understands that, rather than counting everything in the first collection, they can refer to the entire first collection with a single number Student counts on to add, counts back to subtract
Figurative→Countin
g on cognitive reorganization:
a number can be a co
mposite,
“6” includes al
l the numbers tha
t come before it
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Stages of Early Counting:Emergent Counting
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Counting Collections
How many counters are there?
Can you get me 12 counters?
How many counters are there?
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Counting Collections: Cross the River
(front)
(back)
Materials: construction paper
pipe cleaner masking tape
counters
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Counting Collections: Cross the River
(front)
(back)
Materials: construction paper
pipe cleaner masking tape
counters
These “ducks” need to cross the river. Can you help them? When you are done, I want you to tell me how many ducks crossed the river.
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Counting Collections: Cross the River
(front)
(back)
Materials: construction paper
pipe cleaner masking tape
counters
These “ducks” need to cross the river. Can you help them? When you are done, I want you to tell me how many ducks crossed the river.
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Number Row Count
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Number Row Count
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Number Row Count
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Number Row Count
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Number Row Count
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Number Row Count
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Number Row Count
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Number Row Count
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Number Row Count
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Number Row Count
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Stages of Early Counting:Perceptual Counting
Number sequences (verbal)
One-to-one correspondence
Keeping track
One total for two distinct collections
Cardinality: The answer to “How many?” can be a single number word (“5”)
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Domino Addition
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Domino Addition
How many dots do you see?
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Domino Addition
How many dots do you see?
How many dots are under the door?
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Domino Addition
How many dots do you see?
How many dots are under the door?
How many dots do we have all together?
Variations: Conceal only the second card / Conceal both cards
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Stages of Early Counting:Figurative Counting
Progressing from concrete to abstract:~ counts collections that are covered up~ always counts from one (counts everything)
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Addition Spinners
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Addition Spinners
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Addition Spinners
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Addition Spinners
How many counters are there all together?
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Hide & Add
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Hide & Add
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Hide & Add
How much is 7 and 4 more?
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Hide & Add
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Hide & Add
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Hide & Add
How much is 9 and 5 more?
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Features of Early Counting: Counting On
beware of making counting on a procedure: “put the big number in your head”
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Features of Early Counting: Counting On
beware of making counting on a procedure: “put the big number in your head”
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Stages of Early Counting:Counting On / Counting Back
Student understands that, rather than counting everything in the first collection, they can refer to the entire first collection with a single number Student counts on to add, counts back to subtract
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Stages of Early Counting:Where are they in the CCSSM?
The standards address perceptual counting in detail
The standards emphasize solving a variety of problem types (combine, separate, compare, missing addend, missing subtrahend)
Less attention is given to the stages between perceptual counting and solving bare number tasks
Awareness of the stages is important in making decisions about instruction & support for young students
What materials will I use? Will I conceal the materials? Is it time to begin to connect materials to symbolic notation?
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Stages of Early Counting:Kinder CCSSM
Count to tell the number of objects K.CC.4Understand the relationship between numbers and quantities; connect counting to cardinality.
a) When counting objects, say the numbers in the standard order, pairing each object with one number and each number with one object.
b) Understand that the last number said tells the number of items counted, the number of items is the same regardless of arrangement or order in which they are counted.
K.CC.5Count to answer “how many?” questions about up to 20 things; given a number from 1–20, count out that many objects.
Understand addition as putting together & adding to Understand subtraction as taking apart & taking from K.OA.1Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.K.OA.2Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
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Stages of Early Counting:Kinder CCSSM
Count to tell the number of objects K.CC.4Understand the relationship between numbers and quantities; connect counting to cardinality.
a) When counting objects, say the numbers in the standard order, pairing each object with one number and each number with one object.
b) Understand that the last number said tells the number of items counted, the number of items is the same regardless of arrangement or order in which they are counted.
K.CC.5Count to answer “how many?” questions about up to 20 things; given a number from 1–20, count out that many objects.
Understand addition as putting together & adding to Understand subtraction as taking apart & taking from K.OA.1Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.K.OA.2Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
Perceptual Counting
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Stages of Early Counting:Kinder CCSSM
Count to tell the number of objects K.CC.4Understand the relationship between numbers and quantities; connect counting to cardinality.
a) When counting objects, say the numbers in the standard order, pairing each object with one number and each number with one object.
b) Understand that the last number said tells the number of items counted, the number of items is the same regardless of arrangement or order in which they are counted.
K.CC.5Count to answer “how many?” questions about up to 20 things; given a number from 1–20, count out that many objects.
Understand addition as putting together & adding to Understand subtraction as taking apart & taking from K.OA.1Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.K.OA.2Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
Perceptual Counting
figurative counting?
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Allow time for students to become robust perceptual counters
Count collections that are becoming smaller (remember to practice backwards counting sequences)
Consider how to advance student thinking as students become robust perceptual counters
Introduce the idea of counting collections that are covered up
Stages of Early Counting:Mathematizing in Kinder
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Stages of Early Counting:1st Grade CCSSM
Represent and solve problems involving addition and subtraction. 1.OA.1Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem
1.OA.2Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem
Add & subtract within 20 1.OA.5Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
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Stages of Early Counting:1st Grade CCSSM
Represent and solve problems involving addition and subtraction. 1.OA.1Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem
1.OA.2Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem
Add & subtract within 20 1.OA.5Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
(Understand and apply properties of operations and the relationship between addition and subtraction.)
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Stages of Early Counting:1st Grade CCSSM
Represent and solve problems involving addition and subtraction. 1.OA.1Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem
1.OA.2Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem
Add & subtract within 20 1.OA.5Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
(Understand and apply properties of operations and the relationship between addition and subtraction.)
(Work with addition and subtraction equations.)
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Consider how to advance student thinking: How will figurative students develop a counting on strategy?How will your students make sense of symbolic notation (equations)?
Present two collections that are covered up: How many are there in all?
Put a small number of counters in the second collection (students are more likely to realize they can “count on” when there are only one or two more counters to add)
Once students develop a counting on strategy to add collections, begin to present tasks that involve removing counters (subtraction)
Introduce symbolic notation once students have robust counting on / counting back strategies to solve addition & subtractions tasks with counters
Stages of Early Counting:Mathematizing in 1st Grade
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This webinar recording and handouts are available at the MC2 and New Mexico K‐3 PLUS websites.
Please submit any questions or comments to [email protected].
Thank you!