The Learning Trajectory for Counting
Lynn Rule
[email protected]
NAEYC 2015
download at mathrack.com
Imagine
a world without numbers…
3+5=8
Imagine
a world without YOU…
Mathematize the World
Around Us…
•
Childhood teachers need to look to ‘see’ math
around them and be alert to the way that
math and mathematical problem situations
are built into the very fabric of their
students’ lives….Math connects to our
lives!
It begins with YOU
•
Facility with number words and numerals enables students to progress with
further arithmetic knowledge
•
One-to-one correspondence enables students to understand numerosity and to
engage in operations. It is the foundation for children’s early work with numbers
•
Spatial thinking is critical for building subtilizing skills and more advanced
counting strategies.
•
Subitizing supports the development of children’s cardinal understanding.
•
Cardinality is a prerequisite to being able to meaningfully count or carry out
number operations such as addition and subtraction.
•
Understanding threeness, fourness, etc.and how they are different from one
another leads to more proficiency in operations and problem solving
Strong Number Sense in the Early Years
is the Key Building Block of Learning
Arithmetic in the Primary Grades…
•connects
counting to quantities
•solidifies
and refines the
understanding of more and less
•helps
children estimate quantities and
measurements
Why Number Sense Matters…
•
Students will not be mathematically
proficient…fluent…. without number sense.
•
Number sense is to math as phonemic
awareness is to reading.
Context to Develop Children’s
Strong Number Sense
( Big Ideas of Early Mathematics- Erikson Institute) More titles at mathrack.com
•
Anno’s Counting Book by Mitsumo Anno
•
Ten Black Dots by Donald Crews
•
Count and See by Tana Hoban
•
Splash by Ann Jonas
So Where Do We Begin?
Theory and Activities
The Early Math Collaborative
Erikson Institute
Early Counting
Four Interrelated Aspects of Early
Numerical Knowledge-Early Counting
Identify, Instruct, Assess
•
Number Sequence
•
One-to-one correspondence
•
Cardinality
•
Subitizing
Number Sequence
The names and the ordered list of number words
Number Sequence
Includes many interesting patterns that are central to
our understanding of numerosity and place value.
•
Early number learning: 1 to 10 is a rote procedure
but effort should be given to build number
sequence in a meaningful way
•
Early number learning: 11-20 is memorized but
may be introduced as one 10 and 1, one 10 and 2
to be more explicit about the structure of numbers.
Instruction of Number Sequence
•
Vertical number line
•
Counting up and back from a target number
•
Counting circle to a target number
•
Line them up
•
Numeral roll
•
Multi lid screen (math recovery)
•
Activities for Number Sense Development -Big Ideas for Early
Mathematics pg. 42-43
(math recovery)
Assess Number Sequence
•
Have students orally count and /or count from a
given number to a target number
•
Assess knowledge of numerals by presenting
numerals for students to identify, and by asking
students to write particular numerals
One-to-One Correspondence
Counting objects by saying number words in a one-to-one
correspondence with the objects.
one number is named for each object
Common Errors with One-to-One
Correspondence Principal
Correct correspondence
Incorrect sequence
(no stable order)
Correct sequence
Incorrect correspondence
(count too fast)
Correct sequence
Incorrect correspondence
(point too fast)
1
2
3
5
6
4
10
8
1
2,3
4
5,6
7
8
9
10
1
2
3
4
5
Instruction One-to-One
Correspondence
•
Start with small numbers (1-5) (6-10) through many
authentic experiences and mathematical
conversations
•
Daily routines-taking attendance, snacks, lunch
tickets etc.
•
Music movement games as marching to a drumbeat
•
Board games with paths to move along by counting
spaces
Assess One-to-One
Correspondence
•
Observe children count objects-do
they have strategies for keeping
track, like touch-pointing or moving
to another pile?
Cardinality
Understanding that the last number word said when
counting tells how many objects have been counted.
1
2
3
4
5
6
7
8
Instruction Cardinality
Children need experiences and conversations to develop
their understanding that when number words are used to name
‘how many’ the numbers act as attributes .
•
Label the cardinal value of a set after counting ex.
1,2,3,4,….4 books
•
Routines that involve counting out a specified
number such as snack (1,2,3,4,5,6…6 crackers)
Assess Cardinality
Being able to count is not the same as being able to
answer “how many?”
•
Listen to how the child responds when you discuss
counting tasks. After counting a set of objects ask
“How many are here?” If the child recounts the
set, hesitates, or points to the last object counted, it
is likely the child has not constructed the idea of
cardinality. Children with an understanding of
cardinality are apt to emphasize the last count,
will explain that there are ‘nine’ because I counted
them.
Subitizing
Quickly recognizing and naming how many objects
are in a small group without counting.
Subitizing
•
•
•
•
Perceptual subitizing- when the numbers of items is 3 or
less
Conceptual subitizing- helps children know ‘how many’
without counting when the number of items is greater than
3. Example 6 may be seen as two threes
Helps children develop a reliable mental picture of how
quantities relate to one another
Begin with 1-5 and then 6-10
Instruction Subitizing
Quick images, Show me, How do you know?
•
Dot Cards, dice, dominoes
•
Five Frame
•
Ten Frame
•
MathRack
Assess Subitizing
•
Can children quickly say ‘How Many’ are on a dot
card, dice, frame etc. without counting?
•
Can children accurately duplicate the amount
shown to them on their tool?
•
Can children explain their thinking?
Develop Rational Counting Skills
through Authentic Experiences and
Mathematical Conversations
•
Strong grasp of cardinality up to 10 takes 2-3 years.
(preschool)
•
Kindergarten-master good number sense to 20-25
•
Most kindergarteners and first graders do not have a
precise idea of ‘how many’ numbers over 50 and 100
really represent
•
Overemphasizing rote counting to high numbers
before the counting principles are established for
small numbers is counterproductive
Context to Develop Counting
( Big Ideas of Early Mathematics- Erikson Institute) More titles at mathrack.com
•
One Gorilla
Fish Eyes
•
One Stuck Duck
Frog in the Bog
•
Ten in the Bed
Five Little Monkeys
Counting Trajectory
•
Emergent Counter
•
Perceptual Counter
•
Figurative Counter
•
Counting-on Counter
•
Non-Count-By-Ones Counter
Emergent Counter
The child is unable to count the collection of objects. The child
may be unable to coordinate one number word with one object
when counting or may not know the correct number sequence.
*Child may have number sequence but not one-to-one
correspondence
Perceptual Counter
The child can count the collection of objects only if the objects
can be seen. A perceptual counter will count all objects by
counting from the number 1.
*Child has number sequence and one-to-one
correspondence
Figurative Counter
The child can count the collection of objects even if the
objects are blocked from view. The child is able to imagine or
visualize the objects. A figurative counter will count all
imagined objects by counting from the number 1.
Counting-On Counter
A counting-on counter is child who can start counting from a
given number other than 1 and who does not need to see the
objects to count.
6+3=
6 … 7,8,9
Non-Count-by-Ones
Counter
A child who does not use counting by ones but partitions and
combines the numbers involved is a non-count-by-ones
counter. For example, the equation 7 + 6 = , the child may
reason that 7 is 3 from 10, so partition the 6 into a 3 and 3.
Combine 7 and 3 to get 10. Then combine 10 and 3 to get 13.
7+6
(7 + 3) + 3
10 + 3
Leads into relationships
Focus on Relationships
•
When we focus on relationships, it helps
give children flexibility when dealing
with their basic facts and extending
their knowledge to a new task. when we
build a child’s number sense it
promotes thinking instead of just
computing.
Number Relationships
•
Spatial Patterns- Recognizing how many without
counting by seeing a visual pattern-subitizing
•
One/Two More or Less- Knowing which numbers are
one/two more or less
•
Landmarks of 5 and 10- How any number relates to 5
and 10. Numbers important to assist mental computation,
addition and subtraction
•
Part-Part-Whole-Ability to conceptualize a number as
being made up of 2 or more parts
Van de Walle-2013
Benchmarks of 5 and 10
• Help children see how numbers relate to 5 and 10
becomes useful as they start to compute with
numbers. ex. If you know that 7 is 5 + 2 or it is three
less than 10 you could solve:
•
7+8
13 - 7
47 + 6
•
5
2
3 4
5 + (2+8)
(13-3) - 4
•
•
5 + 10
10 - 4
•
Ten Fact
Making Ten
Extending Knowledge
How many dots are there?
How many dots are there?
How many beads? How do you know?
Thinking Flexibly
7+8
7+7+1
One/Two more or less
8+8-1
One/Two more or less
5+2+8
Part-Part-Whole, Benchmarks 5/10
7+3+5
Part-Part-Whole, Benchmarks 5/10
2+5+5+3
Part-Part-Whole, Benchmarks 5/10
Move students’ thinking from
representational to abstract.
8 + 7 = 15
(8 + 2) + 5 = 15
10 + 5 = 15
15 =15
Push students to construct algebra.
Constructing Algebra Fosnot
Four Fact Strategies
•
Plus zero
•
Doubles
•
Make 10
•
Ten Plus Something
Fact Strategies + Relationships = Fluency
What is Fluency?
Fluent-Mathematically Proficient
•
•
•
Accuracy-ability to produce an accurate
answer
Efficiency-ability to choose an
appropriate, expedient strategy
*Flexibility-ability to use number
relationships with ease in computationcompose and decompose numbers
Russell 2000
Fluency and Flexibility
• Fluency - efficient and accurate
• Flexibility - multiple solution strategies determined
by the problem
• Fluency is the by-product of flexibility. Assessing
fluency by occasionally using timed tests is
acceptable. Using timed tests as an
instructional tool to build fluency is
ineffective, inefficient, and damaging to
student learning.
--Henry and Brown
Ways To Help
•
Counting- Counting, and understanding the patterns within
counting, helps lay the foundation for all of mathematics
•
Show Me-showing a given amount, but moving beyond
counting one-by-one
•
Quick Images-Flashing images that can be subitized
•
Relationships NOT Rules- Activities that emphasize how
numbers relate to each other and not just seeing them in
isolation. Discussions that explicitly focus on relationships.
•
AND
Connecting
C-R-A Model
To Math Facts
•
Concrete
•
Representational
•
Abstract
The ideal lesson would introduce the task using a concrete
tool, moving to a representation of the task and ending with an
abstract or symbolic solution of the task.
Number Line
Number Path
Connect to Context
There are 7 passengers on the upper deck and 8
passengers on the lower deck.
Connecting the CRA Model
to Math Facts (7 + 8)
Concrete
5
5
5
+
10
5
2
3
+ 2 +
+
15
5
Representational
3
Abstract
Now you try…
8+6
Why Number Sense Matters
Extending Knowledge to a new task…
8+6
38 + 6
58 + 6
2998 + 146
3.98 + .16
In this session we learned to…
•
Identify WHERE students are in the counting
trajectory
•
ANALYZE students’ progress
•
DEVELOP students’ future instruction
Know where your were,
where you are,
where you are going…
Imagine
a world without YOU…
Resources
• www.mathrack.com
• www.mathematicallyminded.com
• Big Ideas of Early Mathematics, Erikson Institute
• Children’s Mathematics by Carpenter, Fennema, Franke, Levi, Empson
• Contexts for Learning Mathematics by Catherine Fosnot
• Developing Number Knowledge, Rober Wright, David Ellemor-Colling, Pamela Tabor
• Fluency through Flexibility; How to Build Number Sense 0-20 Christina Tondevold
• How the Brain Learns Mathematics by David Sousa
• Last to Finish by Susan Allen and Jane Lindaman
• Mastering the MathRack by Christina Tondevold
• Number Talks by Sherry Parrish
• Teaching Student Centered Mathematics K-3 by John Van de Walle
• Used Any Numbers Lately? Susan Allen and Jane Lineman