The Alignment of
Early Numeracy
Skills
Bethel P-3 Professional Learning
Community
November 3, 2011
Julie Wagner
Elementary Mathematics Specialist
OSPI
Goals
Introduce participants to:
• Current research that supports a
P-3 numeracy alignment
• The Common Core State Standards
domains and alignment of P-3
• Trajectories of learning
• Resources available
The compelling basis for P-3
numeracy alignment
A Meta-Analysis
School Readiness and Later
Achievement
Duncan, et al, Developmental Psychology, 2007.
The strongest predictors of later
achievement are school-entry math, reading,
and attention skills. Early math skills have
the greatest predictive power. By contrast,
measures of socio-emotional behaviors were
generally insignificant predictors of later
academic performance, even among
children with relatively high levels of problem
behaviors.
Review of Research
Mathematics Learning in Early
Childhood: Paths Toward Excellence
and Equity
National Research Council, 2009.
The committee found that, although virtually
all young children have the capability to learn
and become competent in mathematics, for
most the potential to learn mathematics in
the early years of school is not currently
realized. This stems from a lack of
opportunities to learn mathematics either in
early childhood settings or through everyday
activities in homes and in communities.
Article by NCTM President
A Missed Opportunity: Mathematics in
Early Childhood
Henry Kepner, NCTM Summing Up, February 2010.
Prior to kindergarten, many children have the
interest and capacity to learn meaningful
math and acquire considerable mathematical
knowledge. Many early childhood programs
do not extend children’s mathematical
knowledge. Instead, they have these young
students repeat the same tasks in varied
settings without posing challenges that
would push them to the next level.
Social Policy Report
Mathematics Education for Young
Children: What It Is and How to Promote It
Ginsburg, Lee, & Boyd, Society for Research in Child
Development, 2008
Cognitive research shows that young children
develop an extensive everyday mathematics and
are capable of learning more and deeper
mathematics than usually assumed.
Typically, early childhood educators are poorly
trained to teach mathematics, are afraid of it, feel
it is not important to teach, and typically teach it
badly or not at all.
What does all this mean?
• Children can learn much about
mathematics early in their lives.
• The numeracy skills children walk into
the door with in kindergarten predicts
later achievement in math and reading.
• Numeracy skills are often overlooked in
child care settings through lack of
precedence, neglect or fear.
• If kindergarten teachers had students
who had numeracy skills, achievement
would increase dramatically.
• The Common Core State
Standards for Mathematics and a
P-3 alignment
The CCSS Document
Design and Organization
Standards for Mathematical
Practice
 Make sense of problems and persevere in solving
them
 Reason abstractly and quantitatively
 Construct viable arguments and critique the reasoning
of others
 Model with mathematics
 Use appropriate tools strategically
 Attend to precision
 Look for and make use of structure
 Look for and express regularity in repeated reasoning
Mathematical Practices
Graphic
13
Critical Areas of Focus
Insert a K-3 picture
14
Overview Page
Domains, Clusters, Standards
16
Common Core State Standards for
Mathematics - Domain Development
Kindergarten Standards
• The Common Core State Standards in
Mathematics at the kindergarten level
include all of early numeracy…for a
reason.
Common Core State Standards for
Mathematics - Domain Development
Abridged Trajectories of Early Math
Concepts
Pre-Kindergarten Mathematics Standards
Abridged Trajectories of Early Math
Concepts
Number and Operation
What does it mean to count?
Number and Operation
Concepts in counting:
• Recognize counting words
• The sequence of numbers
• One-to-one correspondence
• Cardinality
• Reverse of cardinality
Number and Operation
The sequence of numbers:
1-10
11-13, 15
14, 16-19
20-29
30-39
Number and Operation
One-to-one correspondence
Children have to know sequence of
numbers and remember the sequence
and where they are in the sequence as
they count in order to master one-toone correspondence – assigning one,
and only one, number to each object in
a group.
Number and Operation
Cardinality
Child moves from just saying the
number sequence to understanding that
the last number stated answers the
question, “How many?”
Number and Operation
Reversal of cardinality
Child can be asked to, “Give me eight
blocks,” and is able to count out the
correct amount. Why would this be
more difficult?
Order of Counting
•
•
•
•
•
Small numbers first
In a line
In a circle
In a pattern
Scrambled
Counting Trajectory
Age
late 1
2
Developmental Progression
Chants “sing-song” or sometimes
indistinguishable number words.
Verbally counts with number words, not
necessarily in correct order above “five.”
If knows more number words than number of
objects, rattles them off quickly at end. If more
objects, “recycles” number words.
3
Later 3
4
Verbally counts to ten with some
correspondence
Keeps one-to-one correspondence between
counting words and objects for at least small
groups. May recount if asked a second time,
“How many?’
Accurately counts objects in a line to 5 and
answers the “how many” question with the last
number counted.
Counting Trajectory
Age
Later 4
Learning Progression
Counts arrangements of objects to 10. May
be able to tell the number just after or just
before another number, but only by counting
from 1.
Counts out objects to 5.
5
Counts out objects to 10 and beyond. Keeps
track of objects that have or haven’t been
counted.
Gives next number (usually to 20 or 30).
Separates decade and one part of a number.
Recognizes errors in other’s counting. Add 6
and maybe 7 and 8.
Learning and Teaching Early Math: The Learning Trajectories Approach, Clements & Sarama, 2009
Age
Learning Progression
6
Counts verbally and with objects from numbers other
than 1 (but does not yet keep track of the number of
counts). Skip counts by 10s to 100. Counts to 100,
then 200. Understands place-value for 10s, 100s.
7
Consistently conserves number even in the face of
perceptual distractions. Recognizes that decades
sequence mirrors single-digit sequence.
Number and Operation
Beginning components of operation
• Subitizing
• Comparison words
• Modeling
Number and Operation
Subitizing
Knowing how many are in a collection
without counting.
How valuable is this skill?
What is its role in operation?
Number and Operation
Comparison words
Bigger, smaller
Longer, shorter
Less, more
Lighter, heavier
Comparison Trajectory
Age
Developmental Progression
1
Puts objects, words, actions in one-to-one or oneto-many correspondence.
2
Implicitly sensitive to the relation of “more than/less
than” involving very small numbers.
Compares collections that are quite different in size.
If same size, numbers must be small (one or two).
3
Compares collections of 1-4 items verbally or
nonverbally. Has to be same item.
4
Compares groups of 1-6 by matching. Doesn’t need
the same object.
Accurately counts two equal groups, but more
spread out, or larger will be more.
Learning and Teaching Early Math: The Learning Trajectories Approach, Clements & Sarama, 2009
Comparison Trajectory
Age
5
Developmental Progression
Compares with counting, even when larger
collection’s objects are smaller. Later, figures out
how many more or less. Accurately counts two
equal collections and says they are the same
number, even if one collection has larger blocks.
Names a small number for sets that cover little
space and a “big number.” Names a big number for
sets that are spread out.
6
Uses internal images and knowledge of number
relationship to determine relative size and position.
Orders numerals, and collection. Orders lengths
marked into units.
Learning and Teaching Early Math: The Learning Trajectories Approach, Clements & Sarama, 2009
Comparison Trajectory
Age
Developmental Progression
7
Compares numbers with place value understanding.
Uses internal images and knowledge of number
relationships to determine relative size and position
to 100s.
8
Uses internal images and knowledge of number
relationships, including place value, to determine
relative size and position to 1000s.
Estimating- subitizing is used to quantify a subset
and repeated addition or multiplication used to
produce an estimate.
Learning and Teaching Early Math: The Learning Trajectories Approach, Clements & Sarama, 2009
Operation (+/-) Trajectory
Age
1
2-3
4
Developmental Progression
Sensitivity to adding and subtracting perceptually
combined groups. No formal adding
Adds and subtracts very small collections nonverbally.
Finds sums for joining problems up to 3 + 2 by
counting –all with objects.
4-5
Finds sums for joining and part-part-whole by direct
modeling, counting-all with objects. Solves take-away
problems by separating with objects.
5-6
Finds sums for joining and part-part-whole problems
with finger patterns and/or by counting on
Learning and Teaching Early Math: The Learning Trajectories Approach, Clements & Sarama, 2009
Operation (+/-) Trajectory
Age
Developmental Progression
6
Has initial part-whole understanding. Solve all
previous problems types using flexible strategies.
Sometimes can do start unknown, but only by trial
and error.
6-7
Recognizes when a number is part of a whole and
can keep the part and whole in mind
simultaneously; Solves simple cases of multi-digit
addition by incrementing tens and/or ones.
7
Solves all types of single-digit problems, with
flexible strategies and known combinations.
Multi-digit may be solved by incrementing or
combining tens and ones
Learning and Teaching Early Math: The Learning Trajectories Approach, Clements & Sarama, 2009
Shape (Geometry)
• Recognition of basic shapes
• Application to world around us
• Classification and sorting
Trajectory for Shapes
Age
0-2
Learning Progression
Compares real-world objects and says whether they
are the same or different
Matches familiar shapes
Matches familiar shapes with different sizes
Matches familiar shapes with different orientations
3
Recognizes and names squares and circles, less
often triangles
May rotate shape to mentally match to a prototype
Judges two shapes the same if they are more visually
similar than different.
Learning and Teaching Early Math: The Learning Trajectories Approach, Clements & Sarama, 2009
Trajectory for Shapes
Age
3-4
Learning Progression
Matches a wider variety of shapes with same size
and orientation.
Matches a wider variety of shapes with different
sizes and orientations.
4
Recognizes some less typical triangles, some
rectangles but not rhombuses
Says two shapes are the same after matching one
side on each.
Looks for differences in attributes, but may examine
only part of a shape.
4-5
Recognizes more objects. Looks for differences in
attributes, but may ignore some spatial relaitonships.
Learning and Teaching Early Math: The Learning Trajectories Approach, Clements & Sarama, 2009
Trajectory for Shapes
Age
Learning Progression
5
Recognizes most familiar shapes and typical
examples of other shapes, such as hexagon,
rhombus, and trapezoid.
6
Names most common shapes without making
mistakes such as calling ovals circles. Recognizes
right angles.
7
Identifies shapes in terms of their components.
Determines congruence by comparing all attributes
and all spatial relationships.
8
Refers to geometric properties and explains with
transformations (moves on top of each other to show
congruence).
Represents various angle contexts as two lines and,
at least implicitly, the size of the angle as the rotation
between these lines.
Measurement
• Assigns a number to a measureable
attribute of an object, usually length,
weight, capacity or mass
• In the CCSS, connections between
measurement and whole number
operations and number line
Length Trajectory
Age
Learning Progression
2
Does not identify length as attribute
3
Identifies length/distance as attribute. May understand
length as an absolute descriptor (e.g., all adults are
tall), but not as a comparative.
4
Physically aligns two objects to determine which is
longer or if they are the same length.
Compares the length of two objects by representing
them with a third.
5
Order lengths, marked in 1 to 6 units
6
Lays units end to end. May not recognize the need for
equal length units.
Learning and Teaching Early Math: The Learning Trajectories Approach, Clements & Sarama, 2009
Length Trajectory
Age
Learning Progression
7
Measures by repeated use of a unit. Relates size
and number of units explicitly. Can add up two
lengths to obtain the length of a whole.
8
Considers the length of a bent path as the sum of its
parts. Measures, knowing need for identical units,
parts of unit, and zero point on rulers.
Possesses and “internal” measurement tool.
Mentally moves along an object, segmenting it, and
counting the segments. Estimates with accuracy.
Learning and Teaching Early Math: The Learning Trajectories Approach, Clements & Sarama, 2009
A look at available resources
Resources
 OSPI Website - Common Core State
Standards
•
•
•
•
Common Core State Standards
Transition Documents
Learning Progressions (trajectories)
Arizona examples
 Learning and Teaching Early Math: The
Learning Trajectories Approach, Clements and
Sarama (2009)
 Early learning trajectories, pre-k standards
 Sources of your own
Reflections and Questions
Thank you for undertaking
this important work!
Office of Superintendent of Public Instruction
Julie.Wagner@k12.wa.us
360-725-6233