GRADE BAND: K-2 (BUT THIS
ALSO APPLIES TO K-5)
Domain: Counting & Cardinality and Algebraic
Thinking
Why this domain is a priority for
professional development
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People involved in education must understand that
algebraic thinking involves understanding the
structure of the number system, examining the
behavior of the operations, and articulating and
proving generalizations. (That is, algebra is not
exclusively finding unknowns.) Key to this is
developing models of the operations (e.g.
representations using objects, drawing, story
contexts) and keeping these connected to
interpretation of symbols.
Why this domain is a priority for
professional development

The development of these models starts already
when children are learning to count. Learning to
count is a complex activity that requires staying
connects to objects that are being counted. That is,
counting is a mental activity that involves moving
from the objects to the numbers and from counting
back to the objects.
Why this domain is a priority for
professional development

In both counting and working with the operations,
eventually students can function in an abstract
domain, but they retain the capacity to show how
ideas can be represented in the models (e.g.
pictures, number lines, manipulatives, physical
contexts).
Why this domain is a priority for
professional development
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Many teachers don’t think in terms of models for the
actions of the operations.
Most teachers haven’t worked explicitly on the behavior
of the operations.
Most teachers have not learned to think about how the
symbolic manipulations they teach are based on the
behavior of the operations (that can be understood in
terms of the models).
Teachers must learn to think about the content described
in the previous slides as algebraic.
Many teachers don’t understand the complexity of
learning to count.
Key standards in this domain that pd
should focus on
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KCC5 Count to answer “how many?” questions about as
many as 20 things arranged in a line, a rectangular
array, or a circle, or as many as 10 things in a scattered
configuration; given a number from 1–20, count out that
many objects.
10A3B Apply properties of operations as strategies to
add and subtract.3 Examples: If 8 + 3 = 11 is known,
then 3 + 8 = 11 is also known. (Commutative property
of addition.) To add 2 + 6 + 4, the second two numbers
can be added to make a ten, so 2 + 6 + 4 = 2 + 10 =
12. (Associative property of addition.)
Existing resources that support these
domains and standards
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Kathy Richardson materials
DMI: Building a System of Tens, Making Meaning for
Operations; Reasoning Algebraically about Operations
CGI (books on number and on algebraic thinking)
Sybilla Beckman’s textbook
Singapore Elementary Textbooks
Investigations in Number, Data, and Space
Cathy Fosnot’s Young Mathematicians Series
Connecting Arithmetic to Algebra (to appear next
September)
Domains and standards that are in
need of new resources
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Frequently professional development is vague. It
requires illustrations of specific mathematics, specific
mathematical practices, and specific pedagogical
practices.
If we had time, we would provide illustrations of the
points we are trying to make.