This document discusses goals and essential understandings for teaching number and arithmetic concepts in pre-kindergarten through 2nd grade. It outlines two main goals: 1) for students to understand the roles of numbers and relations among them, and ways of representing numbers; and 2) for students to understand the meanings of operations, how they are related, and to compute fluently. It elaborates on these goals and also discusses topics like counting, number relationships, place value, and addition and subtraction. The document emphasizes that certain understandings are critical milestones in children's mathematical development.

2. Orientation helps
shape decisionmaking in the
moment.
Adapted from Fennema & Franke, 1992

3. In Number and Arithmetic K-2

4. Goals for pre-K-2
Goal 1: Instructions from pre-K to Grade 2 should
enable students to understand the roles of numbers,
relations among them, and both informal and formal
ways of representing number relations.
Goal 2: Instructions from pre-K to Grade 2 should
enable all students to understand the various
meanings of operations, to recognize how the
operations are related, to compute fluently and to
make reasonable estimates.
(Baroody, A. J. (2004). The developmental bases for early childhood number and operations
standards (pp. 173-219). In Clements & Sharma (Eds.) Engaging young children in
mathematics.Mahwah, NJ: Lawrence Erlbaum.)

5. Goals for pre-K-2
Goal 1: Instructions from pre-K to Grade 2 should
enable students to understand the roles of
numbers, relations among them, and both
informal and formal ways of representing number
relations.
(Baroody, A. J. (2004). The developmental bases for early childhood number and
operations standards (pp. 173-219). In Clements & Sharma (Eds.) Engaging young
children in mathematics.Mahwah, NJ: Lawrence Erlbaum.)

6. Elaborate Goal 1
Use numbers to count with understanding – that is, connect
number words to the quantities they represent so as to
recognize how many in a collection or to count out
collections of a particular size.
Use numbers to compare quantities by developing an
understanding of relative position and magnitude of whole
number and the connection between ordinal and cardinal
numbers
Represent collections up to10 [then to 20; then to 100] and
numerical relations by connecting numerals to number
words and the quantities both represent.
(Baroody, A. J. (2004). The developmental bases for early childhood number and operations
standards (pp. 173-219). In Clements & Sharma (Eds.) Engaging young children in
mathematics.Mahwah, NJ: Lawrence Erlbaum.)

7. Goals for pre-K-2
Goal 2: Instructions from pre-K to Grade 2 should
enable all students to understand the various
meanings of operations, to recognize how the
operations are related, to compute fluently and to
make reasonable estimates.
(Baroody, A. J. (2004). The developmental bases for early childhood number and
operations standards (pp. 173-219). In Clements & Sharma (Eds.) Engaging young
children in mathematics.Mahwah, NJ: Lawrence Erlbaum.)

8. Elaborate Goal 2
Understand the [different] meanings of addition and
subtraction of whole numbers and use this knowledge to
make sensible estimates and to develop calculational
proficiency.
(Baroody, A. J. (2004). The developmental bases for early childhood number and
operations standards (pp. 173-219). In Clements & Sharma (Eds.) Engaging young
children in mathematics.Mahwah, NJ: Lawrence Erlbaum.)

9. Listing Core Topics
Number 1- 100
Counting
Number relationships
Number composition and decomposition to 20
Place Value: Number composition and decomposition
of numbers to 100
 numbers as tens and ones;
 adding and subtracting two digit numbers

10. In Number and Arithmetic K-2

11. In Number and Arithmetic K-2

12. Big Ideas
(Schifter & Fosnot, 1993)
Big ideas are “the central, organizing ideas of
mathematics—principles that define
mathematical order” (Schifter and Fosnot 1993,
p. 35)… These ideas are “big” because they are
critical ideas in mathematics itself and because
they are big leaps in the development of the
structure of children’s reasoning.

13. KDUs (Simon, 2006)
Key developmental understandings (KDUs) mark
critical transitions that are essential for students’
mathematical development. Such transitions are
identified by qualitative shifts in students’ abilities
to think about and perceive particular
mathematical relationships.

14. Critical Learning
Phases (Richardson, 2008)
A phase is a particular moment or state in a
process, especially one at which a significant
development occurs or a particular condition is
reached.
Critical learning phases mark essential ideas
that are milestones or hurdles in children’s
growth of mathematics understanding.
(www.didax.com/AMC/: The research basis for Assessing Math ConceptsTM p. 6)

15. Defining Essential
Understandings
The essential ideas that are milestones, hurdles,
big leaps, or critical transitions in children’s
growth of understanding.
The developments that determine the way a
child is able to think with numbers and use
numbers to solve problems.
They are those understandings that are tied to
completing a developmental process and, if not
completed, students will lack a particular
mathematical ability.

16. Defining Essential
Understandings
Students do not tend to acquire essential
understandings as a result of explanation or
demonstration.
These understandings must be in place to ensure
that children are not just imitating procedures or
saying words they do not really understand, thus
creating illusions of learning.
(www.didax.com/AMC/: The research basis for Assessing Math ConceptsTM )

17. Unpacking Essential
Understandings
What essential understandings are critical to
the development of particular core topics?
How do we unpack these essential
understandings in order to focus our
assessment, instruction, and/or interventions?

18. Focus: Counting
You are an early grades elementary school teacher
attending a session intended to engage you in
thinking about what essential understandings are
important for developing knowledge of the core
topic of counting.

19. Focus: Counting
With these “shoes” on, make some notes.
 What does it mean to say that a child
understands counting?
 What do you look for in a child’s
performance and/or language that indicates
“strong” or “weak” knowledge of counting?

21. Development?
What does it mean to say that
learning to count is a
developmental process?

25. In Number and Arithmetic K-2