Adapted from Fennema & Franke, 1992

Orientation helps shape decision-making in the moment.

In Number and Arithmetic K-2

Goals for pre-K-2Goal 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.)

Goals for pre-K-2Goal 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.)

Elaborate Goal 1Use 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.)

Goals for pre-K-2Goal 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.)

Elaborate Goal 2Understand 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.)

Listing Core TopicsNumber 1- 100

CountingNumber relationshipsNumber composition and decomposition to

20Place Value: Number composition and

decomposition of numbers to 100 numbers as tens and ones; adding and subtracting two digit numbers

In Number and Arithmetic K-2

In Number and Arithmetic K-2

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.

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.

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)

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.

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 )

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?

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.

Focus: CountingWith 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?

Development?

What does it mean to say that learning to count is

a developmental process?

In Number and Arithmetic K-2