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  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  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  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 = 11 is known, then = 11 is also known. (Commutative property of addition.) To add , the second two numbers can be added to make a ten, so = = 12. (Associative property of addition.)

Existing resources that support these domains and standards  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  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.