1. Excerpts of Expectations from the Number and Operations Standard Grades Pre-K-5 Principles and Standards for School Mathematics National Council of Teachers of Mathematics 2000

2. Understand meanings of operations and how they relate to one another.  Grades Pre-K-2  Understand various meanings of addition and subtraction of whole numbers and the relationship between the two operations.  Understand the effects of adding and subtracting whole numbers.  Understand situations that entail multiplication and division, such as equal groupings of objects and sharing equally.

3. Understand meanings of operations and how they relate to one another.  Grades 3-5  Understand various meanings of multiplication and division.  Understand the effects of multiplying and dividing whole numbers.  Identify and use relationships between operations, such as division as the inverse of multiplication, to solve problems.  Understand and use properties of operations, such as the distributivity of multiplication over addition.

4. Compute fluently and make reasonable estimates  Grades Pre-K-2  Develop fluency with basic number combinations for addition and subtraction.  Grades 3-5  Develop fluency with basic number combinations for multiplication and division and use these combinations to mentally compute related problems, such as 30 x 50.

5. Components of Model Use 1.Concrete: Use a variety of problem settings and manipulative materials to act out and model the operation. 2.Semi-concrete: Provide representations of objects in pictures, diagrams, and drawings to move a step away from the concrete toward symbolization. 3.Abstract: Use symbols to illustrate the operation

6. Meanings for Operations Addition  Finding how many in all Subtraction  Separation or take away  Comparison or finding the difference  How many more are needed  Multiplication  Equivalent groups of objects or repeated addition  Array or area  Division  Measurement  Partition or sharing

7. When to Memorize  Memorizing can begin as soon as children have a good understanding of the meaning of the operations and the symbols  Children should be able to: – state or write related facts, given one basic fact. – explain how they got an answer, or prove that it is correct. – solve a fact in two or more ways. Ashlock and Washbon (1978)

8. Thinking Strategies for the Basic Facts  Addition-100 facts involving two one-digit addends and their sum – Commutativity – Strategies for 0, 1, and Doubles – Counting On – Adding to 10

9. Thinking Strategies for the Basic Facts  Subtraction-100 facts involving the difference between one addend and the sum for all one-digit addends – Using 0 and 1 – Doubles – Counting Back

10. Thinking Strategies for the Basic Facts  Multiplication-100 facts involving two one-digit factors and their product – Commutativity – Using 0 and 1 – Skip Counting – Repeated Addition – Splitting the Product into Known Parts – Patterns

11. Thinking Strategies for the Basic Facts  Division-90 facts (no division by zero) involving the quotient of one factor and the product for all one-digit factors – Find missing factor in the multiplication problem

12. Principles of Drill  Children should attempt to memorize facts only after understanding is attained.  Children should participate in drill with the intent to memorize. Remembering should be emphasized: This is not a time for explanations.  Drill lessons should be short (5-10 minutes) and should be given almost every day. Children should try to memorize only a few facts to a given lesson and should constantly review previously memorized facts. Davis (1978)

13. Principles of Drill  Children should feel confidence in their ability to memorize and should be praised for good efforts. Records of their progress should be kept.  Drill activities should be varied, interesting, challenging, and presented with enthusiasm. Davis (1978)

14. The Almost Magic Square  Can you change the position of exactly three numbers to form a completely magic square? It would be completely magic if the sum of the numbers in each row, column, and diagonal were the same.  (squares on next slide) 