1. Cognitively Guided Instruction @ Problem Types @ Informal Strategy Types

2. Children’s Informal Strategies @ Direct Modeling -- Children “act out” the problem. They are very literal. They model every number and work the problem in chronological order.

3. Children’s Informal Strategies @ Counting Strategy -- The child does not model every number in the problem. The child “stores” one number and counts up or back to solve the problem.

4. Children’s Informal Strategies @ Derived Fact -- The child uses a known number relationship to figure out the desired relationship.

5. Problem Types Join @ Join -- The action in these problems is a joining of two sets. @ The unknown quantity can be either the result, the change, or the start.

6. Join @ Join -- The action in these problems is a joining of two sets. Result Unknown: Jim had 8 marbles. Sue gave him 4 more. How many marbles does Jim have now? 8 + 4 = ? Change Unknown: Jim had 8 marbles. Sue gave him some more. Now Jim has 12 marbles. How many marbles did Sue give Jim? 8 + ? = 12 Start Unknown: Jim had some marbles. Sue gave him 4 more. Now Jim has 12 marbles. How many marbles did Jim have to start with? ? + 4 = 12

7. Problem Types Separate @ Separate -- The action in these problems is taking a subset out of a set. @ The unknown quantity can be either the result, the change, or the start.

8. Separate @ Separate -- The action in these problems is taking a subset out of a set. Result Unknown: Jim had 12 cookies. He ate 4. How many cookies does Jim have now? 12 - 4 = ? Change Unknown: Jim had 12 cookies. He ate some. Now Jim has 8 cookies. How many cookies did Jim eat? 12 - ? = 8 Start Unknown: Jim had some cookies. He ate 4. Now Jim has 8 cookies. How many cookies did Jim have to start with? ? - 4 = 8

9. Problem Types Part-Part-Whole @ Part-Part-Whole -- There is no action. A set (whole) with defined subsets (parts) is described. @ The unknown quantity can be either one of the parts or the whole.

10. Part-Part-Whole @ Part-Part-Whole -- There is no action. A set (whole) with defined subsets (parts) is described. Whole Unknown: Jim had 8 red marbles and 4 blue marbles. How many marbles does Jim have in all? Part Unknown: Jim had 8 red marbles. The rest are blue. Jim has 12 marbles in all. How many are blue?

11. Problem Types Comparison @ Comparison -- There is no action. The size of two sets is compared. @ The unknown quantity can be either the difference between the sets or one of the sets.

12. Comparison @ Comparison -- There is no action. The size of two sets is compared. Difference Unknown: Jim had 8 marbles. Sue had 12 marbles. How many more marbles does Sue have than Jim? Large Set Unknown: Jim has 8 marbles. Sue has 4 more than Jim. How many marbles does Sue have? Small Set Unknown: Sue has 12 marbles. Jim has 4 fewer marbles than Sue. How many marbles does Jim have?