1. Looking at the Structure of Arithmetic Story Problems Through the Eyes of Children’s Solution Strategies

2. Consider the following problems—How would you solve each? Eliz had 8 cookies. She ate 3 of them. How many cookies does Eliz have left? Eliz has 3 dollars to buy cookies. How many more dollars does she need to earn to have 8 dollars? Eliz has 3 dollars. Tom has 8 dollars. How many more dollars does Tom have than Eliz?

3. Most adults would solve all three of these problems by subtracting 3 from 8. To young children, however, these are 3 different problems, which they solve using different strategies.

4. Research Children can solve many mathematical problems that involve +, -, x or / without being shown how to solve them.

5. Initially, most children use a strategy researchers refer to as “direct modeling.” That is, the children model the action and relationships they see in the story.

6. Additive Structures Four Basic Problem Contexts: –Join, –Separate, –Part-Part-Whole, and –Compare. Although the number size, themes, and contexts of the problems may vary, the basic structure involving actions and relationships within each class remains the same.

7. Join and Separate problems involve action. –In join problems, elements are added to a given initial set. –In separate problems, elements are removed from a given initial set.

8. Part-Part-Whole and Compare-No action Part-Part-Whole problems involve static relationships among a set and its two distinct subsets. Compare problems involve the comparison of two distinct, disjoint sets rather than the relationship between a set and its subsets.

9. Join, Result Unknown Four birds were singing on a branch. Three more flew in to join them. How many birds were singing now? Join (an action), Result Unknown 4 + 3 = __

10. Join, Change Unknown Four birds were singing on a branch. Some more flew in to sing with them. Now there are 7. How many birds are on the branch? Join (an action), Change Unknown 4 + __ = 7

11. Join, Start Unknown Some birds were singing on a branch. Three more flew in to sing with them. Now there are 7. How many birds were already there? Join (an action), Start Unknown __ + 3 = 7

12. Separate, Result Unknown Seven birds were singing on a branch. Three flew away. How many birds are left? Separate (an action), Result Unknown 7 - 3 = __

13. Separate, Change Unknown Seven birds were singing on a branch. Some flew home. Now there are 3. How many birds went home? Separate (an action), Change Unknown 7 - __ = 3

14. Separate, Start Unknown Some birds were singing on a branch. Three flew home. Now there are 4. How many birds were there to begin with? Separate (an action), Start Unknown __ - 3 = 4

15. Part-Part-Whole, Whole Unknown Some birds were singing on a branch. Three were blue and 4 were red. How many birds were on the branch? Part-Part-Whole, Whole Unknown (Note—no action! This is a relationship between 2 parts of one set.) There is no specific number sentence for part- part-whole problems. Can you see why?

16. Part-Part-Whole, Part Unknown Seven birds were singing on a branch. Three were blue and the rest were red. How many birds were red? Part-Part-Whole, Part Unknown (Note— no action! This is a relationship between 2 parts of one set.)

17. Compare, Difference Unknown The oak tree has 7 birds. The pine tree has 4 birds. The oak tree has how many more birds than the pine tree? Compare, Difference Unknown (Note—no action! This is a relationship between 2 distinct, disjoint sets.)

18. Compare, Larger Quantity Unknown The oak tree has 3 more birds than the pine tree. The pine tree has 4 birds. How many birds does the oak tree have? Compare, Larger Quantity Unknown (Note—no action! This is a relationship between 2 different sets.)

19. Compare, Smaller Quantity Unknown The oak tree has 3 more birds than the pine tree. The oak tree has 7 birds. How many birds are in the pine tree? Compare, Smaller Quantity Unknown (Note—no action! This is a relationship between 2 different sets.)

20. As children mature, their strategies become more abstract and efficient. –Direct Modeling strategies are replaced by-- –Counting strategies, which in turn are replaced with-- –Derived Facts strategies (invented strategies with single-digit numbers), which are replaced by-- – Recall of Number Facts strategies.

21. Other Notes Direct Modeling provides a basis for children’s learning of other, more efficient, strategies.

22. Other Notes Strategies at different levels of abstraction are related----which leads to flexible use of strategies. Children in any classroom will be at different levels of understanding and will use different strategies to solve the same problems.