1. Problem Structures and Solution Strategies

2. Solution Strategies Direct Modeling Strategies Counting Strategies Number Facts/Derived Facts

3. Solution Strategies For the most basic strategies, children use physical objects (counters) or fingers to directly model the action or relationships described in each problem.

4. Solution Strategies Over time, children’s strategies become more abstract and efficient. Direct Modeling strategies are replaced by more abstract Counting Strategies, which in turn are replaced with Number Facts.

5. Direct Modeling This strategy is distinguished by a child’s explicit physical represent- ation of each quantity in the problem and the action or relationship involving those quantities before counting the resulting set.

6. Watch Direct Modeling

7. Counting A child essentially recognizes that it is not necessary to actually construct and count sets. The answer can be figured out by focusing on the counting sequence itself.

8. Watch Counting

9. Direct Modeling Physical objects are used to represent objects in a problem. Counting Physical objects are used to keep track of counts. (page 23, 25)

10. Number Facts Children learn number facts and apply this knowledge to solve problems. Children learn certain number combinations before others. Children often use a small set of memorized facts to derive solutions for problems involving other number combinations.

11. Number Facts Children learn doubles before other number combinations. Children learn sums of ten relatively early.

12. Derived Facts Derived Fact solutions are based on understanding relations between numbers. Even without specific instruction, most children use Derived Facts before they have mastered all their number facts at a recall level. When children have the opportunity to discuss alternative strategies, the use of Derived Facts becomes even more prevalent.

13. Watch Derived Facts

14. Levels of Development Children appear to “move through” Direct Modeling, Counting, and Derived Fact strategies when number choices/kinds of quantities are consistent across problems. Direct Modeling strategies are not easily used with some problem types. Mental strategies are extensions of modeling strategies.

15. Levels of Development Cycles of trajectories of strategy use are related to number choices used in problems. Cycling through strategy levels reflects increasing sophistication of knowledge with a particular “range of numbers”.

17. Fig. 3-4, p. 73

18. Fig. 3-5, p. 74

19. Fig. 3-6a, p. 75 Bahr, D. & DeGarcia, L. A. (2008). Elementary mathematics is anything but elementary: Content and methods from a developmental perspective. Cengage Learning.

20. Fig. 3-6b, p. 76 Bahr, D. & DeGarcia, L. A. (2008). Elementary mathematics is anything but elementary: Content and methods from a developmental perspective. Cengage Learning.

21. Fig. 3-7, p. 77 Carpenter, et. al. (1999)

22. Number Choices In story problems, the numbers used should be in accord with the number development of the children (but could include multi- digit numbers if these are numbers children “know” even if they haven’t yet mastered place value). The intent of story problems is to motivate application of numbers in context and not to teach number knowledge. It is important that you know a child’s number knowledge zone! For example, kindergarten children can use numbers as large as they can count meaningfully, which is usually to about 10 to 12 (VandeWalle). Number knowledge and solution strategies interact. If student is direct modeler counting by ones, think about choice of sizes of multi-digit numbers!

23. Number Choices When a child doesn’t seem to be able to solve a problem, change the numbers (i.e., make smaller) and try having the child solve a simpler (i.e., different numbers) problem.

24. Context (with type) promotes strategy choice (adding up) Martha’s goal is to walk 50 laps on the school track. She has already walked 17 laps. How many more laps does Martha need to walk to reach her goal? Paul plans to read 40 pages each day. So far, he has read only 16 pages. How many more pages does Paul need to read today to reach his goal?

25. Bethany has 50 marbles. She decides to give her friend Marco 17 of her marbles. How many marbles will Bethany have left? Makalah saved $182. She bought a video game for $53. How much money does Makalah have left? Context (with type) promotes strategy choice (removal)