1. Cognitively Guided Instruction
Problem Structures and Solution Strategies
CGI Introduction to 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. Direct Modeling
Example: Robin had 4 toy cars. Her friends gave her 7 more toy cars for her birthday. How many toy cars did she have then?
Student makes a set of 4 cubes and a set of 7 cubes. She pushes them together and then counts them, “1,2,3,4,5,6,7,8,9,10,11,” pointing to a cube with each count. Karla then responds, “She had 11 cars.” 6

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. Direct Modeling Counting
Physical objects are used to represent objects in a problem.
Counting
Physical objects are used to keep track of counts.

9. Counting Strategies
Example: Robin had 4 toy cars. Her friends gave her 7 more toy cars for her birthday. How many toy cars did she have then?
Jamie counts, “4 [pause], 5,6,7,8,9,10,11. She has 11 cars. As Jamie counts, he extends a finger with each count. When he has extended seven fingers, he stops counting and gives the answer. 9

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. Derived Facts/Number Facts
Example: 6 frogs were sitting on lily pads. 8 more frogs joined them. How many frogs were there then?
Rudy, Denise, Theo, and Sandra each answer “14,” almost immediately.
Teacher: How do you know there were 14?
Rudy: Because 6 and 6 is 12, and 2 more is 14.
Slide is animated. You will need to click 5 times to get to see all student responses.
Denise: 8 and 8 is 16. But this is 8 and 6. That is 2 less, so it’s 14.
Sandra: 8 and 2 more is 10, and 4 more is 14. 13

14. Using Number Sense to Help
6 + 7
What are some derived facts kids might use to solve this problem?
What relationships (spatial, one/two more and less, benchmarks of 5 & 10, and part-whole) would students need to have before they can use the derived facts for this problem?
Slide is animated.
Possible derived fact: Think double 7 and subtract one.

15. Progression of Development
Strategy Types
7 + 8
Direct Modeler – counts out 7 things, counts out 8, pushes them all together and counts the total.
Counter – holds 7 in their head and counts on 8 more.
Derived Fact – uses a fact they know to help them

16. 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.

17. 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”. (connects to Bahr and de Garcia Numerical Complexity Hierarchy)

18. Strategies for Solving CGI Problems Summary

19. Figure 3.4
Classification of problem types with number sentences. (Adapted from Carpenter et al 1999, 12. Reprinted with permission from Children’s Mathematics. Copyright © 1999 by Thomas Carpenter, Linda Levi, Elizabeth Fennema, Megan Loef Franke, Susan Empson. Published by Heinemann, Portsmouth, NH. All rights reserved.)
Fig. 3-4, p. 73

20. Figure 3.5
Sequence children pass through when developing problem-solving strategies.
Fig. 3-5, p. 74

21. Figure 3.6
Examples of how children use direct modeling and counting strategies for problem types.
Fig. 3-6a, p. 75

22. Figure 3.6
Examples of how children use direct modeling and counting strategies for problem types.
Fig. 3-6b, p. 76

23. Figure 3.7
Strategy sequence chart. (Reprinted with permission from Children’s Mathematics. Copyright © 1999 by Thomas Carpenter, Linda Levi, Elizabeth Fennema, Megan Loef Franke, Susan Empson. Published by Heinemann, Portsmouth, NH. All rights reserved.)
Fig. 3-7, p. 77