1. Cognitively Guided Instruction
Using Story Problems to Develop Mathematics Knowledge

2. Operation Sense Developing meanings for operations
Gaining a sense for the relationships among operations
Determining which operation to use in a given situation
CGI Introduction to Problem Types

3. Operation Sense
Recognizing that the same operation can be applied in problem situations that seem quite different
Developing a sense for the operations’ effects on numbers
Realizing that operation effects depend upon the types of numbers involved
CGI Introduction to Problem Types

4. What factors influence how students reason in story problem contexts?
Types of problem structure
Types of numerical relationships within problems, and
Context of the problem and of the number choices (sizes of numbers/kinds of quantities) used

5. Cognitively Guided Instruction
Looking at Problem Structures and Numerical Relationships

6. Which of These Problems Would be Most Difficult for First-grade Students?
Examine the problems on Appendix B.
Assume the problems are read aloud to the child as many times as needed.
Assume the child has a set of counters they can use to help them.
Assume the child has a much time as they wish to solve the problem.

7. Which of These Problems Would be Most Difficult for First-grade Students?
Examine the problems on Appendix B.
Circle “A” if Problem A is harder than Problem B.
Circle “B” if Problem B is harder than Problem A.
Circle “E” if the problems are of equal difficulty.

8. Which of These Problems Would be Most Difficult for First-grade Students?

9. How are these three problems alike? Different?
Lucy has 8 fish. She wants to buy 5 more fish. How many fish would Lucy have then?
TJ had 13 chocolate chip cookies. At lunch she ate 5 of them. How many cookies did TJ have left?
Janelle has 7 trolls in her collection. How many more does she have to buy to have 11 trolls?

10. Willy has 12 crayons. Lucy has 7 crayons
Willy has 12 crayons. Lucy has 7 crayons. How many more crayons does Willy have than Lucy?

11. 11 children were playing in the sandbox. Some children went home
11 children were playing in the sandbox. Some children went home. There were 3 children still playing in the sandbox. How many children went home?

12. Reflect
How do these last two problems compare in difficulty to the three problems we just saw and discussed?

13. Problem Structures Examine the set of “Marble Problems”.
Sort the “Marble Problems” into sets of problems that seem to be related. Be able to explain how you think the problems are related.

14. Start ------>Change------>Result
JOIN – The action in these problems is a joining of two sets.
Start >Change------>Result
The unknown quantity can be either the result, the change, or the start.
JOIN RESULT UNKNOWN (JRU)
Connie had 5 marbles. Juan gave her 8 more marbles. How many marbles does Connie have altogether?
JOIN CHANGE UNKNOWN (JCU)
Connie has 5 marbles. How many marbles does she need to have 13 marbles altogether?
JOIN START UNKNOWN (JSU)
Connie had some marbles. Juan gave her 5 more marbles. Now she has 13 marbles. How many marbles did Connie have to start with?

15. Start ------>Change------>Result
SEPARATE: The action in these problems is taking a subset out of a set.
Start >Change------>Result
The unknown quantity can be either the result, the change, or the start.
SEPARATE RESULT UNKNOWN (SRU)
Connie had 13 marbles. She gave 5 to Juan. How many marbles does Connie have left?
SEPARATE CHANGE UNKNOWN (SCU) Connie had 13 marbles. She gave some to Juan. Now she has 5 marbles left. How many marbles did Connie give to Juan?
SEPARATE START UNKNOWN (SSU)
Connie had some marbles. She gave 5 to Juan. Now she has 8 marbles left. How many marbles did Connie have to start with?

16. Part-Part Whole: There is no action
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.
Whole Unknown Part Unknown
PPWWU
Connie has 5 red marbles and 8 blue marbles. How many marbles does she have?
PPWPU
Connie has 13 marbles. 5 are red and the rest are blue. How many blue marbles does Connie have?