Cognitively Guided Instruction Using Story Problems to Develop

Cognitively Guided Instruction Using Story Problems to Develop Mathematics Knowledge

Operation Sense • Developing meanings for operations • Gaining a sense for the relationships among operations • Determining which operation to use in a given situation

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

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

Cognitively Guided Instruction Looking at Problem Structures and Numerical Relationships

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.

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.

Which of These Problems Would be Most Difficult for First-grade Students? 1. 2. 3. 4. 5. 6. 7. 8. E B B B A E B E

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?

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

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?

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

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.

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) JOIN CHANGE UNKNOWN (JCU) Connie had 5 marbles. Juan gave her 8 more marbles. How many marbles does Connie have altogether? 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?

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?

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 PPWWU Part Unknown PPWPU Connie has 5 red marbles and 8 blue Connie has 13 marbles. 5 are red marbles. How many marbles does and the rest are blue. How many she have? blue marbles does Connie have?