Carpenter, Franke, & Levi Chapter 2

1. Carpenter, Franke, & Levi Chapter 2 Equality

2. 8 + 4 =  + 5 Table 2.1 from page 9

3. Children’s Conception of the Meaning of the Equal Sign • Lucy: The Answer Comes Next • The equal sign was a command to carry out the calculation • Randy: Use All the Numbers • Did not recognize that where the symbols appeared in the number sentence made a difference • Barb: Extend the Problem • 8 + 4 = 12 + 5 = 17

4. Making Sense • Methods were not unreasonable attempts to deal with an unfamiliar problem • The answer comes right after the equal sign • Order of the symbols in a number sentence is not important • Errors of syntax • 8+4 is an operation to be carried out rather than a way to represent 12

5. Ricardo and Gina: A Related View • Comfortable with expressions involving addition on either side of the equal sign

6. Developing Children’s Conceptions of Equality • Put students in a position to challenge their existing conceptions • Students need to articulate mathematical principles that often are left implicit

7. A Context for Discussing the Equal Sign Appropriately chosen tasks can: • Provide a focus for students to articulate their ideas • Challenge students’ conceptions by providing different contexts in which they need to examine the positions they have staked out • Proved a window on children’s thinking

8. True/False Number Sentences • Provide an example and ask whether the number sentence is true or false • Use relatively simple calculations • Do not require students to supply the answer after the equal sign • Encourage students to examine their conceptions of the meaning of the equal sign

9. Comparing sentences may help students articulate the specific rules they are using to decide whether a sentence is true or false

10. Wording and Notation • Equal sign signifies a relationship between two numbers • “Eight is the same amount as 3 plus 5”

11. Classroom Interactions • The goal is not just to teach students appropriate conceptions of the use of the equal sign • It is equally important to engage them in productive mathematical arguments • Helping students understand the accepted use of the equal sign represents the first step in helping students make and justify generalizations about math.

12. Benchmarks • Getting beyond just comparing the different answers • Children first accept as true some number sentence in the form a + b = c • Children recognize that the equal sign represents a relations between two equal numbers • Compare the mathematical expressions without carrying out the calculations

13. What to Avoid • The equal sign is used as a shorthand for a variety of purposes • A person is not a number • The number of objects in a collection is a number but the set itself is not equal to a number • It is tempting to use the equal sign to represent a series of calculations. This reinforces some of the misconceptions associated with 8 + 4 =  + 5

14. The Use of the Equal Sign is a Convention • The equal sign is used in math is a matter of agreement and convention • It is not possible to justify that the equal sign represents a relation rather than a command to do something

15. Smoothing the Transition to Algebra • Equal sign expresses a relation and that adding or subtracting the same thing to two expressions preserves the equality • Understanding the relationship the equal sign represents opens up the power of algebra for representing problems and performing complex operations on mathematical expressions

16. Where do Children’s Misconceptions About the Equal Sign Come From? • Many children see only examples of number sentences with an operation to the left of the equal sign and the answer on the right • Limiting children’s exposure to a narrow range of number sentences does appear to contribute to students’ misconceptions about the equal sign

17. Calculators may reinforce the notion that the equal sign means “carry out the preceding calculation” • Children may be predisposed to thinking of equality in terms of calculating answers rather than as a relation

18. Children’s Conceptions • Some limits in their understanding of how mathematical ideas are generated and justified • Question why things work out the way they do • We do not anticipate what students are capable of