## Appendix 2: An Outline for the Interviews on Prospective Teachers' Formal and Intuitive Knowledge of Rational Numbers

1.a. Name sets of numbers that you are familiar with.*

1.b. Define these sets.

1.c. Give at least one example of an element for each set.

2. Fill out the following table** :

 The Number Natural Integers Rational Irrational Real 2 -4 0 0.123 0.191991999 ... 0.251 3.14
*Expected answers: Natural numbers, integers, rational numbers, irrational numbers, real numbers

** Discus with the students: inconsistencies between their definitions [question 1b] and the way they completed the table [question 2].

3. What are the relations (e.g. inclusion) between: natural numbers, integers, rational numbers, irrational numbers, and real numbers?

4. Let's assume that we are familiar only with the natural numbers.

What would be the problems that would necessitate the use of other numbers? Please give specific examples.

When would 0 be needed?

When would negative numbers be needed?

When would rational numbers be needed?

5. Look at the following examples:

 5 - = 3 12 x = 6 10 : = 6 10 : = 20 7 : 0 =

In your opinion, what might a young child not understand?

Why?

6. a. Define and give examples of:

1. the commutative law

2. the associative law

3. closure

6. b. A child uses the following procedure to compute 1 + 27 + 99

1 + 27 + 99 =

27 + 1 + 99 =

27 + (1 + 99) =

What laws did he use?

7. Fill out the left side of the following table. Refer to the natural numbers.

Next, refer to the rational numbers. Will there be any difference?

Now fill out the right side of the table. Refer to the rational numbers.

 Natural Numbers Rational Numbers Operation Commutative Associative Commutative Associative + - x ÷

8. Define and give examples of the distributive law.

9. Are there any numbers between 1/4 and 1/5?

No. Why?

Yes. How many? For example?

10. Are there any numbers between 0.23 and 0.24?

No. Why?

Yes. How many? For example?

Return