R ESPONSABILIDAD DEL E STADO

There are limitations to the methodology of this dissertation study. First, as mentioned above, the practice session was not enough to enable participants to encode learning, which is probably the reason for the insignificant difference in participants’ transfer across the three conditions. Follow-up studies are needed to examine this alternative before drawing the final conclusion on the effect of superficial variability of examples on transfer.

Second, even though both participants’ perception and actual performance support the idea that the difficulty level varies among the probability rules, the origin of the difficulty was not controlled. By that, the author means that it was uncertain whether the difficulty stemmed from the problem type or the cover story. In future studies,

researchers need to control for the difficulty of the context of cover stories, if they are interested in how difficulty level for problem types interacts with the effect of superficial variability of examples.

Third, the representativeness of the learner population was not diverse. All participants in this dissertation study were from a selective school. To enhance the generality of the findings, more diverse population samples from various schools should be drawn in future studies.

Finally, the applicability of the findings to other subject domains is worthwhile to investigate. Both this dissertation study and most of previous research focused on

mathematically-based subjects. As schema construction is a generic process (Brewer &

76

and arts subjects to discover how well the instructional strategy fits on a larger scale in future studies.

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Appendix A: Personal Information Survey You are a: Female or Male?

What is your major? _____________________________________ Which school are you from? _____________________________ Have you taken any probability class within past 3 years?

Yes No

How would you describe your familiarity with the statistical probability?

1 2 3 4 5

Not at all Not Really Neutral Somewhat Very Much

How would you describe your mathematical skills?

1 2 3 4 5

Not at all Not Really Neutral Somewhat Very Much

How would you describe your familiarity with poker games?

1 2 3 4 5

Not at all Not Really Neutral Somewhat Very Much

How would you describe your familiarity with the activity of spinning a spinner?

1 2 3 4 5

Not at all Not Really Neutral Somewhat Very Much

Howe would you describe your familiarity with the activity of rolling a dice?

1 2 3 4 5

Not at all Not Really Neutral Somewhat Very Much

Howe would you describe your familiarity with colorful marbles?

1 2 3 4 5

87

Appendix B: Sample Comprehension Questions in the Pretest 1. Please select the correct statement(s) about the Addition Rule:

A. When two events, A and B, are mutually exclusive, the probability that A or B will occur is: P(A or B) = P(A) + P(B) [correct]

B. When two events, A and B, are independent, the probability of both occurring is: P(A and B) = P(A) x P(B)

C. When two events, A and B, are not mutually exclusive, the probability that A or B will occur is: P(A or B) = P(A) + P(B) – P(A and B) [correct]

D. When two events, A and B, are dependent, the probability of both occurring is: P(A and B) = P(A) x P(B|A)

2. Please select the correct statement(s) and formula(s) about the Permutation Rule: A. A permutation is a rearrangement of the elements of an ordered list into a one-

to-one correspondence with itself. [correct]

B. a permutation is a way of selecting items from a collection, such that the order of selection does not matter.

C. [correct] D. nPr= n! (n - r)! nCr= n! r!(n - r)!= nPr r!

88

Appendix C: Sample Application Questions in the Pretest

Q1: Peter is randomly picking a card from a standard deck of 52 playing cards. What is the probability of his choosing a queen or a heart?

A.

B.

[correct]

C.

D.

Q3: Terry is randomly picking two cards, one by one, from a standard deck of 52 playing cards. What is the probability that the first card chosen is a club and the second card chosen is a diamond? A. B. [correct] C. D.

Q5: Mike is drawing three cards from the pile of 52, one by one, what’s the probability of those 3 cards are exactly in the order of King of Club, 2 of Spade, 6 of Spade?

A. B. C. [correct] D. 15 52 16 52 17 52 18 52 13 52´ 1252 13 52´ 1351 13 52´ 1352 13 52´ 1251 1 52C3 1 13´ 112´ 111 1 52P3 1 52´ 250

89 E.

[correct]

Q8: A glass jar contains 3 blue, 2 green, and 2 yellow balls. Choose 3 balls from the jar one by one. What is the probability that those 3 chosen balls include 1 blue, 1 green, and 1 yellow? A. B. [correct] C. [correct] D. E. 1 52´ 151´ 150 (1 7´ 16´ 15) ´ (3´ 2 ´ 2) 1 7C3 ´3´ 2 ´ 2 (3 7´ 26´ 25) ´ (3´ 2 ´1) 1 7P3 ´3´ 2 ´ 2 1 7C3

90

Appendix D: Sample Questions in the Transfer Test

1. On New Year's Eve, the probability of a person having a car accident is 0.09. The probability of a person driving while intoxicated is 0.32 and probability of a person having a car accident while intoxicated is 0.15. What is the

probability of a person driving while intoxicated or having a car accident? A. 0.32 + 0.09

B. 0.32 - 0.09

C. 0.32 + 0.09 – 0.15 [correct] D. 0.32 + 0.09 + 0.15

E. (0.32 + 0.09) x 0.15

2. What is the probability that the word of “SHIFT” is made from distinguishable arrangements of F, H, I, S, and T? These five letters cannot be used

repeatedly. A. B. [correct] C. D. [correct] E. F.

3. A nationwide survey showed that 65% of all children in the United States dislike eating vegetables. If 4 children are chosen at random, what is the probability that all 4 dislike eating vegetables?

A. 0.35 x 0.35 x 0.35 x 0.35 B. 0.65 x 0.35 x 0.65 x 0.35 1 5C5 1 5P5 1 5´ 15´ 15´ 15 1 5´ 14´ 13´ 12 (1 5´ 14´ 13´ 12) ´ (5 ´ 4 ´ 3´ 2) (1 5´ 14´ 13´ 12) ´ (5 + 4 + 3+ 2)

91 C. 0.65 x 0.65 x 0.65 x 0.65 [correct] D. 0.65 + 0.65 + 0.65 + 0.65

92

Appendix E: Feedback Survey For Addition Rule

1. Do you think Addition Rule is difficult for you?

1 2 3 4 5

Not at all Not Really Neutral Somewhat Very Much 2. Do you think the instruction (PowerPoint slides) of Addition Rule is clear?

1 2 3 4 5

Not at all Not Really Neutral Somewhat Very Much 3. Do you think the instruction is helpful for your understanding of Addition Rule?

1 2 3 4 5

Not at all Not Really Neutral Somewhat Very Much 4. How would you rate your understanding of Addition Rule after going through the slides

compared to before you going through the slides?

1 2 3

Worse As well as before Better

For Multiplication Rule 5. Do you think Multiplication Rule is difficult for you?

1 2 3 4 5

Not at all Not Really Neutral Somewhat Very Much 6. Do you think the instruction (PowerPoint slides) of Multiplication Rule is clear?

1 2 3 4 5

Not at all Not Really Neutral Somewhat Very Much 7. Do you think the instruction is helpful for your understanding of Multiplication Rule?

1 2 3 4 5

Not at all Not Really Neutral Somewhat Very Much 8. How would you rate your understanding of Multiplication Rule after going through the

slides compared to before you going through the slides?

1 2 3

93

94

Appendix G: Screen Shots for the Instructional Slides

The following images are screen shots for a worked example of the permutation rule. Each image is for one step of the solution. Each step is triggered by tapping the right- and left-arrow-button on the keyboard.

Step 1

Step 2

95 Step 4

96 Step 6

97 Step 8

98 Step 10

99 Step 12

100

Appendix H: Cover Stories Used in the Pretest and Posttests and Instruction

Pretest/Posttests Instructional Materials

Condition 1 CS Condition 2 VSWR Condition 3 VSBR Additions Cards+Dice Cards Cards+Dice+Marbles+Spinner Dice Multiplications Cards+Marbles Cards Cards+Dice+Marbles+Spinner Cards Permutations Cards+Spinner Cards Cards+Dice+Marbles+Spinner Spinner Combinations Cards+Marbles Cards Cards+Dice+Marbles+Spinner Marbles

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