1. Where does theory enter empirical research? How? What should appear in Theoretical Framework of your research proposal/dissertation/paper? What is meant by “theoretical contribution” of your study? 1

2. What are the requirements from a “good” theory or a model? supports prediction has explanatory power Is applicable to a broad range of phenomena Helps in organizing one’s thinking about complex interrelated phenomena serves as a tool for analyzing data provides a language for communicating ideas about learning that go beyond superficial descriptions. 2 Schoenfeld (2002); Dubinsky & McDonald (2001)

3. Example 1 (Koichu, 2010) N players take part in an individual tennis competition. What is the overall numbers of games, if a player who loses a game leaves the competition? 3

4. How could I miss such a simple solution?! Why do so many mathematics students and teachers miss the immediate solution? Why does the problem frequently appear to be especially difficult for the participants with good mathematical background? Why does the presentation of the immediate solution evoke such a strong emotional reaction? 4

5. Explanation 1 (in spirit of Schoenfeld, 1985) University students tend to reserve little time for planning the solutions. Instead, they quickly delve in executing and writing something down. Critique: This explanation does not elaborate on the cognitive mechanism of the phenomenon. According to Schoenfeld (1992) himself, little is known about how different components of PS come to cohere. 5

6. Explanation 2 (in spirit of Selden, Selden, Hauk & Mason, 2000) Any problem evokes a problem situation image containing tentative solution starts. The students do not retain the most appropriate to the problem's solution knowledge since they possess too few tentative solution starts. Critique: This explanation does not address the fact that so many mathematically students and teachers felt that they could and should had been found the immediate solution. 6

7. Explanation 3: (in spirit of Krutetskii, 1977, or Chi et al., 1981) The immediate solution requires reversing the problem, namely, thinking-from-the-end-to-the- beginning. This strategy belongs to problem solving repertoires of either mathematically gifted individuals or expert problem solvers, and is rarely used by regular students (Koichu, Berman & Moore, 2006). Critique: this explanation leads to the conclusion that there were fewer gifted or experts problem solvers among more mathematically knowledgeable participants than among the rest. 7

8. What are the requirements from a “good” theory or a model? supports prediction has explanatory power Is applicable to a broad range of phenomena Helps in organizing one’s thinking about complex interrelated phenomena serves as a tool for analyzing data provides a language for communicating ideas about learning that go beyond superficial descriptions. 8 Schoenfeld (2002); Dubinsky & McDonald (2001)

9. Example 2 (Span and Overtoom-Corsmit, 1986) The goal: to investigate “the consequences of general intellectual giftedness for the way mathematical problems are coped with” (p. 275). The sample: 14 highly gifted and 14 averagely gifted students chosen from about 400 six-graders by means of a battery of three tests, including Raven’s Advanced Progressive Matrices test (RAPM) and a teacher nomination. The instrument: The participants were individually given seven relatively difficult problems and asked to think aloud while solving them The findings: 1.the “highly gifted” over performed the “averagely gifted” 2.the “highly gifted” students were superior with respect to the quality of their evaluation processes and the amount of time devoted to orientation in the given problems. 3.the students of the averagely gifted group more often tried to solve the problems by trial and error, while the students of the highly gifted group utilized other strategies. 9 Koichu (accepted)

10. The Raven test 10 Taken from the Raven PMT, 1983

11. What does the Raven test measure? "the common ability to decompose problems into manageable segments and iterate through them, the differential ability to manage the hierarchy of goals and subgoals generated by this problem decomposition and the differential ability to form high level abstractions“ (p. 429) Carpenter, Just and Shell (1990) 11

12. In light of the above quotation, one can say that the way of sampling pre-determined that the “gifted” over performed the “good” 12 Koichu (accepted)

13. The threat of circularity Criterion 1: Both the intellectual ability presumably underlying problem-solving competence and problem- solving competence itself need to be explicitly defined and must not overlap at theoretical and/or operational levels... If the latter is not the case, then any attempt to explain problem-solving competence in terms of an underlying intellectual ability is necessarily circular and redundant... Criterion 2: The presumed relations between intellectual ability and problem-solving competence must have a theoretical explanation (Wenke & French, 2003, p. 92). 13 Note that the Span and Overtoom-Corsmit’s (1986) results had not been circular when they were published.

14. The threat of circularity - generalization Criterion 1: Research variables should to be explicitly defined and must not overlap at theoretical and/or operational levels... If the latter is not the case, then any attempt to in terms of an underlying intellectual ability is necessarily circular and redundant... Criterion 2: The presumed relations between the variables must have a theoretical explanation. These conditions are difficult to achieve. What can we do? 14

15. What are the requirements from a “good” theory or a model? supports prediction has explanatory power Is applicable to a broad range of phenomena Helps in organizing one’s thinking about complex interrelated phenomena serves as a tool for analyzing data provides a language for communicating ideas about learning that go beyond superficial descriptions. 15 Schoenfeld (2002); Dubinsky & McDonald (2001)

16. Example 3 (Mestre, 2000) 16 supports prediction has explanatory power Is applicable to a broad range of phenomena Helps in organizing one’s thinking about complex interrelated phenomena serves as a tool for analyzing data provides a language for communicating ideas about learning that go beyond superficial descriptions.