# Creativity understending innovation in problem solving science invention and the arts - Weisberg R.W.

ISBN-10: 0-471-73999-5

**Download**(direct link)

**:**

**141**> 142 143 144 145 146 147 .. 298 >> Next

Table 6.2 Matchstick-arithmetic problems

Problems A and B: Move only one matchstick to produce a true equation.

A. IV = III + III B. III = III + III

Solutions:

A. Move the “I” on the left to the right side of the “V,” so the equation reads

VI = III + III.

B. Move one piece from the + sign to make it an = sign, so the equation reads

III = III = III.

305

Creativity: Understanding Innovation

constraint in arithmetic. In matchstick arithmetic, in contrast, one often changes the value of a number on only one side of an equation. Also, in ordinary arithmetic, an operator (e.g., +, —) cannot be changed arbitrarily (there is an operator constraint), but exactly such a change may be required to solve a matchstick-arithmetic problem.

Knoblich and colleagues (2001) assume that it is harder to relax the operator constraint than the numerical constraint, because the former is wider in its application. Based on that assumption, Knoblich and colleagues made predictions as to the relative difficulty of a set of matchstick-arithmetic problems. When those problems were given to undergraduates, solution rates conformed to predictions, which supports their theory of constraint relaxation and chunk decomposition. This in turn supports the premise that, in response to impasse during a matchstick-arithmetic problem, restructuring occurs along the lines postulated by Ohlsson’s (1992) theory. However, it should be noted that Ohlsson’s theory predicts that restructuring occurs in response to impasse, but Knoblich and colleagues made no attempt to measure whether impasse actually occurred as their participants tried to solve the matchstick-arithmetic problems. Thus, the results obtained by Knoblich and colleagues actually say nothing about restructuring in response to impasse, and therefore they are only indirectly supportive of Ohlsson’s view.

A study by Kaplan and Simon (1990) that used the Mutilated Checkerboard problem (see Figure 6.6) also examined the role of impasse in restructuring. In the Mutilated Checkerboard problem, people initially try to place dominoes in various patterns, but always without success. Thus, the individual will arrive at an impasse, which might lead them to believe that the problem cannot be solved. However, the instructions ask that one prove that there is no solution, which mere belief does not accomplish. According to Kaplan and Simon, in order to prove that the problem cannot be solved, one must move away from simply trying to place dominoes in various configurations and restructure the situation in a way that will allow one to move down the path toward the proof. In the Mutilated Checkerboard problem, the piece of information crucial to restructuring the problem involves the notion of parity: examining the pairing of the squares to be covered by a domino. When one has a way of pairing the squares (e.g., by coloring them alternately black and white), one sees that removing the two diagonally opposite squares disrupts parity, because both removed squares are of the same color. This leaves 32 black squares and 30 white squares, and, since each domino must cover one square of each color, the problem cannot be solved. Thus, the initial problem representation lacks a crucial piece of information: that pointing to parity.

306

The Question of Insight in Problem Solving

A standard 8 x 8 checkerboard has 64 squares. Imagine that you have 32 dominoes; imagine placing them on the board so that one domino covers two horizontally or vertically adjacent squares (not diagonally adjacent squares). It is easy to see that the 32 dominoes will cover all 64 of the squares on the checkerboard. Assume now that two diagonally opposite squares have been removed, leaving 62 squares (see diagram). Now imagine that you have 31 dominoes. Show how those dominoes would cover the remaining 62 squares on the checkerboard, or prove logically that those dominoes cannot cover those remaining squares.

Figure 6.6 Mutilated Checkerboard problem

Kaplan and Simon (1990) collected verbal protocols during their research, which enabled them to determine when their participants reached impasse. They found that people working on the Mutilated Checkerboard problem did not, in response to impasse, discover the importance of parity by themselves. However, when a clue to parity was presented—for example, by directing the participants’ attention to the coloring of the squares—it sometimes brought about a restructuring and an Aha! experience. This result supports the idea that consideration of the possible importance of parity in response to impasse might be crucial in developing insight in the Mutilated Checkerboard problem. Once again, however, questions can be raised about the relevance of this result as support of the neo-Gestalt view. Kaplan and Simon postulate that restructuring occurs in response to impasse, but none of their participants restructured the problem until they were given a hint. The overall results therefore do not support the neo-Gestalt view

**141**> 142 143 144 145 146 147 .. 298 >> Next