The Karplus Islands puzzle, volume with metal cylinders, and other problems
Darrell G. Phillips · Journal of Research in Science Teaching · 1977
A question has been raised (Blake, Lawson, & Nordland, 1976) as to the appropriateness of using the Karplus Islands Puzzle as an instrument to assess formal-operational thought.Such questions need to be asked about many of the socalled "tasks" currently found in the literature. The Karplus Islands PuzzleThe above authors conclude that this puzzle is not measuring the same abilities as their other tasks (it would be surprising if it were), and they question just what the Islands Puzzle does measure.The following comments may provide one answer.A careful analysis of the Islands Puzzle reveals that a correct solution requires the use of a select few of the sixteen binary operations, a Piagetian formal-operational structure.For example, let p = travel between Bean and Fish (Clue l), r = travel between Bird and Snail (Clue 2), 4 = travel between Bean and Bird (Question 1 and Clue 3), and let s = travel between Fish and Bird (Question 2) and t = travel between Fish and Snail (Question Subjects are first given Clues 1 (p) and 2 (presented as not-r, i.e., 0 and then asked Question 1 (i.e., "Is it possible to travel between Bean and Bird?").In effect, what do p and f imply about 4? The subject need only realize that the value of 4 is independent of the values of p and r (i.e., 4 [PI and q [r] ).Clue 3 is then presented and Question 2 is asked (i.e., "Is it possible to travel between Fish and Bird?").Given both p and 4 as true, the subject must now determine the value of s in addition to recognizing that the value of s is independent of the value of r (i.e., s[r]).In effect, the operation of implication is employed again in the form ( p 4) 3 s.Finally, Question 3 is asked (i.e., "Is it possible to travel between Fish and Snail?"), and the subject must determine the value of t based on p . 4, and i, or (p ' 4 + 7) 3 t, F, or neither.This implication is a beginning, but the correct solution is most often derived from the reciprocal operation (i.e., converse implication) by employing the following sequence: if t were true than p , 4, and r must be true, or t 3 (pa 4. r).But r is given as not-r @); therefore, t cannot be true, or (p .4. P) 3 7.Additional operations may be employed.For example, some individuals recognize a form of equivalence (i.e., r and t must share the same value, either F and T, or r and t).The 361 3).