# Development of Reasoning Puzzles Discussion

## Jo Tall and Jo Short Puzzle - Ratio puzzle

*Formal Reasoning* - Each small paper clip can be put into a one-to-one correspondence with the jumbo paper clips or equal multiple clips or fractional parts of those paper clips. Another way to say it - each clip has a relationship equal to the same quantity for all other clips. Further this proportional relationship of small clips to jumbo clips is the same as all the small clips to all the jumbo clips. Reversibility and transformation need to be understood and used to demonstrate this. This value is not given in the information provided with the puzzle and is not explicitly asked for. Once this value (ratio) is known, the answer can be calculated. Alternatively, the student might conceptualize the height ratio, and then reason that this ratio does not change (is invariant) with respect to the units of measurement.

*Concrete Reasoning* - Since the height of Jo Short measures more smaller paper clips than jumbo clips, simply add the extra amount to the height of Jo Tall. Even though the arithmetic difference in units is not stated or asked for; it is a more direct measure of the difference than is the ratio, which is derived by a one-to-one correspondence between individual clips. Another concrete approach makes use of the height difference in clips by looking at one figure and imagining the other with the extra paper clips, in contradiction to the fact that the four clips measuring Jo Short are equal to six and not to four paper clips. This inconsistency is not noticed at the stage of concrete thought, but would be noticed at the formal stage and would lead the student who had originally made this mistake (self-regulation) to reexamine his/her procedure.

## Island Puzzle

*Formal Reasoning* - On Question B, the trip from Island Byll to Island Cyll is conceptualized as possible achieved by a change of planes or stopover at Island Dyll. In other words, the clues about plane routes are not only evaluated in terms of the direct information they provide, but also in terms of the inferences possible by using the information stated in the puzzle. On question C, the formal thinker imagines all possible routes from Island Ayll to Island Cyll in order to discover all information that is provided explicitly or implicitly. In particular, he or she must hypothesize that air travel is possible and evaluate this hypothesis for consistency with the data. Note that most of the sample solutions did not use a the formal approach to the last question (C), but did on B. This mixture of procedures is often observed in practice; a reflection of the fact that the stages of Piaget’s theory are idealizations which help to classify observed behavior, but should not be used to classify people superficially.

*Concrete Reasoning* - Since the clues do not give the answers to the questions directly, a concrete thinker either can’t tell, selects (centers on) certain details from the map (geographical placement, island separation) or postulates properties (invents characteristics) of each island to explain his or her ideas. The properties of a single island (size, topography [natural or physical characteristics]) used in this approach are more familiar to the student than the plane routes between islands and avoids the need to discover the relationship for combining plane routes.

## Water Machine Puzzle

*Formal Reasoning* - The student analyzes the puzzle as a combinatorial one. All the possibilities are merely combinations of different connections from none, one alone, combinations of two, combinations of three, and one combination of four. The solution of all the possible (16) combinations is arrived at in a systematic way. Formal reasoning results in a tidiness, where combinations are not duplicated and are orderly arranged. Students reasoning in this way can generate all the possibilities. This is a hallmark of formal thought – one hypothesizes what could be instead of what is.

*Concrete Reasoning* - Combinations of character are generated by unsystematically and perhaps only in doubles and single. Pieces of a combinatorial reasoning system are evident. However, the full system is not developed. This leads to unsystematic and inexhaustive series of combinations.

## Meal worm Puzzle

*Formal Reasoning* - Variables are held constant while only one is allowed to change. All relevant variables are identified and tested for the hypotheses that light or moisture or both are responsible for the distribution of the meal-worms. The answer will be derived in a systematic manner with each possible conclusion being tested. Probabilistic reasoning is also evidenced by the student’s ability to ignore the few meal-worms in the “wrong” ends of boxes.

*Concrete Reasoning* - An individual using concrete reasoning will center on one variable to the exclusion of others. She or he does not detect the logic of the experiment which allows for variables to be separated and isolated, so they can be explored as independent variables. She or he sees the one-to-one correspondences where one factor causes one response in one of the boxes.

## Frog Puzzle

*Concrete Reasoning* - Difference is focused on rather than ratios. This student assumes constancy of differences and thus reasons as follows: there were 60 more unbanded than banded frogs in the recapture sample, so there are 60 more frogs in the pond as a whole; 60+55=115. How would a person using concrete reasoning apply his reasoning to the following problem? “In a new recapture sample of 50 frogs, how many do you think are banded?” We have observed these responses: (1) Impossible to do; (2) 10; and (3) -10!!

*Formal Reasoning* - Probabilistic reasoning is used. Starting with the relative frequency of banded frogs in the recapture sample (12/72), this student reasons that this ratio is an estimate of the relative frequency of banded frogs in the pond. making it possible to use the proportion of 1/6 with the proportion of 55/x and the answer follows easily. This student is undisturbed by the uncertainty associated with a statistical estimate and realizes that, as an estimation, this procedure is valid.