Transformational and derivational strategies in analogical problem solving

Abstract

Analogical problem solving is mostly described as transfer of a source solution to a target problem based on the structural correspondences (mapping) between source and target. Derivational analogy (Carbonell, Machine learning: an artificial intelligence approach Los Altos. Morgan Kaufmann, 1986) proposes an alternative view: a target problem is solved by replaying a remembered problem-solving episode. Thus, the experience with the source problem is used to guide the search for the target solution by applying the same solution technique rather than by transferring the complete solution. We report an empirical study using the path finding problems presented in Novick and Hmelo (J Exp Psychol Learn Mem Cogn 20:1296–1321, 1994) as material. We show that both transformational and derivational analogy are problem-solving strategies realized by human problem solvers. Which strategy is evoked in a given problem-solving context depends on the constraints guiding object-to-object mapping between source and target problem. Specifically, if constraints facilitating mapping are available, subjects are more likely to employ a transformational strategy, otherwise they are more likely to use a derivational strategy.

Appendices

Appendix 1

Problem: low guidance of mapping

Source: boat

The Johnsons are planning a riverboat tour for their summer holiday visiting five cities: Schwetzingen, Marbach, Blaubeuren, Ludwigsburg and Ulm. The area they will visit is famous for its ancient river locks and they are looking forward to this experience. The Johnsons have heard that each of the eight river locks in this area has its own architectural value and technical concept, so they want to make sure to cross via each of the eight locks. But, as there is a fairly high toll for each lock, they also want to make sure not to travel through any lock more than once. The eight locks are located between the following pairs of cities: Schwetzingen and Blaubeuren, Schwetzingen and Ludwigsburg, Schwetzingen and Marbach, Marbach and Ludwigsburg, Marbach and Blaubeuren, Blaubeuren and Ludwigsburg, Blaubeuren and Ulm, Ulm and Ludwigsburg. The Johnsons plan to start their trip in Schwetzingen. From Schwetzingen, they wish to travel along a route that will enable them to go though each of the eight locks exactly once. Note that their desire to travel through every lock once necessarily means that they will visit some of the cities more than once. Plan a route for the Johnsons so that they travel through every lock exactly once and visit each city as many times as necessary. Feel free to use short notations for the cities, as “B” for “Blaubeuren” and so on.

Solution to boat

For solving this problem, it is very helpful to visualize it with the help of a pen and a sheet of paper. First note each city mentioned in the problem by its first letter. For example, to represent the city “Schwetzingen” you draw a capital “S”, for the city “Blaubeuren” a capital “B” and so forth. When you did this with all of the cities mentioned in the problem, your sheet will look similar to the one displayed below. Do not worry if it does not exactly like yours, because there are several correct possibilities.

Please refer to Fig. 5 for a representation of the solution step.

Fig. 5

Low guidance of mapping solution procedure step 1

After you have represented the cities on your sheet of paper, let us proceed with the river locks between the cities. Each river lock lies on a channel that connects two cities. To represent this channel, and with it also the lock, draw a line between two cities on your sheet of paper. For example, if the text says that a lock is located between the cities “Schwetzingen” and “Blaubeuren”, draw a line between the capital letters “S” and “B” on your sheet of paper. After you have done this with all the connections mentioned in the problem, your sheet will look similar to this one.

Please refer to Fig. 6 for a representation of the solution step.

Fig. 6

Low guidance of mapping solution procedure step 2

Now we have to plan a trip for the Johnsons, as it is mentioned in the problem. The Johnsons want to travel through every lock exactly one time. If we have our representation on the sheet of paper in mind, this means that we have to find a route, where every line is used exactly one time, starting with the city Schwetzingen. Each time we cross a city, we note its capital letter somewhere to keep track of our position. Of course we also have to keep in mind, which lines we already used, because we are not allowed to use them twice. At some place it might be, that our partial solution cannot be completed, because we made an error and end in a city without unused lines attached one to it. Then we have to go back and make another decision at an earlier point of the solving process. Let’s have a closer look at one of the possible correct solutions. We start in Schwetzingen, as mentioned in the problem. From there, we travel over the line to Blaubeuren, further to Ulm, Ludwigsburg, again Blaubeuren, then Marbach, again Schwetzingen, Ludwigsburg and finally again Marbach. If the Johnsons follow this way, they will see each lock (that is, use each line) but no lock twice. Of course they travel through some cities twice, but that is not forbidden by the problem. So one possible solution to the problem would be “SBULBMSLM”—the order of visited cities the Johnson’s could travel, represented by the first capital letters of the cities attached one to an another.

Please refer to Fig. 7 for a representation of the solution step.

Fig. 7

Low guidance of mapping solution procedure step 3

Target: birthday

Five people attended a birthday party: Richard, Eric, Mary, Susan, and Bill. During the course of the evening they played different games. One game they played was a “messenger game” where one person started to write a word on a paper. The paper was then passed to another person who added a second word, and so on. To make things not too simple, the message passing followed a complicated protocol: the message had to be passed between all people knowing each other, but was only allowed to be passed between each acquainted pair of people exactly once. The following pairs and triples of people know each other: Susan, Eric and Bill all know each other; Richard and Susan know each other; Bill, Mary, and Richard all know each other; Eric and Richard know each other. Susan was the person writing the first word. Give the order in which the message was passed person-to-person. Feel free to use short notations for the people, as “S” for “Susan” and so forth.

Problem: high guidance of mapping

Source: boat

The Johnsons are planning a riverboat tour for their summer holiday visiting five cities: Cannenbach, Frankheim, Neustadt, Markburg and Behringen. The area they will visit is famous for its ancient river locks and they are looking forward to this experience. The Johnsons have heard that each of the eight river locks in this area has its own architectural value and technical concept, so they want to make sure to cross via each of the eight locks. But, as there is a fairly high toll for each lock, they also want to make sure not to travel through any lock more than once. The eight locks are located between the following pairs of cities: Behringen is connected to Neustadt and to Markburg via locks. These two cities are connected with all other cities via locks—that is, Neustadt is connected to Behringen, Markburg, Frankheim and Cannenbach; Markburg is connected with Behringen, Neustadt, Frankheim and Cannenbach. Frankheim and Cannenbach are connected with all cities except Behringen—that is, Frankheim is connected with Neustadt, Markburg and Cannenbach; Cannenbach is connected with Neustadt, Markburg and Frankheim. From Cannenbach, they wish to travel along a route that will enable them to go though each of the eight locks exactly once. Note that their desire to travel through every lock once necessarily means that they will visit some of the cities more than once. Plan a route for the Johnsons so that they travel over through every lock exactly once and visit each city as many times as necessary. Feel free to use short notations for the cities, as “N” for “Neustadt” and so on.

Solution to boat

For solving this problem, it is very helpful to visualize it with the help of a pen and a sheet of paper. First note each city mentioned in the problem by its first letter. For example, to represent the city “Cannenbach” you draw a capital “C”, for the city “Frankheim” a capital “F” and so forth. When you did this with all of the cities mentioned in the problem, your sheet will look similar to the one displayed below. Do not worry if it does not exactly like yours, because there are several correct possibilities.

Please refer to Fig. 8 for a representation of the solution step.

Fig. 8

High guidance of mapping solution procedure step 1

After you have represented the cities on your sheet of paper, let us proceed with the river locks between the cities. Each river lock lies on a channel that connects two cities. To represent this channel, and with it also the lock, draw a line between two cities on your sheet of paper. For example, if the text says that a lock is located between the cities “Behringen” and “Neustadt”, draw a line between the capital letters “B” and “N” on your sheet of paper. After you have done this with all the connections mentioned in the problem, your sheet will look similar to this one.

Please refer to Fig. 9 for a representation of the solution step.

Fig. 9

High guidance of mapping solution procedure step 2

Now we have to plan a trip for the Johnsons, as it is mentioned in the problem. The Johnsons want to travel through every lock exactly one time. If we have our representation on the sheet of paper in mind, this means that we have to find a route, where every line is used exactly one time, starting with the city Cannenbach. Each time we cross a city, we note its capital letter somewhere to keep track of our position. Of course we also have to keep in mind, which lines we already used, because we are not allowed to use them twice. At some place it might be, that our partial solution cannot be completed, because we made an error and end in a city without unused lines attached one to it. Then we have to go back and make another decision at an earlier point of the solving process. Let’s have a closer look at one of the possible correct solutions. We start in Cannenbach, as mentioned in the problem. From there, we travel over the line to Neustadt, further to Behringen, Markburg, again Neustadt, then Frankheim, again Cannenbach, Markburg and finally again Frankheim. If the Johnsons follow this way, they will see each lock (that is, use each line) but no lock twice. Of course they travel through some cities twice, but that is not forbidden by the problem. So one possible solution to the problem would be “CNBMNFCMF”—the order of visited cities the Johnson’s could travel, represented by the first capital letters of the cities attached one to an another.

Please refer to Fig. 10 for a representation of the solution step.

Fig. 10

High guidance of mapping solution procedure step 3

Target: birthday

Five people attended a birthday party: Carry, Fred, Ned, Mike, and Babs. During the course of the evening they played different games. One game they played was a “messenger game” where one person started to write a word on a paper. The paper was then passed to another person who added a second word, and so forth. To make things not too simple, the message passing followed a complicated protocol: the message had to be passed between all people knowing each other, but was only allowed to be passed between each acquainted pair of people exactly once. The following pairs of people know each other: Babs knows two people—Ned and Mike. Ned and Mike both know all other people—that is, Ned knows Babs, Mike, Fred, and Carry; Mike knows Babs, Ned, Fred, and Carry. Carry and Fred know all people except Babs—that is, Fred knows Ned, Mike, and Carry; Carry knows Ned, Mike and Fred. Carry was the person writing the first word. Give the order in which the message was passed person-to-person. Feel free to use short notations for the people, as “C” for “Carry” and so forth.

Appendix 2

Strategy assessment questionnaire

Remark

Questions designed to assess transformational strategies: 01, 04, 08, 10, 11, questions designed to assess derivational strategies: 02, 03, 06, 12, 13, all other questions were fillers for checking response reliability.

Instructions

Now you are nearly done. Please answer some final questions for us. After you answered all the questions, please click on “Submit Answers” on the bottom of this page. Describe how you solved the “birthday” problem. Check “yes” if a sentence approximately fits your strategy and “no” if it does not.

01 Yes O No O	It was simple to use the “boat” solution to solve the “birthday” problem by replacing the names of the towns with the names of the people
02 Yes O No O	The “boat” problem and the “birthday” problem seemed similar but I could not figure out how the solutions were related
03 Yes O No O	I remembered how I drew the graph with help of the solution of the “boat” problem. Through this I found the travel relations between the towns and used the same procedure to solve the “birthday” problem
04 Yes O No O	I did not go through the steps I used in the “boat” problem to solve the “birthday” problem because I just replaced the names in the boat problem with the names in the birthday problem
05 Yes O No O	I could not use the boat problem solution to solve the “birthday” problem, but when I finished with the “birthday” solution I realized how they are the same
06 Yes O No O	I used the travel strategy I remembered from the boat problem to solve the “birthday” problem, but I made a few (or one) mistakes as I went along and had to do some of the message route over again
07 Yes O No O	The “boat” problem and the “birthday” problem seemed too dissimilar for me to use the “boat” solution to solve the birthday problem
08 Yes O No O	I did not have to try to remember any of the steps from the “boat” problem solution
09 Yes O No O	I could not solve the “birthday” problem
10 Yes O No O	The solution to the “birthday” problem was obvious from near the beginning because it is just like the “boat” problem
11 Yes O No O	I made a correspondence between the parts of the “boat” problem and the parts of the “birthday” problem and then wrote the solution to the “birthday” problem
12 Yes O No O	I did not make much of a link between the town names in the “boat” problem and the names of the people in the “birthday” problem. I just used the same route of the “boat” problem solution to solve the “birthday” problem
13 Yes O No O	I broke the “birthday” problem into smaller pieces and used the same strategies that were used in the “boat” problem solution
14 Yes O No O	I could not solve the “birthday” problem because I got stuck in the same way as I did trying to understand the solution to the “boat” problem
15 Yes O No O	I did not really use the “boat” problem solution because I already knew the general principle of how to solve problems of this kind