"Life is all about metaphors and personal stories." I wrote that sentence several years ago when I first created my blog website. After more than 5 years and over 500 blog posts, I remain convinced that the use of metaphors, personal stories, and analogies is a great way to teach and to learn. Let me start with, of course, a story that will hopefully illustrate my point.
George de Mestral was a Swiss electrical engineer who invented the hook and loop fastener which he called Velcro®. As the story goes, de Mestral was in the Jura Mountains of Switzerland on a hunting trip with his dog in 1941. Upon his return, he noted that his clothes, as well as his dog's fur were covered in the seed pods (reportedly Xanthium strumarium, the cocklebur plant). He carefully pulled one of the seed pods off his dog and examined it under the microscope, where he noticed that the seed pod was covered in hundreds of tiny hooks. It was a "Eureka" moment for de Mestral. He quickly saw that a similar concept could be used to reversibly fasten different materials together. He would spend the next 10 years perfecting his design, filing for a patent in 1951. He called his invention Velcro, a portmanteau of the French words velours ("velvet") and crochet ("hook"). The rest, as they say, is history.
Undoubtedly, de Mestral greatly benefited from being at the right place at the right time. The invention of Velcro came at the same time as the "space race" in the 1960's, and NASA was one of the first to use de Mestral's product. As I stated earlier, the Velcro story is a great example of analogical thinking or reasoning, which is defined as "the process of finding a solution to a problem by finding a similar problem with a known solution and applying that solution to the current situation."
As it turns out, there are a number of older studies in the cognitive psychology literature on the use of analogical thinking. I've previously written about a few of these studies in the past (see "All the world's a stage..."). Today though, I want to talk about another well-known puzzle that uses analogical thinking, called the "crossing the river problem." Here is the basic concept - a farmer needs to get a chicken, a fox, and a sack of corn to the other side of a river. He has a rowboat, which is just large enough for him and one other passenger or object. If he takes the fox first, he will leave the chicken alone with the corn, and the chicken will eat the corn. If he takes the corn, he will leave the fox alone with the chicken, and the fox will eat the chicken. How does he do it?
I am not going to spoil your fun by giving you the answer! But here's a hint - the man will have to row back and forth across the river a few times! As it turns out, there are several variations of the "crossing the river problem," all of which are variations of the same theme as the chicken, fox, and sack of corn problem. For example, here's one known as the "Hobbits and Orcs" problem:
Once upon a time, in the last days of Middle Earth, three hobbits and three orcs set out on a journey together. They were sent by the great wizard Gandalf to find one of the lost palantiri, or oracle stones. In the course of their journey, they come to a river. On the bank is a small rowboat. All six travelers need to cross the river, but the boat will only hold two of them at a time.
The orcs are fierce and wicked creatures, who will try to kill the hobbits if they get the opportunity. The hobbits are normally gentle creatures, but they are very good fighters if provoked. The orcs know this, and will not try to attack the hobbits unless the orcs outnumber the hobbits. That is, the hobbits will be safe as long as there are at least as many hobbits as orcs on either side of the river.
I realize that fans of the J.R.R. Tolkien's The Lord of the Rings will question the "historical" accuracy of this problem, but try to figure it out! Again, the only hint I will provide is that there will be several boat trips. I've come across different versions of this problem. Substitute missionaries for the hobbits and cannibals for the orcs, and you now have the missionary-cannibal problem. Substitute jealous husbands for the missionaries and wives for the orcs, and you now have the jealous husband-wives problem.
They all sound so familiar. The important question is whether figuring out the solution to one problem - say, the jealous husbands-wives problem - will help individuals find the solution to a similar problem - say the missionary-cannibal problem. This is exactly the question posed by a study conducted in the early 1970's by a group of cognitive psychologists (see "The role of analogy in transfer between similar problem states" by Stephen K. Reed, George W. Ernst, and Ranan Banerji at Case Western Reserve University). The results may surprise you, especially when you look at how similar these different problems really are.
As usual, the study subjects were undergraduate students enrolled in an "Introduction to Psychology" class (remember when you had to do sign up to participate as study subject?). The study utilized the jealous husband-wife problem and the missionary-cannibal problem. Importantly, while the two problems are similar, there are a few subtle yet important differences.
Here is the missionary-cannibal problem:
Three missionaries and three cannibals having to cross a river at a ferry, find a boat but the boat is so small that it can contain no more than two persons. If the missionaries on either bank of the river, or in the boat, are outnumbered at any time by cannibals, the cannibals will eat the missionaries. Find the simplest schedule of crossings that will permit all the missionaries and cannibals to cross the river safely.
Here is a similar problem, known as the jealous husbands problem:
Three jealous husbands and their wives having to cross a river at a ferry, find a boat but the boat is so small that it can contain no more than two persons. Find the simplest schedule of crossings that will permit all six people to cross the river so that none of the women shall be left in company with any of the men, unless her husband is present.
Try to figure each problem out! It's not easy. The investigators conducted a series of experiments in which subjects were presented with each problem in sequence. Transfer of knowledge did occur - in other words, solving one puzzle helped subjects solve the alternative puzzle, with a couple of important caveats. First, subjects were told that the two puzzles were related (if they weren't, solving the first puzzle did not help them to solve the second puzzle). Second, the Jealous Husband problem had to be presented first. Why did the order of sequence matter? The Jealous Husband problem is a little more complicated.
So, as the investigators wrote, "Our attempts to solve problems are greatly influenced by our previous attempts to solve problems." However, it's not quite that simple. Transfer of knowledge does occur, but there's more involved to simply develop an easy model with which to solve similar, related problems. Analogical thinking is far more complex than we realize.
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