This is This and That is That: Isomorphic Representations

Let's play a little game. If you haven't seen it before, it's called the Tower of Hanoi, and the goal is to move all three of the colored disks from the left-most peg to the right-most peg. But here's the catch: you can only move one disk at a time. Oh, and one more thing: you can't place a larger disk on top of a smaller disk [1]. 
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Discs Moves:

In a previous post, we talked about defining a problem space. The problem space for the Tower of Hanoi is shown in Figure 1. At the top of the pyramid is the initial state. All of the colored disks are lined up on the left-most peg. Then each arrow points to a legal move. There is only one operator, which is that you can move one disk to a different peg. There are two constraints. First, you can't just grab the whole stack and move it. You have to move one disk at a time. The second constraint is that you are not permitted to place a larger disk on top of a smaller disk. The goal state is located at the bottom-right side of the pyramid where all of the disks are stacked on the right-most peg. [2]

Figure 1. The problem space for the three-disk version of the Tower of Hanoi

The Tower of Hanoi is like the Drosophila of Cognitive Science. It has been used in countless studies, and is arguably the best understood task in the literature on problem solving. I'm overstating things a bit, but you get the point. It's been around for a while. 

"Play it again Sam"

Since this task is so well known, some creative scientists have toyed around with different versions of the task. Let's play again, but this time, imagine the disks are like different sized cans that have been opened and turned upside-down.

Which version do you like better: the first version with solid disks or the second version with open cans? If you have a preference, what is the reason for your preference? Personally, I prefer the version with the upside-down cans. Why? Because I don't have to remember one of the problem constraints. Instead, the external environment has encoded that information for me. Now, making a move is simply moving something that can withstand the placement of another can. It's so much easier (for me, at least)! 

A Wolf in Wolf's Clothing

The fact that simply changing the representation of the colored bands in the Tower of Hanoi can make the problem easier to solve illustrates an important point about problem solving: the difficulty of a problem can be completely determined by the framing of the problem. Let's take a different example. For this problem, you have to decide if the following rule is valid or not: All cards with a vowel have an even number on the back. I then lay out four cards in front of you. Your task is to flip over the minimum number of cards to test the rule. [3]

Which cards did you turn over? If you're like 90% of the people who have been asked to do this task, then you probably were tempted to turn over "A" and "4." But if you turn over those two cards, then you don't know for sure if the rule is valid. For example, what if you turn over "7" and discover that it has a vowel? Then you know you for a fact that rule is invalid.

When the problem is framed in terms of an abstract (and highly arbitrary!) rule, then most people don't give the correct answer (A and 7). But what happens when we reframe the problem by contextualizing it? 

You're a bouncer at a bar, and you are in charge of finding underage drinkers (i.e., the rule is: If a person is drinking alcohol, then they need to be 21 or over). Which people do you need to interrogate? [4]

1. You need to see the ID of a man drinking a beer.
2. You need to see the ID of a woman drinking a coke.
3. You need to see what a 21 year old is drinking. 
4. You need to see what an 18 year old is drinking. 

Notice that the deep structure of each of these problems is exactly the same, which makes them isomorphic problems. You don't care what the woman is drinking (which maps onto the "K" card), nor do you care what the 21 year old is drinking (which maps onto the "4" card). Instead, to enforce the rule, you need to see the ID of the man drinking a beer (i.e., the "A" card), and you also need to see what the 18 year old is drinking (i.e., the "7" card).

The point is: the difficulty of the task depends on the way it is framed. Isomorphic problems  are easier to solve when framed in terms of familiar, contextualized rules than when  framed as arbitrary and abstract rules. 

The STEM Connection

The educational implication of how isomorphic problems are framed is pretty interesting. In a previous post, we talked about the educational advantage of using multiple representations. For the Tower of Hanoi, the way to solve the problem is to search through a problem space for the path that connects the initial and goal states. We can extend the concept of a problem space to include a space of representations. As we saw above, the Tower of Hanoi can be represented in (at least) two different ways, although many variants exist [5].

Here's another great example of the search for an isomorphic problem representation. As the story goes, John Von Neumann was at a dinner party and someone posed the following problem:

Two cyclist, 100 miles apart, start riding toward one another at a pace of 10 miles per hour. At the same instant they start, a bee flies from one bicycle toward the other at 20 miles per hour. When he reaches the second cyclist, the bee turns around and flies back toward the first cyclist. He does this until the cyclists meet. How far did the bee fly?

The obvious way to represent the problem is through a series of back-and-forth trips by the bee. However, once you start down this path, you quickly realize that the bee is going to take a lot of trips. Mathematically speaking, you would have to sum the distance of an infinite number of trips. That doesn't sound like fun! Instead, you can re-represent the problem as having two parts. First, figure out how long the cyclists will be riding until they meet. Then, use the simple speed = distance / time formula and calculate the distance the bee flew in the time it took the cyclists to meet. That's a much easier problem to solve! 

One of the lessons we need to teach our students is to reframe or re-represent difficult problems in a way that is much more meaningful, concrete, and easier to answer.


Share and Enjoy! 

Dr. Bob

For More Information

[1] Special thanks to Josh Fisher for making the Tower of Hanoi a fully interactive experience. Check out his google+ page for more fun and creative widgets that he has assembled!

[2] Newell, A., & Simon, H. A. (1972). Human problem solving. Englewood Cliffs, N.J.: Prentice-Hall.

[3] Wason, P. C. (1968). Reasoning about a rule. Quarterly Journal of Experimental Psychology, 20, 273 281.

[4] Cheng, P. W., & Holyoak, K. J. (1985). Pragmatic reasoning schemas. Cognitive Psychology, 17, 391–416.

[5] Zhang, J., & Norman, D. A. (1994). Representations in distributed cognitive tasks. Cognitive Science, 18, 87–122.