Confusing problem’s surface structure

To explain why a problem’s surface structure is so confusing and, as a result, how adapting common solutions to new problems is so complicated, let’s first considers how you interpret what is being asked when you get a problem. Everything you hear or read is viewed immediately in the light of what you already know about similar topics. For example, suppose you read these two sentences:  When you hear “corruption,” you wouldn’t dream about eye-patched swabbies screaming “shiver me timbers!”

The filtering of thoughts that happens when learning (or listening) means you tend to focus on the structure of the surface rather than the underlying structure of the problem. For example, a question like this was seen by four people in one experiment: West High School Band members were hard at work preparing for the annual Homecoming Parade. They first attempted to march in 12 columns, but Andrew was left alone to lift the tail.

The director then told the members of the band to march in eight columns, but Andrew was left alone to march. Andrew was left out even when the unit marched in lines of three. Andrew finally told the band manager in exasperation that they should march in lines of five to cover all the columns. He’s got it right. Since there were at least 45 musicians on the field, but less than 200 players, how many students in the West High School Band were there?

The participants had read four issues earlier in the study along with detailed explanations of how to address each, presumably to score them for the quality of the prose. One of the four problems was the number of vegetables to buy from a garden, and it relied on the same type of solution needed to calculate the least common multiple problems for the band. Yet few subjects— only 19 percent— say that the situation with the group was different and they could use the solution to the garden problem. What’s the reason?

When a student reads a word problem, given her prior knowledge, her mind interprets the problem as it happened when you read the two phrases about copyrights and China. The challenge is that the information that seems important applies to the nature of the surface— the reader dredges knowledge about bands, high school, musicians, and so on in this issue. Using the least common multiple, the student is unable to read the question and learn about it in terms of its complex structure. The problem’s surface structure is over, but the problem’s underlying structure is not. Thus, people don’t use the first problem to help them solve the second: the first was about vegetables in a garden in their minds, and the second was about band marchers ‘ rows.

When knowledge of how to solve a problem has never been applied to concerns of new surface systems, schooling would be inefficient or even futile— but, of course, such a transition is taking place. When and why is complex, five but two factors are particularly relevant to educators: familiar with the deep structure of a problem and the knowledge that a profound structure should be looked for. In turn, I’m going to address each. Knowledge about how to solve it translates well when you are familiar with the deep structure of a problem. That familiarity may come from long-term, repeated experience with one problem, or with different manifestations of one problem type (i.e., many problems with different surface structures, but the same deep structure).

The subject perceives the deep structure as part of the problem description following repeated exposure to either or both. Here’s an example: A seeker of gold can discover a cave on a hill close to a beach. He believed that inside the cave there could be many directions so he was scared he might get lost. He did not have a map of the cave, of course; all he had with him were some common items like a torch and a wallet. What could he do to make sure he wasn’t lost trying later to get out of the cave?

The solution is to carry some sand in your bag and leave a trail as you go, so when you’re ready to leave the cave, you can trace your way back. About 75% of American college students thought of this solution — but only 25% of Chinese students solved it.6 The experimenters indicated that Americans solved it because most of them grew up reading Hansel and Gretel’s novel, which involves the idea of leaving a trail as you ride to an unidentified place to find your way back. The experimenters also offered participants another puzzle based on a common folk tale in China and reversed the percentage of solvers from each group.

It takes a lot of practice with a problem type before students know it well enough to recognize its deep structure immediately, regardless of the surface structure, as the Americans did for the problem of Hansel and Gretel. In terms of rocks, caves, and gold, American subjects did not think about the problem; they thought about it in terms of finding something to leave a trail. The problem’s deep complexity is so well reflected in their minds that when they read the problem, they automatically saw that structure.

Would you react to circumstances that are based on your emotions or biases? Looking for ways to communicate better with those around you? Would you like to do more in your career?

You can develop your ability to make rational, meaningful decisions and conclusions by incorporating System Thinking skills. The conclusions can often be one-sided without these capabilities. Criticism may sound like a personal attack on your personality rather than an invitation for discussion and constructive interaction.