8.5 Problem Solving

If you like puzzles and games, you may know the Tower of Hanoi problem (see photo). At the start, disks are stacked on one of three pegs. You have to move them to a different peg while following two rules: (1) Move only one disk at a time, and (2) never put a larger disk on a smaller one. The simple combination of disks, pegs, and rules produces a challenging puzzle.

Tower of Hanoi puzzle Can you move the disks on the left to the peg on the right, moving one disk at a time and never placing a larger disk on a smaller one? Problem-solving tasks such as the Tower of Hanoi puzzle are used in research to study the strategies through which people solve problems.

When working on this sort of task, you’re engaged in problem solving, a thinking process in which people try to reach a solution by working through a series of steps. Problem solving occurs not only when working on puzzles, but when tackling challenges in science, mathematics, or engineering, where you have to figure out how to work through a series of challenging steps to reach a solution.

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The Problem Space and Heuristic Search

Preview Question

Question

How do people solve problems?

To solve a problem, you have to maneuver through its problem space. A problem space is the full set of steps it is possible to take when solving the problem. Suppose the problem is to form an English word out of the three letters tca. The problem space consists of the six possible arrangements of the letters: tca, tac, atc, act, cta, cat.

On simple problems—such as forming a word from tca —you can envision the entire problem space and simply select a solution. However, this is rare; the problem space usually is much bigger. In the Tower of Hanoi problem, there are many ways to move the disks. In a game of chess, the number of possible moves is enormous. You cannot envision the whole problem space; the amount of information exceeds the capacity of short-term memory (see Chapter 7).

How do people solve problems when they can’t envision the entire problem space? As in judgment under uncertainty, heuristics come to the rescue. People use simple strategies, or problem-solving heuristics, to reach a solution.

One such heuristic is means–ends analysis (Newell & Simon, 1972). Means–ends analysis is a problem-solving strategy in which, at each step of a problem, people aim merely to reduce the distance between where they are now and where they want to end up (Simon, 1990). Rather than trying to determine a long series of steps that will lead to a problem solution, people merely look one step ahead; they select a “next step” that will bring them closer to their desired outcome. In the Tower of Hanoi problem, you might select any option that places a disk on the peg you want to reach. In the late stages of a chess game, you might choose any move that gets one of your pieces onto a square from which it can attack your opponent’s king.

WHAT DO YOU KNOW?…

Question 19

When the mwZAKoGdWq0t1eQI space is overwhelming, rather than working through every possible step, sometimes we simply do something that will get us at least a little bit closer to the solution, according to our own rQJoWANLR+VlnlKtO+N+hqcVa2A= analysis.

Problem Solving by People and by Computers

Preview Question

Question

How do computers solve problems?

Two types of evidence indicate that people use means–ends analysis to solve problems. The first comes from research in which psychologists ask people to think out loud while trying to solve problems (see Research Toolkit). Participants’ statements often reveal that they are using means–ends analysis. They frequently mention their current state, desired end point, the difference between the two, and how they might take a step that reduces the difference (Simon, 1990).

When devising study strategies, what does your means–ends analysis look like?

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Herbert Simon, pioneer in the study of human problem solving—plus many other topics. Simon won a Nobel Prize in Economics for his studies of decision making in organizations. With colleagues at Carnegie Mellon University, he wrote some of the world’s first “intelligent” computer programs, that is, programs that could solve problems rather than merely perform simple calculations.

The second source of evidence involves computers. Allen Newell and Herbert Simon (1961) wrote a computer program, the General Problem Solver, that solved a wide variety of problems. It did so through means–ends analysis, manipulating symbols until it reached desired end states. The steps through which the General Problem Solver accomplished tasks often resembled those taken by human problem solvers. “Protocols of human problem-solving behavior in a range of tasks—playing chess, solving puzzles, writing computer programs—contain many sequences of behavior … quite similar to the means-ends analysis of the General Problem Solver” (Newell & Simon, 1961, p. 2014). The convergence between human and computer problem solving suggests that people and the program solved problems in the same way: through means–ends analysis.

WHAT DO YOU KNOW?…

Question 20

m7gnor4VlSPWovs8SKJlV8328fafdOeuohZVbb74kNQ/7UFV/UXx3k4Jtoso4Cpj1mZBZoqQBx3jTyDJnmLWVs2kTSo7Il3NT02UGyleeAg3o3mCjVV0DTsEIl9I6n2ItCmwb8t/3VX3b+Mvv5jIgHlev82IbyM2PjHdrENtEHWJkeSYhCijYJhdIE1FXmb1i6TSIHvZ4Y7scb8x6QEUkb5Fd9tkB21sHjNbqWfUqtHuSGory1+Bgf/zxfNCQNnHjLILn3H6gUy5WWtQXFtB/qtrNxJjmf9BERl9Bamv9OXnG4BpCjKpE7Jpn85efGO8

THINK ABOUT IT

In what ways might information processing by a computer be similar to human thinking? In what ways does it differ from human thinking?

RESEARCH TOOLKIT

Think-Aloud Protocol Analysis

If you watch people work on a problem—a mathematical proof, a chess game, a puzzle in a newspaper—it’s hard to know what they’re thinking. You know something’s “going on in there”; after a while, a solution seems to pop right out. But what exactly were they thinking? And how could you find out?

Finding out is more difficult than it may sound. You could just ask people what they were thinking, but this solution isn’t good enough. Because information in short-term memory decays quickly (Chapter 7), they might simply forget the exact strategy they used.

Fortunately for research on human thinking, psychologists have devised a research tool that overcomes this difficulty. Think-aloud protocol analysis (or “verbal protocol analysis”) is a procedure in which participants verbalize their thoughts while solving a problem (Ericsson & Simon, 1984; Fox, Ericsson, & Best, 2011); that is, they say out loud all the ideas that run through their heads. The experimenter records and analyzes the statements to determine the problem-solving strategy used. Verbalizing thoughts is easy; we often think to ourselves when solving a problem (“3 × 19 = ___? Hmm, let’s see. Three times nine is 27, so there must be a 7 in the ones place….”), so all one needs to do is to give voice to such thinking. With only a bit of practice, verbalizing does not impair people’s normal task performance (Fox et al., 2011).

You can see how this works from the transcription below—it’s a think-aloud protocol recorded while a research participant attempted the Tower of Hanoi problem (from Anzai & Simon, 1979). In the problem, the disks are numbered (1, 2, 3…) and the pegs are identified by letters (A, B, C).

I’ll take 2 from C, and place it on B.

And I’ll take 1 and … place it from A to B.

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So then, 4 will go from A to C.

And then …, um …, oh …, um …

I should have placed 2 on C. But that will take time. But I’ll stay with this a little more.

I’ll take 1 from B and place it on A.

Then I’ll take 2 from B to C.

Oh, this won’t do….

I’ll take 2 and place it from C to B again.

And then, I’ll take 1, and from A….

Oh no! If I do it this way, it won’t work!

The protocol provides a window into the mind of the participant. You can see which strategies the person tried, and even his or her reaction when one of the strategies failed.

Protocol analysis has been applied widely. For example, to find out whether people were giving honest or “fake” answers on a personality test, researchers asked test takers to think out loud while answering test questions (Robie, Brown, & Beaty, 2007). Think-aloud protocols revealed that some participants weren’t describing their actual personality at all; they were trying to figure out which answers would make the best impression on the person scoring the test. For example (from Robie et al., 2007, p. 500):

“I wonder what type of responses they want?”

[Reads item] “Generally yes, so I’ll say probably yes because it’s true and it looks good on the sheet.”

“If I answer definitely yes to all of these, they’ll think I’m lying.”

[Reads item] “I don’t remember reading anything about getting a commission…. However, if I say definitely yes, they are going to be thinking that I am going to be cutthroat.”

Think-aloud protocol analysis, then, is a valuable tool for learning how people solve problems—including the problem of figuring out how to make a good impression on personality tests!

WHAT DO YOU KNOW?…

Question 21

Which of the following statements is true?

  • ngT02kxDkZ19N3hwXelGF72lU9Sq8ejEiuGtIyh24YgNKeWfdfP8yVJe+k2ATEFRq726d5MHhaQZwHDsJCCGPMG2JDnsmbHjv/dyfUNeZbGpo8dvCtxch4KDF2TwMK8YUSo9VQZBK+jRINLlM3smibSNDQRV2jE3C9LM37ckLjywg0Jkh/wo3fFpLhl4BsfzVGl8Fh3IQuzXrPdLTtRVnvYLeArygZzal1+shrd3buYOu3ewWSLzQo9PdYD4IU8Ku+TXXA==
  • QUSTL79OiknveBXuTSWFHJv58YUCqHTmxb2+rUt1ZhfUxdLu0eFLxudVGdgyqfhq95secfdHn9WU+rPQzTDh0JBBpaClH3lvmH/pJBl88a6beCeSXqnuaXEHBoVGuPtMxVt2eNem9f42/fds3GBCCeyTCOVOq/lF/jl/Ag69L39slP3p+BYXwFzTuIHHQ+/JeFLLo19XwsotXc/QnaE0qSVFEAM/PcY0piPdavjGxYsbptUc9jLCPvdGM/mhET6heL2ry2+g8HJ9sqGjkowr4GbrWKKVrLgNJ5RD0V0JY8jE6SOwWbwYXg==
  • gzLHAatfX9XfZsX+UcxMT6BborIaRDjvvIXw7pgQCb/w5UwvsKoi16oZT3nyDkuM0bYUZDgMoBXmtLsC8uaIlGNPEuOsaLFqmMoA3HqlBVw6gKW+j3FTbkivLWwAusKO2CkKOYL+GY8eOYu0v9f6CExKc7IJsP6TS2+KoK5rLrK/015nWC/eUtIhHPUbgGS9FWUFoBz/LyHnRWrLOMCFuhEuQFgtOp9dVnrbuY2n1q+3iW+gHwNCax9g1lE7DsVzKkc8vnJHtS8QF7V1oQv3gAyDZKEj5mXzL+TgCNMQFgPD2ocgOFl0mmzISuPHUuUVtDmvm6wdoUDiie8MtEg82R//f7qHooP26bMxFb3yq6Y=