Introduction
If you are using this book in a class your teacher may ask you to work in pairs as you solve the problems. One partner should read and think aloud, while the other partner listens. On subsequent problems, the partners should change roles, taking turns as problem solver and listener.
You can also use this procedure if you are not in a class, but are working through the book with another person.
Some people find reading and thinking aloud a little awkward at first, but thousands of people have already used this book and have found they adjust to the procedure quickly. Here is the reason that you are asked to read and think aloud.
Thinking is a Hidden Skill
The ability to analyze complex material and solve problems is a skill—just like any other skill such as the ability to play golf or the ability to drive an automobile. However, there is a peculiar difficulty involved in teaching analytical skill. Generally there are two phases to teaching a skill. First, the skill is demonstrated to the student. Then, the student is guided and corrected while practicing. For example, golf is taught by showing the beginner how to grasp the club, how to place the feet, how to move the arms and body as one swings. The beginner can watch a golf pro—can even watch a slow motion film of the pro in action—and in this way learn the pro’s technique. Furthermore, the pro can observe the beginner at practice and point out flaws or demonstrate how to improve.
In contrast to playing golf, analyzing complex material is an activity whcih is generally done inside your head. This makes it somewhat difficult for a teacher to teach and for a learner to learn. In other words, a beginner cannot observe how an expert thinks and solves problems. And the expert has trouble demonstrating technique to a beginning student. There is one way to reduce this difficulty—have people think aloud while they solve problems. If both students and experts vocalize their thoughts as they work through complex ideas and relationships, the steps that they take are open to view and their activities can be observed and communicated.
In this book, the procedure of asking people to think aloud while they solve problems is applied in two ways. Experienced problem solvers (a group of graduate students and professors) were asked to think aloud as they solved the problems that are presented in the book. Their responses were tape-recorded, and then the steps they took in solving a problem were summarized and written out. These summaries are presented under the heading Problem Solution. In other words, the problem solution which follows each problem is a summary of steps taken by an experienced problem solver as he or she worked the problem aloud.
The second application of the procedure consists in asking you, the reader, to think aloud as you work each of the problems. In doing this, you make your thinking visible to other people so that they can observe your attack on a problem. Thus, they can learn the techniques you use; they can help point out any errors you make, and they can compare the steps you take with the steps listed in the problem solution. Furthermore, you will find that by thinking aloud you will be able to look at your own thinking activities more carefully. You will be able to see exactly what strategies you use, and what difficulties you have in solving a problem.
Research has shown that this is an effective way for students to improve their problem-solving skills: work together, think aloud, learn from each other, and read how experienced problem solvers approached the same problems.
Quiz Yourself
1. What are the two phases of teaching a skill such as golf or swimming?
2. What special barrier is met in trying to teach analytical reasoning skill?
3. How has this program attempted to overcome the difficulty encountered in teaching analytical reasoning? How has it handled the two phases of teaching the skill?
Thinking Aloud
In this book, you are asked to do your thinking aloud. Naturally you cannot vocalize all of the mental processing that you do. For example, you cannot explain how you know the meaning of every word you read in a problem. However, when you are unsure about a word or an idea, and you have to stop and think about it, do this thinking aloud. As a rule of thumb, try to think aloud as much as possible while doing the exercises. Spelling out your thoughts—especially at sections of a problem which you find difficult or confusing—is the safest way to insure that you do not skip steps in your reasoning, nor miss facts in drawing conclusions. In other words, you will find that vocalizing your thoughts forces you to be more careful and thorough in analyzing ideas.
Thinking aloud while solving problems requires a certain amount of practice. At first you may find it a little difficult to vocalize your thoughts as you work a problem—to express in words the steps you take in solving the problem. However, research studies show that most students get used to this procedure quickly.
In order to illustrate the procedure of thinking aloud, the response of an experienced problem solver who was asked to think aloud as she worked a problem is reproduced on the following pages.1 As you read this protocol, try to follow all of the steps and activities of the problem solver.
Problem Solver’s Response
Original Problem
If the circle below is taller than the square and the cross is shorter than the square, put a K in the circle. However, if this is not the case, put a T in the second tallest figure.
Problem Solver’s Response
Note: Read both the comments on the left and the problem solver’s report on the right. Quotation marks show when the problem solver read aloud. The absence of quotation marks indicates that the problem solver was thinking aloud.
“If the circle below is taller than the square and the cross is shorter than the square, put a K in the circle.” Let me start again. “If the circle below” … I’ll put my finger on the circle … “is taller than the square” … Yes, the circle is taller than the square. “And the cross is shorter than the square” … I’ll move my finger from the cross to the square and compare them … this part is false … the cross is not shorter than the square. “Put a Kin the circle.” So I shouldn’t put a K in the circle. Part of the statement is false. I would only write K if both the first part and the second part were true. I should read the whole sentence again and see if my conclusion is correct. “If the circle is taller than the square” … Yes … “and the cross is shorter than the square” … No … “Put a K in the circle” … I didn’t. That’s correct. |
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As the Problem Solver reread the sentence, she moved her finger first from the circle to the |
I’ll continue to the next part of the problem. “However, if this is not the case” … and it isn’t the case … “put a T in the second tallest figure.” The second tallest figure is the cross so I’ll put a T in the cross. |
Another Example
Here is the response that an experienced problem solver (a lawyer) gave to another problem. Note how carefully he checks everything, and even stops for a moment in deciding right from left.
Original Problem
If the word sentence contains less than 9 letters and more than 3 vowels, circle the first vowel. Otherwise circle the consonant which is farthest to the right in the word.
The Problem Solver read the entire problem aloud. |
“If the word sentence contains less than 9 letters and more than 3 vowels, circle the first vowel. Otherwise circle the consonant which is farthest to the right in the word.” I’ll start from the beginning. “If the word sentence contains less than 9 letters.” |
The Problem Solver pointed to the letters with his pen as he counted |
I’ll count the letters in sentence. 1, 2, 3, 4, 5, 6, 7, 8. Let me check it. 1, 2, 3, 4, 5, 6, 7, 8. So it does have less than 9 letters. I’ll write the word yes above the problem. That way I’ll remember it. |
The Problem Solver wrote yes over the sentence (see original problem). |
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The Problem Solver resumed reading. |
“And more than 3 vowels.” |
The Problem Solver pointed with his pen as he counted. |
1, 2, 3. Let me check that. 1, 2, 3. It contains exactly 3 vowels, not more than 3 vowels. I’ll write no on the problem to remind me. “Circle the first vowel.” So I won’t do that. “Otherwise circle the consonant which is farthest to the right in the word.” The consonant farthest to the right? Let me see. Which is my right hand? This is my right hand. OK, so the last letter is the one farthest to the right. But the last letter is E. The next letter over is C. So it is the consonant farthest to the right. I’ll circle the C. |
Methods of Good Problem Solvers
The Problem Solvers’ Responses that you just read illustrate several characteristics of good problem solvers. These characteristics have been studied by researchers and they will be summarized here in five sections.
First of all, good problem solvers have a strong belief that academic reasoning problems can be solved through careful, persistent analysis. Poor problem solvers, by contrast, frequently express the opinion that “either you know the answer to a problem or you don’t know it, and if you don’t know it you might as well give up or guess.” Poor problem solvers haven’t learned that a problem may at first appear confusing—that the way to work the problem may not at first be obvious—but that through carefully breaking the problem down, by pinpointing first one piece of information and then another, a difficult problem can be gradually analyzed. Poor problem solvers lack both confidence and experience in dealing with problems through gradual (sometimes lengthy) analysis.
2. Concern for Accuracy
Good problem solvers take great care to understand the facts and relationships in a problem fully and accurately. They are almost compulsive in checking whether their understanding of a problem is correct and complete. By contrast, poor problem solvers generally lack such an intense concern about understanding. For example, good problem solvers sometimes reread a problem several times until they are sure they understand it. Poor problem solvers, on the other hand, frequently miss a problem because they do not know exactly what it states. Quite often they could have found out if they had been more careful, if they had reexamined and thought about the problem analytically. But poor problem solvers have not learned how important it is to try to be completely accurate in understanding all of the ideas of a problem. (Recall how the experienced problem solvers in the last two exercises reread sections of the problem to be sure they understood them fully, and rechecked even their simplest calculations.)
3. Breaking the Problem into Parts
Good problem solvers have learned that analyzing complex problems and ideas consists of breaking the ideas into smaller steps. They have learned to attack a problem by starting at a point where they can make some sense of it, and then proceeding from there. In contrast, poor problem solvers have not learned the approach of breaking a complex problem into subproblems—dealing first with one step and then another. In the problems which follow, you will see many examples of how complex problems can be worked one step at a time.
4. Avoiding Guessing
Poor problem solvers tend to jump to conclusions and guess answers without going through all the steps needed to make sure that the answers are accurate. Sometimes they make intuitive judgments in the middle of a problem without checking to see whether the judgments are correct. At other times they work a problem part of the way, but then give up on reasoning and guess on an answer. Good problem solvers tend to work problems from beginning to end in small, careful steps.
The tendency for poor problem solvers to make more errors—to work too hastily and sometimes skip steps—can be traced to the three characteristics already discussed. First, poor problem solvers do not strongly believe that persistent analysis is an effective way (in fact the only way) to deal with academic reasoning problems. Thus their motivation to persist in working an entire problem precisely and thoroughly—until it is completely solved—is weak.
Second, poor problem solvers tend to be careless in their reasoning. They have not developed the habit of continuously focussing and checking on the accuracy of their conclusions. And third, they have not learned to break a problem into parts and work it step-by-step. As a result of these three characteristics, poor problem solvers have a strong tendency to make hasty responses as they work academic reasoning problems, causing errors in both simple computations and in logic.
5. Activeness in Problem Solving
The final characteristic of good problem solvers is the tendency to be more active than poor problem solvers when dealing with academic reasoning problems. Put simply, they do more things as they try to understand and answer difficult questions. For example, if a written description is hard to follow, a good problem solver may try to create a mental picture of the ideas in order to “see” the situation better. If a presentation is lengthy, confusing, or vague, a good problem solver will try to pin it down in terms of familiar experiences and concrete examples. Furthermore, the problem solver will ask questions about the problem, answer the questions, and “talk to him/herself” to clarify thoughts. The problem solver may also count fingers, point to things with a pen, write on the problem, make diagrams, or use other physical aids to thinking. All in all, good problem solvers are active in many ways which improve their accuracy and help them get a clearer understanding of ideas and problems.
Quiz Yourself
The text listed five areas in which good academic problem solvers differ from poor academic problem solvers. These are: 1. motivation and attitude toward problem solving; 2. concern for accuracy; 3. breaking problems into parts; 4. guessing; and 5. activeness in problem solving. Describe and explain three of these areas in detail.
Role of the Listener When Working with a Partner
As noted earlier, if you are using this book in a class your teacher may ask you to work in pairs. One partner should read and think aloud, while the other partner listens. On subsequent problems the partners should change roles, taking turns as problem solver and listener.
The partner who listens plays an important role in the learning process. The listener should not sit back inattentively with his or her mind elsewhere. Instead, the listener should concentrate on two functions: continually checking accuracy and demanding constant vocalization.
1. Continually Check Accuracy
Because accuracy is all important, the listener should continually check the accuracy of the problem solver. This includes every computation made, every diagram drawn, every conclusion reached. In other words, the problem solver’s accuracy should be checked at every step of the problem, not just at the final answer. For example, if in working the problem shown earlier the problem solver concluded that the word sentence has nine letters, the listener should have immediately caught the error and pointed it out.
Catching errors involves several activities. First, the listener must actively work along with the problem solver. The listener should follow every step the problem solver takes, and should be sure to understand each step. If the listener takes a passive attitude—or does not actively think through each step—he or she won’t know for sure whether or not the problem solver’s steps are totally correct.
Second, the listener should never let the problem solver get ahead of his or her own thinking. This may often mean that the listener will have to ask the problem solver to wait a moment in order to check a conclusion. In this program the emphasis is on accuracy, not on speed. Both the problem solver and the listener should concentrate on accuracy. If the listener needs a moment to verify a conclusion, this will give the problem solver a chance to go over his or her own work and check the thinking. The problem solver should, in the back of the mind, constantly have the thought “is that correct—should I check that?” This will slow down thinking a little so that the listener will be able to keep pace. However, if the problem solver is working too hastily, at the expense of accuracy, the listener should ask him or her to slow down—to follow accurately and analytically. Moreover, even if the problem solver is not working too hastily to be accurate, the listener may still occasionally ask to stop a moment to double-check a point.
Third, the listener should not work the problem separately from the problem solver. When some listeners first learn the procedure used in this program, they turn away from the problem solver and work the problem completely on their own. Occasionally they even finish the problem long before the problem solver. This is incorrect. The listener should listen. This means actively working along with the problem solver, not independently.
Finally, when the listener catches an error, he or she should only point out that an error was made, but should never give the correct answer. By the same token, if the listener sees an answer or a conclusion before the problem solver sees it he or she should keep quiet, not furnish it. The listener should wait for the problem solver to work it out. If the problem solver seems completely stuck, the listener may provide a suggestion on the first step to take, but should not actually take the first step or obtain a partial answer. The problem solver should do all the work.
In summary, the listener should understand that pointing out errors is not being picky or overly critical. The listener is helping to improve scholastic problem-solving skill—a skill that will be useful in all academic courses. The listener should check every step taken and every conclusion reached by the problem solver. The listener should never let the problem solver go on to a second step without checking the first one. When an error is detected, it should be pointed out without actually being corrected.
2. Demand Constant Vocalization
The second function of the listener is to ensure that the problem solver vocalizes all of the major steps taken in solving a problem. Thinking aloud is a primary part of this program. It is the only way to communicate and to monitor thinking. It should not be neglected. Even the solution of simple problems should be vocalized entirely—so that vocalizing can be done easily when difficult problems are met. If the problem solver skips through one or more steps without thinking aloud, the listener should ask the problem solver to back up and explain the steps.
An Example
The roles of the problem solver and listener are demonstrated in the following dialogue between two graduate students working a problem. As you read through the example, notice how the listener helps the problem solver both to be more accurate and to keep talking. The listener is always following what the problem solver is saying and at the same time looking for ways to make the problem solver think harder about the problem.
The Problem Solver begins by reading the problem twice. |
“ ‘Bill, Judy and Sally have the occupations of teacher, plumber, and teamster but not necessarily in that order. Bill is shorter than Judy but taller than Sally. The plumber is the tallest and the teamster is the shortest. What is Judy’s occupation?’ Now you want to see how to solve that? OK. Let’s see. I would read it again. ‘Bill, Judy and Sally have the occupations of teacher, plumber, and teamster but not necessarily in that order. Bill is shorter than Judy but taller than Sally. The plumber is the tallest and the teamster is the shortest. What is Judy’s occupation?’ So I would go back to where it says Bill is shorter than Judy but taller than Sally and I would note that in some way. Um, I guess using lines to indicate, you know, some kind of length. No I wouldn’t. I would just write their names down in the order that it gives. So I’d have Bill is less than Judy but taller than |
Sally < Bill < Judy |
Sally—common inequality.” |
Listener |
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Here the Listener probably understands the Problem Solver’s notation but is looking for a more detailed explanation. It is always a good idea to ask the Listener to explain any new symbol. |
“OK. So you’re using inequality signs here?” |
Problem Solver |
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teamster_plumber |
“Yeah. Just to show and er, well they indicate height; unequal height. So it could be read the height of Bill is less than the height of Judy and greater than the height of Sally. Then it says the plumber is the tallest of these three so I would write down the word plumber and the teamster is the shortest. And so I … where I’d put whatever occupation er, in the middle indicating that the person who is occupied in that manner has the middle height. And then to the left of that I’m going to write teamster.” |
The Listener is trying to get the Problem Solver to explain in diagram in more detail. |
“Ok. So first you wrote plumber and you put that on the right-hand side?” |
Problem Solver |
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“Right. And then I left a space for some other occupation and then I wrote teamster on the left-hand side, indicating that the teamster is the shortest and the plumber is the tallest.” |
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Listener |
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Note that while this may seem like a silly question it does get the Problem Solver to reflect more carefully about the problem. In particular, it forces an explicit statement of the implied connection between the two diagrams. |
“Why do you put the plumber on the right-hand side? Any reason for that?” |
Problem Solver |
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“No, I could have just as easily put the plumber on the left-hand side except that above it I have the inequality going from right to left meaning that Judy who is the tallest and then Bill who is the next tallest, and then Sally who is the next tallest. So that’s probably why I have plumber who is the tallest on the right-hand side under Judy, and teamster who is the shortest on the left-hand side under Sally.” |
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“Do you think that’s why you put it there?” |
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Problem Solver “Yeah, I think that’s why I put it there. Because I had set the precedent with Sally, less than Bill, less than Judy, going from left to right. “Uh, so the final question is what is Judy’s occupation, and I think that Judy is the plumber.” (Pause) |
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Listener |
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Whenever the Problem Solver is quiet for more than a few seconds the Listener should ask for verbalization. |
“What are you thinking now?” |
Problem Solver “That Judy is the plumber. That Judy is the plumber and I’m going to write um, teacher in between teamster and plumber.” |
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Listener |
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Even though the Problem Solver seems to have solved the problem, the listener keeps asking questions to make sure the problem is fully understood. |
“Why?” |
teamster, teacher, plumber |
“Because I think that uh, the teacher is of medium height. But I’m not sure so I’m going to go and check everything now. Bill, Judy, and Sally have the occupations of teacher, plumber, and teamster but not necessarily in that order, meaning that Bill’s not the teacher and Judy’s not the plumber and Sally’s not the teamster.” |
Listener |
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Here the Listener notices that the Problem Solver’s words suggest confusion between the terms “not necessarily” and “not.” A simple question prompts the Problem Solver to be more careful. |
“It means that they’re not that?” |
Problem Solver “It says ‘but not necessarily in that order,’ yeah so Bill does not have to be the teacher. I think when you say ‘that order’ you mean respectively. You’re saying not necessarily respectively. So Bill is not necessarily the teacher.” |
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Listener “Uh-huh.” |
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Problem Solver “Judy’s not necessarily the plumber and Sally’s not necessarily the teamster but it could be that way.” (Pause) |
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Here again the Listener prompts the Problem Solver to keep talking. |
“So what are you thinking now?” |
Problem Solver “So I was thinking that this inequality that I have written down is correct because it says that Bill is shorter than Judy and taller than Sally. So if Bill is shorter than Judy and taller than Sally, then Judy must also be taller than Sally because Bill is shorter than Judy, and Bill is taller than Sally. So if Judy is taller than Bill then she must be taller than Sally. So I’m fine with that particular statement. The plumber is the tallest and the teamster is the shortest. And since Judy is taller than Bill, and Bill is taller than Sally and then Judy is taller than Sally so Judy is the tallest and the plumber is the tallest and so Judy is the plumber. And the teamster is the shortest and since Sally is let’s see—(mumbles to self)—So Judy is shorter than Bill and Judy is shorter than Sally and the teamster is the shortest so I think that Judy is the teamster.” |
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Listener |
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The Listener tries to see if the Problem Solver notices that this contradicts the earlier statement. |
“So you just said Judy is the teamster.” |
Problem Solver “Right. What did I say before, then, she was the plumber?” |
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Listener “Uh-huh.” |
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“Did I say that?” |
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Listener “Yeah.” |
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Problem Solver “And the teamster is the shortest. No, Sally’s the teamster.” |
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Listener “So what were you just looking at?” |
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Problem Solver “Just, you know, momentary dyslexia. Yeah. I’m going to say that Sally’s the teamster, and Bill’s the teacher and Judy’s the plumber.” |
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Listener |
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The Listener should always check to be sure the Problem Solver is confident before the pair moves on to the next problem. |
“Are you sure?” |
The previous example involved two skilled students. It is likely that the first time you work a problem the Problem Solver will have a harder time talking. The following example illustrates how the Listener can help get a stubborn Problem Solver to talk more.
If the second letter in the word west comes after the fourth letter in the alphabet, circle the letter A below. If it does not, circle the B.
A |
B |
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Problem Solver |
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Problem Solver reads the problem. |
“ ‘If the second letter in the word west comes after the fourth letter in alphabet, circle the letter A below. If it does not, circle the B.’ ” |
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Listener “You said ‘in alphabet’ not ‘in the alphabet.’ ” |
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Problem Solver “Oh yeah.” (Pause) |
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Listener “What are you thinking?” |
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Problem Solver “Nothing. I am just looking at it.” (Pause) |
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Listener “What are you looking at?” |
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Problem Solver “It’s A. I circle the A.” |
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Listener “Wait a minute. You just said you weren’t thinking. Now you say it’s A. How did you get that?” |
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Problem Solver “Well, the fourth letter is D.” |
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“Yes.” |
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Problem Solver “And so I circle the A.” |
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Listener “How do you get the fourth letter is D?” |
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Problem Solver “A, B, C, D. I count.” |
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Listener “OK. So how come you circle A?” |
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Problem Solver “Because that’s what it says to do.” |
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Listener |
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Here the Listener lies deliberately. |
“I think you are wrong.” |
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Problem Solver “I am wrong?” (Pause) “You mean it’s B. It can’t be B because the letter E comes after the letter D.” |
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Listener “Yeah?” |
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Problem Solver “And it says that if the second letter in west, which is e, comes after the fourth letter in the alphabet circle the A. So I did.” |
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The Listener tests the Problem Solver’s confidence by trying another interpretation. |
“Yes, but e comes before h, which is the fourth letter in alphabet.” |
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Problem Solver “Hmm. You think they want the fourth letter in alphabet? (Pause) No they don’t, they said the alphabet not in alphabet. You pointed that out to me earlier.” |
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Listener “Oh.” |
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Problem Solver “So it’s A.” |
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Listener “Wait, tell me all over why you think it is A—go slowly.” |
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Problem Solver “Well the problem asks you to circle the A if the second letter in west which is e comes after the fourth letter in the alphabet which is d. And e does come after d so I circle the A.” |
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Listener “Are you sure?” |
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Problem Solver “Yes.” |
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In this example the Listener never gets the Problem Solver to talk in adequate detail, but through a variety of techniques does get the Problem Solver to talk some. With additional practice both students should eventually be able to reach the level of skill demonstrated in the first example.
As you work problems together, you will see places where the problem solver failed to vocalize his or her thoughts, and you will see errors occurring because of hastiness and failure to recheck. The problem solver may neglect to concentrate fully enough on accuracy, or may forget to approach problems in a systematic, step-by-step manner. Sometimes the errors will be minor, sometimes they will set the problem solution on an entirely wrong track. With experience, you should become sensitive to these types of errors and catch them quickly.
Quiz Yourself
1. What are the two roles of the listener?
2. Describe the activities of the listener in checking for the problem solver’s accuracy. According to the text, what are two things the listener should not do?
How Thinking Aloud Pair Problem Solving Works
When you want to learn a new skill there is often a tendency to ask for a complete list of instructions that explains exactly how to perform the task perfectly. But that is not really how learning works. Most learning starts with having a very simple goal in mind and then lots of practice trying to accomplish it. In basketball, you are told the purpose is to get the ball in the basket. After that you get out on the court and learn the details bit by bit through experience. In pottery, someone quickly shows you how to mold clay on a wheel and then you try your hand at it. Over the years you may pick up hundreds of hours in instruction but most of this comes in the form of short hints delivered during the course of your practice. Most of your learning comes not from what other people tell you, but from your own ability to notice what works well and what does not.
Thinking Aloud Fair Problem Solving is a very simple process that will become increasingly complex and sophisticated as your experience with it develops. Initially, all you need to know is that the problem solver must explain every step in her reasoning and that the listener has to understand every step that the problem solver takes. From these two simple goals everything else follows. Teachers and this book can give hints here and there but the vast majority of learning will come from the dynamic of the process itself. Every time you are a listener you are learning about problem solving by paying careful attention to what the problem solver does and does not do. Every time you are the problem solver you are indirectly observing the listener who is listening to you. The process has built into it all the feedback you need.
If in the middle of a basketball game you were to forget that the purpose is to score baskets, you would quickly lose the feedback you need to improve your game. Likewise, in TAPPS if you forget that the purpose is to better understand thinking, then you will cut off your learning feedback and the opportunity to improve your skills. Solving the problems quickly and correctly is not the major purpose of TAPPS. It is only through being able to better understand your own reasoning as well as that of others, that you will become a better thinker and a better problem solver. In the long run being a better problem solver will increase your speed and accuracy but making these short-term goals will only serve to limit your learning.
Thus, there is only one thing you need to remember in Thinking Aloud Pair Problem Solving. The goal is to better understand thinking, your own and other people’s. Keep that goal in mind and everything else will follow.
Problem Solutions
Many of the problems in this book are followed by a section labeled Problem Solution. The problem solution breaks the problem into a series of steps and shows how the problem can be solved in a step-by-step manner. The steps were obtained by asking good problem solvers to think aloud as they worked the problems and then summarizing their steps so they would be easy to read.
After you and your partner agree on the answer to a problem, turn to the problem solution to check that the answer is correct. The steps which are listed may not be identical to the way in which you solved the problem, but the final answer should be the same. If you had any difficulty with the problem, review all the steps in the solution to see whether they differ from the approach you used.
Your instructor may give quizzes in problem solving, asking you to solve problems similar to the ones in the book. Therefore, it will be worth your while to look over the problem solutions carefully to insure you can solve all the problems effectively.
Devising Problems
For homework and additional classwork, your instructor will ask you to devise problems similar to certain of the ones in this chapter. Devising problems based on other problems allows you to see them from the inside out. You come to understand how the various parts of the problems operate and how they relate to each other. Don’t be surprised to find that devising good problems can require much more thinking than solving them. Just take your time changing the various problem elements until you are totally satisfied.
Devise each problem on your own. Then have another student solve it, reading and thinking aloud, and also writing out all the steps in the solution. Check to be sure no steps are skipped. The solution should be as complete as solutions you read in earlier sections of this book. Writing out complete solutions is a powerful exercise for strengthening problem-solving skills as well as writing skills.
Place just one problem per page so that the solution can be written below it. Make the problems difficult enough to be challenging. But be certain they are logically sound and solvable before asking other students to work them.
Awareness and Communication of Thinking: Pre-Test
Two objectives of this workbook are increasing your awareness of the mental activities you use in solving problems and improving your ability to explain these mental activities. Becoming more aware of your mental activities will help you interpret and organize information; keep track of where you are while solving a problem; identify obstacles when you are stuck; and increase your accuracy. Use the diagram below to rate yourself on awareness of mental activities. At the end of the next chapter you will rate yourself again so you can evaluate your progress.
Awareness2
1The problem solver is an outstanding medical student who earned a Master’s degree in Comparative Literature before deciding to enter the field of medicine.
2From The McMaster Problem Soloing Program: Unit I, Developing Awareness by Donald R. Woods. Department of Chemical Engineering, McMaster University, Hamilton, Ontario, Canada, 1985, by permission of the author.