One way to improve your analytical skills is to see the types of errors that people frequently make in solving problems, and then guard against making those same errors yourself.
Various types of errors undoubtedly came to light in your discussion of the WASI. This chapter analyzes a sample of errors made by students in courses that we have taught. Read through these errors and see how they compare to the ones you made.
Occasionally errors are made on the WASI because people don’t have enough information to answer a question. For example, on vocabulary questions (such as question # 15) a person might not know the meaning of the words. But most errors are not of this type. Instead, people have sufficient facts yet miss questions because their analyses and reasoning processes break down. Here are four ways in which the breakdowns frequently occur:
1. Person fails to observe and use all the relevant facts of a problem.
2. Person fails to approach the problem in a systematic step-by-step manner, making leaps in logic and jumping to conclusions without checking them.
3. Person fails to spell out relationships fully.
4. Person is sloppy and inaccurate in collecting information and carrying out mental activities.
These sources of error tend to be interrelated; however, one may be more prominent than the others with some particular person or problem. Examples of all four sources of error follow.
Error Analysis of Sample IQ Questions
In this section we will review errors made on the following questions from the WASI: Questions 7, 9, 10, 12, 18, 22, 27 and 28. These questions are representative of the types of problems that are found on most IQ tests. Let’s begin with question 7.
Question 7
In a different language liro cas means “red tomato,” dum cas dan means “big red barn” and xer dan means “big horse.” What is the word for barn in this language?
a. dum
b. liro
c. cas
d. dan
e. xer
This is a fairly easy question, but one which is often missed by nonana-lytical thinkers. The most common error is to say that dan means barn because dan and barn both occupy the third position in “dum cas dan” and “big red barn.” The error fails to take into account that “xer dan” means “big horse.” It is an example of the one-shot thinking and lack of concern for total accuracy which researchers have observed to be characteristic of nonanalytical thinkers.
Question 9
There are 3 separate, equal-size boxes, and inside each box there are 2 separate small boxes, and inside each of the small boxes there are 4 even smaller boxes. How many boxes are there altogether?
a. 24
b. 13
c. 21
d. 33
e. some other number
This question is especially interesting because its solution is quite straightforward, yet it is often missed. It illustrates the lack of skill which some people have in spelling out ideas fully and accurately.
The simplest way to solve this problem is to draw a diagram representing correctly the relationship of the boxes.
A student who chose answer b described her inadequate reasoning as follows:
I pictured the three boxes and the two smaller boxes inside the three boxes … I added three plus two (which gave five) and counted the four other boxes twice. Five plus eight gave me 13.
This student didn’t spell the diagram out fully. Instead she went ahead and added numbers without carefully considering exactly which numbers should be added and why.
Many people approach mathematics problems in this way. They perform inappropriate numerical operations because they don’t clarify in their own minds the exact relationship of the facts in the problem.
Question 10
Ten full crates of walnuts weigh 410 lb, while an empty crate weighs 10 lb. How much do the walnuts alone weigh?
a. 400 lb
b. 390 lb
c. 310 lb
d. 320 lb
e. 420 lb
This is a conventional math word problem, something which frightens a large percentage of students and adults. They feel that a special inborn ability is required for mathematics, an ability in which they rate a great big zero.
When you look at this problem closely, you see that it doesn’t require any mysterious ability. All that this problem demands is that the facts be spelled out fully and accurately. Once that is done, the remaining arithmetic is simple.
Here is a diagram which spells the facts out fully. It shows that the total weight is composed of 11 parts, the weight of the 10 crates and the weight of the walnuts.
Total Weight: 410 Pounds
You may not have actually visualized this diagram in working the problem, but conceptually you used a similar model. You had to sort the total weight into the various parts shown in the diagram in order to compute the answer.
Students who have trouble with this and other math problems haven’t learned to spell numerical relationships out fully so that correct calculations can be made. For example, a mistake frequently made with this problem is to answer 400 pounds, showing that the person did not take all 10 boxes into account.
A revealing answer to this problem is alternative e, which is 420 pounds. This answer is sometimes selected, even though it doesn’t make sense. The walnuts alone can’t possibly weigh more than the walnuts plus the crates. This answer shows how anxious and flustered some people get in doing math. They see two numbers in the problem, 410 and 10, and immediately add. They read math problems with less care and patient thought than they would the inscriptions on tombstones of strangers in a dark, frightening cemetery. They feel conquered by math, and so they never even begin calling their analytical skills out to battle.
Question 12
The first figure is related to the second figure in the same way that the third figure is related to one of the answer choices. Pick the answer.
This is a fairly easy figurai problem. Still, people make errors because they are inaccurate in observing or using the given information in understanding the analogy and selecting an answer. For example, they may neglect to change the shading and choose answer e. Or they may neglect size and choose c.
Question 18
Cross out the letter after the letter in the word pardon which is in the same position in the word as it is in the alphabet.
This is a fairly difficult verbal reasoning problem. One interesting thing about it is that while on the surface it appears to be very different from question 12 (above), errors on the two questions come about for the very same reasons. People fail to search out and use all the available information.
One frequent error is to cross out the d in pardon, rather than crossing the letter after the d. The person making this error has lost part of the problem.
Another less frequent error is to cross out the d in word, as shown below.
Cross out the letter after the letter in the word pardon which is in the same position in the wor
as it is in the alphabet.
People who make this error haven’t learned to work step-by-step through a complex sentence. They don’t think through the sentence in the following way:
Cross out the letter after the letter (so I have to cross out a letter) in the word (cross out a letter in some word) pardon (so pardon must be the word).
Question 22
The top four figures form a series which changes in a systematic manner according to some rule. Try to discover the rule and choose from among the alternatives the figure which should occur next in the series.
Here is another problem which is missed frequently although it involves no obscure or esoteric ideas. Errors arise from inaccuracies in making observations and building them into a system or rule which leads to the answer. For example, a student who chose answer e explained his thinking as follows:
I noticed that first there are some lines taken away. Then there are more lines taken away going the other way. Then there are more lines taken away going up and down. So I guess the answer should take more lines away. I guess answer e.
I asked him whether he had counted the lines to see exactly how many were deleted in each figure and he answered that he had not, as it seemed too confusing. As you see, this student had never learned to accurately keep track of facts in problem solving.
Question 27
Elephant is to small ________ as is to ________.
a. large: little
b. hippopotamus: mouse
c. turtle: slow
d. lion: timid
Verbal analogies like this one play a large role on tests such as the SAT, GRE, and various other aptitude tests and IQ measures. Analogy questions are widely used because they tap a person’s ability to define ideas and relationships fully and accurately.
With this particular question, students whose leanings are nonanalytical read “elephant is to small” and quickly decide on “large: little” as the answer. Such a student doesn’t spell out in his mind that an “elephant” is an animal and “small” is a quality, whereas “large” and “little” are both qualities. He reaches conclusions on the basis of quick immediate impressions rather than thorough, step-by-step interpretations, and as a result he often misses significant dimensions of relationships.
Question 28
Which word means the opposite of demise?
a. hasty
b. birth
c. accept
d. embrace
This is a vocabulary question, and there is hardly a mental test on the market which doesn’t include at least a few of them. In this question you are given a word and asked to find another word which is opposite in meaning. At other times (as with question 16) you must find the word that is similar in meaning.
You may have noticed that vocabulary questions are different than the other test questions we have looked at. Most test questions require reasoning and problem solving, whereas vocabulary questions are mainly a matter of recall.
This apparent disparity is cleared up when you consider how vocabulary is acquired. A rich vocabulary is the by-product of careful thinking in verbal communication. People who think analytically listen and read for complete understanding of relevant ideas. When they encounter a new word they try to estimate its meaning from the context. If they are still uncertain, they take out the dictionary and study the various entries, interpreting and contrasting ideas and terms until they are sure of the word’s definition. In short, although vocabulary questions do not require problem solving at the time one takes a test, these questions do reflect the precise thinking that people employ in acquiring vocabulary.
Summary
In this chapter we have looked at the types of test questions that are used to measure reasoning ability, and have seen that errors are primarily caused by a lack of accuracy and thoroughness in thinking. Research has shown that accuracy and thoroughness are mental habits which can be cultivated through training and exercise. This book provides some of that training. But you need to go further on your own. With everything you read, practice carefulness in comprehending ideas and relationships. And in solving problems, continually check yourself for accuracy and completeness. Initially this may be difficult and require conscious discipline, just as learning good typing habits or correct swimming movements may at first be difficult. Gradually, however, the attitudes and skills of tight reasoning will become as natural to you as swimming, skating, driving, typing, or any of the various other skills that you have learned with practice and time.
Although accuracy and thoroughness are beneficial to every kind of activity in which we engage, you are likely to be more accurate in some areas than in others. It is a good idea to identify the areas in which you are least accurate and to make special efforts to practice problems in those areas. Errors on the WASI problems (13, 15, 16, 28, 32, 35, 37, 38) suggest a need to practice Verbal Reasoning problems, pages 41–135. You may also need to develop your English vocabulary, which can be done by increasing the amount of reading you do, and by making sure to look up the words you do not know or of which you are unsure. You might also work through a vocabulary development text such as chapter 34 in Linden and Whimbey (1990a). Errors on the WASI problems (9, 10, 17, 18, 20, 22, 23, 34, 36) indicate you need practice on following sequential instructions, problems 32–43 on pages 133–135 and chapter 10 on pages 221–238. Errors on WASI problems (2, 3, 4, 5, 12, 24, 26, 30) indicate you need practice forming Analogies, pages 141–192. Errors on WASI problems (1, 6, 19) can be traced to the specific area of Writing Relationship Sentences, pages 155–169. Errors on WASI problems (8, 11, 14, 21, 25, 29) relate to the Analysis of Trends and Patterns, pages 193–219. Errors on WASI problems 31 and 33 suggest a need to practice Solving Mathematical Word Problems, pages 239–331. But the most important practice is the practice of being thorough and accurate with any kind of problem. The following Checklist provides an introductory guide to pitfalls the thorough problem solver must be careful to avoid.
Checklist of Errors in Problem Solving
Following is a checklist of sources and types of errors in problem solving. Some of the items overlap, referring to different aspects of the same fault in working problems, but this overlap is unavoidable because the various factors that underlie problem-solving skill are interrelated. Read the checklist aloud, discussing any items that are unclear. Then, as you solve problems, be careful not to make these errors. If you recognize some particular error to which you are especially prone, take extra pains to guard against it. Also, when you are listening to another student solve a problem, watch his or her approach for errors of the type listed below.
Inaccuracy in Reading
1. Student read the material without concentrating strongly on its meaning. He/she was not careful to understand the problem fully. He/she read sections without realizing that understanding was vague. Did not constantly ask “Do I understand that completely?”
2. Student read the material too rapidly, at the expense of full comprehension.
3. Student missed one or more words (or misread one or more words) because the material was not read carefully enough.
4. Student missed or lost one or more facts or ideas because the material was not read carefully enough.
5. Student did not spend enough time rereading a difficult section to clarify its meaning completely.
Inaccuracy in Thinking
6. Student did not constantly place a high premium on accuracy—did not place accuracy above all other considerations such as speed or ease of obtaining an answer.
7. Student was not sufficiently careful in performing some operation (such as counting letters) or observing some fact (such as which of several figures is the tallest).
8. Student was not consistent in the way he interpreted words or performed operations.
9. Student was uncertain about the correctness of some answer or conclusion, but did not check it.
10. Student was uncertain about whether a formula or procedure used to solve the problem was really appropriate, but did not check it.
11. Student worked too rapidly, which produced errors.
12. Student was inaccurate in visualizing a description or a relationship described in the text.
13. Student drew a conclusion in the middle of the problem without sufficient thought.
Weakness in Problem Analysis; Inactiveness
14. Student did not break a complex problem into parts. Did not begin with a part of the problem that could be handled in order to get a foothold. Did not proceed from one small step to the next small step, being extremely accurate with each one. Did not use the parts of the material he/she could understand to help figure out the more difficult parts. Did not clarify thoughts on the parts understood and then work from there.
15. Student did not draw upon prior knowledge and experience in trying to make sense of ideas which were unclear. He/she did not try to relate the written text to real, concrete events in making the meaning clear and understandable.
16. Student skipped unfamiliar words or phrases, or was satisfied with only a vague understanding of them, rather than trying to obtain a good understanding from the context and the remainder of the material.
17. Student did not translate an unclear word or phrase into own words.
18. Student did not use the dictionary when necessary.
19. Student did not actively construct (mentally or on paper) a representation of ideas described in the text, where such a representation could have helped in understanding the material.
20. Student did not evaluate a solution or interpretation in terms of its reasonableness, i.e., in terms of his prior knowledge about the topic.
Lack of Perseverance
21. Student made little attempt to solve the problem through reasoning because he/she lacked confidence in ability to deal with this type of problem. Took the attitude that reasoning would not work with this problem. Felt confused by the problem, so didn’t start systematically by clarifying the portions of the problem that were readily understandable, and then attempting to work from there.
22. Student chose an answer based on only a superficial consideration of the problem—on an impression or feeling about what might be correct. Student made only a superficial attempt to reason the problem, then guessed an answer.
23. Student solved the problem in a mechanical manner, without very much thought.
24. Student reasoned the problem partway through, then gave up and jumped to a conclusion.
Failure to Think Aloud
The items above apply to all academic problem solving. The last item refers specially to the procedure used in this course.
25. Student did not vocalize thinking in sufficient detail as he or she worked through the problem. At places he/she stopped and thought without vocalizing thoughts. Student performed a numerical computation or drew a conclusion without vocalizing or explaining the steps taken.