Guess How To Teach . . . Teach How To Guess!

Dan H. Wolaver

One challenge facing every teacher is getting students to participate. Another is getting students to apply what they know rather than just memorize facts. The following exchange shows how guessing can address both these problems.

Q. "Guess how many bookstores there are in the United States."

A. (Silence)

Q. "About how many bookstores would you guess there are?"

A. "I don't know."

Q. "I know; that's why I asked you to guess."

A. "I have no idea."

Q. "Are there less than ten?"

A. "No."

Q. "Are there more than a billion?"

A. "No."

Q. "Then you have some idea. Are there 100?"

A. "Probably more. I know of four or five just in Worcester."

Q. "How many bookstores would you guess are in Worcester?"

A. "Maybe 10 or 15 then."

The accuracy of the student's eventual estimate to the original question depends on knowledge he may have already—the population of the United States, the population of Worcester, etc. But the accuracy is of less importance than the method by which he arrives at the estimate. The student might also be asked where he would look for more information to improve his estimate. The point of this example is that a student can't reasonably refuse to participate if he is asked to guess. He always has something to contribute—probably more than he expects—based on whatever knowledge he has. By guessing, he is applying that knowledge.

We propose that guessing be used as a rallying point—a focus for developing skills of critical and creative thinking. When a student is asked to guess, he must organize facts and make connections between them—recognizing implications, analogies, and patterns. He must weigh plausibility and look for ways to confirm his guesses. This is, in fact, what he must do as a problem solver in the "real world."

"Guessing" is not generally held in esteem in academia; in fact, it is usually discouraged. More dignified terms are "speculation," "prediction," "conjecture," "estimation," and "hypothesis formulation," but we prefer the more prosaic term here. A student knows he can guess but may believe he is not capable of making a Hypothesis. Also, we want to take the curse off guessing. It is certainly true that merely guessing without subsequent refinement and rigorous justification is sloppy thinking, but the rigor is usually emphasized to the point of driving out the proper role of guessing.

Benefits of Guessing

What is the role of guessing in the learning process? Several benefits can be identified.

Benefit 1. As a student sits in lecture or reads a text, he should be anticipating (asking himself, or guessing) what he is about to hear, see, learn, or be asked. This puts his mind into gear, priming him to react quickly in evaluating and ordering the information he is getting. It engages the student as an active participant in the learning process. Russell Stauffer says, "When we predict we establish a reference point from which to measure what the author is telling us. We become critical thinkers, constantly evaluating the author's message against our own information and experience." (Rapid Comprehension Through Effective Reading, LEARN Incorporated, 1969.)

Benefit 2. Guessing at an outcome or answer causes the student to "buy into" an issue. He then has an interest at stake—"Am I right or wrong?" This focuses his attention and makes him a part of the activity.

Benefit 3. Guessing increases participation. By encouraging guessing the teacher assures the student that he needn't be absolutely certain before offering an answer or a solution. Since certainty is impossible, all answers are varying levels of guessing, and wrong guesses are as important as right ones. Jerome Bruner observes: "Very often we are forced, in science and in life generally, to act on the basis of incomplete knowledge; we are forced to guess... It is our feeling that perhaps a student would be given considerable advantage in his thinking, generally, if he learned that there were alternatives that could be chosen that lay somewhere between truth and complete silence." (The Process of Education, Harvard University Press, 1960.)

Benefit 4. Wrong guesses allow the teacher to see where the student needs help. They uncover the misconceptions and false reasoning that need to be addressed. Without this feedback the teacher has no immediate indication whether the students are comprehending. It is tempting to believe that all is going well when there is silence after asking for questions.

Benefit 5. Through guessing, the student comes to see ideas as logical extensions of his own thinking, not as foreign concepts imposed from the outside. By appropriate questions the teacher can lead the student through a series of guesses to the desired result. If this is done properly, the student may feel that he could have reached the result without any help. This may not be far from the truth, in fact. The teacher's aim is to make it eventually be the truth.

Benefit 6. Skill at guessing is central to discovery and creativity, and these should be taught and developed in the schools. In discussing his special area of discovery, Polya says, "The result of the mathematician’s creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing." (Induction and Analogy in Mathematics, Princeton University Press, 1954.)

In preparing to teach a subject, the teacher himself employs guessing to sort through confusion and blind alleys. This is usually hidden from the student, and only the sanitized "bottom line" is presented as the object to be learned. Bypassing the guessing process deprives the student of the opportunity to witness and learn the skill of guessing. In fact it may give the student a sense of inferiority since he is always struggling while the teacher never does, apparently.

Examples of Using Guessing

Some examples will illustrate the benefits of guessing and how it can be effectively incorporated in the classroom.

Example 1. A first-grade teacher brings out a bag with twenty-five marbles in it, some red and some white. A student is asked to take out five marbles. Two are white, and three are red.

Q. From the marbles you see in your hand, would you guess that there are more red marbles or more white marbles in the bag?

A. Red.

Q. Do you think there are a lot more red marbles?

A. Maybe one more.

Q. Why do you think there's just one more red marble?

A. I don't know.

Q. Is it because there's one more red marble than white in your hand?

A. Yes.

Q. Let's see if your right.

The bag is emptied, and the student counts how many red and how many white marbles. He will probably find more red. As the experiment is repeated over the weeks, the students learn something about probability and proportionality, although these terms may not be used.

This example illustrates a couple of points. Students of any age can guess effectively about any topic. They bring some experience in life that gives them a basis for their guesses. Jerome Bruner says, "We begin with the hypothesis that any subject can be taught effectively in some intellectually honest form to any child at any stage of development." (The Process of Education.) The example also suggests that formalities such as definitions, symbols, and equations are not necessary in order to begin thinking, guessing, and understanding. They might actually get in the way by substituting fact memorization for a working sense of the topic.

Example 2. A student asks how to spell "chameleon."

Q. How do you think it's spelled?

A. I already tried "ca" and "ka" in the dictionary, but I couldn't find it.

Q. What other ways are there of spelling the "k" sound?

A. "Ck," but I don't think it would be used at the beginning.

Q. Any other ways?

A. I can't think of any.

Q. There are a few common words that begin with the "k" sound spelled the same as in "chameleon." Why don't you let it simmer on the back burner and see if it comes to you?

The student often wants a quick answer, not appreciating the blessing a struggle brings. The teacher could have given the answer after the evidence of some effort with the dictionary. But there is almost always some further progress the student can make on his own. He needs to discover, appreciate, and use the knowledge he already has.

Example 3. A history class is studying the period between the two World Wars.

Q. What do you think caused the Great Depression of the 1930s?

A. I don't know; we haven't studied that yet.

Q. What would you guess would lead to an economic collapse?

A. I don't know.

Q. Have you never had economic problems yourself?

A. Sure.

Q. What, for example?

A. Well, I borrowed \$50 from my brother to buy a skateboard. Then I lost my job at the grocery store. It took all summer to pay him back.

Q. What was your mistake in all of this?

A. I guess I shouldn't have spent money I didn't have.

Q. What similar mistakes might have been made on a national scale in the 1920s?

Although the student wasn't familiar with the topic at issue, he could relate it to a similar situation he had knowledge of. Nobody lives in a vacuum; every student has experiences or knowledge that can be drawn upon to guess at an answer. The experiences may be rather far removed at first, but the base of experience will grow as he continues guessing and learning.

Example 4. A science teacher shows a jar with two candles fixed inside, a tall one and a short one.

Q. I'm going to light both candles and put a cover on the jar. What do you think will happen?

A. The candles will go out.

Q. Is there an interesting question here?

A. Which one will go out first?

A. Well, the carbon dioxide produced by the burning has a higher molecular weight than oxygen. It should sink to the bottom of the jar. I would guess the short candle goes out first.

The teacher lights the candles and puts the cover on. The tall candle goes out first.

The guess was plausible, but other factors clearly needed to be considered. This example illustrates the need for feedback to the student after a guess. With this additional information he is ready to guess again. Notice that the second question lets the student pose the question for himself—encouraging him to follow this line of thinking on his own.

Example 5. The following example is from a course on electronic circuit design.

Q. What logic gate could we use as a phase detector?

A. I don't know.

Q. Pick one and try it out.

A. An AND gate?

Q. OK, try it out. Apply two square waves to it. Does the average output vary as the phase?

A. Yes, the output rises and falls smoothly with phase.

Q. Anything else you could try?

A. An OR gate... It works too.

Q. Anything else?

A. Well, an XOR gate. They all work!

Q. Does one work better than the others?

A. The output of the XOR gate is more sensitive to phase.

Q. Good! You may find some better circuits later, but the XOR gate works for most applications.

In this example, the student was really guessing blind, but he made progress. Sometimes when we're stuck, it helps to get off dead center and just begin somewhere. Even if the initial guess is totally wrong, the result lends insight that helps in making the next guess.

Example 6. An economics class is discussing supply and demand.

Q. If the price of razor blades doubles, what do you think would happen to the demand?

A. (Silence)

Q. Would the use of razor blades go up or down?

A. Down.

Q. Would it go down a lot? Would it fall to half?

A. I don't think so; you need to use razor blades at a pretty steady rate.

Q. Is it possible that the demand for blades would remain unchanged then?

A. No, you can use each blade a little longer. The last uses just pull a little more.

Q. Are there other ways to avoid using as many blades?

A. I guess you could switch to an electric shaver.

Q. By what percentage do you think your use of razor blades would fall if the price doubled?

A. Maybe ten percent.

One point here is that the first answer doesn't have to be complete. The student can get started by just guessing the direction of change. Then he can guess at whether the change is small or large. Finally he can estimate an actual number. Another point is that the text being used in the course probably tells that elasticity of demand is influenced by alternative supplies. Therefore the student's answer about an electric shaver may have been prompted by reading about the theory of alternative supplies. This is good; it shows an ability to apply what has been read. But guessing is most effective when the student can arrive at some theory on his own. Then he sees that theories originate not just in textbooks, but also in common sense—in extension from his own experience.

A teacher that encourages guessing finds himself sometimes wishing the text didn't lay out so many of the facts—wishing more were left for the student to arrive at by extrapolation. One solution is to have a discussion before the reading assignment. Few students will spoil it by reading ahead.

Guidelines for Guessing

In the examples above the teacher is not only encouraging guessing, he is also guiding guessing. He is suggesting approaches by asking the type of questions the student should learn to ask himself. By example and encouragement he is presenting various techniques that lead to success in guessing:

1. Make an intuitive leap that feels right. Then try to pin down the basis for your feeling.

2. Simplify the question and answer that first. Then address the details of the original question and see how they modify your answer.

3. If the area of the question is unfamiliar, is it at least similar to a question or experience you've dealt with before? Can you extrapolate from analogous situations?

4. If the complete answer isn't apparent, guess at partial answers and rough approximations at first. Each crude step helps support further refinement.

5. If intuition fails and nothing seems promising, try something to get started, even if it seems fruitless. You can't plan serendipity, but serendipity isn't rare if you give it a chance.

6. Look for a way to confirm your guess or at least support its plausibility.

This list of guidelines for guessing is incomplete; every teacher can add to it from his own experience. They are suggestions that the teacher should make either implicitly or explicitly through his questions. In How To Solve It (Princeton University Press, 1945) Polya gives some criteria for such suggestions:

• The suggestions must be simple and natural because otherwise they cannot be unobtrusive. The suggestions must be general, applicable not only to the present problem but to problems of all sorts, if they are to help develop the ability of the student and not just a special technique.
• The list [of suggestions] must be short in order that the questions may be often repeated, unartificially, and under varying circumstances; thus, there is a chance that they will be eventually assimilated by the student and will contribute to the development of a mental habit.
• It is necessary to come down gradually to specific suggestions, in order that the student may have as great a share of the work as possible.
• This method of questioning is not a rigid one; fortunately so, because, in these matters, any rigid, mechanical, pedantical procedure is necessarily bad. Our method admits a certain elasticity and variation,... it can be and should be so applied that questions asked by the teacher could have occurred to the student himself.

These criteria relate to Polya's list of suggestions for problem solving. But, as we have seen, guessing is an integral part of problem solving.

What stands in the way of realizing the benefits of guessing? It doesn't seem to be a failure to recognize their importance. Teachers often complain that too much attention is given to facts rather than methods—to "content" rather than "process." In technical subjects this is put in terms of "too much analysis" and "too little design." There is general acknowledgment that more needs to be done in getting student to really think, yet this seems "more honored in the breach than the observance." Even the most adamant supporters lapse into talking too much and involving the students too little. Why?

Objection 1: "There isn't enough time." Progress is slow when the student is guessing, exploring blind alleys and groping through uncharted territory. The tendency is for the teacher to hurry things up by providing the right answers himself. He has a lot of material to "cover" in the course, and he can't afford to waste time. But the textbook should be able to accomplish the one-way presentation of material, releasing time for the teacher to do what the text can't—exercise students in the skills of guessing. There is no excuse for the teacher wasting valuable contact time by being a talking textbook. The teacher does need to demonstrate how he approaches problems, but a better balance is needed between showing and asking.

Objection 2: "I don't like to see students suffer." This concern is laudable; it is one of the main motivations in becoming a teacher. But in making learning "easier," the teacher can give in too quickly to the student's call for help. The silence after a student has been asked to guess can be painful, and the easiest course is for the teacher to fill the void himself. The student needs to be allowed to guess, confirm or reject the guess, and guess again until he arrives at his goal. This often involves doubt and confusion, but a fair amount of this is necessary in learning to deal with uncertainty. On the other hand, too much frustration can lead to defeat and loss of confidence. A balance is needed, but the desire to be a "good guy" usually tips the balance towards too much help. It is especially difficult to hold the line if the students complain that the teacher isn't doing his job. Here the teacher needs the courage of his convictions, based in part on his own learning experience and in part on past successes in teaching.

Objection 3: "I can't get the students to participate." This complaint is the result of the silence that follows the appeal, "Are there any questions, any comments? Is there any discussion?" There is no excuse for a student’s silence when asked to guess, as the example at the beginning of this article shows. If a student claims an inability to guess, the question needs to be restated and perhaps simplified until the student is able to respond. This requires skill on the teacher's part in letting the student do as much of the work as possible.

Objection 4: "I enjoy lecturing." Some amount of lecturing is valuable and necessary. The difficulty comes in knowing when to curtail a good thing, especially when it is enjoyable. The teacher must be honest in reassessing what is best for the students. One salvation is that the lively exchange resulting from guessing is enjoyable too.

Objection 5: "You can't really teach guessing." It’s possible to teach anything, though the degree of difficulty varies. Guessing skills are more difficult to teach than facts because they require more individual attention, which requires more time. But most of the perceived difficulty stems from lack of practice in teaching guessing. The methods are the same as for any discipline: example, encouragement, critique, practice, and organization.

Objection 6: "The classroom should teach orderliness and rigor; vague methods such as guessing and intuition can be picked up by students on their own." Education could limit itself to the neat and tidy, but this would be a disservice to students. It wouldn’t prepare them for a world of incomplete knowledge, ambiguities, and confusion. There should be a balance in academia that approximates that in the real world. If this had been true in the past, we wouldn't find the word academic defined as "merely theoretical; having no direct practical application." (Webster's New World Dictionary, Simon and Schuster.)

Conclusions

There's nothing really new in the ideas presented here; almost all teachers employ guessing in the classroom to some degree. The thesis here is that a teacher should show cause why he is not employing it almost all the time. In practice, there is a place for some exposition, but if the teacher is always trying to avoid expounding, then the balance comes out about right.

Teaching through guessing is clearly related to the Socratic method of teaching through questions. The emphasis here is on how the student responds to the questions. He should make a beginning, even though he thinks he has little to offer. With the teacher's encouragement (through questions), he will be surprised at what he can contribute.

Guessing is also related to the "scientific method" of making and testing hypotheses. Koen relates guessing to the "engineering method," referring to rules for guessing as "heuristics" (Definition of the Engineering Method, American Society for Engineering Education). But guessing is more than just a formal method applied to a technical field; it applies every day to all of life.

Everything is uncertain to some extent, and we must do the best we can with the knowledge we have. Often the tendency is to lean on others, believing that someone else must be certain of the answers. Students need to be taught to be self-reliant, to value and use their own resources, and to be ready to help others. Sometimes we guess (make a hypothesis) and are wrong. But we learn from it and guess better the next time. The student shouldn't be afraid of being wrong.

Guessing is a treasure hunt—an adventure in discovery through investigating similarities, making intuitive leaps, and recognizing patterns. As a student progresses through schooling, he will need less and less help in this process until he is able to find his own way; he is able to make educated guesses.

To teach guessing, the teacher himself must love discovery through guessing. He must love it enough to invest the great deal of time and patience that this form of teaching requires. The rewards are great.