Prepared by D.H. Wolaver, 1990

We teach because we love to learn ourselves. What is the excitement of learning? It must be more than just learning that d/dx of x2 is 2x, useful as that might be. Learning is exciting only if we learn really new ways of thinking—of organizing and using knowledge. Martin Gardner calls it “aha! Insight.” It’s the excitement of seeing many diverse things as really basically the same thing, such as radio waves and light. It's solving mysteries, such as why gyroscopes don't fall. It’s making the once-complicated simple. It’s the challenge, uncertainty, and joy of wrestling with a puzzle and finally having it yield.

Once something is understood in all its simplicity, there is the illusion that it should be easy to teach it to someone else. But before the simplicity could appear, many old concepts had to be replaced with new ones through long, hard work. A student may not understand the summing of two voltage rises because he doesn't have the analogy of elevation in mind. There is no real shortcut to learning; it requires work if it is to result in new basic ways of thinking.

The task of the teacher is minimize the time required to learn—to make the time spent as effective as possible. Too little help leaves the student to bang his head against a wall until he gives up in discouragement. Too much help deprives the student of becoming accustomed to the agony that goes with uncertainty and the self confidence that comes with success.  Bill Roadstrum calls learning under good instruction “controlled agony.”

What are the methods by which a teacher can help a student to gain insight—to solve new problems?  There are no guaranteed rules because each case is different. But there are some general techniques that are helpful.

Teaching has been going on for thousands of years, but we still don’t know how to do it well. It's an art with many opportunities for discovery in itself. It's rewarding because it involves working directly with people, with the opportunity to profoundly affect their lives. Through new ideas people become creative, eliminate confusion and fear, and discover Truth.



from “How To Solve It by G. Polya, pp 1-22

The students should have some ideas of their own, some initiative. If the teacher, having watched sharply, cannot detect any signs of such initiative, he has to resume carefully his dialogue with the students. He must be prepared to meet often with the disconcerting silence of the student (which will be indicated by dots . . . . .)

Find the diagonal of a rectangular parallelepiped of which the length, the width, and the height are known.

. . . . .

What is the unknown?

The length of the diagonal of a parallelepiped.

What are the data?

The length, the width, and the height of the parallelepiped.

Introduce suitable notation. Which letter should denote the unknown?


Which letters would you choose for the length, the width, and the height?

a, b, c.

Draw a sketch to help you visualize the problem. Incorporate your notation.

Is it a reasonable problem? I mean, are the data sufficient to solve the problem?

Yes they are. If we know a, b, c, we know the parallelepiped. If the parallelepiped is determined, the diagonal x is determined.

Do you know a related problem to help you with this one?

. . . . .

Look at the unknown! Do you know a problem having the same unknown?

. . . . .

Well, what is the unknown?

The diagonal of the parallelepiped.

Do you know any problem with the same unknown?

No. We have not had any problem yet about the diagonal of a parallelepiped.

 Do you know any problem with a similar unknown?

. . . . .

You see, the diagonal is a segment, the segment of a straight line. Did you never solve a problem whose unknown was the length of a line?

Of course, we have solved such a problem. For instance, to find a side of a right triangle.

Good! Here is a problem related to yours and solved before. Could you use it?

. . . . .

You were lucky enough to remember a problem which is related to your present one and which you solved before. Could you introduce some auxiliary element in order to make its use possible?


. . . .

Look here, the problem you remembered is about a triangle. Have you any triangle in your figure? .

No.  Oh, but by adding a line y, like this, the unknown x becomes part of a triangle.

I think that was a good idea. You now have a triangle; but have you the unknown?

The unknown is the hypotenuse of the triangle; we can calculate it by the theorem of Pythagoras.

You can if both legs are known; but are they?

One leg c is given. And the other leg y is not difficult to find. Yes, y is the hypotenuse of another right triangle.

Very good!  Now I see that you have a plan.  Carry it out.

x2 = y2 + c2

y2 = a2 + b2

x2 = a2 + b2 + c2

x = (a2 + b2 + c2)0.5

Can you check the result? (The teacher cannot expect a good answer to this question from inexperienced students. The students, however, should acquire fairly early the experience that...if the problem is given “in letters” its result is accessible to several tests to which a problem “in numbers” is not susceptible at all. The teacher can ask several questions about the result which the students may readily answer with “Yes,” but an answer “No” would show a serious flaw in the results.)

Did you use all the data?

Is the expression you obtained for the diagonal symmetric in a, b, c? Does it remain unchanged when a, b, c are interchanged?

If the height c decreases, and finally vanishes, the parallele­piped becomes a rectangle. If you put c = 0 in your formula, do you obtain the correct formula for the diagonal of the rectangle?

If the height c increases, the diagonal increases. Does your formula show this?

The teacher’s method of questioning in the foregoing...is essentially this: Begin with a general question or suggestion, and, if necessary, come down gradually to more specific and concrete questions or suggestions till you reach one which elicits a response in the student’s mind.

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.

Our method admits a certain elasticity and variation, it admits various approaches, it can be and should be so applied that questions asked by the teacher could have occurred to the student himself. Suppose, for example, with the best intention to help the student, the question may be offered: Could you apply the theorem of Pythagoras? Even if he understands the suggestion the student can scarcely understand how the teacher came to the idea of putting such a question. And how could he, the student, find such a question by himself? It appears as an unnatural surprise, as a rabbit pulled out of a hat; it is really not instructive.

G. Polya



With experience at problem solving, the student will use these questions unconsciously. He will also begin making bigger leaps as he recognizes situations he has seen before. This unconscious behavior is sometimes called “intuition,” but it’s nothing mysterious; it simply comes from experience over time. The job of the tutor is to make that experience be as effective and efficient as possible.



From “The Process of Education by Jerome Bruner (pp 58-64):

The complementary nature of intuitive and analytic thinking should, we think, be recognized. Through intuitive thinking the individual may often arrive at solutions to problems which he would not achieve at all, or at best more slowly, through analytic thinking. Once achieved by intuitive methods, they should if possible be checked by analytic methods... Unfortunately, the formalism of school learning has somehow devalued intuition. It is the very strong conviction of men who have been designing curricula, in mathematics and the sciences particularly, over the last several years that much more work is needed to discover how we may develop the intuitive gifts of our students from the earliest grades onwards. For, as we have seen, it may be of the first importance to establish an intuitive understanding of materials before we expose our students to more traditional and formal methods of deduction and proof...

What variables seem to affect intuitive thinking?... It seems unlikely that a student would develop or have confidence in his intuitive methods of thinking if he never saw them used effectively by his elders. The teacher who is willing to guess at answers to questions asked by the class and then subject his guesses to critical analysis may be more apt to build those habits into his students than would a teacher who analyzes everything for the class in advance...

A heuristic procedure, as we have noted, is in essence a nonrigorous method of achieving solutions of problems... Will the teaching of certain heuristic procedures facilitate intuitive thinking? For example, should the students be taught explicitly, “When you cannot see how to proceed with the problem, try to think of a simpler problem that is similar to it; then use the method for solving the more complicated problem?”... It is difficult to believe that general heuristic rules–the use of analogy, the appeal to symmetry, the examination of limiting conditions, the visualization of the solution–when they have been used frequently will be anything but a support to intuition.


From the preface (p. iii) of “Geometry and the Imaginationby Hilbert and Cohn-Vossen:

In mathematics, as in any scientific research, we find two tendencies present. On the one hand, the tendency toward abstraction seeks to crystallize the logical relations inherent in the maze of material that is being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency toward intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations.


As an example, Hilbert considers an ellipse, defined as the locus of points whose distance to two foci sums to a constant. He proves this definition is identical with the construction by intersecting a cylinder with a plane. Spheres are slid into each end of the cylinder until they touch the plane at F1 and F2. B is any point on the ellipse (the intersection of the cylinder and the plane). A vertical line in the cylinder through B intersects the top circle (where the top sphere touches the cylinder) at P1, and it intersects the bottom circle (where the bottom sphere touches the cylinder) at P2. The line BP1 is the same length as the line BF1 (two lines from B tangent to the top sphere). Similarly, the line BP2 is the same length as the line BF2. But clearly, BP1 + BP2 is a constant, so BF1 + BF2 is a constant.   Therefore the intersection of the cylinder and the plane is shown to be an ellipse with foci F1 and F2. The proof is not rigorous, but it is visual and appeals to the intuition.


From Chapter 20, of Feynman’s “Lectures on Physics”, Vol.I:

The angular momentum [of the gyroscope] does not change in magnitude, but it does change in direction [due to precession]... The torque [which keeps the gyroscope from falling]...is the time rate of change of the angular momentum...

We may now claim to understand the precession of gyroscopes, and indeed we do, mathematically. However, this is a mathematical thing which, in a sense, appears as a “miracle.” It will turn out, as we go to more and more advanced physics, that many simple things can be deduced mathematically more rapidly than they can be really understood in a fundamental or simple sense... What we should try to do is to understand it in a more physical way.

The sketches below illustrate Feynman’s “physical” or intuitive understanding of what keeps a gyroscope from falling. As a particle of the wheel rotates over the top, it follows a curved path (seen from the top) due to the precession of the gyroscope. In so deflecting the particle’s path, the wheel experiences an inward force at the top. This torque counters the downward torque of gravity. (Can you show that particles at the bottom push the bottom of the wheel outward?)



The sum of integers 1 through n can be visualized as a staircase. Gauss saw this at the age of seven. If two of these staircases are put together, they form a rectangle n × (n+1).

The sum we want is half this area:

This approach can be extended to the sum of squares: 1 + 4 + 9 + 16 + · · · n2. Instead of a staircase, we now have a “pyramid” of squares. Three of these pyramids can be put together to form a rectangular block n × n × (n+1) with a staircase left over.

Therefore the volume of the three pyramids is and the sum we want is one-third of this:


Students are given the following problem: Six people are seated at a table. Someone proposes a toast, and everyone clinks glasses with everyone else. How many clinks?

One student says, "Well, each of the six people clinks with five people, so that's 30 clinks."

"No," says another student, "that's double counting because that way two people count the same clink. So the answer is half 30, or 15."

"Maybe," says a third student, "but I have a cleaner way that counts each clink only once. The first person at the table clinks with the five other people and then leaves. The next person clinks with the four other people and leaves. And so on down to the last clink. So the answer is 5 + 4 + 3 + 2 + 1 = 15."

Was it just lucky that these two approaches got the same answer? For n people at the table, one approach

said n(n – 1)/2 is the number of clinks, while the other said . Are these two answers the same for all n?


From the preface (p.5) of Feynman’s Lectures on Physics

When I look at the way the majority of the students handled the problems on the examinations, I think that the system is a failure. Of course, my friends point out to me that there were one or two dozen students who—very surprisingly—understood almost everything in all of the lectures, and who were quite active in working with the material and worrying about the many points in an excited and interested way... But then, “The power of instruction is seldom of much efficacy except in those happy dispositions where it is almost superfluous.” (Gibbons)

Still, I didn’t want to leave any student completely behind, as perhaps I did. I think one way we could help the students more would be by putting more hard work into developing a set of problems which would elucidate some of the ideas in the lectures. Problems give a good opportunity to fill out the material of the lectures and make more realistic, more complete, and more settled in the mind the ideas that have been exposed.

I think, however, that there isn’t any solution to this problem of education other than to realize that the best teaching can be done only when there is a direct individual relationship between a student and a good teacher—a situation in which the student discusses the ideas, thinks about the things, and talks about the things. It’s impossible to learn very much by simply sitting in a lecture, or even by simply doing problems that are assigned. But in our modern times we have so many students to teach that we have to try to find some substitute for the ideal.

Richard P. Feynman

FromHow To Solve It by G. Polya (p.l):

One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles.

The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help or with insufficient help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.

If the student is not able to do much, the teacher should leave him at least some illusion of independent work. In order to do so, the teacher should help the student discreetly, unobtrusively.

The best is, however, to help the student naturally. The teacher should put himself in the student’s place, he should see the student’s case, he should try to understand what is going on in the student’s mind, and ask a question or indicate a step that could have occurred to the student himself.



From "Fuzzy Thinking" by Bart Kosko (p.86):

Inaccuracy pervades science. The goal of science is to remove as much inaccuracy of description as possible, as much as experimental error and good guesswork and physical "laws" permit.

Scientific claims or statements are inexact and provisional. They depend on dozens of simplifying assumptions and on a particular choice of words and symbols and on "all other things being equal." There are just too many molecules involved in a "fact" for a declarative sentence to cover them all. When you speak, you simplify. And when you simplify, you lie.



The following is from “Mathematics and Plausible Reasoning, vol.1 by G. Polya (Preface, p.vi). If the word “electronics” is substituted for “mathematics,” the word “design” for “theorem,” and the word “analysis” for “proof,” the same statement can be applied to electrical engineering.

Mathematics is regarded as a demonstrative science. Yet this is only one of its aspects. Finished mathematics presented in finished form appears as purely demonstrative, consisting of proofs only. Yet mathematics in the making resembles any other human knowledge in the making. You have to guess a mathematical theorem before you prove it; you have to guess the idea of the proof before you carry through the details. You have to combine observations and follow analogies; you have to try and try again. The result of the mathematician’s creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference.

As an example of guessing a theorem, Polya cites the fact that all prime numbers (except 2) are odd; therefore the sum of any two primes is even. This suggests the opposite question: Can every even number be expressed as the sum of two primes? Examination of many even numbers led Goldbach to guess that it is true. The proof of the conjectured theorem is a different process, but it usually involves guessing too. (It is still not proved or disproved today.)


From “The Process of Education by Jerome Bruner (pp 64-65):

There may, indeed, be a kind of guessing that requires careful cultivation. Yet, in many classes in school, guessing is heavily penalized and is associated somehow with laziness. Certainly one would not like to educate students to do nothing but guess, for guessing should always be followed up by as much verification and confirmation as necessary; but too stringent a penalty on guessing may restrain thinking of any sort and keep it plodding rather than permitting it to make occasional leaps. May it not be better for students to guess than to be struck dumb when they cannot immediately give the right answer?.. 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.


From "Guess How to Teach...Teach How to Guess" by Dan Wolaver

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.


From the TECH OLD-TIMERS newsletter of February 21, 1986:

I will not join the world in praising the computer. It takes courage to stand against the crowd, but I know that the computer doesn’t need any support from me, and none will be given, because its reputation far exceeds its true worth. It is fast at computation—period. A computer study of a global model of climatic variations, for instance, is only as good as the quality of program and accuracy of the data input, both of which will always be subject to some question. Yet, many tasks can be performed by the human brain that computers may never learn to do. The challenge seems to be to further the improvement of the computer, to close the gap, whereas the proper course is obviously to widen the gap, by improving the human brain in that field of action where its awesome power often astounds the world’s greatest mathematicians—accurate guesswork. Although all brains are capable of guesswork, very few produce accurate guesses. The accurate guessers are so few because grammar school teachers have discouraged the practice, instead of encouraging it, based on the erroneous notion that it is better to “figure it out.” The Scriptures tell us that “man fell into sin”—which was, through a faulty translation, a crude oversimplification of a rather long passage which stated that man lost his ability to guess accurately. His survival threatened, he was forced to invent such aids as arithmetic (to permit him to quantify), the compass, (to show him the way home), and so on. Before he had learned to count and quantify, “enough was enough”—very simple. Now, no amount is enough. Sadly, man, even with the help of his computers, will never know the whole story until he redevelops that ancient human brain function—accurate guesswork. And I can guess (quite accurately) what you’re thinking of the guy who wrote all this.

M. Leonard Kuniholm W.P.I. ‘38




A large part of the students’ learning takes place in lab. In fact, lab work accounts for from a third to a half of the student’s grade in a course. Lab work provides the following:

The student in a lab desperately needs help to make the most of this valuable, limited time. The most common mistake of a lab instructor is to sit and wait for someone to raise his hand for help. The student usually needs help whether he knows it or not. By strolling around and monitoring the progress of each student, the instructor can catch errors as they occur. Ridiculous results in a lab report are not so much an indictment of the student but of the instructor’s failure to catch the error in the lab.

The attitude should be one of interest: “Do you have the input impedance yet? Does it agree with what you expect?” Lack of this interest on the instructor’s part is almost always the result of not being familiar with the lab. It is essential for the instructor to have performed the lab himself ahead of time so he can anticipate the experiences of the student. The instructor should understand what theory is being reinforced by the lab, know what measurement difficulties are likely to occur, and be familiar with typical results.

The help most often needed by a student is with debugging his circuit. He needs to be taught to trace the signal through stage by stage and isolate the problem. Then the problem must be narrowed to the bad connection or the reversed diode in that stage. This final step in debugging requires that the student understand well how the circuit is supposed to work. Therefore some review of theory is usually necessary. Skill at debugging will eliminate the student’s fear and frustration with lab.

It is possible to help the student too much. If the instructor, in debugging a circuit, makes a few quick measurements and announces to the student where the problem is, he has deprived the student of a learning experience. Let the student make the measurements, and guide him with questions. Another example of too much help is in keeping the student from running into problems. After helping three benches all with the same problem, the tendency is to make a general announcement: “Make sure your scope is set to DC in part 3.” You won’t be bothered with that problem again, but only three benches will understand why the scope must be on DC. If you make the general announcement, “Be sure your supplies are bypassed,” the students will never appreciate that oscillation occurs if they don’t. Remember that learning is controlled agony.  In summary:




Grading a student’s exam, report, or homework is an evaluation of his progress. A good grade tells the student to keep doing whatever he has been doing. A poor grade tells him to change something–his goals, attitude, study habits, style, etc. And the poor grade should be accompanied with an indication of which of these things needs changing. In order to be effective in providing this feedback, the grading must be fair, quick, and specific.

Fair.  The grade should be an accurate measure of the student’s performance. First the grader must know what is important in the performance and what are the relative weightings of the items. This should be made clear by the author of the exam or assignment. It usually amounts to assigning points to the various items. Next the grader must be consistent in taking off points from paper to paper. This isn’t easy because the errors are seldom alike and are hard to compare.

Quick.  With a seven-week term it is especially important that feedback to the student be quick so he has a chance to change his behavior. A graded assignment should be returned to the student well before he submits the next assignment. Sometimes this is impossible, but there should never be a time when the student is waiting for two grades.

Specific.  To learn from his mistakes, a student needs to know specifically what he has done wrong. A common complaint from students is, “I only got 6 out of 10, but there are hardly any red marks. What did I lose points on?” Points deducted should be marked next to the wrong item, sometimes with an explanatory note.


If partial credit is given, then the number of points deducted depends on what type of error led to the wrong answer. Errors usually fall into one of the following categories.

For the sake of consistency, be clear in your mind what weighting you are giving to each type of error. This of course depends on the severity. Sometimes it is difficult to distinguish between two types such as a misunderstanding or a conceptual error.


Too little time spent grading is usually reflected in few marks or comments when there should be more.  If few marks accompany a low grade, the student has little useful feedback, and he usually complains.  Sometimes few marks accompany an undeserved high grade, indicating poor attention on the part of the grader. This leads to grade inflation (but no complaints from the student).

Time can be saved in grading by first going through five or six papers and marking errors without deducting points. Having seen what types of errors are being made, you can assign points to the types of errors more easily and consistently. Then go back and grade with point deductions.

Time can also be saved by establishing a time limit beyond which you won’t attempt to follow messy or poorly documented work. Simply deduct an appropriate number of points and make a note as to the reason.

Trying to grade with too high a resolution is a waste of time. Usually a resolution of 5% is fine enough (e.g., half a point out of ten). It is unrealistic to pretend to be able to assign weighting any more accurately than this. Sometimes many small errors (such as careless mistakes) can be lumped together to count as a half point or one point deducted.


If your discover two papers with very much the same approach, layout, and errors, show the papers to the professor in charge before assigning grades to them.


Despite your best attempts, you are bound to make mistakes in grading. Perhaps the student solved the problem in a different but correct way. Sometimes a problem has been poorly worded, leading to misunderstanding. It is easy to overlook an answer placed out of order or on the back of a paper. It may be that the student thinks you made a mistake when in fact you didn’t. In any case, the student needs to be able to contact the grader of his paper.  Therefore always put your initials on the paper, usually next to the grade at the top of the paper.


It is the grader’s responsibility to keep a record of the grades and to return graded papers to the student boxes.  Meet frequently with the professor in charge of the course to give him feedback as to where the students are making errors.