*Primus*, vol. 9 (December 1999): 339-254.

**ADDRESS:** Department of Mathematics and Computer Science, Emory University, Atlanta GA 30322 USA. wsmOmathcs.emory.edu.

**ABSTRACT**: We describe a "Moore Method" course which we have taught at several universities over the past 30 years. Our course is a modification of one we took from R.L. Moore in 1948. The course is intended to be taken immediately after a one variable calculus course. The purpose of the course is to teach the students to create and present in class mathematically correct proofs of theorems. This, of course, requires training in logic and correct use of language. Our topics are the topology of the line and basic principles of the calculus, including limits, continuity, and differentiability. Our goal is to get the students to prove the fundamental theorem of calculus. We discuss our grading, class discussions, the ways in which we help students, and the extent to which we encourage cooperative learning.

**KEYWORDS:** The Moore Method, discovery learning, linear topology, calculus.

### 1 BACKGROUND

There has been recent interest in what is called Discovery Learning and continuing interest in the Moore method. As a student of R.L. Moore's I have been applying my version of the Moore method in courses from college algebra to graduate topology for over 40 years. I will describe one of these courses in detail in this article. As a result of my attending several Discovery Learning Conferences at The University of Texas in Austin I am convinced that there are many misconceptions about the Moore method. Moore treated different students differently and his classes varied depending on the calibre of his students. The misconception that bothers me most is the idea that the method is a rigid one of providing students with theorems and definitions and expecting them to prove theorems without help. Moore helped his students a lot but did it in such a way that they did not feel that the help detracted from the satisfaction they received from having solved a problem. He was a master at saying the right thing to the right student at the right time. Most of us would not devote the time that Moore did to his classes. This is probably why so many people claim to have tried the Moore method without success.

In what follows I will describe a Moore method course which I taught for over 30 years at The Illinois Institute of Technology, The University of Tennessee, The University of Houston, and at Emory University. I have had enthusiastic students in this course who majored in English, Mathematics, Education and Philosophy. Students from this course have gone into various of Emory's undergraduate honors programs and into graduate programs in mathematics, law and physics. Many of my PhD students started in this course. My primary goal in the course was to get students started creating and presenting correct proofs and to convince them that doing so was fun. As a byproduct I hoped to develop their creative ability. This reflects my own training. My main motivation to work on problems in mathematics is that it is fun. Rather than tell the reader how to teach such a course, I will describe what I consider important in such a course, explain how I try to accomplish my goals and leave it to the reader to determine how to adjust the course to his students and personality.

I do not consider it important in using the Moore method whether one lectures a lot, a little or none at all. It is important that there be regular interaction with the students so that the instructor knows how well the students understand the material. It is also important that they be given challenging problems, be motivated to work on them and to want to report their progress. If one is to teach using the Moore method, one must adjust to new standards of "progress". Most of us try to progress linearly in a mathematics course. This is unreasonable when new concepts are being introduced which take time for a student to fully comprehend. One should expect progress to be very slow at first, but to improve as the students develop. And to slow down again with new concepts. This requires a lot of patience. One must also adjust to new standards of "covering material". One should not consider material covered until the students understand it well enough to apply it in new situations.

I also think the term Discovery Learning is misleading. The student is not likely at an early stage to discover enough on his own to satisfy us in our attempts to cover the syllabus. A better term might be cooperative learning with the instructor taking the lead. In the beginning I give some definitions and ask questions which result in a discussion in class about the meaning of the definitions. The students will ask questions of me and I will direct the questions to the class. There is a cooperative effort to understand the definition of open and closed intervals and of a limit point. I will discuss this more fully in section 2. In the beginning the students help each other, at least in class, in their attempts to understand the concepts being introduced. I do not make any statement at this point either encouraging or discouraging their working together. Moore would likely not have approved this. I often give problems directed at a particular student for various reasons. Perhaps because the student needs help understanding some concept, or perhaps because I think the student has shown insight into some concept or problem and if provided the right incentive or question might actually make some significant discovery with minimal help. Moore certainly did a lot of this. This is not possible without having a good feeling for the student's capabilities and understanding of the concepts.

This article was originally motivated by Roger H. Marty's description of a Moore method course (see [4,5,6]) entitled *Introduction to Mathematical Reasoning* taught at Cleveland State University. His course was intended to develop students' reasoning abilities and to help them learn how to read and write proofs as a prerequisite to their taking a course covering material as found in Bartle's book [1]. My course was modeled after courses I took from R.L. Moore as an undergraduate in the 40's and 50's. I took his analytic geometry, solid analytic geometry, calculus, and advanced calculus courses. I would encourage anyone interested in this type of course to view the MAA video *Challenge in the Classroom* about Moore and his teaching [7].

### 2 THE FIRST FEW DAYS

At the first class meeting I give a brief description of the course indicating that my goal is for them to completely understand some of the basic concepts of the calculus such as limits, continuous functions, derivatives and integrals. I also give a brief description of how I will conduct the course, indicating that I will lecture very little and that there will be a lot of class discussions and student presentations. I explain my grading system which is discussed in Section 4. I indicate that we will be studying properties of numbers, and that we will assume the usual rules about inequalities. I do not usually give a list of axioms to be used, but some properties of numbers will give the students problems and these deserve special treatment. As a result I often state the following as axioms:**AXIOM 1** If *x* and *y* are any two *different* points, there is a point between them,

**AXIOM 2** If *x* is a number, then *x* is an integer or there is an integer *n* such that *x* is between *n* and *n+l*.

We begin with a definition of an open interval and a closed interval. Many students think of a line as a string of points very close together. As a consequence when I define an open interval (*a, b*) as the set of all points between *a* and *b* and not including *a* and *b*, they are likely to ask if this isn't the same as the closed interval [*c, d*] where *c* is the first point to the right of *a* and *d* is the point "just before *b*". At this point we will refer to Axiom 1 which they will agree is a correct statement about numbers and they will admit that there is not a first point after *a*. They are slow to quit making this mistake. I do not bring up this point, but if a student does then it is well worth the class time to discuss in detail the difference between open and closed intervals as sets such that one of them has a last point and the other does not. This is good preparation for a completeness axiom.

One of the problems that students have at this stage is that they will be reading complicated definitions for mathematical objects for which they have no model. The teacher's role is to help the students use the definitions to build models. I have found that students have confidence in their model of the number line so I define an open interval as the set of all points between two points rather than as the set of all numbers *x* such that *a* < *x* < *b*. I have a preference for geometry and another teacher might be more successful with an analytic start. I use the terms point and number interchangeably and refer to order on the number line as left to right. After defining an open interval I define a *limit point p of a point set M* as a point *p* such that every open interval containing *p* contains a point of *M* different from *p*. Then we consider questions requiring use of the definition, for example:

ifMis the point set which contains only the number 0, doesMhave a limit point?

Someone usually provides a proof that 0 is not a limit point of *M* and they may think they are finished because they will assume that if *p* is a limit point of *M*, then *p* must be a point of *M*. I do not fault the student for this "mistake". After all if we have called *p* a limit point of *M*, should it not also be a point of *M*? Why should the student not assume that we mean to define a special kind of point of *M* called a limit *point-of-M*? The longer I teach the better I understand why Moore was so very careful in his use of the English language. I am likely to take the blame and explain that perhaps limit point of *M* is a misleading term and to say that I meant for them to assume nothing about *p* except that each open interval containing *p* contains a point of *M* different from *p*. This is a good time to explain that when they are given a definition, they should not be influenced by the name attached to whatever is being defined. And that they should assume nothing that is not explicitly given by the definition. Then I ask if the number 1 is a limit point of M. Finally we get a proof that no point is a limit point of M by considering where a limit point *p* might be: *p* < 0, *p* = 0 or *p* > 0. All this is likely to take place at the first class meeting, and what I state next depends on what I learn about the students during that meeting. Examples of problems that might come next are:

Show that ifMis a point set that contains only two points, thenMhas no limit point.

Show that every point of every closed intervalIis a limit point ofI.

Show that for every open intervalO, the end points ofOare limit points ofOwhich do not belong toO.

Instead of the previous problem, I may ask for an example of a point set *M* which has a limit point *p* which is not in *M*.

A more challenging and instructive early problem is:

Is it true that for every point setM, and every limit pointpofM, every closed interval containingpcontains a point ofMdifferent fromp?

This is an excellent problem, and students are likely to argue that it is true because every closed interval contains an open interval. It will take a very good student to realize that a closed interval containing *p* might have *p* as one of its endpoints and thus not contain an open interval containing *p*. This again will emphasize the difference between open and closed intervals. This also provides an opportunity to discuss what must be done to show that a statement is false. It also can lead to a discussion of points which are limit points of a set from both sides. Examples like this help the students understand the definitions. However one begins it is important for the teacher to recognize that the student is very likely to have problems with the language. Especially, as R.L. Moore frequently said, with such difficult
to use words as *or*, *and*, *each* or *every*. I try not to use "math-lingo" as it is called in Hersh's excellent article [3]. I recommend this article to anyone interested in teaching mathematics. In the beginning I carefully distinguish between the inclusive and exclusive *or*. As an example I may state a theorem by saying:

Every limit pointThe students are not likely to think of an inclusivepof the point setMis a limit point of points ofMto the right ofpor a limit point of points ofMto the left ofp, possibly both.

*or*unless it is emphasized. It is a big help for the instructor to anticipate some of the problems the students will have with the language.

### 3 WHAT NEXT?

Once I am convinced the students have a good intuitive understanding of a limit point, I define *closed point sets* and state problems which involve this notion. One problem which Hersh [3] mentions will frequently occur. If we tell our students to prove that if *M* is a closed and bounded point set, then *M* has a rightmost point, we should not be surprised if a student goes to the board and says that if *M* is the closed interval *[a, b]*, then *M* is a closed point set and it has a rightmost point, namely *b*. If this happens I am likely to again take the blame from the student by saying that I did not make the problem clear. I explain that I meant for them to show that every closed and bounded point set has a rightmost point. It depends on the class as to whether I state the problem this way in the first place. I may also point out that the student has given an example of a set with a rightmost point and I may ask if someone can give an example of a set which does not have a rightmost point. Then I can point out that this shows that the assumption that *M* is closed or bounded is necessary to the proof. I may then ask if someone can give an example of a set which is not closed but which does have a rightmost point. If some student provides an example then we will discuss converses. My point here is that I let the students determine what is discussed in class by what they contribute. These may seem like minor points but they make a big difference in building the students' confidence and more importantly, convincing them that they can do these problems and that their contributions in class are an important part of the course. I find it will require 6-10 meetings depending on the class before the problems with language, logic and use of definitions are solved so that we can continue with new concepts.

The next important definition I give is that of the *limit of a sequence*. I define a *sequence* as a function whose domain is the natural numbers. I define the *limit of a sequence S* as a point *p* such that if *O* is any open interval containing *p*, then there are at most a finite number of integers *n* such that *S(n)* is not in *O*. I often call *p* a s*equential limit* to emphasize that *p* is associated with a sequence and not a point set. We discuss the facts that a sequence is not a point set, that point sets may have limit points but they do not have sequential limits and that sequences do not have limit points. Then we will investigate relationships between these two concepts, building on the students' understanding of a limit point. Our emphasis now has shifted from understanding basic concepts to developing their ability to discover or prove theorems. Some examples of the kind of problems I give are:

To provide help for this theorem I am likely to first ask them to prove that ifTHEOREMNo sequence has two sequential limits.

*p*and

*q*are two (different) points, then there are two disjoint open intervals, one containing

*p*and the other containing

*q*.

PROBLEMIf the range of a sequence has only one limit pointpmustpbe the sequential limit of the sequence?

THEOREMIfSis a sequence of distinct points which has a sequential limitp, thenpis a limit point of the range ofS.

A more challenging theorem is:

In a very good class I may state these theorems as problems. I soon state a completeness axiom and give them the following theorem.THEOREMIfSis a sequence which has a sequential limitp, then no point other thanpis a limit point of the range of the sequence.

What completeness axiom I state depends on the class. As we progress through the next 10-12 meetings I get a quite good feeling for the abilities of the various students in my class. I have managed class sizes up to about 20 but anything over 15 is a real challenge. Usually by about the 20th meeting we are ready to define a continuous function. At this point the rate at which we cover material is surprisingly fast.THEOREMIfSis a bounded increasing sequence, thenShas a sequential limit.

During the very slow start it is difficult to believe that after about 30-40 meetings they will all be giving correct solutions *of their own creation* for problems which might well be exercises in Bartle's book [1] and the better ones will give proofs which they have discovered for the intermediate value theorem, the Bolzano-Weierstrass theorem, and for Rolle's theorem. Of course this will be with the help of lemmas which I will have stated. As an example, after we have shown that every infinite bounded point set has a limit point, I will often state the following theorem:

Then I will state the following theorem as another step in the direction of proving Rolle's theorem.THEOREMIffis a function defined on a closed interval, and for each positive integern,x, is a number such that_{n}f(x) =_{n}n, then there is a point at whichfis not continuous.

Before stating this theorem we will have established the intermediate value theorem. How many lemmas I give will depend on how good the students are.THEOREMIffis a continuous function defined on a closed interval, thenfis bounded.

My ultimate goal is to get them to prove the fundamental theorem of calculus but I find the definition of an integral slows them down and we rarely make it by the end of my course which lasts about 45 meetings. On the other hand I am confident that at this point they could read and understand a proof. Indeed, I have been asked in more than one class to show them a proof of the fundamental theorem of calculus and I have done so. I have been pleased at the questions I get if I try to omit details, and at their comprehension of the details of the proof.

### 4 GRADING

Grading for a course like this is *very difficult*. Of all the schemes I have tried, I describe the one I dislike least in this section. I require the students to select, write up, and turn in one problem every 2nd or 3rd meeting. It must be written up using complete English sentences and correct grammar and it must be one they have not turned in before. The feedback from these assignments is extremely helpful to both instructor and student. I assign grades as follows: An "A" means they have written a grammatically and mathematically correct proof. A "B" or "C" means that there are logical or grammatical errors in their writeup. A "B" means that in spite of their errors I think they understand how to prove the theorem. A "C" means that I have doubts that they can give a correct proof of the theorem without help. I may give hints on their paper when I return it. A "D" or "F" means that they do not understand the proof and need help. They are free to discuss the problems with other students, but they are to write up their assignments alone. For any grade lower than a "B+" they may resubmit the problem to raise their grade. I count only the best grade. At first I encourage them to write up proofs that have been discussed in class and that they think they completely understand. Later when I know the students better, I suggest for individual students problems that I think will help them with their writing or their mathematical understanding. In his undergraduate courses Moore did this sort of thing. He was a master at understanding what students could do and giving them problems which were difficult enough that one took considerable pride in solving them, but not so hard as to discourage an immature student. I think this is the heart of the Moore method.

There is a 50 minute midterm and a 2½ hour final exam. On these exams they are asked to write proofs of theorems that are similar to ones that they have seen in class. Often I will give special cases of theorems that have been done in class. Even minor changes in the wording of a problem will foil the rare student who tries to memorize proofs. Any attempt at memorization is clear to both the student and me if they write up a long proof for a major theorem when all they were asked was to prove a special case where the proof is short and much simpler. On the other hand should not one be pleased if the student realized that the problem was a special case of a difficult theorem which they knew how to prove? Isn't it true that many of us would be quite happy if our students could write up a grammatically and logically correct proof of, for example, the intermediate value theorem? Indeed, we would likely be pleased if the student could do this even if the proof were memorized. Perhaps the real difficulty with grading is deciding what we want to measure. It is very difficult to make up a reasonable timed exam in a course like this, and I have never been completely satisfied with my exams. I would welcome suggestions from any of my readers.

The students are also required to present proofs of new theorems and problems in class. In the beginning I ask for volunteers, but this usually involves a few students doing most of the work so I assign problems to the students rotating through the class roll, often assigning problems based on my knowledge of the students. When a student is due to present a problem at the board, if they do not have it, they may go to the board to present an idea they have worked on and request help or they may request another problem. They are encouraged to come by my office for help. As Polya says [8] "The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work." From comments made by students at their seats and from presentations at the board, I assign a highly subjective class performance grade. If a student gives up on his problem I may assign it to someone else, ask for volunteers, or just assume it and go on. The final grade is based on all their grades on homework, exams and class participation. Some students are very creative but do a poor job of presenting their proofs while others rarely prove a difficult theorem but do an excellent job of writing up proofs of other students *in their own words*. These students often use verbiage which is quite different from what was presented in class. For this and other reasons, when making up the final grades I often give a higher weight to the best of their three grades than to the others. I tell the students that the final grades are subjective but will not be lower that the average of their three grades.

### 5 HELPING THE STUDENTS

I think the most important single thing one must accomplish in class is to develop good rapport with the students. I work hard to convince them that they can do these problems and that I am willing to spend a lot of time helping them complete a proof if I am convinced they have spent a reasonable amount of time working on it. I encourage students at their seats to ask questions about anything they do not understand and to point out any mistake they think they see. I find it difficult to decide whether to allow students to help someone at the board who seems to be stuck. Again this will depend on the student. Some students welcome help and others would rather continue to work on their own so I make judgement calls based on what I know about the student. I discourage criticism on the grounds that a proof is unnecessarily long. I will tell them that a longer proof often will explain better why a theorem is true, and may be easier to understand than a short one. I may also point out that some proofs which appear shorter are that way only because they make use of other theorems whose proofs are not short. R.L. Moore made this point to me when I was a student. The main point is for them to understand that I will be pleased to see any correct proof of their own creation. My experience is that the students will work very hard on a problem which they consider "their problem" for some reason, and they do not discuss it with other students because they want the satisfaction of knowing they got it on their own. Moore was very skillful at helping students without their realizing how much help thev got. In the rest of this section I will give examples of how I treat different students at the board or in my office.

One weak but hard working student announced in class that he wanted help in proving that if *p*_{1}, *p*_{2}, *p*_{3},... is a Cauchy sequence and the range of the sequence is finite then the sequence converges. I suggested that he assume that the range of the sequence contained exactly two points and sent him to the board to mark two points on a number line. Then I asked him how he thought he should start if he was going to use the definition of a Cauchy sequence. He replied with a somewhat puzzled tone, "A positive number E?" I responded positively and suggested that he look at his picture on the board and try to decide what to pick for E. He asked if he could let it be the distance between his two points I said "Of course!". He then used the definition to show that there is a positive integer *n* such that if *m* is a positive integer and *m* > *n* then *p*_{m} = *p*_{n} so that the sequence converges to *p*_{n}. I said he had done a nice job. At this point only a few of the students understood what he had done. The other students asked a lot of questions which he answered very well. Next I suggested he assume the range of the sequence contained only three points, but he could not decide how to continue. I suggested he think about it after class. Shortly after class he came in my office and asked if he could label the points *a*, *b* and *c *in order so that *a* < *b* < *c* and then just pick the smallest of *b* - *a* and *c* - *b* for E. When I said "Of course" and smiled he said: "If I have a finite set then I can just take the smallest distance between two consecutive ones of them and my proof will work." He had learned the importance of looking at special cases and his work improved in the class. I am reminded of the dictum attributed to Polya to the effect that if you can't solve a problem there is a simpler one you can't solve and you should find it.

As another example, I had a good and hard working student, who was trying to prove that a nested sequence of closed intervals has a common point, ask me for help after class saying she had no idea how to start the problem. I expressed surprise that she needed help and said "Have you considered looking at the endpoints?" She looked surprised and said "Of course, they have to converge." She had a nice proof the following class meeting. I wondered if I should have given her any help at all.

For a third example, I had an excellent student come by my office wanting help in showing that if *G* is a collection of open intervals covering the closed interval *I*, then some finite subset of *G* covers *I*. She explained that she had begun by defining a set *H* to be the set of all points *x* (UNION) [*a, b*] such that the interval [*a,x*] was covered by some finite subset of *G* but she did not know where to go from there. I told her again and again that I did not think she needed my help. She left my office mad and exasperated. The next day in class I asked her: "Are you still mad at me? And even if you are, can you present your theorem?" She snarled "I think so!" and went to the board where she presented a nice proof, showing that the least upper bound of her set was *b*. I made no comment until she asked me after finishing her proof if it was all right. I said it was a beautiful and elegant proof which got a slight smile from her, and an obvious sense of pride and satisfaction. It is quite rewarding to see this sort of thing happen.

### 6 UNDERSTANDING DEFINITIONS

When a definition is introduced it is important to have problems which will help the students understand the definition. For a completeness axiom I usually use the following:IfLater I will call this number the least upper bound ofMis (any) pointset and there is a number larger than every number inM, then there is either a smallest such number, or a largest number inM.

*M*. To help them understand the definition I ask them to tell me what happens if

*M*is a closed interval or an open interval or the set of all negative numbers. I ask them to prove that both cases cannot occur. And that if the least upper bound of a set

*M*is not a member of

*M*, then it must be a limit point of

*M*.

For continuity I usually use the following which is much like that used by R.L. Moore.

The function *f* is continuous at the point (*x, f(x)*) means that:

if and are two horizontal lines with (x, f(x)) between them, then there are two vertical lineshandkwith (x, f(x)) between them such that if (t, f(t)) is a point offbetweenhandk, then (t, f(t)) is between and .

Sometimes I use the following variation of Moore's definition.

If (a, b) is an open interval containing:f(x), there is an open interval.(h, k) containingxsuch that iftis a number in (h, k) and in the domain off, thenf(t)is in (a, b).

Whichever one I use I am likely to state both and have them prove that they are equivalent. Among the questions I would ask them to work on to help them understand the definition would be:

A more demanding question is:

- If
fis the function whose domain and range is the set whose only member is the number O, isfcontinuous at (0, 0)?- Show that if
f(x)= l/xfor each numberx> O, thenfis continuous at (1, 1).- Show that if
fis a function such that if a x 0,f(x)= 1 andf(0)=0, thenfis not continuous at (0,0).

IfThe way the question is phrased is important. Stated this way the students are likely to think this is true and try to prove it. When we discuss their error it helps the class see the ð in the usual definition depends not only on E, but on the point of continuity as well. It will take an exceptional student to find a counterexample if the question is asked this way. Any student who did so I would consider a good candidate for graduate work in mathematics. If I want to encourage them to seek a counterexample I would ask if there is a functionfis a function with domain the interval [-1,1], andfis continuous at (0,f(0)), is there an open interval (h, k) containing 0 such that ifx(h, k),fis continuous at (x,f(x))?

*f*with domain and range the closed interval [0, 1], which is continuous at (0,0) but not continuous at (l/

*n*, f(l/

*n*)) for any positive integer

*n*.

My definition of a derivative is again a modification of the one used by Moore [7]:

We assume that *f* is a function, that *x* is a number in the domain of *f* and that *x* is a limit point of the domain of *f*.

The statement that the numberFor the definition of a derivative, good problems are:mis the derivative of the functionfat the point (x,f(x)) means that if (a, b) is an open interval containingm, there is an open interval (h, k) containingxsuch that iftis a number in the domain offand in (h, k) andt x, then the slope of the line joining (t, f(t)) and (x, f(x)) is in (a, b).

- Show that if
fis a function with domain [-1, 1], and for each numberx[-1, 1], -x^{2}<f(x)<x^{2}, thenf'(0) = 0.- Show that a function cannot have two derivatives at a number
x.

I have a sequence of definitions, theorems, and problems which I use in the course but I do not use the same sequence every year. Problems are given in class, staying several meetings ahead of class presentations. The problems I assign depend on how much trouble the students are having at the time. A good class may get only half as many problems as a weak class, where lots of lemmas are needed. Any reader interested in seeing my list of theorems should feel free to write or email me.

### 7 PERSONAL OBSERVATIONS

Over the years I have seen many people try to teach a course like this without success. Much of the time the problem seems to be a lack of patience. Not only with the pace of the course in the beginning but with the inelegant proofs that will be presented. It is very difficult to resist the temptation to interrupt a student attempting to prove a theorem in an awkward manner and show them a "better" way. This is sure to dampen a student's enthusiasm. To succeed one must believe that the students' proofs will improve in elegance, and that the pace at which material will be covered will increase exponentially. But it will be slow at first and one must be very careful about any criticism of the student's work.

Another problem is the instructor's feeling that not enough material is being covered. As many times as I have taught this way I still have this problem. I must remind myself of my goals and have confidence that this new class also will "catch on" and begin to prove some nice theorems and show some creative originality. The teacher must make a decision as to what is most important in any given course. Is it more important that the student acquire a large body of knowledge, or that he be able to solve a large number of different problems with the knowledge he does acquire? The relative importance of these goals surely depends on the course. I think that it is much easier for students to learn much of the knowledge we try to impart by reading or attending our lectures. I think it is hard, if not impossible, for most students to learn how to speak correctly and how to prove theorems without actually being given the opportunity to do so themselves. And they will need a lot of help from a patient teacher.

Except for introducing new concepts I do no lecturing until there comes a day when no student is prepared to present anything. I always have a lecture prepared for when this happens that will shed some light on where we are heading or hint at some new method of proof they have not yet seen.

There is an advantage to emphasizing acquired knowledge in that it provides a ready source of testable material. In my opinion we often leave the student with only a superficial knowledge of the material and with little training in thinking and with little enthusiasm for mathematics. In a recent book review [10], J.A. Yorke and M.D. Hartl do an excellent job of addressing some of the problems with U.S. teaching in mathematics. They reference an article in *Science News* [9] in which it is reported that East Asian students lead U. S. students in mathematics and that in the U. S. we spend much more time practicing routine problems and we teach many more topics than do the Japanese. In the article Gerald Wheeler of the National Science Teachers Association is quoted as saying: "This study shows we need fewer topics and more depth." I heartily agree.

Whatever else is true about my course, the students leave it with a genuine enthusiasm for mathematics and they know that they are capable of proving and understanding mathematical theorems. They become genuinely excited near the end of the course when they realize that they are going to be asked to prove the fundamental theorem of calculus. It is true that the number of algorithms "covered" in my course is less than that in most courses. I consider this a fair trade for leaving the students with a lasting and deep understanding of the concepts I do cover.

### REFERENCES

- Bartle, R.G. 1976.
*The elements of real analysis, 2nd ed.*New York: Wiley. - Halmos, P.R. 1975. The Teaching of Problem Solving.
*The American Mathematical Monthly.*82(5): 466-470. - Hersh, Reuben. 1997. Math Lingo vs. Plain English: Double Entendre.
*The American Mathematical Monthly.*104(1): 48-51. - Marty, R.H. 1986. Teaching proof techniques.
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### BIOGRAPHICAL SKETCH

The author graduated from The University of Texas at Austin with a BS degree in physics in 1951. He worked at the United States Air Missile Test Center at Point Mugu, California for a year and then returned to The University of Texas to pursue graduate work in physics and mathematics and work at the defense Research Laboratory of The University of Texas. He received his PhD in mathematics under the supervision of R.L. Moore in 1957. He has taught at The University of Texas, Illinois Institute of Technology, the University of Tennessee, the University of Houston, North Texas State University, and is currently Professor of Mathematics and Computer Science at Emery University. He has had 8 PhD students, various MS and undergraduate honors students, and has 12 research publications.* The author would like to express his appreciation to Lida K. Barrett who made many helpful comments about this paper and encouraged the author to publish it.