Celebratio Mathematica

While one can­not say that the “Moore Meth­od” of teach­ing math­em­at­ics has gained wide-spread ac­cept­ance in col­lege and uni­versity circles, it has been and is be­ing suc­cess­fully used by enough people to at­tract at­ten­tion — even out­side aca­demia. Fur­ther­more, there is con­sid­er­able in­terest and curi­os­ity about what it is and how it works. Mainly, for this reas­on, I am go­ing to de­scribe my own ex­per­i­ences in Moore’s classes and in us­ing the meth­od with my own stu­dents, both gradu­ate and un­der­gradu­ate.

While he was still a gradu­ate stu­dent at the Uni­versity of Chica­go (1903–05), R. L. Moore con­ceived the ba­sic ideas that led even­tu­ally to his rather rad­ic­al meth­od of teach­ing. With his quick mind and rest­less spir­it he found the lec­ture meth­od rather bor­ing — in fact, mind dulling. To liven up a lec­ture he would run a race with his pro­fess­or by see­ing if he could dis­cov­er the proof of an an­nounced the­or­em be­fore the lec­turer had fin­ished his present­a­tion. Quite fre­quently he won the race. But in any case, he felt that he was bet­ter off from hav­ing made the at­tempt. So if one could get stu­dents to prove the­or­ems for them­selves, not only would they have a deep­er and longer last­ing un­der­stand­ing, but some­how their abil­ity and in­terest would be strengthened. If the the­or­ems were too dif­fi­cult, then they would have to be broken down in­to easi­er lem­mas.

The more he thought about hav­ing the stu­dents dis­cov­er for them­selves the main­stream of a sub­ject, the more he be­came con­vinced that it not only could be made to work but that it would also be at­tract­ive to stu­dents. He spoke of this plan to one of his pro­fess­ors (pos­sibly Veblen) who ab­ruptly replied: “Ha, let the stu­dents do the work!” But E. H. Moore’s re­ac­tion was more thought­ful — per­haps the ap­proach did have mer­it.

As a be­gin­ning in­struct­or it was not easy for Moore to find a de­part­ment where he had suf­fi­cient free­dom to give the idea a really good try. But when he did, he began (at The Uni­versity of Pennsylvania) to have suc­cess, es­pe­cially in the Found­a­tions of Geo­metry. Here was a fresh, re­l­at­ively new area where Moore had him­self tested the dif­fi­culty of some of the the­or­ems. In the years fol­low­ing his ap­point­ment at The Uni­versity of Texas, he ex­pan­ded the use of his lim­ited lec­ture meth­od to all of his classes: Cal­cu­lus, Ad­vanced Cal­cu­lus, Meas­ure The­ory, Met­ric Dens­ity, etc., as well as Found­a­tions of Geo­metry and (at the gradu­ate level) To­po­logy. The res­ults were, from the stand­point of re­search pro­ductiv­ity after the Ph.D., really phe­nom­en­al. Of the Ph.D’s pro­duced in the U.S. and Canada dur­ing the peri­od 1915–1954, 25% of those from Texas were among the top 15% in the na­tion in pro­ductiv­ity; 5% of those from U.C. (Berke­ley) were in the top 15%; 8% of those from Chica­go were in the top 15%; 16% of those from Har­vard were in the top 15%; 20% of those from Prin­ceton were in the top 15%. (A Sur­vey of Re­search Po­ten­tial and Train­ing in the Math­em­at­ic­al Sci­ences, The Uni­versity of Chica­go, March 15, 1957; The Al­bert Re­port.) And it should be no sur­prise that high qual­ity was as­so­ci­ated with high pro­ductiv­ity. In­cluded in this group from Texas are Moore’s stu­dents R. L. Wilder, G. T. Why­burn, J. H. Roberts, R. H. Bing, E. E. Moise, R. D. An­der­son, and Mary El­len Rud­in (in chro­no­lo­gic­al or­der).

Moore would be­gin his gradu­ate course in to­po­logy by care­fully se­lect­ing the mem­bers of the class. If a stu­dent had already stud­ied to­po­logy else­where or had read too much, he would ex­clude him (in some cases, he would run a sep­ar­ate class for such stu­dents). The idea was to have a class as ho­mo­gen­eously ig­nor­ant (to­po­lo­gic­ally) as pos­sible. He would usu­ally cau­tion the group not to read to­po­logy but simply to use their own abil­ity. Plainly he wanted the com­pet­i­tion to be as fair as pos­sible, for com­pet­i­tion was one of the driv­ing forces. (For the found­a­tions of geo­metry he made no at­tempt to se­lect the stu­dents be­cause all of them, young and old, high school teach­ers or not, were uni­formly ig­nor­ant of the Hil­bert–Veblen–Moore ax­io­mat­ic ap­proach to the sub­ject.)

Hav­ing se­lec­ted the class he would tell them briefly his view of the ax­io­mat­ic meth­od: there were cer­tain un­defined terms (e.g., “point” and “re­gion”) which had mean­ing re­stric­ted (or con­trolled) by the ax­ioms (e.g., a re­gion is a point set). He would then state the ax­ioms that the class was to start with (Ax­ioms 0 and 1 of his book: Found­a­tions of Point Set The­ory, omit­ting part (4) of Ax­iom 1). An ex­ample or two of situ­ations where the ax­ioms could be said to ap­ply (e.g., the plane or Hil­bert space) would be giv­en. He would some­times give a dif­fer­ent defin­i­tion of re­gion for a fa­mil­i­ar space (e.g., Eu­c­lidean 3-space) to give some in­tu­it­ive feel­ing for the mean­ing of an “un­defined term” in the ax­io­mat­ic sys­tem. Of course, this was part of his own per­son­al philo­sophy and he con­sidered it part of the mo­tiv­a­tion of the sub­ject.

After stat­ing the ax­ioms and giv­ing mo­tiv­at­ing ex­amples to il­lus­trate their mean­ing he would then state some defin­i­tions and the­or­ems. He simply read them from his book as the stu­dents copied them down. He would then in­struct the class to find proofs of their own and also to con­struct ex­amples to show that the hy­po­theses of the the­or­ems could not be weakened, omit­ted, or par­tially omit­ted.

When the class re­turned for the next meet­ing he would call on some stu­dent to prove The­or­em 1. After he be­came fa­mil­i­ar with the abil­it­ies of the class mem­bers, he would call on them in re­verse or­der and in this way give the more un­suc­cess­ful stu­dents first chance when they did get a proof. He wasn’t in­flex­ible in this pro­ced­ure but it was clear that he pre­ferred it.

When a stu­dent stated that he could prove The­or­em x, he was asked to go to the black­board and present his proof. The oth­er stu­dents, es­pe­cially those who hadn’t been able to dis­cov­er a proof, would make sure that the proof presen­ted was cor­rect and con­vin­cing. Moore sternly pre­ven­ted heck­ling. This was sel­dom ne­ces­sary be­cause the whole at­mo­sphere was one of a ser­i­ous com­munity ef­fort to un­der­stand the ar­gu­ment.

When a flaw ap­peared in a “proof” every­one would pa­tiently wait for the stu­dent at the board to “patch it up.” If he could not, he would sit down. Moore would then ask the next stu­dent to try or if he thought the dif­fi­culty en­countered was suf­fi­ciently in­ter­est­ing, he would save that the­or­em un­til next time and go on to the next un­proved the­or­em (start­ing again at the bot­tom of the class).

Oc­ca­sion­ally the­or­ems got left over in­def­in­itely but nearly all of these would be proved in some sub­sequent year.

Quite fre­quently when a flaw would ap­pear in a proof every­one would spend some time (pos­sibly in class) try­ing to get an ex­ample to show that it couldn’t be “patched up,” i.e., a counter­example to the ar­gu­ment (even though the the­or­em might be cor­rect). This kind of ex­per­i­ence is sel­dom en­countered in courses or in any place out­side of one’s own re­search work. Yet this kind of activ­ity is vi­tally ne­ces­sary for the re­search work­er.

Oc­ca­sion­ally an im­prove­ment on one of Moore’s the­or­ems would be proved. Moore would then refer to that the­or­em with the stu­dent’s name. (In the re­vi­sion of his book a num­ber of names like this ap­pear in the text and in some cases he made re­marks con­cern­ing the ori­gin of cer­tain proofs and con­cepts in the ap­pendix.) The im­prove­ment might not be very sig­ni­fic­ant but the en­cour­age­ment giv­en by the “pub­lic re­cog­ni­tion” was con­sid­er­able. At the same time, mis­takes were not dis­cour­aged. However, on cer­tain the­or­ems (the ar­c­wise con­nectiv­ity the­or­em, for ex­ample) know­ing from ex­per­i­ence that mis­takes were likely, Moore would in­sist that the stu­dent have the proof writ­ten out be­fore present­ing it. This ten­ded to re­duce the num­ber of false starts (and also gave the stu­dent some use­ful writ­ing prac­tice.)

Prob­ably the most ob­vi­ous dif­fi­culty is the one of class uni­form­ity. If the com­pet­i­tion is not reas­on­ably on an equal foot­ing, then one of the ba­sic drives has been weakened. Ob­vi­ously, in con­trast to the 1920s, little can be done about this now when al­most all un­der­gradu­ates have had some ex­pos­ure to to­po­logy and to­po­lo­gic­al ideas. G. T. Why­burn spoke to me more than once about how to handle this prob­lem. My solu­tion (or par­tial solu­tion) is to be­gin gen­er­al to­po­logy with only the most ba­sic ax­iom: there ex­ists a non-empty set \( S \) of ele­ments called points and a col­lec­tion \( G \) of sub­sets of \( S \) such that \( G \) cov­ers \( S \). Call­ing the ele­ments of \( G \) “re­gions” and us­ing the usu­al defin­i­tions of “lim­it point,” “closed,” and “open,” some of the usu­al pro­pos­i­tions about these no­tions are false but when care­lessly for­mu­lated the usu­al ar­gu­ments seem to work. This forces a kind of uni­form­ity on the class. (I con­tin­ue this in­tro­duct­ory course by sup­pos­ing that the space is semi-met­ric, i.e., met­ric without the tri­angle in­equal­ity, and everything re­turns to nor­mal — de­rived sets are closed, the uni­on and in­ter­sec­tion of closed sets are closed, etc.) However, I do not find the lack of uni­form­ity to be a severe han­di­cap be­cause the class be­comes more ho­mo­gen­ous as to back­ground as time goes on.

There is a prob­lem of what to do with stu­dents who are too tim­id to present their proofs at the board. I gen­er­ally try to draw them in­to the dis­cus­sions, of­fer to do the writ­ing on the board for them while they stay seated, and even­tu­ally after six months or so they get up without real­iz­ing what they are do­ing — es­pe­cially if a subtle ar­gu­ment gets rather heated. But when there are stu­dents who don’t present the­or­ems or counter­examples in class, I simply de­pend upon the fi­nal ex­am­in­a­tion to de­term­ine the course grade. For those not act­ively par­ti­cip­at­ing, the course be­comes a lec­ture course (the lec­tures lack­ing some­what in pol­ish).

There is also the stu­dent whose proof gets “shot down.” People gen­er­ally tend to be em­bar­rassed by mis­takes — es­pe­cially pub­lic mis­takes — and care must be taken not to make them feel that cri­ti­cism is ri­dicule. As work­ing math­em­aticians we have be­come so ac­cus­tomed to mak­ing mis­takes that we are in­clined to for­get the pain ex­per­i­enced by the young.

Then there is the prob­lem of read­ing. After a few months of work­ing out their own proofs it be­comes quite dif­fi­cult to get stu­dents to read math­em­at­ics. They would rather do it them­selves. Nev­er­the­less, read­ing is ne­ces­sary if one is to be­come edu­cated (es­pe­cially in the brief span of gradu­ate school). Moore used to com­plain that when he wanted (fi­nally) the stu­dents to read, they couldn’t. After some tri­al and er­ror I have found the fol­low­ing tech­nique to be ef­fect­ive. Have the stu­dents in the be­gin­ning gradu­ate gen­er­al to­po­logy not read un­til Christ­mas and then buy them­selves a copy of Kel­ley. They can then use Kel­ley for bed­time read­ing to pre­pare for the fi­nal ex­am­in­a­tion (and the qual­i­fy­ing ex­am­in­a­tion — I make them the same) which will be mostly (70%) on Kel­ley. I nev­er dis­cuss Kel­ley in class. The stu­dents are sup­posed to do this read­ing un­as­sisted. Class con­tin­ues as be­fore with the stu­dents work­ing out their own proofs for by then we are go­ing in a dir­ec­tion which over­laps Kel­ley very little.

By far the most dif­fi­cult as­pect of the meth­od is pa­tience. The in­struct­or must not help — must not point out the “ob­vi­ous.” We all know how dif­fi­cult even the ob­vi­ous is be­fore it be­comes ob­vi­ous. The in­struct­or must simply be will­ing to wait for the stu­dents’ men­tal chem­istry to work. It helps if the in­struct­or feels re­war­ded when the stu­dent does fi­nally see how to put to­geth­er a few ideas cor­rectly.

And fi­nally there is the prob­lem of what to do when “no one has any­thing.” One can, of course, start a new top­ic, mo­tiv­ate it, and get the stu­dents to con­struct some ex­amples to fit, or state some new defin­i­tions and the­or­ems. But this kind of thing can­not be done very of­ten for there must not be too many un­settled prob­lems or the stu­dent will be­come dis­trac­ted and un­able to con­cen­trate his at­tack. I have sev­er­al top­ics which can be­come use­ful side is­sues: (1) prob­lems about set the­ory, well-or­der­ing and car­din­al­ity can be taken up and worked out by the stu­dents on the spur of the mo­ment, (2) how some of the the­ory sim­pli­fies if one as­sumes the space to be met­ric, (3) the his­tory of some of the ideas and the per­son­al­it­ies of some of the people in­volved, etc. But gen­er­ally I make up some ques­tions in­volving the ap­plic­a­tion of the­or­ems already proved (or even those yet to be proved) which can be settled in ten or fif­teen minutes each.

At such times one can in­tro­duce the be­gin­nings of no­tions, usu­ally in con­nec­tion with ex­amples, that will be use­ful on the­or­ems to come. That is, one can some­what ran­domly (and out of con­text) put in­to the sub­con­scious minds of the stu­dents pic­tures, ideas and no­tions which will re­sur­face weeks (or even months) later in a proof. There should be no hint as to what the ideas are really for and in fact the stu­dent when he later uses one of the no­tions will have the feel­ing that he dis­covered it him­self. In this way, as well as the ac­tu­al state­ment of ap­plic­able lem­mas, the proofs of quite dif­fi­cult the­or­ems can be made ac­cess­ible.

One fre­quently hears people say: “Yes, it may work in to­po­logy but not in …” I will dis­cuss briefly cer­tain areas where I have found it to work very well.

One of the most re­ward­ing is group the­ory. Here again one may be­gin with rather simple ax­ioms and the cent­ral the­ory can be broken down in­to a rather nice se­quence of in­ter­est­ing but easy the­or­ems. With two dif­fer­ent classes I have used Speiser’s Die The­or­ie der Grup­pen with modi­fic­a­tions (nu­mer­ous il­lus­trat­ive ex­amples should be presen­ted by both the stu­dents and the in­struct­or and The­or­em 40 (Frobeni­us) is too dif­fi­cult without the in­tro­duc­tion of a lemma).

As pre­vi­ously men­tioned, the ax­io­mat­ic de­vel­op­ment of the Hil­bert–Veblen–Moore Found­a­tions of (plane) Geo­metry works ex­tremely well. There are proofs in this sub­ject for stu­dents of widely vary­ing abil­it­ies (the “mid-point” the­or­em: If \( ab \) is an in­ter­val, it con­tains a point \( c \) such that \( ac \) is con­gru­ent to \( cb \), is too dif­fi­cult without the Dede­kind Cut Ax­iom).

An in­tro­duc­tion to Hil­bert Space works very well from von Neu­mann’s 1928 pa­per (see M. H. Stone’s col­loqui­um pub­lic­a­tion — some of the lem­mas may be omit­ted and left to the stu­dents to dis­cov­er). The start of this the­ory makes a nice six to ten week course. I once told von Neu­mann about do­ing this and he re­marked that it would re­quire a very good class. The class was good and be­ing a sum­mer course, the stu­dents were already ex­per­i­enced in “prov­ing things for them­selves.” However, once the ideas are stated and the se­quence of the­or­ems laid out, the rest is much, much easi­er than the situ­ation von Neu­mann faced.

These are areas where a set of ax­ioms can be used for a be­gin­ning. What about oth­er areas? I have per­son­ally used the meth­od in courses in real vari­able the­ory and in com­plex vari­able the­ory. H. S. Wall and some of his stu­dents, MacNer­ney and Por­celli, in par­tic­u­lar, have used the meth­od (with ob­vi­ous suc­cess) in vari­ous courses of ana­lys­is. Some areas re­quire quite some ef­fort in for­mu­lat­ing a good se­quence of the­or­ems. MacNer­ney worked two years to de­vel­op a se­quence (in Com­plex Vari­able The­ory) which would yield the Cauchy In­teg­ral The­or­em in one quarter (one semester is bet­ter).

For the to­po­logy of the line (and the plane) I have found the non-ax­io­mat­ic ap­proach to be more suc­cess­ful (for be­gin­ning col­lege and high school stu­dents). I spend some time be­ing cer­tain that the class is clear on the prop­er­ties of the nat­ur­al num­bers in­clud­ing or­der, well-or­der and in­duc­tion. A “point” is defined to be a func­tion from the nat­ur­al num­bers to the nat­ur­al num­bers. Then giv­ing the set of all such func­tions the lex­ico­graph­ic or­der and us­ing the open in­ter­val to­po­logy (ex­cept at the con­stant func­tion 1 where the cor­rect half-open in­ter­val is ne­ces­sary), one has an in­ter­est­ing be­gin­ning of a de­vel­op­ment that quickly gives all the to­po­lo­gic­al prop­er­ties of the non-neg­at­ive real num­bers. The “sum” and “product” of points giv­en by the usu­al defin­i­tions for func­tions is dis­con­tinu­ous. Hence the dis­cov­ery of a per­fect set which con­tains no in­ter­val presents the stu­dent with something like the situ­ation that Can­tor en­countered.

Of course, there is also a simple ax­io­mat­ic ap­proach to the to­po­logy of the line (which omits arith­met­ic) and Bur­gess has used it quite suc­cess­fully.

Moore of­ten gave a sum­mer course in met­ric dens­ity in the plane and the line where the be­gin­ning was non-ax­io­mat­ic. In fact, a curs­ory know­ledge of Le­besgue meas­ure was re­quired.

I have already men­tioned that the in­struct­or must pos­sess “pa­tience.” But I think it is more than that. It must be “pa­tience” that is born of the con­vic­tion that train­ing a stu­dent to do re­search is im­port­ant — even more im­port­ant than con­vey­ing know­ledge; that try­ing to de­vel­op a stu­dent’s math­em­at­ic­al abil­ity to the lim­it of that abil­ity is im­port­ant.

The in­struct­or should sup­press his own urge to get in­to the act. Even an ugly proof from one of the stu­dents should please the in­struct­or. Only on rare oc­ca­sions (pos­sibly “nev­er” is the bet­ter policy) should he show an el­eg­ant proof. The stu­dent can learn “el­eg­ance” from his read­ing later. Moore once did this to me and his el­eg­ant ar­gu­ment drove my ugly one out of my mind and I have wished many times later that I could re­call it where Moore’s tech­nique would not work. Maybe my ugly one would have. this is not to say that word­ing and clar­ity of thought should not be dis­cussed. But it should be done on the stu­dent’s own ar­gu­ment.

The in­struct­or should work out the se­quence of the­or­ems keep­ing in mind the gen­er­al needs and abil­it­ies of a group as the course pro­ceeds so that ex­tra lem­mas may be in­tro­duced (for a weak group) or some lem­mas may be omit­ted (for a strong group). I even like to leave the gen­er­al dir­ec­tion of the course flex­ible so as to ac­com­mod­ate the in­terests of the group as I be­come bet­ter ac­quain­ted with the vari­ous in­di­vidu­als. The main thing, as of­ten ex­pressed by W. M. Why­burn, is to “give them something they can do.”

Wilder has ex­pressed the opin­ion that Moore was suc­cess­ful in us­ing the meth­od be­cause the stu­dents were prov­ing Moore’s the­or­ems while the de­vel­op­ment of the the­ory was still hot. I ex­pect this may have been a factor in the ob­vi­ous en­thu­si­asm in the Moore School of the 1920s. This was ab­sent later (say in the 1950s or a bit earli­er) and Moore was just as suc­cess­ful. Clearly to be suc­cess­ful, the in­struct­or should give the stu­dent puzzle after puzzle that the stu­dent can­not res­ist. His ap­pet­ite (and abil­ity) in­creases with every solu­tion.

In teach­ing Gen­er­al To­po­logy sev­er­al people have fol­lowed Moore’s tech­nique of simply as­sign­ing the class the se­quence of the­or­ems as they oc­cur in Moore’s book. Most people, however, have used some vari­ant of the meth­od more suit­able to their own ideas of what de­vel­ops the stu­dent. For ex­ample, Bing has some stu­dent in the class act as sec­ret­ary and write up the notes for the whole class. By passing this chore around, some of the stu­dents get some prac­tice at writ­ing. (And it’s a good idea for oth­er reas­ons.) I like to state false pro­pos­i­tions (just as if they were true) for the stu­dents to prove. And quite fre­quently I state a num­ber of defin­i­tions and ask the stu­dents to for­mu­late some the­or­ems us­ing them. I feel that ex­amples (and counter­examples) are very im­port­ant for both un­der­stand­ing and mo­tiv­a­tion. In par­tic­u­lar, one’s in­tu­ition is aided by ex­amples of spaces that do not sat­is­fy the ax­ioms as much as by ex­amples that do: for in­stance, (in my ap­proach to Gen­er­al To­po­logy) to­po­lo­gic­al spaces (even com­pact and Haus­dorff) that are not semi-met­ric and semi-met­ric spaces which are not met­ric.

And it’s nice to have a se­quence of the­or­ems which are use­ful but which can be proved by any­body. Ele­ment­ary prop­er­ties of con­nec­ted point sets can be for­mu­lated in­to a se­quence of this sort.

It is a good plan to en­cour­age stu­dents to change a the­or­em un­til they can prove it: weak­en the con­clu­sion or strengthen the hy­po­thes­is or both. This helps to avoid frus­tra­tion and is good prac­tice.

The in­struct­or should con­vey con­fid­ence, es­pe­cially in the be­gin­ning. The stu­dent should soon learn that some things he can do quickly but oth­ers may take ef­fort and time. Giv­en six months prac­tice, a stu­dent who had nev­er thought of a proof in his life and didn’t know how to start, may de­vel­op to the point where he can settle al­most any­thing you pro­pose. I find this very re­ward­ing and sat­is­fy­ing. It hap­pens of­ten enough to keep one’s en­thu­si­asm for teach­ing vi­tally alive.