Celebratio Mathematica

Andy Gleason was a teach­er in the widest pos­sible sense of the word: he taught us math­em­at­ics, he taught us how to think, and he taught us how to treat oth­ers.

From Andy I learned the im­port­ance of a teach­er see­ing math­em­at­ics through both a math­em­atician’s and a stu­dent’s eyes. Andy’s math­em­at­ic­al breadth is le­gendary; his curi­os­ity and em­pathy about the views of stu­dents, be they first-graders or gradu­ate stu­dents, were equally re­mark­able. I vividly re­mem­ber his con­cern in the early years of the AIDS epi­dem­ic that an ex­ample about the pre­val­ence of HIV in­fec­tions would up­set stu­dents. Equally vivid in my memory is Andy’s de­light when his ap­proach to the def­in­ite in­teg­ral and his in­sight in­to stu­dent un­der­stand­ing came to­geth­er to pro­duce a much bet­ter way to teach in­teg­ra­tion. This was one of dozens of oc­ca­sions when Andy made those around him re­think fa­mil­i­ar top­ics from a fresh view­point. New ideas about teach­ing bubbled out of Andy’s mind con­tinu­ously; he was equally quick to re­cog­nize them in oth­ers. When one of his former Ph.D. stu­dents, Peter Taylor, sent Andy some cal­cu­lus prob­lems, Andy glee­fully sug­ges­ted that we try them. He re­garded teach­ing math­em­at­ics — like do­ing math­em­at­ics — as both im­port­ant and also genu­inely fun.

At Har­vard Andy reg­u­larly taught at every level. He nev­er shied away from large, multi­sec­tion courses with their as­so­ci­ated ad­min­is­trat­ive bur­den. He was al­ways ready to step for­ward in­to the un­charted ter­rit­ory of a new course in real ana­lys­is, cal­cu­lus, quant­it­at­ive reas­on­ing, or the his­tory of math­em­at­ics.

Christine Stevens, one of Andy’s doc­tor­al stu­dents, writes:

I first en­countered Andy in the fall of 1971, when I en­rolled in his course on The Struc­ture of Loc­ally Com­pact To­po­lo­gic­al Groups (Math 232). It goes without say­ing that the course was a mod­el of lu­cid ex­pos­i­tion, but I also re­mem­ber Andy’s en­thu­si­ast­ic and of­ten witty re­sponses to stu­dents’ ques­tions. In­deed, some of them are re­cor­ded in the mar­gins of my note­book, along­side some rather deep math­em­at­ics. I also re­call the cheer­ful en­ergy with which he lec­tured one cold winter day when the heat­ing sys­tem in Sever Hall had giv­en out.

I even­tu­ally wrote my dis­ser­ta­tion on an is­sue that Andy had men­tioned in that course. We mapped out an ap­proach in which the first step in­volved prov­ing something that he deemed “al­most cer­tainly true.” When he com­menced one of our sub­sequent ap­point­ments by ask­ing me how things were go­ing, I replied, “not too well.” I ex­plained that I had proved that the state­ment that was “al­most cer­tainly true” was equi­val­ent to something that we had agreed was prob­ably false. To be hon­est, I was kind of down in the dumps about the situ­ation. Andy’s re­sponse was im­me­di­ate and en­cour­aging. Without miss­ing a beat, he replied, “Well, that’s not a prob­lem. Just change the hy­po­theses!”

In 1964 Andy in­sti­tuted a new course at Har­vard, Math 112, to provide math ma­jors a trans­ition from the three-year cal­cu­lus se­quence to Math 212, the gradu­ate course in real ana­lys­is. It func­tioned as an in­tro­duc­tion to the spir­it of ab­stract math­em­at­ics: first-or­der lo­gic, the de­vel­op­ment of the real num­bers from Peano’s ax­ioms, count­ab­il­ity and car­din­al­ity. This was the first of the “bridge” courses now ubi­quit­ous for math ma­jors, only twenty years be­fore its time. Tom Tuck­er re­calls:

I was a stu­dent in that first Math 112, and it was my first ex­per­i­ence with Andy. He chided me that the course might be too ele­ment­ary for me, since most stu­dents from Math 55 went straight on to Math 212. But I had taken Math 55 as my first course at Har­vard and was still in shock. I needed some en­cour­age­ment, something I really could un­der­stand, and that is ex­actly what Andy gave me. He helped sal­vage my math­em­at­ic­al ca­reer.

Andy’s work in Math 112 led to his only solo text in math­em­at­ics, Fun­da­ment­als of Ab­stract Ana­lys­is.1 In his re­view of the book, Dieud­onné cap­tures the es­sence of Andy’s ped­agogy:2

Every work­ing math­em­atician of course knows the dif­fer­ence between a life­less chain of form­al­ized pro­pos­i­tions and the “feel­ing” one has (or tries to get) of a math­em­at­ic­al the­ory, and will prob­ably agree that help­ing the stu­dent to reach that “in­side” view is the ul­ti­mate goal of math­em­at­ic­al edu­ca­tion; but he will usu­ally give up any at­tempt at suc­cess­fully do­ing this ex­cept through or­al teach­ing. The ori­gin­al­ity of the au­thor is that he has tried to at­tain that goal in a text­book, and in the re­view­er’s opin­ion, he has suc­ceeded re­mark­ably well in this all but im­possible task.

Over the course of his teach­ing ca­reer, Andy wrote hun­dreds of pages of lec­ture notes for his stu­dents, re­work­ing them afresh each year. Some were hand­writ­ten on spir­it du­plic­at­or sheets; some were type­set us­ing mac­ros he de­veloped un­der an early ver­sion of Unix. More than lec­ture notes, these were com­plete with hand-drawn fig­ures and ex­er­cises. His ef­forts in course de­vel­op­ment in the early 1970s in­cluded two com­plete un­pub­lished texts. The first was for a new full-year in­teg­rated lin­ear al­gebra/mul­tivari­able cal­cu­lus course (Math 21) , the second for the his­tory-based gen­er­al edu­ca­tion course Nat­ur­al Sci­ences 1a: In­tro­duc­tion to Cal­cu­lus.

Andy com­bined his in­terest in edu­ca­tion, math­em­at­ics, and his­tory in his design for Nat­ur­al Sci­ences 1a. Noth­ing like a stand­ard treat­ment of the ma­ter­i­al, this course took a his­tor­ic­al ap­proach to the de­vel­op­ment of the ba­sic ideas of cal­cu­lus, be­gin­ning with an ex­plic­a­tion of Archimedes’ The Sand Reck­on­er and cul­min­at­ing with a de­riv­a­tion of Kepler’s laws of plan­et­ary mo­tion from New­ton’s phys­ic­al laws.

Nat­ur­al Sci­ences 1a was in­ten­ded for the non­spe­cial­ist stu­dent with an in­terest in the his­tory of ideas. Andy wanted the stu­dents to grapple with is­sues like ir­ra­tion­al­ity and con­tinu­ity. Many of his as­sign­ments asked stu­dents for non­tech­nic­al es­says in which they ex­plored the math­em­at­ics through per­son­al con­tem­pla­tion. Stu­dents sign­ing up for this course seek­ing an easy way to sat­is­fy a re­quire­ment got a lot more than they bar­gained for.

Andy’s in­quir­ies about learn­ing math­em­at­ics some­times led to rad­ic­al po­s­i­tions. In his art­icle3 “Delay the teach­ing of arith­met­ic” he sug­ges­ted that the usu­al al­gorithms of arith­met­ic not be taught un­til grade 6. He cited work4 of Be­nez­et on just such an ex­per­i­ment in the Manchester, NH, schools in the 1930s. The stu­dents not taught the al­gorithms learned them per­fectly well in sev­enth grade, but their prob­lem-solv­ing abil­ity, their will­ing­ness to “take re­spons­ib­il­ity for their an­swers,” was dra­mat­ic­ally bet­ter than the con­trol group’s. In his pa­per Andy re­calls his own child­hood math classes re­quir­ing four cal­cu­la­tions for each day: a sum of sev­en 6-di­git num­bers, a sub­trac­tion of two 7-di­git num­bers, a product of a 6-di­git num­ber by a 3-di­git num­ber, and a long di­vi­sion of a 6-di­git num­ber by a 3-di­git num­ber; an­swers were graded right or wrong and 75% was passing. Andy es­tim­ates the num­ber of in­di­vidu­al op­er­a­tions for each prob­lem and con­cludes that a stu­dent get­ting each op­er­a­tion cor­rect with 99.5% prob­ab­il­ity would still av­er­age only 73, fail­ing. As Andy re­marked once on long di­vi­sion, get­ting even one prob­lem cor­rect out of ten in­dic­ates suf­fi­cient un­der­stand­ing of the al­gorithm.

Andy was acutely aware of the im­port­ance of stu­dents’ at­ti­tudes to­ward math­em­at­ics, as evid­enced by his re­marks5 in the 1980s to the Na­tion­al Academy of Sci­ences:

Right now there is de­bate ap­par­ently ex­ist­ing as to how math­em­at­ics should re­act to the ex­ist­ence of cal­cu­lat­ors and com­puters in the pub­lic schools. What should be the ef­fect on the cur­riculum?…and so on. Now the un­for­tu­nate point of that is that there is even a very ser­i­ous de­bate as to wheth­er there should be an im­pact on the cur­riculum. That is what I re­gard as ab­so­lutely ri­dicu­lous. Let me just point out that…in this coun­try there are prob­ably 100,000 fifth grade chil­dren right now learn­ing to do long di­vi­sion prob­lems. In that 100,000 you will find very few who are not thor­oughly aware that for a very small sum of money (like \$10) they can buy a cal­cu­lat­or which can do the prob­lems bet­ter than they can ever hope to do them. It’s not just a ques­tion of do­ing them just a little bet­ter. They do them faster, bet­ter, more ac­cur­ately than any hu­man be­ing can ever ex­pect to do them and this is not lost on those fifth graders. It is an in­sult to their in­tel­li­gence to tell them that they should be spend­ing their time do­ing this. We are demon­strat­ing that we do not re­spect them when we ask them to do this. We can only ex­pect that they will not re­spect us when we do that.

About ten years ago Andy gave a talk at the Joint Math­em­at­ics Meet­ings in which he de­scribed how he had, some years pre­vi­ously, spent a sum­mer teach­ing arith­met­ic to young chil­dren. His goal had been to find out how much they could fig­ure out for them­selves, giv­en ap­pro­pri­ate activ­it­ies and the right guid­ance. At the end of his talk, someone asked Andy wheth­er he had ever wor­ried that teach­ing math to little kids wasn’t how fac­ulty at re­search in­sti­tu­tions should be spend­ing their time. Christine Stevens re­mem­bers Andy’s quick and de­cis­ive re­sponse: “No, I didn’t think about that at all. I had a ball!”

Andy led in pro­mot­ing the in­volve­ment of re­search math­em­aticians in is­sues of teach­ing and learn­ing.

He was deeply in­volved with the re­form of the U.S. math­em­at­ics K–12 cur­riculum in the post-Sput­nik era. He chaired the first ad­vis­ory com­mit­tee for the School Math­em­at­ics Study Group (SMSG), the group re­spons­ible for “the new math”. He was a co­dir­ect­or with Ted Mar­tin of the 1963 Cam­bridge Con­fer­ence on School Math­em­at­ics. The re­port of that con­fer­ence pro­posed an am­bi­tious cur­riculum for col­lege-bound stu­dents that cul­min­ated in a full-blown course in mul­tivari­able cal­cu­lus in \( n \)-di­men­sions in­clud­ing the In­verse Func­tion The­or­em, dif­fer­en­tial forms, and Stokes’ The­or­em. Al­though the pro­posed cur­riculum would ap­pear to be far too soph­ist­ic­ated by today’s stand­ards, the space race loomed large in the pub­lic mind and the need for highly trained sci­ent­ists, math­em­aticians, and en­gin­eers be­came a na­tion­al cru­sade. The SMSG pro­gram be­gun in 1959 was aimed at all stu­dents and was roundly cri­ti­cized at the time as be­ing in­ap­pro­pri­ate for av­er­age stu­dents and teach­ers. The Cam­bridge Con­fer­ence ap­peared to be an at­tempt to woo re­search math­em­aticians to school re­form through con­sid­er­a­tion of an “hon­ors” track for the most able stu­dents. In that con­text, some crit­ics com­plained the pro­posed cur­riculum was “tim­id”!

In 1985–89, Andy helped es­tab­lish the Math­em­at­ic­al Sci­ences Edu­ca­tion Board to co­ordin­ate edu­ca­tion­al activ­it­ies for all the math­em­at­ic­al pro­fes­sion­al or­gan­iz­a­tions; his cita­tion for the MAA Dis­tin­guished Ser­vice Award re­cog­nized the im­port­ance of this con­tri­bu­tion. From the 1980s un­til his death, Andy was in­flu­en­tial in cal­cu­lus re­form and the sub­sequent re­think­ing of oth­er in­tro­duct­ory col­lege courses.

That a math­em­atician of Andy’s stature would take the time to think deeply about the school cur­riculum made such work le­git­im­ate.

In the late 1970s Har­vard Col­lege un­der­took a sweep­ing re­or­gan­iz­a­tion of the Gen­er­al Edu­ca­tion re­quire­ments. The new core cur­riculum re­placed ex­ist­ing de­part­ment­al of­fer­ings with spe­cially de­signed courses in a broad vari­ety of areas of dis­course. It was hard to see how math­em­at­ics fit in the new core. Giv­en his ex­tens­ive con­tact with cur­ricular pro­jects and his in­terest in edu­ca­tion, Andy was a nat­ur­al choice to lead an in­vest­ig­a­tion in­to what a math­em­at­ics re­quire­ment might be and how it was to be im­ple­men­ted.

Rather than draw­ing up a check­list of what kinds of math­em­at­ics a Har­vard gradu­ate should know, Andy in­stead star­ted with the idea that at the very least, the core re­quire­ment in math­em­at­ics should pre­pare stu­dents for the kinds of math­em­at­ic­al, stat­ist­ic­al, and quant­it­at­ive ideas they’d be con­front­ing in their oth­er core courses. Work­ing with fac­ulty who were de­vel­op­ing those courses, Andy quickly real­ized that the skills stu­dents re­quired had more to do with the present­a­tion, ana­lys­is, and in­ter­pret­a­tion of data than with any par­tic­u­lar body of math­em­at­ics, such as cal­cu­lus. Thus, the core Quant­it­at­ive Reas­on­ing Re­quire­ment, or QRR, was born.

So, long be­fore quant­it­at­ive lit­er­acy be­came a well-defined area of study with its own cur­riculum and text­books, Andy and Pro­fess­or Fred Mos­teller of the Har­vard stat­ist­ics de­part­ment de­veloped a small set of ob­ject­ives for the QRR. These in­cluded un­der­stand­ing dis­crete data and simple stat­ist­ics, dis­tri­bu­tions and his­to­grams, and simple hy­po­thes­is test­ing. There was no re­li­ance on high school al­gebra or oth­er math­em­at­ics that stu­dents had seen be­fore, since high schools had not yet be­gun of­fer­ing an Ad­vanced Place­ment Stat­ist­ics course. So the re­quire­ment leveled the play­ing field — both math ma­jors and his­tory ma­jors would have to learn something new to sat­is­fy the QRR.

Andy also thought about im­ple­ment­ing the QRR — how to help 1,600 first-year stu­dents meet the re­quire­ment without mount­ing an ef­fort as large, and costly, as fresh­man writ­ing. He de­cided that the ideas stu­dents were be­ing asked to mas­ter, while nov­el, were not very hard and that most stu­dents could learn them on their own, giv­en the ap­pro­pri­ate ma­ter­i­als. For the small num­ber of stu­dents who couldn’t learn from self-study ma­ter­i­als, there would be a semester-long course.

So, in the sum­mer of 1979, Andy gathered a team of about a dozen un­der­gradu­ates (“the Core corps”) who wrote self-study ma­ter­i­als and gathered news­pa­per art­icles for prac­tice prob­lems. These were pub­lished as manu­als and sup­plied to all en­ter­ing stu­dents. Andy in­vited the stu­dent au­thors to his home in Maine that sum­mer, which was typ­ic­al of his friend­li­ness and open­ness. Jeff Te­cosky-Feld­man, then the stu­dent lead­er of the Core corps, helped or­gan­ize the trip to Maine. He re­calls:

The oth­er stu­dents were buzz­ing with the ru­mor that Andy had been in­volved in crack­ing the Ja­pan­ese code in World War II, but were too tim­id to ask him about it them­selves, so they put me up to it. When I asked Andy, his re­sponse was typ­ic­al: “It would not be en­tirely in­cor­rect to say so”, and he left it at that.

In Janu­ary 1986 Andy par­ti­cip­ated in the Tu­lane Con­fer­ence that pro­posed the “Lean and Lively” cal­cu­lus cur­riculum. Oc­to­ber 1987 saw Andy on the pro­gram at the “Cal­cu­lus for a New Cen­tury” con­fer­ence; in Janu­ary 1988 the idea for the Cal­cu­lus Con­sor­ti­um based at Har­vard took shape.

Andy’s role in the Cal­cu­lus Con­sor­ti­um was without fan­fare and without equal. He star­ted by gently turn­ing down my re­quest that he be the PI on our first NSF pro­pos­al and, after a thirty-second si­lence that seemed to me in­ter­min­able, sug­ges­ted we be co-PIs. He then helped build one of the coun­try’s first multi-in­sti­tu­tion col­lab­or­at­ive groups. Now com­mon­place, such ar­range­ments were at the time viewed with some skep­ti­cism at the NSF, whose pro­gram of­ficers wondered wheth­er such a large group could get any­thing done.

Throughout his time with the con­sor­ti­um, Andy’s words, in a voice that was nev­er raised, were the keel that kept us on course. His view of the im­port­ance (or lack of it) of vari­ous top­ics in the cal­cu­lus cur­riculum shaped many of our dis­cus­sions, and his vis­ion in­spired many of our in­nov­a­tions. Andy hated to write — he saw the lim­it­a­tions of any ex­pos­i­tion — so we quickly learned that the best way to get his ideas on pa­per was for one of us to write a first draft. This drew him in im­me­di­ately as he re­shaped, re­ph­rased, and in es­sence re­wrote the piece. That Andy could do this for twenty years without dent­ing an ego is a test­a­ment to his skill as a teach­er. Who else could say, as I re­spon­ded to a flood of red ink by ask­ing wheth­er I’d made a mis­take, “Oh no, much worse than that” and have it come across as a warm in­vit­a­tion to dis­cus­sion? We all re­mem­ber Andy re­mark­ing, “That’s an in­ter­est­ing ques­tion!” and know­ing that we were about to see in an ut­terly new light something we’d al­ways thought we un­der­stood.

The 1988 NSF pro­pos­al led to a plan­ning grant in 1989. The found­ing mem­bers of the con­sor­ti­um met for the first time in Andy’s of­fice. Fac­ulty from very dif­fer­ent schools dis­covered to their sur­prise that stu­dents’ dif­fi­culties were sim­il­ar in the Ivy League and in com­munity col­leges. A mul­ti­year pro­pos­al fol­lowed, with fea­tures now com­mon­place in fed­er­ally fun­ded pro­pos­als but then un­usu­al. Andy was skep­tic­al about some of these and sug­ges­ted we re­move the sec­tion on dis­sem­in­a­tion — after all, he poin­ted out, we didn’t know wheth­er what we’d write would be any good. When the pro­pos­al went to the NSF for feed­back be­fore the fi­nal sub­mis­sion, I got a call from the pro­gram dir­ect­or, Louise Raphael, ask­ing about the miss­ing sec­tion on dis­sem­in­a­tion. When I ex­plained, Louise, who knew how things worked in DC, re­spon­ded by say­ing I should tell Andy “not to be a math­em­atician.” We then un­der­stood our man­date from the NSF to dis­sem­in­ate the dis­cus­sion of the teach­ing of cal­cu­lus to as many de­part­ments and fac­ulty as pos­sible. Over the next dec­ade we gave more than one hun­dred work­shops for col­lege fac­ulty and high school teach­ers, in which Andy played a full part — present­ing, an­swer­ing ques­tions, and listen­ing to con­cerns.

The de­bate about cal­cu­lus be­ne­fit­ted enorm­ously from Andy’s par­ti­cip­a­tion. He be­came a fath­er fig­ure for cal­cu­lus re­form in gen­er­al and the NSF-sup­por­ted pro­ject at Har­vard in par­tic­u­lar. His goal was nev­er re­form per se; it was to dis­cuss openly and ser­i­ously all as­pects of math­em­at­ics learn­ing and teach­ing. In 1997 Hy­man Bass wrote6 “It is the cre­ation of this sub­stan­tial com­munity of pro­fes­sion­al math­em­atician-edu­cat­ors that is the most sig­ni­fic­ant (and per­haps least an­ti­cip­ated) product of the cal­cu­lus re­form move­ment. This is an achieve­ment of which our com­munity can be justly proud and which de­serves to be nur­tured and en­hanced.”

Andy — reasoned, calm, soft-spoken, a gen­tle­man in every sense of the word — was ded­ic­ated to this com­munity throughout his life.

Andy in­spired rather than taught many of us. His trans­par­ent hon­esty and hu­mil­ity were so strik­ing that they were im­possible to ig­nore. For ex­ample, be­fore pub­lish­ing my first text­book, I asked him how au­thors got star­ted, since pub­lish­ers wanted es­tab­lished names. Andy replied mat­ter-of-factly, “Most people nev­er do,” re­turn­ing to me the re­spons­ib­il­ity to achieve this.

Andy’s mor­al in­flu­ence was enorm­ous. Al­ways above the fray and without a mean bone in his body, Andy com­manded re­spect without rais­ing his voice. His mor­al stand­ards were high — very high — mak­ing those around him as­pire to his tol­er­ance, un­der­stand­ing, and ci­vil­ity. Andy’s pres­ence alone forged co­oper­a­tion.

In his com­ment­ary on the first book of Eu­c­lid’s Ele­ments, Pro­clus de­scribed Pla­to as hav­ing “…aroused a sense of won­der for math­em­at­ics amongst stu­dents.” These same words char­ac­ter­ize Andy. Through the courses he taught and the lec­tures that he gave for teach­ers, Andy in­spired thou­sands of stu­dents with his sense of the won­der and ex­cite­ment of math­em­at­ics. Through him, many learned to see the world through a math­em­at­ic­al lens.