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Andy Gleason: teacher
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,, 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.
In the classroom and as an advisor
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!”
Courses, books, and classroom notes
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.
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!”
Education at a national level
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 withof 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.
Quantitative reasoning (QR)
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 andof 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,, 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 6 “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.”wrote
Andy — reasoned, calm, soft-spoken, a gentleman in every sense of the word — was dedicated to this community throughout his life.
Outside the classroom
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.