Niranjan Srinivas: What I learned from an undergraduate education in mathematics

One of the nice things I get out of this blogging gig is to get e-mails from some truly fabulous people either commenting on something they saw here, or sending me a link that might be of interest to us. Sometimes, I even get to meet them in person when they visit our Institute or, more generally, Bangalore. Recently, I had the privilege of meeting one such person: Niranjan Srinivas who graduated with an integrated M. Sc. in Mathematics & Scientific Computing from IIT Kanpur in May 2008, and is currently a Ph.D. candidate in the Computation & Neural Systems program at Caltech.

During our short meeting, Niranjan happened to mention an article he wrote sometime ago; when I asked him for permission to post it here, he readily agreed.

Thanks, Niranjan!

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What I learned from an undergraduate education in mathematics

Niranjan Srinivas

Typically, one’s undergraduate years involve several different kinds of learning: academic growth in the chosen field of study, social development arising from meeting and interacting with a fairly diverse set of people, and personal and emotional growth from teenage adolescence into full-fledged adulthood. In this article, my focus is on academic growth. Specifically, I shall try to articulate what I learned from my undergraduate education in mathematics, with its own particular virtues and weaknesses, and why I think such an education is valuable in the “real world”. I use “mathematics” in a broad sense (which includes theoretical computer science and statistics.)

Although the philosophy of mathematics and its relationship to science and logic is the subject of active debate, much mathematics may be viewed as an exercise in “pure reasoning”. An education in mathematics hence provides training in precise and careful logical reasoning. This logical thought process is a fundamental tool for solving complex problems, whether they arise in mathematics, scientific research, or industry, and is therefore a valuable skill to acquire.

Mathematics is as much a language as it is a field of enquiry. Indeed, it is a powerful language for expressing and investigating complex quantitative relationships between objects of interest. Therefore, it is the language of choice for the physical sciences and engineering. Facility with this language, which is essential to formulate and solve quantitative problems, is another advantage of studying mathematics.

Abstraction is an essential feature of mathematics, as it deals with the properties of and relationships between abstract entities. Although the entities are often inspired by the “real” world, mathematics is concerned with the abstract entities rather than the real world “instances” of those abstract entities. Even the number two is an abstract entity; one may find two birds or two stones or two mountains in the real world, but never just “two”. The ability to reason about abstract entities and formulate general questions is another skill one develops while learning mathematics. This skill is important because generalizing from the specific to the abstract and articulating a precise question that captures the heart of a complex problem is often a formidable challenge in itself.

In my personal experience, my training in mathematics has proven very valuable even though I have been working in very different fields. After I graduated, I spent one year working in the financial industry in a quantitative role. After that, I returned to academia to pursue my Ph. D. My current research interests lie in the intersection of computer science, bioengineering and nanotechnology; my work involves engineering smart molecular systems using synthetic nucleic acids. My colleagues include biochemists, physicists, mathematicians, and theoretical computer scientists, among others. Being trained in mathematics is very useful as it is often a common language between scientists from different backgrounds.

Apart from these specific skills and ways of thinking, an education in mathematics, which is one of the oldest intellectual endeavors, provides a historical perspective on scientific thought. A typical first course in calculus would consist almost entirely of mathematics that would be considered “well-developed” by the nineteenth century. Contrast that with a first course in molecular biology, which would consist almost entirely of ideas developed in the last fifty years. For an academic, this historical perspective is important because it provides context for his or her work. For the non-academic, the perspective provides one possible framework for thinking about modern issues and challenges.

However, majoring in mathematics as an undergraduate is not without its challenges. Although the mathematical education helps one acquire several important skills useful in the “real world”, one could conceivably graduate without any domain-specific knowledge about anything else. Leveraging the skills one acquires to actually solve problems outside mathematics would therefore entail acquiring knowledge about the particular field or problem of interest. However, this is usually not difficult if one has an open mind and is actually motivated to learn about the concerned subject. In my personal experience, acquiring new knowledge about a particular subject is significantly easier and faster than learning to think carefully and precisely in a mathematical way.

Related to this is what I like to think of as the “purist” trap. I think it is frightfully easy, at least if one is interested in “pure” mathematics as an undergraduate, to care only about mathematics for its own sake and not be interested in any “applications”. This is a perfectly admirable attitude after one has learnt a lot of both “pure” and “applied” mathematics, but is probably not the most helpful attitude for an undergraduate since it might bias the student against a lot of beautiful mathematics due to an arbitrary distinction between “pure” and “applied” mathematics. Indeed, historically much “pure” mathematics owes its development to particularly important and interesting real world problems. Personally, I fell into this trap as an undergraduate and it took me about three years of study to realize that a lot of mathematics I had labeled “uninteresting” due to my pre-conceived notions was actually quite fascinating.

Lastly, I think the single most important thing I got out of my undergraduate education is learning how to learn. Nearly everything I did after I graduated required me to learn a variety of new skills, and very few things I learned during my undergraduate education were directly relevant to my work. Indeed, almost every opportunity in an individual’s career would likely be based on what he or she can learn, rather than what he or she already knows. Learning how to learn is more valuable than the actual subject or field of study one chooses to focus on, because it empowers the individual to learn other subjects at will.

Experiments in Higher Ed

As you all know by now, Delhi University has succumbed to the UGC diktat and agreed to bury FYUP, its "flagship" program [See the posts by Rahul Siddharthan, Dheeraj Sanghi and Saroj Giri, and the links within them].

For all its merits, and the support of our science academies, the FYUP at DU never managed to get the support of the people of Delhi.

It is possible that FYUP is not suitable for everyone, and it might have found broader support if it was offered as an option for those who were interested in staying in college for an extra year. In other words, FYUP as an incremental change (or, as an experiment) might have worked better than its introduction as the default option for everyone.

I was, and continue to be, unpersuaded by arguments that complained about the 'haste' with which the FYUP was implemented. Since I see the FYUP as an experiment -- and it will necessarily be a different experiment at each university -- I have always held the view that it is better to take the plunge and implement it. To the extent that an institution has the right processes in place to take care of problems and make mid-course corrections, hasty implementation becomes less of a concern.

This country has unleashed several large scale experiments in the last decade: I am referring here to the creation of IISERs, IITs, NITs, and Central Universities. These institutions are being built from scratch. Each is going about organizing its governance structures and developing its academic programs in its own unique way. In these cases too, critics focused on the lack of preparation and hasty implementation. But people -- and I mean here the broader public -- seem to be fine with the idea that the success of such experiments can be assessed only over a period of years (if not decades).

Since I approach FYUP as an experiment (and let's face it, it's a lot less radical than starting a new institution from scratch), I find it perplexing that the people of Delhi oppose it so vehemently.

So, what is the difference between these two kinds of experiments -- FYUP at DU and creating new institutions all across the country? What makes the public see the former as largely unacceptable and the latter as very desirable?

The Right Response To Ranking Exercises: We are X, And We Should Be The Best X We Can Be

Prof. Peter N. Stearns, the soon-to-be-ex Provost of George Mason University, has what I think is one of the best responses to the ranking exercises that seem to have a nasty effect on the sleep cycles of many university leaders. In his recent post entitled Mason Goals, this is what he has to say (and I hope he won't mind the extended excerpt), and with bold emphasis added by me:

T... [W]here should we be heading? I don’t mean the details of the new strategic plan, which is ambitious and fine, though my comments relate to the overall directions of the plan. I refer more to University identity.

And here we confront a puzzle: George Mason is really hard to categorize. When I first arrived I assumed my job was to help make the University even more recognized in the standard ways – move up in US News rankings, get mentioned more often as a research hub, become more selective, and so on. And we did do some of these things. But we also wanted to keep identities we already had, that were equally valuable: center of student diversity (US News shamefully ignores this in its main ratings); accessible to large numbers of first-generation students – and not just accessible – serving as a means for their academic success; eager to seize new opportunities and innovate where appropriate, without as much traditionalist resistance as is common in many other places.

We wanted, in other words, to be George Mason. I heard talk of earlier goals of becoming the “Harvard of the Potomac”, but this reference has faded partly because we simply lack the means, and partly because that’s not fully what we want to be anyway. Yet at the same time we’re not simply innovative and accessible. We really do want to move meaningful research forward. We really do want to combine opportunity with serious quality standards — otherwise we might have more degrees to brag about, but without the real service to students a good university must pledge. We want to be a distinctive mixture, and that’s what I hope we’ve been accomplishing and will accomplish in the future. We want to maintain an active, creative tension between serious conventional standards and the distinctive flavor we’ve developed as an up-and-comer.

Several years ago, pressed by a Board of Visitors interested in Mason aiming at “world class” standards, we hired a consultant who actually said it most clearly: strive to be the best George Mason we can be. Take pride in the difficulty people have in pinpointing us too easily. Take pride in the combinations. In the process we’ll find, as we already do with some of our international visitors, that other institutions will be seeking to adopt our formula.

The "Price" for Getting into the Top 20

A sobering way of examining university ranking exercises is to put a price on the goal of getting into the Top-20 (or Top whatever) in a certain list. Here's an Inside Higher Ed news story about a recent paper that looked into the price for the University of Rochester, which is "ranked consistently in the mid-thirties" in the US News list, to break into the Top 20 (the paper itself is available at a price that I am not willing to pay ;-):

If it wanted to move into the top 20, Rochester would have to do a lot on several of the various factors U.S. News uses to rank colleges. To move up one spot because of faculty compensation, Rochester would have to increase the average faculty salary by about $10,000. To move up one spot on resources provided to students, it would have to spend $12,000 more per student. Those two things alone would cost $112 million a year.

To get into the top 20, Rochester would also have to increase its graduation rate by 2 percent, enroll more students who were in the top 10 percent of their high school graduating class, get more alumni to give, cut the acceptance rate and increase the SAT and ACT scores of incoming students. Some of those things, like offering aid money to highly qualified students, might further increase the expense.

But that’s not all, the paper argues. Rochester would still have to do well in the rankings magazine’s “beauty contest.”

Because 15 percent of the ranking is based on reputation among other administrators, even massive expenditures year after year and huge leaps in student quality and graduation would not be enough. The reputation score as judged by its peers would need to increase from 3.4 to 4.2 on a scale of 5, something that has only a .01 percent chance of happening, the paper said.

IIT-M Director Bhaskar Ramamurthi on Global University Rankings

It's good to see an institutional leader trying to drill some sense into policy-makers who ought to know better than to go blindly by global ranking exercises. Here's the concluding paragraph from Prof. Ramamurthi's op-ed in The Indian Express:

The contributions of the IITs are to be assessed along several dimensions. Some of these are relevant globally and are used by international ranking agencies, while other important ones are totally ignored. We should reflect on the relative weightage given to the dimensions assessed and not lose sight of those that are not. To the extent that the rankings tell us something about where we stand globally with respect to research, visibility, etc, they are relevant, and the IITs should strive to improve their position. Above all, we should not blindly adopt these rankings as an end in themselves, nor allow ourselves to be railroaded into pursuing select dimensions of performance while neglecting others, especially those that are critical to our national development goals.

News Link

Today's Chennai edition of Deccan Chronicle features a short take on my science outreach work.

Pet-peeve: At least, the titles of the books could have been mentioned...instead of, say, my picture... Well, all of us are learning how to do this anyway...

FYUP Fracas at Delhi

The Four-Year Undergraduate Program (FYUP) introduced last year at the Delhi University has run into some serious trouble, triggered in part by NDA's thumping win in the recent Lok Sabha elections in Delhi, with the repeal of the program a key point in their agenda.

In the process of learning more about the issue, I found the following links useful.

1. First Post has a summary of how DU finds itself in this mess.

2. Business Insider - India also has a primer which goes into some of the specific issues flagged by teachers and students.

3. A Telegraph story from two weeks ago seems to argue that DU's FYUP has some support among students who joined the program last year.

4. The program also finds support among educationists, says this story in the Business Standard.

5. The University Grants Commission has taken a tough stand, ordering DU to admit students this year only to the old three-year bachelor's program. And DU has responded by complying nominally to this order, but keeping the FYUP alive by giving the students an option to pursue an "honours" degree in their fourth year.

FYUP is what we have at IISc (and IIT-K as well, for its science degree programs). Prominent private universities such as the Shiv Nadar University appear to favor FYUP over its conventional three-year variant. So, I think FYUP, as an idea, is quite alive.

Overall, I have always been for the four-year degree program. On paper, DU's program has much to like: its curriculum features a a gen ed component (called "foundation" courses), encourages internships and undergraduate research, and offers exit options at the end of not just the third year with a bachelor's degree, but also at the end of the second year, with an associate degree.

Many of FYUP's troubles at DU appear to be due to the haste with which it was rammed through by Vice Chancellor Dinesh Singh and his crew.

We will have to wait and see how this turbulent phase plays itself out at DU.

How many of us are outliers?

1. Must read article of the day (it appeared quite a while ago, though). Maria Konnikova in the New Yorker: Multitask Masters on those truly extraordinary few who are actually good at multitasking, and who actually become better when the tasking becomes more-multi. This is an apt example of exception proving the rule: Most of us mere mortals, on the other hand, are really, really bad at it, even if we are unwilling to accept this sad reality; the article has tons of links to studies cooroborating this finding. Here's an excerpt from the end of the article:

   The irony of Strayer’s work is that when people hear that supertaskers exist—even though they know they’re rare—they seem to take it as proof that they, naturally, are an exception. “You’re not,” Strayer told me bluntly. “The ninety-eight per cent of us, we deceive ourselves. And we tend to overrate our ability to multitask.” In fact, when he and his University of Utah colleague, the social psychologist David Sanbomnatsu, asked more than three hundred students to rate their ability to multitask and then compared those ratings to the students’ actual multitasking performances, they found a strong relationship: an inverse one. The better someone thought she was, the more likely it was that her performance was well below par.

   At one point, I asked Strayer whether he thought he might be a supertasker himself. “I’ve been around this long enough I didn’t think I am,” he said. Turns out, he was right. There are the Cassies of the world, it’s true. But chances are, if you see someone talking on the phone as she drives up to the intersection, you’d do better to step way back. And if you’re the one doing the talking? You should probably not be in your car.

2. Beckie Supiano in CHE: Smart People Go to College, and Other Twists in Measuring the Value of a Degree. An interview of Douglas Webber, an assistant professor of economics at Temple University, whose recent research has been on teasing out the role of the factors that account for the fairly big effect (in the US) of college education on lifetime earnings. This discussion of people's focus on outliers stands out [with bold emphasis added by me]:

   Now everything I’m talking about, I’m using average returns. When I said that higher-ability people tend to go into certain majors, I’m saying that on average. So there are many, many absolutely brilliant people who major in art history, and there are many not-so-brilliant people who major in engineering.

   A lot of times people put too much weight on outliers. They see someone who is really successful, and they think that’s a good path to take. But if you are an average person, then you should be looking at the average return.

   Mick Jagger—and the world—would be much worse off if he had stayed at the London School of Economics and gotten an econ degree instead of dropping out to hang out with Keith Richards. But you know, he’s an extreme outlier.

3. Inside Higher Ed: Study: Web surfing in class hurts top students too: The study challenges the conventional wisdom which "holds that marginal students may pay more of a price for web surfing during class than top students, who are presumed to be better multitaskers."

Links

1. Richard Van Noorden in Nature: Computer model predicts academic success [of biomedical researchers].

   The mantra 'publish or perish' is drilled into every early-career scientist — and for good reason, a computer model suggests. The most important predictor of success for a young biomedical scientist is the number of first-author papers published in journals with high impact factors early in a researcher's career, according to the formula.

   The model, created by computer scientist Lucas Carey, at Pompeu Fabra University in Barcelona, and his collaborators, also found that, even correcting for publication records, working at a highly ranked university — and being male — are predictors of academic success.

2. Tim Harford: The Four Lessons of Happynomics

3. Kirk Doran and George Borjas in Vox: Which peers matter? The relative impacts of collaborators, colleagues, and competitors.

   Research so far has been inconclusive about the effect of losing and gaining productive peers on one’s own output. This column defines peers in three distinct ways and checks which types of peers matter, focusing on mathematicians shortly after the collapse of the Soviet Union. Losing intellectual competitors results in an increase in one’s output, whereas losing collaborators reduces it. Competition for resources and positive spillovers from high-quality peers are simultaneously at force, explaining the divergent findings in the peer effects literature.

4. Nicholas Thompson in The New Yorker on Tesla CEO Elon Musk's the decision to open up all his patents.

5. SMBC on the true significance of commencement speakers.

6. And, finally, a priceless gem from PhD Comics on whether professors would pass the Turing test. See also: the Professor Turing Test -- Actual Responses from Real Professors.

Fathers Day cartoons in the New Yorker

A lovely slide-show of 14 cartoons from the magazine's archives (12 with sons and 2 with daughters). This one is raucous while this is for academics.

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The New Yorker did a slide-show for Mothers Day last month with a 5:1 skew in favour of sons (and six other cartoons are more generally about mothers -- this one is LOL-funny).

Science in Tamil - My Talk

Sometime back I had mentioned about my popular science books in Tamil language.

Few days back, without much warning, the IIT Madras Muthamil Mandram invited me for a felicitation on this 'precocious punditry', I presume. They also honored the publisher for his derring-do act of investing on such geek-horn verbal gymnastics in vernacular.

They demanded a speech from me as comeuppance, I presume, which I gave -- promising for ten minutes and delivering for forty minutes.

Here is the talk (mostly in 'spoken' Tamil) in two parts, which touches upon the why and how of Science in Tamil -- i.e., my  take of it -- and the contents of two of the books.

part 1 (23 minutes)

part 2 (20 minutes)

 
If the videos don't work here, check them in the related post in Tamil.

Links: Global Ranking of Universities

1. D.D. Guttenplan in NYTimes: Re-evaluating the college ranking game.

   So who will rank the rankings?

   That was the inescapable question when representatives of the four leading ranking organizations sat on the same panel at a conference here last month.

   As Bob Morse, the research director for U.S. News and World Report, pointed out, rankings have become a fiercely competitive global business.

   So the presence of Mr. Morse; Phil Baty, editor of the Times Higher Education Rankings; Nian Cai Lui, originator of the Academic Ranking of World Universities (better known as the Shanghai rankings); and Ben Sowter of the QS World University Rankings would have been enough to make the gathering “a historic event,” Mr. Morse said.

   But in addition to trading friendly digs at their competitors’ methodologies, the rankers had to listen to some stinging criticism — not just from disgruntled academics complaining that their institutions have been undervalued, or education ministers responding to an absence of their country’s universities on a given list, but from their own invited guests. Even the conference host, Michael Arthur, the president of University College London, took a jab.

2. Aisha Labi in NYTimes: E.U. Seeking Better Clarity on Rankings:

   A new international effort to gauge the performance of universities went online last month promising to be a nuanced tool for students and institutions in the contentious field of global rankings.

   The project is U-Multirank, which was announced in 2011, backed by the European Union and aims to foster greater transparency about higher education globally, including in the United States. But while its approach has received praise, some experts say it still has a way to go before achieving its goals, and some higher education groups have already raised questions about its methods.