Mathematical aptitude as Gift

I recently came across an article by Jo Boaler which argued against the usage of mathematics as a “gatekeeping” tool for children, and against thinking of children as having an “innate gift” for mathematics for fear that this would contribute to greater inequity among the populations who dive into mathematics as a profession. This article, especially when read in the context of Boaler’s work in reshaping school education in California, has upset a number of people, including parents of children who are seen as “gifted”.

As someone who was a math major, who did not pursue formal graduate study of mathematics, whose dissertation drew upon a number of abstract mathematical concepts, who continues to study higher mathematics for pure pleasure, and at least one of whose kids is obsessed with numbers, I have a lot to say on this subject! In case you don’t want to read this long, rambling post, here are a number of positions that I think can all be held simultaneously without fear of self-contradiction:

Unsubstantiated conclusions

• There is genuine variation in innate potential for (certain kinds of) mathematical thinking. 
• Consequently, there are indeed certain people who possess this potential to a much greater extent than others.
• Practice and perseverance will allow people to maximize their innate mathematical potential (which may be higher than they think).
• Conversely, some people will not fulfill their innate mathematical potential if they don’t put in this work. This could be due to a whole range of factors, including personal choice, disabilities, or environmental factors.
• Mathematical education at the (American) school level is narrowly focused on a small part of the spectrum of mathematical thinking, so that performance at the school level is not a strong predictor for genuine mathematical potential. 
• In fact, I would assert even more strongly that doing well in school math is neither necessary nor sufficient to do well in college-level math or higher.
• Most people have no real idea of what higher mathematics is actually like, and are quite shocked when they realize that many mathematicians might not actually be great at mental math or other flashy demonstrations of school-level computational performance.
• Mathematical education at the (American) school level is indeed serving a “gatekeeping” function that has deleterious consequences for students.

As a way to make sense of this complex bundle of thoughts, and in order to connect it to my previous musings on this blog on mathematics, meditation and Indian philosophy (specifically the school of Mīmāṃsā), I will posit the following: 

We must distinguish between mathematical study as an end in itself (kim), mathematical study as a means to some other end (kena), and mathematical study as a process or procedure (katham) to be followed to produce some other end.

A biographical interlude

In my undergraduate days, I studied mathematics at one of the most demanding places in the US. In that context, I would rate myself as middle-of-the-road: I did decently, but there were also a number of students who were far ahead. (This was a a humbling lesson for me, and one I very much needed to learn!) I realized that these students could be conveniently classified into two types: 

1. Some students had studied a lot more mathematics than others: their high school math programs had been very intense (often French or Eastern European), or they had had the chance to take a lot of college math while in high school itself. These folks were ahead because they had put in the time and the effort into studying mathematics, and were already familiar with content that was otherwise brand-new to me. To use a commonly-misused term, they were more “mathematically mature”. However, I also noted that some of these students were not necessarily much faster at grasping totally new ideas, or at extrapolating to unusual domains. 
2. And there were others who were just … different. They too had studied a lot more mathematics than others, but you could sense that this was just natural to them. Mathematical facts were obvious to them that were inscrutable to us after hours of bashing our heads against a page of symbols. Where we struggled in the foothills of abstraction, they frolicked freely in the heights, like Sherpas who grew up amidst Himalayan peaks. 

Seeing people with a real mathematical gift was humbling, liberating, but also (in retrospect) constricting for myself. For someone who had always done well in mathematics in school to realize there were people who effectively operated on a different (and yes, non-intersecting) mathematical plane than mine was not easy initially. Once I absorbed that lesson, though, I felt liberated: I didn't need to compete with them! I could do math at my own level, at my own pace, and I could derive pleasure from it. This sense of liberation has allowed me to continue to study math for my own edification, long after departing the hallowed halls of academia. This allows me to enjoy the mysterium tremendum of mathematics: a near-religious sense of awe, marvel, and humility in the face of mathematical structures of immense beauty—which would likely not have been possible had I been trying to grok them to crunch a problem set under time pressure. 

At the same time, in retrospect, I can also see that my recognition of the gift of mathematics kept me from pursuing mathematical study in graduate school. I thought to myself that graduate education in mathematics must surely be the preserve only of those who were truly gifted, with no place for ordinary folks like myself. This, however, was an act of self-limitation, and not necessarily true. 

On an innate Gift for mathematics

It is transparent to me that there is natural variation in people’s affinity for mathematics, and that this often manifests in its extreme forms at very young ages. Consider, as examples of outliers, the extraordinary feats of Carl Friedrich Gauss and Terence Tao in their childhood. Closer to home and a lot more down to earth, I see my son playing day and night with numbers, delighting in their combinations and patterns. (This may perhaps be the mysterium fascinans aspect of mathematics!) 

I want to zero in on this sense of play, for I think it is critical to understanding how real mathematics often operates: It is an autonomous domain, with its own objects and rules, where the only goal of the game is to continue playing the game. (In describing mathematics this way, I do not intend to support either a Platonic or a formalist philosophy of mathematics; I'm merely trying to capture the phenomenology of doing mathematics in a state of flow.) This, as I have said above, has a strongly aesthetic flavor to it: it is enchantingly beautiful to a few humans, baffling to most, and repulsive to some. 

Incidentally, this autonomy of mathematics also poses a danger to those who most love it, in that it might entirely devour a person’s passion, energy and even sanity, if they do not retain a strong tie to the world of concreta. This is why I have called it a “Gift” with a capital ‘G’: to evoke the German word Gift which means “poison”, although it is a cognate of the English gift as well.

Mathematics as kim, kena and katham

The experience of the truly mathematically gifted thus clearly involves treating mathematics as an end in itself, as a source of joy pursued for its own sake. In the language of Mīmāṃsā, this is a kim, an end-goal in itself. Note that this might actually be a real challenge in following the standard school curriculum, where joy is not typically listed as a learning objective! For such students, self-paced self-study may in fact be the right answer.

But there are many who, talented though they may be in mathematics, are interested not in mathematics as an end but in mathematics as a means, as an instrument for some other goal. Scientists, engineers, data scientists, financial analysts, accountants: all of these professionals acquire fluency in some domains of mathematics and apply it to some other problems, creating value for the world and revenue for their bosses and a home with a view for themselves. (Even professional mathematicians who may be doing math as a way to pay the bills would fall into this category.) In the language of Mīmāṃsā, this is the use of mathematics as a kena or as a sādhana, an instrument for realizing a different end-goal. For many (though not all) such students, the structure of math education in school actually works, with its emphasis on procedure and algorithm and its de-emphasis of seemingly unnecessary tools like proofs. This divide often persists through college as well, where courses like calculus, linear algebra and differential equations might be taught both in mathematics departments as well as in engineering departments in wildly different ways.

Finally, we find yet another (mis)use of mathematics, this time as a process and not an outcome, with some other goal in mind (Mīmāṃsā would call this katham, a “how”). Thus, the education system seeks some kind of restriction on student outcomes, whether by design or by accident, and must accomplish this goal through some means of gatekeeping. The process may end up being a sequence of math instructions that feel more like a robotic algorithm, or perhaps like an obstacle course, where students must jump through the hoop of Algebra and across the moat of Geometry and scale the wall of Pre-calculus … all of this just to get into a good college where they in fact plan to study Old Norse sagas. This sort of gatekeeping can and no doubt does damage some students’ self-confidence and motivation to learn. For those who have the potential to utilize mathematics as a tool, such self-limitation and external gatekeeping may well keep them out of potentially rewarding occupations. For others, who may privately enjoy the free play of beautiful mathematics, it may poison the well and damage their associations with this source of great joy.

To me, the use of the Mīmāṃsā framework helps narrow down the problem quite specifically: It is the fetishization of mathematical performance and its misuse as a gatekeeping tool that is the real issue, not the existence of mathematical gifts in some fraction of the population. But it may not be clear why mathematics as gatekeeping is problematic, because modern society has come to privilege performance in mathematics so much.

An analogy to classical music

When I read the article, my first thought was immediately Paul Lockhart’s “A Mathematician’s Lament”, in which he describes the current state of K12 math education by constructing a parallel universe in which writing down sheet music is similarly treated as a gatekeeping tool, without regard to the aesthetic experience of actually listening to, performing, or composing music. I think this analogy is actually a very powerful one and can serve to illuminate the issue at hand.

Because music does not act as a gatekeeping tool in our society, we are much more comfortable with acknowledging that musical genius exists, whether it be as a composer or as a performer. Nevertheless, it is also true that most people can learn to perform some level of music, if taught properly, even if they will never grace Carnegie Hall or feed their families with this skill. (We forget that, for most of human history, when recording and playback devices did not exist, people would have had to sing or hum for themselves if they wanted to hear something musical while walking down the street!)

The analogy to music opens up yet another dimension which is underemphasized in mathematical pedagogy: cultivating taste and developing an aesthetic sense. We cannot all be stage performers; we might not even want to perform any kind of music ourselves; but we may learn to appreciate music and derive deep satisfaction from it. It might not be the same delight that a practitioner enjoys, and it certainly will not be a source of income, but it can be a source of joy. As someone who is neither a professional mathematician nor a professional musician, but who derives comfort and delight and, yes, a sense of proximity to the Divine Infinitude from the consumption of mathematical ideas and musical performances alike, I wish we encouraged all our children and all humans to take the idea of play more seriously and to cultivate a deeper sense of aesthetics. As the Taittirīya Upaniṣad says:

|| raso vai saḥ || 

The Divine truly is aesthetic savoring.

 

 

On goals, systems, and bhāvanā

An article by James Clear called “Forget About Setting Goals. Focus on This Instead” [Oh Upworthy, how I truly hate thee!] has been doing the rounds recently. It has received a lot of attention, but when I finally sat to read it a couple of weeks ago, I found myself deeply bothered by something I couldn’t quite get a handle on. It is only just now that I’ve realized what the problem was, and the answer came to me from Mīmāṃsā.

What on earth does a nearly 3,000-year-old Hindu tradition of ritual hermeneutics have to do with any of this? As it turns out, a lot! Mīmāṃsā’s primary intellectual concern has been with the Vedic sacrifice: how it works, how its descriptions in various ritual texts cohere, how it is organized, and so on. To do so, it has developed a formidable arsenal of techniques and frameworks. One of these, the concept of bhāvanā, was widely used and taken up in disciplines far outside Vedic ritual exegesis, including literary theory and imagination / meditation. (See my prior post on imagination for some other uses of the concept of bhāvanā in South Asia.) The time has now come to apply bhāvanā to yet another problem: motivating people to stay on track with difficult projects!

What is bhāvanā?

To massively oversimplify things, and with apologies to Andrew Ollet’s excellent article, a bhāvanā, a bringing-into-being, is a particular action (or a set of actions) designed to create something, undertaken by an agent. Every such bhāvanā has three essential components to it:

1. The desired end which the agent is trying to bring into being through this operation
2. The instrument using which the agent is carrying out the operation
3. The procedure which the agent is following with the instrument to bring about the desired end

These three are respectively called the kim (the “what”), the kena (the “by what”), and the katham (the “how”). These three things are very different from each other. Confusing them can be fatal to understanding how things are actually supposed to work.

The standard introductory Mīmāṃsā handbooks usually explain bhāvanā with an example from Vedic sacrifice. Here’s a rather different, much more quotidian scenario where the three components are nevertheless clearly distinguishable.

After a long, grueling day at work, you come home utterly famished. You don’t want to go out to get dinner, so you decide to make yourself a quick dinner. You have a microwaveable mac & cheese sitting in the freezer, so you take it out, read the instructions on the packet (you don’t really ever cook), stick it in the microwave, and a few minutes later, satiate your hunger with some piping hot coagulated carbs and fats.

In this scenario, it’s pretty clear that something new was created: the state of the world, and more importantly your own state, was transformed in this scenario. In not-so-technical Mīmāṃsā non-jargon, some sort of bhāvanā thingie occurred here. So what were the components of this bhāvanā? What was created?

It is tempting to think of the mac & cheese dinner as being what is created: after all, before you cooked it, it was just a frozen lump of carbs and dairy and preservatives, and it was your cooking it that transformed it into an (arguably) edible mush. However, this would be a major mistake, according to Mīmāṃsā: the mac & cheese was not the desired end of your actions. It wasn’t why you undertook all these steps. Instead:

1. the real end, the kim, must be the resolution of your hunger. 
2. the mac & cheese is the means, the kena, by which your hunger is resolved. 
3. the way you resolve your hunger, the katham, is by following the procedure outlined on the packet to cook and serve the mac & cheese.

Goals and Systems

If my quick overview of bhāvanā didn’t foreshadow it clearly enough, it should be clear what my beef with Clear’s piece is: he conflates kims and kenas when talking about goals, and therefore overemphasizes the importance of systems (which are not quite kathams). To see how he does this, let’s look at the section where he distinguishes between goals and systems:

What’s the difference between goals and systems?

• If you’re a coach, your goal is to win a championship. Your system is what your team does at practice each day.
• If you’re a writer, your goal is to write a book. Your system is the writing schedule that you follow each week.
• If you’re a runner, your goal is to run a marathon. Your system is your training schedule for the month.
• If you’re an entrepreneur, your goal is to build a million dollar business. Your system is your sales and marketing process.

Look carefully at the four things he describes as goals: winning a championship, writing a book, running a marathon, and building a million-dollar business. And look carefully at the four types of people he describes as having these goals: the coach, the writer, the runner, and the entrepreneur. Now, many of us would agree with these things as being described as “goals” (which to me only reinforces the fact that we live in a deeply instrumentalist society). But are they really goals? Are they more like the mac & cheese or like satiating the hunger for a hungry person?

• For a coach who is hungry for a win, winning a championship will certainly satiate his hunger. But it is not obvious to me that people become coaches in order to win championships. It can be argued that the real purpose of being a coach is to, well, coach a bunch of players to the best of their abilities, so that they can perform superlatively on the field. If the team can do that consistently, then they may very well end up winning a championship. It seems to me that winning the championship is really just a means (a kena) to the real end (kim): the joy that comes from watching people do their best on the field. (It is possible to win a championship and yet be dissatisfied, because perhaps your opponent defaulted; it is possible to lose a championship and yet be pleased, because you did your absolute best and fulfilled your “duties”, so to speak.)
• Writers don’t write in order to create books; they write books in order to do something else: tell a story, persuade their readers to act, create emotional states in their readers, convey some valuable information, or even simply feed their families. The book is clearly just a kena. The kim is whatever motivates the writer to write.
• As with the example of the coach, it may well be that running a marathon is a real kim for some people. However, it is again quite likely that there are other satisfactions here: enjoying the endorphin rush, raising money for a valuable cause, staying in shape, or whathaveyou. In all of those cases, the marathon is just a kena that is subordinated to the more significant kim.
• Again, it certainly is the case that some entrepreneurs are just in it for the money. In that simplest of cases, the business is the kena to their real kim: making a boatload of money. But as Guy Kawasaki has said many a time: “make meaning, not money” is the heart of entrepreneurship. Whether your business is worth a billion dollars or a hundred, the real purpose should be to do something that creates meaning for you and for the people you engage with. In such a view, the mere instrumentality of the business is even more strongly pronounced.

None of the four examples of goals here is clearly and precisely a kim: an end that people strive for and desire. Some of these could possibly be treated as kims, but Mīmāṃsā argues (and Clear would agree with this, as he himself writes) that to do so would be to fundamentally misunderstand the nature of each of these sets of actions.

Clear has correctly identified one problem: the so-called “actionable goals” he describes are notoriously bad at actually getting achieved. The real reason for that is because they aren’t in fact goals: they are means to other ends. These other ends are the real goals we should keep in mind. In the absence of those real goals, people wouldn’t act at all!

The problem isn’t with goals at all; it is with the fact that Clear is using two terms (“goal” and “system”) to describe three distinct pieces (the [real] goal, the means to the goal, and the procedure; to use a different metaphor, the destination, the means of transport, and the particular path you navigate). Because of this conceptual blurring, he emphasizes “systems” more than they can bear: what he calls systems are just glorified procedures. 

The real system is the the whole triplet that Mīmāṃsā describes, and this systems needs all three pieces to succeed: a real goal to motivate us to act, a means by which this goal can be achieved, and a procedure that can be followed (with all the useful tips that Clear provides).  

ADDENDUM

One of the subheadings in Clear’s piece gives the game away to the reader who is keyed into Mīmāṃsā: He describes one of the faults of the goal-based approach as being the fact that “Goals reduce your current happiness.” This is exactly what sets kim apart from kena in Mīmāṃsā! Take the typical injunction that Mīmāṃsā analyzes: yajeta svarga-kāmaḥ, “The heaven-seeker should perform a sacrifice.” Lots of Hindus have desired to perform sacrifices, and still do to this day. But Mīmāṃsā argues that the real end here, the real kim, here has to be heaven. The sacrifice itself is only the means by which this end is brought about. And one of the arguments is this: a sacrifice is a difficult, expensive, resource-intensive, and physically taxing undertaking. No rational (pleasure-maximizing, pain-minimizing) human would perform a sacrifice for its own sake. Therefore, a sacrifice must be the means to some other end: the promised result. To follow Clear’s line of reasoning, you would focus on the individual actions of a sacrifice (its procedure) but ignore the coherence of the sacrifice itself as an instrument, and altogether forget about the real goal, the heaven that is the result of the sacrifice!

“Practical” Universities

David Brooks has a new column out in the New York Times in which he argues that online education will force a transformation of universities. Drawing on Michael Oakeshott, he argues (or really, just states) that universities today offer two kinds of knowledge: technical (the what) and practical (the how). Brooks claims that, because technical knowledge can easily be transmitted online, we will see people gravitating towards MOOCs where they pick up “just the facts, ma’am” from star online teachers. However, since practical knowledge can only be picked up from experience, he thinks that universities will shift increasingly towards offering this sort of irreplaceable knowledge.

Leaving aside the merits and demerits of Brooks’s piece, I am quite intrigued that he ignores another, crucial, kind of knowledge that universities offer: the why. Now sometimes this knowledge seems like anti-knowledge from the outside, because it is about limits, about ends, and about asking the right kinds of questions. But these are critical issues to think about—admittedly, not for everybody, but for society as a whole. A city full of carpenters, or of philosophers, is not a city but an unnatural monoculture.

This is all the more surprising because a threefold distinction of knowledge was known to Aristotle, who called them epistemē, technē, and phronesis. (Of these three, phronesis directly overlaps with Brooks’s practical knowledge; while technē seems to largely make up, but not exactly correspond to, technical knowledge.) A polis needs all three to flourish. I am curious to know where Brooks thinks epistemē will be found in his post-MOOC world.

Human irrationality and the purpose of a liberal arts education

There's a very interesting post by Lane Wallace at the Atlantic, called "All Evidence to the Contrary".

In other words, if people start with a particular opinion or view on a subject, any counter-evidence can create "cognitive dissonance"--discomfort caused by the presence of two irreconcilable ideas in the mind at once. One way of resolving the dissonance would be to change or alter the originally held opinion. But the researchers found that many people instead choose to change the conflicting evidence--selectively seeking out information or arguments that support their position while arguing around or ignoring any opposing evidence, even if that means using questionable or contorted logic.

So where does education come into the picture? Obvious enough:

A liberal education, [UCLA professor Mark] Kleiman says, "ought, above all, to be an education in non-attachment to one's current opinions. I would define a true intellectual as one who cares terribly about being right, and not at all about having been right." Easy to say, very hard to achieve. For all sorts of reasons. But it's worth thinking about. Even if it came at the cost of sacrificing or altering our most dearly-held opinions ... the truth might set us free.

Very, very interesting. Read the whole thing! (Yes, I have more work than I can get done, and I'm procrastinating.)