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.

 

 

Information and Sāṅkhya

Alfred Borgmann’s Holding On To Reality opens with the provocative words:

Information can illuminate, transform, or displace reality. When failing health or a power failure deprives you of information, the world closes in on you; it becomes dark and oppressive. Without information about reality, without reports and records, the reach of experience quickly trails off into the shadows of ignorance and forgetfulness.

This is a very interesting and provocative passage, but I picked it up because of its uncanny resemblance to a famous verse from Īśvarakṛṣṇa’s Sāṃkhyakārikās:

prīty-aprīti-viṣādâtmakāḥ prakāśa-pravṛtti-niyamârthāḥ |
anyônyâ-’bhibhavâ-’’śraya-janana-mithuna-vṛttayaś ca guṇāḥ ||

“The three guṇas [which constitute all non-sentient reality] have the respective natures of joy, non-joy, and sorrow; they act to illuminate, transform, and restrain, respectively; and they have the capacity to ground, produce, combine, and suppress one another.”

I need to bone up on my Sāṃkhya and my Borgmann further before I stretch this analogy, but at least at first glance, it appears as if there is something potentially illuminating lurking in the shadows here. (One thing that strikes me: the three guṇas of Sāṅkhya are regarded as entirely distinct from the individual persons, puruṣas, who alone are conscious observers and actors. This suggests that information itself isn’t enough: it presupposes the existence of conscious observers who can recognize something as being information.)

Open-source software vs open-source religion

It struck me today that there is a duality between the way in which open-source programmers work and the way in which philosophy and theology was done in classical India. (This might seem like a strange duality to bring up, but I do it because people frequently make misleading analogies between computer programming and Sanskrit grammar and other Indian disciplines. Also, because it’s my blog, so there!)

The point of open-source programming is that everyone can edit the source code. You fork a Github repository, make your changes, push them, and perhaps they might get accepted into the main code repository itself. Of course, it’s important that you (broadly) use the same sort of compiler/interpreter that the project is intended to work with, otherwise your code may not quite work the same way on others’ machines. 

In short, for open-source software, the rule is:

change the source, keep the interpreter ( / compiler / whathaveyou).

The structure of classical Indian thought is the precise opposite of this. Most of the oldest philosophical texts were written in laconic, even ambiguous, sūtras. Later philosophers then came to write commentaries on these texts, while their disciples wrote super-commentaries on the commentaries, while their disciples wrote super-super-commentaries on the super-commentaries on the commentaries on the sūtras, and so on.

(Lest you think I am joking, here are just two examples:

1. The foundational text of the Nyāya tradition is the Nyāya-sūtra of Gautama. This gathered the following tower of commentaries:
• Vātsyāyana’s Nyāya-sūtra-bhāṣya
• Uddyotakara’s Nyāya-sūtra-bhāṣya-vārttika
• Vācaspati Miśra’s Nyāya-sūtra-bhāṣya-vārttika-tātparya-ṭīkā
• Udayana’s Nyāya-sūtra-bhāṣya-vārttika-tātparya-ṭīkā-pariśuddhi
2. The foundational text of all Vedāntic schools is the Brahma-sūtra or Vedānta-sūtra of Bādarāyaṇa, sometimes also called Vyāsa. If we look just at one branch of commentaries on this foundational text, the Advaita branch, we get the following picture:
• Śaṅkara’s Advaita Brahma-sūtra-bhāṣya
• Vācaspati Miśra’s Bhāmatī
• Amalānanda’s Vedānta-kalpa-taru
• Appayya Dīkṣita’s Vedānta-kalpa-taru-parimala
• Lakṣmīnṛsiṃha’s Ābhoga
• Akhaṇḍānanda’s Ṛju-prakāśikā
• Padmapada’s Pañcapādikā
• Prakāśātman’s Pañcapādikā-vivaraṇa
• Citsukha’s Pañcapādikā-vivaraṇa-tātparya-dīpikā
• Nṛsiṃhāśrama’s Pañcapādikā-vivaraṇa-prakāśikā
• Nṛsiṃhāśrama’s Vedānta-ratnakośa)

The thing to keep in mind is that each of these commentators took the exact wording of his predecessors extremely seriously. And yet we obtain these complex forking interpretations of these texts, often at loggerheads with each other. The purpose of a commentary was ostensibly to elucidate the meaning of the base-text it commented on—but more often than not, it was to show that the base-text could be read to yield the conclusion you knew it had to yield.

In short, for classical Indian intellectuals, the rule is:

keep the source, change the interpreter.

Reflections on the Revolution in Isla Nublar, epilogue

This is the epilogue to a three-part series of posts, titled “Reflections on the Revolution in Isla Nublar”, interpreting the movie Jurassic World. The whole series of posts in order is the following:

1. An extended look at the role played by clothing in depicting personal transformation in the movie.
2. An expansion of our analysis, from clothing through enclosures to social relations structured by power, understanding, and survival.
3. An examination of survival, evolution, rationality, and the (non-)distinction between the real and the virtual.
4. An epilogue defending my choice of title, and offering some final thoughts.

I realize now that I did not explain at all why I picked the title for this series of posts on Jurassic World. It is of course meant to call to mind Edmund Burke’s classic critique of the French Revolution, which has the much more elaborate title Reflections on the Revolution in France, And on the Proceedings in Certain Societies in London Relative to that Event. In a Letter Intended to Have Been Sent to a Gentleman in Paris.

Burke is sometimes caricatured as being an arch-conservative and stick-in-the-mud, but it is important to realize that in the context of his society, he was actually rather on the progressive wing. Burke was a supporter of the American freedom struggle, a champion of the rights of the Irish (he was of Irish descent himself), and vehemently opposed to the East India Company’s depredations in India. And yet, he was fundamentally opposed to the French Revolution, because he saw in it the release of forces deeper and more dangerous than those unleashed by any other revolution: rationalism, utterly convinced of its own correctness and of the total mistakenness of all other positions, willing to overthrow everything that came before it in the name of a new world order founded on pure reason unencumbered by primitive inherited idiocies.

What happens on Isla Nublar is a revolution in that sense: a full overturning of the existing order (a re-volvo) and the institution of a new system based on different principles.

Reflections on the Revolution in Isla Nublar, part three

This is the third of a three-part series of posts, titled “Reflections on the Revolution in Isla Nublar”, interpreting the movie Jurassic World. The whole series of posts in order is the following:

1. An extended look at the role played by clothing in depicting personal transformation in the movie.
2. An expansion of our analysis, from clothing through enclosures to social relations structured by power, understanding, and survival.
3. An examination of survival, evolution, rationality, and the (non-)distinction between the real and the virtual.
4. An epilogue defending my choice of title, and offering some final thoughts.

In the previous installations of this series, I first wrote about the hugely significant role clothing plays in understanding the transformations of individuals in Jurassic World. I then expanded on the idea of clothing to enclosures in general, which led naturally to relations of power and dominance, which were then contrasted with relations of trust and understanding (familial relations) and with relations of unification for survival (conjugal relations). It is now time to naturally move from conjugal relations to the theme of evolution, which of course underpins the entire Jurassic Park franchise.

Reflections on the Revolution in Isla Nublar, part two

This is the second of a three-part series of posts, titled “Reflections on the Revolution in Isla Nublar”, interpreting the movie Jurassic World. The whole series of posts in order is the following:

1. An extended look at the role played by clothing in depicting personal transformation in the movie.
2. An expansion of our analysis, from clothing through enclosures to social relations structured by power, understanding, and survival.
3. An examination of survival, evolution, rationality, and the (non-)distinction between the real and the virtual.
4. An epilogue defending my choice of title, and offering some final thoughts.

Last time, we took a very close look at some of the significant ways in which clothing works in Jurassic World. I will avoid diving into such detail this time around, as I have a vast range of topics to cover. (Need I say that this will be massively full of spoilers if you haven’t seen the movie?)

2. Enclosures

I’m thinking of enclosures fairly broadly here as any sort of covering that affords protection from the elements. In this sense, clothing is of course a kind of enclosure protecting human skin: the shedding of clothing is the discarding of a layer of insulation. But there are a number of other such enclosures in the film: the gyroball, for instance, literally encloses the two boys in a protective shell as they travel through the park—

Reflections on the Revolution in Isla Nublar, part one

This is the first of a three-part series of posts, titled “Reflections on the Revolution in Isla Nublar”, interpreting the movie Jurassic World. The whole series of posts in order is the following:

1. An extended look at the role played by clothing in depicting personal transformation in the movie.
2. An expansion of our analysis, from clothing through enclosures to social relations structured by power, understanding, and survival.
3. An examination of survival, evolution, rationality, and the (non-)distinction between the real and the virtual.
4. An epilogue defending my choice of title, and offering some final thoughts.

I haven’t blogged for a regrettably long time, so I figured I would mark my return to the scene by reflecting on a movie that hasn’t done too badly for itself this summer: Jurassic World. Given my obsession with dinosaurs (par for the course for anyone who saw Spielberg’s original miracle), this experience was particularly fun for me.

I suppose it needs no spoiler warning to say that Jurassic World is about (yet another) dinosaur going crazy on a theme-park island, with lots of humans dying gruesomely while the protagonists make it through with all body parts intact and with an increase in their wisdom. But what struck me even while watching the movie was that it very artfully sets up a number of structural polarities, only to dissolve some of them—but not all!—into an even more complex jumble. Postmodern neopaleontology? Bring it on!

What I will do now is look at a few motifs that are repeated throughout the film, creating striking parallels and generating resonances.