Can We Radicalize Math Education?
What does “6÷2(1+2)” equal? If you are on social media, you may have seen this piece of engagement-bait. It is ostensibly an exercise in arithmetic: the expression is syntactically valid, but the notation ever-so-slightly ambiguous and unconventional. Applying the standard order of operations, an arbitrary set of rules meant to be unthinkingly obeyed by electronic calculators and elementary school students, gives one answer, while anyone who insists that a mathematical expression must be expressing something will most likely interpret it to give a different one. And if mathematics promises us anything, it’s definite answers and certainty. Such inconsistencies feel like a betrayal to those who believe in this promise. And for those who don’t, like generations of schoolchildren forced to memorize “Please excuse my dear Aunt Sally” and other memory aids, the conviction remains that mathematics is a cunning adversary, out to get you.
Much has been written in recent years about the injustices of the algorithmic age, and plenty of evidence has been presented to justify this conviction. If you are not in the tiny minority who can control the algorithms being rapidly implanted into every aspect of our lives, then you experience these as at best depersonalized and unhelpful, but not infrequently, as an active threat to your well-being and liberty. While earlier discussions of the harm caused by these systems point the finger at bias and ethical failures, in Revolutionary Mathematics, a new book whose short length belies the variety and richness of the ideas contained within, Justin Joque digs deeper, and traces these back to the very nature of the process of objectification and statistical epistemology.
Joque employs the term objectification in the sense of Marx, to mean the process by which “certain things appear objective: the moment when decisions of social actors cease to appear as decisions and begin to appear as natural and unimpeachable”. It applies equally to the process by which knowledge is created as to the production of knowledge workers and their subjectivities, both individual and collective. The algorithm condemns the person who fails to comply with or conform to its opaque and arbitrary criteria of being worthy of risk, threatening them with a range of penalties ranging from bad credit scores to denial of parole. The child who refuses to submit to the vagaries of pedagogical convention is familiar with this logic, having faced the disciplining effect of bad grades, adult disapproval, and in a world of vanishing opportunities for socio-economic mobility, exclusion from a decent life.
One could argue that exclusion is the ultimate end of algorithmic objectification. The reason to render a thing measurable and exchangeable is so that an algorithm can further reduce the complexities of its existence to the point where it can answer any question with either a yes or a no, buy or sell, allow or deny. As long as the decisions made pass certain thresholds of performance with respect to some objective function, often profit, the algorithm has succeeded. Whether they prove correct in some objective sense which reflects human reality or intention is immaterial. The machine does not pretend to get at any universal notion of truth as a human would understand it, which makes it useless for the purposes of advancing our collective understanding and ability to act. The pioneering computer scientist Richard Hamming used to say that "the purpose of computing is insight, not numbers." Information capitalism, as Joque argues, has no use for insight, only for decision making, and the more non-locatable these decisions are within agents who can be held accountable for these decisions, the better.
Abstraction in pure mathematics, “the art of giving the same name to different things”, acts superficially in similar ways. It has, however, retained its claim to universally accessible universal truths. Anyone can follow the proof of a theorem and verify its details for themselves. And if something is wrong, someone will spot it, corrections will be issued, and we will have made progress towards a better, more complete knowledge. Or at least that’s the story we tell ourselves. But as Revolutionary Mathematics describes, the days when this is possible are coming to an end. For one, there is simply too much mathematical output, much of which is hardly read and understood by anyone, let alone used. Meanwhile, anyone hoping to produce new mathematics is required to sift through and claim an understanding of older mathematics, of which there is an ever-increasing amount. Soon the amount of reading one must do in certain areas before embarking on original research will surpass what is possible within the time-frame of a graduate education.
The mathematical profession has responded to this looming crisis in multiple ways. Some are working to develop proof assistants and undertake systematic mechanization, guided by the vision of future mathematicians as hybrid human-computer intelligences. Ultimately the aim would be to train mathematicians of the future to incorporate this kind of computer “support” into their work. If machines can perform our routine calculations without us needing to know the details, then why not have them verify conjectures and flag logical flaws in our arguments? Why not have them prove entire theorems with no human guidance at all?
Computational aids are as old as writing systems. They have freed up our limited working memories and greatly enhanced our ability to think. The new programs aim not just to further this work, but to de-locate mathematical knowledge from its current, distributed form. They want to take the general mathematical intellect, held in and shared among the minds of mathematicians in community and convert it to software. However correct and accessible the resulting systems may be, the process of implementing this new way of doing mathematics is not guaranteed to be straightforward or uncomplicated. Is mathematics simply its formalization? Most mathematicians would vehemently disagree. As the mathematician Michael Harris points out, widespread and radical mechanization of the sort promised by its cheerleaders is likely to change what we mean by mathematics itself. As certain areas lend themselves to mechanization more easily than others, the logic of markets will play no small part in what flourishes and what is left to wither. Then there is the question of what the purpose of mathematical activity is, and whether these programs serve this purpose.
Whatever the contents of future mathematical output, whether by machine or by mind, they will likely be of little to no concern to a human race which so far has shown very little interest in such esoterica. It shows no signs of becoming part of the general intellect. And it is this general intellect that must form a bulwark against capitalist predation, especially in its current form, mediated by algorithmic authority.
The divergence of the general intellect from the development of specialized mathematical knowledge long before the advent of the algorithmic age has been cause for much political and economic concern throughout the western world. It is not just evident in the epistemic crises that seem to have intensified in recent years. The recent repeated failures of technocracy are as much to blame for these as scientific illiteracy and innumeracy, and any real solutions must take the form of collective political action. And the solutions may not be to the taste of the ruling classes.
Another way this concern finds voice is in “national conversations” about falling standards in mathematical education and student achievement that every anglophone country seems to have once every few years. Unfortunately, the only voices heard in these conversations are those of politicians, technocrats, employers and influential academics, who are not so much interested in the general intellect as concerned with reproducing themselves as a class. They vocalize these anxieties through phrases such as “skills gap” and “international competitiveness”. But they do point to a real crisis: it is their refusal to invest in a universal, equitable development of human capacity in the societies which they dominate that is causing stagnation, an inability to innovate solutions to grand challenges and to make progress.
The current model of education, particularly in mathematics and science subjects has been that of weeding out the majority and collecting just enough survivors for the scientific and institutional elites to maintain themselves. They select those who at a young age can negotiate mysterious symbols and arcane formulae, manipulate abstract quantities subject to arbitrary rules, under artificial examination conditions, or in short, make the most of those ghosts of departed learning – grades. The chosen few become mathematicians, statisticians, engineers and scientists, amongst the last well-rewarded professions of our age. The very same people then go on to perpetuate the same systems that produced them and exclude others in turn, maintaining the vast economic and epistemic inequalities that plague us.
This is the perspective that informs the push towards keeping curricula weighed down with an archaic selection of topics which lend themselves to repetition and rote-learning, and to filter those who can jump through these hoops with ever more rigorous standardized testing, before university professors can show them ‘the real stuff’. Efforts to challenge this system from school teachers and their educators are often met with furious backlash. Education researchers analysing traditional practices through a critical pedagogical lens are often at the receiving end of harassment campaigns. Or take, for a recent example, the proposed revisions to California’s school mathematics curriculum and the angry response it has provoked. An open letter criticizing the proposed California Math Framework has been signed by over 1600 self-described ‘STEM professionals and educators’ who claim to be “deeply concerned about the unintended consequences of recent well-intentioned approaches to reform mathematics education”. It was celebrated on the right as resistance to “wokeness” afflicting even mathematics, uppending “universal truths” such as “2+2=4” in service of anti-racism and validating feelings. The letter itself couches its criticisms in the language of “access” and “equality”, even then betraying an almost nostalgic Cold War sensibility. That two seemingly disparate groups, liberal professionals and right-wing provocateurs, find common ground on education becomes less surprising when you consider what they have in common. They are both invested in the status quo, one where failing a majority of children is a fair price to pay for accelerating a select few into elite universities and high-powered careers.
Any education system which aims to help build and serve fairer societies, ones that can withstand the major crises facing us all, must equip students to critically evaluate the technologies that govern them, and to remake these tools to fit the world they must build. In the case of mathematics, developing the better forms of objectification that Joque encourages us to imagine as possible involves a general familiarity with the nature of mathematical abstraction itself. It must be mastered along with the technical skills that pass for all of mathematics in public perception. There is no reason for this understanding to be acquired as part of a hidden syllabus, available only to a select few who choose to make mathematics their lives work and the rest do not even suspect. It can start with asking not what “6÷2(1+2)” equals, but with teaching numbers, notation and arithmetic as powerful and convenient objectifications that allow us to express, communicate and understand truths in a common language. Indeed, where better than the elementary school classroom, where every child’s instinct to count, compare, collect, cluster, distinguish, and spot patterns is reshaped and reified into the supposedly universal language of mathematics? Such a reorientation may help more students to experience the development of their mathematical capabilities as something other than a violent and alien imposition, to cast off the anxieties of older generations weighing like a nightmare on their brains, to become the revolutionary mathematicians of the future.
Sonia Balagopalan is a mathematician and a researcher at Autonomy.